Box and Low-order Models of the
Thermohaline Circulation
MSC. Thesis
Jian Zhang
University of Bremen
Referees: Prof. Dr. Dirk Olbers
Prof. Dr. Gerrit Lohmann
September, 2007
Contents
1 Introduction and Previous Work
1.1 The thermohaline circulation . . . . . . .
1.2 Brief review of box models . . . . . . . .
1.3 Basic equations and scaling . . . . . . .
1.4 Box models . . . . . . . . . . . . . . . .
1.4.1 Stommel’s 2-box Model . . . . . .
1.4.2 Welander’s 3-box Model . . . . .
1.5 Low-order models . . . . . . . . . . . . .
1.5.1 Equations for the low-order model
1.5.2 Stommel case . . . . . . . . . . .
1.5.3 Welander case . . . . . . . . . . .
1.6 Numerical tool . . . . . . . . . . . . . .
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1
1
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2 A coupled system
2.1 Coupled box models . . . . . . .
2.1.1 Configuration of the ocean
2.1.2 Coupled Welander box . .
2.2 Coupled low-order models . . . .
2.2.1 Equations . . . . . . . . .
2.2.2 Welander case . . . . . . .
2.3 Conclusions . . . . . . . . . . . .
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18
18
18
20
23
24
30
37
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basins
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A Mathematical preparation
46
A.1 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
A.3 Numerical tool: BIFURK and the continuation method . . . . . 51
B List of coefficients
54
1
Abstract
The thermohaline circulation (THC) is one of the most important features in
the circulation of the global ocean. Box models are widely used to interpret
and simulate this phenomenon, and a lot of them are based on very simple
configurations of ocean basins. The stability of the configurated system reflected in the bifurcation diagram in variable-parameter space attracts a lot of
interest.
A new method, a low-order model, will be firstly applied to this problem of a
single ocean system, and further to a coupled Atlantic-Pacific ocean system.
The previous work on this subject will be first reviewed in chapter 1, mainly
about the low-order model. The basic equations for this problem and the scaling are in section 1.3, this part follows an unfinished script of Dirk Olbers.
In section 1.4 Stommel’s 2-box model and Welander’s 3-box model are introduced, including the bifurcation diagrams (Dirk Olbers) and the conclusions.
The original equations for the low-order model (Dirk Olbers) are derived in
1.5, the bifurcation diagrams of this model corresponding to Stommel’s 2-box
model and Welander’s 3-box model are produced, the emphasis is put on the
Welander case and the streamfunctions for different stable states are generated. At the end of chapter 1, a very brief introduction of the numerical tool
’BIFURK.M’(Christoph Völker) used in this work is given, and more related
details are in appendix.
In chapter 2 both Welander box model and corresponding low-order model
are developed to apply to coupled Atlantic-Pacific Oceans. The Antarctic Circumpolar Current is indroduced in both models and is presented in terms of
a transport coefficient u. To compensate the salt transport between the two
oceans, new parameters are needed. For the coupled box model, it presents the
difference of precipitation between the equatorial boxes of the two oceans. For
the low-order model, it presents the total salinity differrence between the two
coupled oceans. In section 2.1 analytical solution to the coupled Welander’s
box model is derived and the possible ocean patterns are listed. In section
2.2 the equations for the coupled low-order model is derived at first, then applied to the specified case corresponding to the coupled Welander box model
in 2.2.2, bifurcation diagrams and streamfunction are plotted for results. At
the end, the low-order models are compared with the box model in 2.3.
Chapter 1
Introduction and Previous Work
In this chapter, the concept of the thermohaline circulation (THC) will be
introduced and the following questions will be discussed:
• What is the significance of the thermohaline circulation as part of the
climate system?
• How does it interact with the atmosphere?
• What is its relevance to humans?
• What has been done to explain it?
1.1
The thermohaline circulation
Approximately 70% of the surface of the earth is covered by ocean. The
oceans play a significant role in climate, e.g. through storing, transporting
and thus redistributing heat on a global scale. Besides heat, the circulation of
the ocean carries salt, but also nutrients around, which are essential for the
growth of phytoplankton, which makes it important for planktonic ecosystems
and global elemental fluxes. The exploitation of oceanic resources as well as
the development of sea transport directly influence human society. For those
reasons since the earliest exploration through the first fishermen, investigation
of the mysteries of the ocean has never ended.
Why does the water of the oceans always move, what drives the currents?
Those might be the first questions that arise in our mind. The motion of
the ocean is driven by different mechanisms. At the surface, wind forces the
water to move. Together with the effect of the planetary rotation, this leads
to horizontal currents within a shallow layer near the ocean surface, the socalled Ekman layer. The motion within the Ekman layer establishes pressure
gradients in the interior of the ocean. In geostrophic balance these pressure
1
gradients are balanced by the horizontal component of the Coriolis force. The
circulation that is thus established is called wind-driven circulation, although
it is in fact rather indirectly driven by the wind. A second component of the
oceanic circulation called global thermohaline circulation (THC) is driven by
fluxes of heat and freshwater as indicated by its name. In the high latitudes
dense ocean water forms because of surface cooling as well as the formation of
sea ice. The water sinks to the deep, forming water mass in the Norwegian sea
and drives the circulation consquently. The sinking is strongly preferred in the
northern hemisphere because of the greater continental area there combined
with the slight latitudal asymmetric freshwater flux (Henk. A Dijkstra, 1999).
The thermohaline circulation has interested oceanographers for decades because it is not completely clear whether it will change in the future concerning
about the global warming.
In the zonal average it describes an overturning of the ocean with meridional
transport of water at the surface and depth and downwelling of water in high
latitudes. The concept of thermohaline circulation goes back to Albert Defant
in the 1920s after experiments performed by Johan Sandstroem. He showed
the possibility of thermal forcing of an ocean current and concluded that only
when the vertical mixing of the ocean is strong enough, the current could be
driven by a temperature gradient. The THC is often described as Great Ocean
Conveyor Belt (Kuhl Brodt, 2004) shown in Fig. 1.1. The conveyor begins
with the formation of cool and salty, therefore dense water in high latitudes,
predominantly in the Norwegian Sea, around Greenland and in the Weddell
Sea. After sinking into the deep this water spreads and fills the deep basins
of the world oceans as North Atlantic Deep Water (NADW) and Antarctic
Bottom Water (AABW). In the Indian Ocean, some of the cold salty water
upwells and returns to North Atlantic as surface flow. The rest of the water
continues spilling to the Pacific Ocean where it slowly becomes warmer and
fresher, floats up and moves back to form a great loop between the Atlantic
and Pacific oceans.
1.2
Brief review of box models
Stommel first tried to interpret the THC in 1961 with his original 2-box model,
in which he considered a hemispheric ocean consisting of two homogeneously
mixed boxes for the high-latitude and low-latitude ocean, connected by an
exchange of water between them that is proportional to the density difference between the water in the boxes. The circulation is driven by heat fluxes
and freshwater input in equatorial box and ouput in polar box. Welander’s
3-box model expanded the configuration of ocean basins to allow for interhemispheric circulation with two polar boxes and a equatorial box. In his
2
Figure 1.1: The global THC [Kuhl Brodt, modified after Broecker,1991]
model the sinking region is not determined a priori; the global circulation
could be two symmetric loops in each hemisphere or a great loop connecting the northern and southern ocean, depending on the freshwater input. A
similar model was constructed by Rooth (1982) with the equatorial box divided into two sub-boxes representing the upper and deep parts of the ocean
(Fig. 1.2). This configuration was later used by Rahmstorf (1995) where the
equilibrium strength of the THC was found to be governed uniquely by the
freshwater forcing in the Southern Atlantic. Scott’s model was a further version
of Rooth’s model, compressing the deep equatorial box to a bottom passage
directly connecting the two oceans (Fig. 1.2). While these models all consider a single ocean, other researchers have proposed models that consider a
coupled system of the Atlantic and Pacific ocean basins, connected by the
Antarctic Circumpolar Current (ACC). A typical configuration for this kind
of model is by considering two identical ocean basins and connecting them in
the south . Besides the research on modeling the THC, interpreting the impact of surface buoyancy fluxes on the ocean currents, other studies focus on
ocean-atmosphere interaction and the influence of sudden changes of the THC
on climate (Xiaoli Wang, 1998; Valerio Lucarini, 2005, etc.).
1.3
Basic equations and scaling
Our study of the THC begins with 2-dimensional models applied to an idealized
Atlantic Ocean that is closed by a southern coast and without momentum or
3
Figure 1.2: Rooth’s model and Scott’s model
4
salt transport over lateral boundaries. By averaging all the scalar fields zonally
and integrating the velocities, the dimensionality of the system is reduced and
the equations of motion are:
1
vt = − pφ + Av vzz
a
pz = −gρ
1
(v cos φ)φ + wz = 0
a cos φ
1
Kh
Tt + vTφ + wTz = Kv Tz z + 2
(Tφ cos φ)φ + qT
a
a cos φ
1
Kh
St + vSφ + wSz = Kv Sz z + 2
(Sφ cos φ)φ + qS
a
a cos φ
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
where small subscripts t, z, φ denote partial derivatives, v, h denote vertical and
horizontal direction. T, S, ρ are temperature, salinity and density respectively.
∆p is the pressure over zonal width ∆λ, a is the radius of the Earth, φ is
latitude, K and A are diffusive coefficients. qT and qS are the lateral transport
and are assumed to be 0 in the single ocean model. The zonal momentum
balance can not be considered correctly in a zonally averaged model, as we
have no means to infer the pressure difference across the basin. To close the
system of equations we therefore follow Marotzke (1998) who assumed that the
meridional pressure gradient is balanced by a vertical diffusion of meridional
momentum instead of the correct zonal integral of the meridional Coriolis force.
According to Eq. 1.3, one can define a meridional streamfunction Λ(z, φ):
1 ∂Λ
cos φ ∂z
1 ∂Λ
w=
a cos φ ∂φ
v=−
(1.6)
(1.7)
(1.8)
Now we can write the equations for the 2-dimensional system including salinity,
temperature and overturning:
g
Λzzt − Av Λzzzz = − cos φ(αT − βS)φ
a
1
Kh
χt +
(Λφ χz − Λz χφ ) = Kv χzz + 2
(χφ cos φ)φ
a cos φ
a cos φ
(1.9)
(1.10)
where the linearized state function ρ = −αT + βS is used and χ = (T, S). We
further transform the coordinates with:
z
ζ=
µ = sin φ
H
5
Table 1.1: List of parameters for global oceanic overturning.
where H is the total depth of the ocean. In these (ζ, µ)-coordinates the system
is scaled as in previous work (e. g. Dijkstra and Molemaker 1997) with ψ =
Λa/(HKh ) and τ = tKh /a2 for streamfunction and time. We additionally
redefine the salinity variable to include the buoyancy ratio b = αδT /βδS, then
we arrive at:
P r−1 ψζζτ − ψζζζζ = Ra(1 − µ2 )(T − S)µ
χτ + ψµ χζ − ψζ χµ = [(1 − µ2 )χµ ]µ + κχζζ
(1.11)
(1.12)
The corresponding boundary conditions are
ψ=0
at all boundaries ψζζ = 0 at ζ = −1, 0
2
(1 − µ )χµ = 0
at µ = ±1
χζ = 0
ζ = −1
κTζ = θ(T surf − T )
κSζ = F surf
at
ζ=0
The notations and numbers above are listed in Table. 1.1 following Dirk Olbers work. In this study the surface conditions will be chosen from the Atlantic Ocean. The surface temperature T surf (Fig. 1.3) and the salt flux F surf
(Fig. 1.3) will be presented by the first few Legendre polynomials.
