The generation and interaction of
convection modes in a box of a
saturated porous medium.
Brendan James Florio, BSc (Hons)
This thesis is presented for the degree of Doctor of Philosophy
of The University of Western Australia
School of Mathematics and Statistics
April 30, 2013
ii
Abstract
Inspired by the recent interest in using geothermal energy from the Perth Basin, models
of the underground water flow in the deep sedimentary basin are being developed. If
convection were to occur, then a circulation of fluid assists in bringing hot water closer
to the surface. If such hot upwellings can be located, the cost of extracting water at an
operational temperature can be reduced. This thesis explores issues of modelling convection in porous media, with particular focus on the effect of the domain size on the
observed convection patterns. Convection in an infinite layer of a porous medium occurs
if the dimensionless Rayleigh number exceeds a critical value. This is also true for a box
of a porous medium, however, each discrete modal solution has its own associated critical
Rayleigh number. Usually just one mode will be generated at the onset of convection,
however, there are many critical box dimensions for which up to four modes share the
same critical Rayleigh number and all may be generated at the onset of convection. In
such circumstances there will be a slow interchange of energy between the preferred modes.
A perturbation method is applied to a system where multiple modes are generated at onset to yield a system of ordinary differential equations which govern the evolution of the
amplitudes of the viable modes. For three interacting modes, three unique cases arise,
each with a different phase-space structure. Critical boxes with “moderate” aspect ratios
are systematically categorised into these cases. While two of the examples represent the
usual case where just one mode survives in the final state, the third example is a special
case where it is possible for the three modes to coexist. This procedure is extended to
a case where four modes are generated at the onset of convection, completing all possible scenarios in a moderate box domain. The initial conditions determine which mode(s)
will survive. For non-critical boxes, the bifurcations that occur as the Rayleigh number
increases are analysed and profiled in the weakly-nonlinear regime.
iv
Abstract
Acknowledgements
First of all, I would like to thank my coordinating supervisor, Asst Prof Thomas Stemler.
Thomas is always available to help me with my questions - big or small. The value of his
advice is immeasurable, and never went unappreciated!
I would also like to thank my other supervisors, Prof Kevin Judd and Dr Neville
Fowkes, whose expert knowledge covers a broad field in applied mathematics, and was
integral to different parts of this thesis. Thank you for your time and patience.
I must extend thanks to all those who gave me advice and assistance along the way:
Heather Sheldon, Mike Trefry, Lynn Reid, Frank Horowitz and many others from WAGCoE; and Andrew Bassom, the head of the School of Mathematics and Statistics at UWA.
To my wife, Lauren, I feel I must offer an apology in addition to my thanks: I am
sorry that I was too busy with this thesis to help organise the wedding! Thank you for
your love and support. I truly could not have achieved this without you.
I would like to express my deepest appreciation to my parents, Tony and Julie, who
provided me with the opportunities which enabled me to achieve my goals, and reach
beyond.
Finally I would like to thank my friends who took the stress out of my life: Ryan,
Mitchell, Ben and Aaron. You are the custodians of my sanity.
I appreciate the financial assistance I received from the Robert and Maude Gledden
Scholarship.
vi
Acknowledgements
Statement of candidate
contribution
As the sole author, I have had a paper accepted based on the work in presented Chapter
4 (see Florio (2013)).
A paper authored by me based on the key points in chapters 5 and 6 is in preparation.
My supervisors and I have regularly discussed my research and their supervision has been
invaluable.
viii
Statement of candidate contribution
Contents
1 Introduction
1
1.1
Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Pattern formation of convection modes in porous media . . . . . . . . . . .
3
1.3
Thesis structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2 Modelling convection in porous media
7
2.1
Fluid mechanics in porous media . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
When does convection occur? . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
2.4
2.5
2.2.1
Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2
Numerical verification . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Varying permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1
Anisotropic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2
Other studies on varied permeability . . . . . . . . . . . . . . . . . . 17
Temperature-dependent viscosity . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1
Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2
Numerically modelling a variable viscosity . . . . . . . . . . . . . . . 23
2.4.3
Linear stability analysis based on the harmonic mean value of viscosity 24
2.4.4
Numerical simulations based on harmonic mean value of viscosity . . 26
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Two interacting modes
29
3.1
Introduction to the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2
Perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3
Fixed point and stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Three interacting modes
4.1
4.2
41
Evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.1
Linear stability results . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2
Perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.3
Third order: Secular modes . . . . . . . . . . . . . . . . . . . . . . . 49
Specific Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.1
Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
x
Contents
4.2.2
4.3
4.4
Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Structure of the separatrix manifolds . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1
Exploring the parameter space . . . . . . . . . . . . . . . . . . . . . 58
4.3.2
Classification of the critical boxes . . . . . . . . . . . . . . . . . . . . 61
Special case of the dynamical system . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1
Two interacting modes . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.2
Three interacting modes . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5
Dynamics of the special case. (Example 3) . . . . . . . . . . . . . . . . . . . 65
4.6
Hidden symmetries in rectangular domains . . . . . . . . . . . . . . . . . . 69
4.7
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 A perturbed box: Splitting the bifurcations
5.1
5.2
73
2D box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.1
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.2
Dynamical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3D box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.1
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.2
Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.3
Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.4
Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.5
Example 3: The cusp bifurcation . . . . . . . . . . . . . . . . . . . . 115
5.3
Comparisons to previous work . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.4
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6 Four interacting modes
123
6.1
Perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2
Unperturbed box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3
Perturbed box
6.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3.1
Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3.2
Quaternary bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . 133
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7 Conclusion
143
A Calculation of polynomial coefficients
151
B Bifurcation Trees
155
B.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
B.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
List of Figures
1.1
Temperature contour lines under the effects of convection . . . . . . . . . .
2
2.1
The classic Horton-Rogers-Lapwood problem . . . . . . . . . . . . . . . . . 10
2.2
Convection cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3
Numerical results for the fluid flow and temperature . . . . . . . . . . . . . 15
2.4
Dynamic viscosity data (dots) and the reciprocal linear approximation (solid
line) between T = 40◦ and T = 100◦ . . . . . . . . . . . . . . . . . . . . . . 19
2.5
The change in the critical Rayleigh number as a function of φ̂∆T , for α = π 24
2.6
Dynamic viscosity data (dots) and the reciprocal linear approximation about
the harmonic mean (solid line) between T = 40◦ and T = 100◦ . . . . . . . 25
2.7
The change in the critical Rayleigh number as a function of θ̂∆T . . . . . . . 27
3.1
The critical Rayleigh number as a function of L for the one-cell, (p = 1),
and two-cell, (p = 2), modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2
Streamlines of the single- and double-cell modes. . . . . . . . . . . . . . . . 31
3.3
The phase-space of the dynamical system. . . . . . . . . . . . . . . . . . . . 38
4.1
The preferred mode, (p, q), at the onset of convection . . . . . . . . . . . . . 43
4.2
The three modes which share the same critical Rayleigh number for example
1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3
The 3D phase-space for example 1 . . . . . . . . . . . . . . . . . . . . . . . 54
4.4
Two manifolds, each sharing the trajectory shown in red . . . . . . . . . . . 55
4.5
The convection pattern of the (1, 1) mode in the box, with Lx = Ly = 21/4 . 55
4.6
The 3D phase-space for example 2 . . . . . . . . . . . . . . . . . . . . . . . 57
4.7
The 2D quotient system (the 2D unstable invariant manifold of GABC ) . . . 58
4.8
The ellipse formed by the intersection of the non-spherical nullclines . . . . 60
4.9
Categorisation of each system of three modes . . . . . . . . . . . . . . . . . 62
4.10 A schematic of the boundaries of the basins of attractions and the fixed
points in the C ≥ 0, B ≤ A quadrant. . . . . . . . . . . . . . . . . . . . . . 67
4.11 The 2D quotient system, (the 2D unstable separatrices of H4+ , H5+ and
HA+B− ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1
The critical Rayleigh number as a function of the horizontal aspect ratio, L
74
xii
List of Figures
5.2
The location of the fixed points and nullclines in the 2D phase-space . . . . 79
5.3
The bifurcation diagrams for, (a), ∆ = −1; and, (b), ∆ = 1 . . . . . . . . . 80
5.4
The bifurcation diagram for the unperturbed case, ∆ = 0 . . . . . . . . . . 80
5.5
The horizontal dimensions of the box are perturbed
5.6
Example 1 in Lx –Ly space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.7
Example 1: A schematic of the bifurcation diagram as R1 increases for θ = 0 87
5.8
Example 1: A schematic of the bifurcation diagram as R1 increases for θ = 4 89
5.9
Example 1: A schematic of the quotient system for θ = 4 as R1 increases . . 90
. . . . . . . . . . . . . 81
5.10 Example 1: Bifurcation set diagram in R1 − θ space . . . . . . . . . . . . . 92
5.11 Example 2 in Lx –Ly space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.12 Example 2: The bifurcation set diagram in R1 − θ space . . . . . . . . . . . 97
5.13 Example 2: Bifurcation tree for θ < 0.32 or θ < 5.96 . . . . . . . . . . . . . 98
5.14 Example 3 in Lx –Ly space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.15 Example 3: Bifurcation set diagram in R1 − θ space . . . . . . . . . . . . . 104
5.16 The ellipse nullcines on the C = 0 plane for θ = 2.5 . . . . . . . . . . . . . . 105
5.17 The ellipse nullcines on the C = 0 plane for θ = 3.75 . . . . . . . . . . . . . 105
5.18 Bifurcations on the r = 0 “plane” . . . . . . . . . . . . . . . . . . . . . . . . 108
5.19 C Secondary bifurcation: Nullclines for θ = 2.8 . . . . . . . . . . . . . . . . 109
5.20 Saddle node bifurcations for θ = 2.8 . . . . . . . . . . . . . . . . . . . . . . 111
5.21 Example 3: The bifurcation lines in R1 − θ space . . . . . . . . . . . . . . . 112
5.22 Example 3: The bifurcation diagram for θ = 0 . . . . . . . . . . . . . . . . . 113
5.23 Example 3: The evolution of the fixed points, manifolds, and basins of
attraction for θ = 0 on the quotient system . . . . . . . . . . . . . . . . . . 114
5.24 Cusp bifurcation: breaking the symmetry . . . . . . . . . . . . . . . . . . . 116
5.25 Cusp bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.26 Bifurcation lines along Lx = Ly as approximated by perturbing about the
Lx0 = Ly0 = 21/4 critical box . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.1
Example 4 in Lx –Ly space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2
This 3D quotient system represents the 4D dynamics . . . . . . . . . . . . . 127
6.3
Example 4: Bifurcation set diagram in R1 –θ space . . . . . . . . . . . . . . 134
6.4
Example 4: Quaternary and tertiary bifurcation set diagram in R1 –θ space 137
6.5
Example 4: Bifurcation diagram for θ > 6.25 or θ < 0.46 as R1 increases . . 138
6.6
Continued in figure 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.7
Example 4: The evolution of the fixed points on the quotient system in the
positive hyperoctant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.1 −0.64 < θ < 0.79 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.2 −0.79 < θ < 1.57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B.3 1.57 < θ < 2.36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B.4 2.36 < θ < 3.63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.5 3.63 < θ < 3.93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
List of figures
xiii
B.6 3.93 < θ < 4.32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B.7 4.32 < θ < 4.71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B.8 4.71 < θ < 5.50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.9 5.50 < θ < 5.64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.10 −0.32 < θ < 0.32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
B.11 −0.32 < θ <
B.12
π
4
π
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
< θ < 1.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
B.13 1.25 < θ < 1.89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
B.14 1.89 < θ < 2.82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5π
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
< θ < 5.03
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.15 2.82 < θ <
B.16
5π
4
B.17 5.03 < θ < 5.96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
B.18 −0.32 < θ <
B.19 θ =
B.20
π
4
π
4.
π
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
At the “cusp” a pitchfork bifurcation is observed. . . . . . . . . . . . 166
< θ < 1.89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
B.21 1.89 < θ < 2.73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.22 2.73 < θ < 2.82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.23 2.82 < θ < 2.90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B.24 2.90 < θ < 3.44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5π
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
< θ < 4.42
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B.25 3.44 < θ <
B.26
5π
4
B.27 4.42 < θ < 4.95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.28 4.95 < θ < 5.04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.29 5.04 < θ < 5.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.30 5.12 < θ < 5.96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.31
5π
4
< θ < 4.31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B.32 4.31 < θ < 4.82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.33 4.82 < θ < 4.83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.34 4.83 < θ < 5.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.35 5.10 < θ < 5.28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B.36 5.28 < θ < 5.30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B.37 5.30 < θ < 5.55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B.38 5.55 < θ < 5.70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.39 5.70 < θ < 6.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.40 6.17 < θ < 6.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.41 θ > 6.25 or θ < 0.46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.42 0.46 < θ <
π
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
xiv
List of figures
List of Tables
3.1
Properties of the fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1
Example 1: All fixed points and their stabilities . . . . . . . . . . . . . . . . 53
4.2
Example 2: All fixed points and their stabilities . . . . . . . . . . . . . . . . 57
4.3
Conditions for which a special case occurs for two modes . . . . . . . . . . . 63
4.4
Conditions for which a special case occurs for three modes . . . . . . . . . . 64
4.5
Example 3: All fixed points and their stabilities . . . . . . . . . . . . . . . . 66
5.1
The bifurcation Rayleigh numbers for the cube box . . . . . . . . . . . . . . 120
5.2
The bifurcation Rayleigh numbers for the “stretched” box . . . . . . . . . . 121
6.1
Example 4: The fixed points and their stabilities in the positive hyperoctant 126
6.2
Example 4: The fixed points and their stabilities on the representative 3D
manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
xvi
List of tables
Chapter 1
Introduction
The current drive for renewable resources has inspired interest in geothermal energy. Perth,
Western Australia, is not known to be volcanically active, and would not immediately be
considered as a prime location to use geothermal energy if not for the Perth Basin. This
deep sedimentary basin spans approximately 1000km parallel to the coast, in the southwest area of Western Australia (Playford et al., 1976). The Perth Basin provides both the
metropolitan and surrounding rural areas with a wealth of groundwater (Davidson, 1995).
It has been proposed that the Perth Basin can be used to provide a low-temperature
geothermal energy. Geothermal energy can be obtained by extracting heat from the fluid
drawn from the depths with a bore. While the target fluid is not hot enough for viable
electricity production, it can be efficiently used for direct-heat purposes. For example, such
a system could directly power central air-conditioning units and desalination plants, with
an operational temperature above 65◦ (Regenauer-Lieb et al., 2009). Low-scale geothermal projects requiring an operational temperature of approximately 40◦ have already been
implemented, including year-round geothermally heated swimming pool at Challenge Stadium, and at St Hilda’s Anglican School for Girls.
Groundwater is able to flow in the sedimentary deposits, and can be modelled using
Darcy’s law for fluid flow in porous media (Darcy, 1856). As this fluid is free to flow, (albeit very slowly), there is a possibility of convection occurring, where water geothermally
heated from below becomes less dense, and is driven upwards by buoyancy. If convection
were to occur, this assists in transporting heat from the depths to the surface. Identifying where the “upwellings” of hot fluid occur can substantially reduce the cost of the
project; higher temperatures are found closer to the surface, so the bore depth required
to obtain the operational temperature of fluid, (and thus cost), is reduced. The presence
of convection would also assist in the replenishment of hot fluid to the local extraction
region. Measurements taken from existing bores within the Perth Basin show a temperature difference, which may be the result of convection (Sheldon et al., 2012). Theoretical
models give insight into the behaviour of the fluid and temperature profiles, assisting in
the predictions of where the upwellings occur. A schematic of the temperature contour
lines under the effect of convection is shown in figure 1.1.
2
Chapter 1. Introduction
A
B
Figure 1.1: Temperature contour lines under the effects of convection. Plant A, situated above a cold downwelling, must bore to a greater depth to obtain the operational
temperature, increasing project cost.
Theoretical modelling is important in the geothermal industry, as data is scarce
and expensive to obtain from fresh bores. Bore data is dense in the vertical direction,
but localised in the horizontal directions, so spatial knowledge of the aquifer is limited.
An understanding of the theoretical principles underlying the groundwater behaviour will
supplement the physical measurements.
1.1
Convection
Convection is a process of heat transfer, where heat energy is transported by the motion of
particles in a fluid. This heat also diffuses to neighbouring particles by conduction. If such
mass transport is the result of a pump, wind, or any other external effects, it is known as
forced convection. Natural convection arises where a density gradient in the presence of
an external field causes fluid motion. For example, a density gradient in a system subject
to gravity may experience buoyancy forces (Kays et al., 2005). A well-studied problem is
Rayleigh-Bénard convection, where a horizontal layer of fluid is heated from below and a
density inversion occurs; hot fluid at the bottom of the layer becomes less dense than the
fluid above it, and seeks to rise under the buoyancy forces. By this instability, convection
patterns are generated, where fluid circulation transports heated mass upwards, and colder
mass downwards.
Convection in a saturated porous medium is a much simpler process, in the sense that
the no-slip boundary condition is not required to model fluid flow in porous media (Nield &
Bejan, 2006). Boundary layer phenomena are not seen, except in extreme circumstances.
As well as being able to model geothermal systems, convection in porous media problems
give insight into the behaviour of their non-porous equivalents without being mired in
boundary-layer details.
One of the simplest approaches to convection in porous media is known as the HortonRogers-Lapwood problem. This problem was first addressed by Horton & Rogers (1945)
and independently by Lapwood (1948). Consider a saturated porous layer of infinite horizontal extent, where the top and bottom are held at a constant temperature, such that
the bottom is hotter. This is the porous media equivalent of Rayleigh-Bénard convection.
1.2. Pattern formation of convection modes in porous media
3
Darcy’s law, coupled with an energy conservation equation, is used to model the fluid
flow and heat distribution in the system. The fluid is considered as incompressible, and a
nonlinear conservation of energy equation models how heat is distributed by conduction
and convection. This is a simple model; all physical parameters, such as the permeability
of the rock matrix, fluid density, and fluid viscosity, are considered constant throughout
the entire system. While fluid viscosity and density are well known to change with temperature, and must therefore change throughout the system, this approximation leads to
a tractable solution. Physically, it is the buoyancy forces which drive convection. To
appropriately model the convection phenomenon, the density of the fluid must be allowed
to vary in the gravity term of Darcy’s law.
Linear stability analysis can determine the conditions for which convection assists in
the transportation of heat energy, or when this energy is transported by conduction alone.
The Rayleigh number is a measure of many different physical parameters of the system
including those mentioned above. If the Rayleigh number is above a certain critical value,
then convection theoretically occurs. Details on the analysis of this problem are found in
Chapter 2.
Physical systems are not of infinite extent, so convective systems in finite domains
should be considered. By restricting the domain to a rectangular box, only discrete modal
convection patterns (“modes”) can be seen. For example, in the case of a two-dimensional
(2D) box, any integer number of cells may be aligned vertically or horizontally. Each of
these convection patterns have upwellings in different places, so identifying which convection pattern occurs will assist in finding a suitable location for an extraction bore. From
one point of view the problem can be seen as a system of coexisting competing modes.
These modes interchange energy over a period of time until a steady solution is reached.
This steady solution may be a single mode by itself, or a sum of many viable modes.
In most domains, there is a clearly favoured mode, with a low critical Rayleigh number.
By carefully choosing the domain such that multiple modes are viable, the interaction
of these modes can be studied in a weakly nonlinear regime. These particular domains,
where multiple modes are equally viable and share the same critical Rayleigh number, are
called “critical boxes”.
1.2
Pattern formation of convection modes in porous media
The aim of this thesis is to provide a theoretical investigation of the type of interaction
and competition seen between convection patterns. Similar studies have been performed
in the past, both analytical and numerical, and are noted here.
Many numerical studies of pattern selection in rectangular boxes have been documented
in literature (see Horne & O’Sullivan (1978), Horne (1979), Horne & Caltagirone (1980),
Straus & Schubert (1979), Straus & Schubert (1981), Caltagirone et al. (1981), Riley &
Winters (1991)) and deal with high Rayleigh number flows. Many analytical studies are
4
Chapter 1. Introduction
also available (see Kordylewski & Borkowska-Pawlak (1983), Néel (1990a), Néel (1990b)).
By comparing the heat transfer rates (Nusselt numbers) of three different modes, Zebib
& Kassoy (1978) showed that 2D “roll” patterns are preferred over 3D “cell” patterns for
low Rayleigh numbers.
Riley & Winters (1989), studied the bifurcation process of interacting modes in a 2D
box as the Rayleigh number is increased, using both analytical and numerical methods.
The analytical investigation is conducted using a perturbation method. From the governing fluid dynamics partial differential equations (PDEs), a system of ordinary differential
equations (ODEs) is derived which governs the growth and decay of the two possible convection modes. The results are identical to the system discussed in chapter 3 of this thesis.
In a further collaboration, the authors introduce sidewall heat imperfections, which breaks
the symmetry of the pitchfork bifurcations (Impey et al., 1990). In this paper, the authors
claim that the perturbation method is inadequate to use in special cases, as the symmetrybreaking terms are not identified. It will be shown in chapter 4 that with appropriate care,
the perturbation method can resolve such behaviour, (see section 4.1.3). Due to the ease
of which the perturbation method can be applied to many examples, and will be shown
to be appropriate, it is used exclusively in this thesis. Borkowska-Pawlak & Kordylewski
explored the effect of changing the Prandtl values in a square box, (see Borkowska-Pawlak
& Kordylewski (1982), Kordylewski et al. (1983)), and showed the bifurcations that occur
in a 3D box using a Galerkin method, (Borkowska-Pawlak & Kordylewski, 1985). They
found that solutions which arise can be quite sensitive to the box geometry. Physical
effects also influence the pattern selection. Vincourt studied the effect of a non-uniformly
heated system, (Vincourt, 1989a), and a system with heterogeneous porosity (Vincourt,
1989b) while Impey & Riley (1991) explored the effect of a tilted domain.
Steen (1983) used an eigenfunction expansion technique to determine the eventual
steady-state solution of a 3D box with Lx = Ly = 21/4 . A box with these aspect ratios
is a critical case for which three eigenfunction solutions share the same critical Rayleigh
number. Other examples are also considered, including a cubic box, and a near-cubic box
elongated in the y-direction. For these examples, the critical Rayleigh numbers of each
mode are no longer same, and the eigenvalue is said to be ‘split’. After reducing the governing equations to a system of ordinary differential equations, Steen (1983) demonstrated
that each eigenfunction has a finite-amplitude stable solution. The eigenfunction which
is realised depends on the initial perturbation. Other solutions were found, including the
conduction solution and states for which multiple eigenfunctions exist but all of these are
unstable. The results of Steen (1983) cannot be used to infer the behaviour of all degenerate cases because in his study, symmetry between positive and negative amplitudes
(direction of flow) is assumed, and the evolution equations are given in terms of the amplitude squared. This hides the fact that special cases may occur where the direction of
flow affects the exchange between eigenfunctions. An example of this symmetry-breaking
case is analysed in section 4.5. Such an example was actually sought by Steen (1986), as a
necessary (but not sufficient) condition, to see time-dependent convection solutions close
1.3. Thesis structure.
5
to the onset of convection.
While the interaction of modal solutions has been analysed for particular examples in
the literature, a complete story is still missing; in this thesis, the behaviour close to all 3D
critical boxes with moderate aspect ratios are considered, and the qualitative bifurcation
structure outlined. Each of the examples can be categorised into one of four equivalence
classes, and one example from each equivalence class is analysed in detail. For future
research, this categorisation provides a quick reference for which the type of interactions
seen between the modes can be found.
1.3
Thesis structure.
A basic introduction to fluid dynamics in porous media is given in chapter 2. A model of
the heat and mass transfer in porous media is presented, and the possibility of convection
occurring in the Horton-Rogers-Lapwood problem is reviewed. Furthermore, other notable
issues with the simplified model are reviewed and re-examined, where physical parameters
are allowed to vary, as opposed to the approximation they are constant. A temperaturedependent viscosity and varying permeability (ease of which the fluid travels through the
medium) is discussed.
The generation and interaction of convection modes is introduced in chapter 3, where
a simple 2D box with two viable modes is examined. The 2D example is an intuitive
introduction to the analytical procedures and behaviour which will be expanded upon in
further chapters, where 3D boxes are considered.
In chapter 4, 3D boxes for which three modes are viable are analysed. It is found that
different critical box examples show different interchanges between the modes. Each of
these critical boxes are systematically categorised into 2 classes, each of which shows a
different relationship between the modes, and are two of the aforementioned equivalence
classes. There also exists a symmetry-breaking special case, where the convection dynamics
interact in a unique way, not seen in any other three-mode examples at the onset of
convection. This is the only example of a system belonging to the third equivalence class
and is an important example to study, as it has not been discussed in the literature.
The “critical” boxes have exact aspect ratios, and are unlikely to be found in a realworld situation. By investigating examples arbitrarily close to these critical boxes, the
modes will no longer share a critical Rayleigh number. If the example box is close enough
to a critical box, then the critical Rayleigh numbers of each mode will be close together,
and the behaviour of the solutions will remain weekly nonlinear. As the Rayleigh number
is increased, the existence and stability of solutions are seen to change in stages. This
process is known as bifurcation, and is investigated in chapter 5.
In a highly symmetrical domain, it is possible for up to four modes to share a critical
Rayleigh number at the onset of convection. The behaviour of such an example is analysed
in chapter 6, and belongs to the fourth equivalence class.
All of the analysis is valid close to the onset of convection. This may not always
6
Chapter 1. Introduction
be the case in the real world. The analysis shown here, however, provides a guide to
the qualitative behaviour that can be seen for interacting modes, where the problem is
nonlinear and analytical solutions are not available.
Chapter 2
Modelling convection in porous
media
The Horton-Rogers-Lapwood problem was defined in the previous chapter. Here, the
problem is analysed, and conditions for which convection occurs are sought. Other key
issues in modelling convection in porous media are addressed. The fundamental equations
for heat and mass transfer in a saturated porous media are examined in section 2.1. The
standard problem, where all physical parameters are constant, is analysed in section 2.2.
In practise, this is not always the case; permeability can vary greatly depending on the
rock matrix location, and viscosity is known to be temperature-dependent, and must vary
in systems with a temperature gradient. Literature on convection in porous media with
varying permeability and viscosity are reviewed in sections (2.3) and (2.4), respectively.
The necessity of increasing the complexity of the model by introducing temporal and
spatially dependent parameters, is investigated in both cases.
2.1
Fluid mechanics in porous media
A brief summary of the equations of state used to model fluid flow in porous media is
given below. Unless otherwise stated, the arguments are those presented in Nield & Bejan
(2006), and the reader is directed to this text for a greater level of detail.
At a fundamental level, modelling the flow of fluid through a porous medium is quite
complicated. Depending on the porosity, the structure and the connectivity of the pores,
individual fluid particles may take wild and erratic paths as they travel through the
medium. To avoid these complexities, a continuum approach is sought, where local properties of the rock matrix, and fluid streamlines are averaged. The length-scale of the
averaging procedure must be dealt with carefully. Consider the porosity of the rock matrix, measured as the relative volume of the voids (pore space), compared to the overall
volume of the rock matrix and voids. One can find the average porosity by measuring
the relative volume over the entire domain, however, this procedure may ignore useful
detail. Conversely, averaging over an infinitesimal region will yield a porosity value of 0
8
Chapter 2. Modelling convection in porous media
or 1, depending on whether the region in question lies inside a void or in the rock matrix,
respectively. The perfect scale size would be as small as possible, but much larger than
the size of the pores. A volume of this scale size is known as a representative elementary
volume (REV). For more detail on the technical considerations, please see chapter 1 in
Bear (1972).
This averaging procedure introduces the concept of permeability: the ease of which
the medium allows fluid to traverse. This is often related to the porosity, as a higher pore
space allows more fluid to flow. This is not always the case: permeability is also highly
sensitive to the geometry of the pores. To illustrate this point, Nield & Bejan (2006) use
the following example: Consider a series of parallel, horizontal pipes. While fluid can
easily travel horizontally through the pipes, there will be no vertical flux, regardless of the
porosity of the pipes. Thus, porosity and permeability are distinct, but related concepts.
In this continuum model, the true fluid velocity is no longer considered; the fluid velocity averaged over the REV is used, and is known as the Darcy velocity. The true velocity
will no longer be discussed, so the term “velocity” is used interchangeably with “Darcy
velocity”.
Three equations are used to model the fluid and heat flow in a porous medium, and all
derive from the conservation equations in mass, momentum and energy. The conservation
of mass equation is given by
ϕ
∂
ρf + ∇ · (ρf v̂),
∂ t̂
(2.1)
where ϕ, t̂, ρf , and v̂ are the porosity, time, fluid density and Darcy velocity, respectively.
For the conservation of momentum, Darcy’s law is used. Darcy’s law is based on
the experimental work of Darcy (1856), and is considered the porous media analogue of
the Navier-Stokes equation. Based on empirical data, Darcy developed a model which
describes how the velocity flows with respect to pressure gradient. The relationship is
given by
∇P̂ = −
µ
v̂,
K
(2.2)
where P̂ , µ and K are the pressure, dynamic viscosity and permeability, respectively.
Darcy’s law can be extended to include acceleration, and is given by
ρf ĉa ·
∂
µ
v̂ = −∇P̂ − v̂,
K
∂ t̂
(2.3)
where ĉa is a tensor used to describe the nature of the pores, and how fast local transients
decay (Nield & Bejan, 2006). While Darcy’s law was based on empirical data, it can be
derived from the Navier-Stokes equation by an averaging process. Darcy’s law is a good
model to use for low values of v̂. For large values of v̂, where the local Reynolds number is
greater than unity, quadratic drag effects cannot by ignored, and an inertia term should be
added to Darcy’s law (see Joseph et al., 1982). For the purpose of this thesis, the dynamics
at the onset of convection are studied, and the Darcy velocity is small, and Darcy’s law
(2.3) is valid.
2.2. When does convection occur?
9
The energy equation can be found by considering the transport of heat in both the
solid rock matrix and fluid. Using the subscripts “s” and “f ”, to denote the solid and
fluid properties, the two energy equations are given by
∂
T̂s = (1 − ϕ)∇ · (ks ∇T̂s ) + (1 − ϕ)qs000 ,
∂ t̂
(2.4)
∂
T̂f + (ρcP )f v̂ · ∇T̂f = ϕ∇ · (kf ∇T̂f ) + ϕqf000 ,
∂ t̂
(2.5)
(1 − ϕ)(ρc)s
and
ϕ(ρc)f
where k is the thermal conductivity, q 000 is the heat input per unit volume, and c and
cP are the specific heat parameters, with the “P ” subscript denoting constant pressure.
Assuming the rock matrix and fluid are in local thermal equilibrium (within a REV), such
that Ts = Tf , the total energy equation can be given by adding equations (2.4) and (2.5)
to give
(ρc)m
∂
000
T̂ + (ρcP )f v̂ · ∇T̂ = ∇ · (km ∇T̂ ) + qm
,
∂ t̂
(2.6)
where the properties of the solid/fluid mixture, subscript m, are averaged for a quantity,
X, by
Xm = (1 − ϕ)Xs + ϕXf .
(2.7)
The assumption of thermal equilibrium is valid where the Darcy velocity is slow, and thermal communication between the solid matrix and fluid occurs at a much faster timescale.
This is consistent with the assumption made for the validity of Darcy’s law, and is appropriate in the geothermal context.
The equations, (2.1), (2.3) and (2.6), are used as the governing partial differential
equations for the model. Assumptions and boundary conditions are introduced in the
relevant chapters.
2.2
When does convection occur?
In saturated porous media systems, heat can be transported throughout the layer by conduction alone. For systems with uneven heating, where density gradients are generated,
convection may occur, assisting in the distribution of heat energy. For geothermal systems, heated from below, it has been observed experimentally that convection may not
necessarily occur, and all heat is transferred via conduction. If the temperature gradient
is large enough, however, convection is observed. There is a variety of tools available to
investigate this problem. Where the buoyancy forces are sufficiently high, the viscous drag
is overcome, and fluid is able to flow. Convection assists in the transfer of heat from the
bottom of the layer, to the top. Both a linear stability analysis technique, and a finite
elements package are used to model the problem, (see sections 2.2.1 and 2.2.2 respectively).
10
Chapter 2. Modelling convection in porous media
T̂ = T0
ẑ
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Saturated Porous Media
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H
T̂ = T0 + ∆T
x̂
Figure 2.1: The classic Horton-Rogers-Lapwood problem. The temperature at the bottom
boundary of the layer is greater than the temperature at the top. The temperature
inversion allows for convective flows under certain circumstances.
2.2.1
Linear stability analysis
To determine the conditions for which the conduction solution is viable, a linear stability
analysis is performed. This problem was first investigated by Horton & Rogers (1945),
and independently by Lapwood (1948).
The linear stability analysis of this problem is covered quite succinctly in chapter 6 of
Nield & Bejan (2006). This work is the basis for further research in this thesis, and as
part of this literature review, the details are included below.
A three-dimensional (3D) saturated layer of a porous medium of infinite horizontal
extent is considered. The domain has finite vertical height, H, with the top and bottom
boundaries held at a constant temperatures, T0 and T0 + ∆T respectively, with ∆T > 0.
The top and bottom boundaries are closed, and do not permit transmission of fluid. Figure
2.1 shows a vertical 2D slice of the problem.
The continuity equations, (2.1), (2.3) and (2.6), are used to model this system. For
an incompressible fluid, with no internal heat generation, these equations are given by
∇·v̂ = 0,
ĉa ρ0
(ρĉ)m
∂ v̂
µ
= −∇P̂ − v̂ + ρf g,
K
∂ t̂
∂ T̂
+ (ρĉP )f v̂·∇T̂ = km ∇2 T̂ .
∂ t̂
(2.8)
(2.9)
(2.10)
Under the Oberbeck-Boussinesq approximation, (Oberbeck, 1879; Boussinesq, 1903), all
physical parameters are assumed to be constant, with one exception; for convection to
occur, the density of the fluid must be allowed to vary with temperature in the gravity
term of (2.9), so that a buoyancy force exists. If density were held constant everywhere,
2.2. When does convection occur?
11
no convection would occur in this model. As a first approximation, the density of the fluid
is given by
ρf = ρ0 [1 − β(T̂ − T0 )],
(2.11)
where β is the thermal expansivity.
A steady-state solution, (v b , Tb , Pb ), exists where all heat is transfered via conduction.
This solution is given by
v b = 0,
Tb = T0 + ∆T (1 −
(2.12)
ẑ
),
H
1
ẑ 2
Pb = P0 − ρ0 g[ẑ + β∆T ( − 2ẑ)],
2
H
(2.13)
(2.14)
where x̂ = (x̂, ŷ, ẑ), with ẑ being the vertical coordinate. To explore the stability of the
conduction solution and find the form of the instabilities, the variables are perturbed:
v̂ = v b + v 0 ,
T̂ = Tb + T 0 ,
(2.15)
P̂ = Pb + P 0 ,
where the primed variables are relatively small. Arranging equations (2.8)-(2.10) in terms
of the perturbed variables gives
∇ · v 0 = 0,
ĉa ρ0
(2.16)
∂v 0
µ
= −∇P 0 − v 0 − βρ0 T 0 g,
K
∂ t̂
(2.17)
∂T 0
∆T 0
− (ρcP )f
w = km ∇2 T 0 ,
H
∂ t̂
(2.18)
(ρc)m
where v 0 = (u0 , v 0 , w0 ) and second order perturbation terms are ignored. The variables are
non-dimensionalised by applying the characteristic scales used by Nield & Bejan (2006):
x̂
,
H
Hv 0
v=
,
αm
x=
Here αm =
km
(ρĉP )f ,
and σ =
(ρĉ)m
(ρĉP )f .
αm t̂
,
σH 2
T0
T =
,
∆T
KP 0
P =
.
µαm
t=
(2.19)
Note also that x = (x, y, z) and v = (u, v, w). Using
values from the Perth Basin, the characteristic velocity is the order of centimeters per
year. By non-dimensionalising, equations (2.16)-(2.18) become
∇ · v = 0,
(2.20)
12
Chapter 2. Modelling convection in porous media
γa
∂v
= −∇P − v + RaT k,
∂t
(2.21)
∂T
− w = ∇2 T.
∂t
(2.22)
k is the unit vector in the z–direction, such that g = −gk, and the dimensionless parameters of the problem are given by
Ra =
ρ0 gβKH∆T
,
µαm
γa =
ca K
,
σP rm H 2
P rm =
µ
.
ρ0 αm
(2.23)
The Rayleigh number, Ra, is critical in determining the stability of the system. It quantifies the balance between buoyant forces and viscous drag effects. The parameter, γa , is
typically very small for slow moving fluids, and is set to zero. Consequently, the Prandtl
number, P rm , is does not contribute. This leaves the Rayleigh number as the only dimensionless parameter in the system. While many different physical parameters describe the
problem, spanning a multidimensional parameter space, one only needs to calculate the
dimensionless Rayleigh number to characterise the system.
It is well known that in these convection systems, the three dimensional problem in
(x, y, z) can be reduced to a two dimensional problem in (x, z) without loss of generality
(Nield, 1996). This allows the introduction of the two dimensional stream function, ψ,
such that v = (ψz , 0, −ψx ); the subscripts denote partial differentiation with respect to
that variable. The incompressibility equation, (2.20) is identically true, while Darcy’s law,
(2.22), becomes
Tt + ψx = ∇2 T.
(2.24)
Equation (2.21) is separated into its components and a cross product is used to eliminate
the ∇P term, resulting in the scalar equation,
− RaTx = ∇2 ψ.
(2.25)
Equations (2.24) and (2.25) form a set of linear, coupled PDEs. As such, the temporal
growth of the solutions is expected to take the form, exp(ŝt), where ŝ may be complex.
Due to the translational symmetry of the infinite layer, solutions are not expected to grow
in the x direction, and can be assumed to be a periodic function. Due to linearity, the
variables can be separated, and the solutions are assumed to take the form
ψ(ẑ, t) = F (z) exp(ŝt + iαx)
(2.26a)
T̂ (ẑ, t) = G(z) exp(ŝt + iαx).
(2.26b)
Substituting these solutions into equations (2.24) and (2.25) result in a pair of ODEs in
z:
ŝG(z) + iαF (z) = (D2 − α2 )G(z),
(2.27)
− iαRaG(z) = (D2 − α2 )F (z),
(2.28)
2.2. When does convection occur?
where D =
∂
∂z ,
13
ŝ is some growth factor, and α is the horizontal wavenumber. It can be
clearly seen from assumption (2.26), that if the real part of ŝ, <(ŝ), is greater than zero,
the perturbation variables ψ and T will grow larger, implying that the conduction solution
is no longer stable. Similarly the conduction solution is stable for <(ŝ) < 0 and convection
cannot occur. The critical value of ŝ for which the stability changes is at <(ŝ) = 0. By
substituting ŝ = ωi into equations (2.27) and (2.28), it is found that ω = 0. Thus, the
solutions are non-oscillatory in nature, and the principle of exchange of stabilities applies.
Hence, setting ŝ = 0 allows one to determine the critical Rayleigh number, Rac , for which
convection occurs, giving
iαF (z) = (D2 − α2 )G(z),
(2.29)
− iαRac G(z) = (D2 − α2 )F (z).
(2.30)
On the top and bottom boundaries, T = 0. This gives G(0) = G(1) = 0. Similarly, for
a closed system with impermeable boundaries at the top and bottom, ψ = 0. This gives
F (0) = F (1) = 0. G(z) can be eliminated to give the fourth order differential equation:
(D2 − α2 )2 F (z) = α2 Rac F (z),
(2.31)
with F (0) = F (1) = F 00 (0) = F 00 (1) = 0. The non-trivial solutions to this eigenvalue
problem are given by
F (ẑ) = Asin(jπz),
with
Rac =
((jπ)2 + α2 )2
,
α2
(2.32)
(2.33)
where A is an arbitrary constant, and j is the vertical mode number, (j = 1, 2, 3, . . .).
The minimum possible critical Rayleigh number is Rac = 4π 2 for j = 1, and α = π. This
corresponds to a horizontally-aligned series of convection rolls, each with a width and
height equal to the height of the layer (see figure 2.2). The overall form of the instabilities
is given by
ψ(x, z) = A sin(jπz)eiαx ,
(jπ)2 + α2
T (x, z) = A
sin(jπz)ei(αx−π/2) .
Rac
(2.34a)
(2.34b)
The imaginary part of the solutions (2.34) are plotted in figure 2.2. Taking the real part
is equally valid, resulting in a translated solution, which is invariant in the infinite layer.
Note that the streamlines for fluid flow are the contour lines of ψ.
2.2.2
Numerical verification
To use a finite elements package to model this convection problem, the features of the linear
stability analysis should be seen in the results. That is, the numerical results should show
14
Chapter 2. Modelling convection in porous media
ψ = Tx = 0
ψ
-
+
-
+
-
+
T
z
x
ψ = Tx = 0
Figure 2.2: Convection cells. The top and bottom layers show the fluid streamlines and
T contours respectively. While arrows denote direction of fluid flow, ‘+’ and ‘-’ symbols
denote hot and cold spots respectively. The solutions periodically satisfy ψ = Tx = 0.
that convection occurs only for Rac > 4π 2 . The solution should also take the form given
by equation (2.32).
The aim here is to numerically model the scaled governing equations, (equations (2.20)–
(2.22)), in their nonlinear form. Recall that in an earlier step, the nonlinear terms were
discarded, as they are very small. The scaled nonlinear governing equations are given by
− RaTx = ∇2 ψ,
(2.35)
Tt + ψz Tx − ψx Tz + ψx = ∇2 T.
(2.36)
Equations (2.35) and (2.36) are numerically solved using FlexPDE 5.1.0s 3D. For these
simulations, side boundaries must be imposed. This influences the horizontal wavelength
of the convection cells. The appropriate choice is that the side boundaries are a scaled
distance of 1 apart, which corresponds to the cell width for α = π, (for which the minimum
critical Rayleigh number occurs). The box walls are assumed to be impermeable:
ψ(x, 0, t) = ψ(x, 1, t) = ψ(0, z, t) = ψ(1, z, t) = 0.
(2.37)
This condition implies that all flow at the boundaries must be parallel to that boundary.
The side-walls are considered to be heat-insulating. The temperature boundary conditions
are:
T (x, 0, t) = T (x, 1, t) = Tx (0, z, t) = Tx (1, z, t) = 0.
(2.38)
Introducing these boundary conditions does not change the linear stability problem. The
solution form given in the infinite layer case, (2.34), satisfies these boundary conditions
2.3. Varying permeability
1.
15
1.
x
n
m
a
d
0.8
0.8
h
o
g
j
n
d
c
i
z
0.6
s
j
o
g
p
0.4
b
f
0.2
d
r
l
f
j
h
r
l
u
o
b
m
p
i
k
i
0.2
b
x
s
f
c
e
n
l
k
k
g
a
a
t
a
0.4
q
a
u
z
o
h
m
q
0.6
o
e
c
e
a
0.
0.
0.
0.2
0.4
0.6
0.8
1.
0.
0.2
0.4
0.6
0.8
1.
x
x
(a) Fluid streamlines
(b) Temperature contour lines
Figure 2.3: Numerical results for the fluid flow and temperature. The slight asymmetry
in the temperature contour lines is due to small nonlinear effects.
by definition. Assuming j = 1, the imaginary part of ψ and T is given by
ψ(x, z) = A sin(πz) sin(πx),
A
T (x, z) = − sin(πz) cos(πx).
2π
(2.39a)
(2.39b)
It can be seen that ψ(n, z) = 0 = Tx (n, z) = 0, for any integer, n. Hence, the boundary
conditions (2.37) and (2.38), do not change the essence of the problem at the onset of
convection, (see figure 2.2 for a graphical description).
From these simulations, it is seen that the critical Rayleigh number is 39.481 (compared
with the analytical result of 4π 2 = 39.478...). This shows that the numerical method can
produce accurate results for the critical Rayleigh number. The numerical solutions for
Ra= 40 are shown in figure 2.3.
2.3
Varying permeability
One of the most variable parameters in the field of groundwater flow, is permeability.
Permeability measurements within a domain, can physically vary by several orders of
magnitude. This occurs in the Perth basin, for example (see Smith (1967) and McKibbin
et al. (2011)). Coupled with the fact that the permeability within a domain is very difficult
to measure, it can often be misrepresented in the model. Another property of the rock
matrix, which is difficult to measure in the entire domain, is the thermal conductivity, km ,
though it does not vary to a degree that the permeability does.
One way of allowing more detail in the permeability, is to use an anisotropic model.
The permeability is considered as a tensor, and has different values for horizontal and
vertical transmission. Such a system was investigated by Castinel & Combarnous (1975),
and the results agreed well with their experiments. Epherre (1975) performed a linear
16
Chapter 2. Modelling convection in porous media
stability analysis on a system, which also included anisotropy in the thermal conductivity.
This is further discussed in section 2.3.1.
Due to the stratified nature of geological features, the permeability and conductivity
can be modelled as piecewise-constant functions with depth. While the overall domain
is heterogeneous, composed of multiple layers, each horizontal layer in the domain is
considered as a homogeneous and isotropic. If the difference in the properties is not
too great, the heterogeneous structure can be modelled as an anisotropic, homogeneous
structure, where the appropriate property tensors can be determined by an averaging
process (Wooding, 1978). For a greater disparity between layers, the anisotropic model is
not appropriate, as significant local flow can occur (see McKibbin & Tyvand, 1982).
2.3.1
Anisotropic model
Epherre (1975) performed a linear stability analysis of the anisotropic model. The Rayleigh
number is calculated from the vertical permeability and conductivity, given by KV and
kV , respectively, as opposed to the horizontal values, KH and kH . Hence, the Rayleigh
number is given by
Ra =
