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The purpose of the research was to investigate the exact solutions of nonlinear shallow water equations associated with planetary equatorial waves corresponding to the Cauchy-Poisson free boundary problem describing the nonstationary motion of an incompressible perfect fluid propagating around a solid circle. We consider water waves for which the ratio of the depth of fluid above the circular bottom to the radius of the circle is small (shallow water).

Planetary waves naturally occur in rotating fluids largely due to the rotation of the planet. These waves affect the planet’s weather and climate by transferring heat towards the poles, and cold air towards the equator. Equatorial planetary waves are a key part of the tropical climate system and they are associated with large-scale perturbations of the atmospheric motion extending coherently around a full longitude circle [

It is also believed that planetary waves are responsible for widespread changes in the climate system, especially in the ozone variation (see e.g. [

It has been discussed in [

In this research, we aim to visualize the set of exact invariant solutions of nonlinear shallow water equations associated with planetary equatorial waves

corresponding to the Cauchy-Poisson free boundary problem describing the nonstationary motion of an incompressible perfect fluid propagating around a solid circle.

We consider a two-dimensional motion of an incompressible perfect fluid which has a free boundary η and a solid bottom represented by a circle of radius R. The fluid is circulating around a solid circle and confined by a free boundary. For modeling purposes, we can assume that the motion of the fluid is irrotational and the pressure on a free boundary is constant. It is postulated that that the fluid depth is small compared to the radius of the circle, as shown schematically in

We introduce polar coordinates x = r cos ( θ ) , y = r sin ( θ ) , where θ is a polar angle, r is the distance from the origin, and we use the following notation: R is the radius of the circle representing the radius of a planet, h ( t , θ ) = h 0 + η ( t , θ ) , where t is time, h 0 is a constant undisturbed level of the equatorial atmospheric layer from the center of the planet, and η ( t , θ ) is the unknown level of disturbance of a free boundary. Hence, irrotational motion of a perfect fluid is confined in the domain

Ω η = { ( r , θ ) : 0 ≤ θ ≤ 2 π , R ≤ r ≤ R + h ( t , θ ) } ,

which is bounded by a solid circular boundary r = R and a perturbed free boundary r = R + h ( t , θ ) . For the sake of simplicity, the atmospheric flow is supposed to be irrotational and pressure on a free boundary r = R + h ( t , θ ) is constant. It is postulated that the unperturbed level of atmospheric “depth” h 0 is small compared to the radius R and the homogeneous gravity field is given by the vector g , which is assumed to be a constant and directed to the center of the circle.

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

v r = − 1 r ∂ ψ ∂ θ , v θ = ∂ ψ ∂ r . (1)

Notice that ψ is a harmonic function in Ω η , since we assumed that the flow is irrotational.

As has been shown in [

∂ 2 ψ ∂ θ 2 + r 2 ∂ 2 ψ ∂ r 2 + r ∂ ψ ∂ r = 0 ( R < r < R + h ) , (2)

ψ ( R , θ , t ) = 0 , (3)

ψ ( R + h , θ , t ) = u ( θ , t ) h , (4)

∂ 2 ψ ∂ t ∂ r − 1 r 2 ∂ h ∂ θ ∂ 2 ψ ∂ t ∂ θ + 1 2 r ∂ ∂ θ [ 1 r 2 ( ∂ ψ ∂ θ ) 2 + ( ∂ ψ ∂ r ) 2 ] + g r ∂ h ∂ θ = 0 , ( r = R + h ) , (5)

r ∂ h ∂ t + ∂ ∂ θ ( u h ) = 0 , ( r = R + h ) , (6)

where the average velocity u ( θ , t ) is defined as

u ( θ , t ) = 1 h ∫ R R + h v θ ( r , θ , t ) d r = 1 h ψ ( R + h , θ , t ) . (7)

Shallow Water ApproximationIn order to reduce the number of parameters, we introduce dimensionless variables:

θ = θ ^ , r = R + h 0 r ^ , h = h 0 h ^ , t = R t ^ g h 0 , ψ = h 0 g h 0 ψ ^ , u = g h 0 u ^ . (8)

We next introduce the parameter

ε = h 0 R . (9)

Of course, water is shallow if the parameter ε is small. So, in the present model (2)-(6), the functions η ( θ , t ) and ψ ( r , θ , t ) are two unknown functions whereas 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 R as a natural physical scale since, in the frame of the present model. We consider waves with wavelengths of the order of the radius of a planet.