T surf (µ) =
2
X
Tn Pn (µ); F surf (µ) =
n=0
2
X
Fn Pn (µ)
(1.13)
n=0
The net salt flux F0 must be zero for salt conservation in a single ocean and
the constant T0 is dynamically irrelevant. As P2 has a central minimum the
coefficient T2 must be negative to represent a warm equatorial and cool polar
region. With our scaling, T2 ≈ −1 corresponds to a 28 degrees drop (the range
of P2 is 2.3, δT = 12) in temperature between equator and the poles. For S2 ,
if we take it negative, salt is assumed to be added in the equatorial region
6
Figure 1.3: Left:Three estimates of P-E for the Atlantic Ocean. Right: surface
temperature for the Atlantic Ocean (from Levitus, 1994).
and removed in high latitudes (i.e. an excess evaporation in low latitudes and
an excess precipitation in high latitudes) and the central maximum shown
in Fig. 1.3 is ignored. This case has also been implemented in the idealized
models of Marotzke (1988) and Dikjstra and Molemaker (1997). With our
scaling (see Table. 1.1), the range of F surf is 7 which implies an amplitude of
S2 = F2 /κ = 7/2.3/0.25 = 12.
1.4
Box models
Box models consider the balance of properties such as heat or salinity in a small
number of boxes which are assumed to be homogeneously mixed in terms of external fluxes over the box boundaries and transport between boxes. The state
variables used in these models are the box mean values of the property fields,
assuming a fast mixing process within the boxes, both vertically and horizontally. This effective mixing, however, is assumed to vanish at the boundaries
between two adjacent boxes to avoid a totally homogeneous system. The discrepancy can be reduced by dividing the entire ocean domain into more boxes
or using more complex dynamics within a box. The latter can be achieved by
low-order models where the strong mixing is relaxed and the property field is
expressed by the mean value restrained by the boundary conditions.
Stommel’s 2-box model (1961) and Welander’s 3-box model (1986) will be introduced in this section. Basic results will accompany each model as well as
some conclusions.
7
Figure 1.4: Stommel’s 2-box model. T1 , T2 , S1 , S1 are mean values of properties;
F denotes the freshwater input into the box while −F the output. The clockwise
direction of circulation in the figure corresponds to a positive q
1.4.1
Stommel’s 2-box Model
Here we analyze Stommel’s box model (1961) consisting of a polar box and an
equatorial box as shown in Fig. 1.4. The governing equation (Dirk Olbers) for
this model is
d∆S
= 2F − 2|q|∆S − 2K∆S
dt
q = C(∆T − ∆S)
(1.14)
(1.15)
where C is a positive constant depending on the friction coefficients of the
system that is of order 10, ∆S = S1 − S2 , ∆T = T1 − T2 are the salinity and
temperature differences between the boxes, K is a diffusion determined by the
width of boxes. The forcing is an asymmetric
salinity (or freshwater) flux
q
surf
F
= αF P1 (µ), α is obtained as 2 2/3 when the integral salt flux in the
surf
boxes
q is ±Fq. As the range of F q is 7 (see Table. 1.1), the amplitude of F is
7/2 3/2/2 (2/3) = 1.75, since 2 (2/3) is the range of P1 . The temperature
difference between low latitudes and high latitudes is assumed to be constant
at first for simplicity, in order to study the change of ∆S and q induced only by
the salt flux F . The only relevant variable is the salinity difference in this case.
The analytical steady-state solution of Stommel’s model equations involving
both temperature and salinity as variables is worked out in G. Lohmann and
J. Schneider’s work (1999) in which the parameter sensitivity, dynamics, and
error growth of this model are studied . The steady-state solutions are classified
according to the sign of q: When q > 0, the circulation is driven mainly by
thermal contrast. This state is called TH state in some publications. In this
case
√
1
∆S = ∆T + [1 ∓ 1 − 4F + ]
2
8
(1.16)
ΔS
Δ T = −1
ΔT=1
1.5
2
0.5
1
1.5
0
0.5
1
−0.5
0
0.5
−1
−0.5
0
−1.5
−1
−0.5
−2
−20
q
ΔT=0
1
−10
0
10
20
−1.5
−20
−10
0
10
20
−1
−20
10
15
20
5
10
15
0
5
10
−5
0
5
−10
−5
0
−15
−10
−20
−20
−15
−20
−10
0
F
10
20
−10
0
10
20
−10
0
F
10
20
−5
−10
0
F
10
20
−10
−20
Figure 1.5: Bifurcation of the Stommel case for C = 10, K = 1 and various
∆T . Blue:thermal, black:haline, dashed:unstable. ∆S is the salinity difference
between the two boxes and q the transport variable.
When q < 0, the haline contrast is dominant in driving the current, so this
case is called SA state. In this case we have
√
1
∆S = ∆T − [1 ± 1 + 4F − ]
2
(1.17)
In the above equations, F ± = F/(∆T ± )2 , with ∆T ± = ∆T ± K/C. There
exist stable and unstable steady solutions. Fig. 1.5 shows the resulting q for
C = 10, and ∆T = −1, 0, 1 respectively, indicating also the stability of the
solution branches.
Looking into the right two panels where ∆T = 1, q is linearly related with
∆S as we have the hydraulic relation Eq. 1.15. There are 3 different regimes in
each panel corresponding to currents driven by different mechanisms. When
−20 < F < 0, from the bifurcation diagram of ∆, the ocean water in the
polar box is saltier than that in the equatorial box. However the temperature
difference is dominant (q > 0) in driving the current. This is aTH state :
the ocean surface water is heated at the equatorial region, flows toward high
latitudes, causing upwelling at the equator. At the pole the water is cooled
and sinks to the bottom. When 2.5 < F < 20, the direction of the circulation
is anti-clockwise and the current is driven predominantly by haline contrast.
When 0 < F < 2.5, the system has multiple equilibria. Thermal and haline
9
Figure 1.6: Welander’s 3-box model
contrast compete with each other, both TH and SA states can be stable in
this regime.
1.4.2
Welander’s 3-box Model
For Welander’s 3-box model the Atlantic is considered as a symmatric basin
around the equator, which is closer to the real ocean. In Fig. 1.6 box 1 and 3 are
polar boxes and box 2 stands for the equatorial box. As in the Stommel case
we limit our analysis there to the case where the temperatures for the boxes
are prescribed, and a symmetric freshwater flux is applied. The governing
equations for the system are then:
DS˙1
DS˙3
DS1 + 2(1 − D)S2 + DS3
q1
q3
=
=
=
=
=
−F + |q1 |(S2 − S1 )
−F + |q3 |(S2 − S3 )
2S0 = const
−C[(T1 − T2 ) − (S1 − S2 )]
−C[(T3 − T2 ) − (S3 − S2 )]
(1.18)
(1.19)
(1.20)
(1.21)
(1.22)
where D is the width of the polar box and 2S0 is the total amount of salt in the
Atlantic. The signs of q1 and q3 and therefore the direction of the currents are
indicated in figure 5. Solutions for the steady states are more complicated and
can be classified into 9 cases according to the sign of q1 and q3 . Fig. 1.7 shows
an example for ∆T = 2(∆T = T2 − T1 = T2 − T3 ), S0 = 2, F = 5, C = 10.
From the lowest right panel we can find out that when F varies between 0
and 10 in this condition, the symmetric circulations in each hemisphere break
into one large loop. Welander’s model thus allows for both one-cell and two-cell
overturning. There exists therefore the possibility of an asymmetric pattern
of circulation even for a symmetric forcing.
10
2
salinity σ3
salinity σ1
16
1
0
−20
−10
0
10
forcing F
flux q1
flux q3
−10
0
10
forcing F
0
10
forcing F
20
−10
0
10
forcing F
20
−10
0
10
forcing F
20
0
−1
−20
20
1
flux q1 − q3
1
flux q1 + q3
−10
1
0
0
−1
−2
−20
1
0
−20
20
1
−1
−20
2
−10
0
10
forcing F
0
−1
−20
20
Figure 1.7: Results for Welander 3-box model, where σi = Si /S0
1.5
Low-order models
In the low-order method, the solution to a particular problem is represented
by a function series, in most cases polynomials or trigonometric functions.
Orthogonality of the chosen base functions is used to arrive at ordinary differential equations for the time-dependent coefficients of the base functions.
Finally, the dimensionality of the problem is reduced by restricting the expansion to a very limited number of the base functions. In this paper Legendre
polynomials Pn were chosen to expand the latitude dependency of the solution.
The depth-dependency of the solution components were solved as eigenfunction
problems to meet the boundary conditions at the surface and bottom of the
ocean. In the low-order model, only the first three meridional modes P0 , P1 , P2
are used. This truncation allows for a constant value (which is dynamically
irrelevant for the streamfunction) and one-cell and two-cell circulations. For
the vertical, the first two eigenfunctions with the lowest eigenvalues (only for
salnity variation because S00 refers to a spatially constant field) are used. The
surface forcing terms are also restricted to the first two meridional base functions, T surf = Ta P1 + Ts P2 , Fsurf /κ = Sa P1 + Ss P2 . Here the index a, s means
asymmetric and symmetric, respectively. Prognostic equations were finally obtained by projection. In this study only the steady states of the system are
11
investigated, the time evolution is not considered. The continuation method
(Khibnik et al., 1993, see also Appendix) is used to numerically follow the
steady states with varying parameter values, using a matlab program ’BIFURK’ (Christoph Voelker, 2002).
1.5.1
Equations for the low-order model
The derivativation of the equations for the low-order model will be shown here,
the reader who is more interested in the application may skip this section. The
result applied to different cases will be shown in next subsection.
Starting with the linear stationary case, we noticed that the Legendre polynomials are the solutions for the meridional component of Eq. 1.24 if we perform
a coordinate separation on it.
−ψζζζζ = Ra(1 − µ2 )(T − S)µ
[(1 − µ2 )χµ ]µ + κχζζ = = 0
(1.23)
(1.24)
For salinity, the solution has the form
S(µ, ζ) =
X
Sn Pn (µ)hn (ζ)
(1.25)
n
where S0 must be 0 as discussed before for the conservation of salt and the
vertical function hn (ζ) (n > 0) should meet the boundary conditions hnζ = 1
at the top and hnζ = 0 at the bottom. The solution for n > 0 is thus
hn (ζ) =
cosh ξn (ζ + 1)
ξn sinh ξn
(1.26)
where ξn2 = νn2 /κ and νn2 = n(n + 1). Similar solutions can be obtained for the
temperature with different vertical functions
gn (ζ) =
cosh ξn (ζ + 1)
cosh ξn
(1.27)
The solution for streamfunction is derived from Eq. 1.24. Expressing ψ in the
form
X
(1.28)
ψ(µ, ζ) = Ra(1 − µ2 )
ψn Pnµ (µ)fn (ζ)
n
with
Tn ξn − Sn coth ξn
ξn5
cosh ξn (ζ + 1)
fn (ζ) =
+ α0 + α1 ζ + α2 ζ 2 + αζ3
cosh ξn
ψn = −
12
(1.29)
(1.30)
Table 1.2: The vertical structure functions
Eq. 1.30 is the solution for the vertical function of streamfunction which satisfies the boundary conditions ψ = ψζζ = 0 at top and bottom. The α coefficients
are listed in appendix.
For time dependent problem we add to the particular solution of the inhomogeneous problem Eq. 1.25 a general solution of the homogeneous problem, and
we have
X
X
(1.31)
S(µ, ζ, τ ) =
[Sn hn (ζ) +
Snq (τ )Hq (ζ)]Pn (mu)
n
q
the vertical eigenfunctions Hq (with eigenvalues ηq ) have zero derivative at
top and bottom to ensure that the previous boundary conditions still hold.
