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LongitudinalDimensionsofPolygon-shaped
PlanetaryWaves
ArticleinJournalofAppliedNonlinearDynamics·June2015
DOI:10.5890/JAND.2015.06.005
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Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
Journal of Applied Nonlinear Dynamics
https://lhscientificpublishing.com/Journals/JAND-Default.aspx
Longitudinal Dimensions of Polygon-shaped Planetary Waves
Ranis N. Ibragimov1†and Guang Lin2
1 GE
Global Research 1 Research Circle, Niskayuna, NY 12309, USA
Mathematics, and School of Mechanical Engineering, Purdue University, West Lafayette,
IN 47907, USA
2 Department of
Submission Info
Communicated by Lev Osctrovsky
Received 9 July 2014
Accepted 1 March 2015
Available online 1 July 2015
Keywords
Shallow water approximation
Planetary hexagon-shaped waves
Free boundary problem
Atmospheric modeling
Abstract
Polygon-shaped longitudinal large-scale waves are described by means
of higher-order shallow water approximation corresponding to the
Cauchy–Poisson free boundary problem on the stationary motion of
a perfect incompressible fluid circulating around a circle. It is shown
that there are four basic physical parameters, which exert an influence
on a wave number (or wave length), which is one of the basic values
used to characterize the planetary flow pattern in mid-troposphere.
Some analogy with the jet-stream following hexagon-shaped path at
Saturn’s north pole is observed.
©2015 L&H Scientific Publishing, LLC. All rights reserved.
1 Introduction
Planetary waves are large-scale perturbations of the atmospheric dynamical structure that extend
coherently around a full longitude circle. They are important because they have significant influence on
the wind speeds, temperature, distribution of ozone, and other characteristics of the middle atmosphere
structure. They also play an important role in global climate control and weather prediction [1–3].
In oceanographic applications, understanding of the atmospheric processes mechanisms have greatly
increased due to microstructures measurements over the past two decades. In terms of mathematical
modeling, the large-scale atmospheric dynamics is usually described by moving air masses on a sphere or
circle by means of three and two dimensional Navier–Stokes or Euler equations a thin rotating spherical
shell (see e.g. [4–12] ) or within the theory of shallow water approximation [13–20]. Particularly, a largescale two-dimensional modeling with the inclusion of a spherical shape can be associated e.g. with the
eastward moving, wave of warm water, known as a Kelvin wave that can be seen traveling eastward
along the equator as shown in the left Panel of Figure 1 (see also [21–23]). Another spectacular
example of circulating waves is demonstrated on the right panel of Figure 1 showing a jet stream that
follows a hexagon-shaped path at the north pole of Saturn. The hexagon was hidden in darkness
† Corresponding
author.
Email address: [email protected]
ISSN 2164 − 6457, eISSN 2164 − 6473/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved.
DOI : 10.5890/JAND.2015.06.005
154
Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
Fig. 1 Left: Sea-level height data from November 2009 showing the dynamics of warm water known as Kelvin
waves that can be seen traveling eastward along the equator (black line) in Nov. 01, 2009 image. El Ninos form
when trade winds in the equatorial western Pacific relax over a period of months, sending Kelvin waves
eastward across the Pacific like a conveyor belt. Image credit: NASA/JPL. Right: Image from Cassini, made
possible only as Saturn’s north pole emerged from winter darkness, shows new details of a jet stream that
follows a hexagon-shaped path and has long puzzled scientists (source:
http://saturn.jpl.nasa.gov/video/videodetails/?videoID=200)
during the winter of Saturn’s long year, a year that is equal to about 29 Earth years. But as the planet
approached its August 2009 equinox and signaled the start of northern spring, the hexagon was revealed
to Cassini’s cameras. This is the first time the whole hexagonal shape has been mapped out in visible
light by Cassini, and these images show unprecedented details of Saturn’s high northern latitudes. The
hexagon was originally discovered in images taken by Voyager spacecraft in the early 1980s. Since
2006, the Cassini Visual and Infrared Mapping Spectrometer (VIMS) instrument has been observing
the hexagon at infrared wavelengths, but at lower spatial resolution than these visible light images.