ρ0 gβKV H∆T
,
µαV
(2.40)
where αV = kV /(ρcP )f . The analysis shows that the critical Rayleigh number, is given by
Rac =
π 2 (ξ + L2 )(η + L2 )
,
ξL2
(2.41)
where ξ and η denote the ratios, KV /KH and kV /kH respectively, and L the width of a
cell (half wavelength). The anisotropy also affects the preferred cell width. The L which
minimises (2.41) is given by
Lc = (ξη)1/4 .
(2.42)
Where the anisotropy model is used to describe a stratified system, the appropriate directional averages can be found by taking the arithmetic and harmonic means. This method
is derived from considering the average fluid and heat flux across the stratified media
parallel and perpendicular, (in series), to the layers (Wooding, 1978). The suitability of
modelling a stratified system with an anisotropic model is investigated by McKibbin &
Tyvand (1982). Thus, the horizontal permeability and conductivity in a system with N
layers are given by
KH
kH
=
=
N
X
i=1
N
X
i=1
Ki hi ,
(2.43a)
ki hi ,
(2.43b)
2.3. Varying permeability
17
where Ki ,ki and hi are the permeability, conductivity and height of the ith layer, respectively. Note that in a scaled box, the total height is assumed to be unity, such that
1=
N
X
hi ,
(2.44)
i=1
though any height box can be considered with the appropriate weighting. Similarly, the
vertical values are given by
1
KV
1
kV
N
X
hi
=
,
Ki
=
i=1
N
X
i=1
hi
.
ki
(2.45a)
(2.45b)
Due to the nature of these averages, the vertical component of the permeability and
thermal conductivity are greater than their horizontal counterparts. This leads to ξ and
η values larger than 1, and the convection cells tend to be elongated in the horizontal
direction by equation (2.42).
One should be wary about making bold statements about the effect of anisotropy.
Consider ξ, the permeability ratio, which is typically larger than η due to the increased
variability of K. As ξ increases, the critical Rayleigh number decreases. One may naı̈vely
claim that anisotropy in layers encourages convection, however, one should also consider
that the Rayleigh number of the system, equation (2.40), is determined using KV ; this
is the lesser measure of the system’s permeability, and the Rayleigh number will be low.
These considerations effectively cancel out. In fact, by using an effective Rayleigh number,
(proposed by Nield (1994) and applied to the anisotropic case by Nield (1997)), the critical
Rayleigh number remains at 4π 2 , where an appropriate average of the permeability is used.
The effective Rayleigh number is also used in section 2.4, where a temperature-dependent
viscosity is considered.
2.3.2
Other studies on varied permeability
McKibbin and O’Sullivan studied a fully stratified model by considering each layer as a
separate domain, and then matching boundary conditions at the interfaces. The onset of
convection, (see McKibbin & O’Sullivan, 1980), and overall heat transfer, (see McKibbin
& O’Sullivan, 1981), in two and three layer systems was investigated. It is found that the
layered structure affects the critical Rayleigh number, the preferred width of the cells, and
the heat transfer from top to bottom of the system. It can be seen that there is a significant
change in the critical Rayleigh number, but this must be taken with some reservation. The
overall Rayleigh number is determined by the properties of the bottom layer of the system.
If the other layers of the system have a higher permeability, then convection is more likely,
yet the Rayleigh number remains the same. This manifests as a decrease in the critical
Rayleigh number. The difficulty here is that it is unclear whether the change in Rac is
18
Chapter 2. Modelling convection in porous media
due to the raw change in permeability, or the geometrical arrangement. The concept of
an “effective Rayleigh number” could be applied here to isolate the effects of the layered
geometry, from the effects of simply changing the permeability.
McKibbin & Tyvand (1983) attempted to create a layered system which cannot be
modelled accurately using the homogeneous anisotropic model. To do this, the system
is considered as alternating thick and thin layers which are infinitesimally small. Due
to the small local effects, and high contrast between the permeabilities of the thick and
thin layers, local flow effects can be seen. As the contrast in permeabilities is increased
further, convective flow can be seen to be restricted between the highly impermeable, thin
sheets. Clearly this is a completely different system to an elongated cell, which would be
the result of an anisotropic approximation. This study is further extended by McKibbin
& Tyvand (1984), where thin, highly permeable layers are considered, known as “cracks”.
The presence of these thin cracks promotes a large amount of horizontal flow through
these thin layers. The convection cells are not confined to any particular layer, but the
local behavior of the flow can be quite detailed.
A recent series of publications by Nield and Kuznetsov have explored horizontal and
vertical heterogeneity in both permeability and conductivity. Nield & Kuznetsov (2007)
investigated a box separated into four quadrants, each with a different constant value for
K and k, using a Galerkin method. They use the approximation that the parameters are
only slightly different, so this is considered “weak heterogeneity”. It is found that the
heterogeneity in both permeability and conductivity have approximately the same effect.
The horizontal and vertical variations are independent at first order, and only slightly
affect the onset of convection. These conclusions are not revised by Nield & Kuznetsov
(2008), who extend the Galerkin method to accommodate “moderate heterogeneity”.
For systems with a strong heterogeneity, large changes in the material properties can
cause local destabilisation to occur (see Nield & Simmons, 2007). Such local effects beg
the question: How useful is a global Rayleigh number? Nield et al. (2009) address this
issue by dividing the domain into grid cells, and identifying the local Rayleigh number of
rectangles formed by the grid. Such rectangles may be formed by a single grid cell, or larger
rectangles can be formed by multiple cells. If any rectangle exceeds a Rayleigh number of
41, then local destabilisation occurs, evolving to local, or possibly global convection. Many
types of heterogeneity were considered: A 2D quartered square, a 3D octered cube, linear
and quadratic variation in both permeability and conductivity. In further papers, the
authors apply this method to various types of heterogeneity: Local and periodic variation,
(Kuznetsov et al., 2010), and spatially correlated random permeability, (Nield et al., 2010).
2.4
Temperature-dependent viscosity
The fluid viscosity is a temperature dependent physical parameter, and can be represented as such in the model. For example, in the Perth basin, where temperatures may
range between 40◦ C and 100◦ C, the dynamic viscosity can vary between 6.53 and 2.83
2.4. Temperature-dependent viscosity
19
0.65
0.60
0.55
µ
(10−3 N s/m2 )
0.50
0.45
0.40
0.35
50
60
T̂
70
80
90
100
(◦ C)
Figure 2.4: Dynamic viscosity data (dots) and the reciprocal linear approximation (solid
line) between T = 40◦ and T = 100◦ . The linear approximation is given by equation 2.46,
with µ0 = 0.653 · 10−3 N s/m2 and φ̂ = 0.02K −1
(10−4 N s/M 2 ), throughout the system. The following discussion explores the issue of
whether a model with a temperature-dependent viscosity is required, or if the constant
viscosity model is adequate. Kassoy & Zebib (1975) argued that allowing the viscosity to
vary can significantly reduce the critical Rayleigh number required for convection to occur.
Nield (1994) countered this claim. He argued that Kassoy & Zebib (1975) calculated the
Rayleigh number with reference to the viscosity at the top of the domain, (where the fluid
is coldest), thereby introducing a bias. Of course, if one allows the viscosity to vary, it can
only decrease, facilitating easier flow, and hence lowering the critical Rayleigh number.
It is unclear whether the change in Rac is due to the overall decrease in viscosity, or the
temperature dependency. When calculating the Rayleigh number of a physical system,
one should use an appropriate mean for the physical parameters and an effective Rayleigh
number, rather than the value of that parameter at the top of the domain. Nield (1996)
investigated a model with a temperature dependent viscosity.
In the following sections, two temperature-dependent viscosity modes are investigated.
In section 2.4.1, the Rayleigh number is calculated based on the viscosity at the top of
the domain. A numerical model is used to verify these findings in section 2.4.2. In section
2.4.3, an appropriate mean value is used. Again, this is numerically verified in section
2.4.4. In both cases, the aim is to determine the effect of temperature-dependent viscosity
on the critical Rayleigh number.
2.4.1
Linear stability analysis
The following section is a detailed reexamination of the work by Nield (1996). Nield
approximates µ(T )−1 with a linear profile, as a model of this type can fit the data of
water, which is the most applicable fluid in a geothermal context, (Kassoy & Zebib, 1975).
A fit of this approximation to viscosity data is shown in figure 2.4. Where Nield used a
specific example and set a value to the reference viscosity, here the model is more general
20
Chapter 2. Modelling convection in porous media
by keeping the viscosity as a constant parameter. Let
µ0
µ(T̂ ) =
1 + φ̂(T̂ − T0 )
,
(2.46)
where µ0 is the viscosity of the fluid at the temperature of the upper boundary, T0 ,
and φ̂ is some viscosity change coefficient. Note that here, Nield’s suggestion of using
an intermediate value for viscosity is ignored. Using the above perturbation and scaling
process outlined in section 2.2.1, but scaling with µ0 now instead of µ, the Rayleigh number
is now defined as
Ra =
ρ0 gβKH∆T
.
µ0 α m
(2.47)
In terms of the dimensionless variables,
µ0
µ(T, z) =
1 + φ̂∆T (T + 1 − z)
.
(2.48)
As T is a small variable, it is dropped, giving (µ)−1 as a linear function of z:
µ(z) =
µ0
1 + φ̂∆T (1 − z)
.
(2.49)
Equations (2.20) - (2.22) become:
∇ · v = 0,
(2.50)
0 = −∇P − µ̂(z)v + RaT k,
(2.51)
∂T
− w = ∇2 T.
∂t
(2.52)
Here µ̂ is the dimensionless viscosity,
µ
µ0 .
The time derivative in Darcy’s law is dropped,
as γa is small. To solve for the critical Rayleigh number, the time derivative in the energy
equation is set to 0, which represents marginal stability. Letting v = (ψz , 0, −ψx ), and
taking the cross-product of equation (2.51), the incompressibility condition, (2.50), is
identically true and equations (2.51) and (2.52) become:
∇2 ψ +
µ̂z
Rac
ψz = −
Tx ,
µ̂
µ̂
(2.53)
ψx = ∇2 T,
(2.54)
at critical stability, (s = 0, Ra = Rac ). The governing equations are now in the form of
a variable coefficient linear problem. To simplify equation (2.53) let C =
define
µ̂ =
1−C
,
z−C
η=−
µ̂z
1
=
.
µ̂
ẑ − C
1
φ̂∆T
+ 1, and
(2.55)
Note that C ∈ (1, ∞) and z ∈ [0, 1], so µ̂ and η are never undefined. Equations (2.53) and
(2.54) become:
∇2 ψ − ηψz = −
Rac (z − C)
Tx ,
1−C
(2.56)
2.4. Temperature-dependent viscosity
21
ψx = ∇2 T.
(2.57)
To solve equations (2.56) and (2.57), let ψ = F (z) exp(iαx), and T = G(z) exp(iαx) as in
the previous section. This results in the ordinary differential equations,
0 = (D2 − α2 )F (z) − ηDF (z) +
Rac (z − C)
iαG(z),
1−C
(2.58)
iαF (z) = (D2 − α2 )G(z),
(2.59)
with the boundary conditions, F (0) = F (1) = G(0) = G(1) = 0. Solving in terms of G(z)
keeps the boundary conditions simple. Thus, F (0) = F (1) = 0 =⇒ G00 (0) = G00 (1) = 0.
This gives the fourth order equation:
0 = (D2 − α2 )2 G(z) − η(D3 − α2 D)G(z) −
Rac (z − C) 2
α G(z).
1−C
(2.60)
Equivalently, one can multiply by the non-zero factor, (z − C), to give
0 = (z − C)(D2 − α2 )G(z) − (D3 − α2 D)G(z) −
Rac (z − C)2 2
α G(z).
1−C
(2.61)
This is exactly the fourth order ODE solved in Nield (1996), however, here C varies with
dimensionless group, φ̂∆T , whereas in Nield (1996), φ̂∆T is fixed. Nield solves this using
a polynomial series, which gives the four solutions of the form:
f (i) (z) =
n=∞
X
n
b(i)
n z ,
(2.62)
n=0
(i)
for i = 0, 1, 2, 3, corresponding to each of the four independent solutions with bn = δni for
n < 4. While these polynomials are not orthogonal, the derivation is simple, and many
terms can be generated for increased accuracy. After rigorous calculation, values of bn for
n ≥ 4 follow this general rule:
b(i)
n
=
(i)
(i)
(n − 5)(n − 1)!bn−1 + 2α2 C(n − 2)!bn−2
(i)
−α2 (2n − 9)(n − 3)!bn−3
α2 Rac C 2 (i)
− α4 C +
(n − 4)!bn−4
1−C
2α2 Rac C (i)
+ α4 +
(n − 4)!bn−5
1−C
α2 Ra (i)
c
−
(n − 4)!bn−6 /Cn!,
1−C
(2.63)
where b−1 = b−2 = 0 such that this rule is valid for n = 4, 5. This recurrence relation can
be used to exactly determine the coefficients to any order. The corresponding solution can
be regarded as an exact solution, providing the sequence converges. It should be noted
22
Chapter 2. Modelling convection in porous media
(i)
here that Nield (1996) states that the coefficient of bn−3 is given by
−α2 (2n − 7)(n − 3)!,
which differs from the relation obtained above. This is only a small error, and is only
noticeable when a large number of terms are taken, due to the recurrent nature of the
definition. Correcting this error does not significantly alter the results, and thus, does not
change Nield’s conclusion. The comparison between the two analytical solutions can be
seen in figure 2.5. See appendix A for detailed calculation.
This series must be tested for convergence. In order of increasing powers of ẑ, let each
term of the power series be Sn , with n being the power of ẑ. Using the ratio test for
convergence:
L =
=
=
=
=
Sn ,
lim
n→∞ Sn−1 (n − 5)(n − 1)!z ,
lim
n→∞ Cn!
n − 5 lim z ,
n→∞ Cn z
5 1−
.
lim
n→∞ C
n z
.
C
(2.64)
It is known that
C>1
∀ φ̂∆T, and
z ∈ [0, 1].
(2.65)
Hence, L < 1, and the series is absolutely convergent for the solution range. The usefulness
of the series is shown by approximating the solution with both a 15 and 17 term polynomial.
P3
(i)
The general solution of (2.61) is G(z) =
i=0 ci f (z). The boundary conditions
G(0) = 0 and G00 (0) = 0 imply that c0 = c2 = 0. Imposing G(1) = G00 (1) = 0 results in
the equation:
f (1) (1)D2 f (3) (1) − f (3) (1)D2 f (1) (1) = 0,
(2.66)
which can be rearranged such that
Rac = g(α, C, l),
(2.67)
where l is the number of terms taken in the polynomial approximation, (2.62). For a
given C and α, the critical Rayleigh number can be calculated. In general, α should be
chosen such that Rac is minimised. For simple comparison to the numerical results, the
wavenumber is set as α = π. Nield (1996) reports only negligible change in the preferred
α value. The critical Rayleigh number as a function of φ̂∆T is shown in figure 2.5. The
results were obtained using a polynomial of order l = 15 and l = 17.
2.4. Temperature-dependent viscosity
2.4.2
23
Numerically modelling a variable viscosity
The nonlinear governing equations are numerically modelled to compare the critical
Rayleigh numbers calculated from the analytical theory. The nonlinear equivalent of
equations (2.50) - (2.52) are
∇ · v = 0,
(2.68)
0 = −∇P − µ̂(T, z)v + RaT k,
(2.69)
∂T
− w + v·∇T = ∇2 T,
∂t
(2.70)
where
µ̂(T, z) =
1
1 + φ̂∆T (T + 1 − z)
,
(2.71)
introduces a nonlinearity into Darcy’s law. Introducing the stream function, and crossdifferentiating to eliminate the pressure term results in the equations,
2
1 + (φ∆T )(T + 1 − ẑ) RaTx = φ∆T (Tz ψz − ψz + Tx ψx )
− 1 + (φ∆T )(T + 1 − z) ∇2 ψ,
Tt + ψz Tx − ψx Tz + ψx = ∇2 T.
(2.72)
(2.73)
These equations are numerically solved with FlexPDE, using the same boundary, and
initial conditions used in the earlier numerical models, (equations (2.37) and (2.38)).
The analytical and numerical results are plotted in figure 2.5. The critical Rayleigh
number decreases quite substantially over this region of φ∆T . For example, if the viscosity
of water at the top boundary is double the viscosity of water at the lower boundary,
φ∆T = 1, and the critical Rayleigh number is reduced by almost 15. The analytic solution,
(2.62) matches closely to the numerical results. The polynomial solution is shown to be
useful, and both the 15 and 17 term solutions match well.
These results seem to suggest that the Rayleigh number does not provide a very
good global measure of the system, if the viscosity of the fluid has a large range of values
throughout the system; if one were to apply the Oberbeck-Boussinesq approximation and
assume that the viscosity were constant, then the quoted critical Rayleigh number of 4π 2
is no longer relevant. This leads back to the “effective Rayleigh number”, proposed by
Nield (1994). Taking the viscosity at top boundary to measure the Rayleigh number is
not appropriate, as it does not adequately represent the viscosity throughout the system.
Nield proposed that the appropriate physical constants to use are the arithmetic and
harmonic means for the constants that appear in the numerator and denominator of the
Rayleigh number (2.23), respectively. To verify this, an alternate model with temperaturedependent viscosity is used. This new model uses the harmonic mean of the viscosity as
the reference viscosity, from which the effective Rayleigh number is calculated.
24
Chapter 2. Modelling convection in porous media
2
Legend
0
1.
2.
3.
3.
−2
−4
Numerical
Analytical (Nield) Nield (1996)
Equation (2.67), l = 15
Equation (2.67), l = 17
−6
Rac − 4π
2
−8
−10
−12
−14
−16
−18
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
φ̂∆T
Figure 2.5: The change in the critical Rayleigh number as a function of φ̂∆T , for α = π.
As the change in viscosity increases, the critical Rayleigh number greatly decreases. The
numerical and corrected analytical results are closely matched.
2.4.3
Linear stability analysis based on the harmonic mean value of
viscosity
The effect of a temperature-dependent viscosity is better determined by choosing a base
viscosity which represents the viscosity of the entire system. For µ inversely linear to
temperature, the harmonic mean, µ̄, occurs at the arithmetic mean of the temperature.
An approximation of this type is plotted alongside data in figure 2.6. Assuming the
conduction solution, where temperature increases linearly in z, µ̄ is the viscosity that
occurs at the middle temperature, T0 + ∆T /2:
µ̄ = µ(T̄ ) = µ(T0 + ∆T /2).
(2.74)
Hence, an appropriate model would take the form:
µ=
µ̄
1 + θ̂(T̂ − (T0 + ∆T /2))
,
(2.75)
where θ̂ is some viscosity change coefficient. The Rayleigh number is defined as
Ra =
ρ0 gβKH∆T
.
µ̄αm
(2.76)
In terms of dimensionless parameters, µ is given by
µ(T, z) =
µ̄
1 + θ̂∆T (T + 1/2 − z)
.
As T is a small variable, it is dropped, giving µ−1 as a linear function of z:
(2.77)
2.4. Temperature-dependent viscosity
25
0.6
µ
0.5
−3
(10 N s/m2 )
0.4
50
60
70
80
90
100
◦
T
( C)
Figure 2.6: Dynamic viscosity data (dots) and the reciprocal linear approximation about
the harmonic mean (solid line) between T = 40◦ and T = 100◦ . The linear approximation
is given by equation 2.46, with µ̄ = 0.405 · 10−3 N s/m2 and θ̂ = 0.013K −1
µ(z) =
µ̄
1 + θ̂∆T (1/2 − z)
.
(2.78)
Equations (2.53) and (2.54) are again used to find the critical Rayleigh number, however,
η and µ̂ are now defined as:
µ̂ =
where B =
1
θ̂∆T
µ
1/2 − B
=
,
µ̄
z−B
η=−
µ̂z
1
=
,
µ̂
z−B
(2.79)
+ 12 . The dimensionless group, θ̂∆T , may take values between 0 (constant
viscosity), and 2 (the viscosity at the top of the layer tends to ∞). Following the same
steps as in equations (2.58) - (2.60), the analogue of equation (2.61) is
0 = (z − B)(D2 − α2 )2 G(z) − (D3 − α2 D)G(z) −
Rac (z − B)2 2
α G(z),
1/2 − B
(2.80)
with the same boundary conditions as (2.61). This can be solved again in terms of polynomial solutions:
f
(i)
(z) =
n=∞
X
n=0
an(i) z n ,
(2.81)
26
Chapter 2. Modelling convection in porous media
(i)
for i = 0, 1, 2, 3, corresponding to each of the four independent solutions with an = δni
for n < 4. After further calculation, values of an for n ≥ 4 follow this rule:
a(i)
=
n
(i)
(i)
(n − 5)(n − 1)!an−1 + 2α2 B(n − 2)!an−2
(i)
−α2 (2n − 9)(n − 3)!an−3
α2 Rac B 2 (i)
− α4 B +
(n − 4)!an−4
1/2 − B
2α2 Rac B (i)
+ α4 +
(n − 4)!an−5
1/2 − B
α2 Ra (i)
c
(n − 4)!an−6 /Bn!,
−
1/2 − B
(2.82)
where a−1 = a−2 = 0 such that this rule is valid for n = 4, 5. As this result can be used
to evaluate the coefficients to any order, the solution can be regarded as being exact. The
ratio test for convergence (as used in (2.64)) yields
L=
z
.
B
(2.83)
For this problem, the polynomial converges ∀ẑ ∈ [0, 1] if 0 ≤ θ̂∆T < 2, which is precisely
the prescribed values that θ̂∆T can take. Convergence is slower than in the previous
problem, and more terms are needed for the same degree of accuracy.
The boundary conditions of equation (2.80) imply that G(0) = G00 (0) = 0 and also
imply equation (2.66). This can be rearranged to calculate the critical Rayleigh number:
Rac = g(α, B, l), .
(2.84)
where l is the number of terms taken in the series. Again it should be noted that to
compare the results to the numerical simulations, α is forced to be π. The critical Rayleigh
number, as a function of θ̂∆T , is shown in Figure 2.7. These results were obtained by
using polynomials of order l = 20 and l = 22, both of which match well. The results will
be discussed after the numerical model results are obtained.
2.4.4
Numerical simulations based on harmonic mean value of viscosity
The complete scaled nonlinear equations are solved with FlexPDE to compare the critical
Rayleigh number. Equations (2.68) - (2.70) are used, where,
µ̂(T, z) =
1
1 + θ̂∆T (T + 1/2 − z)
,
(2.85)
2.4. Temperature-dependent viscosity
27
1.5
Legend
1. Numerical
2. Equation (2.84), l = 20
3. Equation (2.84), l = 22
1
Rac − 4π 2
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ̂∆T
Figure 2.7: The change in the critical Rayleigh number as a function of θ̂∆T . The critical
Rayleigh number increases slightly as the variation in viscosity (θ̂∆T ) increases.
such that the reference viscosity occurs at the mid-point temperature. The analogue of
equation (2.72) becomes
2
1 + (θ̂∆T )(T + 1/2 − z) Rac Tx = θ̂∆T (Tz ψz − ψz + Tx ψx )
− 1 + (θ̂∆T )(T + 1/2 − z) ∇2 ψ, (2.86)
while the conservation of energy equation remains the same as equation (2.73):
Tt + ψz Tx − ψx Tz + ψx = ∇2 T.
(2.87)
Equations (2.86) and (2.87) were numerically solved using FlexPDE, with boundary conditions, (2.37) and (2.38). The critical Rayleigh number plotted against θ̂∆T can be seen
in figure 2.7. As θ̂∆T increases, and the variation of viscosity across the system becomes
greater, the critical Rayleigh number increases, stabilising the conduction solution. The
increase in critical Rayleigh number is so small, that it seems that one may use a simple
constant viscosity model without significant error, as long as the appropriate reference
viscosity is used. Even for a two-fold difference in the viscosity in the system, where
θ̂∆T ≈ 0.67, the critical Rayleigh number only deviates by about 0.5, a mere 1.3%.
This research supports the findings of Nield (1994) and Nield (1996); an appropriate
average of the varying viscosity can be used to adequately model a convection in porous
media system. One needs to be careful, however, as a large enough variation in the viscosity would cause a constant viscosity model to be no longer appropriate. For values
of θ̂∆T > 1 (corresponding to a three-fold increase in viscosity), the change in critical
Rayleigh number starts to become substantial.
28
Chapter 2. Modelling convection in porous media
2.5
Conclusion
The linear stability analysis of the Horton-Rogers-Lapwood problem has been reviewed.
The results indicate that the instabilities take the form of spatially oscillating solutions.
While the analysis was performed specifically in 2D, to take advantage of the streamfunction, ψ, the solutions generalise to 3D with an overall horizontal wavenumber, α. With an
extra degree of freedom, these 3D solutions can take many forms, including y-independent
rolls and 3D hexagonal cells (Nield & Bejan, 2006). This linear stability analysis has been
re-examined in such detail to familiarise the reader with the structure and concepts that
will be used for the nonlinear perturbation analysis in the following chapters.
The effect of a varying permeability in porous media systems is a well-studied topic,
and the significant results have been reviewed. It is found that a stratified, heterogeneous
system can generally be approximated by an isotropic, homogeneous system as long the
heterogeneity is not too strong; that is, the local change in permeability must not change
a large amount over a small spatial scale.
A modification of the linear stability analysis, including the effects of a temperaturedependent viscosity has been re-examined. Exact solutions have been derived for a linearly
varying dynamic viscosity. It is found that large variations in viscosity can occur throughout the domain, without changing the effective Rayleigh number by a significant amount.
This examination reveals that one must be careful how the Rayleigh number of a system
is measured, as relaxing the Oberbeck-Boussinesq approximation can cause ambiguities,
and false conclusions can be reached.
Chapter 3
Two interacting modes
While convection in an infinite layer is well understood in terms of linear stability analysis, in reality, convection domains are of finite horizontal extent. When imposing side
boundaries, linear stability analysis shows that multiple modes are viable solutions, each
with its own critical Rayleigh number. The “preferred mode”, which has the lowest critical Rayleigh number, usually takes the form of an almost-square shape, where the cell
has width and height dimensions which are roughly the same. For example, in a square
box, a single cell mode is preferred over a double cell mode, which would be compressed
horizontally.
As the Rayleigh number of the system becomes larger, more of these modal solutions
become viable. From linear stability analysis, it is not clear how the system is resolved.
Which mode takes precedence? Do solutions exist which are superpositions of multiple
modes? How does one determine the most likely solution?
In the following sections, a nonlinear perturbation method is used to answer these questions. In section 3.1, the model is introduced mathematically. A perturbation procedure
is used in section 3.2 to reduce the governing PDEs to a system of ordinary differential
equations (ODEs) with respect to time. In section 3.3 this system of ODEs is analysed using dynamical systems theory, to find the behaviour of the system as it approaches stable
solutions.
3.1
Introduction to the problem
Consider a scaled 2D box of a saturated porous medium with scaled height, 1, and length,
L. Assume that the side-walls are impermeable to fluid, and heat insulating. These boundary conditions are the same as those used in the numerical models in section 2.2.2 except
that the side boundaries are L units apart, instead of 1. These boundary conditions are
natural, in that they allow the fluid and heat solutions to obey the same phase relationship
seen in the infinite layer case. Recall that the solutions in the infinite layer periodically
satisfy these boundary conditions, as shown in figure 2.2. The effect of imposing these
sidewalls forces the convection cells into a particular size, rather than taking the ‘square’
30
Chapter 3. Two interacting modes
form seen in the infinite layer.
In an infinite layer, the critical Rayleigh number was shown to be dependent on the
vertical mode number, j, and the horizontal wavenumber, α (equation (2.33)),
Rac =
(j 2 π 2 + α2 )2
.
α2
(3.1)
For the finite box case, the critical Rayleigh number has the same dependence on the
solution width, however, the solutions do not belong to a continuous spectrum. The
horizontal wavenumber, α, is discretised by
α=
pπ
,
L
(3.2)
where p = 1, 2, 3, . . . , is the horizontal mode number (Beck, 1972). The critical Rayleigh
number for the mode (p, j) becomes
(j 2 + P 2 )2
,
P2
(3.3)
p
,
L
(3.4)
Rac (P, j) = π 2
where
P =
(Beck, 1972). As Rac (p, j) >Rac (p, 1) for j > 1, the onset of convection always occurs
for j = 1. It is difficult to investigate the interaction of two modes that become viable
at Rayleigh numbers much larger than the onset of convection as an analytical solution
is unavailable. The global convection behavior would be dominated by the mode with
the lowest critical Rayleigh number, and it is assumed that this mode would be highly
nonlinear. Hence, all future examples occur at the onset of convection, where the only
modes that are stable are represented in the phase-space. Modes that are not explicitly
analysed decay quickly.
In the following analysis, it is assumed that j = 1. Figure 3.1 shows the critical
Rayleigh number for the one-cell (p = 1) and two-cell (p = 2) convection pattern as a
function of the box length, L. The two critical curves divide the plane into four regions.
In region 0, the conduction solution is stable; no convection occurs. In regions 1 and 3, the
conduction solution is unstable, and the convection will occur in a one- or two-cell pattern
respectively. In region 2, both convection patterns are available, and it is not clear from
the linear stability analysis what occurs here. Figure 3.2 shows the streamlines of the two
viable modes.
Ideally, to use a perturbation expansion, the nonlinear effects should be small. Both
the temperature and fluid flow solutions are small in the regions immediately above the
critical Rayleigh lines. The only condition where both modes exist and are weakly non√
linear is close to where the two critical Rayleigh lines intersect at L = 2, Rac = 9π 2 /2.
A perturbation expansion is used to investigate the interaction of the two modes at the
onset of convection for this condition.
3.2. Perturbation analysis
31
48
47
2
46
45
Single cell (p = 1)
1
Rac
3
44
43
0
42
Double cell (p = 2)
41
40
39
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
L
Figure 3.1: The critical Rayleigh number as a function of L for the one-cell, (p = 1),
and two-cell, (p = 2), modes. At the point of intersection it is unclear which mode will
be preferred, as each mode has the same critical Rayleigh number. In region 0, only
the conduction solution is present. In regions 1 and 3, the one- and two cell modes,
respectively, are present. It is not clear from linear stability analysis what occurs in region
2, where both modes are viable.
A
B
Figure 3.2: Streamlines of the single- and double-cell modes. The single-cell “A” mode
is elongated horizontally, while the double-cell “B” mode is compressed horizontally. Solutions for which the flow is reversed are also available, and are represented by “negative
amplitude” solutions.
3.2
Perturbation analysis
This section follows closely the perturbation method in chapter 14 of Fowler (1998). Fowler
performs this perturbation analysis on a square box, where only the one-cell mode is viable
at the onset of convection. The nonlinear interaction of the mode with itself causes the
dampening required to reach a finite-amplitude steady-state. Here, the procedure has
been adapted for a box where two modes are viable at the onset of convection, as the
interaction between the modes is considered.
The nonlinear scaled governing equations are
∇2 ψ = −RaTx ,
Tt + ψx + ψz Tx − ψx Tz = ∇2 T,
(3.5a)
(3.5b)
32
Chapter 3. Two interacting modes
(see equations (2.35) and (2.36)). The boundary conditions are given by
0 = ψ(0, z) = ψ(L, z) = ψ(x, 0) = ψ(0, 1),
(3.6a)
0 = Tx (0, z) = Tx (L, z) = T (x, 0) = T (x, 1).
(3.6b)
Following closely the procedure in chapter 14 of Fowler (1998), it is assumed that the
system is close to the critical Rayleigh number, so
Ra = Rac 1 + R1 ε2 + . . . ,
(3.7)
where ε 1 and R1 ∈ {−1, 1}. Negative values of R1 correspond to precritical conditions
and positive values to postcritical conditions. The timescale for modal interactions is of
order 1/ε2 , so it is convenient to rescale time using
t=
1
τ.
ε2
(3.8)
An expansion in increasing powers of ε is applied to the streamfunction and temperature
solutions, as per Fowler (1998), and is given by
ψ = εψ1 + ε2 ψ2 + . . . ,
(3.9a)
= εT1 + ε2 T2 + . . . .
(3.9b)
T
Substituting these expansions into the conservation equations, (3.5), yields the following
PDEs at each order of ε:
O(ε) : ∇2 ψ1 + Rac T1x = 0,
(3.10)
2
∇ T1 − ψ1x = 0,
O(ε2 ) : ∇2 ψ2 + Rac T2x = 0,
(3.11)
∇2 T2 − ψ2x = ψ1z T1x − ψ1x T1z ,
O(ε3 ) : ∇2 ψ3 + Rac T3x = −Rac R1 T1x ,
(3.12)
∇2 T3 − ψ3x = T1τ + ψ2z T1x + ψ1z T2x − ψ2x T1z − ψ1x T2z .
For each PDE at any order, the boundary conditions are the same as in equation (3.6).
To investigate the system where the two modes are both viable at the onset of convection,
√
2
L = 2 and Rac = 9π2 . As both the single- and double-cell solutions solve the linear
homogeneous first order PDE, the solution is the sum of the two modes. The O(ε) solution
3.2. Perturbation analysis
33
is therefore given by:
√
√
√
πx
ψ1 = −3 2πA(τ ) sin( √ ) sin(πz) − 3 2πB(τ ) sin(π 2x) sin(πz),
2
√
πx
T1 = 2A(τ ) cos( √ ) sin(πz) + 2B(τ ) cos(π 2x) sin(πz),
2
(3.13a)
(3.13b)
where A and B are unknown functions of τ . The modes that solve the first order PDE are
referred to as the primary modes. To solve the PDEs at higher orders of ε, it is convenient
to express the problem in a matrix operator form. The O(εn ) PDE is given by
M u n = hn ,
(3.14)
where
M=
∇2
∂
Rac ∂x
∂
− ∂x
∇2
!
,
(3.15)
un = (ψn , Tn )T ,
(3.16a)
h1 = 0,
(3.16b)
T
h2 = (0, ψ1z T1x − ψ1x T1z ) ,
−Rac R1 T1x
h3 =
.
T1τ + ψ2z T1x + ψ1z T2x − ψ2x T1z − ψ1x T2z
(3.16c)
(3.16d)
Note that if u is in the eigenfunction form,
u = eipπx/L
a
b
!
sin jπz,
(3.17)
then
M u = M p,j u,
where
− ( Lp )2 + j 2 π 2
M p,j =
−ipπ
L
−
(3.18)
ipπRac
L
( Lp )2 + j 2 π 2
!
.
(3.19)
In matrix form, the O(ε2 ) PDE , (3.11), is given by
ψ2
0
M
=
.
T2
ψ1z T1x − ψ1x T1z
(3.20)
Substituting the first order solutions, ψ1 and T1 , into the right hand side, gives
ψ2
M
T2
0
sin(2πz).
πx
))
3π 3 (A2 + 2B 2 + 3AB cos( √
2
=
(3.21)
34
Chapter 3. Two interacting modes
The right hand side can be expressed in eigenfunction form (3.17), to give
0
ψ2
=
sin(2πy) +
M
T2
3π 3 (A2 + 2B 2 )
iπx
0
√
e 2 sin(2πy) + (complex conjugates).
9 3
2 π AB
(3.22)
Without loss of generality, the homogeneous solution can be equated to zero; any such
term can be absorbed into the O(ε) solution, equation (3.13). The particular solution is
assumed to be of the form
!
ψ2
=
T2
a
b
!
sin(2πz) +
c
d
!
e
iπx
√
2
sin(2πz) + (CC),
(3.23)
where a, b, c and d are unknown complex numbers, and (CC) represents the complex
conjugates. Substituting the assumed solution, (3.23), into the left hand side of (3.22)
gives
a
c iπx
√
LHS(3.22) = M
sin(2πz) + M
e 2 sin(2πz) + M (CC),
(3.24a)
b
d
a
c iπx
√
= M 0,2
sin(2πz) + M 1,2
e 2 sin(2πz) + M p,j (CC), (3.24b)
b
d
after using the eigenvalue equation, (3.18). By collecting coefficients of the two independent modes in equation (3.22) and premultiplying by the appropriate M −1
p,j , the a, b, c
and d coefficients are evaluated to give
and
a
0
0
−1
= M0,2
=
,
b
3π 3 (A2 + 2B 2 )
− 43 π(A2 + 2B 2 )
(3.25)
2
9iABπ
− 8 √2
c
0
−1
= M1,2 9 3
=
,
d
− 9 ABπ
2 π AB
(3.26)
8
as M0,2 and M1,2 are full rank. Hence, the solution to the second order expansion is
2
9iABπ
− 8√2
iπx
ψ2
0
√
2 sin(2πz) + (CC).
sin(2πz)
+
e
=
3
9
T2
− 4 π(A2 + 2B 2 )
− ABπ
(3.27)
8
The two modes in the second order solution are known as the secondary modes. The
critical Rayleigh number of these modes are much greater than the primary modes, and
linear stability analysis shows that these modes decay to zero. These secondary modes
can only be sustained via the nonlinear interaction of the primary modes.
The O(ε3 ) PDE, (3.12), is given by
ψ3
−Rac R1 T1x
=
M
T3
T1τ + ψ2z T1x + ψ1z T2x − ψ2x T1z − ψ1x T2z
(3.28)
3.2. Perturbation analysis
where
35
√
πx
T1τ = 2Ȧ cos( √ ) sin(πz) + 2Ḃ cos(π 2x) sin(πz),
2
and the dot notation denotes differentiation with respect to τ . The right hand side of
the O(ε3 ) PDE, (3.28), contains many modes, including secular modes which solve the
homogeneous problem. These secular terms give rise to growing solutions which are unacceptable, as they do not fit the adopted scheme. To eliminate these growing solutions, the
secular terms must be suppressed by a solvability condition. Evaluating the right hand
side, and explicitly showing the secular terms, the O(ε3 ) PDE, (3.28), can be expressed as
ψ3
M
T3
−iπRa
√ c R1 A
iπx
√
2
2
e
9 3 4
45
2
4
A
π
+
AB
π
8
8
=
sin(πz)
Ȧ +
√
√
−i 2πRac R1 B
+
ei 2πx sin(πz)
2 Bπ 4 + 9 B 3 π 4
Ḃ + 63
A
16
2
+(non-secular modes) + (CC).
(3.29)
The particular solution will be of the form
√
ψ3
c i 2πx
a iπx
√
2
e
sin(πz) + (NSM) + (CC),
=
e sin(πz) +
T3
d
b
(3.30)
where, (NSM) represents the non-secular modes. By following the same procedure that
yields the second order solution, the following relations are obtained for the secular modes:
−iπRa
√ c R1 A
a
2
M 1,1
,
=
2 4
b
Ȧ + 98 A3 π 4 + 45
8 AB π
√
c
−i 2πRac R1 B
M 2,1
=
9 3 4 .
2
4
d
Ḃ + 63
16 A Bπ + 2 B π
(3.31)
(3.32)
By definition, both M 1,1 and M 2,1 are singular matrices, so a, b, c and d cannot be determined. For the (1, 1) and (2, 1) modes to satisfy the homogeneous PDE, M 1,1 and M 2,1
must each have a non-trivial nullspace. For equations (3.31) and (3.32) to make sense, a
solvability condition must be applied; the right hand side of these equations must lie in
the eigenspace of the corresponding M p,j matrix. Equivalently, they must be orthogonal
to the nullspace of the M Tp,j matrix. The nullvectors of the two matrices are equal, and
given by
i√2 − 3π
Nullvector[(M 1,1 ) ] = Nullvector[(M 2,1 ) ] =
.
1
T
T
(3.33)
36
Chapter 3. Two interacting modes
Orthogonality implies that
√
−iπRa
√ c R1 A
i 2
2
(−
, 1) ·
= 0,
2 4
3π
Ȧ + 98 A3 π 4 + 45
8 AB π
√
√
−i 2πRac R1 B
i 2
, 1) ·
= 0.
(−
63 2
3π
Ḃ + 16
A Bπ 4 + 29 B 3 π 4
By rearranging the solvability condition, the following system of ordinary differential equations (ODEs) for the evolution of the amplitudes is obtained:
3
9
45
Ȧ = π 2 A( R1 − π 2 A2 − π 2 B 2 ),
2
8
8
63
9
Ḃ = π 2 B(3R1 − π 2 A2 − π 2 B 2 ).
16
2
(3.35a)
(3.35b)
The governing fluid dynamics PDEs have been reduced to a system of ODEs governing
the growth and decay of the amplitudes of the two primary modes. Note that applying
the solvability condition to the complex conjugate terms yields the same system of ODEs.
This system can be analysed using dynamical systems theory. In particular, the form and
stability of steady-state solutions are sought.
3.3
Fixed point and stability analysis
For any initial values of A and B, these amplitudes evolve as a function of time, τ ,
according to the dynamical system (3.35). A time-dependent solution, (A(τ ), B(τ )), is
known as a trajectory in the A–B phase-space.
Zero-isoclines (nullclines) are surfaces where Ȧ = 0 or Ḃ = 0. Steady solutions, or fixed
points, occur where Ȧ = Ḃ = 0 at the intersection of the two nullclines. By linearising
about these fixed points, the local stability can be determined by the Hartman–Grobman
theorem, (see Guckenheimer & Holmes, 1983). Fixed points which are locally stable
are likely candidates for the asymptotic solution of the system. If a fixed point has an
instability in any direction, small fluctuations will deflect trajectories away from the fixed
point.
The phase-space has symmetry about the A– and B–axes:
Ȧ(−A, B) = −Ȧ(A, B),
(3.36a)
Ȧ(A, −B) = Ȧ(A, B),
(3.36b)
Ḃ(−A, B) = Ḃ(A, B),
(3.36c)
Ḃ(A, −B) = −Ḃ(A, B).
(3.36d)
This symmetry derives from the fact that the reverse flow solutions are equally viable. As
each quadrant is equivalent, the focus of investigation will be on the A, B ≥ 0 quadrant
3.3. Fixed point and stability analysis
37
where there are four fixed points:
q
q
2 R1 4 R1 0
3π
3
A
0
3
q
, 1 q 2R1 ,
=
, π
,
R
2
B
0
0
1
π
3
3π
3
which are labelled 0∗ , A∗ , B ∗ and AB ∗ respectively. 0∗ represents the conduction solution,
where both modal amplitudes are zero and there is no fluid flow. Note that A∗ , B ∗ and
AB ∗ only exist for R1 ≥ 0. AB ∗ lies on the intersection of the two elliptical nullclines.
The only solution for precritical Rayleigh numbers is the conduction solution. It becomes
obvious here why only the values of 1 or −1 are used for R1 , rather than being a continuous
bifurcation parameter: There is only one bifurcation, which occurs at R1 = 0, and it is
degenerate. Allowing R1 to continuously vary merely rescales the locations of the fixed
√
points by the scale factor, R1 ; the stability analysis remains the same. This issue is
explored further in chapter 5, where the degeneracy is broken by introducing another
bifurcation parameter.
To linearise about each fixed point, the Jacobian matrix,
∂
∂A Ȧ
∂
∂A Ḃ
J=
∂
∂B Ȧ
∂
∂B Ḃ
!
(3.37)
is evaluated. The sign of the real part of the eigenvalues of the Jacobian evaluated at
each fixed point yields stability information. Negative eigenvalues imply stability, while
positive eigenvalues imply instability in the direction of the corresponding eigenvector. At
0∗ , J is given by:
∗
J(0 ) =
which has eigenvalues
3π 2 R1
2 ,
3π 2 R1
2
0
0
3π 2 R1
!
,
(3.38)
3π 2 R1 and eigenvectors (1, 0), (0, 1) respectively. This fixed
point is clearly stable if the convective system is precritical (R1 = −1) but is unstable if
the system is postcritical (R1 = 1). In the precritical case, the system is simple: there is
only one solution (the conduction solution), and it is stable. The postcritical system is
investigated by setting R1 = 1. The Jacobian matrices at A∗ , B ∗ , and AB ∗ are given by
J(A∗ ) = π 2
−3
0
0
− 49
!
,
J(B ∗ ) = π 2
π2
J(AB ∗ ) = −
3
4 10
7
4
− 49
0
0
−6
!
,
!
.
(3.39)
From the eigenstructure analysis, the fixed points are categorised in table 3.1.
The A∗ stable node corresponds to a steady-state solution of the system which is
single cell only. The amplitude of the B mode tends to zero. Conversely, the B ∗ stable
node corresponds to a solution which is strictly double cell, with no contribution from the
38
Chapter 3. Two interacting modes
Fixed Point
A∗
B∗
AB ∗
Eigenvalues
2
(−3π 2 , − 9π4 )
2
(− 9π4 , −6π 2 )
√ 2
− π3 4 ± 70
Signature
−−
−−
−+
Eigenvectors
(1, 0), (0, 1)
(1,q
0), (0, 1)
(±
Classification
Stable node
Stable node
10
7 , 1)
Saddle point
Table 3.1: Properties of the fixed points. The sign of the eigenvalues determines the
stability of the fixed point. Both A∗ and B ∗ are stable nodes, while AB ∗ is a saddle point.
B
B∗
S
AB ∗
S
0∗
A∗
A
Figure 3.3: The phase-space of the dynamical system. There are two stable nodes, A∗
and B ∗ . S is the stable manifold of the AB ∗ saddle point, and separates the basins of
attraction for the two stable nodes. The black lines represent some trajectories.
lower harmonic, as A tends to zero.
The stable A∗ and B ∗ fixed points are possible steady-state solutions which trajectories
asymptotically approach. The fixed point that the solution tends towards is determined
by which basin of attraction the initial point lies in. The two basins of attraction are
separated by the stable separatrix of AB ∗ , labelled S. Figure 3.3 shows the phase-space
of this system, showing the fixed points, basins of attraction and sample trajectories.
While it is technically possible in this model for the solution to come to rest at AB ∗ ,
if the initial coordinates lie on the stable separatrix, S, or on the fixed point itself, any
small fluctuations will push the phase into the basin of attraction for A∗ or B ∗ . Hence,
AB ∗ is not considered as a candidate solution to be realised after sufficient time.
3.4
Conclusion
Where the linear stability analysis shows that two modes are equally viable at the onset
of convection, a nonlinear perturbation method has been used to investigate further. By
reducing the governing PDEs to a set of ODEs, stability analysis of the existing solutions
3.4. Conclusion
39
is performed. It is found that either of the two modes are possible steady-state solutions.
The initial conditions determine which of the two solutions remain. This behaviour has
been replicated in numerical experiments using FlexPDE. By changing the initial conditions, the solution approaches either of the two modes.
There is also the presence of a mixture solution, AB ∗ , which is a sum of both modes.
This solution, however, is unstable. As the solution approaches either A∗ or B ∗ , the second order correction term for the stream function decays as one of the amplitudes tends
to zero. The temperature solution does, however, always have a second order correction
of the form sin(2πz).
While only one particular example has been analysed, any other system of two interacting modes shows a topologically equivalent phase-space. The only exceptions occur
for symmetry-breaking special cases, where the direction of one mode’s flow affects how
it interacts with the other mode. The four quadrants of the phase-space are no longer
equivalent. No examples of this symmetry-breaking occur at the onset of convection, and
only occur in the presence of another, nonlinear mode. As such, an example of this type
is not analysed. The conditions for which symmetry-breaking effects occur is discussed
further in section 4.4.
40
Chapter 3. Two interacting modes
Chapter 4
Three interacting modes
An extra degree of freedom is introduced when modelling 3D systems. As well as the
natural extension of the 2D circular form to 3D cylindrical rolls, there can also appear 3D
convection “cells”, where the fluid flow is dependent on both horizontal coordinates. A
new mode number, q, is introduced to denote the mode number in the other horizontal
direction, y. Beck (1972) performs a linear stability analysis on convection in a 3D box,
and derived the critical Rayleigh surface for each solution with mode numbers, (p, q), as
a function of the horizontal aspect ratios, Lx and Ly .
It is observed that there are certain box dimensions for which three or even four unique
modes share the same critical Rayleigh number at the onset of convection. These box
dimensions are ideal locations to explore the nonlinear interaction of multiple modes. In
the following section, a general perturbation procedure is outlined to derive the evolution
equations for three interacting modes. The interaction of four unique modes is investigated
in chapter 6. This procedure is analogous to the procedure in chapter 3, however, it has
been generalised to three-dimensional flow, and derives the evolution equations for any
size box where three modes are equally viable.
Two specific examples are investigated in section 4.2, both of which exhibit a different
phase-space topology. Other examples are categorised into either of these two cases in
section 4.3. There are special circumstances where the general dynamical system derived
in section 4.1 is not general enough, and a new description is required. These special
cases are derived in section 4.4, and the dynamics of a particular example of this special
case is shown in section 4.5. The literature on symmetry-breaking in modal solutions is
reviewed in 4.6. The different classes investigated here form a complete characterisation
of all three-mode interactions that occur in a reasonable box (up to four times wider than
it is tall). The contents of this chapter form the core of a recently accepted paper (see
Florio (2013)).
42
Chapter 4. Three interacting modes
4.1
Evolution equations
For three interacting modes in a 3D box of a porous medium, the fluid dynamics equations
are reduced to a dynamical system which governs the evolution of the amplitudes of the
three modes. In this section, the perturbation technique is developed as an extension of
the 2D method seen in chapter 3. In section 4.1.1, linear solutions are obtained, and are
consistent with the solutions used in a 2D box. Following this, in section 4.1.2, the perturbation method is described, giving a dynamical system which leads to finite-amplitude
solutions.
4.1.1
Linear stability results
A 3D finite box of a saturated porous medium is considered. As in previous chapters,
this box has impermeable boundaries at the top and bottom, and are held at a constant
temperature of T0 and T0 + ∆T respectively. The box is confined by side boundaries in the
x and y directions by impermeable, heat insulating sides. This domain is a direct analogy
to the 2D Box considered in chapter 3.
The linear stability analysis performed by Beck (1972), shows that the same critical
Rayleigh number given by equation (3.1) is expected here. The act of imposing these side
boundaries again forces the horizontal wavenumber, α, to take discrete values, however,
compared to the 2D case, α has an extra degree of freedom. In analogy to the 2D example,
(equation (3.2)), α is discretised by
α2 =
p2 π 2 q 2 π 2
+ 2 ,
L2x
Ly
(4.1)
where p, q = 0, 1, 2, . . .. Thus, the critical Rayleigh number, as a function of the mode
numbers and box dimensions, is given by
Rac (p, q, j) = π 2
where
P =
(j 2 + P 2 + Q2 )2
,
P 2 + Q2
p
q
, and Q =
.
Lx
Ly
(4.2)
(4.3)
Each mode, denoted by the mode numbers (p, q, j), has a unique critical Rayleigh surface
in Lx –Ly space. By determining the mode which has the lowest critical Rayleigh number
for a given box dimension, Beck (1972) produced a bifurcation set diagram showing the
preferred mode as a function of box size. This diagram is reproduced in figure 4.1, with
particular examples highlighted. Note that at the onset of convection, j = 1 for every
mode, and so the modes are only denoted by the horizontal mode numbers, (p, q).
√
√
For example, the box with dimensions, (Lx , Ly ) = ( 2, 1/ 2) has three viable modes
at the onset of convection; the (1, 0), (2, 0) and (0, 1) modes share a critical Rayleigh
4.1. Evolution equations
43
3
(0,3)
(0,3)
(2,2)
(1,3)
(1,2)
(1,3)
(2,0)
(2,1)
(0,3)
(0,3)
(3,0)
(1,3)
(1,3)
(2,2)
(3,0) (3,1)
(0,3) (0,3) (0,3)
(3,0)
(1,0)
(2,1)
(0,2)
(0,2)
(1,0)
(2,0)
(1,1)
(0,2)
(1,2)
(0,2)
(4,0)
(1,2)
(0,2)
(4,0)
(3,0)
(2,1)
(1,1)
(3,0)
(3,1)
(0,1)
(0,1)
(2,1)
(1,1)
(0,1)
(2,0)
(0,1)
(2,2)
(1,2)
(0,2)
(3,1)
X2
1
(4,0)
(2,2)
(3,1)
(1,2)
(0,2)
X3
Ly
− Example 1
− Example 2
− Example 3
− Example 4
(3,0)
(1,2)
(2,0) (2,1)
(0,2)
(2,2)
X4
(1,1)
(3,1)
(2,2)
(2,0)
(1,2)
2
X1
X2
X3
X4
(4,0)
(4,0)
(2,1)
(1,1)
(0,1)
X1
(1,0)
(3,0)
(2,0)
1
(4,0)
3
2
(5,0)
4
Lx
Figure 4.1: The preferred mode, (p, q), at the onset of convection, for a box size, (Lx , Ly ),
(After Beck (1972)). Examples analysed in further detail are highlighted.
number of
Rac =
9π 2
.
2
(4.4)
This is labelled example 1, in figure 4.1. To investigate the nonlinear interaction of these
modes, a perturbation expansion is used.
4.1.2
Perturbation analysis
The perturbation procedure employed in this section is a 3D extension of the procedure
used in chapter 3. The procedure is developed in general for the interaction of N viable
modes, so that the applicable dynamical system can be easily generated for any example.
The governing equations can no longer by simplified by utilising the stream function.
Therefore, the governing equations are not completely analogous to those seen for the 2D
case in chapter 3. The scaled linear governing equations (2.20) - (2.22) are used, however,
the nonlinear term in the energy equation is retained, (previously it was discarded). These
governing equations are given by
∇ · v = 0,
(4.5a)
−∇P − v + RaT k = 0,
∂T
+ v · ∇T − w − ∇2 T = 0.
∂t
(4.5b)
(4.5c)
The pressure term in (4.5b) can be eliminated by operating twice with curl. Taking each
element of the resulting vector as a separate equation, the five governing equations are
44
Chapter 4. Three interacting modes
given by:
∂
∂
∂
u+
v+
w
∂x
∂y
∂z
∇2 w − Ra∇2H T
∂ ∂
∇2 v + Ra
T
∂y ∂z
∂ ∂
∇2 u + Ra
T
∂x ∂z
∂T
∂
∂
∂
+ u T + v T + w T − w − ∇2 T
∂t
∂x
∂y
∂z
= 0,
(4.6a)
= 0,
(4.6b)
= 0,
(4.6c)
= 0,
(4.6d)
= 0.
(4.6e)
Though there are five equations in four unknown variables, it will be shown later that
the system is not over-specified. The boundary conditions for fluid flow in a box with
insulated sides are applied. The box is confined to the region 0 < x < Lx , 0 < y < Ly and
0 < z < 1. On all sides, top and bottom of the box,
v · n = 0,
(4.7)
where n is a unit vector perpendicular to the boundary. On the top and bottom of the
box,
T (x, y, 0) = T (x, y, 1) = 0,
(4.8)
∇T · n = 0.
(4.9)
and on the sides,
To understand the behaviour of the solution close to the the onset of convection, where
|R − Rac | is small, a perturbation expansion is used. The expansion is given by
R = Rac (1 + R1 ε2 + . . .),
1
t = 2 τ,
ε
(T, v) = ε(T1 , v 1 ) + ε2 (T2 , v 2 ) + . . . ,
(4.10a)
(4.10b)
(4.10c)
where ε 1, and R1 ∈ {−1, 1} denotes the sign of the perturbation, (precritical or
postcritical). This expansion is directly analogous to the 2D expansion in (3.9). By
substituting these expansions into the governing equations, (4.6e), the PDEs for the O(εk )
expansion are given by
M u k = hk ,
(4.11)
where