Projection yields
Ṡnq = −(νn2 + κηq2 )Snq
(1.32)
And for temperature and streamfunction
Ṫnq = −(νn2 + κγq2 )Tnq
1 X
P r−1 ψ̇nq + λ2q ψnq = − 2
(Tnp {Gp Fq } − Snp {Hp Fq })
λa p
(1.33)
(1.34)
where the curly brackets abbreviate the ζ-integration. The vertical eigenfunctions G, H and F and their eigenvalues are listed in Table. 1.2. They are to
be taken normalized. For the complete nonlinear system, Eq. 1.32 is found as
Ṡnq = −(νn2 + κηq2 )Snq − < J(ψ, S)Pn Hq >
(1.35)
where J denote the (µ, ζ)-Jacobian and integration over µ and ζ is abbreviated
by the cornered brackets. The final solution for this model result a 8-variable
complex system which is
U̇ = −k(M + U )
V̇ = −k(N + V )
Ẋ = −(U − cPa )Y + (3V − dPs )Z
+(aaX U + baX Pa )Ta + (asX V + bsX Ps )Ts − rκ0 X
4
Ẏ = (U − cPa )X − √ (U − cPa )Z
5
13
(1.36)
(1.37)
(1.38)
(1.39)
Ż =
L̇ =
Ṁ =
Ṅ =
+(asY U + bsY Pa )Ys + (aaY V + baY Ps )Ta − r(2 + κ0 )Y
4
√ (U − cPa )Y + (3V − dPs )L
5
+(aaZ U + baZ Pa )Ta + (asZ V + bsZ Ps )Ts − r(6 + κ0 )Z
√
√
2 2(U − cPa )M + 2 2(3V − dPs )N
+(aaL U + baL Pa )Sa + (asL V + bsL Ps )Ss − 4rκ0 L
√
−2 2(U − cPa )L
+(asM U + bsM Pa )Ss + (aaM V + baM Ps )Sa − 2rκ0 M
√
−2 2(3V − dPs )L
+(aaN U + baN Pa )Sa + (asN V + bsN Ps )Ss − 6rκ0 N
(1.40)
(1.41)
(1.42)
(1.43)
The dynamical variables are defined as
U = π 5 ψ10
V = π 5 ψ20
X = 8T
√ 00 /3 Y = 8T
√10 /3 Z = 8T
√20 /3
L = 2 2S01 M = 2 2S10 N = 2 2S20
X, Y, Z are the thermal and L, M, N are the haline variables and U represents
the great overturning and V the hemispheric overturning (all the variables are
only the nonlinear part of the solution). Further
4
(Ta ξ1 − Sa coth ξ1 )
Ta = T1 Sa = S1 Pa = 2π 4 ψ1 = − 2π
ξ5
1
4
Ts = T2 Ss = S2 Ps = 2π 4 ψ2 = − 2π
(Ts ξ2 − Ss coth ξ2 )
ξ5
2
The complicated coupling coefficients
are listed in the appendix, κ0 =
π 2 κ/4, the time derivative indicated by the dot is now with respect to τ 0 = τ /r
with the redefined inverse Rayleigh number r = 2π 4 /Ra. The derivation of
the equations in this section is following an unfinished script of Dirk Olbers.
aaX ...c, d
1.5.2
Stommel case
As was discussed in section 1.4.1, the Stommel case only allows for an asymmetric circulation, symmetric modes thus disappear here and we have 3 variables
and heat/salt fluxes as parameters. The temperature forcing is first spinned
up to a value which indicates the temperature difference between two boxes.
Equations for the model are:
U̇ = −k(M + U )
√
L̇ = 2 2(U − cPa )M + (aaL U + baL Pa )Sa − 4rk 0 L
√
Ṁ = −2 2(U − cPa )L − 2rM
(1.44)
(1.45)
(1.46)
where U stands for the one-cell overturning, L, M are haline variables, Pa is the
asymmetric forcing term consisting of Ta and Sa , all the other are coefficients.
14
0.5
0
−0.5
4
2
−1
−1
−0.5
Ta
0
0
1
0
−1
U
1
6
U
8
U
U
1
0.5
0
0
5
10
Sa
15
−4
−1
−1
20
0
−2
−3
−0.5
−0.5
Ta
0
−5
−20
2
−15
−10
Sa
−5
0
−15
−10
S
−5
0
6
0
−2
−2
M
L
−4
−4
−6
−6
−8
−10
4
−2
−4
M
L
0
0
5
10
S
a
15
20
−8
2
−6
0
5
10
S
15
−8
−20
20
a
−15
−10
S
a
−5
0
0
−20
a
Figure 1.8: Bifurcations of the Stommel case. Notice that M = −U .
The bifurcation diagram is shown in Fig. 1.8. Similar with that of Stommel’s
box model, multiple equilibrium regimes can be observed.
1.5.3
Welander case
For the Welander case, both symmetric and asymmetric terms are allowed, but
the forcing set for the model is symmetric, Ta = 0, Sa = 0, so the equations
are:
U̇
V̇
L̇
Ṁ
Ṅ
=
=
=
=
=
−k(M + U )
−k(N + V )
√
√
2 2U M + 2 2(3V − dPs )N + (asL V + bsL Ps )Ss − 4rκ0 L
√
−2 2U L + asM U Ss − 2rM
√
−2 2(3V − dPs )L + (asN V + bsN Ps )Ss − 6rN
(1.47)
(1.48)
(1.49)
(1.50)
(1.51)
where V is the amplitude of the two-cell overturning, M corresponds to the
asymmetric variable S1 − S3 and N corresponds to the symmetric variable
S1 + S3 . The result for this case is shown in Fig. 1.9. For steady states
the left-hand sides of the equations vanish and we can see from the first two
equations above that M = −U, N = −V . In the upper right panel, starting
with Ss = 0, the first bifurcation point is detected at Ss = 0.53. The thermally dominated solution, a two-cell symmetric structure with upwelling in
equatorial region, loses stability at this pitchfork bifurcation (see appendix).
Two mirror-image solution branches appear that have asymmetric circulation
patterns. The upper branch presents SPP (southern sinking pole to pole circulation) states and the lower branch presents NPP (northern sinking pole to
pole circulation) states. Depending on the initial conditions, a time-dependent
solution will be attracted to one of the NPP or SPP steady states, except for
inital coditions directly at the unstable steady state. As the salt flux increases,
15
1
0.5
U
L
1
0.5
0
−0.5
0
−0.5
−1
−1.5
−1
−0.5
−1
−25
0
−20
−15
Ts
−10
−5
0
−10
−5
0
−10
−5
0
Ss
0.5
2
0
−2
L
V
0
−0.5
−1
−4
−1.5
−2
−25
−6
−20
−15
−10
−5
−8
−25
0
−20
−15
S
S
s
s
1
2
1.5
0.5
N
M
1
0
0.5
−0.5
−1
−25
0
−20
−15
−10
−5
−0.5
−25
0
Ss
−20
−15
Ss
Figure 1.9: Bifurcations for the Welander case. Upper left panel shows thermal
spinup, the others show the continuation with haline forcing. Notice that
M = −U, N = −V .
the circulation of the ocean finally breaks into one great overturning cell. The
streamfunctions are shown in Fig. 1.10. Between the Hopf bifurcation points,
−6.2 < Ss < −1.4, there is no stable steady state.
1.6
Numerical tool
The continuation method (T. Parker, et al.,1989; A. Khibnik, et al., 1993)is
used to calculate the steady states. BIFURK written by Christoph Voelker
follows the state branch and tries to find the bifurcation points when varying
parameter. It also notices the location where the bifurcation occurs and can
classify saddle-node, pitchfork, transcritical and Hopf bifurcations. To use
this tool two matlab files are required: one provides the time derivatives, U̇ for
instance, of the the aimed variables in the system, the Jacobian matrix, and
the derivaives of U̇ with respect to the external parameter; the other provides
the initial condition of the system which must be a solution for steady state.
Details of this code are introduced in the appendix.
16
total ψ
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9
−1
−1
−0.5
0
−10
−5
0.5
0
1
5
10
total ψ
total ψ
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
−0.6
−0.6
−0.7
−0.7
−0.8
−0.8
−0.9
−1
−1
−0.9
−0.8
−0.6
−10
−9
−0.4
−8
−0.2
−7
0
−6
−5
0.2
0.4
−4
−3
0.6
−2
0.8
−1
−1
−1
1
0
−0.8
−0.6
−2.5
−0.4
−0.2
−2
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
−0.6
−0.6
−0.7
−0.7
−0.8
−0.8
−0.9
−0.9
−0.6
0
−0.4
2
−0.2
0
4
0.4
0.6
−1
0.8
1
0.8
1
−0.5
total ψ
0
−0.8
0.2
−1.5
total ψ
0
−1
−1
0
0.2
0.4
6
0.6
8
0.8
−1
−1
1
10
−0.8
−0.6
0.5
−0.4
−0.2
1
0
0.2
1.5
0.4
0.6
2
Figure 1.10: Streamfuctions for different stable states ploted in latitude (x axis) and
depth (y axis). Top: TH state, upwelling in equatorial region (Ss = −0.23). Middle:
SPP states, the bigger loop in the left panel circulates anti-clockwise (Ss = −1.19),
as the salt flux increases (Ss = −6.54) the big overning appears as shown in the
right panel, where circulation is anti-clockwise. Bottom: NPP states, the bigger
loop in the left panel circulates clockwise (Ss = −1.19), as the salt flux increases
(Ss = −6.54) the big overning appears as shown in the right panel, where circulation
is clockwise.
17
Chapter 2
A coupled system
The models mentioned in last chapter are obviously not enough to handle
a global ocean because the lateral exchange between ocean basins qT , qS are
nonzero for real oceans. The Antarctic Circumpolar Current (ACC) connects
the Atlantic, Indian and Pacific oceans (Fig. 2.1) in the southern hemisphere,
transporting significant amounts of salt and heat between them. This current is
the basic feature for the coupling in my study and a parameter u is introduced
to indicate the inter-ocean transport coefficient. Besides that, the Bering Strait
which connects the Atlantic and the Pacific is also a possible factor influencing
the THC (Hasumi, H., 2002). Although it might be important, I will not
further considerit in this paper.
2.1
Coupled box models
In this section analytical solutions for a coupled Welander model will be shown
and the circulation patterns will be classified as well.
2.1.1
Configuration of the ocean basins
The basic idea of extending a Welander-type model into a model for the global
ocean is to double the number of boxes into corresponding boxes for the Atlantic and Indo-Pacific ocean, and to connect them in the southern hemisphere.
The coupling if the basins is done by introducing a transport coefficient u which
basically arises from the Antarctic Circumpolar Current. A transport coefficient u which basically arises from the Antarctic Circumpolar Current. The
passage where the ocean water flows through could be the Drake Passage. Heat
and freshwater fluxes applied as surface forcing should be considered carefully.
The ocean basins are configurated as shown in left panel of Fig. 2.2. The flow
directions correspond to positive q. The fact that the Atlantic Ocean has a
higher salinity than the Pacific Ocean (right panel in Fig. 2.2) is because that
18
Figure 2.1: Antarctic Circumpolar Current connecting Atlantic, Indian and
Pacific Oceans
19
Figure 2.2: Left: Configuration of coupled Welander box model. Right: Global
ocean salinity distribution
the water vapor transported from Atlantic to Pacific by the trade winds does
not return back to the Atlantic Ocean through precipitation. The freshening
of the two oceans is thus not identical. This property should be represented by
our configuration. In Fig. 2.2, four polar boxes have the same surface forcing
condition, two equatorial boxes have flux into the oceans. A flux difference dF
is used for the middle boxes to balance the transport between the two oceans.