As has been shown in [

series expansion: ψ = ∑ n ε n ψ ( n ) and application of the Lagrange’s method allows

one to reduce the Cauchy-Poisson free boundary problem (2)-(6) to the following nondimensional system of nonlinear shallow water equations, which are a higher-order analogue of the Su - Gardner equations ( [

u t + u u θ + h θ + ε 2 ( 3 h u t − u h t − u 2 h θ + 2 h h θ ) = 0 , (10)

h t + u h θ + h u θ + ε h h t = 0 , (11)

where the subscripts denote partial derivatives in which independent variables t and θ denote the time and the polar angle, respectively, the dependent variables are the average velocity u and the level ε ≪ 1 of the atmosphere perturbed from h 0 , whereas ε ≪ 1 is a small parameter.

We eliminate u t and h t from the terms of the Equations (10)-(11) and ignore the terms with ε 2 . This leads to the following system:

u t + u u θ + h θ − ε h ( u u θ + h θ 2 ) = 0 , (12)

h t + u h θ + h u θ − ε h ( u h θ + h u θ ) = 0. (13)

Due to the fact that ε is a small parameter, the terms of order 0 ( ε ) in (12)-(13) can be considered as small perturbation to the zeroth order terms (unperturbed model). Our main concern is a simplified version of the model, in which the perturbations (nonlinear terms at ε ) are ignored and we consider the following unperturbed system:

u t + u u θ + h θ = 0 , (14)

h t + u h θ + h u θ = 0. (15)

It can be checked by direct substitution that one particular exact solution of the unperturbed shallow water system (14)-(15) is

h 0 = ε , u 0 = − Γ 2 π ε ln ( 1 + ε ) , (16)

where Γ = const . is the intensity of the vortex (source) localized at the center of the planet and is related with the rotation rate of the planet (for example, the angular velocity of the earth is Ω = 2 π rad / day ≈ 0.73 × 10 − 4 s − 1 ) by the equation Γ = 2 π Ω R 2 . The solution (16) is visualized in

The exact solution ( h 0 , u 0 ) given by (16) is shown in

the free surface h and the fixed origin and the angle θ , as demonstrated by ^{8}. Since Γ is a constant, without loss of generality we set it to make u 0 to be normalized to one. At certain extent, the above ansatz (16) can be associated with the polar vortex, which represents a very powerful whirlpool swirling steadily around the planet’s poles at all times. The term polar vortex is used to describe several different features in the atmosphere. It most commonly refers to a planetary-scale mid- to high-latitude circumpolar circulation (see [

than the diameter of Earth (about 12,700 km), which is also shown schematically in this Figure.

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 phenomenon showing waves that can be seen traveling along hexagon remains unexplained.

Understanding and predicting break downs and overall dynamic structure of the polar vortex is important for improving seasonal forecasting (see also [

Up to the present days, our knowledge of gravity wave sources and properties in the polar region is very much limited because collecting the observations is generally difficult because of harsh natural environments ( [

Computation of invariant solutions of the shallow water system (14)-(15) is based on finding the infinite-dimensional Lie algebra.^{1}

Basic concepts from Lie group analysis of differential equations used in the present paper are assembled here. If the reader wants to have more information about Lie groups and their applications in the theory of differential equations he has a big choice of classical and modern texts in this field, e.g. [

Definition of one-parameter groups. Let

z ¯ i = f i ( z , a ) , i = 1 , ⋯ , N , (17)

be a one-parameter family of invertible transformations of points z = ( z 1 , ⋯ , z N ) ∈ ℝ N into points z ¯ = ( z ¯ 1 , ⋯ , z ¯ N ) ∈ ℝ N . Here a is a real parameter from a neighborhood of a = 0 , and we impose the condition that Transformation (17) is an identity if and only if a = 0 , i.e.,

f i ( z , 0 ) = z i , i = 1 , ⋯ , N . (18)

The set G of transformations (17) satisfying Condition (18) is called a (local) one-parameter group of transformations in ℝ N if the successive action of two transformations is identical to the action of a third transformation from G, i.e., if the function f = ( f 1 , ⋯ , f N ) satisfies the following group property:

f i ( f ( z , a ) , b ) = f i ( z , c ) , i = 1 , ⋯ , N , (19)

where

c = φ ( a , b ) (20)

with a smooth function φ ( a , b ) defined for sufficiently small a and b. The group parameter a in the transformation (17) can be changed so that the function (20) becomes c = a + b . In other words, the group property (19) can be written, upon choosing an appropriate parameter a (called a canonical parameter) in the form

f i ( f ( z , a ) , b ) = f i ( z , a + b ) . (21)