This image also shows another unexplained phenomena such as waves that can be seen traveling along
hexagon. Scientists think the hexagon is a meandering jet stream at 77 degrees north latitude, but
they don’t know what controls the path the stream takes.
Multiple images acquired by the VIMS instrument over a 12-day period showed that the feature
is nearly stationary and is likely an unusually strong pole-encircling planetary wave that extends deep
into the atmosphere. Scientists had speculated that a large vortex seen outside the hexagon during
the Voyager observations exerted forces on the jet stream making it adopt a hexagonal pattern in a
manner similar to how jet streams on Earth divert around high-pressure systems. However, in these
new images, the vortex is notably absent while the hexagon persists almost 30 years after it was first
seen. The images were taken in visible light with the Cassini spacecraft wide-angle camera on Jan.
3, 2009. The images were obtained at a distance of approximately 764,000 kilometers (475,000 miles)
from Saturn. The smallest resolved features at the latitude of the hexagon have a horizontal scale
of approximately 100 kilometers. Recent laboratory experiments in [24] suggest that the observed
Saturn’s North Polar Hexagon might result from the stabilization of a standing waves caused by the
difference in angular velocity. However, because of the complex atmospheric structure in Saturn, the
provided experiments do not provide the clear answers and these waves and the six-sided shape of the
Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
155
jet stream remain a mystery up to the date. The information about this phonemenon can be fouind at
http://saturn.jpl.nasa.gov/video/videodetails/?videoID=200.
The primary focus of this paper is to a show that longitudinal large-scale waves within a central
gravity field might also provide a similar polygon-shaped structure when looking from above the North
Pole, as shown in Figure 1 by means of two-dimensional free boundary large-scale shallow water approximation describing a simple atmospheric motion around an equatorial plane (see also [25]). As
has been discussed in [26] and [27], the mathematical model can be derived from the assumption that
the atmosphere is approximated by a perfect fluid and its motion is irrotational and pressure on a free
boundary is constant. It is also postulated that the fluid depth is small compared to the radius of the
circle and the gravity vector is directed to the center of the circle. In the first approximation, shallow
water equations represent the mathematical theory that can be used to investigate the fluid flows in
channels (see e.g. [28–30]). However, as has been discussed in [27], this theory does not reveal the role
the role of an undisturbed level of the fluid surface which is needed to determine the precision of the
first approximation. A higher order approximation is derived in this work.
2 The model
We introduce polar coordinates x = r cos θ , y = r sin θ and use the following notation: R is the radius of
the Earth, θ is a polar angle, r is the distance from the origin, h = h0 + η (t, θ ) , where h0 is undisturbed
level of atmosphere above the Earth and η (t, θ ) is the level of disturbance of a free boundary, as shown
schematically in Figure 2. It is supposed in what follows that θ ∈ [0, 2π ] while r ∈ [R, h (t, θ )] . The
→
homogeneous gravity field −
g is assumed to be a constant and directed to the center of the Earth. The
restriction θ ∈ [0, 2π ] appears for the following reason: the velocity potential ϕ (ς ) can be introduced
by the analyticity of the complex potential ϑ (ς ) = ϕ + iψ , where ς = reiθ is the independent complex
variable and ψ (ς ) is the stream function. Correspondingly, the complex velocity d ϑ /d ς is a singlevalued analytic function of ς , although ϑ is not single-valued. In fact, when we turn around the bottom
2´π
r = R once, ϕ increases by − ∂∂ψr (R, θ ) d θ which has a positive sign by the maximum principle (Hopf’s
0
lemma). Hence, if we remove the width of annulus region θ = 0, r ∈ [R, R + h0 ] , then at every point
(r, θ ) , the complex potential ϑ (ς ) is a single-valued analytic function.