In consequence the total amount of salt is not conserved for an individual
ocean but for the whole system.
2.1.2
Coupled Welander box
For a two-basin ocean model, the total amount of salt in the ocean is conserved.
In addition,for in steady state the salinity change for an individual ocean box
should be 0. Consider the 4 polar boxes and the Atlantic equatorial box and
the salt conservation:
DṠ1A
DṠ3A
DṠ1P
DṠ3P
2(1 − D)Ṡ2A
S0
=
=
=
=
=
=
−F + |q1A |(S2A − S1A )
−F + |q3A |(S2A − S3A ) + u(S3P − S3A )
−F + |q1P |(S2P − S1P )
−F + |q3P |(S2P − S3P ) − u(S3P − S3A )
2F + dF − |q1A |(S2A − S1A ) − |q3A |(S2A − S3A )
D(S1A + S3A + S1P + S3P ) + 2(1 − D)(S2A + S2P )
20
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
With q1 = −C(S2 − S1 − ∆T ), q3 = −C(S2 − S3 − ∆T ), and ∆T = T2 −
T1 = T2 − T3 . When qi > 0 the respective ocean circulation is driven by the
temperature difference between two adjacent boxes. The conservation of salt
in each ocean in steady state requires the sum of time derivatives of salinity
differences to vanish. Adding up Eq. 2.2, Eq. 2.3 and Eq. 2.6, the following
relation between dF and u is obtained.
−dF = u(S3P − S3A )
(2.7)
Alternatively, this can also derived by simply summing the flux terms and
avective terms in the Atlantic ocean. According to Eq. 2.7, neither dF nor u
can be 0 unless a the solution is totally symmetric (meaning the salinity of the
two oceans are identical) between the oceans with requires dF = 0 . Solving
for the steady state, I get
S2A − S1A = a1,2,3
S2A − S3A = b1,2,3
S2P − S1P = a1,2,3
S2P − S3P = c1,2,3
(2.8)
(2.9)
(2.10)
(2.11)
where
q
q
1
1
(∆T + ∆T 2 + 4F/C), (∆T ± ∆T 2 − 4F/C)
2
2
q
q
1
1
[∆T + ∆T 2 + 4(F + dF )/C], [∆T ± ∆T 2 − 4(F + dF )/C]
=
2
2
q
q
1
1
=
[∆T + ∆T 2 + 4(F − dF )/C], [∆T ± ∆T 2 − 4(F − dF )/C]
2
2
a1,2,3 =
b1,2,3
c1,2,3
By now the inter-ocean circulation can be determined because the flow is
dependent only on the salinity difference when temperature is prescribed. The
circulation patterns are classified into 16 types according to the signs of the
qi which tell the direction of flow. All patterns are listed in Table. 2.1.
Some of these patterns include a global conveyor belt circulation (see below).
The pattern in the ith row and jth column will be mentioned as (i, j) for
convenience.
These 16 patterns can be sorted into 4 groups.
1-cell overturning The great conveyor belt (4,3) is found when q1A > 0, q3A <
0; q1P < 0, q3P > 0. In the north Atlantic the current is predominantly driven by
thermal contrast, cold water sinks to the deep part of the ocean, flows southward, and enters the Pacific basin. Another circulation type in this group is
an ’anti-conveyor belt’ (3,4), whose flow direction is opposite to the actual
circulation, with cold bottom water forming in the north Pacific Ocean.
21
Table 2.1: The 16 patterns of circulation for the coupled Welander box model.
The positions of the Atlantic and Pacific oceans are according to the basin
configuration in Fig.2.2.
22
2-cell overturning The first type (including (1,3), (3,1), (2,4), (4,2)) in this
group has a large inter-ocean circulation and a small independent circulation
in rest of the ocean. For the other type ((3,3) and (4,4)) there is no connection
between the great loops in each ocean, bottom water is still forming in high
latitudes. If u gets larger, meaning a stronger inter-ocean transport, these two
cells might break into one. That is because of the value of dF which influences
the signs of qi is related to u.
3-cell overturning When u or dF is large, the southern oceans circulate
as an entity in a cell and the northern oceans separately as there is no interocean exchange allowed for the northern part of the ocean basins. This is the
condition for patterns (1,2) and (2,1).
4-cell overturning All 4 hemi-oceans circulate individually.
The six saline variables can be also sorted out through Eq. 2.9-Eq. 2.11,
together with Eq. 2.6 and Eq. 2.7. Solving the linear problem AS = B, where
S is a vector comprising all variables, B = [ai bj am cn − dF/u S0 ]T , and





A=




−1
1
0
0
0
0
0
1
−1 0
0
0 


0
0
0 −1
1
0 

0
0
0
0
1
−1 


0
0
−1 0
0
1 
D 2(1 − D) D D 2(1 − D) D

34 = 81 possible solutions might be obtained by the different combinations, but
only some of them stand for stable steady states, depending on the external
parameters, F and dF or F and u. Again, u cannot be 0, otherwise −dF/u
will be infinity.
2.2
Coupled low-order models
In the following a coupled 2-ocean version of the low-order model will be studied. Especially, the bifurcation behavior of the system to different parameters
(the salt fluxes and the strength of the inter-ocean coupling F, u) is studied.
The derivation of the low-order model equations will be shown in 2.2.1. Then,
the special case of a coupled Welander case will be discussed with respect to
its solutions and their bifurcations in 2.2.2.
We envision two 3-dimensional ocean boxes coupled by a gap (Drake Passage,
Fig. 2.1) in the southern hemisphere and a zonal current UACC (µ, ζ) flowing
through this gap, transporting salt and heat between the ocean basins. By
23
1
∂(UACC S)/∂λ we get for the
integrating the associated flux divergence − a cos
φ
inter-ocean transport qS
qS = −(UACC S)east + (UACC S)west = UACC (S P − S A )
(2.12)
The superscripts A, P denote the Atlantic and Pacific, respectively. The typical
annually averaged volume transport of the ACC flowing through Drake Passage
is on the order of 130 Sv with an error up to 27 Sv (S. A. Cunningham, 2003).
Drake passage is located between −55◦ S to −62◦ S, and is about 800km wide.
With our scaling in the first chapter, an average depth depth of the ocean
of H = 4000m, this flow yields a transport coefficient of the magnitude of
1.3 × 108 m3 /s/4000m/8 × 105 m ≈ 0.05m/s. However, for understanding the
influence of the ACC on the global thermohaline circulation we also allow for
larger values of the transport coefficient in the following.
2.2.1
Equations
For coupled oceans, we start with the exact zonally averaged time-dependent
nonlinear equations for the salt balance in both oceans (analogous to Eq. 1.12
for a single ocean).
A
SτA + ψµ SζA − ψζ SµA = [(1 − µ2 )SµA ]µ + κSζζ
+ qS
(2.13)
A
SτP + ψµ SζP − ψζ SµA = [(1 − µ2 )SµA ]µ + κSζζ
− qS
(2.14)
where couples the two ocean basins,
qS = u(S P − S A )G(µ)
(2.15)
with u denoting the zonal current which flows through the passage between
Atlantic and Pacific. The function G(µ, ζ) is defined as
(
G(µ, ζ) =
1 µ ∈ [µS , µN ]
0 elsewhere
(2.16)
The vertical boundary conditions are
κSζ = F surf at ζ = 0
κSζ = 0 at ζ = −1
We consider the Atlantic ocean first and expand the solution S A and the surface
forcing with Legendre polynomials because of the completeness of the latter
in the interval [−1, 1].
SA =
F surf (µ) =
∞
X
n=0
∞
X
n=0
24
SnA (ζ, τ )Pn (µ)
(2.17)
Fn Pn (µ)
(2.18)
Inserting Eq. 2.18 into Eq. 2.14, and using the orthogonality of Pn , we get the
equation shown below that will be used to arrive at expressions for the SnA (ζ, τ ),
n = 0, 1, 2.... In the low-order model, only the first 3 orders (n = 0, 1, 2) are
taken into account. We see from Eq. 2.18 that only the 0th-order component
contributes to the total net salt flux over the ocean surface. To simplyfy the
notation, the superscript ’A’ will be dropped and ’S n ’ will be used instead of
’Sn ’.
n
(2.19)
Sτn + < J(ψ, S)Pn >= −νn2 S n + κSζζ
+ < q S Pn >
Here J(ψ, S) denotes the (µ, ζ)-Jacobian and integration over µ is abbreviated
by the cornered brackets. The boundary conditions for Eq. 2.19 are
κSζn = Fn at ζ = 0
κSζn = 0 at ζ = −1
• For n 6= 0
We split the solution into two parts where the first part is the solution
for the time-indepedent linear problem of a single ocean.
S n = Sn hn (ζ) + Ŝ n (ζ, τ )
where hn (ζ) are the vertical eigen functions as derived in 1.5.1. and the
second part satisfies
n
Ŝτn + < J(ψ, S n )Pn >= −νn2 Ŝ n + κŜζζ
+ < q S Pn >
(2.20)
with boundary conditions
κŜζn = 0 at ζ = 0, −1
To treat this problem we expand Ŝ n with vertical eigenfunctions Hq (ζ) =
cos(ηq ζ) (where ηq = qπ), so we have
Ŝ n =
∞
X
Snq (τ )Hq (ζ)
(2.21)
q=0
By now we find that
Snqτ = −(νn2 + κηq2 )− < J n (ψ, S n )Pn Hq > + < qS Pn Hq >
(2.22)
where the cornered bracket is now the integration over µ and ζ. Recalling
the expression for qS , in this low-order model we only take S10 and S20
25
for n 6= 0. Using cij =
µRN
µS
Pi Pj dµ, hij =
R1
hi Hj dζ and Hij =
−1
R1
Hi Hj dζ,
−1
we get
ZµN Z0
< q S Pn H 0 > = u
= u
+ u
+ u
µS −1
ZµN Z0
(S P − S)Pn H0 dµdζ
(S P 0 − S 0 )Pn H0 dµdζ
µS −1
2
X
P
(Sm
− Sm )cmn hm0
m=1
2
X
P
(Sm0
− Sm0 )cmn H00
(2.23)
m=1
• For n = 0
Eq. 2.19 still holds, except that ν = 0 now
0
Sτ0 = − < J(ψ, S)P0 > +κSζζ
+ u(S P 0 − S 0 )c00
+ u
∞
X
(S P m − S m )cm0
(2.24)
m=1
We write the solution as S 0 (ζ, τ ) = S̃ 0 (ζ) + Ŝ 0 (ζ, τ ), so that the first
term is the solution of the following boundary condition problem.
0
κS̃ζζ
− uc00 S̃ 0 = 0
(2.25)
κS̃ζ0 |ζ=0 = F0
(2.26)
κS̃ζ0 |ζ=−1 = 0
(2.27)
Solving for S̃ 0 , we get
S̃ 0 (ζ) = S0 h0 (ζ)
with
F0
cosh[ξ0 (ζ + 1)]
S0 =
, h0 =
, ξ0 =
κ
ξ0 sinh ξ0
(2.28)
r
uc00
κ
Then the second part Ŝ 0 is the solution for the Eq. 2.29 with a homogeneous boundary condition
0
Ŝτ0 = − < J(ψ, S)P0 > +κŜζζ
− uc00 Ŝ 0 + uc00 Ŝ P 0 + uc00 S0P h0
+ u
∞
X
(S P m − S m )cm0
m=1
26
(2.29)
Similar to Eq. 2.21, Ŝ 0 =
∞
P
q=0
S0q (τ )Hq (ζ), and for S0q
S0qτ = − < J(ψ, S)P0 Hq > −κηq2 S0q
P
− S0q ) + uc00 h0q S0P
+ uc00 Hqq (S0q
+ u
+ u
∞
X
m=1
∞
X
P
(Sm
− Sm )hmq cm0
P
(Sm0
− Sm0 )H0q cm0
(2.30)
m=1
So far we can write balances for all the haline variables of the lowest few orders.
