Lie group method for solving viscous barotropic vorticity equation in ocean climate models

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Lie group method for solving viscous barotropic vorticity equation in ocean climate models Mina B. Abd-el-Malek *, Amr M. Amin Department of Engineering Mathematics and Physics, Alexandria University, Alexandria 21544, Egypt

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Article history: Received 1 August 2017 Received in revised form 27 October 2017 Accepted 12 November 2017 Available online xxxx Keywords: Viscous barotropic non-divergent vorticity equation Rossby wave Rossby–Haurwitz wave Relative vorticity Absolute vorticity

a b s t r a c t In studying the problem of the nonlinear viscous barotropic non-divergent vorticity equation on f - and β - planes, the method of Lie group has been applied. The method reduces the number of independent variables by one, and consequently, for the case of three independent variables we applied the method successively twice and the nonlinear partial differential equation reduces to ordinary differential equation. Investigation of exact solutions of the viscous barotropic non-divergent vorticity equation on f - and β - planes, via the application of Lie group, provides large classes of new exact solutions which include both Rossby and Rossby–Haurwitz waves as special cases. Also, The Lie symmetries of the viscous barotropic non-divergent vorticity equation with two parameters F and β , are determined. The possible reductions of the viscous barotropic vorticity equation with two parameters F and β have been investigated by means of one- dimensional Lie subalgebras. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction It is very interesting to obtain the exact solutions of nonlinear partial differential equation (PDE). The exact solutions may well describe various phenomena in our life. Also, these exact solutions when they exist can help to understand the dynamical processes that are modeled by the corresponding nonlinear evolution equations (NLEEs). Ocean is a major component of the earth climate system, since ocean has large heat storage. Roughly, three meters of sea water has about the same heat capacity as the whole atmospheric column above it, ocean heat storage modulates diurnal and seasonal cycles and climate variations. Waves in fluids can be simulated with mathematical models based on the equations for conservation of mass, momentum and energy. These equations are the starting point for modeling the propagation of Rossby waves and gravity waves in the atmosphere and ocean. The derivation of the basic equations can be found in any fluid dynamics text such as Batchelor (1967), Pond and Pickard (1983), Pedlosky (1992), Kundu and Cohen (2004), Marshall and plumb (2008), Vallis ( 2011) and Talley, Pickard, Emery and Swift (2011) [1–8]. In geophysical fluid dynamics, the effects of the earth’s rotation and the density-stratification of the basic flow are taken into consideration. There are various types of waves in geophysical flows in the atmosphere and ocean [9,10]. These include internal gravity waves which result from the effects of gravity and the buoyancy of the fluid flow, and Rossby waves which result from the effects of the earth’s rotation. The driving mechanism of Rossby waves is the interaction of the flow with meridional variations of the Coriolis parameter f . Considering f = 2Ω sin ϕ [3–5], which, due to spherical shape of the Earth, depends on latitude ϕ . We can expand f as a Taylor series about the reference latitude ϕ0 , and approximate it by the first two terms. Hence, f = f0 + β y where f0 = 2Ω sin ϕ0 , β = (2Ω /r ) cos ϕ0 , r is the earth’s radius, and Ω the earth’s angular velocity. Typical mid-latitude values are f0 = 8 × 10−5 s−1 and β = 1 × 10−11 m−1 s−1 . From a

*

Corresponding author. E-mail address: [email protected] (M.B. Abd-el-Malek).

https://doi.org/10.1016/j.camwa.2017.11.016 0898-1221/© 2017 Elsevier Ltd. All rights reserved.

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mathematical perspective, Rossby waves can be studied in 2 D using Cartesian coordinates in a plane tangent to the surface of the earth, where x for the West–East (zonal) coordinate and y for the South–North (meridional) coordinate, the domain in this case should be assumed to be infinite allowing the waves to propagate indefinitely southwards. If this approximation of f is applied in a Cartesian framework, this framework is called the f -plane if the β term is neglected, and the β - plane if the beta term is retained. 1.1. Barotropic ocean climate model The viscous barotropic non-divergent vorticity equation model is very important in the field of meteorological science and applied mathematics. Many scientists pay attention to the research of numerical methods of this equation. Most of them mainly apply the finite difference methods [11–13]. The analytical solutions of barotropic vorticity equation early obtained by Ekman (1905), Rossby (1937, 1938), Sverdrup (1947), Stommel (1948), Munk (1950), Charney (1955) and Morgan (1956) [14–21], who among them identified the main dynamical features of the large-scale ocean circulation, previous models details have either considered linear frictional model, e.g., Munk, or nonlinear inviscid models, e.g., Charney and Morgan. In (1963) Moore constructed a model of the general circulation in an ocean basin which includes viscous effects and inertial effects through an Oseen linearization [22]. The nonlinear viscous barotropic non-divergent vorticity equation in stream function form with β -plane approximation reads [3,8]

∇ 2 ψt − ψy ∇ 2 ψx + ψx ∇ 2 ψy + β ψx − ν ∇ 4 ψ = 0,

(1.1)

where ψ is the steam function, the velocity field {u , v} is linked to stream functions by u = −ψy and v = ψx , ∇ 2 ≡

∂2

∂ x2 ∂4 . ∂ y4

+ ∂∂y2 , 2

+ 2 ∂ x2∂ ∂ y2 + Eq. (1.1) describes the fluid motion on a β -plane with viscosity effect in terms of the stream function ψ and the term ξ = ∇ 2 ψ which is known as the relative vorticity, measures the amount of rotation that the fluid undergoes in the x − y

ν is the kinematic viscosity, β =

df dy

is the meridional change of the Coriolis parameter and ∇ 4 ≡

∂4 ∂ x4

4

plane and is given by →

→

ξ = e z · ∇ × u = ∇ 2 ψ,

(1.2)

→

where e z is the unit vector in the z direction. By substituting (1.2) in (1.1), we get

ξt + ψx ξy − ψy ξx + β ψx − υ ∇ 2 ξ = 0.

(1.3)

We notice that value of the Coriolis frequency f0 has dropped entirely out of the equation; this means that any of the results discussed in this paper are independent of f0 and hence independent of the Rossby number. Eq. (1.3) can be written as Dξ Dt

= −β v + υ ∇ 2 ξ .