→
We start with the usual assumption that the velocity field −
v = (vr , vθ ) satisfies the Euler’s equations
and the no-leak condition vr = 0 on a solid bottom r = R. We also assume the kinematic condition on
the free boundary. Namely, the velocity on the free boundary r = R + h (t, θ ) is tangential to the free
boundary. We define the free boundary by equation f = r − h (θ ,t) = 0 so that the kinematic condition
is written as
∂f −
df
=
+→
v ∇ f = 0,
(1)
dt
∂t
where
∇=
∂ 1 ∂
,
∂r r ∂θ
.
(2)
In what follows, it is assumed that the fluid motion is potential in the domain of the motion which
allows to introduce the stream function ψ (t, r, θ ) via
vr = −
1 ∂ψ
,
r ∂θ
vθ =
∂ψ
.
∂r
(3)
So that the no-leak condition on the solid boundary can be written as ψ (R, θ ,t) = 0 whereas the
kinematic condition (1) takes the form
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Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
h (θ,t)
R
g
θ
r
h0
0
Fig. 2 Schematic showing a longitudinal atmospheric motion circulating around the Earth.
∂h 1 ∂ψ 1 ∂ψ ∂h
+
+
= 0.
∂t r ∂θ r ∂r ∂θ
Since
1
vr = −
r
R+h
ˆ
R
∂ vθ
dr,
∂θ
(4)
(5)
we can also write Eq. (4) at the free boundary r = R + h as the mass balance equation. i.e.,
1 ∂
∂h
+
∂t R + h ∂θ
R+h
ˆ
vθ dr = 0.
(6)
R
Following [27], we next define the average velocity u (θ ,t) as
1
u (θ ,t) =
h
R+h
ˆ
R
1
vθ (r, θ ,t) dr = ψ (R + h, θ ,t) .
h
(7)
In terms of the average velocity, the kinematic condition (4) is written as
∂h 1
+
∂t r
R+h
ˆ
R
1 ∂ψ ∂h
∂ vθ
dr +
= 0.
∂θ
r ∂r ∂θ
Finally, the dynamic condition is obtained from the requirement that the pressure p is constant at the
free boundary r = R + h (θ ,t). Thus, projecting of the impulse equation
→
1 →2
1
∂−
v
→
+ ∇( |−
v | ) + ∇p = −
g,
∂t
2
ρ
(8)
Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
on the tangential vector
157
1 ∂h
−
→
τ =(
, 1)
r ∂θ
to the free boundary yields
1 ∂ 1 2
1 1 ∂h ∂ p 1 ∂ p
1 ∂ h ∂ vr ∂ vθ 1 ∂ h ∂ 1 2
+
+
vr + v2θ +
vr + v2θ + (
+
) = 0,
r ∂θ ∂t
∂t
r ∂θ ∂r 2
r ∂θ 2
ρ r ∂θ ∂r r ∂θ
(9)
where ρ is a constant fluid density.
Thus, since p|r=R+h = const. and ψ is the harmonic function at the domain of the fluid motion, the
model describing a longitudinal atmospheric motion around the Earth can be written as the following
free boundary problem:
2
∂ 2ψ
∂ψ
2∂ ψ
= 0 (R < r < R + h),
(10)
+
r
+r
2
2
∂θ
∂r
∂r
ψ (R, θ ,t) = 0,
(11)
ψ (R + h, θ ,t) = u (θ ,t) h,
(12)
1 ∂ h ∂ 2ψ
1 ∂ 1 ∂ψ 2
∂ 2ψ
∂ψ 2 g ∂h
− 2
) ]+
+
[ 2(
) +(
= 0, (r = R + h) ,
(13)
∂ t ∂ r r ∂ θ ∂ t ∂ θ 2r ∂ θ r ∂ θ
∂r
r ∂θ
∂h
∂
+
(uh) = 0, (r = R + h).
(14)
r
∂t ∂θ
One can check by direct differentiation that there exists an exact stationary solution to the model
(10)–(14) given by
Γ
r
(15)
h0 = 0, ψ0 = − log( ),
2π
R
where Γ = const.is intensity of the vortex (source) localized at the center of the earth and is related with
the the rotation rate of the earth (angular velocity Ω = 2π rad/day ≈ 0.73 × 10−4 s−1 ) by the equation
Γ = 2π ΩR2 .