The remaining task is to evaluate the (µ, ζ)-Jacobian:
S00τ = − < J(ψ, S)P0 H0 >
P
+uc00 H00 (S00
− S00 ) + uc00 h00 S0P
P
P
+uc10 H00 (S10
− S10 ) + uc20 H00 (S20
− S20 )
2
S01τ = − < J(ψ, S)P0 H1 > +κη1 S01
P
−uc00 H11 (S01
− S01 ) + uc00 h01 S0P
P
P
+uc10 H01 (S10
− S10 ) + uc20 H01 (S20
− S20 )
2
S10τ = − < J(ψ, S)P1 H0 > −ν1 S10
+uc01 h00 (S0P − S0 )
P
P
+uc01 H00 (S00
− S00 ) + uc01 H10 (S01
− S01 )
P
P
+uc11 H00 (S10 − S10 ) + uc21 H00 (S20 − S20 )
S20τ = − < J(ψ, S)P2 H0 > −ν22 S20
+uc02 h00 (S0P − S0 )
P
P
+uc02 H00 (S00
− S00 ) + uc02 H10 (S01
− S01 )
P
P
+uc12 H00 (S10 − S10 ) + uc22 H00 (S20 − S20 )
(2.31)
(2.32)
(2.33)
(2.34)
The (µ, ζ)-Jacobian is written as
J(ψ, S) = ψµ Sζ − Sµ ψζ
= J(ψnlin , Snlin ) + J(ψnlin , Slin )
+J(ψlin , Snlin ) + J(ψlin , Slin )
(2.35)
As both P0 and H0 are constant,
< J(ψ, S)P0 H0 >= P0 H0 < J(ψ, S) >= 0
(2.36)
Comparing the set of equations for the components of S’ for S we have got
for Atlantic in a coupled-ocean model with that obtained in a single ocean, we
27
now have additional terms S0 h0 (ζ) in Slin and S00 H0 in Snlin . All equations
for ψ remain the same as before. To make the procedure clear, one can write
each part of the chosen modes in Eq. 2.35 separately.
ψµlin = −Ra
ψµnlin = −Ra
2
X
n=1
2
X
νn2 ψn fn Pn
νn2 ψn0 F0 Pn
n=1
2
X
ψζlin = −Ra(1 − µ2 )
ψζnlin = −Ra(1 − µ2 )
n=1
2
X
ψn fn0 Pn0
ψn0 F00 Pn0
n=1
Sµlin =
Sµnlin =
Sζlin =
Sζnlin =
2
X
n=1
2
X
n=1
2
X
n=1
2
X
Sn hn Pn0
Sn0 H0 Pn0
Sn h0n Pn +S0 h00 P0
Sn0 H00 Pn + S01 H10 P0 +S00 H00 P0
n=1
Only the terms marked in boldface above are new contributions to the change
in the projection of the (µ, ζ)-Jacobian on µ and ζ. Notice that because H0
is constant the last term involving S00 vanishes. Therefore only J(ψnlin , Slin )
and J(ψlin , Slin ) change.
The contributions of the additional term to the Jacobian are
∆J(ψnlin , Slin ) = −Ra
∆J(ψlin , Slin ) = −Ra
2
X
n=1
2
X
νn2 ψn0 F0 Pn · S0 h00 P0
(2.37)
νn2 ψn f0 Pn · S0 h00 P0
(2.38)
n=1
< J(ψ, S) >01 remains as before, and the new parts for < J(ψ, S) >10 and
< J(ψ, S) >20 are directly visible from the equations above. Conservation of
salt requires that the total change with time of the two coupled oceans must be
P
zero, which means Ṡ00
+ Ṡ00 = 0, so the difference of the total salinity between
P
Atlantic and Pacific, i. e. K = Ṡ00
− Ṡ00 , is used as the 11th variable in this
coupled system. The final equations are straightforward now:
U̇ = −k(M + U )
(2.39)
28
V̇ = −k(N + V )
√
√
L̇ = 2 2(U − cPa )M + 2 2(3V − dPs )N
+(aaL U + baL Pa )Sa + (asL V + bsL Ps )Ss − 4rκ0 L
+ur[−eL S0 + eLL (LP − L)]
√
Ṁ = −2 2(U − cPa )L
+(asM U + bsM Pa )Ss + (aaM V + baM Ps )Sa − 2rκ0 M
4
− esM S0 U − 2eaM Pa S0
π
N
M
+ur[−2eM S0 + eK
M K + eM (MP − M ) + eM (NP − N )]
√
Ṅ = −2 2(3V − dPs )L
+(aaN U + baN Pa )Sa + (asN V + bsN Ps )Ss − 6rκ0 N
12
− eaN S0 V − 6esN Ps S0
π
M
N
+ur[−2eN S0 + eK
N K + eN (MP − M ) + eN (NP − N )]
U̇ P = −k(M P + U P )
V̇ P = −k(N P + V P )
√
√
L̇P = 2 2(U P − cPa )M P + 2 2(3V P − dPs )N P
+(aaL U P + baL Pa )Sa + (asL V P + bsL Ps )Ss − 4rκ0 LP
−ur[−eL S0 + eLL (LP − L)]
√
Ṁ P = −2 2(U P − cPa )LP
+(asM U P + bsM Pa )Ss + (aaM V P + baM Ps )Sa − 2rκ0 M P
4
+ esM S0 U P + 2eaM Pa S0
π
M
N
−ur[−2eM S0 + eK
M K + eM (MP − M ) + eM (NP − N )]
√
Ṅ P = −2 2(3V P − dPs )LP
+(aaN U P + baN Pa )Sa + (asN V P + bsN Ps )Ss − 6rκ0 N P
12
+ eaN S0 V P − 6esN Ps S0
π
M
N
−ur[−2eN S0 + eK
N K + eN (MP − M ) + eN (NP − N )]
M
P
N
P
K̇ = 2ur[eK S0 − eK
K K − eK (M − M ) − eK (N − N )]
(2.40)
(2.41)
(2.42)
(2.43)
(2.44)
(2.45)
(2.46)
(2.47)
(2.48)
(2.49)
where the symbols denote the same variables as for a single ocean and the
Pacific variables are denoted by the superscript P . K is the total salinity
difference as discussed above. The values of the coefficients are listed in the
appendix at the end. Remember that in our single ocean model, only S1 , S2
are taken as salt flux because the salt in the ocean must be conserved. Here,
in contrast, the net flux (S0 integrated over latitude) into an individual ocean
does not need to vanish. Only for the total salt flux into the two oceans
29
S0P + S0 = 0 is demanded.
The water vapor transport carried by the trade winds from Atlantic is about
0.3Sv. The net δ(P − E) is calculated by deviding this number with the area
of the Atlantic Ocean, about 100 million km2 , so δ(P − E) = 3 × 10−9 m/s. As
the salt flux is proportional to P − E and a the difference of P − E has a value
of 50 × 10−9 m/s (approximately, from Fig. 1.3, ignoring the central maximum)
√
surf
2 ≈ 2.5.
this corresponds
to
a
range
of
F
of
7,
S
=
F
/κ
=
7×3/50/0.25×
0
0
√
The factor 2 arises from the normalized Legendre polynomial P0 .
2.2.2
Welander case
In Welander case, we have 11-variable system without the asymmetric forcing
terms, so Ta , Sa , Pa are all 0. The governing equations are:
U̇ = −k(M + U )
V̇ = −k(N + V )
√
√
L̇ = 2 2U M + 2 2(3V − dPs )N + (asL V + bsL Ps )Ss
−4rκ0 L + ur[−eL S0 + eLL (LP − L)]
√
4
Ṁ = −2 2U L + asM U Ss − 2rκ0 M − esM S0 U
π
M
+ur[−2eM S0 + eK
K
+
e
(M
−
M ) + eN
P
M
M
M (NP − N )]
√
Ṅ = −2 2(3V − dPs )L + (asN V + bsN Ps )Ss
12
−6rκ0 N − eaN S0 V − 6esN Ps S0
π
M
N
+ur[−2eN S0 + eK
N K + eN (MP − M ) + eN (NP − N )]
U̇ P = −k(M P + U P )
V̇ P = −k(N P + V P )
√
√
L̇P = 2 2U P M P + 2 2(3V P − dPs )N P + (asL V P + bsL Ps )Ss
−4rκ0 LP − ur[−eL S0 + eLL (LP − L)]
√
4
Ṁ P = −2 2(U P − cPa )LP + asM U P Ss − 2rκ0 M P + esM S0 U P
π
K
M
N
−ur[−2eM S0 + eM K + eM (MP − M ) + eM (NP − N )]
√
P
Ṅ
= −2 2(3V P − dPs )LP + (asN V P + bsN Ps )Ss − 6rκ0 N P
12
+ eaN S0 V P − 6esN Ps S0
π
M
N
−ur[−2eN S0 + eK
N K + eN (MP − M ) + eN (NP − N )]
M
P
N
P
K̇ = 2ur[eK S0 − eK
K K − eK (M − M ) − eK (N − N )]
(2.50)
(2.51)
(2.52)
(2.53)
(2.54)
(2.55)
(2.56)
(2.57)
(2.58)
(2.59)
(2.60)
3 experiments are performed on this system:
1. Starting with a zero point where all the variables are 0, Ss varies as parameter for S0 = 0 and a reasonable u. In this case there’s no net salt flux into
30
the two oceans which are identical in salinity, but the current is forced flow
between them with no salt transport. The results show symmetric structure
in bifurcation diagrams.
2. The results generated from experiment 1 give the same figure for the first
10 variables as shown in Fig. 1.9, all the 5 variables of Atlantic and Pacific
ocean looking the same. 4 points are selected from the stable branches of U ,
for each S0 is spinned up to 2.5 and then change Ss , so we get a branch/loop
from a perturbation of S0 on the stable state.
3. From the initial point first spinup S0 to 2.5, then vary Ss . The symmetry
in experiment 1 breaks into two part because of inducing of S0 .
The experiments are repeated for different u to see its influence on the circulations. From these experiments some results can be obtained:
Bifurcation diagrams for two identical oceans Fig. 2.3 and Fig. 2.4
show the bifurcationa of variables with respect to Ss for u = 0.05 and u = 0.1,
the net forcing S0 is set to 0, so there’s no salt transport between the Atlantic
and Pacific Oceans. They show the similar feature with Fig. 1.9 which is
for single ocean. The amplitudes of variables are different now because the
temperature is spinned up to -1 (in a single ocean we have -1.5). In this
case the flow connecting the two oceans doesn’t influence the circulation and
salinity because the system starts at zero points and there’s no salinity gradient
between the two from beginning, so no diffence shows when u varies. The two
oceans can be treated as identical in this condition. The total salinity difference
between the two oceans is not 0, indicating the currents for the two oceans
are not the totally the same. The symmetry of K is more complicated than
that of U , because many combinations of the circulations of the two oceans
are possible. Stable branches exist only when both oceans circulates stably.
As observed the stable branches of K are quite similar with that of U , or are
proportional to U when there’s no net salt flux put into individual ocean.