(1.4)

If there is no rotation, i.e. β = 0, (1.4) resembles the energy conservation equation. In the context of rotating flows is called ξ the potential vorticity or the relative vorticity. The total vorticity η = ξ + f0 + β y, is called the absolute vorticity. Eq. (1.4) represents the conservation of potential vorticity. The generalization of (1.3), it has been successfully used both for theoretical consideration and practical numerical weather predication since it is capable of describing some prominent features of mid-latitude weather phenomena such as the well known Rossby waves and blocking regimes. In non-dimensional form it reads [3]

ξt − F ψt + ψx ξy − ψy ξx + β ψx − υ ∇ 2 ξ = 0,

(1.5)

where F represents the ratio of characteristics length scale to the Rossby radius of deformation. The Lie symmetry of (1.5) arises since different values of F and β leads to different Lie symmetry properties of (1.5), there are only three combination of the values of two parameter, given by F = 0, β = 0; F = 0, β ̸ = 0 and F ̸ = 0. The first combination leads to the vorticity form of Euler’s equations, which has been discussed in [23]. The second combination leads to the barotropic vorticity equation, which has been completely discussed here. For F ̸ = 0 we can set β = 0, then F can be scaled to ±1. For this consideration, we recomputed the symmetries of (1.5) for the case F ̸ = 0 and β arbitrary. Hence, we investigate interesting group invariant reductions for (1.5) based on the obtained symmetries. The boundary condition plays an important role in conception of geophysical fluid dynamics. With 2D ocean models, since the water coming in contact with a solid boundary, therefore it subjects to a no-slip condition, in practice the no slip condition very difficult to resolve in ocean models, so many different approximation applied to boundary condition such that free slip (ξ = 0) on the boundary where is possible and hyper slip (∂n ξ = 0) which has boundary layers that are similar in nature to free slip boundary layers. Munk [18] introduced the solution for (1.3) by considering β v ≈ ν ξxx , Stommel [17] introduced the solution for (1.3) by β considering β ν ≈ −r ξ . Finally Charney [19] introduced the solution for (1.3) by considering ψxx + β y ≈ − U ψ . 0

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Several problems are faced by the scientists who wish to solve the barotropic vorticity equation. Since some of the parameters are due to physical processes which are not well-understood or well defined and others come in because of the simplifying procedure of constructing the model, it is not clear what values one can assign to these parameters a priori. However, the specific values which one chooses will determine the model behavior. The only significant assumption required involves replacing the kinematic viscosity with eddy viscosity appropriate to ocean climate model. The kinematic viscosity of water is about ≈10−6 m2 s−1 for ocean surface temperatures. Unfortunately eddy viscosity numbers are not well known for the ocean. It is common to use vertical eddy viscosities up to the order of 102 m2 s−1 and horizontal eddy viscosity up to the order of 108 m2 s−1 [18–21], depending on the scale of motion. The Lie group method is a powerful and fundamental tool for obtaining the invariant solutions of differential equations. The Lie group method is applicable to both linear and nonlinear differential equations. The mathematical technique in the present analysis is the one-parameter group transformation. The Lie group method expresses the infinitesimals of the group in terms of one or more functions, called the infinitesimal functions, where each of the infinitesimal functions depends on independent and dependent variables. The procedure for finding the infinitesimals then reduces to finding the auxiliary equation which can be obtained by solving a system of coupled linear partial differential equations called the determining equations that arise as a result of invoking invariance of the partial differential equations and its auxiliary conditions. Hence, the Lie group method is applicable to solve a wider variety of nonlinear problems. In this paper we introduce new family of explicit solutions for viscous barotropic non-divergent vorticity equation in β -plane. Using Lie group method; our model considers the general model for all models stated above. We are able to find a complete study of the viscous barotropic non-divergent vorticity equation in β -plane. The results are found analytically and illustrated graphically. Also, we introduce the possible reduction of the viscous barotropic non-divergent vorticity equation with two parameters F and β . 2. Lie symmetry group method Consider the viscous barotropic non-divergent vorticity equation (β BVE ) in Eq. (1.1) as follows:

( ) ( ) ( ) ψxxt + ψyyt + ψx ψxxy + ψyyy − ψy ψxxx + ψyyx + β ψx = υ ψxxxx + 2ψxxyy + ψyyyy .

(2.1)

To specify the symmetry algebra of the differential (2.1), we use the Lie symmetry method. For a general introduction to the subject we refer to [24–28] specifically [25]. Let a partial differential equation contain p independent variables x = (x1 , . . . , xp ), and q dependent variables u = (u1 , . . . , xq ). The one-parameter Lie group of transformation is X =

p ∑ i=1

q

∑ ∂ ∂ ξ (x, u) i + ηλ (x, u) λ . ∂x ∂u i

(2.2)

λ=1

The corresponding one-parameter Lie group of infinitesimals transformations is given by xi∗ = exp(ε X ) = xi + ε ξ i (x, u) + O(ε 2 ), , uλ∗ = exp(ε X ) = uλ + ε ηλ (x, u) + O(ε 2 )

}

where ξ i =

⏐

∂ xi∗ ⏐ ∂ε ⏐ε=0

for i = 1, . . . , p, and ηi =

(2.3)

∂ uλ∗ ⏐ ∂ε ⏐ε=0

⏐

for λ = 1, . . . , q.

on the space of independent and dependent variables. Therefore, the characteristic of the vector field X given by (2.2) is the function Q λ x, u(1) = ηλ (x, u) −

(

)

p ∑

ξ i (x, u)

i=1

∂ uλ , λ = 1, . . . , q. ∂ xi

(2.4)

Let X be a vector field given by (2.2), and let Q = (Q 1 , . . . , Q q ) be its characteristic, as in (2.4). The nth prolongation is given explicitly by Pr(n) X =

p ∑ i=1

ξ i (x, u)

q n ∑ ∑ ( ) ∂ ∂ + ηJλ x, u(j) , i ∂x ∂ uλJ

(2.5)

λ=1 ̸=J =j=0

with the coefficient ηJλ = DJ Q λ + i=1 ξ i uλJ ,i , and uλJ , i = Di (uλJ ), and Di represents total derivative operator described in [25]. The infinitesimal generator of the symmetry group admitted by (2.1) is given by:

∑p

X ≡ ξ1

∂ ∂ ∂ ∂ + ξ2 + ξ 3 + η1 . ∂x ∂y ∂t ∂ψ

(2.6)

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Table 1 Table of commutators of the basis operators. Ad

Γ1

Γ2

Γ1 Γ2 Γ3 (f4 ) Γ4 (f3 )

0 0

0 0 0 ( ) Γ3 f3′

( ) −Γ3 (f4′ ) −Γ4 f3′

Γ3 (f2 ) ( ) Γ3 f2′ 0 0 0

Γ4 (f1 ) ( ) Γ4 f1(′ ) −Γ3 f1′ 0 0

Where f1 , f2 , f3 , f4 are arbitrary smooth functions in t and Γ4 f1′ = f1′ ∂∂x −

( )

∂ y f1′′ ∂ψ

.