The solution (15) corresponds to the singular constant flow with an undisturbed free surface with
the vortex localized at the origin. However, the vortex is isolated since R represents a solid boundary.
Thus the exact solution (15) can be visualized as a flow whose streamlines are concentric circles with
the common center at the origin. Understanding of singular flows were conduced in [13,31,32] and [14].
As has been remarked in [33], the computational experiments in the latter papers provide a credible
evidence to support the conclusion that singular solutions may exist on a stationary sphere in terms of
shallow water approximation. We remark that, in terms of physical interpretation, the fluid particles
at the North and South Poles spin around themselves at a rate Ω = 2π rad/ day, whereas fluid particles
in the
polar
domain θ ∈ [θ0 , π − θ0 ] do not spin around themselves but simply translate provided
π
θ0 ∈ 0, 2 . Thus the physically possible atmospheric motion rotating around the poles correspond to
the flows that are being translated along the equatorial plane (Ibragimov, [9, 10]).
At certain extent, the above ansatz (15) can also be associated with such atmospheric phenomena
as illustrated in Figure 3 which is used to show the NASA images of a polar vortex on Venus (Left
panel) and clouds circling over Saturn’s north pole (Right panel).
Particularly, it is believed that the polar vortex as shown in the left panel of Figure 3 is a very
powerful whirlpool swirling steadily around the planet’s poles at all times. It might be caused by a
gigantic hurricane with two calm, dark eyes. This double-eyed feature, dubbed the “dipole of Venus,”
was thought to form when warm air from the planet’s equator rose and traveled toward the pole, where
it cooled and sank to form a deep, swirling atmospheric pit. For decades, astronomers expected to find
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Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
Fig. 3 Left: Recent pictures of a polar vortex on Venus which is attributed to cloud formations on the palnet
leaving an unexplained dark hole. It has been iscovered at Venus’ north pole by the Pioneer Venus spacecraft in
1979. Credit: ESA/Virtis/INAF-IASF/Obs. de Paris-LESIA Right: Picture taken by the Cassini spacecraft of
clouds circling over Saturn’s north pole (source: http://www.wired.com/2010/09/venus-polar-vortex/)
a similar vortex at Venus’ south pole. While Venus itself rotates slowly, just once every 117 Earth days,
its atmosphere whips around the planet once every four Earth days. This “super-rotating” atmosphere
ought to form massive storms at both poles, astronomers reasoned [34, 35]. The image of the clouds
circling over Saturn’s north pole shown in the right panel of Figure 3 are taken by the Cassini spacecraft
from a distance of about 380,000 kilometers and represents the stunning detail in Saturn’s atmosphere.
Clouds rise and sink and get stretched out, forming long valleys and ridges, streamers circling the
planet’s pole. This vortex is over 2000 kilometers; that’s far bigger than a fully mature hurricane on
Earth, but unlike a terrestrial cyclone, this may be a permanent feature in Saturn’s atmosphere (the
source: http://www.wired.com/2010/09/venus-polar-vortex/; see also [36]).
3 Shallow water approximation
It is useful to recast the model in nondimensional form by introducing the following dimensionless
variables:
R
t
θ = θ, r = R + h0
r, h = h0 ,
h, t = √
gh0
, u = gh0 u.
ψ = h0 gh0 ψ
We next introduce the parameter
(16)
h0
.
(17)
R
Of course, water is shallow if the parameter ε is small. So, in the present model (10)–(14), the
functions η (θ ,t) and ψ (r, θ ,t) are two unknown functions whereas the parameter ε is a given parameter.
Although shallow water theory is usually related to the case when the water depth is small relative
to the wavelengths of the waves, we find it more appropriate to choose the radius of the earth R as a
ε=
Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
159
natural physical scale since, in the frame of the present model, we consider waves with wavelengths of
the order of the radius of the Earth.