Streamfunctions for S0 = 0 The streamlines are calculated and shown for 4
different stable steady states in Fig. 2.5 and Fig. 2.6. Where U, UP , V, VP have
different meanings with which appeared before, they are the total solution
which are the sum of the nonlinear parts of the solution solved from above
equations and the linear parts which annihilated when take the derivative with
respect to time. ’Force Ψ’ shows the influence of symmetric forcing terms Ts , Ss
on the forced part of circulation, equatorial upwelling appears for both two
values of Ss . As we set a cool pole and warm equator system, the temperature
driven circulation must have equatorial upwelling, the salt flux put into the
equator (negative Ss ) however weakens this phenomenon, so the magnitude of
two-cell circulation is in fact the competition of these two forcing. In Fig. 2.5,
Ss = −3.97 for both (’forced Ψ’ is weak), with opposite U , we have northern
31
UA
VA
0.4
0.4
0.3
0.2
0.2
LA
0.5
0
0
−0.5
0.1
−0.2
0
−1
−0.4
−0.1
−0.2
−2
−0.8
−0.3
−0.4
−15
−1.5
−0.6
−10
−5
0
−1
−15
−10
Ss
−5
−2.5
−15
0
−10
Ss
UP
VP
0.4
0.4
0.3
0.2
0.2
−5
0
−5
0
Ss
LP
0.5
0
0
−0.5
0.1
−0.2
0
−1
−0.4
−0.1
−0.2
−2
−0.8
−0.3
−0.4
−15
−1.5
−0.6
−10
−5
0
−1
−15
Ss
−10
−5
−2.5
−15
0
Ss
−10
Ss
11 equations model
1.5
1
K
0.5
0
−0.5
−1
−1.5
−15
−10
−5
0
S
s
Figure 2.3: Bifurcation for S0 = 0, u = 0.05. For steady states, M = −U, N =
−V, MP = −UP , NP = −VP
32
UA
VA
0.4
0.4
0.3
0.2
0.2
LA
0.5
0
0
−0.5
0.1
−0.2
0
−1
−0.4
−0.1
−0.2
−2
−0.8
−0.3
−0.4
−15
−1.5
−0.6
−10
−5
0
−1
−15
−10
Ss
−5
−2.5
−15
0
−10
Ss
UP
VP
0.4
0.4
0.3
0.2
0.2
−5
0
−5
0
Ss
LP
0.5
0
0
−0.5
0.1
−0.2
0
−1
−0.4
−0.1
−0.2
−2
−0.8
−0.3
−0.4
−15
−1.5
−0.6
−10
−5
0
−1
−15
Ss
−10
−5
−2.5
−15
0
Ss
−10
Ss
11 equations model
1.5
1
K
0.5
0
−0.5
−1
−1.5
−15
−10
−5
0
S
s
Figure 2.4: Bifurcation for S0 = 0, u = 0.1. For steady states, M = −U, N =
−V, MP = −UP , NP = −VP
33
sinking in Atlantic and southern sinking in Pacific when U < 0 and vice versa
for U > 0, the former condition could present the Great Conveyor Belt. Great
overturning is true for both oceans, but in opposite directions. In Fig. 2.6,
Ss = −1.25, a small loop appears in high latitude region, either north pole or
south pole for each ocean. The circulation directions for the two oceans are
the same in polar regions which means this type can’t break into the great
conveyor belt. When u = 0.1 the same features remain as mentioned before it
doesn’t affect the circulation when S0 = 0.
Symmetry breaking Once S0 is nonzero, etc. spinned up to a certain value,
the symmetry of the system no longer remain. The state Ss = −1.25, U =
−0.34 corresponding to the upper panel in Fig. 2.6 is studied for this change.
Fig. 2.7 shows two types of spinup of S0 . First starting with the chosen state,
we can easily noticed when S0 increases the bifurcation curve quickly loses
stablity and then gets back stable again. On the stable branch, there exists
a critical value near S0 = 0.3 from where the magnitude of U tends to be
constant (approximately -0.01). Below this value U is strongly increased with
S0 , above this value the change of U is limited by the transport coefficient
u. The net salt flux and the salt transport balance each other to maintain
the stability of the circulation of the oceans. Then increase S0 from the zero
point. After changing S0 in two ways, Ss is changed from 0 to -15 for these two
conditions respectively and the bifurcation curves are found the same (lower
left panel in Fig. 2.7). However this is only true for small u. For large u, u = 1
for instance, the mirror-image branches starting from the pitchfork breaks into
a independant loop and a smooth branch which goes to 0 in the end.
Impact of net salt flux Different S0 are used to see its influence on the
bifurcation. The results are shown in Fig. 2.8. To compare with the result for
S0 = 0, streamfunctions for states with Ss ≈ −3.97 are also shown in Fig. 2.8.
As shown in Fig. 2.5 the two oceans are in either NPP or SPP states, circulating
oppositely, this pole-to-pole circulation also appears when S0 = 0.1 for both
oceans and S0 = 0.2 for the Atlantic Ocean only. However these two states
is no longer stable states now. The big overturning of the Pacific Ocean first
breaks into two parts, a water mass forms in deep southern polar region. As S0
increases further, the patterns show a thermally dominated trend gradually.
Influence of inter-ocean salt transport When S0 6= 0, there is salinity
difference between the two oceans, salt is transport from one ocean to the
other by the ACC. To find out the influence of salt exchange on the ocean
circulation, we compare the bifurcation and stream function of the model when
u = 0.1 with the results for u = 0.05. S0 , Ss are the same as for u = 0.05.
From Fig. 2.9, the bifurcation diagrams do not show obvious change, which
34
U
Up
0
0
−0.5
−0.5
−1
−1
−0.5
0
0.5
0.5
1
forced ψ
1.5
1
−1
−1
2
−2
0
0
−0.5
−0.5
−0.5
−0.5
−1
0
−1.5
V
0
−1
−1
−0.5
0.5
0
1
−1
−1
1
−0.5
0
−1
0.5
0
1
−1
−1
1
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
0
0.5
1
1
1.5
−1
−1
2
−0.5
−2
0
−0.5
−0.5
−0.5
−2
−1.5
forced ψ
0
0.5
−1
V
−0.5
1
−1
−1
0
−0.5
−0.5
−0.5
−1
−1
−0.5
0
−1
0.5
−1
1
−0.5
−0.5
0
1
0.5
1.5
1
2
Vp
0
1
1
0
−1.5
0.5
0
0
1
Up
0
−1
−1
−1
0.5
0
U
1
0
−0.8
−0.5
0.5
0.5
−0.5
1
total ψp
0
−1
−1
0
−1
Vp
−1
0
−0.8
−0.5
0.5
−0.5
total ψ
−1
−1
0
0.5
0
1
−1
−1
1
−0.5
0
−1
total ψ
0.5
0
1
1
total ψp
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−0.5
−2
0
−1.5
0.5
−1
−0.5
1
−1
−1
−0.5
0.5
0
1
0.5
1.5
1
2
Figure 2.5: Streamfunctions for u = 0.05, S0 = 0, Ss = −3.97.
U (nonlinear) ≈ −0.15, lower: U (nonlinear) ≈ 0.15
35
Upper:
U
Up
0
0
−0.5
−0.5
−1
−1
−0.5
1
0
2
forced ψ
0.5
3
1
−1
−1
4
−4
0
−0.5
−0.5
−0.5
−5
0
−3
0.5
0
1
−1
−1
−0.5
5
0
−2
0.5
0
1
−1
−1
−0.5
2
−2
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.5
0
0
0.5
2
4
1
−1
−1
6
−0.5
−6
0
−0.5
−0.5
−0.5
−4
0
−3
forced ψ
0.5
−2
1
−1
−1
1
−0.5
−0.5
−0.5
−0.5
5
0
−2
−2
1
0
0
2
0.5
3
1
4
Vp
0
−1
−1
0.5
−4
−0.5
−1
0
0
2
0
V
0
−5
1
Up
0
−1
−1
1
0.5
0
U
0.5
−1
−0.8
−1
−1
0
1
total ψp
0
−0.8
−0.5
−2
0
total ψ
−1
−1
0.5
Vp
0
−0.5
0
V
0
−1
−1
−0.5
0.5
0
1
−1
−1
−0.5
2
0
−2
0.5
0
total ψ
1
2
total ψp
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−0.5
−6
0
−4
0.5
−2
1
0
−1
−1
−0.5
0
0
2
0.5
4
1
6
Figure 2.6: Streamfunctions for u = 0.05, S0 = 0, Ss = −1.25.
U (nonlinear) ≈ −0.34, lower: U (nonlinear) ≈ 0.34
36
Upper:
11 equations model
0.5
0.1
0
U
U
−0.1
0
−0.2
−0.3
−0.4
−0.5
−15
−10
−5
−0.5
0
0
0.5
1
Ss
1.5
2
2.5
1.5
2
2.5
S0
0.5
1
U
U
0.5
0
0
−0.5
−0.5
−15
−10
−5
−1
0
Ss
0
0.5
1
S0
Figure 2.7: Left: bifurcations of U when S0 = 0 and S0 6= 0. Upper right:
spinup S0 when Ss = −1.25, U = −0.34, lower right: spinup S0 from zero point
might be because that the change in u is not big enough. But in the panels
for stremfunctions, difference can be found. The amplitude of total stream
funtion is smaller although there’s no qualitative change. We can still know
that increasing u can cancel part of the impact of S0 . The net salt flux and
the salt transport compensate each other in the ocean.
2.3
Conclusions
The box model and low-order model are two different approaches to study
the thermohaline circulation in a simplified way. Both of them can find important characteristics of the THC, such as multiple equilibrium states, symmetry
breaking, etc. In box models, the fluxes are added over boxes to meet the conservation of salt: for a single ocean the sum must be 0, but for a coupled
ocean, we can introduce a net flux dF from one equatorial box to the other to
compensate for the salt transport between the Atlantic and Pacific oceans by
the Antarctic Circumpolar Current (except in the totally symmetric solution).
In the low-order model, the conservation is obtained by choosing the right
meridional eigen functions. The integration of Legendre polynomials P1 , P2
over latitude yields 0, which means they do not introduce a net flux into the
oceans. Sa , Ss are chosen according to the range of these two function in the
37
11 equations model
0
−0.5
−15
−10
−5
0
−0.5
0
0.5
U
0.5
U
U
0.5
0
0.05
S0
Ss
0
0.05
0.1
S
0.15
0
0.5
S
0
−0.5
−15
−10
0
0.5
1
−5
0
1.5
2
2.5
S
0
0.5
U
U
U
−0.5
1
0.5
0
0
0
0
0.5
−5
0.5
0
−0.5
0.2
−10
Ss
U
0
−0.5
−0.5
−15
0.1
0.5
U
U
0.5
0
0
−0.5
−15
−10
Ss
−5
0
0
−0.5
−15
−10
Ss
−5
0
Ss
total ψ
0
2.5
−0.2
2
−0.4
1.5
−0.2
2.5
−0.6
1
−0.4
2
−0.8
0.5
−0.6
total ψ
0
3
1.5
1
−1
−1
−0.8
−0.5
0
0.5
0.5
1
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
total ψp
total ψp
0
0
−0.5
0.1
0
−0.2
−0.1
−0.4
−0.5
−1
−0.2
−0.3
−0.6
−0.4
−0.8
−0.5
−1.5
−1
−1
−0.5
0
0.5
−0.6
−1
−1
1
−0.8
−0.6
−0.4
−0.2
total ψ
0
0.2
0.4
0.6
0.8
1
total ψ
0
5
−0.2
0
8
−0.2
6
4
−0.4
4
−0.4
3
2
−0.6
2
−0.6
−0.8
1
−0.8
0
−2
0
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
−1
−1
1
−0.8
−0.6
−0.4
−0.2
total ψp
0
0.2
0.4
0.6
0.8
1
total ψp
0
0
1
0.5
−0.2
−0.2
0.5
−0.4
−0.4
0
−0.6
0
−0.6
−0.5
−0.5
−0.8
−1
−1
−1
−0.8
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−1.5
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 2.8: Bifurcation diagrams of U when S0 = 0.1, 0.2, 1, 2.5 for u = 0.05.