Since (β BVE ) equation has at most fourth-order derivatives, we prolong the vector field X to the fourth order. The action of Pr(4) (X ) on Eq. (2.1) must vanish, where ψ is the solution of Eq. (2.1), and then we find the following determining equations:

ξx1 = ξy1 = ξψ1 = 0,

⎫ ⎪ ⎪ ⎪ ⎪ 2 2 2 2 ξx = ξy = ξt = ξψ = 0,⎬ , ξx3 = ξy3 = ξt3 = ξψ3 = 0,⎪ ⎪ ⎪ ⎪ ⎭ ηx1 = 0, ηy1 = −ξt1

(2.7)

Solving (2.7), we get

ξ 1 = f1 (t), ξ 2 = C1 , ξ 3 = C2 , and η1 = −y f1′ (t) + f2 (t),

(2.8)

where f1 (t), f2 (t) are arbitrary smooth functions of t and C1 , C2 are arbitrary constants. Unfortunately, in studying (2.1) by using the generalized equivalence transformation method which discussed in [29], or by using the transformation of dependent and independent variables by scaling, no extra symmetry can be added. Hence, the viscous barotropic non-divergent vorticity equation (2.1) admits the infinite dimensional Lie symmetry algebra β0∞ , the basis generators of β0∞ are

∂ ∂ ∂ ∂ ∂ , Γ2 ≡ , Γ3 (f2 ) ≡ f2 , Γ4 (f1 ) ≡ f1 − y f1′ . (2.9) ∂t ∂y ∂ψ ∂x ∂ψ The physical significance of these generators is as follows: Γ1 generates time translations, Γ2 generates translations in y direction (i.e., serves as translations in the N-S-direction), Γ3 represents gauging the stream function, and Γ4 represents the Γ1 ≡

infinitesimal of generalized transformations on a time-dependent moving coordinate system in x directions. The problem becomes more difficult since we have to deal with an infinite dimensional Lie algebra as two of the infinitesimal generators contain arbitrary functions of the independent variable t. Instead of this, we have only computed solutions that are invariant under a certain one parameter group of transformations. Accordingly, it remains to give the optimal system of one-parameter group-invariant solutions of the viscous barotropic non-divergent vorticity equation by classifying the one-dimensional subalgebras. For the one-dimensional optimal systems, the technique of Olver has been used. Olver [25] constructed a table of adjoint operators to simplify a general element in Lie algebra as much as possible. The additional restriction is that the simplified element of each resulting set of operators must be closed under commutation. Thus, besides simplification by means of the adjoint actions, closure of the Lie algebra structure has to be assured. Ovsiannikov [28] demonstrated the construction of group foliation method using infinite dimensional Lie algebra. The method of group foliation provides a reduction of the original system of PDEs by constructing a set of equations identifying the individual orbits. Solutions found in the space of the reduced coordinates can then be returned to the full space by solving another system of PDEs. It is clear that the symmetry obtained using group foliation method inherits the finite dimensional part of Lie algebra of the original equation [30]. For this reason it is convenient to use Olver method with the restriction stated above. The commutator table of the symmetries is given in Table 1, where the entry in the ith row and jth column is defined as:

[Γi , Γj ] = Γi Γj − Γj Γi

(2.10)

2.1. One-dimensional optimal system of subalgebras of the symmetry group ‘‘Opt’’ We try to get invariant solutions under the linear combination of the operators given by (2.9). By determining the onedimensional optimal system of subalgebras of the given partial differential equations, all of these solutions can be obtained. Olver’s approach [25] starts out by computing the commutators of the symmetry Lie algebra (2.9), which we got in Table 1, and then obtaining the adjoint representations. The adjoint action on Lie algebras is defined by the adjoint operator given by Adexp(ε Γi ) ⟨Γj ⟩ = e−ε Γi Γj e ε Γi ,

(2.11)

where ε is a small parameter. Please cite this article in press as: M.B. Abd-el-Malek, A.M. Amin, Lie group method for solving viscous barotropic vorticity equation in ocean climate models, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.11.016.

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Table 2 Table of adjoint representations. Ad

Γ1

Γ2

Γ3 (f2 )

Γ4 (f1 )

Γ1 Γ2 Γ3 (f4 ) Γ4 (f3 )

Γ1 Γ1 ( ) Γ1 + ε Γ3 (f4′ ) Γ1 + ε Γ4 f3′

Γ2 Γ2 Γ2 ( ) Γ2 − ε Γ3 f3′

( ) Γ3 − ε Γ3 f2′ Γ3 Γ3 Γ3

( ) Γ4 − ε Γ4 (f1′ ) Γ4 + ε Γ3 f1′ Γ4 Γ4

In terms of Lie brackets using Campbell–Baker–Hausdorff theorem [28,31], this operator can be rewritten as

] ε2 [ [ ]] Γi , Γj + Γi , Γi , Γj − · · · (2.12) 2! In our problem ⟨Γ1 , Γ2 , Γ3 , Γ4 ⟩ is the Lie algebra associated with the symmetry group. The calculations of the adjoint Adexp(ε Γi ) ⟨Γj ⟩ = Γj − ε

[

action are summarized in Table 2. Now we will construct the one-dimensional optimal system of ‘‘Opt’’. Consider a general element of ‘‘Opt’’ given by E = a1 Γ1 + a2 Γ2 + Γ3 + Γ4 ,

(2.13) ′

for some constants a1 and a2 , and search for E that can be transformed to a new element E under the general adjoint action, where E′ takes a simpler form than E [28,29]. Let, E′ = Adexp(ε Γi ) ⟨E ⟩ = a′1 Γ1 + a′2 Γ2 + Γ3 + Γ4 .

(2.14)

We make appropriate choice of ai such that a1 and a2 can be made 0 or 1. We end up with simpler forms of E that will constitute the one-dimensional optimal system. Case (i): By setting a1 ̸ = 0, scaling E if necessary, we can assume that a1 = 1, so we get E = Γ 1 + a2 Γ 2 + Γ 3 + Γ 4 .

(2.15)

By substituting Γi = Γ4 (f3 ) in (2.14) E ′ = Γ1 + ε Γ4 f3′ + a2 Γ2 − ε Γ3 f3′

( )

( ))

(

+ Γ3 (f2 ) + Γ4 (f1 ) ,

(2.16)

we can cancel the Γ4 terms by setting ε Γ4 f3 + Γ4 (f1 ) = 0 in (2.16), from which we get

( ′)

E ′ = Γ1 + a2 Γ2 − ε Γ3 f3′

(

( ))

+ Γ3 (f2 ) .