The the dynamic condition (13) is then nondimensionalized as follows:
1
1
∂ 2ψ
ε2
∂ 2ψ ∂ h
∂
ε2
∂ψ 2
∂ψ 2
∂h
−
) )+
+
(
(
) +(
= 0.
2
2
∂ t ∂ r (1 + ε h) ∂ t ∂ θ ∂ θ 2 (1 + ε h) ∂ θ (1 + ε h) ∂ θ
∂r
(1 + ε h) ∂ θ
(18)
Following the Lagrange’s method we represent the stream function ψ by the following series expansion:
ψ = ∑ε n ψ (n) . Then the Laplace equation (10) takes the form
n
∂ 2 ψ (0)
∂ 2 ψ (1)
∂ 2 ψ (0) ∂ ψ (0)
)
+
ε
(
+
2r
+
∂ r2
∂ r2
∂ r2
∂r
2 (0)
∂ 2 ψ (0) ∂ 2 ψ (2)
∂ 2 ψ (1)
∂ ψ (1)
∂ ψ (0)
2∂ ψ
+
r
) + 0(ε 3 ) = 0.
+
+
2r
+
r
+
+ε 2 (
∂θ2
∂ r2
∂ r2
∂ r2
∂r
∂r
(19)
A comparison of the terms with the same order ε in equation (19) yields a recurrent system of differential
equations for the determination of all functions ψ (n) , i.e. the Lagrange method consists in presentation
of ψ as the solution of the Cauchy problem with boundary conditions (11)–(12) for ψ (0) and zero
boundary conditions for ψ (1) and ψ (2) .
Thus, up to the order ε 2 , the function ψ is determined as follows:
ψ = ur + ε (u
r
r2
r2
r3
r
r
− uh ) + ε 2 (uh − uθ θ + uθ θ h2 − uh2 ).
2
2
4
6
6
4
(20)
Note that the unknowns u and h are related by the dynamic and conditions (18) and the kinematic
condition (14) which is written in nondimensional form as follows:
∂
1∂ 2
ε h + 2h +
(uh) = 0.
2 ∂t
∂θ
(21)
(1 + ε h)−1 = 1 − ε h + ε 2h2 + 0 ε 3 ,
(22)
Using the Taylor series expansion
and keeping the terms 0 ε 2 , we write the dynamic condition (18) as
∂ 2ψ 1 ∂ 2 ∂ ψ 2
∂ψ 2
∂ 2ψ ∂ h
+
) ) − ε2
(ε (
) +(
∂ t∂ r 2 ∂ θ
∂θ
∂r
∂ t∂ θ ∂ θ
εh ∂ ∂ ψ 2
∂h
∂h
) + ε h )(ε h − 1) +
(
= 0.
(23)
2 ∂θ ∂r
∂θ
∂θ
Substituting ψ given by (20) into equation (23), we arrive at the following system of nonlinear
shallow water equations (higher-order analogue of the Su - Gardner equations [37]):
+(
5h
u
u2
ut − ht + 2huuθ − hθ + hhθ − 2h [ut + uuθ + hθ ])
2
2
2
2
2
h
h
1
2
+ε 2 ( ut − uθ θ t + uθ θ hht − ut θ hhθ + h2 uθ uθ θ
4
3
3
3
3 2
1
h2
+ h uuθ + hhθ uuθ θ − uuθ θ θ + h2 hθ ) + o ε 3 = 0,
4
3
3
∂
1∂ 2
ε h + 2h +
(uh) = 0.
2 ∂t
∂θ
ut + uuθ + hθ + ε (
(24)
(25)
160
Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
Application of Lie Group Analysis ( [12, 38, 39]) shows that there exists an exact singular nonstationary one-parameter invariant solution (an invariant solution of a differential equation is a solution
of the differential equation which is also an invariant curve (surface) of a group admitted by the differential equation. Such solutions can be found without determining its general solution.) of the model
(24)–(25) at zeroth order epsilon, which can be written as
2θ
+ α,
3t
u=
h=(
θ
− α )2 ,
3t
(26)
where α is an arbitrary constant. Sophus Lie proposed for the first time to study the symmetries of
differential equations and use them for constructing solutions at the end of nineteenth century. By a
symmetry we mean a continuous group of transformations acting on the dependent and independent
variables of the system of differential equations so that the system stays unchanged ( [40, 41]). Since
the effects of rotation are not included in this work, the invariant solution (26) is different from the set
of invariant solutions obtained in [40].