Stream functions for the Atlantic and Pacific Ocean , (from left to right) when
Ss = −3.97; S0 = 0.1, 0.2, 1, 2.5
38
11 equations model
0
−0.5
−15
−10
−5
0
−0.5
0
0.5
U
0.5
U
U
0.5
0
0.05
S0
Ss
0
0.05
0.1
S
0.15
0
0.5
S
0
−10
1
−5
0
2
−0.5
−15
−10
−5
0
−0.5
−15
−10
−5
0
Ss
total ψ
total ψ
2.5
−0.2
0
1.5
−0.4
−0.6
1
−0.6
−0.8
0.5
−0.8
0.5
2.5
−0.2
2
−0.4
1.5
−1
−1
1
2
1
0.5
−0.5
total ψp
0
0.5
1
total ψp
0
0
−0.2
−0.5
−0.5
−0.4
−0.5
−1
−1
−1
−0.6
−1.5
−0.5
0
0.5
1
−1
−1
−0.8
−0.5
0
0.5
1
total ψ
total ψ
0
5
−0.2
4
−0.4
0
8
−0.2
6
−0.4
4
3
−0.6
2
−0.8
1
−1
−1
0
−0.5
0
0.5
1
2
−0.6
0
−0.8
−1
−1
−2
−0.5
0
0.5
1
total ψ
total ψp
p
0
0
1
0.5
0
−0.5
0
−0.5
−0.5
−1
−1
2.5
0
Ss
0
1.5
0
0
0
−0.5
0.5
0.5
Ss
−1
−1
0
S
U
U
U
−0.5
−15
−0.5
1
0.5
0
0
0
0
0.5
−5
Ss
0
−0.5
0.2
−10
0.5
U
0
−0.5
−0.5
−15
0.1
0.5
U
U
0.5
0
−1
−0.5
0
0.5
−1
−1
−1
1
−0.5
0
0.5
1
Figure 2.9: Bifurcations of U when S0 = 0.1, 0.2, 1, 2.5 for u = 0.1. Stream
functions for the Atlantic and Pacific Ocean , (from left to right) when Ss =
−3.97, S0 = 0.1, 0.2, 1, 2.5
39
inteval [-1,1]. The 0-order Legendre polynomial
P0 is different, it is constant
√
over all latitudes and yields a factor of 2 when integrated. So for a single
ocean, the amplitude of P0 must be zero for vanishing net flux, but for coupled
ocean, P0 is necessary to drive the system. The treatment of temperature in
low-order models is also different from that in box models. In the box models,
the temperature difference between boxes is considered as constant, but not as
a parameter. In the low-order model the temperature is increased to a certain
value (-1.5 or -1) to represent ∆T before varying any other parameters. The
thermal variables are not considered here further for simplicity. However a
more complicated and complete system can be obtained with the low-order
model if we include thermal varibles and induce Ta , Ts as control parameters.
The symmetry breaking of the thermohaline circulation is studied with a loworder model in both single ocean and coupled Atlantic-Pacific ocean configurations. Similar to box models, NPP and SPP branches appear after a pitchfork
bifurcation, as studied before already by H.A. Dijkstra (2003) in box models.
The bifurcation point which connects symmetric and asymmetric solutions
breaks up when S0 is not 0. Considering the Atlantic and Pacific Oceans, the
low-order model is used to study the influence of salt exchange where we use
that S0 and inter-ocean salt transport must balance. This conclusion is presented by Eq. 2.7 in the low-order model. The super-symmetric structures of
K in Fig. 2.3 and Fig. 2.4 indicatethat the system could have various steady
states (as shown in Table. 2.1 for Atlantic-Pacific coupled Welander box model)
although only few of them are stable . K is the total salinity difference between
the two oceans, its bifurcation behavior is qualitatively similar with q1 − q3
(the difference of flow, proportional to difference of salinity)in Fig. 1.7.
Both the box model and the low-order model indicate the system is sensitive
to the freshwater flux into the system. The nonlinear response of the circulation to the freshwater flux enables the existence of multiple equilibria of the
THC. For a single ocean basin, bifurcations of the Atlantic thermohaline circulation have been verified by ocean general circulation models (S. Rahmstorf,
1995). For a coupled Atlantic-Pacific ocean, the Great Conveyor Belt is found
in our coupled Welander box model and the coupled low-order model (with
symmetric freshwater flux). With the parameter we used only one branch in
the bifurcation diagram is obtained (Fig.2.8 and Fig.2.9), only Hopf bifurcations appear and no multiple equilibria are observed. This is contrast with the
results in some OGCMs (H. A. Dijkstra, 2004) where saddle-node bifurcations
exist and a mutiple-equilibria regime is found. This might be due to the parameter values that we choose for the transport coefficient u and for S0 . As we
derived for the coupled Welander box model, Eq. 2.7 determines the relation
between these two parameters. If we use the average salinity of the Atlantic
(34.92) and Pacific (34.6) ocean (E. Fahrbach, and J. Meincke, 1989) for the
right hand side of this equation, we can get u ≈ 0.75 for S0 = 2.5 (scaled).
40
With larger u (up to 1), the system does have a multiple-equilibria regime in
the bifurcation diagram. This still requires further study. The thermal variables are not considered in this thesis and will be included in the low-order
model in further studies as well.
41
Bibliography
[1] The derivation of the 8-variable equations in Chapter 1 is basically from
an unfinished script of Dirk Olbers.
[2] A. Khibnik, Y. Kuznetsov, V. Levitin, and E. Nikolaev,1993: Continuation techniques and interactive software for bifurcation analysis of ODEs
and iterated maps, Physica D, 62, 360-371.
[3] Broecker W. S., 1991: The great ocean conveyor. Oceanography, 4, 79-89.
[4] Cunningham, S. A., S. G. Alderson, B. A. King, and M. A. Brandon,
2003: Transport and variability of the Antarctic Circumpolar Current in
Drake Passage, J. Geophys. Res., 108(C5), 8084.
[5] Dijkstra, H. A.,2000: Nonlinear Physical Oceanography: A Dynamical
Systems Approach to the Large Scale Ocean Circulation and El Nino,
Springer, Bremen.
[6] –, and M. J. Molemaker, 1997: Symmetry breaking and overturning oscillations in thermohaline-driven flows. J. Fluid Mech., 331, 195-232.
[7] –, and J.D. Neelin, 1999; Imperfections of the thermohaline circulation:
Latitudinal asymmetry and preferred northern sinking. J. Climate, 13,
366-382.
[8] –, and W. Weijer, 2003: Stability of the global ocean circulation: The
connection of equilibria within a hierarchy of models. J. Marine Research,
61, 725-743
[9] –, W. Weijer and J. D. Neelin, 2003: Imperfections of the threedimensional thermohaline ocean circulation: Hysteresis and unique state
regimes. J Phys. Oceanogr., 33, 2796-2814.
[10] Elmer, F. J., 2002: The pendulum lab. Online laboratory.
42
[11] Fahrbach, E., Meincke, J., 1989: Temperature-salinity characteristics
of world ocean waters, Landolt-Boernstein: Numerical data and functional relationshps in science and technology, Springer, Berlin,V/3b,
Oceanography,15-58.
[12] Hasumi, H., 2002: Sensitivity of the global thermohaline circulation to
interbasin freshwater transport by the atmosphere and the Bering Strait
throughflow, J. Climate, 15 (17), 2516-2526.
[13] Latif, M., Roeckner, E., Mikolajewicz, U. and Voss, R., 2000: Tropical stabilization of the thermohaline circulation in a greenhouse warming
simulation, J. Climate, 13, 1809-1813.
[14] Levitus, S., R. Burgett and T. Boyer, 1994; World Ocean Atlas 1994, 3:
Salinity. NOAA Atlas NESDIS 3, US Department of Commerce, Washington DC; 0-99.
[15] Lohmann, G., and J. Schneider, 1999: Dynamics and predictability of
Stommel’s box model. A phase-space perspective with implications for
decadal climate variability. Tellus, 51A, 326-336.
[16] Marotzke, J., Welander P, Willebrand J., 1988: Instability and multiple
steady states in a meridional - plane model of the thermohaline circulation,
Tellus, 40A, 162-172.
[17] Marotzke, J., 1996: Analysis of thermohaline feedbacks. In: Decadal Climate Variability: Dynamics and Predictability. D.L.T. Anderson and J.
Willebrand, eds., NATO ASI series, I44, 333-378.
[18] –, 2000: Abrupt climate change and thermohaline circulation: Mechanisms and predictability, PNAS 97,4, 1347-1350.
[19] –, and Klinger, B. A., 2000: The dynamics of equatorially asymmetric
thermohaline circulations, J. Phys. Oceanogr, 29, 2962-2970
[20] –, and J. Willebrand, 1991: Multiple equilibria of the global thermohaline
circulation. J. Phys. Oceanogr., 21, 1372-1385.
[21] Parker, T. and Chua, L., 1989: Practical Numerical Algorithms for
Chaotic Systems, Springer, New York.
[22] Rahmstorf, S., 1995: Bifurcation of the Atlantic thermohaline circulation
in response to changes in the hydrological cycle. Nature, 378, 145-149.
[23] –, 1996: On the freshwater forcing and transport of the Atlantic thermohaline circulation. Climate Dyn., 12, 799-811.
43
[24] –, and M. England, 1997: Influence of Southern Hemisphere winds on
North Atlantic deep water flow. J. Phys. Oceanogr., 27, 2040-2054.
[25] –, 2003: The concept of the thermohaline circulation. Nature, 421, 699.
[26] Rooth, C., 1982: Hydrology and ocean circulation. Progress in Oceanography, Vol. 11, 131-149.
[27] Wang, X., P. H. Stone, and J. Marotzke, 1999a: Thermohaline circulation.
Part I: Sensitivity to atmospheric moisture transport. J. Climate, 12, 7182.
[28] –, –, and –, 1996b: Thermohaline circulation. Part II: Sensitivity with
interactive atmospheric transport. J. Climate, 12, 83-92.
[29] Scott, J., J. Marotzke, and P. Stone, 1999: Interhemispheric thermohaline
circulation in a coupled box model. J. Phys. Oceanogr., 29, 351-365.
[30] Seydel, R. (Ed.), 1999: World of Bifurcation. Online Collection and Tutorials of Nonlinear Phenomena. WOB99.
[31] Stommel, H., 1961: Thermohaline convection with two stable regimes of
flow. Tellus, 13, 224-230.
[32] Stone, P. H., and Y. P. Krasovskiy, 1999: Stability of the interhemispheric
thermohaline circulation in a coupled box model. Dyn. Atmos. Oceans.,
29, 415-435.
[33] V. Lucarini, and P. H. Stone, 2005: Thermohaline circulation stability: a
box model study. Part I: uncoupled model. J. Climate, 18, 501-513.
[34] –, –, 2005: Thermohaline circulation stability: a box model study. Part
II: coupled atmosphere-ocean model, J. Climate, 18, 514-529.
[35] Weijer. W., W. De Ruijter and H. A. Dijkstra., 2001: Stability of the Atlantic overturning circulation: competition between Bering Strait freshwater flux and Agulhas heat and salt sources. J. Phys. Oceanogr., 31,
2385-2402.