(2.17)

By substituting Γi = Γ3 (m(t)) in (2.14), we get

+ Γ3 (f2 ) , ( ) we can cancel the Γ3 terms by setting ε Γ3 m − a2 ε Γ3 f3′ + Γ3 (f2 ) = 0 in (2.18), from which we get E ′′ = Γ1 + ε Γ3 m′ + a2 Γ2 − ε Γ3 f3′

( )

(

( ))

(2.18)

( ′)

E ′′ = Γ1 + a2 Γ2 ,

(2.19)

where a2 ∈ {−1, 0, 1}, since the viscous barotropic vorticity equation admits the discrete symmetries (t , x , y , ψ ) → (t , x , −y , −ψ ). Hence, we can assume a2 ∈ {0, 1}, the optimal in this case are Γ1 , Γ1 + Γ2 . Case (ii) By setting a1 = 0 and a2 ̸ = 0, scaling E if necessary, we can assume that a2 = 1, so we get E = Γ2 + Γ3 + Γ4 .

(2.20)

By substitution Γi = Γ4 (f3 ) in (2.14) we get E ′ = Γ2 − ε Γ3 f3′ + Γ3 (f2 ) + Γ4 (f1 ) .

( )

(2.21)

We can cancel the Γ3 terms, as we did before, so we get E ′ = Γ2 + Γ4 .

(2.22)

Case (iii): By setting a1 = 0, a2 = 0 and f1 (t) ̸ = 0, we get E = Γ3 + Γ4 .

(2.23)

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By substitution Γi = Γ2 in (2.14) we get E ′ = Γ3 (f2 ) + Γ4 + ε Γ3 f1′ .

( )

(2.24)

We can cancel the Γ3 terms, as we did before, so we get E ′ = Γ4 .

(2.25)

Case (iv) By setting a1 = 0, a2 = 0 and f1 (t) = 0, one gets E = Γ3 (f2 ).

(2.26)

The optimal system of one-dimensional subalgebras of the nonlinear viscous non divergent barotropic vorticity equation on the f - and β -planes consists of the one-dimensional subalgebras generated by the following five vector fields: Opt = {Γ1 , Γ1 + Γ2 , Γ2 + Γ4 , Γ3 , Γ4 } .

(2.27)

The auxiliary equation in general form will be: dx

=

f1 (t)

dy C1

=

dt

=

C2

dψ ′

−yf1 (t) + f2 (t)

.

(2.28)

3. Invariant solutions Case (1): Solution under invariant symmetry Γ1 : We start to solve the governing equation (2.1) under the invariant symmetry Γ1 which was obtained in optimal onedimensional Lie subalgebra, and describe the model as follows to construct the boundary condition. The ocean basin is idealized as rectangular and of uniform depth, H, the fluid is of uniform density and uniform viscosity, and the flow is parallel to the surface. The origin of a Cartesian coordinate system is placed in the southwestern corner of the basin, y-axis running northwards along a rigid wall and the x-axis eastwards; a uniform flow of speed U approaches the wall from the East. The northwards velocity along the x-axis is assumed to vanish. The governing equations (2.1) will be solved subject to the following boundary conditions:

} ψy (0, y) = ψx (0, y) = 0, , ψx (x, 0) = 0

(3.1)

and when x/lx → ∞

} ψy → U , , ψx → 0

(3.2)

where lx is the width of the stream. The invariant solution under Γ1 = dx 0

=

dy 0

=

dt C2

=

dψ 0

∂ ∂t

leads to C2 ̸ = 0, and all others are zeros in (2.28), the auxiliary equation will be

.

(3.3)

Solving (3.3) we get q1 = x, q2 = y, χ (q1 , q2 ) = ψ

} ,

(3.4)

we used shorthand notation χ (q1 , q2 ) = χ . By substituting (3.4) in (2.1) we get

( ) ( ) ( ) χq1 χq1 q1 q2 + χq2 q2 q2 − χq2 χq1 q1 q1 + χq1 q2 q2 + β χq1 = ν χq1 q1 q1 q1 + 2χq1 q1 q2 q2 + χq2 q2 q2 q2 ,

(3.5)

from (3.4) we can assume

χ = g0 (q2 )g(α ),

(3.6)

where g0 (q2 ) is arbitrary function of q2 , g(α ) is arbitrary function of q1 , α = k0 q1 and k0 is arbitrary constant. By substituting (3.6) in (3.5) and using (3.4) to transforming back again to original variables we get:

ψ=

(

ν2β k21 k42

)1/3

(y + k3 ) g (α) ,

(3.7)

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where k1 , k2 and k3 are arbitrary constants, α =

(

β k22

)1/3

ν k21

)

By setting k1 = 1,

1 R

=

7

x, and the equation satisfied by g is:

− g ′′′ g + g ′′ g ′ + k21 g ′ = k22 g (iv) . k22

–

(3.8)

in (3.8), we get 1

g (iv) . R By substituting (3.7) in (3.1) and (3.2) we get

− g ′′′ g + g ′′ g ′ + g ′ =

(3.9)

g (0) = 0, g (0) = 0, g (∞) = 0, g (∞) = (

U

′

′

3 /2 U √

By setting the Reynolds number, R =

βν

βν

2 1/3

)

( )2/3 1

R

.

(3.10)

in (3.10) we get

g (0) = 0, g ′ (0) = 0, g ′ (∞) = 0, g (∞) = 1.

(3.11)

Eq. (3.7) can be simplified to

ψ = U y g (α) where α =

x Lx

(3.12)

√

and Lx =

U

β

.

Integrating (3.9) from zero to α we get

− g ′′ g + g ′2 + g =

1 R

1

g ′′′ −

R

g ′′′ (0).

(3.13)

By setting g ′′′ (0) = −R in (3.13), we get 1

g ′′′ + 1. R We wish to study the explicit solution of (3.14) for large α , we assume

− g ′′ g + g ′2 + g =

(3.14)

g(α ) = 1 + µ(α ).

(3.15)

By substituting (3.15) in (3.14), we get 1 ′′′ µ + µ′′ + µ′′ µ − µ′2 − µ = 0. (3.16) R By assuming the solutions of (3.16) small and are of the form em α , requires that m have a negative real part and satisfy 1 R

m3 + m2 − 1 = 0

(3.17)

√

√

when R > 3 2 3 , the decaying root is complex and a countercurrent exists, and if when R < 3 2 3 , the decaying root is real and no countercurrent exists. So, depending on the value of R, a countercurrent exist at some times and none at the other. The nature of the explicit solution depends on the values assigned to U and υ , there is not much freedom with respect to the choice of U, which must be of order 10 m s−1 . But the values assigned to the eddy viscosity coefficient υ have varied 3 /2 greatly [18,21]. Considering U = 10 m s−1 , β = 10−11 m−1 s−1 and υ = 108 m2 s−1 . Using R = U√βν , therefore R = 0.1, which is similar to that discussed by Munk [18] (see Fig. 1). Munk’s solution [18] cannot obtained directly from (3.14), since when R → 0, therefore g(α ) → 0. Hence, by assuming ( )1/3 αm = R1/3 α , then αm = βυ x, and (3.14) becomes d3 g 3 dαm

−R

3/2

((

dg

)2

dαm

−g

d2 g

)

2 dαm

− g + 1 = 0.