Group Generator. Let G be a group of transformations (17) satisfying the condition (18) and the group property (21). Expanding the functions f i ( z , a ) into Taylor series near a = 0 and keeping only the linear terms in a, one obtains the infinitesimal transformation of the group G:

z ¯ i ≈ z i + a ξ i ( z ) , (22)

where

ξ i ( z ) = ∂ f i ( z , a ) ∂ a | a = 0 , i = 1 , ⋯ , N . (23)

The first-order linear differential operator

X = ξ i ( z ) ∂ ∂ z i (24)

is known as the generator of the group G.

Invariants. A function J ( z ) is said to be an invariant of the group G if for each point z = ( z 1 , ⋯ , z N ) ∈ ℝ N is constant along the trajectory determined by the totality of transformed points z ¯ : J ( z ¯ ) = J ( z )

The function J ( z ) is an invariant of the group G with Generator (24) if and only if

X ( J ) ≡ ξ i ( z ) ∂ J ∂ z i = 0. (25)

Hence any one-parameter group has exactly N − 1 functionally independent invariants (basis of invariants). One can take them to be the left-hand sides of N − 1 first integrals J 1 ( z ) = C 1 , ⋯ , J N − 1 ( z ) = C N − 1 of the characteristic equations for linear partial differential Equation (25). Then any other invariant is a function of J 1 ( z ) , ⋯ , J N − 1 ( z ) .

Invariant equations. We say that a system of equations

F k ( z ) = 0 , k = 1 , ⋯ , s (26)

is invariant with respect to the group G (or admits the group G) if the transformations (17) of the group G map any solution of Equation (26) into a solution of the same equations, i.e.,

F k ( z ¯ ) = 0 , k = 1 , ⋯ , s (27)

whenever z solves Equation (26). The group G with the generator (24) is admitted by Equation (26) if and only if

X ( F k ) | ( 26 ) = 0 , k = 1 , ⋯ , s , (28)

where the symbol | ( 26 ) means evaluated on the solutions of Equation (26).

If z is a collection of independent variables x = ( x 1 , ⋯ , x n ) , dependent variables u = ( u 1 , ⋯ , u m ) and partial derivatives u ( 1 ) = { u i α } , u ( 2 ) = { u i j α } , ⋯ of u with respect to x up to certain order, where

u i α = ∂ u α ∂ x i , u i j α = ∂ 2 u α ∂ x i ∂ x j , ⋯ (29)

then (26) is a system of partial differential equations

F k ( x , u , u ( 1 ) , ⋯ ) = 0 , k = 1 , ⋯ , s . (30)

Furthermore, if the transformations (17) are obtained by the transformations of the independent and dependent variables

x ¯ = f ( x , u , a ) , u ¯ = g ( x , u , a ) (31)

and the extension of (31) to all derivatives u ( 1 ) , etc. involved in the differential equations (30), then Equation (27) define a group G of transformations (31) admitted by the differential Equation (30). In other words, an admitted group does not change the form of the system of differential Equation (30). The generator of the admitted group G is termed an infinitesimal symmetry (or simply symmetry) of the differential Equation (30). Equation (28) serves for obtaining the infinitesimal symmetries and is known as the determining equations. These equations are linear and homogeneous and therefore the set L of its solutions is a vector space. Integration of determining equations often provides several linearly independent infinitesimal symmetries. Moreover, the determining equations have a specific property that guarantees that the set L is closed with respect to the commutator [ X 1 , X 2 ] = X 1 X 2 − X 2 X 1 . Due to this property L is called a Lie algebra. If the dimension of the vector space L is equal to r, the space is denoted by L_{r} and is called an r-dimensional Lie algebra. An r-dimensional Lie algebra L_{r} generates a group depending on r parameters which is called an r-parameter group.