∂
∂x
∂
∂y

0
 0


M =  0 ∇2
 2
 ∇
0
0
0
∂
∂z
∇2
0
0
1
0


−Rac ∇2H 

∂ ∂ ,
Rac ∂y
∂z 
∂ ∂ 
Rac ∂x
∂z 
∇2
(4.12)
4.1. Evolution equations
45
uk = (uk , vk , wk , Tk )T ,
(4.13)
h1 = 0,
(4.14)
h2 = (0, 0, 0, 0, v 1 · ∇T1 )T ,

0

Rac R1 ∇2H T1


∂ ∂
h3 = 
−Rac R1 ∂y

∂z T1

∂ ∂
−Rac R1 ∂x ∂z T1

∂
∂τ T1
(4.15)




.



(4.16)
+ v 1 · ∇T2 + v 2 · ∇T1
The boundary conditions for the O(εk ) equation are given by v k · n = 0 on all boundaries,
Tk (x, y, 0) = Tk (x, y, 1) = 0 and ∇Tk · n = 0 on the sides. The first order solutions are
given by
u1 =
N
X
l=1
pl
pl πx
ql πy
−2Rac
Al (τ ) sin(
) cos(
) cos(πz),
2
L
L
Ly
ql
x
x
+ Ly + 1
N
X
ql
−2Rac
pl πx
ql πy
Al (τ ) cos(
) sin(
) cos(πz),
2
L
L
Ly
ql
p
y
x
+
+
1
l=1 Ll
Ly
x
2 2
pl
N
+ Lqly
X
Lx
ql πy
pl πx
) cos(
) sin(πz),
=
2Rac Al (τ ) cos(
2 2
Lx
Ly
pl
ql
+ Ly + 1
l=1 L
x
v1 =
w1
pl
Lx
2
T1 =
N
X
l=1
2
2Al (τ ) cos(
pl πx
ql πy
) cos(
) sin(πz),
Lx
Ly
(4.17a)
(4.17b)
(4.17c)
(4.17d)
where N is the number of viable modes, and the “Al ”s the unknown amplitudes. This is
a 3D generalisation of the result found for the 2D convection modes in chapter 3. In fact,
√
substituting in y = 0, Lx = 2, Rac = 9π 2 /2, N = 2, p1 = 1 and p2 = 2 into solutions
(4.17) gives the 2D solutions (3.13). Recall that (u, w) = (ψz , −ψw ) for the comparison to
be valid.
Note that if u takes the form,

a

 
 b  pπix/L qπiy/L jπiz
x
y

u=
e
e
,
 c e
 
d
(4.18)
then
M u = M p,q,j u,
(4.19)
46
Chapter 4. Three interacting modes
for all a, b, c, d, p, q, j, where

M p,q,j
P πi
Qπi
jπi

0

0
0
−π 2 σ 2 Rac π 2 (P 2 + Q2 )


=
0
−π 2 σ 2
0
−Rac π 2 Qj


0
0
−Rac π 2 P j
 −π 2 σ 2
0
0
1
−π 2 σ 2
P =
p
,
Lx
Q=
q
,
Ly



,



(4.20)
and σ 2 = P 2 + Q2 + j 2 .
As the matrix, M p,q,j , has 5 rows and 4 columns, the rows are not linearly independent;
explicitly, it can be shown that
iπσ 2 (row 1) = j (row 2) + Q (row 3) + P (row 3).
(4.21)
This implies that for solutions of the form (4.18), the conservation of momentum equations ensure that the incompressibility condition is satisfied. Hence, the incompressibility
equation is discarded in the differential operator, M , and the matrices are redefined as

0
0

 0 ∇2
M =
 ∇2 0

0
0
and

M p,q,j
0
0


0
−π 2 σ 2
=
 −π 2 σ 2
0

0
0
∇2 −Rac ∇2H

∂ ∂ 

Rac ∂y
∂z 
,
∂ ∂ 
Rac ∂x

∂z
∇2
0
0
1
(4.22)
−π 2 σ 2 Rac π 2 (P 2 + Q2 )
0
−Rac π 2 Qj
0
−Rac π 2 P j
1
−π 2 σ 2



.


(4.23)
The necessary tools are now available to solve the O(ε2 ) PDE, given by

u2

 v2
M ·
 w
 2
T2


0

 
 
=
 
 
0


.


0
∂
∂
∂
u1 ∂x
T1 + v1 ∂y
T1 + w1 ∂z
T1
(4.24)
u1 , v1 , w1 and T1 are known trigonometric functions from the linear analysis, and so the
above PDE is inhomogeneous. The right hand side can be evaluated as a series of trigonometric functions of the form
X
n
r̂n cos(pn πx/Lx ) cos(qn πy/Ly ) sin(jn πz),
(4.25)
4.1. Evolution equations
47
where r̂n , pn , qn and jn are known. This series can be converted to exponential form using
the identity,
cos(pn πx/Lx ) cos(qn πy/Ly ) sin(jn πz) =
1 X X X
sj eisp pn x/Lx eisq qn y/Ly eisj jn z ,
8i
sj ∈S sq ∈S sp ∈S
(4.26)
where S = {−1, 1}. The O(ε2 ) PDE is now expressed as

u2

 v2
M
 w
 2
T2


0




 X  0  ip πx/L iq πy/L ij πz
x
y
=
 n

e n
e n .

 0 e



n
rn
(4.27)
The set of all (pn , qn , jn ) contains both the unique second order modes, and the non-unique
combinations which differ only by sign. A particular solution is sought, of the form

u2

 v2

 w
 2
T2


an
 X

 bn
=




n  cn
dn


 ip πx/L iq πy/L ij πz
y
x
e n
e n ,
e n


(4.28)
where an , bn , cn and dn are unknown complex numbers, and pn , qn , jn are chosen to match
the right hand side of equation (4.27). Substituting this solution into the PDE (4.27), and
distributing the matrix M over all terms gives

X
n
an

 bn
M
 c
 n
dn


0




 ip πx/L iq πy/L ij πz X  0  ip πx/L iq πy/L ij πz
x
n
y
n
x
y
e n

 n
e
e
=
e n
e n .

 0 e


n 
rn
(4.29)
Using equation (4.19):

X
n
an

 bn
M pn ,qn ,jn 
 c
 n
dn


0




 ip πx/L iq πy/L ij πz X  0  ip πx/L iq πy/L ij πz
x
y
x
n
y
n
e n

 n
e
e n
e n .
e
=

 0 e



n
rn
(4.30)
By collecting like coefficients,

an

 bn
M pn ,qn ,jn 
 c
 n
dn


0

 

  0 

=
  0 ,
 

rn
(4.31)
48
Chapter 4. Three interacting modes
for each mode, (pn , qn , jn ). Each matrix, M pn ,qn ,jn , is of full rank, and can be inverted
to solve for an , bn , cn , and dn . For M pn ,qn ,jn to be singular, jn must be equal to 1, as
only the lowest vertical mode number solves the homogeneous problem, however, all of the
secondary modes seen on the right hand side of the PDE (4.24), only take values of jn = 2,
and no secular terms are present. Vertical mode numbers of jn = 0 are also calculated,
however, these occur as sin(0) terms and so are identically zero.
The second order solutions take the following form,
pn πx
qn πy
) cos(
) cos(2πz),
Lx
Ly
n
X
pn πx
qn πy
=
C2n cos(
) sin(
) cos(2πz),
Lx
Ly
n
X
pn πx
qn πy
C3n cos(
=
) cos(
) sin(2πz),
Lx
Ly
n
X
pn πx
qn πy
=
C4n cos(
) cos(
) sin(2πz),
L
L
x
y
n
u2 =
v2
w2
T2
X
C1n sin(
(4.32a)
(4.32b)
(4.32c)
(4.32d)
(4.32e)
where the amplitudes, Cln , are dependent on the unknown primary amplitudes, and known
primary mode numbers, aspect ratios and critical Rayleigh number, but are tediously long
to express here.
∂
∂
T1 , v1 ∂y
T1 , and
The types of secondary modes seen here are generated by the u1 ∂x
∂
w1 ∂z
T1 terms in the PDE (4.24). These terms are the product of primary modes with
themselves and with each other. For the product of two primary modes, n and m, (where
n may be equal to m), the secondary modes generated are the 8 different combinations of
(|pn + sp pm |, |qn + sq qm |, |jn + sj jm |),
(4.33)
where sp , sq , sj ∈ {−1, 1}. Furthermore, for N primary modes, there are
N (N + 1)
,
2
(4.34)
unique primary mode interactions including self-interaction. Hence, there are potentially
4N (N + 1) unique secondary modes. For three interacting modes, there may be as much
as 48 unique secondary modes. This number can be reduced by considering the fact
that the jn = 0 modes disappear, as discussed above, halving the number of modes seen.
Further non-uniqueness arises; consider the secondary modes generated by subtractive
self-interactions, (sp = sq = −1), of the n and m modes, given by
qn − qn
pn − pn
πx) cos(
πy) sin(2πz) = cos(0) cos(0) sin(2πz) = sin(2πz), (4.35a)
Lx
Ly
pm − pm
qm − qm
cos(
πx) cos(
πy) sin(2πz) = cos(0) cos(0) sin(2πz) = sin(2πz). (4.35b)
Lx
Ly
cos(
4.1. Evolution equations
49
These secondary modes are identical, regardless of the individual mode numbers, and are
always observed. Additive self-interaction secondary modes (sp = sq = 1), also do not
occur, as they cancel each other out. In general, for 3 modes interacting in a 3D box of a
porous medium, up to 19 unique secondary modes are observed. The symbolic package,
Mathematica, was used to assist in the manipulation of the large number of modes.
4.1.3
Third order: Secular modes
The O(ε3 ) PDE is given by
M u 3 = h3 .
(4.36)
By seeking a particular solution, the same method is used as in the O(ε2 ) PDE. Again it
is found that