(3.18)

Solution of (3.18) is: 2

− α2m

g(αm ) = 1 − √ e 3

(√ cos

3

2

αm −

π 6

) .

(3.19)

By transforming back (3.19) to original variables we get

[

2

ψ (x, y, t) = U y 1 − √ e 3

( ) 1 /3 x − βυ 2

(√ ( ) )] 1/3 3 β π cos x− . 2 υ 6

(3.20)

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Fig. 1. (a) Represents g(α ), and (b) Represents g ′ (α ) for β = 10−11 m−1 s−1 , U = 10 m s−1 and R = 0.1, 1, 2.

Case (2): Solution under the invariant Γ4 − Γ1 : We start by solving the governing equation (2.1) under the invariant symmetry Γ4 − Γ1 , although the symmetry in this case is not included in the optimal one dimensional Lie algebra but we study this case as it leads to interesting solutions. The invariant solution under Γ4 − Γ1 leads to

Γ4 − Γ1 ≡ f1 (t)

∂ ∂ ∂ − − y f1′ (t) . ∂x ∂t ∂ψ

(3.21)

The auxiliary equation will be dx f1 (t)

dy

=

0

=

dt

=

−1

dψ

−yf1′ (t)

.

(3.22)

Solving (3.22) we get q1 = x + F (t), q2 = y, , χ (q1 , q2 ) = ψ − y f1 (t)

}

(3.23)

where F (t) = f1 (t)dt and we used shorthand notation χ (q1 , q2 ) = χ . By substituting (3.23) in (2.1) we get

∫

) ( ) ( ) ( χq1 χq1 q1 q2 + χq2 q2 q2 − χq2 χq1 q1 q1 + χq1 q2 q2 + β χq1 = ν χq1 q1 q1 q1 + 2χq1 q1 q2 q2 + χq2 q2 q2 q2 .

(3.24)

Apply Lie group again on (3.24) and the generator will be X ≡ ξ 11

∂ ∂ ∂ + ξ 22 + η33 . ∂ q1 ∂ q2 ∂χ

(3.25)

Repeating as procedure (2.2)–(2.8) we get

ξ 11 = C11 , ξ 22 = C22 , η33 = C33

(3.26)

where C11 , C22 , and C33 are arbitrary constants. The auxiliary equation will be dq1 C11

=

dq2 C22

=

dχ C33

.

(3.27)

The invariant solutions of (3.24) will be obtained from the following subcases: Subcase (2.1): C22 ̸ = 0, and all others are zeros in (3.27). The auxiliary equation will be dq1 0

=

dq2 C22

=

dχ 0

.

(3.28)

Solving (3.28), we get

χ = χ (q1 ).

(3.29)

By substituting (3.29) in (3.24) we get

χ (iv) −

β ′ χ = 0. ν

(3.30)

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9

Solution of (3.30) is:

χ = m1 + e

(√

[

q

− 2δ1

m2 cos

m

3q1

(√

) + m3 sin

2δm

3q1

)]

2δm

where m1 , m2 , m3 , m4 are arbitrary constants, and δm =

√ 3

q1

+ m4 e δm ,

(3.31)

ν . β

Since the solution is bounded, therefore as x → ∞, we find q1 → ∞, from which χ → 0, then m4 = 0, from which we get:

χ = m1 + e

(√

[

q

− 2δ1

m2 cos

m

3q1

(√

) + m3 sin

2δm

3q1

)] .

2δm

(3.32)

By substituting (3.32) in (3.23) and transforming back to the original variables we obtain:

ψ (x, y, t) = f1 (t)y + m1 + e

F (t) − x+ 2δ

(√

[

m

m2 cos

3 (x + F (t))

(√

)

3 (x + F (t))

+ m3 sin

2δm

)] ,

2δm

(3.33)

∫

where F (t) = f1 (t)dt. By setting f1 (t) = c then F (t) = c t, by substituting in (3.33) we get

ψ (x, y, t) = c y + m1 + e

ct − x2+ δ m

(√

[ m2 cos

3 (x + ct ) 2δm

(√

) + m3 sin

3 (x + ct )

)]

2δm

.

(3.34)

Solution (3.34) is well-known one-dimensional damped Rossby wave solution. Actually (3.34) is considered as the general form of the solution obtained by Munk [18]. Hence Munk introduced steady state solution for ocean model by considering linear approximation β v ≈ ν ψxxxx for (2.1), subjected to the following boundary conditions:

∂ψ =0 ∂x ∂ψ ψ = ψ0 , =0 ∂x

ψ = 0,

at

x = 0 (no slip)

(3.35)

as

x → ∞.

(3.36)

By applying (3.35) and (3.36) to (3.34), and by using u = −ψy , v = ψx , we get

[ ψ (x, y, t) = ψ0 1 − e

v (x, y, t) = √

2

ψ0 e

3δm

− 2δx

cos

m

− 2δx

m

(√

(

)

2δm

(√ sin

3x

3x

2δm

(√

1

+ √ sin 3

3x

))] (3.37)

2δm

) ,

(3.38)

which is the same as the obtained solution by Munk [18]. Also (3.37) can be considered as the solution of (2.1) subject to linear approximation. Subcase (2.2): C11 and C22 ̸ = 0 and C33 = 0. The auxiliary equation (3.27) will be dq1 C11

=

dq2 C22

=

dχ 0

.

(3.39)

Solving (3.39) we get

} χ (q1 , q2 ) = χ (γ ) , γ = a ( q1 + b q2 )

(3.40)

where a and b are arbitrary constants. √ By substituting (3.39) in (3.24), by setting b = ± a − 1, and a ≥ 1 to simplify the form of solution, therefore the general solution of (3.24) √

χ = m1 + e

−

(q1 ±(

) )

a−1 q2

2δm a2/3

[ m2 cos

(√ (

3 q1 ±

(√

)

a − 1 q2

2δm a2/3

(√ (

)) + m3 sin

3 q1 ±

(√

)

a − 1 q2

2δm a2/3

) )] .