These solutions for different values of time t are plotted in Figure 4 versus the polar angle θ , in
which we set α = 10. For example, the exact solution for h at t = 0.1 can be associated with a single
wave oscillating around the equatorial plane, as shown in Figure 5.
Finding the invariant solutions of the complete shallow water system (24)– (25) with ε = 0 will
be the task of the forthcoming project. Unfortunately, any small perturbation of an equation breaks
the admissible group of transformations and reduces the applied value of these ”refined” equations and
group theoretical methods in general. Therefore, development of methods of group analysis stable
with respect to small perturbations of differential equations has become vital. As is discussed in the
Conclusion section, the methods of Approximate Group Analysis will be applied to the given model
(24)–(25).
120
Exact solution
80
100
Exact solution
80
θ=π
U (t =0.1)
H (t = 0.1)
U (t = 0.3)
H (t = 0.3)
U
60
40
H
20
0
0.2
0.4
0.6
0.8
1
time
60
40
20
0
0
1
2
3
θ
4
5
6
Fig. 4 Exac solution of the shallow water model, that results in the limiting case of the system
system(24)–(25) at ε = 0.
Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
161
r
h(θ,t)
R
θ
Fig. 5 Schematic presentation of the invariant solution of the zeroth-order model (26) for h(θ ,t) at t = 0.1.
4 Stationary waves
In order to investigate the polygon-shaped waves having the structure similar to the hexagon as shown
in Figure 1, we analyze the model (10)–(14) in the reference frame moving with a wave so that the
unknown functions do not depend on time. In this case, the two-dimensional model (10)–(14) for two
unknowns ψ (r, θ ) and h (θ ) > 0 is written in non-dimensional variables (16) as
ε2
2
∂ 2ψ
∂ψ
2∂ ψ
= 0 (0 < r < h) ,
+
(1
+
ε
r)
+ ε (1 + ε r)
2
2
∂θ
∂r
∂r
ψ (0, θ ) = 0,
ψ (h, θ ) = Q,
(27)
(28)
ε2
∂ψ 2
∂ψ 2
) + 2gh = 2b, (r = h),
(
) +(
2
(1 + ε r) ∂ θ
∂r
(29)
∂h
∂
+
(uh) = 0, (r = h) ,
∂t ∂θ
(30)
r
where b is the Bernoulli’s constant, Q = h0 u0 is the constant representing a flow rate and, as follows
from (15),
Γ
ln (1 + ε ) .
(31)
u0 =
2π h0
We first observe that the model (27)–(30) can be reduced to the following relation evaluated at the free
boundary:
∂
∂θ
ˆh
[(1 + ε r)(
0
∂ψ 2
ε2 ∂ ψ 2
∂ψ 2
ε2
∂ ψ 2 dh
) −
(
) +
) ] = −(1 + ε h)[(
(
) ] .
2
∂r
1 + εr ∂ θ
∂r
(1 + ε h) ∂ θ d θ
(32)
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Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
In the stationary case, for the equation (25), we have uh = Q = const. Using this relation, we can exclude
u from (24), integrate (32) and substitute the expression for ψ given by (20) to arrive to the following
first-order differential equation for the unknown h (θ ) :
2
1 2 2 dh 2
ε2
ε2
ε Q ( ) = − ε h4 + (ε b − 1)h3 + (2b + Q2 )h2 + ( Q2 − c)h + Q2 ,
3
dθ
3
4
2
(33)
where c is an integrating constant.