an

 bn
M pn ,qn ,jn 
 c
 n
dn



 = rn ,


(4.37)
for all tertiary modes, (pn , qn , jn ), with amplitude vectors, r n . As with the 2D case,
secular modes appear at the third order of expansion. The tertiary terms are generated
by the product of the secondary terms with the primary terms. Consider, for example,
the subtractive self-interaction secondary for mode n, equation (4.35a). By multiplying
with any primary mode, m, (which may be equal to n), the two tertiary modes generated
are
pm
qm
πx) cos( πy) sin(πz),
Lx
Ly
pm
qm
cos( πx) cos( πy) sin(3πz),
Lx
Ly
cos(
(4.38a)
(4.38b)
the first of which is a secular mode. Thus, these expected secular modes always appear in
the O(ε3 ) PDE. Again, a solvability condition must be imposed on the secular terms. For
the secular modes, r n must lie in the eigenspace of the corresponding M pn ,qn ,jn matrix.
Equivalently, r n must be perpendicular to the nullspace of the transpose. This is expressed
mathematically by
0 = m · rn ,
(4.39)
for all secular modes, where m = Nullvector[M Tpn ,qn ,1 ]. To find the secular terms from
the right hand side of equation (4.36), the orthogonality of the exponential terms is used.
Taking the inner product of the right hand side with e
of the e
pn
πx
Lx
e
qn
Ly
πy πz
e
−pn
πx
Lx
e
−qn
πy
Ly
e−πz gives the amplitude
secular term. The secular modes are known, as these modes are
solutions of the first order, homogeneous equation. The use of the inner product here
identifies the problem-causing secular terms, whilst ignoring the terms for which a third
50
Chapter 4. Three interacting modes
order solution can be easily found. Algebraically:
1
rn =
8Lx Ly
1
Z
Z
Ly
Z
Lx
(e
−1
−Ly
−pn
πx
Lx
e
−qn
πy
Ly
e−πz h3 )dx dy dz.
(4.40)
−Lx
Substituting the projection, (4.40), into equation (4.39), gives
1
0=
8Lx Ly
Z
1
Z
Ly
Z
Lx
(e
−1
−Ly
−pn
πx
Lx
e
−qn
πy
Ly
e−πz (m · h3 )dx dy dz.
(4.41)
−Lx
It is quite important to identify the secular modes using equation (4.41). Using the
orthogonality of the different modes, eliminates the technically difficult task of identifying
the generated secular terms. Impey et al. (1990) identify the secular modes that are
expected to occur (equation (4.38a)). Unfortunately, there is every possibility that secular
modes form unexpectedly, depending on the mode numbers (as occurs in the symmetrybreaking case). The method of Impey et al. (1990) does not capture these terms, and
they discard the perturbation method and claim it cannot resolve such systems. By using
the projection method, (4.40), unexpected secular terms are captured when they occur.
Thus, the perturbation procedure is appropriate to use, even for symmetry-breaking cases.
Cases where unexpected terms occur will be known as special cases, and are discussed in
section 4.4.
The general result of applying equation (4.41) for each secular mode, is a system of time
evolution equations for the amplitudes of the modes. For simplicity, let (A1 , A2 , A3 ) =
(A, B, C). The basic set of equations takes the following form:
∂
A = Ȧ = A(a − aa A2 − ab B 2 − ac C 2 ),
∂τ
∂
B = Ḃ = B(b − ba A2 − bb B 2 − bc C 2 ),
∂τ
∂
C = Ċ = C(c − ca A2 − cb B 2 − cc C 2 ),
∂τ
(4.42a)
(4.42b)
(4.42c)
where a, b and c are real and linear functions of R1 , while all other coefficients are real,
positive and independent of R1 . Thus, a general method for deriving the dynamical system
which describes the time evolution of the amplitudes of the three modes which share the
same critical Rayleigh number at the onset of convection has been outlined. This procedure
will be used throughout the thesis. Equation (4.41) is evaluated for each secular term.
In the following section, specific examples of three-mode interactions are studied, each of
which has different topological structure.
4.2
Specific Examples
In this section, two different specific examples are analysed. These examples represent the
two different types of dynamics (and thus, two equivalence classes) seen for systems of the
form (4.42).
4.2. Specific Examples
51
z
z
1
1
y
y
√
√1
2
2
√1
2
x
(a) The (1, 0) mode
√
2
x
(b) The (2, 0) mode
z
1
y
√1
2
√
2
x
(c) The (0, 1) mode
Figure 4.2: The three modes which share the same critical Rayleigh number for example
1.
4.2.1
Example 1
The following section is focussed on the analysis of the dynamical system which governs
the evolution of the amplitudes of the (p, q) = (1, 0), (2, 0) and (0, 1) modes in a box with
√
√
dimensions Lx = 2 and Ly = 1/ 2, which is an example of where three regions meet
(example 1) in figure 4.1. All three of these modes correspond to 2D convection rolls,
which are independent of one horizontal coordinate, (see figure 4.2). These modes share
a critical Rayleigh number of
Rac =
9π 2
.
2
(4.43)
The first order solution is given by
√
√
√
πx
u1 = −3 2π 2 A sin( √ ) cos(πz) − 3 2π 2 B sin( 2πx) cos(πz),
2
√ 2
√
v1 = −3 2π C sin( 2πy) cos(πz),
√
πx
w1 = 3π 2 A cos( √ ) sin(πz) + 6π 2 B cos( 2πx) sin(πz)
2
√
2
+6π C cos( 2πy) sin(πz),
√
πx
T1 = 2A cos( √ ) sin(πz) + 2B cos( 2πx) sin(πz)
2
√
+2C cos( 2πy) sin(πz),
(4.44a)
(4.44b)
(4.44c)
(4.44d)
52
Chapter 4. Three interacting modes
where A, B and C are the time-dependent amplitude variables for the (1, 0), (2, 0) and
(0, 1) modes respectively. The dynamical system for this example is given by
3R1 9 2 2 45 2 2 2061 2 2
− π A − π B −
π C ,
2
8
8
496
63 2 2 9 2 2 279 2 2
2
Ḃ = π B 3R1 − π A − π B −
π C ,
16
2
46
153 2 2 279 2 2 9 2 2
2
π A −
π B − π C ,
Ċ = π C 3R1 −
62
46
2
Ȧ = π 2 A
(4.45a)
(4.45b)
(4.45c)
as derived by the process outlined in section 4.1.2. Like all dynamical systems of the
form given in the general system, (4.42), this example contains the reflection symmetry
about the A = 0, B = 0, and C = 0 planes. That is, the dynamical system is Z2 × Z2 ×
Z2 equivariant, (the 3D equivalent of the 2D symmetry properties observed by Riley &
Winters (1989) and Impey et al. (1996)) .
Each of the 8 octants in the phase-space has equivalent dynamics, so it is sufficient to
focus the analysis on the positive octant, where A, B, C ≥ 0. Fixed points occur where
Ȧ = Ḃ = Ċ = 0 and those which are stable under small perturbations are possible
candidates for the steady-state solution of the convection problem.
Steen (1983) utilises a useful transformation, and expresses system (4.42) in terms of
squares of the amplitudes. This assumes Z2 × Z2 × Z2 equivariance holds, but this is not
always the case. The special case presented in section 4.5 cannot be represented in this
way, so this transformation is not used here, to keep this systematic approach consistent.
The geometric advantages of the transformed phase-space are used in section 4.3, without
explicitly performing the transformation.
For a precritical system, where Ra < Rac and R1 = −1, there is only one fixed
point, located at (A, B, C) = (0, 0, 0), which is a stable node. This solution, for which
all amplitudes are zero, corresponds to the conduction solution. For a postcritical system
however, where R1 = 1, there are six more fixed points found in the positive octant. These
fixed points and their stabilities are listed in table 4.1.
As with the precritical system, there exists a conduction solution, F0 , however, in
the postcritical case, it is an unstable node. The candidates for the steady-state solution
are the stable nodes that lie on the coordinate axes of each amplitude variable, labelled
FA , FB and FC . These stable nodes correspond to solutions where only one mode has a
non-zero amplitude. Depending on the initial conditions, the system will approach one of
the three stable nodes. Lastly, there are three saddle points which correspond to solutions
of two modes with non-zero amplitudes, labelled FAB , FAC and FBC . For future reference,
note that FAB has a different stability signature to FAC and FBC . This has an important
effect on the structure of the phase-space.
For these “critical boxes”, the bifurcation that occurs is degenerate. All fixed points
are created at (0, 0, 0) for R1 = 0. Increasing R1 beyond 0 merely spatially rescales all of
the nullclines and consequently the fixed points; no qualitatively new dynamics is seen.
4.2. Specific Examples
53
Fixed point
F0
Position (A, B, C)
(0, 0, 0)
Approximate position
Stability
Unstable node
FA
( √23π , 0, 0)
≈ (0.368, 0, 0)
Stable node
FB
2
(0, √3π
, 0)
≈ (0, 0.260, 0)
Stable node
FC
2
(0, 0, √3π
)
≈ (0, 0, 0.260)
Stable node
FAB
( 3√43π , 3√23π , 0)
≈ (0.245, 0.123, 0)
Saddle point (2U)
FAC
2170
, 0, 3π2 √31
)
( 3π√
219
219
≈ (0.334, 0, 0.080)
Saddle point (1U)
FBC
(0,
≈ (0, 0.170, 0.170)
Saddle point (1U)
√
√
√
√
√
√
23
23
,
9π
9π )
Table 4.1: Example 1: All fixed points and their stabilities for those which lie in the
positive octant, for the postcritical, R1 = 1, dynamical system. 1U indicates the saddle
point has 1 direction of instability.
Hence, it is appropriate to set R1 to −1 or 1, and not treat it as a continuous parameter.
In the phase-space, one can identify the basins of attraction which are associated with
each stable node. Unlike the two-mode example, where a 1D separatrix divides the 2D
basins of attraction, here 2D separatrix surfaces serve as boundaries between the 3D basins
of attraction. Before addressing the 3D structure, consider the dynamics on the C = 0
plane, given by
3
9
45
Ȧ(A, B, 0) = π 2 A( R1 − π 2 A2 − π 2 B 2 ),
2
8
8
63
9
Ḃ(A, B, 0) = π 2 B(3R1 − π 2 A2 − π 2 B 2 ).
16
2
Ċ(A, B, 0) = 0.
(4.46a)
(4.46b)
(4.46c)
This plane is an invariant set; trajectories on this plane never leave. As one would expect,
system (4.46) is equivalent to the 2D system (3.35), (The A and B modes have the same
wavenumber in the same size box as in the 2D case). The dynamics on this surface are
known, as shown in figure 3.3, and can be used to infer the behaviour off of this surface.
The A = 0 and B = 0 planes are also invariant sets, and it is found that the 2D dynamics
on these planes is qualitatively equivalent to system (4.46). That is, each 2D system contains four fixed points; an unstable node at the origin, 2 stable nodes on each of the axes,
and one saddle point. In each system, the stable separatrix of the saddle point passes
through the origin, and separates the two basins of attraction.
Now that the dynamics on the plane walls of the octant are known, the 3D basins of
attraction can be deduced. In the full three dimensional system, (4.45), the 2D stable
separatrix manifolds of FAC and FBC divide the octant into three basins of attraction by
54
Chapter 4. Three interacting modes
C
FC
F BC
1U
B
2U
FB
F AB
1U
F AC
FA
A
F0
3U
Figure 4.3: The 3D phase-space for example 1. The 2D stable separatrices of FAC and
FBC divide the octant into 3 basins of attraction. 1U indicates the fixed point has 1
direction of instability.
both folding onto the 1D stable separatrix of FAB . The intersection of these manifolds
with the walls of the octant lie along the separatrix line of each 2D system. This is shown
in figure 4.3. This apparent asymmetry in the structure of the basins occurs because FAB
has an extra degree of instability, compared to the other saddles.
Rather than examining the full 3D phase-space, the topology of the basins of attraction can be shown on a simple 2D space for ease of comprehension. This reduction
eliminates information about the simple growth and decay of the modes, to focus on the
interchange of energy between the modes. Take a planar slice through the positive octant,
which separates the origin from infinity. Regardless of which plane is chosen, the same
topology is seen; there are three basins, where the FA and FB basins are not connected.
Here, rather than use an arbitrary plane, a 2D invariant surface is used which contains
all fixed points except for F0 . The surface chosen is the 2D unstable separatrix of FAB
which separates trajectories which originate at the origin, from those that originate at
infinity. The dynamics off of the surface which closes the origin off from infinity will be
the same, but with an additional vector component pointing towards the separatrix. As
such, this surface is a sufficient representation of the entire phase-space. Each trajectory
on the surface represents a family of trajectories, all approaching the same fixed point.
Technically, the 2D unstable separatrix of FAB represents a quotient of the phasespace, where the family of trajectories which approach a trajectory on the separatrix are
considered equivalent.
Figure 4.4(a) shows the basins of attraction on the separatrix surface (quotient space).
Figure 4.4(b) shows an equivalence set of all trajectories, and is a 2D invariant manifold.
The relative size of each of the basins provides a measure for the probability of
the survival of the associated modes. The volume of each basin is calculated by using a
Runge-Kutta method in MATLAB R2009b to numerically integrate the dynamical system,
4.2. Specific Examples
55
FC
FBC
FAC
+
FC basin
+
+
FB
FAB
FA basin
FB basin
++
FB
FA
FAB
(a) The 2D quotient space (the unstable separatrix of FAB ). The
topology seen here is representative
of the full 3D space.
++ FO
(b) A 2D invariant manifold from
the origin to infinity of equivalent trajectories. Trajectories approach the separatrix, shown in
red. The entire phase-space can be
constructed from the family of all
such manifolds.
Figure 4.4: Two manifolds, each sharing the trajectory shown in red. Each trajectory in
figure 4.4(a) represents a corresponding family of trajectories, such as figure 4.4(b). +
signs indicate instability.
z
1
y
21/4
21/4
x
Figure 4.5: The convection pattern of the (1, 1) mode in the box, with Lx = Ly = 21/4 .
(4.45), for a grid of initial conditions in the domain, A, B, C ∈ [0, 0.4]. The separatrices
are identified numerically in the phase-space. This provides a direct comparison between
the amplitudes of the temperature function of the viable modes. It has been found that
the (0, 1) mode is by far the most likely, with a volume of 57.9%, followed by the (2, 0)
mode with 38.7% and the (0, 1) mode with a slim 3.5%.
4.2.2
Example 2
A second example is used to show a system that exhibits different phase-space structure
to that seen in example 1. A box with dimensions, Lx = Ly = 21/4 , is used, for which the
(1, 0), (0, 1), and (1, 1) modes share the same critical Rayleigh number,
3
√
,
Rac = π 2 +
2
2
(4.47)
56
Chapter 4. Three interacting modes
and so are all viable at the onset of convection. This example is circled in figure 4.1. This
example has been studied by Steen (1983) using a Galerkin method, and is compared to
the behaviour obtained here by the perturbation method. Unlike the previous example,
one of the competing modes is a three dimensional square convection cell, rather than a
two dimensional roll, (see figure 4.5). The first order solution is given by
u1 = −Lµπ 2 A sin
π π π x cos(πz) − Lλπ 2 C sin
x cos
y cos(πz),
L
L
L
(4.48a)
v1 = −Lµπ 2 B sin
π π π y cos(πz) − Lλπ 2 C cos
x sin
y cos(πz),
L
L
L
(4.48b)
π π x sin(πz) + µπ 2 B cos
y sin(πz)
L L
π
π
+2λπ 2 C cos
x cos
y sin(πz),
L
L
w1 = µπ 2 A cos
π π x sin(πz) + 2B cos
y sin(πz)
L
L
π π +2C cos
x cos
y sin(πz),
L
L
(4.48c)
T1 = 2A cos
(4.48d)
where A, B and C are the amplitude variables for the (1, 0), (0, 1) and (1, 1) modes
√
√
respectively, µ = 2 + 2, λ = 1 + 2 and L = Lx = Ly = 21/4 . The dynamical system
obtained takes the general form given in (4.42), with
√
1
3+2 2 4
2
a=b= 1+ √
π R1 , aa = bb =
π ,
(4.49a)
4
2
√
√
1225 + 148 2 4
6332 + 4571 2 4
ab = ba =
π , ac = bc =
π ,
(4.49b)
204
4416
√
√
√ 2407 + 1768 2 4
11 + 7 2 4
c = 1 + 2 π 2 R1 , ca = cb =
π , cc =
π . (4.49c)
1104
12
In addition to the reflective symmetry properties about the A = 0, B = 0 and C = 0
planes, this dynamical system also has symmetry about the A = B and A = −B planes.
This symmetry comes from the fact that the (1, 0) and (0, 1) modes in a box of equal
horizontal lengths have the same horizontal wavenumber, so one mode can be obtained
from the other by a π/2 rotation.
For the precritical system where R1 = −1, the conduction solution is again the only
fixed point found, and it is stable. The fixed points and the stabilities found in the positive
octant of the postcritical system are listed in table 4.2. As in the previous example, three
stable nodes are found, each of which corresponds to a solution where only one mode is
present in the final state. In this case, however, there exists a saddle point, GABC , which
corresponds to a solution where all three modes coexist.
As there are three stable nodes, the octant is again divided into three basins of
attraction. The 2D manifolds on the plane walls of the positive octant are dynamical
systems which are topologically equivalent to the 2D example, (4.46). The boundaries of
4.2. Specific Examples
Fixed point
G0
GA
GB
GB
GAB
GAC
GBC
GABC
57
Approximate position
(0, 0, 0)
≈ (0.344, 0, 0)
≈ (0, 0.344, 0)
≈ (0, 0, 0.375)
≈ (0.219, 0.219, 0)
≈ (0.198, 0, 0.200)
≈ (0, 0.198, 0.2)
≈ (0.145, 0.145, 0.184)
Stability
Unstable node
Stable node
Stable node
Stable node
Saddle point
Saddle point
Saddle point
Saddle point
Signature
1D
1D
1D
2D
unstable
unstable
unstable
unstable
Table 4.2: Example 2: All fixed points and their stabilities for those which lie in the
positive octant, for the postcritical, R1 = 1, dynamical system.
C
GC
1U
B
G BC
G ABC
1U
2U
GB
G0
1U
3U
G AB
G AC
GA
A
Figure 4.6: The 3D phase-space for example 2. The 2D stable separatrix surfaces of GAB ,
GAC and GBC divide the octant into 3 basins of attraction.
these basins of attraction are formed by the 2D stable separatrices of the GAB , GAC , and
GBC saddles. In contrast to the previous example, these separatrices do not fold onto
one of the plane walls of the octant, but approach the 1D stable manifold of GABC . The
basins of attraction are shown in figure 4.6. The 2D stable separatrix formed by GAB is
trivial to locate, as by symmetry it must lie on the plane, A = B. The nullclines of this
example have the same qualitative structure seen in figure 3(c) of Steen (1983). Similarly,
the boundaries of the basins of attraction in figure 4.6 are equivalent to those shown by
Steen (1983) in his figure 5.
The basins of attraction in this example can also be represented by an appropriate
2D quotient surface. In this case, the 2D unstable manifold of GABC , contains all fixed
points except for the origin and contains all equivalence classes regarding the exchange
between the modes. The quotient system on this surface is given in figure 4.7.
The volume of each basin of attraction has been measured in the domain A, B, C ∈
[0, 0.4], to compare how likely it is that each stable node is the steady-state solution. The
(1, 0) and (2, 0) modes are equally likely to occur with a relative volume of 37.4%, while
58
Chapter 4. Three interacting modes
GC
GBC
+
GC basin
++
GB basin
GAC
+
GABC
GA basin
+
GB
GAB
GA
Figure 4.7: The 2D quotient system (the 2D unstable invariant manifold of GABC ). The
topology seen here is representative of the full 3D space.
the (1, 1) mode is less likely with a volume of 25.1%.
The two examples above in sections 4.2.1 and 4.2.2 each exhibit a different structure of
the phase-space. It is not immediately obvious why this is so, as they both have the same
form of dynamical system, given by (4.42), but with different coefficients. In the following
section, the coefficient space is explored to determine the possible types of phase-space
topology.
4.3
Structure of the separatrix manifolds
In this section the parameter values for which the dynamical system for a critical box will
contain the structure seen in the first example, (4.45), or contain the structure seen in
the second example, (4.49) are derived. The existence of the GABC fixed point is the key,
as this saddle point provides the 1D stable manifold which the separatrices tend towards.
The conditions for which GABC occurs is obtained by taking advantage of the ellipsoidal
nature of the nullclines.
4.3.1
Exploring the parameter space
Fixed points occur at the intersection of the Ȧ, Ḃ and Ċ nullclines, as the motion in all
directions is zero. The three nullclines for the postcritical, general system, (4.42), are
• Ȧ = 0 ⇒ A = 0 ∪ a = aa A2 + ab B 2 + ac C 2 ,
• Ḃ = 0 ⇒ B = 0 ∪ b = ba A2 + bb B 2 + bc C 2 ,
• Ċ = 0 ⇒ C = 0 ∪ c = ca A2 + cb B 2 + cc C 2 .
Each nullcline has a plane wall and an ellipsoidal component. A fixed point like GABC does
not lie on any plane wall, and can only exist if there is a point where all three ellipsoidal
nullclines intersect. All three ellipsoid nullclines share the following properties:
• they have (0, 0, 0) as the origin,
4.3. Structure of the separatrix manifolds
59
• they are symmetric about the A = 0, B = 0 and C = 0 planes and
• they contain semi-principal axes which are parallel to the coordinate axes.
It is easy to see that these ellipsoidal nullclines are linear planes in the squared amplitude
phase-space, (i.e A0i = A2i ). This implies that as one varies the parameters, the critical
intersection of two ellipsoidal nullclines must along a coordinate axis, and the critical
intersection between all three ellipsoids must occur on the octant walls.
For the purpose of this problem it is assumed that the axial fixed points, denoted by the
A, B and C subscripts, are stable nodes. For this to occur, the X ellipsoid nullcline must
be the furthest away from the origin along the X axis, where X could be any variable, A,
B or C. For example, for the Ȧ = 0 ellipsoid must have a larger semi-principal axis along
the A-coordinate axis, than the Ḃ = 0 and Ċ = 0 ellipsoids. This imposes the parameter
inequalities:
aa
ba
< ,
a
b
aa
ca
< .
a
c
(4.50)
Similar conditions are applied for the other two axes:
bb
ab
< ,
b
a
bb
cb
< ,
b
c
(4.51)
cc
ac
< ,
c
a
cc
bc
< .
c
b
(4.52)
To determine if an intersection of all ellipsoids occurs, the problem is simplified by scaling
the system so that one of the ellipsoids is transformed into a sphere centered at the origin.
The intersection of the other two ellipsoids is an ellipse, (call it I). The distance from the
origin to this ellipse is optimised to find the maximum and minimum distance. The radius
of the spherical nullcline must lie in between this maximum and minimum distance for an
intersection of all three ellipsoids to occur. To scale, let
Ā
A= √ ,
ca
B̄
B=√ ,
cb
C̄
C=√ .
cc
(4.53)
The scaled ellipsoids are defined by:
1 = āa Ā2 + āb B̄ 2 + āc C 2 ,
(4.54a)
1 = b̄a Ā2 + b̄b B̄ 2 + b̄c C̄ 2 ,
(4.54b)
c = Ā2 + B̄ 2 + C̄ 2 ,
(4.54c)
where
āi =
ai
,
aci
b̄i =
bi
,
bci
(4.55)
for all i ∈ {a, b, c}. The parameter, c, is the radius squared of the spherical nullcline. The
conditions, (4.50) - (4.52), become
āa < b̄a ,
1
āa < ,
c
(4.56)
60
Chapter 4. Three interacting modes
C̄
P3
a¯c = b¯c
P5
a¯c < b¯c
I
a¯c > b¯c
B̄
P6
I
I
P4
P4
Ā
P4
Figure 4.8: The ellipse formed by the intersection of the non-spherical nullclines, for
different values of ac and bc . The minimum distance to the origin may occur on either the
Ā = 0 plane, the B̄ = 0 plane, or both.
b̄b < āb ,
1
< āc ,
c
1
b̄b < ,
c
1
< b̄c .
c
(4.57)
(4.58)
These will be referred to as the “axial conditions”. The distance from the ellipse, I, to the
origin is optimised subject to the constraint that the I lies on the intersection between the
two non-spherical ellipsoids. A Lagrange multiplier method is used for the parameters λ1
and λ2 . The Lagrangian,
L(Ā, B̄, C̄, λ1 , λ2 ) = Ā2 + B̄ 2 + C̄ 2
+λ1 (āa Ā2 + āb B̄ 2 + āc C̄ 2 − 1)
(4.59)
+λ2 (b̄a Ā2 + b̄b B̄ 2 + b̄c C̄ 2 − 1),
is optimised by finding the coordinates where
∂
∂
∂
∂
∂
L=
L = 0.
L=
L=
L=
∂λ1
∂λ2
∂ Ā
∂ B̄
∂ C̄
(4.60)
Figure 4.8 shows the relevant optimal points. P4 is the maximum point, while P3 ,
P5 and P6 are all possible minimum points. These optimal points all lie on the octant
walls, as expected by the geometrical arguments from earlier, where the ellipsoid nullclines
are planes after the amplitude squared transformation. Only one minimum point exists
for each set of parameters. For the spherical nullcline to intersect I, its radius squared,
c, must lie between the magnitude squared of the minimum and maximum points. This
gives a set of three parameter inequalities which must all be satisfied for an intersection
4.4. Special case of the dynamical system
61
to occur:
c(ab ba − bb aa ) < b(ca ab − cb aa ) + a(cb ba − ca bb ),
(4.61a)
a(bc cb − cc bb ) < c(ab bc − ac bb ) + b(ac cb − ab cc ),
(4.61b)
b(ca ac − aa cc ) < a(bc ca − ba cc ) + c(ba ac − bc aa ).
(4.61c)
No more than one condition can fail for any given set of parameters. If all conditions are
true, then an intersection occurs and the dynamics obtained are equivalent to example
2. If (4.61a) is false, then no intersection occurs and the system is equivalent to example
1, where the C–mode is favoured. If (4.61b) or (4.61c) are false, then the dynamics are
equivalent to those seen in example 1 given a permutation of the coordinate axes, and the
A–mode or B–mode are favoured.
It has been shown that for a dynamical system of the form given in (4.42), where stable
nodes are assumed to exist on each axis of the phase-space, that there are two distinct
classes; systems belonging to class 2 contain an extra floating fixed point in each octant
and the boundaries of the basins of attraction intersect on the stable manifold of this fixed
point, (e.g example 2). For class 1, the boundaries of the basins of attraction intersect on
one of the plane walls, A = 0, B = 0 or C = 0, usually resulting in one mode having a
favourably larger basin of attraction (e.g. example 1). Thus, the simple parameter test,
(4.61), not only determines which class occurs for a given dynamical system of the appropriate form, but in the first class, shows the orientation of the structure.
4.3.2
Classification of the critical boxes
The dynamical system of three interacting modes for each critical box with moderate
aspect ratios (Lx , Ly < 3.5) have been derived. Each critical box has been categorised
into either of the two classes listed above, using test (4.61). The results are shown in
figure 4.9. Neither of the two classes is particularly prevalent; they have almost equal
occurrences. These results are important as they offer a quick reference for the underlying
manner in which the three modes interact.
4.4
Special case of the dynamical system
In general, the dynamical system takes the form shown in (4.42), however, there may be
special cases where symmetry-breaking terms appear. This can occur where a primary
viable mode can be reinforced by the nonlinear interaction of the other viable modes.
Usually the secondary and tertiary modes generated by the interaction of two primary
modes decay rather quickly, as the critical Rayleigh number of the higher order modes
are much higher than the critical Rayleigh number of the primary modes, which Ra is
assumed to be close to. In special cases, however, some non-obvious tertiary modes are
62
Chapter 4. Three interacting modes
(2,2)
(1,3)
(1,3)
(1,2)
(2,0)
(2,1)
3
(0,3)
(0,3)
(0,3)
(0,3)
(3,0)
(1,3)
(1,3)
(2,2)
(3,0) (3,1)
(0,3) (0,3) (0,3)
(2,1)
(4,0)
(2,2)
(3,0)
(1,1)
(1,2)
(2,0) (2,1)
(0,2)
Class 2.
(2,2)
(1,0)
(0,2)
(3,1)
(2,2)
(2,0)
(1,2)
2
Legend
(3,0)
(0,2)
(3,1)
(1,2)
(0,2)
(0,2)
Class 1
(2,2)
(1,2)
(0,2)
(1,2)
(0,2)
(4,0)
(1,2)
(0,2)
Special case.
(4,0)
(2,0)
Ly
(1,0)
(3,0)
(2,1)
(3,1)
(1,1)
(3,1)
(2,1)
(1,1)
(1,1)
(0,1)
(2,0)
1
(0,1)
(0,1)
(0,1)
(1,0)
1
(3,0)
(2,0)
(4,0)
(2,1)
(1,1)
(0,1)
(4,0)
3
2
(4,0)
(3,0)
(5,0)
4
Lx
Figure 4.9: Categorisation of each system of three modes. Each critical box has been
categorised into either of the two classes, and the results are overlaid onto figure 4.1. For
class 1, the favoured mode is indicated by the shaded segment of the circle.
viable solutions, and do not necessarily decay. These special cases occur when either:
• the tertiary interaction of a mode with itself reinforces one of the other primary
modes, or
• the tertiary interaction of two modes reinforces the third primary.
If either of these occur, extra inhibitive cubic terms appear in the dynamical system. In
this section, the mode numbers for which these special cases occur are identified.
4.4.1
Two interacting modes
The simplest non-trivial case where symmetry breaking can occur is found in the interactions between two modes. Consider the two modes, with mode numbers, kA = (p1 , q1 , j1 )
and kB = (p2 , q2 , j2 ). The multiplication of these two modes generates a set of higher-tier
modes given by
kA × kB = {(|p1 + sp p2 |, |q1 + sq q2 |, |j1 + sj j2 |) |sp , sq , sj ∈ {−1, 1}},
(4.62)
(similar to (4.33) ). Note that this operation is commutative and associative. For the
primary modes at the onset of convection, j = 1. As such, it is ignored, except for noting
that secular modes can only appear at the ith order expansion, where i is odd.
Unusual cubic terms that could appear in the Ȧ equation, (4.42a), from the interaction
of 2 modes, are A2 B or B 3 . These would occur if kA ∈ kA ×kA ×kB , or kA ∈ kB ×kB ×kB ,
respectively.
To determine the conditions for which the A2 B term appears in the Ȧ equation, it is
noted that the set, kA × kB is the only set of modes for which multiplying by kB generates
4.4. Special case of the dynamical system
A2
kA × kA
(2p1 , 2q1 )
(2p1 , 2q1 )
(2p1 , 2q1 )
(2p1 , 2q1 )
(2p1 , 0)
(2p1 , 0)
(2p1 , 0)
(2p1 , 0)
(0, 2q1 )
(0, 2q1 )
(0, 2q1 )
(0, 2q1 )
AB
kA × kB
(p1 + p2 , q1 + q2 )
(p1 + p2 , |q1 − q2 |)
(|p1 − p2 |, q1 + q2 )
(|p1 − p2 |, |q1 − q2 |)
(p1 + p2 , q1 + q2 )
(p1 + p2 , |q1 − q2 |)
(|p1 − p2 |, q1 + q2 )
(|p1 − p2 |, |q1 − q2 |)
(p1 + p2 , q1 + q2 )
(p1 + p2 , |q1 − q2 |)
(|p1 − p2 |, q1 + q2 )
(|p1 − p2 |, |q1 − q2 |)
63
Equal if
p1 = p2
p2 = p1
p2 = 3p2
p2 = 3p1
p2 = p1
p2 = p1
p2 = 3p1
p2 = 3p1
p2 = p1 = 0
p2 = p1 = 0
p2 = p1
p2 = p1
q1 = q2 (trivial)
q2 = 3q1
q2 = q 1
q2 = 3q1
q2 = q1 = 0 (trivial)
q2 = q1 (trivial)
q2 = q1 = 0
q2 = q1
q2 = q1 (trivial)
q2 = 3q1
q2 = q1 (trivial)
q2 = 3q1
Table 4.3: Conditions for which a special case occurs for two modes. The bold conditions
are non-trivial.
the kA mode. Therefore the set, (kA × kA ) × kB , can contain kA iff
kA × kA ∩ kA × kB 6= ∅.
(4.63)
Table 4.3 compares the mode numbers of the two sets in (4.63). It is found that for any
two modes, k1 and k2 , a special case will occur if the mode numbers obey any of the
following conditions:
p2 = p1 ∧ q2 = 3q1 ,
(4.64a)
p2 = 3p1 ∧ q2 = q1 ,
(4.64b)
p2 = 3p1 ∧ q2 = 3q1 ,
(4.64c)
By a similar argument, the A3 term also appears in the Ḃ equation, (4.42b), if (4.63)
is non-empty. Therefore, if any of conditions (4.64) are satisfied for the interaction of any
two modes, then the dynamical system takes the form,
Ȧ = A aR1 − aa A2 − ab B 2 − ac C 2 − ad AB ,
Ḃ = B bR1 − ba A2 − bb B 2 − bc C 2 − bd A3 ,
Ċ = C cR1 − ca A2 − cb B 2 − cc C 2 .
(4.65a)
(4.65b)
(4.65c)
The mode labels, A, B, and C can be always be chosen such that the dynamical system
can be written in the form of (4.65).
While there are many examples of two primary modes which satisfy one of these
conditions, for the purpose of this study it is only if these modes share the same critical
Rayleigh number at the onset of convection that they are of interest. Modes that satisfy
conditions (4.64) are of the form (p, q)×(3p, 3q), (p, q)×(3p, q), and (p, q)×(p, 3q). It can be
64
Chapter 4. Three interacting modes
C2
kC × kC
(2p3 , 2q3 )
(2p3 , 2q3 )
(2p3 , 2q3 )
(2p3 , 2q3 )
(2p3 , 0)
(2p3 , 0)
(2p3 , 0)
(2p3 , 0)
(0, 2q3 )
(0, 2q3 )
(0, 2q3 )
(0, 2q3 )
AB
kA × kB
(p1 + p2 , q1 + q2 )
(p1 + p2 , |q1 − q2 |)
(|p1 − p2 |, q1 + q2 )
(|p1 − p2 |, |q1 − q2 |)
(p1 + p2 , q1 + q2 )
(p1 + p2 , |q1 − q2 |)
(|p1 − p2 |, q1 + q2 )
(|p1 − p2 |, |q1 − q2 |)
(p1 + p2 , q1 + q2 )
(p1 + p2 , |q1 − q2 |)
(|p1 − p2 |, q1 + q2 )
(|p1 − p2 |, |q1 − q2 |)
Equal if
2p3 = p1 + p2
2p3 = p1 + p2
2p3 = |p1 − p2 |
2p3 = |p1 − p2 |
2p3 = p1 + p2
2p3 = p1 + p2
2p3 = |p1 − p2 |
2p3 = |p1 − p2 |
p1 = p2 = 0
p1 = p2 = 0
p1 = p2
p1 = p2
2q3 = q1 + q2
2q3 = |q1 − q2 |
2q3 = q1 + q2
2q3 = |q1 − q2 |
q1 = q2 = 0
q1 = q2
q1 = q2 = 0
q1 = q2
q 3 = q1 + q2
q3 = |q1 − q2 |
q3 = q 1 + q2
q3 = |q1 − q2 |
Table 4.4: Conditions for which a special case occurs for three modes. The bold conditions
are non-trivial.
shown that in each case the intermediate mode, ( (2p, 2q), (2p, q) and (p, 2q) respectively),
has a lower critical Rayleigh number than the shared critical Rayleigh number of the
two modes. Hence, conditions (4.64) never occur at the onset of convection, and the
corresponding extra terms will never occur at a vertex in figure 4.1.
4.4.2
Three interacting modes
The remaining cubic terms that may appear in the Ȧ equation, (4.42b), are BC 2 , CB 2
and ABC. All other cubic terms always occur (from (4.42)), or can appear from the
interaction of two modes as discussed in section 4.4.1. Consider the conditions for which
the BC 2 term appears in the Ȧ equation. As stated earlier, a secondary set of modes can
only generate the kA mode when multiplied by kB , iff the set shares an element with the
set kA × kB . Therefore, the set, (kC × kC ) × kB , only contains kA iff
kC × kC ∩ kA × kB 6= ∅.
(4.66)
Table 4.4 compares these secular modes to determine under what conditions they are
equal, and unexpected secular terms appear. It is found that a secular term appears if
any of the following conditions are true:
(2p3 = p1 + p2 ∨ 2p3 = |p1 − p2 |) ∧ (2q3 = q1 + q2 ∨ 2q3 = |q1 − q2 |) , (4.67a)
(2p3 = p1 + p2 ∨ 2p3 = |p1 − p2 |) ∧ q1 = q2 ,
p1 = p2 ∧ (2q3 = q1 + q2 ∨ 2q3 = |q1 − q2 |) ,
(4.67b)
(4.67c)
By a similar argument, if any of conditions (4.67) occur, then an AC 2 term and an
ABC term also appear in the Ḃ and Ċ equations, respectively. Therefore, if any set of
4.5. Dynamics of the special case. (Example 3)
65
three modes satisfy any of the conditions in (4.67), then the dynamical system will take
the form,
Ȧ = A aR1 − aa A2 − ab B 2 − ac C 2 − ad BC 2 ,
Ḃ = B bR1 − ba A2 − bb B 2 − bc C 2 − bd AC 2 ,
Ċ = C cR1 − ca A2 − cb B 2 − cc C 2 − cd AB .
(4.68a)
(4.68b)
(4.68c)
A permutation of this dynamical system may be found if a permutation of the modenumbers of the conditions in table 4.4 are met. All three-mode intersections in figure 4.1 have
been tested with conditions (4.67). By permutations of these conditions, all possible cubic
terms have been discussed. There is only one example at the onset of convection for which
the three viable modes satisfy these conditions. For a box of size Lx = Ly = 81/4 , the
three modes, (2, 0), (0, 2) and (1, 1), share a critical Rayleigh number of
Rc = π
2
√ !
3+2 2
√
,
2
(4.69)
at the onset of convection. It is seen in this example that the tertiary interaction of the
(1, 1) mode with itself and the (2, 0) mode generates the third primary mode, (0, 2). This
example is analysed in section 4.5.
Conditions for which the dynamical system obtained does not take the general form
given in (4.42) have been identified. For reasonable box sizes, where (Lx , Ly ≤ 4), a
three-mode intersection of figure 4.1 that satisfies these conditions has been found. This
example is a special case, and the dynamics are unique.
4.5
Dynamics of the special case. (Example 3)
An analysis of this particular example has not been found in the literature. As the only
example of symmetry-breaking at the onset of convection, the analysis is important: it
is the only available guide to the behaviour of symmetry-breaking interacting modes at
higher Rayleigh numbers, where an analytical solution is not possible. The dynamical
system obtained for the interaction of the (2, 0), (0, 2) and (1, 1) modes takes the form of
equations (4.68), with the coefficients,
√
√
3+2 2 4
13 + 8 2 4
a = b = (1 + 2)π R1 , aa = bb =
π , ab = ba =
π ,
4
6
√
√
872 + 649 2 4
663 + 470 2
ac = bc =
π , ad = bd =
1104
2208
√
1
1079 + 787 2 4
2
c= 1+ √
π R1 , ca = cb =
π
552
2
√
√
189 + 125 2 4
2016 + 1423 2 4
cc =
π , cd =
π
408
1104
√
2
(4.70a)
(4.70b)
(4.70c)
(4.70d)
66
Chapter 4. Three interacting modes
Fixed point
H0
HA+
HB−
H1+−+
HC+
HA+B+
HA−B−
H2+−+
H3+−+
HA+B−
H4+
H5+
Approximate position
(0, 0, 0)
≈ (0.290, 0, 0)
≈ (0, −0.290, 0)
≈ (0.152, −0.152, 0.288)
≈ (0, 0, 0.439)
≈ (0.188, 0.188, 0)
≈ (−0.188, −0.188, 0)
≈ (0.197, −0.133, 0.223)
≈ (0.133, −0.197, 0.223)
≈ (0.188, −0.188, 0)
≈ (0.092, 0.092, 0.289)
≈ (−0.092, −0.092, 0.289)
Stability
Unstable node
Stable node
Stable node
Stable node
Saddle point
Saddle point
Saddle point
Saddle point
Saddle point
Saddle point
Saddle point
Saddle point
Signature
1D
1D
1D
1D
1D
2D
2D
2D
unstable
unstable
unstable
unstable
unstable
unstable
unstable
unstable
Table 4.5: Example 3: All fixed points and their stabilities for those which lie in the C ≥ 0,
B ≤ A quadrant, for the postcritical, R1 = 1, special case dynamical system. The “+”
and “−” symbols indicate which octant the fixed point lies in.
and where A is the amplitude of the (2, 0) mode, B the amplitude of the (0, 2) mode and
C the amplitude of the (1, 1) mode. This example is circled in figure 4.1 as “example
3”. In contrast to the previous examples, this dynamical system no longer has reflective
symmetry about the A = 0, B = 0 and C = 0 planes. Due to the symmetry of the
box, which implies the (2, 0) and (0, 2) modes are rotations of the same solution, there is
reflective symmetry about the A = B and A = −B planes, as seen in the second example.
For the precritical system, R1 = −1, the conduction solution, (0, 0, 0), is the only fixed
point. For the postcritical system, R1 = 1, 27 fixed points are found. The A = 0 and
B = 0 planes are not 2D invariant manifolds in this system, and trajectories are no longer
confined to a single octant. Due to the symmetry induced by the square box, the phasespace can, however, be separated into four equivalent quadrants. The C ≥ 0, B ≤ A is
used as the base quadrant. The 12 fixed points in the C ≥ 0, B ≤ A quadrant are listed
in table 4.5.
Unlike the previous examples, there are stable nodes like H1+−+ which correspond
to solutions where all three modes coexist. It is also strange that there is no stable node
which corresponds to the existence of the (1, 1) mode in the absence of the other two.
Such a fixed point exists (HC+ ), however, it is a saddle point.
Of the 3 stable nodes in the C ≥ 0, B ≤ A quadrant, each one has a associated basin
of attraction. The entire A = B plane is a boundary of the basins, and is formed by the
2D stable separatrices of HC+ , HA+B+ and HA−B− . The other boundaries are formed by
the 2D stable manifolds of H2+−+ and H3+−+ . These form a ‘valley’ that runs along the
A = −B plane. These boundaries along with the listed fixed points, are shown in figure
4.10. The A = B surface is plotted as a white plane.
The 3D dynamics can again be reduced to a 2D quotient surface, which shows only
the interactions between the modes, rather than the overall growth and decay. Due to
4.5. Dynamics of the special case. (Example 3)
67
C
1U
2U
2U
H5+
1U
HA-B-
HC+
H1+-+
H4+
H3+-+
1U
HA+B+
H0
1U
H A+B-
H2+-+
3U
1U
2U
-B
HB-
HA+
A
Figure 4.10: A schematic of the boundaries of the basins of attractions and the fixed points
in the C ≥ 0, B ≤ A quadrant.
68
Chapter 4. Three interacting modes
H5+
HA−B−
+
H4+
HC+
++
+
HA+B+
++
+
H1+−+
+
HB−
H1+−+ basin
+
H2+−+
H3+−+
HA+
HB− basin
HA+ basin
++
HA+B−
Figure 4.11: The 2D quotient system, (the 2D unstable separatrices of H4+ , H5+ and
HA+B− ). The boundaries of the surface are the A = B plane, (top straight line), and the
C = 0 plane, (bottom half circle).
the different symmetries, this surface is not restricted to one octant, and is best restricted
to the C ≥ 0, B ≤ A quadrant. The quotient surface is defined as the union of the 2D
unstable separatrices of H4+ , H5+ and HA+B− . The dynamics and basins of attraction
for the quotient system are shown in figure 4.11. Both figures, 4.10 and 4.11, show that
the HA+ and HB− basins are not connected.
The relative size of each basin of attraction is measured to determine how likely each
steady-state solution is, given a random initial perturbation. Due to the symmetry in the
system, measuring the relative volume of the basins of attraction in the C ≥ 0, B ≤ A
quadrant represents the relative volume in the whole space. The (2, 0) and (0, 2) modes
are both equally likely to be the final state solution, with a volume of 41.4% each. The
chance that the solution will arrive at the state where all three modes coexist, (H1+−+ ),
is smaller, at 17.2%.
While many parallels exist between the fixed points in the first two examples, this
special case is completely different. There is no stable node which corresponds to the lone
survival of the (1, 1) mode, but there is a stable node which corresponds to the survival
of all three modes. Further properties in this special case are as follows:
• The (2, 0) and (0, 2) modes can grow from zero amplitude, purely via the nonlinear
interaction of the other two modes.
• Some trajectories evolve in such away that the amplitudes, A and B, switch from
positive to negative (or vice-versa) values. This results in the applicable mode having
its direction of flow reversed.
• The presence of the (1, 1) mode breaks the symmetry of the interactions between the
other two modes; the direction of flow matters. This results in different dynamics in
the A > 0, B > 0 quadrant than the A < 0, B > 0 quadrant.
• There is a stable solution which is not a pure mode solution, but a superposition of
all three. This is the only case where a natural solution of the system is not a pure
mode.
4.6. Hidden symmetries in rectangular domains
69
While this special case is the only example of these dynamics close to the critical Rayleigh
number, this type of interaction amongst modes is common amongst higher Rayleigh
numbers; as more and modes become viable, it is more likely that an interaction between
modes creates symmetry-breaking effects. While this perturbation analysis cannot derive
dynamical systems for strongly nonlinear cases, it shows that it is possible that stable
solutions other than the modal solutions exist.
The mixed mode solution is also interesting in that the existence of a stable mixture
solution is a necessary (but not sufficient) condition for the system to undergo a bifurcation
to a time-dependent final state solution at relatively low Rayleigh numbers (see Steen,
1986). Steen concluded that as there were no systems where such a stable fixed point
exists, there can be no bifurcation to a time-dependent final state. The current model is
unable to explore this issue further. One may increase the Rayleigh number by increasing
R1 , however, due to the degeneracy of the critical box cases, no bifurcations other than
those already discussed will occur. While the conclusion of Steen (1986) cannot be revised
with this model, the problem is reopened.
4.6
Hidden symmetries in rectangular domains
Symmetry-breaking in modal interactions has been observed in other applications. Crawford (1991) observed that in surface wave experiments in square domains, there were
hidden symmetries that were not immediately obvious from the symmetry of the domain.
These hidden symmetries occur where the mathematical solution may be extended from a
finite square domain to an infinite domain, where the solution satisfies periodic boundary
conditions. These symmetries occur in Lapwood convection. Note that the first order
solutions (4.17) satisfy the first order PDEs and boundary conditions if they are translated by a half wavelength in any direction. Bifurcations in such systems were analysed
by Crawford (1994).
The consequence of the effect of hidden symmetries on modal interactions, is studied
by Gomes & Stewart (1994). The results of Gomes & Stewart (1994) are applied to Lapwood convection by Impey et al. (1996).
By using the method of Gomes & Stewart (1994), any symmetry-breaking effects are
identified. The method outlined in sections 4.4.1 and 4.4.2, however, is specific to convection modes at the onset of convection, and only identifies symmetry-breaking terms at
third order of expansion. Less physically relevant symmetry-breaking terms can be found
by extending the perturbation method to a higher order. As such, the conditions found
are a subcase of those stated by Gomes & Stewart (1994). A summary of the key points
in Gomes & Stewart (1994) is given below.
For two interacting modes in an n-dimensional domain, with mode numbers, k1 =
(k11 k21 , . . . , kn1 ) and k2 = (k12 k22 , . . . , kn2 ), a special case occurs if all k1j have the same parity,
and all k2j have the same parity, (but not necessarily the same parity as each other). Note
70
Chapter 4. Three interacting modes
that this assumes that
hcf(kj1 , kj2 ) = 1
for j = 1, 2, . . . , n.
(4.71)
If they are not coprime, then by the geometrical symmetries in the rectangle, the problem
can be reduced to an equivalent problem where the coprime condition, (4.71), holds. Impey
et al. (1996) investigated two examples, both of which are classified as special cases: The
interaction of the (3, 1) and (1, 1) modes, as well as the interaction of the (3, 1) and (2, 2)
modes. Note that while the first example is also classified as a special case by conditions
(4.64), the second example is not. This is due to the fact that the symmetry-breaking
terms in the interaction of the (3, 1) and (2, 2) modes occur at fourth order, and are not
identified by conditions (4.64). These examples do not occur at the onset of convection,
and are not considered in this thesis, as the convection behaviour will be dominated by a
highly nonlinear mode.
Gomes & Stewart (1994) also generalise to the special case of m interacting modes.
Consider the modes, k1 , k2 , . . ., km . A special case will occur if aij can be identified such
that
0 =
m
X
aij kji
for
1 ≤ j ≤ n,
(4.72a)
i=1
ai1 ≡ ai2 ≡ . . . ≡ ain (mod 2)
for
1 ≤ i ≤ m.
(4.72b)
Again, this technique identifies the symmetry-breaking terms of special cases at any order. The conditions identified earlier for tertiary secular modes, (4.67), are a sub-case of
conditions, (4.72) for m = 3.
One may question whether it is appropriate to truncate examples 1, (4.45), and 2,
(4.49), at third order, as there is every possibility that symmetry-breaking terms can appear at higher order. It can be shown, however, that by applying the conditions of Impey
et al. (1996), equations (4.72), examples 1 and 2 do not have symmetry-breaking terms at
any order. Hence, it is appropriate to truncate them at third order.
While the method of Gomes & Stewart (1994) is complete, in that it captures any
symmetry-breaking effects, the perturbation method used in this thesis helps to identify
how strong these effects are. The symmetry-breaking effects may be very small, and only
occur at a large order expansion.
4.7
Conclusion
A perturbation technique has been used to systematically categorise each and every example of a critical box (of reasonable size) where three modes interact. The resulting
structure of the phase-space for three different examples has been displayed. The dynamics of the interaction between the modes is simplified by taking advantage of a quotient
system, which ignores the overall growth and decay of the modes. These examples are
4.7. Conclusion
71
representative of all structures that may be seen at the onset of convection, for small to
moderate mode numbers. Most examples belong to just two classes, however, there is a
unique symmetry-breaking example belonging to a third class. In each of these examples,
the probability of arrival at each of the stable steady-state solutions was markedly different. Each critical box case has been classified according to a simple parameter test, and
will have a phase-space that is topologically equivalent to one of the examples shown.
The third example breaks the symmetry of the positive and negative amplitudes, and
has not been described in the literature. In this symmetry-breaking case, there is a stable solution where all three modes coexist in the final state. While many occurrences of
this type can be found for Rayleigh numbers well above the critical value, this case is
interesting in that it occurs at the onset of convection.
72
Chapter 4. Three interacting modes
Chapter 5
A perturbed box: Splitting the
bifurcations
Chapters 3 and 4 focussed on critical box dimensions where multiple modal solutions
become stable for the same critical Rayleigh number, Rac . The bifurcation that occurs is
degenerate, with one stable fixed point suddenly becoming multiple fixed points as Ra is
increased beyond Rac . It is quite unrealistic to expect that a given physical system will
have the precise length, width and height dimensions that are required for such degenerate
bifurcations to occur.
One would ideally like to observe a progression of non-degenerate bifurcations, however,
the analytical model used in earlier examples is not valid for Rayleigh numbers much
larger than critical. This means that for the model to be accurate, the critical Rayleigh
numbers of the observed modes must be close together. This condition is satisfied if the
box dimensions are chosen such that it is almost, but not quite, critical.
Riley & Winters (1989) used a perturbation method similar to that used chapter 3, to
analyse the case where the p = 1 and p = 2 modes interact in a 2D box. This method
accommodates a perturbation in the aspect ratio of the box, causing the two modes to
have unique critical Rayleigh numbers. In section 5.1 this perturbation method is applied
to the same example, with aim to replicate the results of Riley & Winters (1989). In
section 5.2, the perturbation method used in chapter 4 is modified to accommodate a
length perturbation in a 3D box, extending the principles of Riley & Winters (1989).
This simple 2D example is used to introduce the concept of splitting the bifurcations,
allowing an easier understanding of the 3D cases. The behaviour of the special case is
particularly important, as it is the only example of three-mode symmetry-breaking at the
onset of convection. It provides insight to the behaviour of such interacting modes at
higher Rayleigh numbers where they are unable to be modelled analytically due to the
strongly nonlinear effects.
74
Chapter 5. A perturbed box: Splitting the bifurcations
48
47
46
45
Rac
Rp2
Single cell (p = 1)
44
Rp1
43
42
Double cell (p = 2)
41
ε2 ∆
2
40
39
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
L
Figure
√ 5.1: The critical Rayleigh number as a function of the horizontal aspect, L. Near
L = 2, the two modes have similar critical Rayleigh numbers, but do not become stable
simultaneously
5.1
2D box
Consider the 2D case where the scaled box width is given by L. To examine the behaviour
√
√
close to the critical value of 2, a perturbation is given by L = 2(1 + 12 ε2 ∆ + . . .), where
ε2 is the size of the perturbation from the critical value, and ∆ = ±1 denotes whether the
√
width is greater than or less than the critical value. For L < 2, the single cell (p = 1)
√
modes have a lower critical Rayleigh number and are preferred, whereas for L > 2, the
double cell, (p = 2), mode is preferred.
The modified perturbation method is applied in section 5.1.1, and the resulting dynamical system is investigated in section 5.1.2.
5.1.1
Governing equations
The scaled governing equations of the temperature and the stream function (from (3.5)),
are given by,
∇2 ψ = −RaTx ,
Tt + ψx + ψz Tx − ψx Tz = ∇2 T,
(5.1a)
(5.1b)
where the subscripts x, z and t, denote a partial derivative with respect to that subscript.
It is convenient to ‘fix’ the boundaries of the box by rescaling x. Let x = Lx̄. The
governing equations become
ψx̄x̄
Ra
+ ψ1 zz = − Tx̄ ,
2
L
L
ψx̄ ψz Tx̄ ψx̄ Tz
Tx̄x̄
Tt +
+
−
=
+ Tzz .
L
L
L
L2
(5.2a)
(5.2b)
5.1. 2D box
75
The perturbation expansions, (3.9), are applied, as in addition to the perturbation of
√
horizontal length of the box, L, away from L = 2. The perturbations are given by
ψ = εψ1 + ε2 ψ2 + . . . ,
T
(5.3a)
2
= εT1 + ε T2 + . . . ,
(5.3b)
2
Ra = Rac (1 + ε R1 + . . .),
√
1
L =
2(1 + ε2 ∆ + . . .),
2
1
t = 2 τ,
ε
(5.3c)
(5.3d)
(5.3e)
where ∆ can take the value of 1 or −1 to denote a positive or negative perturbation of the
horizontal length. L appears in the governing equations in its inverse form,
expansion of
1
L
about ε = 0 gives an expression for
1
L
1
L.
A Taylor
in terms of increasing powers of ε:
∆ε2
1
1
= √ − √ + ....
L
2 2 2
(5.4)
The PDEs for the first three orders, O(εk ), are given by
M u k = hk ,
(5.5)
where
M=
ˆ2
∇
− √12 ∂∂x̄
Ra
√c ∂
2 ∂ x̄
ˆ2
∇
!
,
(5.6)
uk = (ψk , Tk )T ,
(5.7a)
h1 = 0,
1
h2 = √ (0, ψ1z T1x̄ − ψ1x̄ T1z )T ,
2
Ra
√ c ∆ − R1 + ∆ ψ1x̄x̄
2
2
2
h3 =
T1τ + ψ1z T2x̄ + ψ2z T1x̄ − ψ1x̄ T2z − ψ2x̄ T1z +
(5.7b)
(5.7c)
∆
2 T1x̄x̄
−
∆
√
ψ
2 2 1x̄
, (5.7d)
(5.7e)
ˆ2 =
and ∇
1 ∂2 2
2 ∂ x̄
+
∂2
.
∂z 2
Note that the first and second order equations are equivalent to
the first and second order equations in the unperturbed box, (3.14). They only differ by
√
the effect of the horizontal scale transformation, x = 2x̄. Therefore the first and second
order solutions are also equivalent, and need not be explicitly calculated. The first order
solutions, as before, are:
√
√
ψ1 = −3 2πA sin(πx) sin(πz) − 3 2πB sin(2πx) sin(πz),
(5.8a)
T1 = 2A cos(πx) sin(πz) + 2B cos(2πx) sin(πz),
(5.8b)
76
Chapter 5. A perturbed box: Splitting the bifurcations
where the bar notation has been dropped. The second order solutions, as before, are:
9π 2
√ AB sin(πx) sin(2πz),
4 2
9
3
= − π(A2 + 2B 2 ) sin(2πy) − πAB cos(πx) sin(2πz).
4
4
ψ2 =
(5.9a)
T2
(5.9b)
It is in the third order equations that the perturbation of the horizontal aspect of the box
causes additional terms to appear. If u takes the form,
a
u=
!
eipπx sin(jπz),
b
(5.10)
where a and b are complex numbers, then
M u = M p,j u,
where

p2
2
2 +j
− √π2 ip
−π 2
M p,j = 
(5.11)
Ra
√ c iπp
22
−π 2 p2 + j 2 .