(3.41)

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Fig. 2. (a) Represents v (x, y, 0) for c = 1 m s−1 , a = 10, m2 = 100 m2 s−1 , m3 =

v (x, 0, 0) for c = 1 m s−1 , m2 = 100 m2 s−1 , m3 =

100 √ 3

u(x, 0, 0) for c = 1, m2 = 100 m s

−1

, m3 =

100 √ 3

2

m s

−1

–

m2 s−1 , β = 10−11 m−1 s−1 , ν = 400 m2 s−1 and (b) Represents

m2 s−1 , ν = 400 m2 s−1 , ν = 400 m2 s−1 and a = 10, 100, 1000.

Fig. 3. (a) Represents u(x, y, 0) for c = 1 m s−1 , a = 10, m2 = 100 m2 s−1 , m3 = 2

100 √ 3

)

, β = 10

−11

−1 −1

m

s

100 √ m2 3 2 −1

, ν = 400 m s

s−1 , β = 10−11 m−1 s−1 , ν = 400 m2 s−1 and (b) Represents and a = 10, 100, 1000.

By substituting (3.41) in (3.23) we get

( √ ((

⎡ ψ (x, y, t) = f1 (t)y + m1 + e

−

((x+

∫

x+

∫

)

f1 (t)dt ±

(√

) )) ⎤

a−1 y

m2 cos ⎥ ))⎢ 2δm a2/3 ⎥ ⎢ ⎢ ( ) √ (( ∫ ) (√ ) ) ⎥ ⎥. ⎢ 3 x + f1 (t)dt ± a−1 y ⎦ ⎣ + m3 sin 2δm a2/3

√

) (

f1 (t)dt ± 2δm a2/3

3

a−1 y

(3.42)

The familiar two-dimensional damped Rossby wave solution can be obtained by setting f1 (t) = c in (3.42). Edmo and Donald [32] have been assumed that the guessing solution for (2.1) in the form ψ (x, y, t) = −U y + A ei (k x+l y−ω t ) . Clearly, this solution could be obtained by setting f1 (t) = −U in (3.23) and assuming χ (x, y, t) = A ei (k x+l y−ω t ) in (3.40). Hence, the values of the parameters which presented in [32] have been used for constructing Figs. 2–4. Clearly, the solution in (3.42) is new and interesting and has not been considered before. Now we can expand (3.37) by using (3.42), we get

[

√

−

ψ = ψ0 1 − e

(x±(

√ 2 a−1

u = −√

3δm a2/3

))

a−1 y 2δm a2/3

ψ0 e

−

(x±(

v= √

3δm a2/3

ψ0 e

−

(x±(

3 x±

cos

√

(√ ( sin

3 x±

) ))

a−1 y

sin

1

+ √ sin 3

(√

3 x±

(√

3 x±

(√

) ) ))]

a−1 y

2δm a2/3

,

(3.43)

,

(3.44)

) ))

a−1 y

2δm a2/3

(√ (

) ))

a−1 y

2δm a2/3

(√ (

))

a−1 y 2δm a2/3

(√

2δm a2/3

))

a−1 y 2δm a2/3

√

2

(√ (

(

.

(3.45)

Solutions (3.43)–(3.45) are considered as an extension to Munk’s solution in two dimensions. Please cite this article in press as: M.B. Abd-el-Malek, A.M. Amin, Lie group method for solving viscous barotropic vorticity equation in ocean climate models, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.11.016.

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)

Fig. 4. (a) Represents v (x, 0, t) and (b) Represents u(x, 0, t) for c = 1 m s−1 , a = 10, m1 = 100 m2 s−1 , m3 =

ν = 400 m2 s−1 .

Case (3): Solution under the invariant Γ2 + Γ4 : The invariant solution under Γ2 + Γ4 = f1 (t) ∂∂x + Lie subalgebra. The auxiliary equation will be dx f1 (t)

=

dy 1

=

dt 0

=

∂ ∂y

–

100 √ 3

11

m2 s−1 , β = 10−11 m−1 s−1 and

∂ − y f1′ (t) ∂ψ , where Γ2 + Γ4 is obtained in optimal one dimensional

dψ . −yf1′ (t)

(3.46)

Solving (3.46) we get q1 = x − f1 (t)y, q2 = t ,

⎫ ⎪ ⎬

1 ⎪ ψ = χ (q1 , q2 ) − f1′ (t)y2 ⎭

.

(3.47)

2 By Substituting (3.47) in (2.1) we get

( ) χq1 q1 q2 + 2f1 f1′ χq1 q1 + f12 χq1 q1 q2 + β χq1 − f1′′ = ν 1 + 2f12 + f14 χq1 q1 q1 q1 .

(3.48)

Eq. (3.48) is not easy to solve but we will solve it under some special forms for the functions. Note that we have transformed f1 (t) to f1 (q2 ) = f1 . Subcase (3.1): f1 = s, s ∈ R. By substituting in (3.48) we get

)2

1 + s2 χq1 q1 q2 + β χq1 = ν 1 + s2 χq1 q1 q1 q1 .

(

(

)

(3.49)

Repeating as procedure (3.25)–(3.27), from which the solution of (3.49) will be in the form

χ = g1 (q2 ) + A exp (n1 q1 + n2 q2 )

(3.50)

where A, n1 , and n2 are arbitrary constants, and g1 (q2 ) is arbitrary function of q2 . The relation between n1 , n2 is obtained by substituting (3.50) in (3.49). Finally, the general solution of (3.49) will be

( χ = g1 (q2 ) + A exp

( n1

( q1 +

( )2 ) )) −β + ν n31 1 + s2 ( ) q2 n21 1 + s2

(3.51)

where n1 is arbitrary constant, g1 (q2 ) is arbitrary function of q2 , and s ∈ R. By setting A = ψ0 , n1 = i k in (3.51) and by using (3.47) to transforming back to original variables we get

( ( ( ) ) ψ (x, y, t) = g1 (t) + ψ0 exp −υ k2 1 + s2 t × exp

( k

( (x − s y) +

βt ( ) k2 1 + s2

))) .

(3.52)

Consider l s=− . k By substituting (3.53) in (3.52) we get

( ( ) ) ψ (x, y, t) = g1 (t) + ψ0 exp −υ k2 + l2 t × exp

(3.53)

( ( i

kx + ly +

β kt k2 + l2

))

.

(3.54)

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Fig. 5. Represents ψ (x, y, t) for β = 10−11 m−1 s−1 , ν = 108 m2 s−1 , k = 2 × 10−7 m−1 , l = 5 × 10−7 m−1 , t = 0 and ψ0 = 100 m2 s−1 .

Fig. 6. Represents ψ (x, y, t) for β = 10−11 m−1 s−1 , ν = 108 m2 s−1 , k = 2 × 10−7 m−1 , l = 5 × 10−7 m−1 , t = one day and ψ0 = 100 m2 s−1 .