Invariant solutions. Let the differential Equation (30) admit a multi-parameter group G, and let H be a subgroup of G. A solution

u α = h α ( x ) , α = 1 , ⋯ , m (32)

of Equation (30) is called an H-invariant solution (termed for brevity an invariant solution) if Equation (32) are invariant with respect to the subgroup H. If H is a one-parameter group and has the generator X, then the H-invariant solutions are constructed by calculating a basis of invariants J 1 , J 2 , ⋯ .

The functionally independent invariants of the operator

X = t ∂ ∂ θ + ∂ ∂ u (33)

provide the following particular exact solution of the system (14)-(15):

u ( t , θ ) = 1 t ( θ + B ) , h ( t , θ ) = A t , (34)

where A andB are arbitrary constants.

The exact solution (34) is visualized in ^{8} attained at θ = 2 π . Then, due to periodic nature of the flow, the velocity goes to the minimum value right after θ = 2 π and increases as values of θ increase.

The term similarity solution refers to invariant solutions based on scaling symmetries [

X = θ ∂ ∂ θ + 2 h ∂ ∂ h + u ∂ ∂ u ,

which provides the following functionally independent invariants:

J 1 = t , J 2 = u θ , J 3 = h θ 2 . (35)

Accordingly, the system (14)-(15) admits the invariant solution in the form

u = θ U ( t ) , h = θ 2 H ( t ) . (36)

Direct substitution of the presentation (36) in the system (14)-(15) yields the following nonlinear ordinary differential equations:

( d U d t ) 2 + 2 H = 0 , (37)

( d H d t ) 2 + 3 U H = 0. (38)

In case when H ≠ 0 , the Equation (38) can be written as the following coupled equations

H = e − 3 W , U = d W d t , (39)

where W satisfies the nonlinear differential equation of the second order,

d 2 W d t 2 + ( d W d t ) 2 + 2 e − 3 W = 0. (40)

Integration of the Equation (40) yields

d W d t = 2 e − 3 W 2 1 + k e W , (41)

where k is a constant and so W is given implicitly by equation

∫ e 3 W / 2 1 + k e W d W = ± 2 ( t − t 0 ) , (42)

where t 0 is an arbitrary constant.

The equation (42) provides an implicit representation of the function W ( t ) and hence the functions U ( t ) and H ( t ) is (36) due to the Equation (39). We calculate the integral and distinguish the following three cases:

When k = 0 the Equation (42) has the form

F 0 ( W ) : = e 3 W 2 = ± 3 ( t − t 0 ) . (43)

In this case, the Equation (36) provide us with the solution

H = 1 9 ( t − t 0 ) 2 , U = 2 3 ( t − t 0 ) (44)

and thus the exact solution of the system (14)-(15) is given by

u = 2 θ 3 ( t − t 0 ) , h = θ 2 9 ( t − t 0 ) 2 ⋅ (45)

The exact solution (45) is visualized in _{0}, functions u and h depends on θ only. Correspondingly, for the fixed values of t and t_{0}, the solution (34) is visualized as θ -dependent flow confined in an θ -dependent unperturbed circular layer. We visualize the flow in the same color scheme as in

We observe the same qualitative behavior of u and h as in

When k > 0 , the functions U and H are given by

U = ± 2 e − 3 W 2 1 + k e W , H = e − 3 W , (46)

where k > 0 and W is given implicitly by equation

F + ( W ) : = e W 2 1 k + e W − 1 k ln ( e W 2 + 1 k + e W ) = ± 2 ( t − t 0 ) . (47)

In particular, if the function H is known, the function u can be expressed in terms of H as

u = ± 2 θ H 1 + k H − 1 / 3 . (48)

When k < 0 , the functions U and H are given by

U = ± 2 e − 3 W 2 1 + k e W , H = e − 3 W , (49)

where k < 0 and W is given implicitly by equation

F − ( W ) : = − e W 2 − 1 k − e W − 1 k arcsin ( − k e W 2 ) = ± 2 ( t − t 0 ) . (50)

The function F − ( W ) is defined subject to constraint

| k | < e − W . (51)

For example, the function F − ( W ) is not defined for k = 0.1 .