[36] –, W. P. M. De Ruijter, H. A. Dijkstra. and P. J. Van Leeuwen, 1999:
Impact of interbasin exchange on the Atlantic overturning circulation. J.
Phys. Oceanogr., 29, 2266-2284.
[37] Welander, P, 1986: thermohaline effects in the ocean circulation and related simple models, in Large-Scale Tranport Processes in Oceans and
Atmosphere, J. Willebrand and D. L. T. Anderson, eds., D. Reidel, 163200.
44
[38] Wunsch, C, 2002: What is the thermohaline circulation? Science, 298,
1179-1180.
45
Appendix A
Mathematical preparation
A.1
Legendre polynomials
The Legendre polynomials, sometimes called Legendre functions of the first
kind, are solutions to the Legendre differential equation which has the form
dy
d
[(1 − x2 ) ] + l(l + 1)y = 0
dx
dx
(A.1)
The Legendre differential equation has regular singular points at -1, 1, and ∞.
They obey the orthogonality
Z
1
−1
Pn (x)Pm (x)dx =
2
δm n
2n + 1
And The Legendre polynomials satisfy the recurrence relation
(n + 1)Pn+1 = (2n + 1)xPn − nPn−1
The first few Legendre polynomials Pn (x) are
P0 (x) = 1
P1 (x) = x
1
P2 (x) =
(3x2 − 1)
2
1
P3 (x) =
(5x3 − 3x)
2
1
P4 (x) =
(35x4 − 30x2 + 3)
8
1
P5 (x) =
(63x5 − 70x3 + 15x)
8
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
Normalized Legendre polynomials Pn (x) for n = 0, ..., 5 and derivatives of Legendre polynomials are illustrated in Fig. A.1. Integrals of the first 3 normalized
46
2.5
2.5
P0
P1
2
P3
P2
2
1.5
P4
1.5
P5
1
0.5
Pn(x)
Pn(x)
1
0.5
0
−0.5
−1
0
−1.5
−0.5
−2
−1
−1
−0.5
0
x
0.5
−2.5
−1
1
4
−0.5
0
x
0.5
4
dP1
dP0
3
(1−x )dPn/dx
0
2
(1−x2)dPn/dx
5
2
1
−1
1
0
−1
−2
−2
−3
−3
0
x
3
dP
dP4
−0.5
dP
3
dP2
2
−4
−1
1
0.5
−4
−1
1
−0.5
0
x
0.5
1
Figure A.1: The normalized Legendre polynomials Pn (x), n = 0, ..., 5 and (1 −
x2 )dPn /dx
polynomials are
Z
s
1
−1
Z
1
√
1
P0 (x)dx = 2
2
s
−1
Z
1
−1
s
(A.8)
3
P1 (x)dx = 0
2
(A.9)
5
P2 (x)dx = 0
2
(A.10)
As eigenfunctions the Legendre polynomials are also complete, meaning any
function f (x) that is continuous in interval [-1, 1] can be written as a sum over
Lengendre polynomial.
f (x) =
∞
X
cn Pn (x)
(A.11)
n=0
where the coefficient cn can be calculated from the orthogonality of the polynomials.
2n + 1 Z 1
cn =
f (x)Pn (x)dx
(A.12)
2
−1
47
A.2
Bifurcations
A continuous-time dynamical system can be described by a general system of
ordinary differential equations (ODEs)
dx
= f (x, λ)
dt
(A.13)
where x is the n-dimensional state vector, f is a smooth vector field, λ is a
parameter and t is time. In steady state, the solution x which is a fixed point
for a certain value of λ should satisfy
(A.14)
f (x, λ) = 0
If the parameter does not change, a trajectory with this fixed point as initial
condition will remain unchanged as time evolves. Adding a small perturbation
δx to x, the linearization of Eq. A.13 around x gives
dδx
= J(x, λ)δx
dt
(A.15)
where J is the Jacobian matrix given by

∂f1
∂x1
J =
 ...
∂f1
∂x1
...
...
...
∂f1
∂xn

... 

(A.16)
∂f1
∂xn
Eq. A.15 has solutions δx = x0 eσt , for which we get an eigenvalue problem for
the complex factor σ = σr + iσi
σδx = J(x, λ)δx
(A.17)
The eigenvalue changes with the control parameter, which may lead to changes
in type or number of solutions. If this happens, we say that the dynamic
system undergoes a bifurcation. Bifurcations always coincides with a change
of local stability. A bifurcation diagram is a graph in which the variation of
the solutions of a particular problem is displayed in the state viariable-control
parameter space. A bifurcation that needs at least m parameters to occur is
called a codimension-m bifurcation. In this paper although we have more than
one parameter in a system, the parameters are changed one by one and the
bifurcation diagrams are shown with respect to only one control parameter
each time, so the study is still focused on codimension-1 bifurcations. The
type of codimension-1 bifurcations depends on how the eigenvalue cross the
imaginary axis. When J has real coefficients, either real eigenvalues or complex
conjugate pairs of eigenvalues cross this axis. There are 3 different types of
48
Figure A.2: Eigenvalues cross the imaginary axis, F. J. Elmer, 1998
stationary bifurcations: saddle-node, transcritical, and pitchfork bifurcations.
Fig. A.2 shows different conditions, in the left panel a real eigenvalue changes
sign which is the case of saddle-node, transcritical and pitchfork bifurcations; a
complex conjugate pair of eigenvalues cross the imaginary axis at the same time
which causes Hopf bifurcation. The saddle-node and transcritical bifurcations
occur in the Stommel 2-box model and pitchfork bifurcation in the Welander
3-box model. There are also Hopf bifurcations in Welander model, but in
this study only the locations of this kind are noted, and we cannot follow the
periodic orbits with our numerical tool.
• Saddle-node bifurcation
The prototype function for this condition is
dx
= λ − x2
dt
(A.18)
√
The fixed point exists only when λ ≥ 0, with which x = ± λ. At λ = 0,
these two branches collide with each other, see Fig. A.3. If we perturb
the system by adding δx to a fixed point and linearize
Eq. A.18, one
√
can find that the points on the upper branch x = λ are stable and the
lower ones unstable. The zero point (0, 0) at which the stable point and
unstable point annihilate is the bifurcation point.
• Transcritical bifurcation
The prototype function for this case is
dx
= x(λ − x)
dt
(A.19)
The trivial fixed point of this system is λ = 0, x = 0. When λ 6= 0, there
are two branches (x = 0, x = λ) for the steady states, of which one is
49
Figure A.3: Bifurcation diagram of the saddle-node (left) and transcritical
(right) bifurcations, F. J. Elmer, 1998
stable and the other is unstable. These two branches change stabilities
when they collide at the fixed point, and the system is said to undergo
a transcritical bifurcation.
• Pitchfork bifurcation
Pitchfork bifurcations are possible in dynamical systems with a reflection
symmetry. The prototype function for this condition is
dx
= λx − x3
dt
(A.20)
or
dx
= λx + x3
(A.21)
dt
Notice that both equations remain unchanged if we change the sign of
x. For Eq. A.20 , three fixed points exist at any , i. e, x = 0, x = ±λ
at any positive λ. In a bifurcation diagram [Fig. A.4] the latter two
branches are stable; while there is only one fixed point x = 0 at negative
λ, this branch is also stable. The bifurcation of Eq. A.20 is said to be
supercritical and in Welander 3-box model a similar structure can be
observed. Eq. A.21 describes the subcritical pitchfork bifurcation and
there is only one stable branch at which λ < 0, x = 0.
• Hopf bifurcation
The Hopf bifurcation requires at least a 2 × 2 system to appear because
there are must be at least two eigenvalues of J. Similar to the pitchfork bifurcations, there exist supercritical and subcritical types. The
50
Figure A.4: Bifurcation diagram of the pitchfork bifurcation, F. J. Elmer, 1998
bifurcation happens when the solution for a particular system undergoes
a transition from a fixed point to a periodic orbit. In the supercritical Hopf bifurcation the oscillation is stable while it is unstable for the
subcritical case. The prototype equations for this case are
dx
= λx − y − δx(x2 + y 2 )
dt
dx
= λx + y − δx(x2 + y 2 )
dt
A.3
(A.22)
(A.23)
Numerical tool: BIFURK and the continuation method
The MATLAB function, BIFURK. M, numerically calculates the changes of
location of bifurcation points and number of steady states for a system which
is defined by a ordinary differential equations
dx
= F (x, λ)
dt
(A.24)
where x is the state vector of the system and F is a vector of function depending
on a control parameter λ. As we aim for the steady states, the starting point
must satisfy
F (x0 , λ0 ) = 0
(A.25)
Some times the initial point is just a zero vector, but that is not always the
case. So it is quite important to check if the point you select fulfills Eq. A.25.
From the right initial condition, BIFURK calculates points on the curve of
steady states, finds bifurcation points, distinguishes pitchfork, saddle-node,
Hopf conditions, and follow the new branches which split from a bifurcation
point. BIFURK also notes the stability of steady states. The continuuation
51
algorithm is relatively simple but sufficient to settle our problems concerned in
this paper. The process consists basically of two steps: find the predictor, add
the corrector to the first step. To find a predictor we construct a tangent to
the one-dimensional manifold in the (x, λ)-space that is given Eq. A.25. The
differential of this equation gives
Dx F · dx + Dλ F · dλ = 0
(A.26)
∂Fi
and
where Dx F is the Jacobian, a n × n matrix with elements Dx Fij = ∂x
j
Dλ F is the vector of the derivatives of F with respect to the parameter λ, i.
i
. The predictor can be easily found through
e. Dλ Fi = ∂F
∂λ
dx = (Dx F )−1 · Dλ F · dλ
(A.27)
with the assumption that (Dx F )−1 is of full rank. Under this condition the
inverse (Dx F )−1 does exist. Then the algorithm can take a step of certain
length h along this tangent by x0i+1 = xi + hdx and λ0i+1 = λi + hλ. The point
reached will in general not lie on the curve F (x, λ) = 0, so we find back to the
solution curve by a few Newton-Raphson iterations. The corrector is found by
Dx F · δx + Dλ F · δλ = −F (x, λ)
(A.28)
To determine the corrector uniquely, we force the corrector to be perpendicular
to the step along the tangent, thus we get an additional equation
dxT · δx + dλ · δλ = 0
(A.29)
Solving these n + 1 equations, the correctors are found and will be added to
the predictor and (λi+1 , xi+1 ) is calculated.
Bifurcation points are found when the real part of eigenvalues of Jacobian
change sign, i.e. a real eigenvalue or a pair of conjugate complex eigenvalues
cross the imaginary axis as mentioned in last section. The version of BIFURK
detects this by checking the sign of determinant of Jacobian which encounters
problem when two eigenvalues change sign at the same time (not for Hopf
bifurcation) and the detection will fail as a result. To avoid this, I changed
the detecting rule in this function as following:
i. Check if the system undergoes a Hopf bifurcation. The code first finds the
conjugated pairs of eigenvalues and then check the sign for both to see if they
change the sign of real parts when the parameter varies from λi to λi+1 ;
ii. When no Hopf bifurcation is found, find out the eigenvalues whose real parts
change sign whenever they exist and notice a bifurcation there. We establish
three files: ’run. m’ calling BIFURK to calculate, ’function. m’ defining the
system with all equations, the Jacobian and the derivatives with respect to the
parameters, and ’ini. m’ setting the step size, maximum branch number, etc.
52
Figure A.5: predictor-corrector principle, R. Seydel (Ed.), 1999
53
Appendix B
List of coefficients
54
55