Fig. 7. Represents ψ (x, y, t) for β = 10−11 m−1 s−1 , ν = 108 m2 s−1 , k = 2 × 10−7 m−1 , l = 5 × 10−7 m−1 , y = 0 and ψ0 = 100 m2 s−1 .

Interpreting k = 2π/Lx and l = 2π/Ly as the zonal and meridional wave number, respectively, while Lx and Ly are the wavelengths of the wave in zonal and meridional directions. A typically midlatitude disturbance might have a half wavelength of 5000 km in x-direction and 2000 km in y-direction. Therefore, k ≈ 1/5000 km ≈ 2 × 10−7 m−1 and l ≈ 1/2000 km ≈ 5 × 10−7 m−1 . The solution (3.54) obviously represents damped two-dimensional Rossby wave (see Figs. 5–7). ( ( )) By setting l = 0 (i.e. f1 = 0) in Eq. (3.54), we find that ψ (x, y, t) = g1 (t) + ψ0 exp −υ k2 t × exp

(

)

i

kx +

β kt k2 +l2

, the

solution in this case is simply damped one dimensional Rossby wave. Subcase (3.2): f1 = s1 q2 + s2 , where s1 , s2 ∈ R. By substituting in (3.48), we get

)2

2s1 (s1 q2 + s2 ) χq1 q1 + 1 + (s1 q2 + s2 )2 χq1 q1 q2 + β χq1 = ν 1 + (s1 q2 + s2 )2 χq1 q1 q1 q1 .

(

)

(

(3.55)

The solution of Eq. (3.55) is in the form

χ = g2 (q2 ) +

A 1 + (s1 q2 + s2 )

2

exp

(

n3 q1 + n4 r(q2 )tan−1 (s1 q2 + s2 )

)

(3.56)

where g2 (q2 ), r(q2 ) are arbitrary functions of q2 , while n3 and n4 are arbitrary constants. By Substituting (3.56) in (3.55), and repeating the same steps as we did in subcases (3.1), the solution will be A exp n23 ν

[

χ = g2 (t) +

(1

s2 t 3 3 1

( ))] + s1 s2 t 2 + t 1 + s22

1 + (s1 t + s2 )2

[ exp

n3 (x − (s1 t + s2 ) y) −

β n3 s1

]

tan−1 (s1 t + s2 ) .

(3.57)

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13

Fig. 8. Represents ψ (x, y, t) for β = 10−11 m−1 s−1 , ν = 108 m2 s−1 , k = 2 × 10−7 m−1 , s1 = 10−8 s−1 , s2 = 2, t = 0 and ψ0 = 100 m2 s−1 .

Fig. 9. Represents ψ (x, y, t) for β = 10−11 m−1 s−1 , ν = 108 m2 s−1 , k = 2 × 10−7 m−1 , s1 = 10−8 s−1 , s2 = 2, t = one day, and ψ0 = 100 m2 s−1 .

Fig. 10. Represents ψ (x, 0, t) for β = 10−11 m−1 s−1 , ν = 108 m2 s−1 , k = 2 × 10−7 m−1 , s1 = 10−8 s−1 , s2 = 2, and ψ0 = 100 m2 s−1 .

By setting A = ψ0 , and n3 = i k in (3.57) we get

( ( ( ) )) ⎧ ⎫ exp −k2 ν 13 s21 t 3 + s1 s2 t 2 + 1 + s22 t ⎪ ⎪ ⎪ ⎪ ⎨ ψ0 ⎬ 2 1 1 + s t + s ( ) 2 1 2 ψ (x, y, t ) = g2 (t) − s1 y + [ ( )] . ⎪ ⎪ 2 ⎪ ⎩× exp i k (x − (s1 t + s2 ) y) + β tan−1 (s1 t + s2 ) ⎪ ⎭ 2

(3.58)

s1 k

This solution (3.58) is non separable, with the factor e−i k s1 t y , also, the solution is non-singular, even as t → ∞. The viscous solution (3.58) decays in time and approaches the trivial solution as t → ∞ (see Figs. 8–10). The solution in (3.58) may be interpreted as a damped wave solution with a time-dependent meridional wave number. It is clear that our solutions presented in this section are new solutions and more general, while the solutions obtained by [33] are special cases when ν = 0. Case (4): Solution under the invariant Γ3 : The invariant solution under Γ3 which obtained in optimal one dimensional Lie( subalgebra does ) ∂ not seem to be very useful to be constructed alone. So, the invariant solution under Γ3 + Γ4 = f1 (t) ∂∂x + f2 (t) − y f1′ (t) ∂ψ , when f1 (t) ̸ = 0 and all others are zeros in Eq. (2.28). Please cite this article in press as: M.B. Abd-el-Malek, A.M. Amin, Lie group method for solving viscous barotropic vorticity equation in ocean climate models, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.11.016.

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The auxiliary equation will be dx

dy

=

f1 (t)

dt

=

0

dψ

=

0

′

−f1 (t)y + f2 (t)

.

(3.59)

Solving (3.59) we get q1 = y, q2 = t ,

⎫ ⎪ ⎬ f1′ (t)

ψ = χ (q1 , q2 ) −

f1 (t)

f2 (t) ⎪ x⎭ f1 (t)

xy +

.

(3.60)

Note that we transformed f1 (t) and f2 (t) to f1 (q2 ) and f2 (q2 ) respectively. By substituting (3.60) in (2.1), we get

χq1 q1 q2 +

(

f2 (q2 )

−

f1 (q2 )

f1′ (q2 ) f1 (q2 )

) q1

χq1 q1 q1 + β

(

f2 (q2 ) f1 (q2 )

−

f1′ (q2 ) f1 (q2 )

) q1

= ν χq1 q1 q1 q1 .

(3.61)

Eq. (3.61) can be written, by using δ (q1 , q2 ) = χq1 q1 , in the form

δq 2 +

(

f2 (q2 ) f1 (q2 )

f1′ (q2 )

−

f1 (q2 )

) q1

δq1 + β

(

f2 (q2 ) f1 (q2 )

−

f1′ (q2 ) f1 (q2 )

) q1

= ν δq1 q1 .