We remark that in the limiting case ε → 0 (which corresponds to the case of the flat bottom when
R → ∞), we obtain the cubic Bouusinesq-Rayleigh equation [42]. We denote by h j ( j = 1, 2, 3, 4) the
roots of the polynomial (33). Then the Viète theorem [43] yields the following relation between the
parameters Q, c and b are the roots h j ( j = 2, 3, 4) :
Q2 =
− 23ε + ε2 (h2 h3 + h2 h4 + h3 h4 ) − h2 − h3 − h4
2
ε
− 23ε h2 h13 h4 + 34 H − 16
;
1
1
1
2ε
ε
c = Q2 [ + + + ] − h2 h3 h4 ;
2 h2 h3 h4
3
ε
1
b = Q2 H + (h2 h3 + h2 h4 + h3 h4 ) ,
2
3
where
H=
1
1
1
+
+
.
h3 h4 h2 h4 h2 h3
(34)
(35)
(36)
(37)
Additionally, a simple perturbation analysis shows that formulae (37) imply that h1 represents a nonphysical solution with the following asymptotic: h1 → ∞ as R → ∞. Namely,
h1 = −
3
1
.
2ε h2 h3 h4
(38)
and the roots h2 , h3 and h4 are also the roots of the Bouusinesq-Rayleigh equation in the limiting case
when ε → 0. The existence of non-trivial wave-like solutions corresponds to the case when all the roots
of the polynomial (33) are real and have the values in the interval 0 < h2 h3 < h < h4 . Particularly,
we also remark that, since, according to its physical meaning, h is positive and continuous function,
the domain of the admissible solution is the interval [h3 , h4 ].
Implicitly, the shape of the free boundary is given by the quadrature
εQ
θ=√
3
ˆh
h3
ds
,
(ε s + λ )(s − h2 ) (s − h3 ) (s − h4 )
(39)
where λ > 0 is a constant. Namely, according to the relation (38) for h1 ,
λ=
3 1
.
2 h2 h3 h4
(40)
We next introduce a small finite Jacobi amplitude a = h4 − h3 and assume that a is of order ε . In
this approximation, the change of variable of integration s = h3 + ξ reduces the integral in (39) to the
following asymptotic form:.
ˆh
h3
ds
˜
(ε s + λ )(s − h2 ) (s − h3 ) (s − h4 )
ˆξ
0
dξ
,
(ε h3 + λ ) (h3 − h2 ) ξ (1 − ξ )
(41)
Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
163
which can now be evaluated in terms of elementary functions.
As follows from (41), the polygon-shaped wave structure is described by a 2π −periodic nonlinear
wave and the wavenumber n is determined by the relation
√
3
(ε h3 + λ )(h3 − h2 ) .
(42)
n=
ε
For example, in order to visualize the particular hexagon-shaped path as shown in in Figure 1, we set
n = 6 and visualize the gravity waves by detecting the wave trougs and crests by locating the points of
minimum and maximum of points of minimum of the integral in (39) as shown in Table 1.
Table 1
trougs
θ =0
crests
π
6
θ=
θ=
θ=
π
3
π
2
θ=
θ=
2π
3
5π
6
θ =π
θ=
7π
6
θ=
θ=
4π
3
9π
6
θ=
θ=
4π
3
11π
6
θ = 2π
Schematically, the resulting waves for n = 6 can be visualized as shown in Figure 6, where the scales
are chosen arbitrarily but the points of trougs and crests correspond to values of θ in Table 1.
In terms of the mathematical modeling presented here, the value R should not represent necessarily
the radius of the planet, it can be just a radial scale satisfying the relation R/h0 << 1 so that the waves
can be observed rotating not only around the planet but also around the polar axis, in clockwise or
anticlockwise sense (depending on the sing of Γ) looking above the North Pole.
π/2
5π/6
π/6
R
θ
11π/6
7π/6
9π/6
Fig. 6 Visualization of the hexagon-shaped shallow gravity longitudinal waves.