Approximation of the similarity solution

We note that the function F + ( W ) represents a better approximation of F 0 ( W ) for larger values of | k | whereas the function F − ( W ) represents a better approximation of F 0 ( W ) for smaller values of | k | , which is demonstrated in

This observation hallows us to approximate the implicit presentations (47) for F + and (50) for F − by their linear approximations in k in the vicinity of the

point k = 0 at which, according to (43), e 3 W 0 2 = ± 3 ( t − t 0 ) , where W 0 = W | k = 0 is constant for some fixed values of t_{0} and t. Thus

W 0 = 2 3 ln ( t − t 0 ) 3 , t > t 0 . (52)

Consider the case when k > 0 . We denote

m 1 = ± 2 P , α = 1 P ( ∓ 2 t 0 − a 1 e W 0 2 ) + W 0 , (53)

where

P = 1 4 e W 0 2 a 1 ( − e W 0 2 + ( k e W 0 + 2 k 2 e W 0 ) [ a 2 + e W 0 2 ] ) ( 1 + k e W 0 ) ( e W 0 2 + a 2 ) a 1 k + 1 2 e W 0 2 a 1 , (54)

a 1 = 1 + k e W 0 − ln ( e W 0 2 + 1 + k e W 0 k ) k (55)

and W 0 is given by (52).

Thus, expanding F + ( W ) in Taylor series in the vicinity of the point k = 0 , we get the following linear approximation for W:

W = m 1 t + α , k > 0. (56)

Now we consider the case when k < 0 . We denote

m 2 = ± 2 T 2 , β = 1 T 2 ( ∓ 2 t 0 + T 1 ) + W 0 , (57)

where

T 1 = e W 0 2 b + 1 k arcsin [ − k e W 0 2 ] , (58)

T 2 = 1 2 e W 0 2 ( 1 + k [ b 2 + e W 0 ] + 2 k 2 b 2 e W 0 ) ( − k ) 1 4 b 3 2 k , (59)

in which

b = − 1 + k e W 0 k . (60)

and W 0 is given by (52).

Thus, expanding F − ( W ) in Taylor series in the vicinity of the point k = 0 , we get the following linear approximation for W:

W = m 2 t + β , k < 0. (61)

In this paper, we have visualized some exact invariant solutions of the nonlinear simplified version of the shallow water equations

u t + u u θ + h θ = 0 , (62)

h t + u h θ + h u θ = 0 , (63)

which are being used to simulate equatorial atmospheric waves of planetary scales. Our model is represented by the Cauchy-Poisson free boundary problem on the nonstationary motion of a perfect incompressible fluid circulating around a vortex field approximated by a circle of a large radius and the gravity is directed to the center of the circle. We also found some additional invariant solutions based on similarity solutions. In particular, the nontrivial forms of the motion can be represented by the following linear combination of the Galilean and scaling infinite-dimensional Lie algebra of operators:

X = ( 2 θ − 6 t u ) ∂ ∂ t + ( 6 h t − 3 t u 2 ) ∂ ∂ θ + 4 h u ∂ ∂ h + ( 4 h + u 2 ) ∂ ∂ u . (64)

The operator (64) provides a more complicated form of h than that given in (36), namely for

h = m H 1 / 3 + θ 2 H ( t ) , m = const . (65)

We will study more complicated forms of similarity solutions based on the operator (64) in the forthcoming studies.

We also remark that the exact solutions (34) and (45) represent only “local” physically relevant solutions, i.e. they valid for small variations of θ and t. Also, these solutions are associated with the unperturbed version of the complete

system (12)-(13). So, this might result in solutions that look “non-physical”. Our goal for the forthcoming studies is to investigate the more complex forms of the similarity solution like in (65) and we also aim to study numerically the complete nonlinear model (12)-(13) by using the perturbation analysis, in which the invariant solutions obtained in this paper will serve at the unperturbed solutions. We hope to find the perturbed solutions that could be associated with more physically relevant solutions like in schematic shown in

We would like to thank Professor Derin Wysham (Department of Mathematics, Wenatchee Valley College) for his great help in attracting undergraduate students to this research project and useful discussions. Our grateful thanks are also extended to the Dean of STEM & Social Sciences at Wenatchee Valley College, Holly Bringman for providing us with all the necessary facilities for the research, great support, and encouragement.

The authors declare no conflicts of interest regarding the publication of this paper.

Ibragimov, R.N., Mongrain, L.D., Stimmel, B., Trozkaya, O., Lin, G., Zhang, S., Vatchev, V. and Stankiewicz, D. (2021) Visualization of Exact Invariant Solutions Associated with Atmospheric Waves in a Thin Circular Layer. Journal of Applied Mathematics and Physics, 9, 901-919. https://doi.org/10.4236/jamp.2021.95061