.
(5.12)
Using the first- and second-order solutions, h3 is calculated to be
3
3π
√
2 2
h3 =
1
8
+
3R1 −
∆
2
!
iA
9π 4 A3 + 45π 4 AB 2 + 2π 2 ∆A + 8Aτ
√3 π 3 3R1 + ∆ iB
2
2
1
16
63π 4 A2 B
+
72π 4 B 3
−
8π 2 ∆B
eiπx sin(πz)
+ 16Bτ
!
ei2πx sin(πz)
(5.13)
+ (CC) + (NSM),
where the secular terms are explicit. Using the same arguments as in chapter 3, the
amplitude vector of the eipπx sin(πz) must lie in the eigenspace of M p,1 , for p ∈ {1, 2}.
Applying these restrictions gives the system,
9
45
π2
(3R1 − ∆)A − π 4 A3 − π 4 AB 2 .
2
8
8
63 4
9
2
2
Ḃ = π (3R1 + ∆)B − π AB − π 4 B 3 .
16
2
Ȧ =
(5.14a)
(5.14b)
Each of these restrictions is an ordinary differential equation which describes the evolution of the amplitude of one of the two viable modes. This system of coupled ordinary
differential equations is treated as a dynamical system. System (5.14) is equivalent to the
evolution equations derived by Riley & Winters (1989). They only differ by a rescaling of
A and B, which arises from the free choice of definition.
5.1. 2D box
5.1.2
77
Dynamical system
The dynamical system for this problem is given by
9 2 2 45 2 2
1
(3R1 − ∆) − π A − π B ,
Ȧ = π A
2
8
8
63 2 2 9 2 2
2
Ḃ = π B (3R1 + ∆) − π A − π B .
16
2
2
(5.15a)
(5.15b)
As one would expect, by setting the length perturbation, ∆, to zero, the dynamical system
for the unperturbed box, (3.35), is recovered. Furthermore, it can be seen that for relatively
large values of R1 , the length perturbation becomes less important. In fact, as R1 → ∞,
the dynamical system asymptotically approaches the unperturbed box system:
3
9
45
Ȧ = π 2 A( R1 − π 2 A2 − π 2 B 2 ),
2
8
8
63 2 2 9 2 2
2
Ḃ = π B(3R1 − π A − π B ),
16
2
(5.16a)
(5.16b)
which is equivalent to system (3.35). This indicates that as R1 increases, the fixed points
must approach the geometrical location of the already established unperturbed box system
(up to a rescaling). Furthermore, the number and stabilities of each fixed point should
match those seen in the unperturbed box system.
The additional ∆ terms change the Rayleigh number for which each primary mode is
created. These bifurcations no longer happen simultaneously. The A primary bifurcation
occurs when the linear coefficient of A in (5.15a) is zero. This occurs for
R1 =
∆
.
3
(5.17)
Similarly the B primary bifurcation occurs when the linear coefficient of B in (5.15b) is
zero, where
R1 = −
∆
.
3
(5.18)
When the box is perturbed such that it is smaller (larger), where ∆ = −1, (∆ = 1),
the single-cell (double-cell), A (B) mode is preferred and bifurcates for a lower Rayleigh
number. The primary mode which bifurcates at a higher Rayleigh number is not stable
upon creation; it is stabilised via a secondary pitchfork bifurcation. In the ∆ = −1 case,
this secondary bifurcation occurs at R1 =
secondary bifurcation occurs at R1 =
11
9
7
9
on the B ∗ fixed point. In the ∆ = 1 case, the
on the A∗ fixed point. In both cases, the order of
bifurcations progresses from a first primary bifurcation at R1 = Rp1 to the second primary
bifurcation at R1 = Rp2 , followed by a secondary bifurcation at R1 = Rs . The progression
of the location of the nullclines and fixed points for both cases is shown in figure 5.2. As
R1 increases, these figures asymptotically approach each other.
Using the above calculations, the bifurcation diagrams in figure 5.3 can be constructed. These bifurcation diagrams clearly show the progression from a stable conduction
78
Chapter 5. A perturbed box: Splitting the bifurcations
solution (O∗ ), to a system where the two modal solutions are possible. For comparison,
the degenerate bifurcation tree for the unperturbed, ∆ = 0, case is shown in Figure 5.4.
Note that the underlying stability exchanges are not seen in the degenerate bifurcation.
For Rp2 < R1 < Rs , both modal solutions exist, however, only one is stable. While
the traditional linear stability analysis shows that both modes are viable, the interaction
between the modes affects the linear stability. Linear stability analysis may show that a
mode is stable, however, the entire phase-space may be a basin of attraction for a second
viable mode (except where the second mode has zero amplitude). All initial points that
do not lie on the axes of the phase-space will evolve toward either the stable mode or its
counter-rotating twin.
This analysis has provided insight into how two modes interact for a non-degenerate
bifurcation. It is not simply a case where the two modes are independently favoured or
disfavoured by a change in the box dimensions, and the critical Rayleigh numbers change
accordingly. Due to the presence of an already stable convection pattern, the stability
of a second mode occurs for Rayleigh numbers larger than the linear stability analysis
indicates. It is unfortunate that this analysis does not extend to any arbitrary box size,
as it is only accurate while the viable modes are only weakly nonlinear, where Ra−Rac is
small.
5.1. 2D box
79
∆ = −1
B
0.6
B
0.6
0.4
0.4
0.2
0.6
0.4
∆=1
0.2
0∗
0.2
0.2
0.4
0.6
A
0.6
0.4
0.4
0.4
0.4
0.6
0.6
B
0.6
B
0.6
0.4
0.4
0.2
0.4
0.6
A
0.6
0.4
0.2
0.2
0.4
0.4
0.6
0.2
0.2
0.4
0.6
A
0.6
0.4
0.2
A∗
0.2
0.2
0.2
0.4
0.4
0.6
0.6
A
(f) R1 = 0.5
B
0.6
0.4
0.4
AB ∗
0.2
0.6
0.4
B∗
B
0.6
0.4
0.4
A
B
0.6
(e) R1 = 0.5
0.6
0.6
0.6
0.4
0.2
0.4
A
(d) R1 = 0
B
0.6
0.4
0.2
0.2
0.2
0.6
0.2 B ∗
A∗
0.2
0.4
(b) R1 = −0.5
(c) R1 = 0
0.6
0.2
0.2
0.2
0.6
0.2
0.2
(a) R1 = −0.5
0∗
0.2
AB ∗
0.2
0.4
0.6
A
0.6
0.2
0.2
0.6
A
0.2
0.4
0.6
(g) R1 = 1
0.4
0.6
(h) R1 = 1.5
Figure 5.2: The location of the fixed points and nullclines in the 2D phase-space as R1
increases for ∆ = −1, (left column), and ∆ = 1 (right column). Square fixed points are
unstable.
80
Chapter 5. A perturbed box: Splitting the bifurcations
∆ = −1
R1
+
+
+
+
++
+
+
+
Rp1 = − 31
Rp2 =
1
3
Rs =
∆=1
A∗
+ AB ∗
B∗
+ AB ∗
++O∗
+ AB ∗
B∗
+ AB ∗
A∗
R1
+
+
+
(a)
+
+
+
Rp1 = − 13
7
9
++
+
Rp2 =
1
3
Rs =
B∗
+ AB ∗
A∗
+ AB ∗
++O∗
+ AB ∗
A∗
+ AB ∗
B∗
11
9
(b)
Figure 5.3: The bifurcation diagrams for, (a), ∆ = −1; and, (b), ∆ = 1. The number of ‘+’
symbols indicates the number of unstable directions of that fixed point. Regardless of ∆,
the final number and stability of each fixed point remains the same as in the unperturbed
box case.
∆=0
R1
A∗
+ AB ∗
B∗
+ AB ∗
++O∗
+ AB ∗
B∗
+ AB ∗
A∗
R1 = 0
Figure 5.4: The bifurcation diagrams for the unperturbed case, ∆ = 0. The number of ‘+’
symbols indicates the number of unstable directions of that fixed point. This bifurcation
is degenerate.
5.2. 3D box
81
y
Ly0
θ
∆y
ε2
2
∆x
Critical Box
Perturbed Box
x
Lx0
Figure 5.5: The horizontal dimensions of the box are perturbed. The magnitude of the
2
perturbation is ε2 regardless of the direction, θ.
5.2
3D box
In this section, the perturbation method used in Chapters 4 is modified to accommodate
a perturbation in the horizontal aspect ratios of a 3D box. In the three archetypal 3D
examples, the scaled horizontal dimensions, Lx and Ly , are perturbed away from the
critical values, (Lx0 , Ly0 ), where a degenerate bifurcation occurs. The aspect ratios are
perturbed in the horizontal directions and are given by
(∆x , ∆y ) 2
(Lx , Ly ) = (Lx0 , Ly0 ) 1 +
ε + ... .
2
(5.19)
As the box may be perturbed in two dimensions, ideally one would like the magnitude of
the perturbation to remain of order ε2 , regardless of the perturbation direction. To achieve
this, the horizontal perturbations are set as ∆x = cos(θ), ∆y = sin(θ), where θ ∈ [0, 2π).
The angle of perturbation is continuously described, where θ = 0 corresponds to a positive
x-direction change, (with no change in y), and θ =
π
2
corresponds to a positive y-direction
change. The geometry of the perturbation is shown in figure 5.5.
82
Chapter 5. A perturbed box: Splitting the bifurcations
5.2.1
Governing equations
The governing equations (4.6) are again used. These five equations are given by
∇ · v = 0,
2
∇ w−
Ra∇2H T
∂ ∂
T
∂y ∂z
∂ ∂
∇2 u + Ra
T
∂x ∂z
∇2 v + Ra
∂T
+ v·∇T − w − ∇2 T
∂t
(5.20a)
= 0,
(5.20b)
= 0,
(5.20c)
= 0,
(5.20d)
= 0.
(5.20e)
The domain is a rectangular box with the x, y and z bounds given by [0, Lx ], [0, Ly ],
and [0, 1] respectively. The boundary conditions for the enclosed box with no heat flux
escaping the sides are again given by:
v · n = 0,
(5.21)
on every surface, (where n is the normal vector),
T (x, y, 0) = T (x, y, 1) = 0,
(5.22)
∇T · n = 0,
(5.23)
and
on the sides.
The horizontal coordinates are scaled such that Lx and Ly are explicit in the equations,
not the boundary locations. Let x = Lx x̄ and y = Ly ȳ. The governing equations become
¯ · v,
0 = ∇
(5.24a)
¯ 2 w − Ra∇
¯ 2 T,
0 = ∇
H
∂
∂
Ra
¯ 2v +
0 = ∇
T,
Ly ∂ ȳ ∂z
¯ 2 u + Ra ∂ ∂ T,
0 = ∇
Lx ∂ x̄ ∂z
∂
¯ ·v−w−∇
¯ 2 T,
0 =
T + ∇T
∂t
(5.24b)
(5.24c)
(5.24d)
(5.24e)
where
1 ∂ 1 ∂ ∂
,
,
,
Lx ∂x Ly ∂y ∂z
1 ∂2
1 ∂2
∂2
=
+
+
,
L2x ∂x2 L2y ∂y 2 ∂z 2
1 ∂2
1 ∂2
=
+
.
L2x ∂x2 L2y ∂y 2
¯ =
∇
¯2
∇
¯ 2H
∇
(5.25)
(5.26)
(5.27)
5.2. 3D box
83
The scaled rectangular box takes the x̄, ȳ, z̄ domain of [0, 1], [0, 1], [0, 1]. While the boundaries have changed, the boundary conditions remain as they are expressed in equations
(5.21 - 5.23).
Following chapter 4, the perturbations are given by
Ra = Rac (1 + R1 ε2 + . . .),
1
t = 2 τ,
ε
(T, v) = ε(T1 , v 1 ) + ε2 (T2 , v 2 ) + . . . ,
(∆x , ∆y ) 2
(Lx , Ly ) = (Lx0 , Ly0 ) 1 +
ε + ... ,
2
(5.28a)
(5.28b)
(5.28c)
(5.28d)
where Lx0 and Ly0 are the dimensions for a “critical box”, where degenerate bifurcations
occur. The length parameters, Lx and Ly , only appear as a reciprocal in the governing
equations, and so are generally expressed as
1 1
,
Lx Ly
=
1
1
,
Lx0 Ly0
(∆x , ∆y ) 2
1−
ε + ... .
2
(5.29)
For each order of ε, a different partial differential equation is obtained. The O(εk ) order
PDE is given by:
M u k = hk ,
(5.30)
where, (dropping the bar notation),

1 ∂
Lx0 ∂x
1 ∂
Ly0 ∂y
0
0
ˆ
∇2




M =



0
ˆ2
∇
0
∂
∂z
ˆ2
∇
0
0
0
0
1
0


ˆ2 
−Rac ∇
H 
Rac ∂ ∂  ,
Ly0 ∂y ∂z 

Rac ∂ ∂ 
Lx0 ∂x ∂z 
ˆ2
∇
(5.31)
uk = (uk , vk , wk , Tk )T ,
(5.32)
h1 = 0,
(5.33)
ˆ 1 )T ,
h2 = (0, 0, 0, 0, v 1 · ∇T

h3




= 




∆y ∂
∆x ∂
2Lx0 ∂x u1 + 2Ly0 ∂y v1
2
∆ Ra ∂ 2
c ∂
ˆ 2 T1 + ∆2x ∂ 22 w1 + ∆2y ∂ 22 w1 − ∆x Ra
T − Ly 2 c ∂y
Rac R1 ∇
2 T1
H
Lx0 ∂x
Ly0 ∂y
L2x0 ∂x2 1
y0
2
2
∆
Ra
∆
y ∂
c y ∂ ∂
∆x ∂
∂ ∂
− RaLcy0R1 ∂y
∂z T1 + L2x0 ∂x2 v1 + L2y0 ∂y 2 v1 + 2Ly0 ∂y ∂z T1
∆y ∂ 2
∆x ∂ 2
Rac ∆x ∂ ∂
∂ ∂
− RaLcx0R1 ∂x
∂z T1 + L2x0 ∂x2 u1 + L2y0 ∂y 2 u1 + 2Lx0 ∂x ∂z T1
∆y ∂ 2
∆x ∂ 2
∂
ˆ
ˆ
∂τ T1 + v 1 · ∇T2 + v 2 · ∇T1 + L2x0 ∂x2 T1 + L2y0 ∂y 2 T1
(5.34)





,




(5.35)
84
Chapter 5. A perturbed box: Splitting the bifurcations
ˆ =
∇
,
(5.36)
1 ∂2
1 ∂2
∂2
+
+
L2x0 ∂x2 L2y0 ∂y 2 ∂z 2
!
1 ∂2
1 ∂2
+
.
L2x0 ∂x2 L2y0 ∂y 2
ˆ2 =
∇
ˆ2
∇
H
1 ∂
1 ∂ ∂
,
,
Lx0 ∂x Ly0 ∂y ∂z
=
!
,
(5.37)
(5.38)
The first and second order equations are equivalent to those seen in the unperturbed box,
given the horizontal rescaling by Lx and Ly . Hence, the first and second order solutions
to any of the examples can be found by referring to chapter 4. Before addressing the third
order solutions, note that if u is of the form,
u = reipπx eiqπy eijπz ,
(5.39)
with complex vector r and real p, q, and j, then
M u = M p,q,j u,
(5.40)
where,

M p,q,j
ip
Lx0 π


 0

=
 0


 σ2
0
iq
Ly0 π
ijππ
0
σ2
σ2
0
0
0
0
1
Rac
π2
σ = −π
2
p2
L2x0
+
Rac
Ly0
Rac
Lx0
∂ ∂
∂y ∂z
∂ ∂
∂x ∂z
σ2
and
2

0
p2
q2
+
+ j2
L2x0 L2y0
q2
L2y0




,




(5.41)
!
.
(5.42)
M p,q,j is a 5 × 4 matrix with a maximum rank of 4. The first row is linearly dependent on
the second, third and fourth rows, and brings no new constraints. Hence, it is discarded.
For solutions of the form given in equation (5.39), the incompressibility equation is implied
by the conservation of momentum equations and M p,q,j is defined as:

M p,q,j
0
0
σ 2 Rac π 2


2

= 0 σ
 2
 σ
0
0
0
1
0
0
p2
L2x0
Rac
Ly0
Rac
Lx0
+
∂ ∂
∂y ∂z
∂ ∂
∂x ∂z
σ2
q2
L2y0




.


(5.43)
For secular modes, the Mp,q,j matrix is singular, and secular terms must be suppressed to
avoid growing solutions. By applying a solvability condition for each secular mode, the
dynamical system which governs the growth and decay of the amplitudes is found. If the
5.2. 3D box
85
(2,2)
(1,3)
(1,3)
(1,2)
(2,0)
(2,1)
3
(0,3)
(0,3)
(0,3)
(0,3)
(3,0)
(1,3)
(1,3)
(2,2)
(3,0) (3,1)
(0,3) (0,3) (0,3)
(3,0)
(1,2)
(2,1)
(2,2)
(1,0)
(1,2)
(2,0) (2,1)
(0,2)
(0,2)
(4,0)
(2,2)
(3,0)
(1,1)
2
(3,1)
(2,2)
(2,0)
(0,2)
(3,1)
(1,2)
(0,2)
(2,2)
(1,2)
(0,2)
(0,2)
(1,2)
(0,2)
(4,0)
(1,2)
(0,2)
(4,0)
(2,0)
Ly
(1,0)
(3,0)
(2,1)
(3,1)
(1,1)
(3,1)
(2,1)
(1,1)
(1,1)
(0,1)
(2,0)
1
(0,1)
(0,1)
(0,1)
(1,0)
1
(3,0)
(2,0)
(2,1)
(4,0)
(1,1)
(0,1)
(4,0)
3
2
(4,0)
(3,0)
(5,0)
4
Lx
Figure 5.6: Example 1 in Lx –Ly space.
modes are not a special case, then the dynamical system is given by
Ȧ = A a − aa A2 − ab B 2 − ac C 2 ,
Ḃ = B b − ba A2 − bb B 2 − bc C 2 ,
Ċ = C c − ca A2 − cb B 2 − cc C 2 ,
(5.44a)
(5.44b)
(5.44c)
where the aa , ab , . . . , ba , . . . , ca , . . ., are positive real numbers, while a, b and c are dependent on R1 , ∆x and ∆y . The three archetypal examples are analysed in the following
sections, including the special case, (example 3), which does not take the form shown in
equations (5.44).
5.2.2
Example 1
The first typical example shown is for a box perturbed about the critical dimensions,
√
Lx0 = 2, Ly0 = √12 (See Example 1, section 4.2.1, and circled in figure 5.6), representing
equivalence class 1. For this box, there are three viable modes which share the same
Rayleigh number of Rac =
9π 2
2 .
These modes are denoted by (p, q) = (1, 0), (2, 0) and
(0, 1).
Aside from the horizontal scale factors, the first order solutions are equivalent to
those seen in the unperturbed box in section 4.2.1, and are given by
√
√
u1 = −3 2π 2 A sin(πx) cos(πz) − 3 2π 2 B sin(2πx) cos(πz),
√
v1 = −3 2π 2 C sin(πy) cos(πz),
(5.45a)
(5.45b)
w1 = 3π 2 A cos(πx) sin(πz) + 6π 2 B cos(2πx) sin(πz) + 6π 2 C cos(πy) sin(πz), (5.45c)
T1 = 2A cos(πx) sin(πz) + 2B cos(2πx) sin(πz) + 2C cos(πy) sin(πz),
(5.45d)
86
Chapter 5. A perturbed box: Splitting the bifurcations
where A, B, and C are the amplitudes of the (1, 0), (2, 0), and (0, 1) modes respectively.
The dynamical system obtained by imposing solvability conditions on the third order
equation is given by
1
9
45
2061 2 2
(3R1 − ∆x ) − π 2 A2 − π 2 B 2 −
π C ,
2
8
8
496
63 2 2 9 2 2 279 2 2
2
π C ,
Ḃ = π B (3R1 + ∆x ) − π A − π B −
16
2
46
153 2 2 279 2 2 9 2 2
Ċ = π 2 C (3R1 + ∆y ) −
π A −
π B − π C .
62
46
2
Ȧ = π 2 A
(5.46a)
(5.46b)
(5.46c)
If there is no perturbation of the horizontal dimensions, where ∆x = ∆y = 0, the dynamical
system becomes the unperturbed example 1, system (4.45). Using a unit perturbation in
direction θ, the dynamical system becomes
9 2 2 45 2 2 2061 2 2
1
(3R1 − cos(θ)) − π A − π B −
π C ,
Ȧ = π A
2
8
8
496
63
9
279 2 2
Ḃ = π 2 B (3R1 + cos(θ)) − π 2 A2 − π 2 B 2 −
π C ,
16
2
46
153 2 2 279 2 2 9 2 2
2
π A −
π B − π C .
Ċ = π C (3R1 + sin(θ)) −
62
46
2
2
(5.47a)
(5.47b)
(5.47c)
The nullclines of this system still retain the same structure as the unperturbed box. The
nullclines, AN , BN and CN , each consist of the union of a plane and an ellipsoid. The key
difference in this case is that the three different ellipsoids no longer come into existence
for the same Rayleigh number. The creation of an ellipsoid as R1 increases corresponds
to a primary bifurcation of that mode where its corresponding single mode fixed point,
(such as FA , FB and FC ), is created.
The Rayleigh number, R1 , at which the three primary bifurcations occur depends on
the angle of the box perturbation, θ. The ellipsoid nullclines only exist if the coefficient of
the corresponding linear term in (5.47) is greater than zero. For example, the A ellipsoid
nullcline exists if 3R1 − cos(θ) > 0, (from equation (5.47a)). The Rayleigh numbers at
which the primary bifurcation occurs as a function of θ are given by
1
cos(θ),
3
1
= − cos(θ),
3
1
= − sin(θ).
3
A primary :
R1 =
(5.48)
B primary :
R1
(5.49)
C primary :
R1
(5.50)
The order in which these bifurcations occur is highly dependent on θ, and every permutation of the order can be observed. Unlike the unperturbed box case, not all single-mode
fixed points are stable nodes at their creation. This is only true for the first primary
bifurcation to occur. The second primary mode must experience a secondary bifurcation
to stabilise the nodes, while the third primary bifurcation to occur requires two secondary
5.2. 3D box
87
FB+
FB+C+
FC+
+ FB−C+
++ FA+B+
+ FA+C+
FA+
+ FA+C−
++ FA+B−
+++ FO
++ FA−B+
+ FA−C+
FA−
+ FA−C−
++ FA−B−
+ FB+C−
FC−
+ FB−C−
FB−
+
T1.1
+
+
++
+
+
+
++
+
+
++
+++
++
++
++
+
+
+
+
+
++
+
R1
Figure 5.7: Example 1: A schematic of the bifurcation diagram as R1 increases for θ = 0.
The number of +’s indicates the number of unstable directions. While this is not a true
projection, the fixed points have been labelled by location. FA−B+ lies in the A < 0,
B > 0 quadrant on the C = 0 plane. Solid lines indicate that the fixed point is a stable
node.
bifurcations to stabilise. These secondary bifurcations correspond to the creation of fixed
points which lie on either of the A = 0, B = 0 or C = 0 planes, such as the FAB , FAC
and FBC fixed points seen in the unperturbed box case. This can be seen in a schematic
bifurcation diagram shown in figure 5.7.
Secondary bifurcations create fixed points which correspond to mixed solutions of
two modes. While the primary bifurcations occur when the ellipsoid nullclines come into
existence, the secondary bifurcations occur when two ellipsoid nullclines intersect along
an axis. The progression leading up to the AB secondary bifurcation has been described
in the 2D example and is seen in figure 5.2.
It is simple to find the R1 values for which the ellipsoid nullclines intersect as they
must first intersect along an axis. The R1 values as a function of θ for which a secondary
bifurcation occurs can be found by solving for when two ellipsoids share a semi-principal
axis length along a coordinate axis. For example, the length of the A and B ellipsoid
nullclines along the A-axis are equal if
r
or
a
=
aa
r
b
,
ba
a
b
= ,
aa
ba
(5.51)
(5.52)
(assuming the lengths must be real). If this were to occur, it would indicate a secondary
AB bifurcation on the A-axis, on the FA fixed point. It is found that this occurs for
R1 =
11
π
3π
cos(θ), for θ ≤ or θ ≥
.
9
2
2
(5.53)
88
Chapter 5. A perturbed box: Splitting the bifurcations
The remaining secondary bifurcation equations are shown below. AB secondary bifurcations occur for
R1 =

 11 cos(θ),
9
for θ ≤
− 7 cos(θ), for
9
π
2
π
2
or θ ≥
3π
2 , (on
3π
2 , (on
≤θ≤
A axis),
(5.54)
B axis).
The AC secondary bifurcations occur for
R1 =

 1 (31 sin(θ) + 34 cos(θ)),
for θ ≤
− 1 (229 sin(θ) + 124 cos(θ)),
315
for
9
3π
4
3π
4
or θ ≥
≤θ≤
7π
4 , (on
7π
4 , (on
A axis)
(5.55)
C axis).
The BC secondary bifurcations occur for

 1 (23 sin(θ) − 31 cos(θ)),
R1 = 24
 1 (23 cos(θ) − 31 sin(θ)),
24
for
π
4
5π
4 , (on B axis)
π
5π
4 or θ ≥ 4 , (on C axis).
≤θ≤
for θ ≤
(5.56)
Each of the three secondary bifurcations will occur for any value of θ. For θ values where
the two different branches of each bifurcation intersect, a degenerate bifurcation occurs
where the two associated primary bifurcations equations also intersect.
It is observed in figure 5.7, that as R1 becomes large, 19 fixed points exist each with
the corresponding stability signature to those seen in the unperturbed box case. As R1
increases, the fixed point locations approach those found in the unperturbed box case. As
R1 approaches infinity, the dynamic system is given by
3R1 9 2 2 45 2 2 2061 2 2
− π A − π B −
π C ,
lim Ȧ = π A
R1 →∞
2
8
8
496
63 2 2 9 2 2 279 2 2
2
lim Ḃ = π B 3R1 − π A − π B −
π C ,
R1 →∞
16
2
46
153 2 2 279 2 2 9 2 2
2
lim Ċ = π C 3R1 −
π A −
π B − π C ,
R1 →∞
62
46
2
2
(5.57a)
(5.57b)
(5.57c)
which is equivalent to the unperturbed box, system (4.45). As R1 increases, the nullclines
and fixed points approach the equivalent location as in the unperturbed box case, bar a
rescaling. This assists in the bifurcation analysis, as the high Rayleigh number system is
known from the analysis in section 4.2.1.
One might expect that changing the angle of perturbation, θ, would simply change the
order of bifurcations that are seen in figure 5.7, and that a series of different permutations
of the bifurcation diagram would be seen. Due to the inherent asymmetry however, this
cannot occur. As seen in the unperturbed box case, the signature of the FAB fixed points
have a different signature to the FAC and FBC fixed points; they have two directions of
instability. The perturbed case approaches the unperturbed case as R1 is increased, so
regardless of the angle of perturbation, the final set and stability of fixed points will be
the same. This poses a problem for some values of θ. The FAB fixed points attain their
5.2. 3D box
89
++
T1.6
++
+
++
++
+
++
+
++
+
+
++
++
+
+
+
++
++
++
+
+
+
+
++
+
++
++
+++
+
+
++
+
++
++
+
R1
++
++
FA+
++ FA+B+
FB+
++ FA−B+
+ FB+C+
+ FA+C+
FC+
+ FA−C+
+ FB−C+
+++ FO
+ FB+C−
+ FA+C−
FC−
+ FA−C−
+ FB−C−
++ FA+B−
FB−
++ FA−B−
FA−
Figure 5.8: Example 1: A schematic of the bifurcation diagram as R1 increases for θ = 4.
twice unstable signature by being the first stabilising secondary bifurcation of the fixed
points created by the third primary bifurcation. This is possible as long as either the A or
B primary bifurcations occur last. Some other mechanism must occur for θ values where
the C primary bifurcation occurs last.
For the θ values where the FAC or FBC fixed points are created with a twice unstable
signature, an exchange of stability must occur with the FAB fixed points. This occurs via a
stabilising supercritical pitchfork bifurcation off FAC (or FBC ). This “tertiary bifurcation”
creates a floating fixed point in each octant with a 2+ signature. The structure of the
phase-space and each of the 27 fixed points resembles the unperturbed box example 2.
To conclude the exchange of stability, these floating fixed points experience a subcritical
pitchfork bifurcation on the FAB fixed points. A schematic bifurcation tree for this process
can be seen in figure 5.8 .
During the exchange of stability, the structure of the phase-space is seen to take the
form of three of the four possible states outlined in section 4.3. These four states are:
1. A floating fixed point exists with a 2+ signature, the secondary fixed points are all
1+,
2. A floating fixed point does not exist, FAB is 2+, the other secondaries are 1+,
3. A floating fixed point does not exist, FAC is 2+, the other secondaries are 1+,
4. A floating fixed point does not exist, FBC is 2+, the other secondaries are 1+.
For θ = 4, as R1 increases, the structure changes from state 3 to state 1, then to state
2. the 2D invariant surface discussed in section 4.2.1, and shown in figure 4.4, is used
to show the progression of the phase-space structure and fixed points as the bifurcations
occur (see the progression of the quotient systems in figure 5.9).
Note that the 4 states above are equivalent to the cases discussed in section 4.3. In
90
Chapter 5. A perturbed box: Splitting the bifurcations
FC
FBC
FC basin
++
FC
FBC
FAC
+
+
FA basin
FB
+
FAC
+
FC basin
FA
(a) Before supercritical pitchfork
FB
FA basin
FB basin
FA basin
+
FAB
+
FABC
FB basin
+
FBC
FAC
FC basin
++
FB basin
FC
++
FA
FAB
FB
(b) After supercritical pitchfork
FA
FAB
(c) After subcritical pitchfork
Figure 5.9: Example 1: A schematic of the quotient system for θ = 4 as R1 increases.
During the exchange of stability between FBC and FAB , the system passes through three
unique states of the separatrix structure.
fact, the tertiary bifurcation lines are defined by the boundaries of the parameter space
defined by (4.61), which define the region for which a floating fixed point exists. As many
of the parameters in (4.61) are dependent on both R1 and θ, the region in R1 –θ space, for
which a floating fixed point exists, is given by
754
2139
cos(θ) −
sin(θ),
12303
1367
−52855 cos(θ) − 99889 sin(θ)
,
136824
266564 cos(θ) − 666283 sin(θ)
.
622557
R1 <
R1 >
R1 >
(5.58a)
(5.58b)
(5.58c)
The upper and lower boundaries correspond to sub- and super-critical pitchfork bifurcations respectively. From (5.58a), the upper boundary is defined from by
2139
754
cos(θ) −
sin(θ),
12303
1367
R1 =
(5.59)
for θ values that satisfy
arccos
69
√
6130
31
+ π ≤ θ ≤ arccos − √
+ π,
1490
(3.63 / θ / 5.64).
From (5.58b) and (5.58c), the lower boundaries are
R1 =
−52855 cos(θ) − 99889 sin(θ)
,
136824
(5.60)
for
arccos
69
√
6130
+ π ≤ θ ≤ arccos
(3.63 / θ / 4.32),
1423
√
13632578
+ π,
5.2. 3D box
91
and
R1 =
266564 cos(θ) − 666283 sin(θ)
,
622557
(5.61)
for
arccos
1423
√
13632578
31
+ π ≤ θ ≤ arccos − √
+ π,
1490
(4.32 / θ / 5.64),
respectively.
Bifurcations that occur in this system have been identified in terms of the two bifurcation parameters, R1 and θ. A bifurcation set diagram in R1 − θ space is shown in figure
5.10. The entire collection of bifurcation trees for all values of θ, (of which figure 5.7 and
figure 5.8 belong), can be found in appendix B.1. Each bifurcation diagram is representative of the bifurcation structure within a θ interval. A new diagram is not shown if the
order of unrelated bifurcations, (on different branches of the tree), switch.
The dynamics of the interacting (1, 0), (2, 0) and (0, 1) modes have been completely
√
described for low Rayleigh numbers and box dimensions close to Lx = 2, Ly = √12 . It is
found that as the Rayleigh number increases, (R1 increases), the phase-space approaches
the topology seen for the unperturbed box, regardless of the direction of perturbation of
the box dimensions. The key difference is at lower Rayleigh numbers, and the bifurcation
route taken as R1 increases. In this example, there are 9 equivalence classes of bifurcation
routes that are taken to achieve the same topology.
This study is much more physically realistic than the unperturbed box, as a continuous
region of figure 4.1, has been described, rather than a critical point, which was the case in
the unperturbed box analysis. Hidden in the highly degenerate bifurcation seen before is a
complex set of interactions, dynamics and bifurcations when the degeneracy is overcome.
The bias created by the box perturbation becomes less important as the Rayleigh number
increases.
Chapter 5. A perturbed box: Splitting the bifurcations
92
R1
2.5
2.0
1.5
1.0
0.5
T1.1
1
T1.2
2
T1.3
3
T1.4
4
(8)
(7)
T1.5 T1.6 T1.7
(9)
5
T1.8
T1.9
6
T1.1
(5)
(4)
(6)
(1)
(3)
(2)
θ
Legend
(1) ‘A’ Primary bifurcation.
(2) ‘B’ Primary bifurcation.
(3) ‘C’ Primary bifurcation
(4) ‘AB’ Secondary bifurcation.
(5) ‘AC’ Secondary bifurcation.
(6) ‘BC’ Secondary bifurcation.
(7) Subcritical Tertiary bifurcation.
(8) Supercritical Tertiary
bifurcation off ‘FBC ’.
(9) Supercritical Tertiary
bifurcation off ‘FAC ’.
Equation
Equation
Equation
Equation
Equation
Equation
Equation
(5.48)
(5.49)
(5.50)
(5.54)
(5.55)
(5.56)
(5.59)
Equation (5.60)
Equation (5.61)
Figure 5.10: Example 1: Bifurcation set diagram in R1 − θ space. Three primary and three secondary bifurcations will occur as R1 increases
for all values of θ. In regions T1.5 - T1.8, a tertiary bifurcation occurs to facilitate an exchange of stability between the secondary fixed points.
5.2. 3D box
93
(2,2)
(1,3)
(1,3)
(1,2)
(2,0)
(2,1)
3
(0,3)
(0,3)
(0,3)
(0,3)
(3,0)
(1,3)
(1,3)
(2,2)
(3,0) (3,1)
(0,3) (0,3) (0,3)
(3,0)
(1,2)
(2,1)
(2,2)
(1,0)
(1,2)
(2,0) (2,1)
(0,2)
(0,2)
(4,0)
(2,2)
(3,0)
(1,1)
2
(3,1)
(2,2)
(2,0)
(0,2)
(3,1)
(1,2)
(0,2)
(2,2)
(1,2)
(0,2)
(0,2)
(1,2)
(0,2)
(4,0)
(1,2)
(0,2)
(4,0)
(2,0)
Ly
(1,0)
(3,0)
(2,1)
(3,1)
(1,1)
(3,1)
(2,1)
(1,1)
(1,1)
(0,1)
(2,0)
1
(0,1)
(0,1)
(0,1)
(1,0)
1
(3,0)
(2,0)
(4,0)
(3,0)
(1,1)
(0,1)
(4,0)
3
2
(2,1)
(4,0)
(5,0)
4
Lx
Figure 5.11: Example 2 in Lx –Ly space.
5.2.3
Example 2
The second example is for a box with length dimensions, Lx0 = Ly0 = 21/4 , (see figure
5.11). The three viable solutions are the (p, q) = (1,0), (0, 1) and (1, 1) modes, which
2
share a critical Rayleigh number of Rac = π 2 + √32 . A denotes the amplitude of the
(1, 0) mode, B, the amplitude of the (0, 1) mode, and C the amplitude of the (1, 1) mode.
Again, as the first order equations are the same in the perturbed an unperturbed cases,
the first order solution is the same, and is given by
√ u1 = −2(1/4) 2 + 2 π 2 A sin (πx) cos(πz)
√ −2(1/4) 1 + 2 π 2 C sin (πx) cos (πy) cos(πz),
(5.62a)
√ v1 = −2(1/4) 2 + 2 π 2 B sin (πy) cos(πz)
√ −2(1/4) 1 + 2 π 2 C cos (πx) sin (πy) cos(πz),
(5.62b)
w1 =
√ 2 + 2 π 2 A cos (πx) sin(πz)
√ + 2 + 2 π 2 B cos (πy) sin(πz)
√ +2 1 + 2 π 2 C cos (πx) cos (πy) sin(πz),
(5.62c)
T1 = 2A cos (πx) sin(πz) + 2B cos (πy) sin(πz)
+2C cos (πx) cos (πy) sin(πz).
(5.62d)
94
Chapter 5. A perturbed box: Splitting the bifurcations
Imposing the solvability condition on the third order equations gives the dynamical system
(4.49), except the linear coefficients are given by:
√
√
2)R1 + ( 2 − 2) cos(θ) ,
√
√
b =
(2 + 2)R1 + ( 2 − 2) sin(θ) ,
!
√
√
6073 2 − 8584
2
c = π (1 + 2)R1 +
(cos(θ) + sin(θ)) .
23
a =
π2
2
π2
2
(2 +
(5.63a)
(5.63b)
(5.63c)
Again, it can be seen that if the perturbation of the box is zero, and the sin(θ) and cos(θ)
terms are removed, the dynamical system becomes the unperturbed example 2, system
(4.49). Similar to example 1, the nullclines retain their structure as the union of a plane
and an ellipsoid. Introducing a perturbation to the box dimensions results in the ellipsoid
nullclines being generated at unique Rayleigh numbers. In the unperturbed box case,
all bifurcations were degenerate and occurred at R1 = 0. The creation of these ellipsoid
nullclines as R1 is increased, corresponds to a primary bifurcation where the existence of
a single mode solution is created. As R1 becomes large, the perturbation terms become
less important, and the phase-space approaches the geometry seen in the unperturbed box
case.
The primary bifurcations occur when the coefficient of the linear terms become positive,
and depends on R1 and θ. The values of R1 as a function of θ for which these occur is
given by
√ 3 − 2 2 cos(θ),
√ =
3 − 2 2 sin(θ),
3 √
=
− 2 (cos(θ) + sin(θ)) .
2
A primary :
R1 =
B primary :
R1
C primary :
R1
(5.64)
(5.65)
(5.66)
Note the symmetry properties of these primary bifurcations. The C primary is symmetric about θ =
π
4,
other about θ =
while the A primary and B primary lines are also reflections of each
π
4.
The same symmetry also applies about θ =
5π
4 .
These two values
are special, they correspond to a horizontally square box. By setting θ =
π
4
the box is
perturbed by increasing the x and y dimensions by the same amount, keeping the box
square. Similarly, θ =
5π
4
corresponds to decreasing the x and y dimensions by the same
amount, also keeping the box square. This symmetry is the consequence of the C mode
having equal wavelength in both the x and y direction, and the A and B modes being
reflections of each other about the y = x line. This symmetry can easily be seen as the
sine and cosine functions are interchanged, i.e. f (cos(θ), sin(θ)) = f (sin(θ), cos(θ)) implies
f has symmetry about
π
4
and
5π
4 .
Three secondary bifurcations also occur, creating the GAB , GAC and GBC fixed points.
The ellipsoidal nullclines have the same properties as those seen in example 1, and these
5.2. 3D box
95
secondary secondary bifurcations occur when two ellipsoids intersect along one coordinate
axis. The AB secondary bifurcation may occur on either the A-axis or the B-axis, and
the switch between the two is seen as a cusp in figure 5.12.
Using condition (5.51), and the corresponding condition on the B-axis, the AB secondary bifurcation is found to occur at for the following R1 values:
R1 =

1



28







√ √ 192 − 125 2 cos(θ) + 3 −36 + 23 2 sin(θ) ,



1


 28




√ √ 192 − 125 2 sin(θ) + 3 −36 + 23 2 cos(θ) ,
for θ ≤
π
4
or θ ≥
5π
4
(on A axis),
(5.67)
for
π
4
≤θ≤
5π
4 ,
(on B axis).
The boundaries of each piecewise region have been calculated analytically, however for
simplicity, the approximate value is stated. The AC secondary bifurcations occur for:
R1 =

1



3017







√ √ 42891 − 29488 2 cos(θ) + 6 1880 − 1303 2 sin(θ) ,
for θ ≤ 1.89 or θ ≥ 5.03,
(on A axis),
(5.68)


√ √ 
1


3 1555 − 1146 2 cos(θ) + 841 − 670 2 sin(θ) ,

714




for 1.89 ≤ θ ≤ 5.03, (on C axis).
The BC secondary bifurcations occur for:
R1 =

1



3017







√ √ 42891 − 29488 2 sin(θ) + 6 1880 − 1303 2 cos(θ) ,
for θ ≤ 2.82 or θ ≥ 5.96,
(on B axis),
(5.69)


√ √ 
1


3 1555 − 1146 2 sin(θ) + 841 − 670 2 cos(θ) ,

714




for 2.82 ≤ θ ≤ 5.96, (on C axis).
The symmetry of the modes also affects the secondary bifurcations. For square boxes,
where θ = θs such that θs ∈ { π4 , 5π
4 }, the (1, 1) mode interacts with the (1, 0) and (0, 1)
modes in an equivalent way, and the critical R1 values of the AC and BC secondary bifurcations are equal. Any change in θ would favour one or the other mode equally, depending
on the direction. Hence the AC and BC secondary bifurcation values, (equations (5.68)
and (5.69)), are reflections of each other about θ = θs . Similarly, changing the θ value
from
π
4
equally affects the AB secondary, no matter which direction is changed. This is
reflected in the symmetry about θs in the AB secondary bifurcation, equation (5.67).
As in example 1, this dynamical system approaches the unperturbed box system as R1
grows large. This implies that as R1 increases larger still, 27 fixed points with the same
stability signatures seen in the unperturbed box need to appear. A tertiary bifurcation is
96
Chapter 5. A perturbed box: Splitting the bifurcations
required to produce the GABC fixed points. A tertiary fixed point exists if all equations
in (4.61) are satisfied. Substituting the parameter values specific to this example, the
conditions become:
R1 >
R1 >
R1 >
√ 1 (5.70a)
−848395 + 600058 2 (cos(θ) + sin θ) ,
686
√
√
1
−237999 + 50300 2 cos(θ) + 1251103 − 806172 2 sin(θ) ,
131376
(5.70b)
√
√
1
−237999 + 50300 2 sin(θ) + 1251103 − 806172 2 cos(θ) .
131376
(5.70c)
The highest valued expression on the right hand side of these inequalities serve as the
critical tertiary bifurcation line. Thus, the tertiary bifurcation line is piecewise defined
by:

√ √ 1


−237999 + 50300 2 sin(θ) + 1251103 − 806172 2 cos(θ) ,

131376





for θ ≤ 0.32 or θ ≥ 5π
(on AC plane),

4 ,









 1 −848395 + 600058√2 ,
R1 = 686


for 0.32 ≤ θ ≤ 1.25, (on AB plane),









√ √ 
1


 131376 −237999 + 50300 2 cos(θ) + 1251103 − 806172 2 sin(θ) ,




for 1.25 ≤ θ ≤ 5π , (on BC plane).
(5.71)
4
The three different piecewise regions represent bifurcations which occur on each of the
three possible planes, A–B, A–C and B–C. The expected symmetries occur, with the AB
tertiary line symmetric about θ = θs , while the AC and BC tertiary lines are reflections of
each other about the same values. The bifurcation set diagram for this example is shown
in figure 5.12.
Figure 5.12 does not have a bound region for which a tertiary fixed point exists, as
seen in figure 5.10 for example 1. Once a tertiary bifurcation occurs, (it is guaranteed to
occur for any θ,) there will be no subcritical tertiary bifurcation. Once a fixed point is
created, there is no stability exchange mechanism, and there are no interactions between
the branches of the bifurcation tree.
A sample bifurcation tree of region T2.1 can be seen in figure 5.13. The bifurcation
trees in each region of figure 5.12 are structurally equivalent. The only difference between
the trees is a permutation of the bifurcation order. The other bifurcation trees for this
example can be found in appendix B.2. The bifurcation structure in examples 1 and 2
differ at the tertiary bifurcation level. These two examples are typical of the two different
types of bifurcation structure seen close to all critical points where three modes are viable,
T2.1 T2.2 T2.3
1
T2.4
2
T2.5
3
T2.6
4
T2.7
5
T2.8
(2)
6
T2.1
(7)
(3)
(1)
(6)
(5)
(4)
θ
(1)
(2)
(3)
(4)
(5)
(6)
(7)
‘A’ Primary bifurcation.
‘B’ Primary bifurcation.
‘C’ Primary bifurcation.
‘AB’ Secondary bifurcation.
‘AC’ Secondary bifurcation.
‘BC’ Secondary bifurcation.
‘ABC’ Tertiary bifurcation.
Legend
Equation
Equation
Equation
Equation
Equation
Equation
Equation
(5.64)
(5.65)
(5.66)
(5.67)
(5.68)
(5.69)
(5.71)
Figure 5.12: Example 2: The bifurcation set diagram in R1 − θ space. Three primary, three secondary and one tertiary bifurcation will occur
as R1 increases for all values of θ. Due to the high degree of symmetry, each line is either symmetric, or the reflection of another line, about
θ = θs .
0.5
1.0
1.5
R1
5.2. 3D box
97
98
Chapter 5. A perturbed box: Splitting the bifurcations
GC+
GB+C+
GB+
+ GB+C−
++ GA+B+C+
+ GA+C+
++ GA+B−C+
+ GA+B+
GA+
+ GA+B−
++ GA+B+C−
+ GA+C−
++ GA+B−C−
+++ GO
++ GA−B+C+
+ GA−C+
++ GA−B−C+
+ GA−B+
GA−
+ GA−B−
++ GA−B+C−
+ GA−C−
++ GA−B−C−
+ GB−C+
GB−
+ GB−C−
GC−
+
T2.1
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+++
++
+
++
+
++
+
++
++
+
+
+
++
+
+
++
+
++
R1
Figure 5.13: Example 2: Bifurcation tree for θ < 0.32 or θ < 5.96. As R1 increases, a
series of pitchfork bifurcations occur. Every bifurcation tree for example 2 has 3 primary,
3 secondary and 1 tertiary set of bifurcations.
5.2. 3D box
99
(2,2)
(1,3)
(1,3)
(1,2)
(2,0)
(2,1)
3
(0,3)
(0,3)
(0,3)
(0,3)
(3,0)
(1,3)
(1,3)
(2,2)
(3,0) (3,1)
(0,3) (0,3) (0,3)
(3,0)
(1,2)
(2,1)
(2,2)
(1,0)
(0,2)
(0,2)
(3,1)
(1,2)
(1,2)
(2,0) (2,1)
(0,2)
(4,0)
(2,2)
(3,0)
(1,1)
2
(3,1)
(2,2)
(2,0)
(0,2)
(2,2)
(1,2)
(0,2)
(0,2)
(1,2)
(0,2)
(4,0)
(1,2)
(0,2)
(4,0)
(2,0)
Ly
(1,0)
(3,0)
(2,1)
(3,1)
(1,1)
(3,1)
(2,1)
(1,1)
(1,1)
(0,1)
(2,0)
1
(0,1)
(0,1)
(0,1)
(1,0)
1
(3,0)
(2,0)
(4,0)
(2,1)
(1,1)
(0,1)
(4,0)
3
2
(4,0)
(3,0)
(5,0)
4
Lx
Figure 5.14: Example 3 in Lx –Ly space. This particular box is the only case where the
nonlinear interactions between two modes produces the third viable mode.
except for the special case. The bifurcation structure for the special case is analysed in
the following section.
5.2.4
Example 3
The following example is the special case where a third viable mode can be reinforced by
the nonlinear interaction of another two viable modes. The phase-space is very different,
as shown in section 4.5, and many geometrical arguments used in examples 1 and 2, are no
longer available. Phasespace flow is no longer restricted to one octant, as certain primary
modes can be generated from zero amplitude.
While a “special case” occurs regularly for high Rayleigh numbers, there is only one
instance for a reasonable box size where this occurs very close to the onset of convection:
The example where A, B, and C are the amplitudes of the (2, 0), (0, 2) and (1, 1) modes,
respectively. This intersection is highlighted in figure 5.14. The scaled box dimensions
are
√ given by Lx0 = Ly0 = 81/4 , with a shared critical Rayleigh number of Rac = π 2 2 +
3 2
2
.
Analysis of this special case is important, as it is an opportunity to analytically investigate the interaction of three modes where the nonlinear action of two modes can create
the third. This provides insight into the behaviour of the interacting modes for higher
Rayleigh numbers, where analytic results are unavailable.
Imposing the solvability condition at third order gives the dynamical system, (4.70),
except the linear coefficients are given by:
√ √ 2 R1 + −1 + 2 cos(θ) ,
√ √ b = π 2 1 + 2 R1 + −1 + 2 sin(θ) ,
√ √ π2 c =
2 2 + 2 R1 + −2 + 2 (cos(θ) + sin(θ)) .
4
a = π2
1+
(5.72a)
(5.72b)
(5.72c)
100
Chapter 5. A perturbed box: Splitting the bifurcations
As in the previous two examples, if one sets the perturbation to zero, where the cos(θ)
and sin(θ) terms are removed, the dynamical system is equivalent to the unperturbed
system, (4.70). In addition, the effect of the box perturbation is reduced as R1 becomes
large, and the system asymptotically approaches the system for the unperturbed box.
The consequence of introducing a length perturbation to the box is similar to the previous
examples. It is immediately obvious that the coefficients of the linear term are no longer
0 at R1 = 0, and so the primary bifurcations also occur for non-zero R1 values. The
primary bifurcation lines in R1 − θ space are derived simply by solving for where the
linear coefficients are equal to zero. They are given by
√
2 2 − 3 cos(θ),
√
=
2 2 − 3 sin(θ),
3 √
=
− 2 (cos(θ) + sin(θ)) .
2
A primary :
R1 =
(5.73)
B primary :
R1
(5.74)
C primary :
R1
(5.75)
As with example 2, this example is a perturbed square box, and symmetries about θ = θs
occur. The modes (2, 0) and (0, 2), are reflections of each other about y = x. The
(1, 1) mode is symmetric about y = x. Hence, The A and B primary bifurcation lines
are reflections of one another about θs , and the C primary bifurcation line has lines of
symmetry at θs .
The higher-order bifurcations are no longer simple to analyse in the example. In fact,
it is somewhat difficult to categorise bifurcations into secondary and tertiary, as the simple
geometry is no longer available. The C nullcline is still the union of a plane and an ellipsoid
surface, however, it no longer has semi-principal axes parallel to the A and B coordinate
axes. The A and B nullclines are no longer the unions of an ellipsoid surfaces and planes,
but are general cubic surfaces.
A few of the simple techniques used to find the higher order bifurcations lines can still
be used in this example. Consider the dynamics on the C = 0 plane, which is the only 2D
invariant set on a linear plane. The dynamical system becomes
√
√
3 + 2 2 4 2 13 + 8 2 4 2 Ȧ = A a −
π A −
π B ,
2 √
6 √
13 + 8 2 4 2 3 + 2 2 4 2 Ḃ = B b −
π A −
π B ,
6
2
Ċ = 0.
(5.76a)
(5.76b)
(5.76c)
On this 2D plane, the A and B nullclines take their usual form: The union of the surface of
an ellipse and a line. These ellipses also have semi-principal axes along the two coordinate
axes. This allows the secondary AB bifurcation line to be calculated in the same way as
the other two examples. It is simply a case of finding the R1 values for which the ellipses
are the same length along either of the coordinate axes. The AB secondary bifurcation
5.2. 3D box
101
occurs for

 3 2 − √2 cos(θ) + 1 −18 + 11√2 sin(θ), for θ ≤ π or θ ≥ 5π (on A axis),
4
4
4
R1 = 4
 3 2 − √2 sin(θ) + 1 −18 + 11√2 cos(θ), for π ≤ θ ≤ 5π (on B axis).
4
4
4
4
(5.77)
The symmetry is again seen here; the two ‘arms’ of this bifurcation line are reflections of
one another about θ = θs .
The AC and BC secondary bifurcation lines cannot completely be described in the
usual way. Fortunately, it is quite simple to define these bifurcations where they occur on
the A or B axis, due to the dynamics on the C = 0 plane being very familiar. The Ċ = 0
nullcline on the C = 0 plane is the boundary of an ellipse. This ellipse is centered on
A = B = 0, and has semi-principal axes along the A = B and A = −B lines. Regardless
of the unusual orientation of the ellipse, the same principle as the previous two examples
applies; if the length of the C ellipse along the A-axis (B-axis) is the same as the length
of the A (B) ellipse along the A-axis (B-axis) then a secondary AC (BC) bifurcation will
occur. The AC bifurcation on the A-axis occurs for
R1 =
√ √ 3 1 −746 + 527 2 sin(θ) +
−6783 + 4789 2 cos(θ),
23
23
for 1.89 ≤ θ ≤ 5.03.
(5.78)
The BC bifurcation occurs on the B-axis for
R1 =
√ √ 1 3 −746 + 527 2 cos(θ) +
−6783 + 4789 2 sin(θ),
23
23
for 2.82 ≤ θ ≤ 5.96.
(5.79)
As expected, these bifurcation lines are reflections of one another about θ = θs . The terms
‘AC secondary’ and ‘BC secondary’ are used quite loosely. The fixed points generated are
not restricted to the B = 0 or A = 0 planes like the previous examples, but tend to float
about in the octants. These are merely convenient terms to tie these fixed points to those
that are familiar from the previous examples, based on how they are created.
The final bifurcation that occurs on the C = 0 plane is a “tertiary bifurcation”. This
bifurcation occurs as the C ellipsoid nullcline intersects the AB secondary fixed points.
The C ellipsoid is no longer symmetrical about the A = 0 and B = 0 lines, so the
intersection with the AB fixed points in different quadrants occur for different R1 values.
For ease, the A–B phase-space quadrants are labelled by number:
1.
2.
3.
4.
A > 0,
A < 0,
A < 0,
A > 0,
B
B
B
B
> 0,
> 0,
< 0,
< 0.
Consider the 3 ellipse nullclines on the C = 0 plane:
AN :
a = aa A2 + ab B 2 ,
(5.80a)
BN :
b = ba A2 + bb B 2 ,
(5.80b)
CN :
c = ca A2 + cb B 2 + cab AB,
(5.80c)
102
Chapter 5. A perturbed box: Splitting the bifurcations
where the general parameter terminology has been used for simplicity, and are known from
system (4.70). Note that a, b, and c are the only parameters that depend on R1 and θ.
Using the symmetry properties, some parameters are equal. The ellipses reduce to:
AN :
a = a a A2 + a b B 2 ,
(5.81a)
BN :
b = ab A2 + aa B 2 ,
(5.81b)
CN :
2
2
c = ca A + ca B + cab AB,
(5.81c)
These three equations in four variables, (A, B, C, R1 and θ), can be solved for a relationship between R1 and θ. This relationship is the bifurcation line of interest. First, a
π
4
rotation change of coordinates is introduced, so that the C ellipse now has both semi-
principal axes parallel to the coordinate axes. Let A =
√1 (B 0 + A0 )
2
and B =
√1 (B 0 − A0 ).
2
The A, B and C nullcines become:
Ȧ :
2a = (aa + ab )A02 + (aa + ab )B 02 + 2(aa − ab )A0 B 0 ,
(5.82a)
Ḃ :
2b = (aa + ab )A02 + (aa + ab )B 02 − 2(aa − ab )A0 B 0 ,
(5.82b)
Ċ :
02
02
2c = (2ca − cab )A + (2ca + cab )B .
(5.82c)
√
The coordinates are then scaled such that C ellipse becomes a circle. Let  = σ− A0 ,
√
and B̂ = σ+ B 0 , where σ± = 2ca ± aab . Note that 2ca > cab , so the coefficient of A0 is
real.
aa + ab 2 aa + ab 2
aa − ab
 +
B̂ + 2 √
ÂB̂,
σ−
σ+
σ+ σ−
aa + ab 2 aa + ab 2
aa − ab
 +
B̂ − 2 √
ÂB̂,
σ−
σ+
σ+ σ−
Ȧ :
2a =
Ḃ :
2b =
Ċ :
2c = Â2 + B̂ 2 .
(5.83a)
(5.83b)
(5.83c)
Subtracting equation (5.83b) from equation (5.83a) gives B̂ in terms of Â:
√
1 (a − b) σ+ σ−
B̂ =
.
 2(aa − ab )
(5.84)
Substituting equation (5.84) back into equation (5.82a) gives the  coordinates, Âs , for
which an intersection between the A and B nullclines occur.
v
q
u
u (a + b) + s (a − b)2 − (aa +ab )2 (a − b)2
u
2
(aa −ab )2
Âs = s1 t
,
aa +ab
2 σ−
(5.85)
where s1 , s2 ∈ {−1, 1}. Each As has a corresponding Bs by equation (5.84). There are
four solutions to the intersections of these two ellipsoids. They may be separated into
pairs which share the same distance to the origin, (an s2 = 1 and an s2 = −1 pair). As
the C nullcline is a circle, the (As , Bs ) intersections lie on this nullcline iff their distance
5.2. 3D box
to the origin is equal to
103
√
2c. That is,
√
q
2
2
2c = Âs + B̂s .
(5.86)
It is found that the s2 = −1 pair, (now labelled ABN pair for convenience), never lie on
the C nullcline. Equation (5.86), can only be solved for the s2 = 1 pair, (now labelled
ABI pair). This solution gives a relationship between R1 and θ, for which an intersection
between all three nullclines occurs on the C = 0 plane, causing a bifurcation. This bifurcation line is analytically derived, but tediously long. This bifurcation line is seen in figure
5.15, labelled, “tertiary off HA±B± or HA±B∓ ”. This line has the expected symmetry
about θ = θs . Note that the bifurcation line contains cos(2θ) and sin(2θ) terms.
An interesting feature of this bifurcation line is that it passes through an the intersection of the AB and AC secondary lines, as well as passing through an intersection
of the AB and BC secondary lines. In each of these cases, it is tangential to the the
AB secondary line. It seems as though an exchange of stability is occurring that is not
immediately obvious from the previous calculations. It just so happens that while the
ABI pair of fixed points is the only pair of fixed points which may intersect the C ellipse
nullcline, this pair may be the HA±B± or the HA±B∓ pair, depending on θ. For the large
majority of θ values, the ABI pair are the AB fixed points which lie in quadrants 2 and
4, (HA±B± ). For 3.44 < θ < 4.42, however, the ABI pair is actually the AB fixed points
in quadrants 1 and 3, (HA±B∓ ).
The reason for this is simple. The nullclines approach the locations seen in the unperturbed box as R1 becomes large; it is known that the quadrant 1 and 3 fixed points lie
outside the C nullcline, while the quadrant 2 and 4 AB fixed points lie inside. Note that
all AB fixed points are formed for the same R1 value, and are either formed on the A– or
B–axis and both pairs are either formed inside, or both are formed outside the ellipse (as
the magnitude of the C ellipse along an axis is independent of the sign of the axis). In one
case, for some R1 value, both pairs are formed outside the C ellipse. As R1 increases, it is
required that there will be an intersection of the C ellipse and the quadrant 2 and 4 fixed
points. Conversely, the other case occurs where the AB fixed points are formed inside the
C ellipse, and it is required that the ellipse intersects the quadrant 1 and 3 fixed points
as R1 increases. This is illustrated in figures 5.16 and 5.17, which show the progression of
the ellipses as R1 increases, for different values of θ.
The fixed points generated by this bifurcation are loosely labelled as tertiary fixed
points, as it is not clear where they end up for high Rayleigh numbers. In fact, it is found
that the final fixed point that is associated with this tertiary fixed point is not always the
same, but dependent on θ.
All previous bifurcation lines have been solved for analytically. The following bifurcations occur on a very complicated geometry, and the intersection of the nullclines can not
easily be analytically calculated. The existence of the appropriate fixed points is solved
for numerically, and the bifurcation lines are numerically defined.
Chapter 5. A perturbed box: Splitting the bifurcations
104
R1
2.0
1.5
1.0
0.5
−0.5
T3.1
(9)
T3.2
1
T3.3
2
(5)
T3.4
(9)
T3.7
(8)
3
T3.5 T3.6
T3.8
(9)
4
T3.9
(8)
T3.10
5
(9)
T3.11 T3.12
(6)
T3.13
6
(2)
(1)
(3)
(4)
(7)
T3.1
‘A’ Primary bifurcation.
‘B’ Primary bifurcation.
‘C’ Primary bifurcation.
‘AB’ Secondary bifurcation.
‘AC’ Secondary bifurcation off A.
‘BC’ Secondary bifurcation off A.
Tertiary bifurcation off HA±B± or
Saddle-node bifurcation.
‘C’ Secondary bifurcation.
Legend.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
θ
Equation
Equation
Equation
Equation
Equation
Equation
HA±B∓ .
(5.73)
(5.74)
(5.75)
(5.77)
(5.78)
(5.79)
Figure 5.15: Example 3: Bifurcation set diagram in R1 − θ space. Line (8) extends beyond the scale of the figure, and is plotted to scale in
figure 5.21. Due to the high degree of symmetry, each line is either symmetric, or the reflection of another line. about θ = θs .
5.2. 3D box
B
Q2
105
Q2
Q1
HB+
HB+
HA−B+
HA+
HA−
B
A
HA+
HA−B−
Q3
HB−
Q3
Q4
(a) R1 = 0.74: The HAB fixed
points form outside the C ellipse.
HA+B+
HA−
HA+
A
A
HA+B−
HA−B−
Q3
Q4
HB−
Q1
HA−B+
HA+B+
HA−
B HB+
Q2
Q1
Q4
HA−B+
HB−
(b) R1 = 1.19: The ABI ,
(HA±,B∓ ), fixed points intersect
the C ellipse in quadrants 2 and
4.
(c) R1 = 2: The fixed points are
in the expected position
Figure 5.16: The ellipse nullcines on the C = 0 plane for θ = 2.5. The HAB fixed points
form outside the C ellipse. The fixed points in quadrant 2 and 4 intersect the C nullcine.
Q1 denotes quadrant number 1.
Q2
B
Q2
Q1
B
HB+
HB+
HA−B+
HA+
HA−
A
HA+
HA−
Q3
Q4
(a) R1 = 0.25: The HAB fixed
points form inside the C ellipse.
Q3
A
HA+B−
HB−
B
HB+
HA−B+
HA+B+
HA−B−
HB−
Q2
Q1
Q4
(b) R1 = 0.46: The ABI ,
(HA±,B± ), fixed points intersect
the C ellipse in quadrants 1 and
3.
Q1
HA+B+
HA−
HA−B−
Q3
HA+
A
HA+B−
HB−
Q4
(c) R1 = 0.66: The fixed points
are in the expected position.
Figure 5.17: The ellipse nullcines on the C = 0 plane for θ = 3.75. The HAB fixed points
form inside the C ellipse. The fixed points in quadrant 1 and 3 intersect the C nullcine.
106
Chapter 5. A perturbed box: Splitting the bifurcations
There are two bifurcations that have not been covered, one of which can be visually
seen when plotting the nullclines in 3D. It is a saddle-node bifurcation which occurs in
quadrants 2 and 4. This bifurcation occurs in the vicinity of the region where the H1 fixed
points lie and is (usually) associated with the creation of it.
The other is a “secondary” bifurcation which is not immediately obvious when plotting
the nullclines but must occur to account for the resulting fixed points. This bifurcation is
located on the C-axis where A = B = 0, on the HC fixed points. This axis is problematic
when observing bifurcations, as the C-axis belongs to both the nullclines of Ȧ and Ḃ, but
has the unfortunate position to also belong to the asymptote of both nullclines. To unravel
the singularity on the C–axis, and observe a bifurcation occurring, cylindrical coordinates
are used. Let
A = r cos(φ),
(5.87a)
B = r sin(φ),
(5.87b)
C = C.
(5.87c)
The dynamical system in cylindrical coordinates is given by:
3aa + ab 2
aa − ab 2
2
2
ṙ(r, φ, C) = r rab − (
r + ac C +
r cos(4φ) + abc C sin(φ)) ,
4
4
(5.88a)
b−a
ab − aa 2
φ̇(r, φ, C) =
sin(2φ) −
r sin(4φ) + abc C 2 cos(2φ) , for r 6= 0 (5.88b)
2
4
cab 2
Ċ(r, φ, C) = C c − (ca r2 + cc C 2 +
r sin(2φ) ,
(5.88c)
2
where rab = a cos(φ) + b sin(φ). The R1 and θ dependence of this system comes into the
equations through the parameters a, b and c. There are many advantages to the cylindrical
coordinate system:
• Due the symmetry of the A and B modes, the (b − a) term, (and thus φ̇ ), is
independent of R1 .
• The only simple ellipsoid nullcline, the C ellipsoid, remains an ellipsoid.
• The r nullcine is well behaved with no asymptotes.
• The singular behaviour of the nullclines on the C-axis is unravelled.
Note that the “C–axis” refers to the r = 0 line, regardless of the value of φ. To observe
the C axis bifurcation, the r, φ and C nullcines are analysed on the r = 0 “plane”, where
φ is considered as a variable. The definitions of the nullcines are arranged such that the C
coordinate is given as a function of the r and φ coordinates. Although the φ dynamics are
not defined on r = 0, for bifurcation purposes, it is the limiting behaviour of the nullclines
as r → 0 that is of interest, (i.e the extrusion of the φ nullcline away from the C–axis).
5.2. 3D box
107
Hence, φ̇(0, φ, C) is used to examine this behaviour. The nullclines on r = 0 are given as
C(φ):
q
√
√
(1 + 2)R1 + ( 2 − 1)(cos(θ) cos2 (φ) + sin(θ) sin2 (φ))
q
, (5.89a)
√
√ π 1744 + 1298 2 + 663 + 470 2 sin(2φ)
q
√ r
−1603
+
1133
2 (cos(θ) − sin(θ)) sin(2φ)
3
p
,
(5.89b)
=4
97
π cos(2φ)
=
√ √ √ 2 7030 + 4971 2 R1 − 1206 + 853 2 cos(θ) − 1206 + 853 2 sin(θ)
√
,
π(44303 + 31327 2)
(5.89c)
rN :
√
C = 4 138
φN :
C
CN :
C
q
6
where rN , φN and CN are the ṙ = 0, φ̇ = 0 and Ċ = 0 surfaces, respectively. As one
would expect, the C nullcline is independent of φ, as this is just the intersection of the
C ellipse with the C–axis. An example plot of these nullclines for large R1 is found on
the right hand side of figure 5.18. The φ nullclines asymptote at φ =
π
4
+ n π2 , where n is
an integer. Depending on the sign of (cos(θ) − sin(θ)), the nullclines curve away from the
asymptote towards 0 + n π2 or towards
π
2
+ n π2 . In the case where (cos(θ) − sin(θ)) = 0,
(θ = θs ), the φ nullclines are vertical, and take the form φ =
π
4
+ n π2 . In the “stretched
out” r = 0 plane, it is important to note that the entire C nullcline is a fixed point.
The important intersections are the those between rN and φN . These are not fixed
points themselves, but belong to the intersection of the 2 nullclines which extend beyond
the C–axis. When these intersections also lie on the C nullcline, a bifurcation occurs. Depending on the θ value, there are four different types of bifurcation progression seen. For
θ < 1.89 or θ > 5.96, the C nullcline forms below rN ∩ φN , by the C primary bifurcation.
As the nullclines tend toward the form seen in the unperturbed box case, it is known that
the C nullcline must lie in between the upper and lower rN ∩ φN points. Accordingly, as
R1 increases, the C nullcline moves upwards and finally intersects the lower intersection of
the other nullclines, and a bifurcation occurs. This is shown in figure 5.18 (top). Similarly,
for 2.82 < θ < 5.04, the upper rN ∩ φN points form below the C nullcline. As R1 increases,
the C nullcline moves below the upper intersections, and a bifurcation occurs. This can be
seen in figure 5.18 (bottom). For all other values of θ, the C nullcline forms in between the
formation of the upper and lower intersections. This can be seen in figure 5.18 (middle).
Note that the two upper (lower) rN ∩ φN points are π apart, and correspond to each arm
of the pitchfork bifurcation. The two fixed points created travel away from the C–axis in
opposite directions.
There is a small range of θ, (2.23 < θ < 2.82 or 5.04 < θ < 5.12), for which the
C nullcline lies in between the upper and lower intersections, and as R1 increases, briefly
moves above the upper intersection before moving back down below. The two bifurcations
that occur are pitchfork bifurcations. While one may expect that these would be counter-
108
Chapter 5. A perturbed box: Splitting the bifurcations
C
0.4
φN
φN
φN
φN
0.3
rN
0.2
CN
0.1
1
2
3
4
5
φ
6
Bifurcation
θ = 1.5, R1 = 0.2
C
C
0.35
0.6
φN
φN
φN
φN
φN
φN
φN
φN
0.5
0.30
CN
0.25
0.4
CN
rN
0.20
0.3
No
Bifurcation 0.2
0.15
0.10
rN
0.05
1
2
3
4
5
0.1
φ
6
0
1
2
3
4
5
6
φ
θ = 2.5, R1 = 0.2
C
0.5
φN
φN
φN
φN
Bifurcation
0.4
CN
0.3
0.2
rN
0.1
φ
1
2
3
4
5
6
θ = 3.5, R1 = 0.5
R1 increasing.
Figure 5.18: Bifurcations on the r = 0 “plane”. As R1 increases, the nullclines approach
the same configuration. Depending on the initial formation, pitchfork bifurcations may
be required to reach this final state. A cross is marked on rN ∩ φN .
5.2. 3D box
109
C
C
φN
0.4
rN ∩ φ N
0.3
0.2
rN
0.1
CN
CN
1
2
3
4
5
φ
6
(a) r = 0 “plane”, R1 = −0.01.
(b) Side view along the dotted line.
C
C
0.25
φN
φN
φN
φN
rN ∩ φN
0.20
CN
0.15
CN
0.10
rN
0.05
1
2
3
4
5
φ
6
(c) r = 0 “plane”, R1 = 0.1.
(d) Side view along the dotted line.
C
C
0.6
φN
φN
φN
φN
CN
0.5
rN ∩ φN
0.4
rN
0.3
0.2
CN
0.1
1
2
3
4
5
(e) r = 0 “plane”, R1 = 1.
6
φ
(f) Side view along the dotted line.
Figure 5.19: C Secondary bifurcation: Nullclines for θ = 2.8. Left: Each time the C
nullcline crosses the upper intersection points a pitchfork bifurcation occurs. Right: Side
perspective, (along the dotted line), of the intersection of the r and φ nullclines, and the C
nullcline ellipse. The cross indicates rN ∩ φN on the C–axis. Boxes indicate fixed points.
110
Chapter 5. A perturbed box: Splitting the bifurcations
acting bifurcations, they are not, and 4 additional fixed points are generated from the 2
pitchfork bifurcations. These fixed points soon experience a saddle-node bifurcation with
each other, and are destroyed, while the C–axis fixed point, HC , survives. This saddlenode bifurcation is the final bifurcation to be addressed. A schematic for this process is
shown in figure 5.19.
The R1 –θ relationship for which these intersections and bifurcations occur can, in
principle, be analytically derived, as it is a simple geometry problem. In practise, however,
the result is difficult to obtain, even with the help of a symbolic package like Mathematica.
Consequently, the relation plotted on the bifurcation regions diagram, figure 5.15, is the
result of a numerical solution to the intersection of analytic surfaces. This relationship is
labelled as “C secondary bifurcation”.
The fixed points generated from this bifurcation are unlike any seen in the previous
examples. They cannot even be classified by their final position, as this depends on θ.
They can be related to the AC or BC secondary bifurcation off the C-axis, as seen by the
exchange of stability that occurs with the other “arms” of the AC and BC secondaries
at θ = 3.46 and θ = 1.89. These fixed points often end up in the “trio” of floating fixed
points in quadrants 2 and 4. For some values of θ, these fixed points become the stable
mixture solution.
The final bifurcation process to be analysed in this example is a saddle-node bifurcation. This bifurcation can usually be observed creating two of the three fixed points in the
“floating trio”. Cylindrical coordinates are used to analyse this bifurcation. It is not immediately clear why this is helpful, however, one finds that the bifurcation is constrained
to occur within small domains of φ.
The intersection of CN and rN is solved for as a function of φ. Using φ to parameterise
this intersection line, the C and φ nullclines, (CN I and φN I ), are defined by the functions,
C1 (φ) and C2 (φ).Equivalently,
rN I = CN I ,
(5.90)
by definition. The φN I has asymptotes at 2.02, 2.69, 2.03+π and 2.69+π , (exact numbers
are known). It is observed that a saddle-node bifurcation only occurs for 2.02 < φ <
2.69, (quadrant 2), or, by symmetry, 2.02 + π < φ < 2.69 + π, (quadrant 4). The
intersection of C1 (φ) and C2 (φ) is numerically calculated in the domain, 2.02 < φ < 2.69,
for tangent intersections, indicating a saddle-node bifurcation. There are two main saddlenode bifurcations observed as R1 increases: destructive and constructive, where fixed
points are destroyed and created, respectively. The destructive saddle node bifurcation
occurs for 2.73 < θ < 2.90 and 4.95 < θ < 5.12, and involves the destruction of the fixed
points generated by the double C-axis bifurcation. This can be seen in figure 5.20(a) 5.20(d).
The constructive saddle node bifurcation occurs over a much wider domain of θ,
including the domains where a destructive saddle node bifurcation occurs. That is, if a
destructive saddle-node bifurcation occurs as R1 increases, then a constructive bifurcation
will occur for a higher R1 . This can also be seen in figure 5.20(c) - 5.20(f). A constructive
5.2. 3D box
111
C
C
φN I
0.35
rN ∩ φN
0.30
2
1
2
CN I
0.25
1
1
2
CN
0.20
2.10
2.15
2.20
φ
(a) R1 = 0.34.
(b) For this θ value, the saddle-node bifurcation involves the C-axis pitchfork fixed
points.
C
C
rN ∩ φ N
0.35
φN I
0.30
CN I
0.25
CN
0.20
2.10
2.15
2.20
φ
(c) R1 = 0.40.
(d) Fixed points are destroyed
C
C
φN I
1.2
φN I
rN ∩ φN
1.0
0.8
2
1
CN I
3
0.6
2
0.4
0.2
1
1
2
CN
2.2
2.3
2.4
2.5
2.6
2.7
φ
(e) R1 = 6. The third fixed point of the
“trio” moves closer.
(f) It is not obvious from these diagrams
why the constructive saddle-node bifurcation creates a stable fixed point.
Figure 5.20: Saddle node bifurcations for θ = 2.8. As R1 increases, a destructive saddlenode bifurcation occurs, followed by a constructive saddle-node bifurcation.
Chapter 5. A perturbed box: Splitting the bifurcations
112
R1
12
10
8
6
4
2
T3.1
(9)
T3.2
1
T3.3
(7)
2
T3.4
(8)
3
T3.5 T3.6
T3.7
T3.8
(9)
4
T3.9
T3.10
5
(8)
T3.11 T3.12
T3.13
6
(4)
T3.1
‘A’ Primary bifurcation.
‘B’ Primary bifurcation.
‘C’ Primary bifurcation.
‘AB’ Secondary bifurcation.
‘AC’ Secondary bifurcation off A.
‘BC’ Secondary bifurcation off A.
Tertiary bifurcation off HA±B± or
Saddle-node bifurcation.
‘C’ Secondary bifurcation.
Legend.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
θ
Equation
Equation
Equation
Equation
Equation
Equation
HA±B∓ .
(5.73)
(5.74)
(5.75)
(5.77)
(5.78)
(5.79)
Figure 5.21: Example 3: The bifurcation lines in R1 − θ space. This diagram is identical to figure 5.15, but for larger R1 values to see the
entire saddle-node bifurcation line.
5.2. 3D box
113
HA+
HA+B+
HB+
+ H3−++
H1−++
+ H2−++
++ HA−B+
+ H2−+−
H1−+−
+ H3−+−
++ H5+
+ HC+
++ H4+
+++ HO
++ H4−
+ HC−
++ H5−
+ H3+−+
H1+−+
+ H2+−+
++ HA+B−
+ H2+−−
H1+−−
+ H3+−−
HB−
+ HA−B−
HA−
+
T3.1
7
+
+
+
++
6
+
3
+
+
+
++
5
2
1
+
+
++
4
++
+++
++
++
++
+
++
+
+
+
++
+
+
R1
+
+
Figure 5.22: Example 3: The bifurcation diagram for θ = 0. The circled numbers show
the order of bifurcation, as quoted in figure 5.23
114
Chapter 5. A perturbed box: Splitting the bifurcations
H0
+
H0
HA+
(a) Pre-bifurcation. The origin is a stable
fixed point.
(b) Bifurcation 1. The HA fixed point
forms, and is stable.
H0
HA−B−
+
HA+B+
+
++
H0
++
HB−
HA+ basin
HB− basin
+
HA+ basin
HB−
HA+
HA+
+
HA+B−
(c) Bifurcation 2. The HB fixed point is
formed.
HC+
HA−B−
+
HA+B+
+
++
(d) Bifurcation 3. The HAB fixed points
are formed, stabilising HB .
HA−B−
H5+
+
HB−
+
HA+
+
HA+B+
+
+
H4+
HC+
++
HA+
HA+B−
(e) Bifurcation 4. The HC fixed point is
formed. This surface is now the 2D manifold which separates the origin from infinity.
H5+
HA+ basin
HB− basin
HA+B−
+
++
HB−
HA+ basin
HB− basin
HA−B−
H4+
HC+
++
+
++
HA+B+
+
(f) Bifurcation 5. H4 and H5 are formed
via the ‘C’ secondary, partially stabilising
HC .
HA−B−
H5+
+
H4+
HC+
++
+
++
HA+B+
+
H1+−+
+
HB−
+
HB−
H3+−+
HB− basin
HA+ basin
HA+
++
HA+B−
(g) Bifurcation 6. A tertiary bifurcation on
HA+B− occurs, forming H2 .
H1+−+ basin
+
H2+−+
H3+−+
HB− basin
HA+ basin
HA+
++
HA+B−
(h) Bifurcation 7. The saddle node bifurcation occurs, forming H1 and H3 . As R1
increases, the phase-space resembles figure
4.11.
Figure 5.23: Example 3: The evolution of the fixed points, manifolds, and basins of
attraction for θ = 0 on the quotient system. The other fixed points exist on symmetrical
manifolds. The bifurcation numbers refer to those labelled in figure 5.22.
5.2. 3D box
115
saddle node bifurcation occurs for θ < 2.90 or θ > 2.73. This constructive saddle-node
bifurcation is always associated with the creation of two of the three fixed points in the
“floating trio”, one of which being the mixed stable solution. For θ values used in figure
5.20, the saddle-node bifurcation is closely related to the C secondary pitchfork bifurcation.
The bifurcation line in R1 –θ space, for which the saddle node bifurcation occurs, attains
quite high R1 values, (up to R1 = 11). Ideally, one would like to have R1 O(1), without
compromising the perturbation technique. R1 = 11 would be pushing close to the limits
of the suitability of the model. A rescaled bifurcation set diagram is found in figure 5.21,
which includes the full saddle-node bifurcation line.
A sample bifurcation diagram for θ = 0 is shown in figure 5.22. The evolution of fixed
points and basins of attraction quotient system is shown in figure 5.23.
5.2.5
Example 3: The cusp bifurcation
The cusp in the saddle-node bifurcation line at θ = π4 , is an exchange of stability between
two constructive saddle-node bifurcations. On one side of the cusp, the stable mixed
mode solution and a saddle point are created, while the other saddle point in the trio
already exists. On the other side of the cusp, the saddle point that is created, and the
saddle point that has long existed, swap roles. At the critical point, θ =
π
4,
the saddle-
node bifurcation is a symmetric pitchfork bifurcation, which stabilises the mixed-mode
solution. A schematic of this process can be seen in figure 5.24.
This type of bifurcation is called a “cusp bifurcation”, and is specific to dynamical systems
with two bifurcation parameters. The bifurcation is formally defined in Kuznetsov (2004),
and it can be shown that this bifurcation satisfies the criteria. Let the point (x∗ ,γ ∗ ), be
the location of the non-hyperbolic fixed point in question, at the phase location, x∗ , for the
bifurcation values, γ ∗ . The bifurcation parameters in question are analytically calculated
to be:
∗
γ : (θ, R1 ) =
π 2637330
3637563
√
,
−
4 13177
13177 2
≈ (0.785, 4.947),
(5.91)
which corresponds to the “cusp” location seen in figure 5.21. For these parameter values,
one of the four non-hyperbolic fixed points is located at H1 , given by the phase coordinates,
q
q
q
√
√
√ 
6(−26134+22217 2)
6(−26134+22217 2)
6(115443−81520 2)
2
16
2
92239
92239
92239