As seen from the relation (42), the nature of the polygon-shaped free boundary (i.e., the wave
number or wave length) is detected by three physical roots h j ( j = 2, 3, 4) of the polynomial equation
(33). These roots are also the roots of the Bouusinesq-Rayleigh equation that is obtained in the limiting
case when R → ∞. In turns, the four basic parameters, which exert an influence on the roots h j are
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the Bernoulli constant b, flow rate Q, constant of integration c and the parameter ε . The question
naturally arises to whether these parameters can be explained physically. We remark that the values ε
represents the altitude of the unperturbed atmospheric layer (e.g. for the Earth, it can be associated
with troposphere), the value Q is determined by the unperturbed value h0 and the intensity of the vortex
localized at the pole of the planet, the Bernoulli constant b can be detected by evaluating the specific
atmospheric properties (such as the pressure, density and the velocity field) at certain altitude. Finally,
since the parameter c was obtained after the integration of the relation (32), physically it represents the
horizontal component of the impulse flux that is also dependent on the specific atmospheric properties.
Understanding the structure of longitudinal planetary waves depending of the atmospheric properties
have been investigated earlier in meteorological sciences ( [44, 45]).
5 Discussions
In summary, we have identified four basic parameters that control the polygon-shaped stationary planetary longitudinal waves. The analysis is based on the higher-order shallow water approximation, which
allows to reveal the role of the unperturbed level h0 , which is needed to determine the precision of the
first approximation [28,37]. We did not include this analysis in the frame of the present work. However,
the preliminary analysis shows that, at the second -order approximation, splitting phenomenon of the
shallow water system (24)–(25) takes place. Namely, there are two different approaches in deriving the
equation (23). Correspondingly,
there are two different forms of the shallow water systems and the
2
difference is observed at 0 ε terms. This analysis will be presented in our forthcoming work and will
be published elsewhere.
Permanent water waves have been considered in a large number of papers. However, most researchers are concerned with fluid motion which is infinitely deep and extends infinitely both rightward
and leftward (see e.g. [46–48] or [49] for the history). In this work, we focused on the deriving the model
that uses a curved solid bottom and the circular shape of the unperturbed atmospheric layer in order
to visualize a polygon-shaped structure of longitudinal planetary waves. For visualization purposes,
Figure 7 presents the same Saturn’s jet stream that follows a hexagon-shaped path at the north pole
but from the different perspective.
We hope that a polygon-shaped structure and especially the detection of the wave number by the
four physical parameters found is this work might also be of interest for scientists who are trying to
figure out what causes the hexagon, where it gets and expels its energy and how it has stayed so
organized for so long.
Finally, we are interested to find approximately invariant solutions of the higher-order shallow
water model (24)–(25). We found the exact invariant solution (26) for the limiting case when ε = 0.
Methods of classical group analysis allow to single out symmetries with remarkable properties among
all equations of mathematical physics. Unfortunately, any small perturbation of an equation breaks
the admissible group of transformations and reduces the applied value of these ”refined” equations and
group theoretical methods in general. Therefore, development of methods of group analysis stable
with respect to small perturbations of differential equations has become vital. In order to find both
the approximate symmetries and approximately invariant solutions of the model (24)–(25) with ε = 0,
governing equations in question.
In order to find approximately invariant solution, we will use the ASP symmetry packages that
has been developed originally in [50] and employed in our earlier work [38], in which invariant and
approximately invariant solutions of non-linear internal gravity waves forming a column of stratified
fluid affected by the Earth’s rotation were found. The ASP symmetry packages is based on the analytic
algorithm developed in [51] . The same package has also been incorporated in finding the new class
of exact solutions of the Navier-Stokes equations in a thin spherical shell that were associated with
Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167
165
Fig. 7 This view is centered on Saturn’s north pole. North is up and rotated 33 degrees to the left. The image
was taken with the Cassini spacecraft wide-angle camera on June 14, 2013 using a spectral filter.
nonlinear atmospheric flows with west-to-east jets perturbations that can also be visualized as Kelvin
wave that can be seen traveling eastward along the equator as shown in the left panel of Figure ??
(see also [52] ). The goal of these forthcoming studies is to get an explicit analytic expression for h (θ )
without a small-amplitude assumption as has been used in this work.
Acknowledgments
This research was supported in part by an appointment to the U.S. Department of Energy’s Visiting
Faculty Program.
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