,
,
(A, B, C) = −
π
π
π

≈ (−0.373, 0.373, 0.514),
(5.92)
while the other three are trivially found by symmetry. A cusp bifurcation is said to occur
if the following properties are true at (x∗ ,γ ∗ ):
1. The Jacobian matrix has only one eigenvalue equal to zero. No other eigenvalue can
have zero real parts,
116
Chapter 5. A perturbed box: Splitting the bifurcations
C
C
C
φN I
φN I
0.8
0.8
0.6
φN I
φN I
0.8
0.6
0.6
φN I
φN I
0.4
0.4
CN I
0.2
2.2
2.3
(a) θ =
2.4
π
4
2.5
2.6
0.4
CN I
0.2
2.7
φ
2.2
−δ
2.3
2.4
(b) θ =
2.5
2.6
CN I
0.2
2.7
φ
2.2
π
4
2.3
(c) θ =
2.4
π
4
2.5
2.6
2.7
φ
+δ
R1 = 4.5 (Pre-bifurcation)
C
C
C
φN I
0.8
0.8
0.7
0.7
0.7
φN I
0.6
0.6
0.5
0.5
0.5
CN I
CN I
0.4
2.2
2.3
(d) θ =
2.4
π
4
2.5
−δ
2.6
φN I
φN I
φN I
0.6
φN I
0.8
CN I
0.4
0.4
2.7
φ
2.2
2.3
2.4
(e) θ =
2.5
2.6
2.7
φ
2.1
π
4
2.2
2.3
(f) θ =
2.4
2.5
π
4
+δ
2.6
2.7
φ
R1 = 6 (Post-Bifurcation)
Figure 5.24: Cusp bifurcation: breaking the symmetry. For θ = π4 , a pitchfork bifurcation
occurs, stabilising the mixed mode solution. On either side of this critical value, the
symmetry of the bifurcation is broken, and it becomes a saddle-node bifurcation. Note
that δ = 0.01.
2. The quadratic coefficient, Q must be zero,
3. The cubic coefficient, P must be non-zero,
4. The dynamical system and its first and second order derivatives must be regular
maps.
If this is the case, then the dynamical system is locally topologically equivalent to the
normal form of a cusp bifurcation, given by:
ẋ = β1 + β2 x + σx3 ,
(5.93)
where σ = SIGN(P ).
Condition 4. is trivially satisfied. Each vector field is of polynomial form, except for
the θ terms, which are simple trigonometric functions. Hence, the dynamical system is
infinitely differentiable, and the derivatives are continuous functions.
1. The Jacobian matrix evaluated at H1 , for the parameters, γ ∗ , denoted Jc , is analytically evaluated. The eigenvalues are (−223.53, −11.64, 0). Clearly there exists a zero
eigenvalue, and the other two eigenvalues have non-zero real parts.
2. In a multidimensional dynamical system, this condition must be satisfied along
the center manifold. That is, the manifold upon which the bifurcation is occurring. To
first order, the centre manifold is approximated by the eigenvector corresponding to the
zero eigenvalue. Hence, the centre manifold is approximated by the nullvector of Jc . The
5.2. 3D box
117
quadratic coefficient 3-vector is given by Q(q, q), where
3
X
∂ 2 fj (ξ, γ ∗ ) Qj (v , v ) =
v1v2,
∂ξk ∂ξl ξ=x∗ k l
1
2
(5.94)
k,l=1
q is a nullvector of Jc , and ξ1 , ξ2 , and ξ3 are A, B, and C respectively. The quadratic
effect perpendicular to the eigenspace of Jc is calculated by the directional derivative,
1
Q = p · Q(q, q),
2
(5.95)
where p is the scaled adjoint nullvector, such that JcT p = 0, and p · q = 1. For the
quadratic coefficient to be zero along the centre manifold, equation (5.95) must be zero.
The nullvector of the Jc is calculated to be,
q = (1, 1, 0)T .
This can be easily inferred by symmetry. For θ =
(5.96)
π
4,
the x and y lengths of the box
are equal, and there is symmetry between the A and B modes. In the phase space, this
causes reflective symmetry about the A = B and A = −B planes. As the fixed point,
H1 , lies on the A = −B plane, and it is known that the fixed points created from this
bifurcation do not lie on this plane as R1 becomes large, it follows that the centre manifold
must be perpendicular to the A = −B plane to preserve symmetry. Hence, q, the first
approximation to the centre manifold, must also be perpendicular to the A = −B plane.
As the nullspace spanned by q is perpendicular to the eigenspace of Jc , (which spans
the A = −B plane), the adjoint nullvector, p is parallel to q, and is scaled to be:
1
p = (1, 1, 0)T ,
2
(5.97)
such that q·p = 1. The quadratic effect along the centre manifold is analytically calculated,
but expressed numerically as:
Q(q, q) = (341.099, −341.099, −1159.66).
(5.98)
1
Q = p · Q(q, q) = 0,
2
(5.99)
This gives
so condition 2 is satisfied.
3. To calculate the cubic coefficient, the centre manifold is approximated to second
order. The details are found in Kuznetsov (2004). The cubic coefficient is given by
1
P = p · (P (q, q, q) + 3Q(q, w2 )) ,
6
(5.100)
118
Chapter 5. A perturbed box: Splitting the bifurcations
where,
1
2
n
X
3
Pj (v , v , v ) =
k,l,m=1
∂ 3 fj (ξ, γ ∗ ) v1v2v3 ,
∂ξk ∂ξl ∂ξm ξ=x∗ k l m
(5.101)
and w2 is the unique solution to
0 = p · w2 ,
Q(q, q) = Jc w2 + αq.
(5.102a)
(5.102b)
Note that α takes the value which ensures equation (5.102) is consistent, as α must be
non-zero if Q(q, q) does not lie in the eigenspace of Jc . It is found that
P (q, q, q) = (−4071.6, −4071.6, 0),
(5.103)
w2 = (−10.9289, 10.9289, −63.2164).
(5.104)
and
Hence the cubic coefficient is calculated to be
P = 4485.9.
(5.105)
As P > 0, this implies that the system is locally topologically to equation (5.93), where
σ = 1. This is consistent with the observation that the bifurcation stabilises the centre
fixed point, H1 .
Conditions 1 - 4 show that a pitchfork bifurcation occurs at H1 , for the bifurcation
values, γ ∗ . As γ ∗ is the intersection of two saddle-node bifurcation branches in a dynamical system with two bifurcation parameters, this is sufficient to conclude that a cusp
bifurcation occurs (Kuznetsov, 2004).
One way to visualise this bifurcation is by plotting the locations of the fixed points on
the centre manifold, u, as a function of the bifurcation parameters. Figure 5.25 shows a
surface in 3D space, where the amount of fixed points that exists for the bifurcation values is equal to the number of intersections a vertical line has with the fixed point surface.
When this surface is projected onto the R1 –θ plane, the folds form the distinctive “cusp”
from which the bifurcation gets its name. This is the cusp seen in figure 5.21. On this
plane, there are 3 fixed points that exist in region (1), and a single fixed point in region
(2).
All of the bifurcations occurring in example 3 have been described. As opposed to
the other examples, this symmetry-breaking case includes saddle-node bifurcations. Two
saddle-node bifurcation lines intersect at a cusp in R1 –θ space, where a pitchfork bifurcation occurs. This is known as a cusp bifurcation and is found in dynamical systems with
at least two bifurcation parameters.
5.3. Comparisons to previous work
119
(ξ ∗ , γ ∗ )
u
R1
γ∗
θ
Figure 5.25: Cusp bifurcation diagram. The position of the fixed points on the centre
manifold, u is plotted against the two bifurcation values. 3 fixed points exist for parameter
values in region (1), and a single fixed point exists for parameter values in region (2).
5.3
Comparisons to previous work
This perturbation analysis has been done with the aim of obtaining qualitatively useful
information close to the “critical” box dimensions. The results can, however, be applied
to any box with varying degrees of success. By straying too far from “critical” boxes, the
suitability of the model is decreased.
Steen (1983) analysed two different box examples, using an eigenvalue-splitting technique. Here, the results of Steen (1983) are compared to those found by the above perturbation technique. The two examples are the cube, where Lx = Ly = 1, and the stretched
box, where Lx = 1, Ly = 1.2. Assuming ε4 ε2 , The Rayleigh number and horizontal
aspect ratios can be approximated by
Ra = Rac (1 + R1 ε2 ),
cos(θ) 2
Lx = Lx0 (1 +
ε ),
2
sin(θ) 2
Ly = Ly0 (1 +
ε ).
2
(5.106a)
(5.106b)
(5.106c)
Any box, (Lx , Ly ), can be considered as a perturbation away from the critical boxes,
(Lx0 , Ly0 ), though for the best accuracy, the closest critical box is considered. Hence,
the cubic box is considered as a perturbation away from (Lx0 , Ly0 ) = (21/4 , 21/4 ), with
θ = 5π/4 and ε2 ≈ 0.45. Clearly ε is not small here; the perturbation method is not
appropriate, and large errors are expected. The perturbation method is only used here
for the sake of comparison to previous results. The R1 values for which bifurcations occur
for this value of θ are known, (see figure 5.12), and can be used to find the corresponding
Rayleigh numbers for bifurcation by using (5.106a). The cubic box has symmetry between
x and y dimensions, and hence symmetry between the (1, 0) and (0, 1) modes. This symmetry causes some bifurcations to be degenerate, and these modes share the same critical
Rayleigh number. A comparison of the results calculated here, and by those found by
Steen (1983) is shown in table 5.1. Note that the A, B, and C modes denote the (1, 0),
(0, 1), and (1, 1) modes respectively.
120
Chapter 5. A perturbed box: Splitting the bifurcations
Bifurcation
A primary / B primary / AB secondary
C primary
AC secondary / BC secondary / ABC tertiary
Ra
38.46
42.90
46.18
Ra (Steen (1983))
4π 2 ≈ 39.48
44.41
48.06
Table 5.1: The bifurcation Rayleigh numbers for the cube box, compared with the results
of Steen (1983).
Ra
60
Legend
(1) A Primary (approximate)
(2) A Primary
(3) C Primary (approximate)
(4) C Primary
(5) AC Secondary (approximate)
55
50
(4)
(5)
45
(3)
40
35
0.8
(2)
(1)
0.9
1.0
1.1
1.2
1.3
Lx
Figure 5.26: Bifurcation lines along Lx = Ly as approximated by perturbing about the
Lx0 = Ly0 = 21/4 critical box. The approximated lines consistently underestimate the
bifurcation Rayleigh numbers, derived from linear stability analysis for a cubed box, (Lx =
Ly = 1). The B bifurcation lines are equal to the A bifurcation lines.
The results obtained from the perturbation method consistently undervalue the critical Rayleigh numbers found by Steen. The A and C primary bifurcation values found by
Steen both match what is expected by the linear stability analysis, (by equation (4.2)).
The perturbation method gives an inaccurate result here due to the fact that all bifurcation
lines are approximated by linear functions about the critical box. The critical Rayleigh
surfaces have positive curvature, and any linear approximation will underestimate the
value. This is shown in figure 5.26, which shows the actual primary bifurcation lines from
linear stability analysis (see section 2.2.1 and figure 3.1), and all approximated bifurcation
lines.
The “stretched box” can also be considered as a perturbation of the (21/4 , 21/4 ) critical box, with θ ≈ 3.08, ε2 ≈ 0.31. There is no longer horizontal symmetry in this box,
and no bifurcations are degenerate. The Rayleigh numbers for bifurcation from both the
perturbation method, and from Steen (1983) are shown in table 5.2.
Again, most Rayleigh numbers derived by the perturbation method, are below those
found by Steen. The tertiary bifurcation, however, occurs for a higher Rayleigh number.
There is no simple explanation for this. It could be the case that the tertiary bifurcation
line has a detailed structure, which could account for the linear approximation to overestimate, rather than underestimate. A fifth order perturbation expansion would be required
to see the tertiary bifurcation line in greater detail.
Although this perturbation was designed to glean qualitative information about the
5.4. Conclusion
Bifurcation
A primary
B primary
C primary
BC secondary
AC secondary
AB secondary
ABC tertiary
121
Ra (perturbation)
38.46
40.80
41.72
42.40
44.14
45.89
57.74
Ra (Steen (1983))
4π 2 ≈ 39.48
40.81
42.29
43.45
44.65
46.86
54.83
Table 5.2: The bifurcation Rayleigh numbers for the “stretched” box, compared with the
results of Steen (1983)
bifurcation structure, the Rayleigh numbers for which bifurcations occur hold up well to
the examples analysed by Steen. Neither of these examples is particularly close to the
critical box, so other boxes which are closer will have increased accuracy.
5.4
Conclusion
These examples represent the types of interactions between three convection modes in a
box of a porous medium. While the analytical results obtained are only applicable in the
weakly nonlinear regime, where the Rayleigh number is very close to critical, very similar
interactions occur for higher Rayleigh numbers, and the results obtained here can be used
as a baseline for comparison. Even the special case, example 3, occurs regularly for higher
Rayleigh numbers, and it is fortunate that analytical results are able to be obtained by
exploiting this particular example. On the other hand, there are other potential special
cases that have been identified, where there are, unfortunately, no examples occurring for
low Rayleigh numbers.
The dynamics of the interacting modes have been analysed for dimensions close to the
critical boxes. This analysis is more physically relevant than the previous chapter, which
relies on examples having exact specifications. The perturbations away from the critical
dimensions reveal a bifurcation structure which is highly dependent on the direction of
perturbation. While some examples show pure cascading pitchfork bifurcations, for other
examples, an exchange of stability must occur between branches of the bifurcation diagram. As the Rayleigh number is increased, the direction of perturbation ceases to be
important, and the system approaches the critical systems seen in previous chapters.
The 3D dynamical systems for each box example can be easily generated by this perturbation method. Each example of three convecting modes at the onset of convection,
the bifurcation behaviour can be categorised into the three cases typified by examples 1,
2 and 3.
This analysis highlights the sensitivity of the system to changes in the box dimensions.
Obviously these “critical” boxes are exploited here and the dynamics are extremely sensitive, but the demonstration still applies; the interaction between modes is complicated. At
122
Chapter 5. A perturbed box: Splitting the bifurcations
higher Rayleigh numbers, more primary modes come into play, and the system becomes
quite complicated indeed.
Chapter 6
Four interacting modes
In this chapter, highly symmetric cases where four modes share the same critical Rayleigh
number are studied. Each of the two examples in figure 6.1, occurs for a square box, where
Lx = Ly . Of the four viable modes, they are naturally paired into couplets; one mode
can be rotated by π/2 to get the other mode in the couplet. The smallest box is chosen
as an example to analyse. This is where the (2, 0), (0, 2), (2, 1) and (1, 2) modes interact.
The box dimensions for this example is Lx0 = Ly0 = 201/4 , with convection occurring for
√ Rac = ( 20 + 9 5 π 2 )/10. The (0, 2) and (2, 0) modes belong to one couplet, while the
(2, 1) and (1, 2) modes belong to the other.
6.1
Perturbation analysis
The perturbation method for a perturbed 3D box from the previous chapter applies. Thus,
the governing PDEs at each order for convection in a perturbed box of a saturated porous
medium are given by equations (5.30) - (5.38). For this example, there are four solutions
to the first order PDE. This has a cascading effect on the higher order solutions; there are
now 33 second order solutions. Applying the solvability condition as before, a system of
four ODEs is obtained, each corresponding to one of the four secular solutions:
Ȧ = A(a − aa A2 − ab B 2 − ac C 2 − ad D2 ),
(6.1a)
Ḃ = B(b − ba A2 − bb B 2 − bc C 2 − bd D2 ),
(6.1b)
Ċ = C(c − ca A2 − cb B 2 − cc C 2 − cd D2 ),
(6.1c)
2
2
2
2
Ḋ = D(d − da A − db B − dc C − dd D ),
(6.1d)
where A, B, C and D are the amplitudes of the (2, 0), (0, 2), (2, 1) and (1, 2) modes
respectively. The parameters are calculated to be:
a=
√ √ π 2 5 + 2 5 R1 + −5 + 2 5 ∆x ,
5
124
Chapter 6. Four interacting modes
(2,2)
(1,3)
(1,3)
(1,2)
(2,0)
(2,1)
3
(0,3)
(0,3)
(0,3)
(0,3)
(3,0)
(1,3)
(1,3)
(2,2)
(3,0) (3,1)
(0,3) (0,3) (0,3)
(3,0)
(1,2)
(2,1)
(2,2)
(1,0)
(1,2)
(2,0) (2,1)
(0,2)
(4,0)
(2,2)
(3,0)
(1,1)
2
(3,1)
(2,2)
(2,0)
(0,2)
(0,2)
(3,1)
(1,2)
(0,2)
(2,2)
(1,2)
(0,2)
(0,2)
(1,2)
(0,2)
(4,0)
(1,2)
(0,2)
(4,0)
(2,0)
Ly
(1,0)
(3,0)
(2,1)
(3,1)
(1,1)
(3,1)
(4,0)
(2,1)
(1,1)
(3,0)
(0,1)
(0,1)
(0,1)
(1,0)
(0,1)
(4,0)
3
2
(2,1)
(1,1)
(3,0)
(2,0)
1
(4,0)
(1,1)
(0,1)
(2,0)
1
(5,0)
4
Lx
Figure 6.1: Example 4 in Lx –Ly space. At the highlighted intersections, four modes are
equally viable.
b=
c=
√ √ π 2 5 + 2 5 R1 + −5 + 2 5 ∆y ,
5
√ √ π2 5 2 + 5 R1 + −2 + 5 (4∆x + ∆y ) ,
10
d=
ab = ba = −π
ad = bc = −π
4
cb = da = −π
4
√ π4 9+4 5 ,
10
√ √ π2 5 2 + 5 R1 + −2 + 5 (4∆y + ∆x ) ,
10
√ 3463 + 1536 5
,
2670
√ 46750 + 20983 5
,
58000
4
aa = bb = −
√ 86975 + 39066 5
,
58000
cd = dc = −π
4
ac = bd = −π
√ 781315 + 350329 5
,
536800
4
ca = bd = −π
√ 289041 + 129664 5
,
107360
4
cc = dd = −π
4
√ 97321 + 43184 5
,
130800
√ 26908252445 + 11939619048 5
.
29643934800
For simple notation, let (A, B, C, D) = (A1 , A2 , A3 , A4 ). This dynamical system takes the
usual form. Ȧi = 0 for Ai = 0. This implies that each of the four walls of the hyperoctant
in phase-space are invariant manifolds. Trajectories on these hyperplanes do not leave
the hyperplane. In addition, trajectories within the hyperoctant are constrained to that
hyperoctant. Furthermore, the dynamics in each hyperoctant are equivalent due to the
6.2. Unperturbed box
125
symmetry:
Ȧi (Ai ) = −Ȧi (−Ai ),
(6.2)
Ȧi (Aj ) = Ȧi (−Aj ), for i 6= j.
(6.3)
Hence, it is only necessary to focus analysis on one hyperoctant (e.g Ai ≥ 0 ∀i) of the
phase-space, to determine the global behaviour.
The dynamical system for each amplitude is very similar to its partner in the couplet.
The coefficients of the nonlinear terms are interchangeable, however, the coefficients of the
linear terms are only interchangeable between couplets if the box dimensions are square
(∆x = ∆y ). If this were the case, then the additional symmetry properties occur:
Ȧ(B, A, D, C) = Ḃ(A, B, C, D),
(6.4)
Ċ(B, A, D, C) = Ḋ(A, B, C, D).
(6.5)
These symmetries derive from the fact that the A-mode is a rotation of the B-mode, and
the C-mode is a rotation of the D-mode in a square box.
6.2
Unperturbed box
Analysis will proceed as in the previous examples. First, the fixed points and their stabilities of the dynamical system of the critical box are investigated. That is, the dynamical
system where ∆x = ∆y = 0. Also let R1 = 1, to represent the post-bifurcation behaviour
for R1 > 0. The dynamical system for the unperturbed box takes the form of system (6.1),
with all parameters the same, except for a, b, c and d, which are given by:
a=b=
√ π2 5+2 5 ,
5
c=d=
√ π2 2+ 5 .
2
For a set of four polynomials of order 3, there 81 possible fixed points. The number of
fixed points that need to be analysed is greatly reduced due to the symmetry between the
hyperoctants. The stability signatures of the fixed points found in the positive hyperoctant
are shown in table 6.1.
As per the previous examples (excluding the special case), the only stable nodes exist
on the axes of the phase-space, and correspond to solutions of a single mode in the absence
of others. There are 6 secondary fixed points, all of which are saddle points. The FAC and
FBD fixed points are odd in that they have a different signature to the other four. This
seems to suggest that the (2, 0) and (2, 1) modes do not mix well together, in the presence
of the other modes. A similar argument is made for the (0, 2) and (1, 2) modes. It makes
sense that these two secondary fixed points are odd; in figure 6.1, it can be seen that the
(2, 0) and (2, 1) modes do not share an edge (likewise for the (0, 2) and (1, 2) modes.
One may immediately notice the absence of tertiary fixed points (solutions of three
mixed modes). Up to four tertiary fixed points are possible, yet none exist. If one considers
the 3D dynamical systems on each of the four hyperoctant wall manifolds, they all belong
126
Chapter 6. Four interacting modes
Fixed Point
FO
FA
FB
FC
FD
FAB
FAD
FBC
FCD
FAC
FBD
FABCD
Location
Origin
A–axis
B–axis
C–axis
D–axis
A–B plane
A–D plane
B–C plane
C–D plane
A–C plane
B–D plane
Floating
Stability signature
Unstable node (4u)
Stable node (4s)
Stable node (4s)
Stable node (4s)
Stable node (4s)
Saddle (1u 3s)
Saddle (1u 3s)
Saddle (1u 3s)
Saddle (1u 3s)
Saddle (3u 1s)
Saddle (3u 1s)
Saddle (2u 2s)
Table 6.1: Example 4: The fixed points and their stabilities in the positive hyperoctant
for the postcritical, unperturbed box, (R1 > 0, ∆x = ∆y = 0).
to class 1 as no tertiary fixed points exist (similar to example 1, section 4.2.1). Due to
the degeneracy of the bifurcation at R1 = 0, it is not clear how a quaternary mode can
be present without the existence of any tertiary modes. This will become clear in section
6.3, where the box dimensions are perturbed, and the bifurcation structure is revealed.
It is much more difficult to explore the basins of attraction of each fixed point in a
four dimensional phase-space. One may reduce the dimensions of the problem, however, by
appealing to the same argument used in the 3D cases. There is a surface in the phase-space
which contains all fixed points except for FO , and completely separates the origin from
infinity. This quotient system contains all qualitative information about the structure of
the separatrices and basins of attraction, and ignores the almost-radial overall growth and
decay of the modes. In this 4D phase-space, a 3D hypersurface is required to separate the
origin from infinity. This hypersurface must separate two regions where:
1. trajectories come from the origin,
2. trajectories come from infinity.
Hence, it must be a 3D unstable manifold of a saddle point. The possible candidates
are the FAC and FBD saddle points. The hypersurface in question is the union of the
3D unstable manifolds of FAC and FBC . These two manifolds both approach a (2u 1s)
manifold of FABCD . Outside of this manifold, the dynamics are qualitatively the same,
however there is an additional vector component pointing towards this manifold.
The dynamics on this 3D manifold must be consistent with a 3D dynamical system.
As there are four stable nodes, this 3D space is divided into four basins of attraction,
which are separated by 2D stable separatrices. The signatures of the fixed points on this
manifold are found in table 6.2.
The 3D dynamics on this manifold are shown in figure 6.2. Note that the faces of
this tetrahedral 3D space are the 2D representations of a 3D dynamical system, (such as
figure 4.4). The 2D manifolds which separate the four basins of attraction are shown in
6.3. Perturbed box
127
Fixed Point
FA
FB
FC
FD
FAB
FAD
FBC
FCD
FAC
FBD
FABCD
Location
A–axis
B–axis
C–axis
D–axis
A–B plane
A–D plane
B–C plane
C–D plane
A–C plane
B–D plane
Floating
Stability signature
Stable node (3s)
Stable node (3s)
Stable node (3s)
Stable node (3s)
Saddle (1u 2s)
Saddle (1u 2s)
Saddle (1u 2s)
Saddle (1u 2s)
Unstable node (3u)
Unstable node (3u)
Saddle (2u 1s)
Table 6.2: Example 4: The fixed points and their stabilities on the representative 3D
manifold in the positive hyperoctant for the postcritical, unperturbed box, (R1 > 0,
∆x = ∆y = 0).
D∗
AD∗
CD∗
BD∗
ABCD∗
A∗
C∗
AC ∗
BC ∗
AB ∗
B∗
Figure 6.2: This 3D quotient system represents the 4D dynamics. The basins of attraction
for FA and FC are not adjacent. Similarly, the FB and FC basins are also not adjacent.
blue. Note that the A and C basins are not adjacent to each other.
The fact that the A and C basins of attraction are not adjacent seems consistent
with figure 6.1, where the (2, 0) and (2, 1) regions are also not adjacent. By reducing the
4D dynamics to a 3D quotient system, the overall dynamics has been represented by a
lower dimension system. This has allowed the easy determination and visualisation of the
topological structure of the separatrices and, thus, the 4D basins of attraction.
6.3
Perturbed box
To see the progression of bifurcations as R1 increases, the box must be perturbed away
from the critical dimensions. As in section 5.2, let ∆x = cos θ and ∆y = sin θ. The
magnitude of the perturbation remains constant, and the direction of the perturbation is
given by θ, which acts as a bifurcation parameter.
The dynamical system remains as defined in system (6.1), however the linear coeffi-
128
Chapter 6. Four interacting modes
cients are now defined as:
a=
√ √ π 2 5 + 2 5 R1 + −5 + 2 5 cos θ ,
5
b=
√ √ π 2 5 + 2 5 R1 + −5 + 2 5 sin θ ,
5
c=
√ √ π2 5 2 + 5 R1 + −2 + 5 (4 cos θ + sin θ) ,
10
d=
√ √ π2 5 2 + 5 R1 + −2 + 5 (4 sin θ + cos θ) .
10
Like all previous perturbed box cases, as R1 grows large, the perturbation terms have
less significance, and the system approaches the unperturbed box system. As there are
no tertiary fixed points seen in the unperturbed example, the creation of the quaternary
fixed point is of particular interest.
There is also symmetry in the bifurcation parameter, θ. As with the other square box
(Lx = Ly ) examples, there is reflective symmetry about θs , where θs ∈ {π/4, 5π/4}. The
dynamics of the A mode is equivalent to the dynamics of the B mode under a reflection
about θ = θs , (a similar relationship exists for the C and D modes.). This can be defined
by:
6.3.1
Ȧ(A, B, C, D, R1 , −(θ − θs )) = Ḃ(B, A, C, D, R1 , (θ − θs ))
(6.6)
Ċ(A, B, C, D, R1 , −(θ − θs )) = Ḋ(A, B, D, C, R1 , (θ − θs ))
(6.7)
Bifurcations
The following analysis will consist of deriving the bifurcation lines which exist in the R1 –θ
parameter space. The primary bifurcation lines are easy to derive; it is simply a calculation
of where the linear coefficients are equal to zero. The A primary bifurcation occurs for
a = 0. This gives the equation:
√ R1 = 9 − 4 5 cos θ.
(6.8)
The B primary bifurcation occurs for b = 0, which implies that
√ R1 = 9 − 4 5 sin θ.
(6.9)
The C primary bifurcation occurs for c = 0, where
√
−9 + 4 5
R1 =
(4 cos θ + sin θ).
5
(6.10)
6.3. Perturbed box
129
The D primary bifurcation occurs for d = 0, which implies that
√
−9 + 4 5
R1 =
(4 sin θ + cos θ).
5
(6.11)
The secondary bifurcations occur where one nullcline ellipsoid intersects another along one
of the axes associated with the nullcines. For example, the AB secondary occurs for:
r
aa
=
a
r
ab
=
a
r
ba
,
b
(on the A–axis),
(6.12)
bb
,
b
(on the B–axis).
(6.13)
or,
r
Solving these conditions gives the following piecewise defined function for the AB secondary bifurcation.
R1 =

1



80







√ √ 1515 − 671 5 cos θ + −795 + 351 5 sin θ ,



1



80




√ √ 1515 − 671 5 sin θ + −795 + 351 5 cos θ ,
for θ ≤
π
4
or θ >
5π
4 ,
(on A–axis),
(6.14)
for
π
4
<θ≤
5π
4 ,
(on B–axis).
Similarly, the 5 other secondary bifurcation lines can be defined. The AC secondary
bifurcation line is defined by:
R1 =

1



616645










√ √ 34836525 − 15547612 5 cos θ + 8 406760 − 181681 5 sin θ ,
for θ ≤ 1.68 or θ > 4.82,
(on A–axis),
√ 1



1153982309 61748554599 − 27680993705 5 cos θ+


√ 
3



1153982309 1902322734 − 854261647 5 sin θ,




for 1.68 < θ ≤ 4.82,
(on C–axis).
(6.15)
The θ values for the boundaries of the piecewise defined function are analytically calculated, however, a rounded solution has been shown for simplicity. The AD secondary
130
Chapter 6. Four interacting modes
bifurcation line is defined by:
R1 =

1



7139










√ √ 1620171 − 723956 5 cos θ + 80 12966 − 5795 5 sin θ ,
for θ ≤ 2.16 or θ > 5.30,
(on A–axis),
√ 1


1692360657 − 767446204 5 cos θ+

74595655


√  12


56722209 − 26059088 5 sin θ,

74595655




for 2.16 < θ ≤ 5.30,
(on D–axis).
(6.16)
The BC secondary bifurcation line is given by:
R1 =

1


 7139










√ √ 1620171 − 723956 5 sin θ + 80 12966 − 5795 5 cos θ ,
for θ ≤ 2.55 or θ > 5.70,
(on B–axis),
√ 1


1692360657
−
767446204
5 sin θ+

74595655


√

12



74595655 56722209 − 26059088 5 cos θ,




for 2.55 < θ ≤ 5.70,
(on C–axis).
(6.17)
The BD secondary bifurcation line is defined by:
R1 =

1



616645










√ √ 34836525 − 15547612 5 sin θ + 8 406760 − 181681 5 cos θ ,
for θ ≤ 3.03 or θ > 6.17,
(on B–axis),
√ 1



1153982309 61748554599 − 27680993705 5 sin θ+


√ 
3


1902322734
−
854261647
5 cos θ,

1153982309




for 3.03 < θ ≤ 6.17,
(on D–axis).
(6.18)
Finally, the CD secondary line is given by:
R1 =

√ 1


91601172941217 − 40706581049240 5 cos θ

4106599654190


√ 
1


+ 4106599654190
−128560569828927 + 57132979666000 5 sin θ ,






for θ ≤ π4 or θ > 5π
(on D–axis),

4 ,



√ 
1


91601172941217 − 40706581049240 5 sin θ

4106599654190


√ 
1


+ 4106599654190
−128560569828927 + 57132979666000 5 cos θ ,





for π4 < θ ≤ 5π
(on C–axis).
4 ,
(6.19)
6.3. Perturbed box
131
Although no tertiary fixed points exist for a large R1 , tertiary bifurcations are sought
regardless. The tertiary bifurcations can be calculated as before; the tertiary fixed point,
FABC , exists if the following conditions are satisfied:
c(ab ba − bb aa ) < b(ca ab − cb aa ) + a(cb ba − ca bb ),
(6.20a)
a(bc cb − cc bb ) < c(ab bc − ac bb ) + b(ac cb − ab cc ),
(6.20b)
b(ca ac − aa cc ) < a(bc ca − ba cc ) + c(ba ac − bc aa ).
(6.20c)
The existence of tertiary fixed points on the other hyperoctant walls can be found by a
permutation of conditions (6.20). Therefore the tertiary bifurcations occur for
c(ab ba − bb aa ) = b(ca ab − cb aa ) + a(cb ba − ca bb ),
(6.21a)
a(bc cb − cc bb ) = c(ab bc − ac bb ) + b(ac cb − ab cc ),
(6.21b)
b(ca ac − aa cc ) = a(bc ca − ba cc ) + c(ba ac − bc aa ),
(6.21c)
where these lines are a boundary of the region which satisfies conditions (6.20).
There is a problem with applying conditions (6.20) in this particular example. These
equations were derived from scaling the geometry such that one ellipsoid nullcline is spherical, then optimising the radius of the intersection of the two remaining ellipsoidal nullclines.
The spherical nullcline intersects with the other nullclines if:
2
2
rmin
< c < rmax
,
(6.22)
2
2
are the optimal points.
and rmax
where c is the radius squared of the sphere, and rmin
This process was first derived under the assumption that all bifurcations have occurred,
and so assumes complete stability of the axial fixed points. This gives rise to the axial
conditions (equations (4.50) - (4.52)). If all bifurcations have not occurred, it may be the
case that the optimal points rmax and rmin are found such that rmax < rmin . If this is so,
then a tertiary fixed point exists for if
2
2
rmax
< c < rmin
,
(6.23)
is satisfied. for such cases, FABC exists if
c(ab ba − bb aa ) > b(ca ab − cb aa ) + a(cb ba − ca bb ),
(6.24a)
a(bc cb − cc bb ) > c(ab bc − ac bb ) + b(ac cb − ab cc ),
(6.24b)
b(ca ac − aa cc ) > a(bc ca − ba cc ) + c(ba ac − bc aa ).
(6.24c)
All four hyperoctant walls are similar to example 1 of the 3-mode examples. As R1
increases, any tertiary fixed points created soon disappear. In the R1 –θ space, there is
a closed region where the tertiary fixed point exists. While the bifurcation lines have
been analytically calculated, they become excessively complicated. Hence, approximate
132
Chapter 6. Four interacting modes
amplitudes are shown. For the ABC tertiary, the upper boundary, which corresponds to
a subcritical tertiary bifurcation on the A–C plane, is approximately given by:
R1 = −0.014 cos θ + 0.262 sin θ,
for 0.46 ≤ θ ≤ 2.76.
(6.25)
The lower boundary of the region, which corresponds to a supercritical bifurcation, is
given by:

0.024 cos θ + 0.188 sin θ, for 0.46 ≤ θ < 1.61, (on A–B plane),
R1 =
−0.045 cos θ + 0.185 sin θ, for 1.61 ≤ θ ≤ 2.76, (on B–C plane).
(6.26)
For the ABD tertiary, The upper boundary, which corresponds to a subcritical bifurcation
on the B–D plane is given by:
R1 = 0.262 cos θ − 0.014 sin θ,
for θ ≤ 1.10 or θ ≥ 5.10.
(6.27)
The lower boundary, which corresponds to a supercritical bifurcation, is given by:

0.185 cos θ − 0.045 sin θ,
R1 =
0.188 cos θ + 0.024 sin θ,
for 5.10 ≤ θ < 6.25
(on A–D plane),
for θ ≤ 1.10 or θ ≥ 6.25,
(6.28)
(on A–B plane).
Note that the ABC and ABD bifurcation regions are reflections of each other about θs .
2
2 , so (the ACD equivalent of)
The ACD bifurcation occurs where rmax
< c < rmin
conditions (6.24) are used to determine the region where FACD exists. The lower boundary,
which corresponds to a supercritical bifurcation on the A–C plane, is given by:
R1 = −0.058 cos θ − 0.186 sin θ,
for 3.54 ≤ θ ≤ 5.55
(6.29)
The upper boundary, which corresponds to a subcritical pitchfork bifurcation, is given by:

−0.045 cos θ − 0.218 sin θ,
R1 =
−0.092 cos θ − 0.223 sin θ,
for 3.54 ≤ θ < 4.83,
(on C–D plane),
for 4.83 ≤ θ ≤ 5.55,
(on A–D plane).
(6.30)
This region is unusual in the sense that the upper boundary is piecewise defined, where
normally it is the lower boundary which is piecewise defined. This is a characteristic of the
2
2 . This unusual situation also occurs for the BCD bifurcation.
fact that rmax
< c < rmin
The lower boundary, which corresponds to a supercritical pitchfork bifurcation on the
B–D plane, is given by:
R1 = −0.186 cos θ − 0.058 sin θ,
for 2.31 ≤ θ ≤ 4.31
(6.31)
6.3. Perturbed box
133
The upper boundary, which corresponds to a subcritical pitchfork bifurcation, is given by:

−0.223 cos θ − 0.092 sin θ,
R1 =
−0.218 cos θ − 0.045 sin θ,
for 2.31 ≤ θ < 3.03,
(on B–C plane),
for 3.03 ≤ θ ≤ 4.31,
(on C–D plane).
(6.32)
A figure with every single bifurcation line in R1 –θ space is quite busy. For simplicity,
four different bifurcation set diagrams are drawn; each one corresponds to the bifurcations
that occur on one of the side walls of the hyperoctant. This is shown in figure 6.3. These
figures can be superimposed to form the overall bifurcation diagram, with the exception
of the quaternary bifurcation line.
6.3.2
Quaternary bifurcation.
To determine the quaternary bifurcation line, the conditions for which a floating fixed
point exists are derived. That is, if there is an intersection between all four ellipsoid nullclines off the side walls, then a floating fixed point exists, and a quaternary bifurcation
has occurred.
An extension of the process used to derive the tertiary bifurcation can be used. Consider the four ellipsoid nullclines, defined by:
AN :
a = a a A2 + a b B 2 + a c C 2 + a d D 2 ,
(6.33a)
BN :
b = ba A2 + bb B 2 + bc C 2 + bd D2 ,
(6.33b)
CN :
DN :
2
2
2
2
2
2
c = ca A + cb B + cc C + cd D ,
2
2
d = da A + db B + dc C + dd D .
(6.33c)
(6.33d)
These ellipsoids can be rescaled such that DN is a hypersphere, and unnecessary coefficients are scaled out to give:
AN :
1 = a¯a Ā2 + a¯b B̄ 2 + a¯c C̄ 2 + a¯d D̄2 ,
(6.34a)
BN :
1 = b¯a Ā2 + b¯b B̄ 2 + b¯c C̄ 2 + b¯d D̄2 ,
(6.34b)
CN :
1 = c¯a Ā2 + c¯b B̄ 2 + c¯c C̄ 2 + c¯d D̄2 ,
(6.34c)
DN :
d = Ā2 + B̄ 2 + C̄ 2 + D̄2 ,
(6.34d)
where,
Ā
A= √ ,
da
and
āi =
B̄
B=√ ,
db
ai
,
adi
C̄
C=√ ,
dc
bi
,
b¯i =
bdi
c¯i =
C̄
D=√ ,
dd
ci
,
cdi
(6.35)
(6.36)
for all i ∈ {a, b, c, d}. To determine if an intersection between all four ellipsoids exists, an
optimisation problem is constructed. That is, the magnitude squared of any coordinate,
(A, B, C, D), is optimised subject to the constraints that the coordinate must lie on all
Chapter 6. Four interacting modes
134
Legend
(1) A Primary, eq (6.8)
(2) B Primary, eq (6.9)
(3) C Primary, eq (6.10)
(4) D Primary, eq (6.11)
(5) AB Secondary, eq (6.14)
(6) AC Secondary, eq (6.15)
(7) AD Secondary, eq (6.16)
(8) BC Secondary, eq (6.17)
(9) BD Secondary, eq (6.18)
(10) CD Secondary, eq (6.19)
(11) ABC Sub-critical tertiary, eq (6.25)
(12) ABC Super-critical tertiary, eq (6.26)
(13) ABD Sub-critical tertiary, eq (6.27)
(14) ABD Super-critical tertiary, eq (6.28)
(15) ACD Sub-critical tertiary, eq (6.30)
(16) ACD Super-critical tertiary, eq (6.29)
(17) BCD Sub-critical tertiary, eq (6.32)
(18) BCD Super-critical tertiary, eq (6.31)
R1
0.4
0.3
0.2
0.1
R1
0.4
0.3
0.2
0.1
4.11
4.11
4.12
4.12
1
1
(11)
2
2
(12)
D=0
3
B=0
3
4
4
4.1
(15)
(16)
4.1
4.2
4.2
(1)
θ
(3)
(2)
(7)
(6)
(1)
4.11
4.10
θ
4.11
4.10
(5)
(6)
6
4.9
(4)
6
(3)
4.9
(10)
4.6 4.8
4.7
(15)
4.6 4.8
4.7
4.4 4.5
5
4.4 4.5
5
(8)
4.3
(5)
4.3
R1
0.4
0.3
0.2
0.1
R1
0.4
0.3
0.2
0.1
4.11
4.11
(13)
(14)
4.12
4.12
1
1
2
2
(5)
(17)
3
3
C=0
A=0
(17)
(18)
4.1
(9)
4
4
4.1
(7)
4.2
4.2
(4)
4.3
4.3
5
(13)
(14)
6
(2)
4.9
(1)
θ
4.11
4.10
θ
(3)
4.11
4.10
(8)
(10)
6
4.9
(9)
4.6 4.8
4.7
4.4 4.5
(4)
5
4.6 4.8
4.7
4.4 4.5
Figure 6.3: Example 4: Bifurcation set diagram in R1 –θ space on each of the four hyperoctant walls. Quaternary bifurcations are not yet
plotted, (see figure 6.4).
6.3. Perturbed box
135
three ellipsoidal nullclines, (6.34a) - (6.34c). If the radius squared of the spherical ellipsoid,
d, lies in between the maximum and minimum value, then an intersection occurs, and a
floating fixed point exists.
The optimisation problem is stated with Lagrange multipliers as:
L(Ā, B̄, C̄, λ1 , λ2 , λ3 ) = Ā2 + B̄ 2 + C̄ 2 + D̄2
+λ1 (a¯a Ā2 + a¯b B̄ 2 + a¯c C̄ 2 + a¯d D̄2 − 1)
+λ2 (b¯a Ā2 + b¯b B̄ 2 + b¯c C̄ 2 + b¯d D̄2 − 1)
+λ3 (c¯a Ā2 + c¯b B̄ 2 + c¯c C̄ 2 + c¯d D̄2 − 1).
(6.37)
The optimal points occur for
∂
L
∂ Ā
∂
0=
L
∂ B̄
∂
L
0=
∂ C̄
∂
L
0=
∂ D̄
0=
= Ā 1 + λ1 a¯a + λ2 b¯a + λ3 c¯a ,
(6.38a)
= B̄ 1 + λ1 a¯b + λ2 b¯b + λ3 c¯b ,
(6.38b)
= C̄ 1 + λ1 a¯c + λ2 b¯c + λ3 c¯c ,
(6.38c)
= D̄ 1 + λ1 a¯d + λ2 b¯d + λ3 c¯d ,
(6.38d)
and must lie on the ellipsoids (6.34a) - (6.34c). Rather than systematically calculating
each critical point, the behaviour of the 3D system is used to assume the results here. It is
assumed that the tertiary bifurcations occur on the hyperoctant walls, where exactly one
amplitude is zero, (and for particular parameter values, two amplitudes are zero). From
equations (6.38a) - (6.38d), a system of three equations in three unknown λl . There are
four such optimal points, O, each of which lies on one of the four hyperoctant walls.
The cases where two amplitudes are zero, and the bifurcations occur on a 2D plane
wall are seen as a cusp on the quaternary bifurcation line, where a degenerate bifurcation
occurs.
For each of the four optimal points, O, the coordinates can be calculated from the
ellipsoid equations, (6.34a) - (6.34c). For example, the optimal point that lies on the
D = 0 hyperplane is located at:
−a¯c b¯b + a¯b b¯c + a¯c c¯b − b¯c c¯b − a¯b c¯c + b¯b c¯c
,
a¯c b¯b c¯a − a¯b b¯c c¯a − a¯c b¯a c¯b + a¯a b¯c c¯b + a¯b b¯a c¯c − a¯a b¯b c¯c
a¯c b¯a − a¯a b¯c − a¯c c¯a + b¯c c¯a + a¯a c¯c − b¯a c¯c
= − ¯
,
a¯c bb c¯a − a¯b b¯c c¯a − a¯c b¯a c¯b + a¯a b¯c c¯b + a¯b b¯a c¯c − a¯a b¯b c¯c
¯
a¯b b¯a − a¯a b¯b − a¯b c¯a + b¯b c¯a + a¯a c¯b − b¯a cb
= −
.
−a¯c b¯b c¯a + a¯b b¯c c¯a + a¯c b¯a c¯b − a¯a b¯c c¯b − a¯b b¯a c¯c + a¯a b¯b c¯c
Ā2 = −
(6.39a)
B̄ 2
(6.39b)
C̄ 2
(6.39c)
Each of the other four optimal points can be similarly calculated. Substituting equations
(6.39) into (6.34), possible bifurcation lines in R1 –θ space are found. This expression can
be rearranged to:
R1 = f1 (θ).
(6.40)
136
Chapter 6. Four interacting modes
From the other three optimal points, another three bifurcation lines are also derived:
C=0:
R1 = f2 (θ),
(6.41)
B=0:
R1 = f3 (θ),
(6.42)
A=0:
R1 = f4 (θ),
(6.43)
(6.44)
The bifurcation lines fn are analytically known, but tediously complicated. They are all of
the form µ1 sin θ + µ2 cos θ. These four bifurcations separate the R1 –θ space into multiple
regions. In one of these regions, the quaternary fixed point exists. It is easy to see the
region for which this occurs. In the critical box case, with no length perturbation, the
quaternary fixed point exists. As with previous examples, the nullclines in the phase-space
of the perturbed box approach a cubic rescaling of unperturbed box nullclines. Hence,
as R1 becomes large, a quaternary fixed point exists. Therefore, the region where the
quaternary fixed point exists is the region where all of the following are true:
R1 > fn (θ),
(6.45a)
(6.45b)
for all n. Therefore, the bifurcation line is given by:
R1 = maxn (fn (θ)).
(6.46)
The quaternary bifurcation line, alongside the tertiary bifurcation lines, is shown in figure
6.4. All four functions, fi (θ), contribute to the quaternary bifurcation line. The quaternary
bifurcation line lies in the “envelopes” for which tertiary fixed points exist.
For a fixed θ value as R1 increases, all 4 primary and all 6 secondary bifurcations
occur. At least 1 set of tertiary bifurcations occur, which constitutes a supercritical
bifurcation followed by a subcritical bifurcation. For many θ values, a second set of
tertiary bifurcations occur. During the existence of one of the tertiary fixed points, the
quaternary bifurcation occurs. For an example bifurcation tree, refer to figure 6.5, which
shows the bifurcation profile as R1 increases, for θ = 0.
This progression of bifurcations can also be shown topologically. The advantage to
this method is that the progression of important features, such as basins of of attraction
and manifold surfaces, can be shown. Figures 6.6 and 6.7 show the bifurcation behaviour
as R1 increases, for θ = 0, to accompany figure 6.5. These are drawn as the net of a
tetrahedron, as most of the bifurcations occur on the faces.
The first few subfigures in figure 6.6, (up to figure 6.6(e)), ignore the dynamics of the
amplitudes which have not undergone a primary bifurcation. Once all primary bifurcations
have occurred, (figure 6.6(f) and onwards), the important dynamics can be reduced to a
3D space. Recall that in section 6.2, a manifold surface was identified, which contains all
(13)
4.11
(14)
4.12
1
(12)
(19)
(11)
2
3
(18)
(17)
4
4.1
(19)
4.2
4.3
(16)
(15)
6
4.6 4.8
4.7
4.4 4.5
4.9
5
(14)
θ
4.11
4.10
(19)
(13)
Legend
11. ABC Sub-critical tertiary, eq (6.25)
12. ABC Super-critical tertiary, eq (6.26)
13. ABD Sub-critical tertiary, eq (6.27)
14. ABD Super-critical tertiary, eq (6.28)
15. ACD Sub-critical tertiary, eq (6.30)
16. ACD Super-critical tertiary, eq (6.29)
17. BCD Sub-critical tertiary, eq (6.32)
18. BCD Super-critical tertiary, eq (6.31)
19. Quaternary bifurcation, eq (6.46)
Figure 6.4: Example 4: Quaternary and bifurcation set diagram in R1 –θ space. This figure can be superimposed onto the graphs in figure 6.3,
to give the complete bifurcation set diagram.
0.1
0.2
0.3
(19)
R1
0.4
6.3. Perturbed box
137
138
Chapter 6. Four interacting modes
FC
4.11
FCD
+
+
R1
FD
8
FBC
+
6
FB
+
4
13
3+
++
12
++
+
++
10
++
7
++
+
3+
+
1
2
3+
3+
++
3
4+
+
FBD
3+
FABCD
++
FAB
+
FAD
+
3+
++
9
+
+
11
FA
FAC
3+
FO
4+
5
Figure 6.5: Example 4: Bifurcation diagram for θ > 6.25 or θ < 0.46 as R1 increases.
The circled numbers show the specific order of bifurcation for θ = 0. All bifurcations
are pitchforks, however, one symmetric branch of each bifurcation is hidden to prevent
unnecessary detail.
6.3. Perturbed box
139
FO
FC
FO
+
(a) Pre-bifurcation. The origin is a
stable fixed point.
(b) Bifurcation 1. The FC fixed
point forms, and is stable.
FC
FD
FC
FO
FD
+
FB
++
+
FC
The FD fixed
(c) Bifurcation 2.
point is formed.
FBD
FC
FBD
FB
+
FC
++
(e) Bifurcation 4. The FBD fixed
point is formed, partially stabilising FB
FBD
++
FBD
FC
++
+
FBD
FB
FD
+
FD
+
FB basin
FBC
FBC
+
+
FC basin
FC basin
FC
FC
(f) Bifurcation 5. The FA fixed
point is formed. The hypersurface
drawn now ignores dynamics in the
direction of the origin.
FBC
FB basin
++
FD
+
FA
3+
+
FB
+
FBD
FC
FBC
FBD
FBD
FC
FB
+
FC
FC
(d) Bifurcation 3. The FB fixed
point is formed. The 3D space required to show the necessary features is drawn as the net of a tetrahedron. Each fixed point has 3 independent manifold sets, (ignoring
the A dynamics).
FD
+
FO
3+
FC
FO
3+
++
FA
3+
FC
(g) Bifurcation 6. The FBC fixed
point is formed, stabilising FB .
The boundary of the FB and FC
basins of attraction are shown by
the dotted lines.
FC
FAC
3+
FA
++
FAC
3+
FC
(h) Bifurcation 7. The FAC fixed
point is formed, partially stabilising FA .
Figure 6.6: Continued in figure 6.7
140
Chapter 6. Four interacting modes
FBD
FC
FBC
+
++
FBC
FCD
+
FB
FBD
FB basin
FBD
FB
FD
FCD
+
FB basin
FAB
FBC
+
FD
FD basin
FCD
+
FC basin
FAC
3+
FC
FAC
3+
FA
++
FC
(a) Bifurcation 8. The FCD fixed point is
formed, stabilising FD . There are now three
basins of attraction.
FBD
FB
FBC
+
FBD
FAC
3+
FAC
3+
FA
+
FA basin
FC
FC
(c) Bifurcation 10. The FABD tertiary fixed
point is formed, partially stabilising FAB .
++
FBD
FAB
+
FABD
3+
(unstable perpendicular
to ABD face)
FAD
+
FCD
+
FC
++
(stable perpendicular
to ABD face)
FABCD
FBC
+
FBD
FB
FBC
+
FB basin
FAB
FA basin
FC
FAC
3+
FA
FC
(e) Bifurcation 12. The FABCD Quaternary fixed point is formed, lying within the
tetrahedron. FABD is partially destabilised
FC
FBD
FAB
FAD
FC
++
(stable perpendicular
to ABD face)
FABCD
FBC
+
FBC
+
FC basin
FAC
3+
FC basin
FAC
3+
FA
FB basin
FAB
FAC
3+
FD
FD basin
FAD
FA basin
FC
FA
3+
+
+
FCD
+
FBD
FB
FCD
+
FCD
+
(d) Bifurcation 11. The FAD fixed point is
formed, stabilising FA .
FD
FD basin
FABD
FAD
FAC
3+
FD
FD basin
FABD
FAD
FB basin
FAB
FC basin
FC
FCD
+
FB
FD
FCD +
FC
FBD
++
FAB
+
FABD ++
(stable perpendicular
to ABD face)
+
FAD
FC
FBC
+
FD basin
FABD
FB basin
FAB
FAC
3+
FA
+
(b) Bifurcation 9. The FAB fixed point is
formed, partially stabilising FA .
FCD
+
FBC
+
FBC
+
FC basin
FAC
3+
FC
++
FBD
FAB
+
++
FABD
(stable perpendicular
to ABD face)
FC
++
++
FCD
+
+
FD basin
FBC
+
FBD
FAB
FC
FCD
+
FC basin
FAC
3+
FC
(f) Bifurcation 13. The FABD fixed point
is destroyed, destabilising FBD
Figure 6.7: Example 4: The evolution of the fixed points on the quotient system in
the positive hyperoctant, including manifolds and basins of attraction for θ = 0. The
bifurcation numbers refer to those labelled in figure 6.5.
6.4. Conclusion
141
fixed points except for the origin (see figure 6.2). Any parallel surface in the phase-space
would exhibit equivalent dynamics, except for an additional vector component pointing
toward this surface. This reduction method can be applied here, however, the reduced
system is a 3D hypersurface. The faces of the tetrahedron are, in fact, the equivalent
reduced system in its corresponding 3D case, (see figure 4.4 ).
The phase-space shown in the final state, figure 6.7(f), is equivalent to that seen in
figure 6.2, although the manifolds twist and curve to approach fixed points by approaching
along the eigenvector with the lowest eigenvalue. As R1 becomes large, the curvy basins
of attraction seen in figure 6.7, become straighter, approaching figure 6.2. This is due to
the forced A/B and C/D symmetry, which brings eigenvalues close together.
The reduction to a 3D quotient phase-space from a 4D phase-space loses very little
qualitative information. The 4D basins of attraction appear as radial extensions of the
simpler 3D basins shown in figures 6.6 and 6.7.
The fixed points, stabilities and bifurcations for four eigenmodes interacting in a box
with horizontal aspect ratios close to Lx = Ly = 201/4 , has been described. As with all
previous examples, as R1 becomes large, the length perturbation has less effect, and all
systems approach the same configuration. All four possible tertiary fixed points exist for
some region in the R1 –θ space.
The interaction of four modes at the onset of convection only occurs in these very
symmetric cases. In this example, it is the squareness of the horizontal domain that
allows such an interaction. The only other example of four modes interacting at the onset
of convection for low mode numbers, is the other intersection highlighted in figure 6.1. The
(3, 0), (0, 3), (3, 1) and (1, 3) modes are all viable at the onset of convection in a square box
√
with horizontal aspect ratio, L = 3(101/4 ). An analysis of the dynamical system formed
in the unperturbed box case reveals the exact same set of fixed points and stabilities as
the previous example, and the same qualitative behaviour is assumed to occur.
Technically, example 4 is a symmetry-breaking case, according to the rules set by
Gomes & Stewart (1994). The perturbation method would need to be taken to fifth order
to see these symmetry-breaking effects. The (3, 0)-(0, 3)-(3, 1)-(1, 3) example however,
will not show symmetry-breaking effects, and shares the qualitative behaviour seen in the
above analysis.
6.4
Conclusion
The bifurcation behaviour of four modes at the onset of convection in a 3D box of a porous
medium has been described. Both examples show the same behaviour; As R1 becomes
large, no tertiary fixed points exist. That is, all four of the 3D dynamical systems on
the walls of the hyperoctant belong to case 1 described in section 4.3. In general, a four
mode interacting system may take many configurations. Consider the hypothetical case
where one of the 3D dynamical systems on the walls belongs to case 2; a tertiary fixed
point survives as R1 is increased. If more than one of the 3D systems belongs to case
142
Chapter 6. Four interacting modes
2, then multiple tertiary solutions survive. There is also the possibility that quaternary
fixed points do not survive as R1 is increased, (this case being the higher order equivalent
of cases 1 and 2 of the 3D examples). While it is acknowledged that any combination of
these cases can occur, providing many qualitatively different bifurcation sets, no examples
of these are seen at the onset of convection for low mode numbers.
The interaction of a large number of modes becomes increasingly complicated, though
only a maximum of four modes can be simultaneously generated at the onset of convection.
An increasing number of modes are generated at higher Rayleigh numbers, but their impact
is dominated by the strongly nonlinear modes.
Chapter 7
Conclusion
The Horton-Rogers-Lapwood problem has been examined in finite rectangular domains,
where multiple modes are generated and nonlinearly interact. Investigating the effects of a
variable viscosity or permeability gives a finer detail to the model, though a high variation
is required to observe a large change in behaviour. On the other hand, the convection behaviour can be very sensitive to the geometry of the domain, particularly where multiple
modes are viable.
The interaction of modal solutions in critical box dimensions has been thoroughly
investigated. Particular examples of such modal interactions has been studied in the literature, but here a systematic approach is used to describe the behaviour for all cases
where multiple modes are viable for a box with moderate aspect ratios (Lx , Ly < 3.5).
For any two interacting modes, the phase-space is topologically equivalent to figure 3.3,
unless a symmetry-breaking special case occurs. It is noted, however, that these special
cases only occur for Rayleigh numbers higher than the onset of convection, where other,
highly nonlinear modes are present and dominate the global behaviour.
For three interacting modes, two unique classes are identified in addition to a third,
symmetry-breaking special case. For moderate box sizes, critical box examples have been
categorised into either of the two classes, where the “favoured” mode is also identified for
class 1 examples. A special case is identified at the onset of convection (example 3), and
the stability of possible solutions is examined. In contrast to the previous examples, one
of the stable solutions is not a pure mode, but a superposition of all three modes. This
example of symmetry-breaking in three-mode interactions has not been discussed in the
literature. To focus on the interaction between the modes, a quotient system is used which
ignores the overall growth and decay of the modes. A method has been derived by which
dynamical systems can be easily generated for any example of multiple interacting modes.
By using box dimensions close to the critical dimensions, the bifurcation structure can
be analysed as viable modes are generated, and become stable, in stages. As the Rayleigh
number increases, a series of pitchfork bifurcations occur. The bifurcation structure can
also be categorised into class 1 or class 2. The only exception to this, is in the special case,
(example 3), where a saddle-node bifurcation was seen to occur. In this example, a cusp
144
Chapter 7. Conclusion
bifurcation can also be observed. This bifurcation is specific to systems with two bifurcation parameters. This analysis shows just how complex the interaction between various
modes becomes. The bifurcation diagrams of the three specific examples have been shown.
Other three-mode examples at the onset of convection can be categorised into the same
equivalence class of one of these examples, and the same qualitative behaviour is observed.
There exists an example where four modes are generated at the onset of convection.
The existence, stability, and bifurcation behaviour of this example is studied. By considering 3 of the four interacting modes, these interactions belong to class 1, regardless
of which three modes are chosen. As such, while there exists quaternary solutions, no
tertiary solutions are present. Each bifurcation tree shows the creation and destruction
of at least 1 tertiary fixed point, via super- and sub-critical pitchfork bifurcations respectively. These examples are important, as the largest number of interacting modes which
are simultaneously generated is four modes.
The four-mode example is the final class of behaviour observed at the onset of convection; the behaviour of all moderate critical box example has been described. All two-mode
examples are typified by the behaviour shown in section 5.1. All three-mode examples are
equivalent to either of examples 1 or 2. The two four-mode examples are both equivalent
to the example used in chapter 6. As such, there is a complete picture available for the
behaviour of interacting modes close to the onset of convection.
An analytical description of the dynamics at all reasonable critical box dimensions
has been obtained. This thesis allows one to quickly reference the type of interactions
seen near these critical examples, without the need for nonlinear analysis. The qualitative
behaviour seen in these examples is invaluable to modelling fluid flows at higher Rayleigh
numbers, where the problem is highly nonlinear and an analytical solution is not available.
Such theoretical understanding is quite important in the groundwater field, as a complete
understanding of the system is immeasurable. Where a physically realistic model is required, these results can be used to supplement detailed numerical analysis. Identifying
where these upwellings occur with the least amount of costly measurements is a primary
goal.
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Appendix A
Calculation of polynomial
coefficients
By replicating the procedure of Nield (1994), a solution to the viscosity-dependent model
is sought. The fourth order ODE is given by
0 = (z − C)(D2 − α2 )G(z) − (D3 − α2 D)G(z) −
Rac (z − C)2 2
α G(z),
1−C
(A.1)
(see equation (2.61)). This can be rearranged to give
Rac (z − C)2 0 = (z−C)G(4) (z)−G(3) (z)−2(z−C)α2 G(2) (z)+α2 G0 (z)+ (z−C)α4 −
G(z).
1−C
(A.2)
A polynomial solution is sought. Let
G(z) =
=⇒ G0 (z) =
=⇒ G(2) (z) =
=⇒ G(3) (z) =
=⇒ G(4) (z) =
∞
X
n=0
∞
X
n=1
∞
X
n=2
∞
X
n=3
∞
X
n=4
bn z n
nbn z n−1
n!
bn z n−2
(n − 2)!
n!
bn z n−3
(n − 3)!
n!
bn z n−4 .
(n − 4)!
152
Chapter A. Calculation of polynomial coefficients
Substituting these polynomials into equation (A.2) gives
0 = z
∞
X
∞
n=4
∞
X
−
n=3
X
n!
n!
bn z n−4 − C
bn z n−4
(n − 4)!
(n − 4)!
n=4
∞
X
n!
n!
bn z n−3 − 2α2 z
bn z n−2
(n − 3)!
(n − 2)!
n=2
+2Cα2
∞
X
∞
X
n!
bn z n−2 + α2
nbn z n−1
(n − 2)!
n=2
n=1
∞
Rac C 2 α2 X
bn z n
+ − Cα4 −
1−C
n=0
∞
2CRac α2 X
bn z n
+ α4 +
z
1−C
n=0
−
α2
Rac
z2
1−C
∞
X
bn z n .
n=0
The indices are transformed to compare powers of z:
0 =
∞
X
(n + 3)!
(n − 1)!
n=1
∞
X
−
n=0
n
bn+3 z − C
∞
X
(n + 4)!
n!
n=0
bn+4 z n
∞
X (n + 1)!
(n + 3)!
bn+3 z n − 2α2
bn+1 z n
n!
(n − 1)!
n=1
+2Cα
2
∞
X
(n + 2)!
n!
n=0
n
bn+2 z + α
2
∞
X
(n + 1)bn+1 z n
n=0
∞
Rac C 2 α2 X
+ − Cα4 −
bn z n
1−C
n=0
∞
2CRac α2 X
bn−1 z n
+ α4 +
1−C
n=0
−
∞
α2 X
Rac
1−C
bn−2 z n .
n=0
The last two terms are true if b−1 and b−2 are set as 0. For the four different solutions,
f
(i)
=
∞
X
n=0
n
b(i)
n z ,
153
(i)
for i = 0, 1, 2, 3, let bn = δni . These solutions are independent, but not orthogonal. For
each power of z (≥ 1), the coefficients must sum to 0. This gives the equation
0 =
(n + 3)!
(n + 4)!
(n + 3)!
bn+3 − C
bn+4 −
bn+3
(n − 1)!
n!
n!
(n + 1)!
(n + 2)!
−2α2
bn+1 + 2Cα2
bn+2
(n − 1)!
n!
Rac C 2 α2
2
4
bn
+α (n + 1)bn+1 + − Cα −
1−C
2CRac α2
Rac α2
+ α4 +
bn−1 −
bn−2 .
1−C
1−C
By shifting the indices and rearranging the equation, bn is given as a recurrence relation:
C
n!
bn =
(n − 4)!
(n − 1)! (n − 1)!
bn−1
−
(n − 5)! (n − 4)!
(n − 2)!
+2Cα2
bn−2
(n − 4)!
(n − 3)!
bn−3
+ α2 (n − 3) − 2α2
(n − 5)!
Rac C 2 α2
4
bn−4
− Cα +
1−C
2CRac α2
4
+ α +
bn−5
1−C
Rac α2
bn−6 .
−
1−C
Further simplification yields
bn =
h
(n − 1)!(n − 5)bn−1
+2Cα2 (n − 2)!bn−2
2
2
+ α (n − 3)(n − 4)! − 2α (n − 3)!(n − 4) bn−3
Rac C 2 α2
4
− Cα +
(n − 4)!bn−4
1−C
2CRac α2
4
+ α +
(n − 4)!bn−5
1−C
i
Rac α2
−
(n − 4)!bn−6 /(Cn!).
1−C
154
Chapter A. Calculation of polynomial coefficients
The coefficient of bn−3 is simplified in detail:
α2 (n − 3)(n − 4)! − 2α2 (n − 3)!(n − 4)
= α2 (n − 3)! − 2α2 (n − 3)!(n − 4)
= α2 (n − 3)!(1 − 2(n − 4))
= −α2 (n − 3)!(2n − 9)).
Therefore, bn is given by
bn =
h
(n − 1)!(n − 5)bn−1
+2Cα2 (n − 2)!bn−2
−α2 (n − 3)!(2n − 9))bn−3
Rac C 2 α2
4
− Cα +
(n − 4)!bn−4
1−C
2CRac α2
+ α4 +
(n − 4)!bn−5
1−C
i
Rac α2
(n − 4)!bn−6 /(Cn!),
−
1−C
as stated in equation (2.63).
Appendix B
Bifurcation Trees
The θ values for which the bifurcation diagrams change form have been analytically derived, however, an approximate region is defined. The example 1, 2, 3 and 4 diagrams are
shown in the following sections.
B.1
Example 1
Most bifurcation trees in this example are a series of cascading pitchfork bifurcations. For
some θ values, there is an exchange of stability between branches of the bifurcation tree
via super- and sub-critical pitchfork bifurcations.
FB+
FB+C+
FC+
+ FB−C+
++ FA+B+
+ FA+C+
FA+
+ FA+C−
++ FA+B−
+++ FO
++ FA−B+
+ FA−C+
FA−
+ FA−C−
++ FA−B−
+ FB+C−
FC−
+ FB−C−
FB−
+
T1.1
+
+
++
+
+
+
++
+
+
+++
++
++
++
++
+
+
+
++
+
+
+
R1
Figure B.1: −0.64 < θ < 0.79
156
Chapter B. Bifurcation Trees
FC+
FB+C+
FB+
+ FB+C−
++ FA+B+
+ FA+C+
FA+
+
FA+C−
++ F A + B−
+++ FO
++ FA−B+
+ F A − C+
FA−
+ FA−C−
++ FA−B−
+ FB−C+
FB−
+ FB−C−
FC−
+
T1.2
+
+
++
+
+
+
++
+
+
+++
++
++
++
++
+
+
+
+
+
++
+
R1
Figure B.2: −0.79 < θ < 1.57
FC+
FA+C+
FA+
+ FA+C−
++ FA+B+
+ FB+C+
FB+
+ FB+C−
++ FA−B+
+++ FO
++ FA+B−
+ FB−C+
FB−
+ FB−C−
++ FA−B−
+ FA−C+
FA
+ FA−C−
FC−
+
T1.3
+
+
++
+
+
+
++
+
+
+++
++
++
++
++
+
+
+
++
+
+
+
R1
Figure B.3: 1.57 < θ < 2.36
B.1. Example 1
157
FA+
FA+C+
FC+
+ FA−C+
++ FA+B+
+ FB+C+
FB+
+ FB+C−
++ FA−B+
+++ FO
++ FA+B−
+ FB−C+
FB−
+ FB−C−
++ FA−B−
+ FA+C−
FC−
+ FA−C−
FA−
+
T1.4
+
+
++
+
+
+
++
+
+
+++
++
++
++
++
+
+
+
+
+
++
+
R1
Figure B.4: 2.36 < θ < 3.63
FA+
FA+C+
FC+
+ FA−C+
+ FB+C+
++ FA+B+
FB+
++ FA−B+
+ FB+C−
+++ FO
+ FB−C+
++ FA+B−
FB−
++ FA−B−
+ FB−C−
+ FA+C−
+
T1.5
+
++
++
+
+
++ +
++
++
+
++
+
++
++
+
++
+
+
++
+
+
+
+++
++
++
++
++
+
+
+
+
++
++
++
R1
++
++
+
+
Figure B.5: 3.63 < θ < 3.93
FC−
FA−C−
FA−
158
Chapter B. Bifurcation Trees
++
T1.6
++
+
++
++
+
+
++
+
++
+
+
++
++
+
+
++
++
++
+
+
+
+
++
++
+++
++
+
+
++
+
++
++
+
+
++
R1
++
FA+
++ FA+B+
FB+
++ FA−B+
+ FB+C+
+ FA+C+
FC+
+ FA−C+
+ FB−C+
+++ FO
+ FB+C−
+ FA+C−
FC−
+ FA−C−
+ FB−C−
++ FA+B−
FB−
++ FA−B−
FA−
Figure B.6: 3.93 < θ < 4.32
++
T1.7
++
+
++
+
++
+
+
+
++
+
++ +
+
+++
++
++
++
++
+
+
+
+
++
+
+
++ +
++
++
++ +
++
++
R1
+
++
++
Figure B.7: 4.32 < θ < 4.71
FA+
++ FA+B+
FB+
++ FA−B+
+ FA+C+
+ FB+C+
FC+
+ FB−C+
+ FA−C+
+++ FO
+ FA+C−
+ FB+C−
FC−
+ FB−C−
+ FA−C−
++ FA+B−
FB−
++ FA−B−
FA−
B.1. Example 1
159
++
T1.8
++
+
++
+
+
++
+
++
+
++
+
+
+
+++
++
++
++
+
+
++
++
+
+
++
++
++
+
+
++
+
++
++
+
+
++
R1
++
FB+
++ FA+B+
FA+
++ FA+B−
+ FA+C+
+ FB+C+
FC+
+ FB−C+
+ FA−C+
+++ FO
+ FA+C−
+ FB+C−
FC−
+ FB−C−
+ FA−C−
++ FA−B+
FA−
++ FA−B−
FB−
Figure B.8: 4.71 < θ < 5.50
FB+
FB+C+
FC+
+ FB−C+
+ FA+C+
++ FA+B+
FA+
++ FA+B−
+ FA+C−
+++ FO
+ FA−C+
++ FA−B+
FA−
++ FA−B−
+ FA−C−
+ FB+C−
FC−
+ FB−C−
FB−
+
T1.9
+
++
++
+
++
+
++ +
++
++
+
+
++
+
+
+
+++
++
++
++
++
++
+
+
+
++
++
+
++
+
++
+
++
++
+
+
++
R1
Figure B.9: 5.50 < θ < 5.64
160
B.2
Chapter B. Bifurcation Trees
Example 2
Each bifurcation diagram for this example are a series of cascading pitchfork bifurcations.
Each diagram is a permutation of every other diagram.
GC+
GB+C+
GB+
+ GB+C−
++ GA+B+C+
+ GA+C+
++ GA+B−C+
+ GA+B+
GA+
+ GA+B−
++ GA+B+C−
+ GA+C−
++ GA+B−C−
+++ GO
++ GA−B+C+
+ GA−C+
++ GA−B−C+
+ GA−B+
GA−
+ GA−B−
++ GA−B+C−
+ GA−C−
++ GA−B−C−
+ GB−C+
GB−
+ GB−C−
GC−
+
T2.1
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+++
++
+
++
+
++
+
++
++
+
+
+
++
+
+
++
+
++
R1
Figure B.10: −0.32 < θ < 0.32
B.2. Example 2
161
GC+
GB+C+
GB+
+ GB+C−
++ GA+B+C+
+ GA+B+
++ GA+B+C−
+ GA+C+
GA+
+ GA+C−
++ GA+B−C+
+ GA+B−
++ GA+B−C−
+++ GO
++ GA−B+C+
+ GA−B+
++ GA−B+C−
+ GA−C+
GA−
+ GA−C−
++ GA−B−C+
+ GA−B−
++ GA−B−C−
+ GB−C+
GB−
+ GB−C−
GC−
+
T2.2
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+
+++
++
++
+
++
+
++
++
+
+
+
++
+
+
++
+
++
R1
Figure B.11: −0.32 < θ <
π
4
GC+
GA+C+
GA+
+ GA+C−
++ GA+B+C+
+ GA+B+
++ GA+B+C−
+ GB+C+
GB+
+ GB+C−
++ GA−B+C+
+ GA−B+
++ GA−B+C−
+++ GO
++ GA+B−C+
+ GA+B−
++ GA+B−C−
+ GB−C+
GB−
+ GB−C−
++ GA−B−C+
+ GA−B−
++ GA−B−C−
+ GA−C+
GA−
+ GA−C−
GC−
+
T2.3
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+
+++
++
++
+
++
+
++
++
+
+
+
+
++
+
++
+
++
R1
Figure B.12:
π
4
< θ < 1.25
162
Chapter B. Bifurcation Trees
GC+
GA+C+
GA+
+ GA+C−
++ GA+B+C+
+ GB+C+
++ GA−B+C+
+ GA+B+
GB+
+ GA−B+
++ GA+B+C−
+ GB+C−
++ GA−B+C−
+++ GO
++ GA+B−C+
+ GB−C+
++ GA−B−C+
+ GA+B−
GB−
+ GA−B−
++ GA+B−C−
+ GB−C−
++ GA−B−C−
+ GA−C+
GA−
+ GA−C−
GC−
+
T2.4
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+
+++
++
++
+
++
+
++
++
+
+
++
+
+
+
++
+
++
R1
Figure B.13: 1.25 < θ < 1.89
GA+
GA+C+
GC+
+ GA−C+
++ GA+B+C+
+ GB+C+
++ GA−B+C+
+ GA+B+
GB+
+ GA−B+
++ GA+B+C−
+ GB+C−
++ GA−B+C−
+++ GO
++ GA+B−C+
+ GB−C+
++ GA−B−C+
+ GA+B−
GB−
+ GA−B−
++ GA+B−C−
+ GB−C−
++ GA−B−C−
+ GA+C−
GC−
+ GA−C−
GA−
+
T2.5
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+
++
+++
++
+
++
+
++
++
+
+
+
++
+
+
++
+
++
R1
Figure B.14: 1.89 < θ < 2.82
B.2. Example 2
163
GA+
GA+B+
GB+
+ GA−B+
++ GA+B+C+
+ GB+C+
++ GA−B+C+
+ GA+C+
GC+
+ GA−C+
++ GA+B−C+
+ GB−C+
++ GA−B−C+
+++ GO
++ GA+B+C−
+ GB+C−
++ GA−B+C−
+ GA+C−
GC−
+ GA−C−
++ GA+B−C−
+ GB−C−
++ GA−B−C−
+ GA+B−
GB−
+ GA−B−
GA−
+
T2.6
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+
+++
++
++
+
++
+
++
++
+
+
++
+
+
+
++
+
++
R1
Figure B.15: 2.82 < θ <
5π
4
GB+
GA+B+
GA+
+ GA+B−
++ GA+B+C+
+ GA+C+
++ GA+B−C+
+ GB+C+
GC+
+ GB−C+
++ GA−B+C+
+ GA−C+
++ GA−B−C+
+++ GO
++ GA+B+C−
+ GA+C−
++ GA+B−C−
+ GB+C−
GC−
+ GB−C−
++ GA−B+C−
+ GA−C−
++ GA−B−C−
+ GA−B+
GA−
+ GA−B−
GB−
+
T2.7
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+
+++
++
++
+
++
+
++
++
+
+
+
++
+
+
++
+
++
R1
Figure B.16:
5π
4
< θ < 5.03
164
Chapter B. Bifurcation Trees
GB+
GB+C+
GC+
+ GB−C+
++ GA+B+C+
+ GA+C+
++ GA+B−C+
+ GA+B+
GA+
+ GA+B−
++ GA+B+C−
+ GA+C−
++ GA+B−C−
+++ GO
++ GA−B+C+
+ GA−C+
++ GA−B−C+
+ GA−B+
GA−
+ GA−B−
++ GA−B+C−
+ GA−C−
++ GA−B−C−
+ GB+C−
GC−
+ GB−C−
GB−
+
T2.8
++
+
++
+
+
++
+
+
+
+
++
++
++
+
++
+
+++
++
++
+
++
+
++
++
+
+
+
++
+
+
++
+
++
R1
Figure B.17: 5.03 < θ < 5.96
B.3. Example 3
B.3
165
Example 3
The bifurcation diagrams for this example vary in terms of structure. The symmetrybreaking causes saddle-node bifurcations be present.
HA+
HA+B+
HB+
+ H3−++
H1−++
+ H2−++
++ HA−B+
+ H2−+−
H1−+−
+ H3−+−
++ H5+
+ HC+
++ H4+
+++ HO
++ H4−
+ HC−
++ H5−
+ H3+−+
H1+−+
+ H2+−+
++ HA+B−
+ H2+−−
H1+−−
+ H3+−−
HB−
+ HA−B−
HA−
+
T3.1
+
+
+
+
++
+
+
+
++
+
++
+
++
+++
++
++
++
+
++
+
+
+
++
+
+
+
+
R1
Figure B.18: −0.32 < θ <
π
4
166
Chapter B. Bifurcation Trees
HA+
HA+B+
HB+
+ H3−++
H1−++
+ H2−++
++ HA−B+
+ H2−+−
H1−+−
+ H3−+−
++ H5+
+ HC+
++ H4+
+++ HO
++ H4−
+ HC−
++ H5−
+ H3+−+
H1+−+
+ H2+−+
++ HA+B−
+ H2+−−
H1+−−
+ H3+−−
HB−
+ HA−B−
HA−
+
T3.2
+
+
+
++
+
+
+
+
++
+
+
++
++
+++
++
++
++
+
+
++
+
+
+
++
+
+
+
+
R1
Figure B.19: θ = π4 . At the “cusp” a pitchfork bifurcation is observed.
HB+
HA+B+
HA+
+ H3+−+
H1+−+
+ H2+−+
++ HA+B−
+ H2+−−
H1+−−
+ H3+−−
++ H5+
+ HC+
++ H4+
+++ HO
++ H4−
+ HC−
++ H5−
+ H3−++
H1−++
+ H2−++
++ HA−B+
+ H2−+−
H1−+−
+ H3−+−
HA−
+ HA−B−
HB−
+
T3.3
+
+
+
++
+
+
+
+
++
+
++
+
++
+++
++
++
++
+
++
+
+
+
+
++
+
+
+
R1
Figure B.20:
π
4
< θ < 1.89
B.3. Example 3
167
T3.4
+
+
+
+
++
++
+
+
+
++
++
+
++
+
+++
++
+
++
+
+
+
++
++
+
+
+
+
R1
HB+
+ HC+
++ H4+
+ HA+B+
HA+
+ H3+−+
H1+−+
+ H2+−+
++ HA+B−
+ H2+−−
H1+−−
+ H3+−−
++ H4−
+++ HO
++ H5+
+ H3−++
H1−++
+ H2−++
++ HA−B+
+ H2−+−
H1−+−
+ H3−+−
HA−
+ HA−B−
++ H5−
+ HC−
HB−
Figure B.21: 1.89 < θ < 2.73
+
T3.5
+
+
+
+ ++
++
+
+
++
++
+
++
+
++
+
+
+
++
+++
++
+
++
+
+
++
++
+
+
++
+
+
++
+
++
+
++
+
+
R1
+
Figure B.22: 2.73 < θ < 2.82
HB+
H3−++
H1−++
+ HC+
H1+−+
+ H3+−+
++ H4+
+ HA+B+
HA+
+ H2+−+
++ HA+B−
+ H2+−−
++ H4−
+++ HO
++ H5+
+ H2−++
++ HA−B+
+ H2−+−
HA−
+ HA−B−
++ H5−
+ H3−+−
H1−+−
+ HC−
H1+−−
+ H3+−−
HB−
+
168
Chapter B. Bifurcation Trees
+
+
T3.6
+
+
+
+
+
+
+
++
+
+
+
++
+
++
+
++
++
+
++
+++
++
+
+
+
+
++
+
++
+
+
+
R1
+
H3−++
H1−++
+ HC+
H1+−+
+ H3+−+
HB+
++ H4+
+ HA+B+
HA+
+ H2+−+
++ HA+B−
+ H2+−−
++ H4−
+++ HO
++ H5+
+ H2−++
++ HA−B+
+ H2−+−
HA−
+ HA−B−
++ H5−
HB−
+ H3−+−
H1−+−
+ HC−
H1+−−
+ H3+−−
Figure B.23: 2.82 < θ < 2.90
+
T3.7
+
+
+
+
+
+
+
++
+
+
++
+
++
+
++
++
+
++
+++
++
+
+
+
+
+
++
+
++
+
+
+
R1
Figure B.24: 2.90 < θ < 3.44
H3−++
H1−++
+ HC+
H1+−+
+ H3+−+
HB+
++ H4+
+ HA+B+
HA+
+ H2+−+
++ HA+B−
+ H2+−−
++ H4−
+++ HO
++ H5+
+ H2−++
++ HA−B+
+ H2−+−
HA−
+ HA−B−
++ H5−
HB−
+ H3−+−
H1−+−
+ HC−
H1+−−
+ H3+−−
B.3. Example 3
169
+
T3.8
+
+
++
+
++
+
++
+
+
++
+
+
+
++
+
++
++
+
++
+
+++
++
+
+
++
+
+
++
+
++
+
+
R1
Figure B.25: 3.44 < θ <
H3−++
H1−++
+ HC+
H1+−+
+ H3+−+
HB+
++ H4+
+ HA+B+
++ H4−
+ H2+−+
HA+
+ H2+−−
++ HA+B−
+++ HO
++ HA−B+
+ H2−++
HA−
+ H2−+−
++ H5+
+ HA−B−
++ H5−
HB−
+ H3−+−
H1−+−
+ HC−
H1+−−
+ H3+−−
5π
4
+
T3.9
+
+
++
+
++
+
+
++
+
+
+
+
++
+
++
++
+
++
+
++
+++
++
+
+
++
+
+
++
+
++
+
+
R1
Figure B.26:
5π
4
< θ < 4.42
H2+−+
H1+−+
+ HC+
H1−++
+ H2−++
HA+
++ H4+
+ HA+B+
++ H4−
+ H3−++
HB+
+ H3−+−
++ HA−B+
+++ HO
++ HA+B−
+ H3+−+
HB−
+ H3+−−
++ H5+
+ HA−B−
++ H5−
HA−
+ H1+−−
H2+−−
+ HC−
H2−+−
+ H1−+−
170
Chapter B. Bifurcation Trees
+
T3.10
+
+
+
+
+
+
+
++
+
+
+
++
+
++
+
++
++
+
++
+++
++
+
+
+
+
++
+
++
+
+
+
R1
H2+−+
H1+−+
+ HC+
H1−++
+ H2−++
HA+
++ H4+
+ HA+B+
HB+
+ H3−++
++ HA−B+
+ H3−+−
++ H4−
+++ HO
++ H5+
+ H3+−+
++ HA+B−
+ H3+−−
HB−
+ HA−B−
++ H5−
HA−
+ H2+−−
H1+−−
+ HC−
H1−+−
+ H2−+−
Figure B.27: 4.42 < θ < 4.95
+
+
T3.11
+
+
+
+
+
+
++
+
+
+
++
+
+
++
+
++
++
+
++
+++
++
+
+
+
+
++
+
++
+
+
+
R1
+
Figure B.28: 4.95 < θ < 5.04
H2+−+
H1+−+
+ HC+
H1−++
+ H2−++
HA+
++ H4+
+ HA+B+
HB+
+ H3−++
++ HA−B+
+ H3−+−
++ H4−
+++ HO
++ H5+
+ H3+−+
++ HA+B−
+ H3+−−
HB−
+ HA−B−
++ H5−
HA−
+ H2+−−
H1+−−
+ HC−
H1−+−
+ H2−+−
B.3. Example 3
171
+
T3.12
+
+
+
+ ++
++
+
+
++
++
+
++
+
++
+
+
+
++
+++
++
+
++
+
+
++
++
+
+
+
++
++
+
++
+
+
++
+
+
R1
+
HA+
H2+−+
H1+−+
+ HC+
H1−++
+ H2−++
++ H4+
+ HA+B+
HB+
+ H3−++
++ HA−B+
+ H3−+−
++ H4−
+++ HO
++ H5+
+ H3+−+
++ HA+B−
+ H3+−−
HB−
+ HA−B−
++ H5−
+ H2+−−
H1+−−
+ HC−
H1−+−
+ H2−+−
HA−
+
Figure B.29: 5.04 < θ < 5.12
T3.13
+
+
+
+
++
++
+
+
++
++
+
+
+
++
+++
++
++
+
+
+
+
++
++
+
+
+
R1
Figure B.30: 5.12 < θ < 5.96
+
HA+
+ HC+
++ H4+
+
HA+B+
HB+
+
H3−++
H1−++
+
H2−++
++ HA−B+
+
H2−+−
H1−+−
+
H3−+−
++ H4−
+++ HO
++ H5+
+
H3+−+
H1+−+
+
H2+−+
++ HA+B−
+
H2+−−
H1+−−
+
H3+−−
HB−
+
HA−B−
++ H5−
+
HC−
HA−
172
B.4
Chapter B. Bifurcation Trees
Example 4
Bifurcation diagrams for example 4. Due to the symmetry between hyperoctants, one
branch of each pitchfork bifurcation is not shown, as it’s progression can be inferred by
the symmetry. This eliminates unnecessary clutter. Like example 1, sub- and super-critical
tertiary bifurcations occur, exchanging stability between branches of the bifurcation tree.
For different values of θ, either one or two sets of tertiary bifurcations occur.
FB
4.01
FAB
+
+
FA
R1
+
FBC
+
FC
+
3+
++
FAC
3+
FABCD
++
FCD
+
FBD
3+
FAD
+
++
++
+
+
++
++
3+
3+
+
++
++
+
+
3+
++
3+
+
++
4+
Figure B.31:
FD
FO
5π
4
< θ < 4.31
4+
B.4. Example 4
173
FB
4.02
FAB
+
+
FA
R1
+
FBC
+
FC
+
3+
++
FAC
3+
FABCD
++
FCD
+
FAD
+
++
++
+
+
++
+
+
++
++
+
+
FD
3+
3+
3+
++
4+
FBD
3+
FO
4+
Figure B.32: 4.31 < θ < 4.82
FB
4.03
FBC
+
+
FC
R1
+
FAB
FA
+
3+
++
+
+
++
+
+
++
FABCD
++
FCD
+
FAD
+
FD
3+
3+
++
3+
3+
+
+
FAC
++
++
++
+
4+
FBD
3+
FO
4+
Figure B.33: 4.82 < θ < 4.83
FB
4.04
FBC
+
+
FC
R1
+
FAB
+
3+
++
+
+
++
+
+
++
3+
++
3+
FABCD
++
FAD
+
FCD
+
FD
3+
+
+
FAC
++
++
++
+
FA
3+
4+
Figure B.34: 4.83 < θ < 5.10
FBD
3+
FO
4+
174
Chapter B. Bifurcation Trees
FB
4.05
FBC
+
+
FC
R1
+
FAB
FA
+
3+
++
+
+
++
3+
3+
3+
++
+
+
3+
3+
++
+
++
FAC
3+
FABCD
++
FAD
+
FBD
3+
FCD
+
++
++
++
+
+
FD
4+
FO
4+
Figure B.35: 5.10 < θ < 5.28
FB
4.06
FBC
+
+
FC
R1
+
FAB
FA
+
++
++
++
3+
++
3+
+
+
++
3+
3+
++
3+
FAD
+
FABCD
++
FBD
3+
FCD
+
++
++
+
FAC
3+
+
++
+
+
3+
FD
4+
FO
4+
Figure B.36: 5.28 < θ < 5.30
FB
4.07
FBC
+
+
FC
R1
FCD
+
+
3+
++
++
++
3+
++
+
3+
+
++
+
+
++
3+
++
3+
FBD
3+
FABCD
++
FAD
+
FAC
3+
FAB
+
3+
++
++
+
FD
+
4+
Figure B.37: 5.30 < θ < 5.55
FA
FO
4+
B.4. Example 4
175
FB
4.08
FBC
+
+
FC
R1
FCD
+
+
FD
+
3+
++
FBD
3+
FABCD
++
FAD
+
FAB
+
3+
++
++
+
++
+
+
++
FA
++
+
+
3+
3+
3+
++
4+
FAC
3+
FO
4+
Figure B.38: 5.55 < θ < 5.70
FC
4.09
FBC
+
+
FB
R1
FCD
+
+
FD
+
3+
++
FBD
3+
FABCD
++
FAD
+
FAB
+
3+
++
++
+
++
+
+
++
FA
++
+
+
3+
3+
3+
++
4+
FAC
3+
FO
4+
Figure B.39: 5.70 < θ < 6.17
FC
4.10
FCD
+
+
FD
R1
FBC
+
+
3+
++
FBD
3+
FABCD
++
FAD
+
FAB
+
3+
++
++
+
++
+
+
++
+
FB
FA
++
+
+
3+
++
3+
3+
4+
Figure B.40: 6.17 < θ < 6.25
FAC
3+
FO
4+
176
Chapter B. Bifurcation Trees
FC
4.11
FCD
+
+
FD
R1
FBC
+
+
FB
+
3+
++
FBD
3+
FABCD
++
FAB
+
FAD
+
3+
++
++
+
++
+
+
++
FA
++
+
+
3+
3+
3+
++
4+
FAC
3+
FO
4+
Figure B.41: θ > 6.25 or θ < 0.46
FC
4.12
FCD
+
+
FD
R1
FBC
+
FB
+
3+
++
++
++
+
3+
3+
3+
+
++
+
+
++
3+
++
3+
FBD
3+
FABCD
++
FAB
+
FAC
3+
FAD
+
3+
++
++
+
+
FA
4+
Figure B.42: 0.46 < θ <
FO
π
4
4+
B.4. Example 4
177