(3.62)

The stream function ψ (x, y, t ) is obtained in this case by obtaining δ and integrating it twice with respect to q1 and by substituting (3.60) to transforming back the original variables: ∫

ψ (x, y, t ) =

Ae

−ν f12 (t)dt

(

( d0 sin

f1 (t)y −

)

∫ f2 (t)dt

( + d1 cos

f12 (t) f ′ (t) 1 f2 (t) − β y3 + h0 (t)y + h1 (t) − 1 x y + x 6 f1 (t) f1 (t)

))

∫ f1 (t)y −

f2 (t)dt (3.63)

where h0 (t), h1 (t), f1 (t) and f2 (t) are arbitrary smooth functions of t. Case (5): Solution under the invariant Γ1 + Γ2 : The invariant solution under Γ1 + Γ2 = ∂∂t + ∂∂y , and all others are zeros in Eq. (2.28). The auxiliary equation will be dx 0

=

dy C1

=

dt C2

=

dψ 0

(3.64)

where C1 and C2 are arbitrary constants. Solving (3.64) we get q1 = x, q2 = y − c t , , ψ = χ (q1 , q2 )

}

(3.65)

where c arbitrary constant. By substituting (3.65) in (2.1), we get

( ) ( ) ( ) −c( χq1 q1 q2 + χq2 q2 q2 + χq1 χq1 q1 q)2 + χq2 q2 q2 − χq2 χq1 q1 q1 + χq1 q2 q2 + β χq1 = ν χq1 q1 q1 q1 + 2χq1 q1 q2 q2 + χq2 q2 q2 q2 .

(3.66)

By applying Lie group method on (3.66) as we did (3.25)–(3.27), the auxiliary equation will be: dq1 C111

=

dq2 C222

=

dχ C333

.

(3.67)

The invariant solutions of (3.67) will be obtained for some special forms for the functions. Subcase (5.1): C222 ̸ = 0, all others are zeros in Eq. (3.67). The auxiliary equation will be dq1 0

=

dq2 C222

=

dχ 0

.

(3.68)

Solving (3.68) we get

χ = χ (q1 ).

(3.69)

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15

By substituting (3.69) in (3.66) we get

χ (iv) −

β ′ χ = 0. ν

(3.70)

Solution of Eq. (3.70) is similar to (3.30) by using (3.65) to transforming back to original variables we get:

ψ = m1 + e

− 2δx

(√

[ m2 cos

m

3x

(√

)

2δm

+ m3 sin

3x

)] .

2 δm

(3.71)

Subcase (5.2): C111 ̸ = 0, all others are zeros in Eq. (3.67). The auxiliary equation will be dq1 C111

=

dq2 0

=

dχ 0

.

(3.72)

Solving (3.72) we get

χ = χ (q2 ).

(3.73)

By repeating (3.70) and (3.71) we get:

ψ = m1 + m2 (y − c t ) + m3 (y − c t )2 + m4 e

−c (y−c t ) ν

.

(3.74)

Subcase (5.3): C111 , C222 ̸ = 0 and C333 = 0 in Eq. (3.67). The auxiliary equation will be: dq1 C111

=

dq2 C222

=

dχ 0

.

(3.75)

Solving (3.75) we get

χ = χ (γ1 ) γ1 = a1 (q1 + b1 q2 )

} (3.76)

where a1 and b1 are arbitrary constants. By substituting (3.76) in (3.66) we get

χ (iv) +

cb1 β ′ ) χ ′′′ − ( ( )2 χ = 0. 3 ν a1 1 + b21 ν a1 1 + b21

(3.77)

The exact solution of (3.77) can be obtained according to specific values of constants. In what follows, we give the list of explicit invariant solutions derived from the different cases considered above. Clearly, the solutions listed in Table 3 are completely new. 4. Symmetry analysis of barotropic vorticity equation with two parameters F and β In this section, we have investigated the symmetry properties of the barotropic vorticity equation with two parameters F and β presented in (1.5). Also, it is shown that for the case of F ̸ = 0 there exist a point of transformation to set β = 0. This means that by studying reductions using Lie symmetries, it is possible to consider the reduced case β = 0 initially and finally obtain a solution of the equation with β ̸ = 0 by applying the point transformation relating these two cases. We consider the viscous barotropic non-divergent vorticity equation with two parameters F and β in equation (1.21) as follows:

( ) ( ) ( ) ψxxt + ψyyt − F ψt + ψx ψxxy + ψyyy − ψy ψxxx + ψyyx + β ψx = υ ψxxxx + 2ψxxyy + ψyyyy .

(4.1)

We compute the symmetries of (4.1) for the case F ̸ = 0 and β arbitrary, considering the infinitesimal generator of the symmetry group admitted by Eq. (4.1), given by: X ≡ ξ1

∂ ∂ ∂ ∂ + ξ2 + ξ 3 + η1 . ∂x ∂y ∂t ∂ψ

(4.2)

By repeating the same procedure presented in (2.2)–(2.8), then (4.1) admits the Lie symmetry

ξ 1 = −C1 y + C4 , ξ 2 = C1 x +

C1 β t F

+ C2 , ξ 3 = C3 , and η1 =

C1 β F

x+

C1 β 2 F2

t + C5 ,

(4.3)

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Table 3 Invariant solutions for ∇ 2 ψt − ψy ∇ 2 ψx + ψx ∇ 2 ψy + β ψx − ν ∇ 4 ψ = 0. Invariant symmetry

Γ1 ≡

Invariant solution

∂ ∂t

ψ (x, y, t) = U y g(α ), where α =

, The solution obtained in this case is new and more general.

, Lx =

√

U

3/2 U √

,R=

β

, and g(α )

βν

1 ′′′ g R

′2

′′

x Lx

+ 1. Seeking the explicit solutions must satisfy −g g + g + g = of g(α ) are of the form em α . So, we defined g(α ) = 1 + µ(α ), µ(α ) and m must satisfy R1 µ′′′ + µ′′ − µ = 0 and R1 m3 + m2 − 1 = 0. Munk solution [18] obtained if we change the independent variables α to αm with αm = R1/3 α .

Γ4 − Γ1 ≡ f1 (t) ∂∂x −

∂ ∂ − y f1′ (t) ∂ψ ∂t The solution in this case completely is new and has not been considered before.

ψ (x, y, t) = f1 (t)y + m⎡1 + e

−

((

( √ (( 3

x+

∫

)

f1 (t)dt ±

(√

) )) ⎤

a−1 y

⎥ ) ) ⎢m2 cos 2δm a2/3 ⎥ ⎢ ( √ (( ∫ ) (√ ) ) )⎥ . ⎢ 3 x + f1 (t)dt ± a−1 y ⎦ ⎣

∫ √ a−1 y x+ f1 (t)dt ± 2δm a2/3

) (

+m3 sin

2δm a2/3

The familiar two-dimensional damped Rossby wave solution can be obtained by setting f1 (t) = c. Also, Munk’s solution [18] obtained by setting f1 (t) = 0 and a = 1 in the above solution.

Γ2 + Γ4 = f1 (t) ∂∂x +

∂ ∂ − y f1′ (t) ∂ψ ∂y The explicit solutions in this case are obtained under some special forms for the function f1 (t). The solutions presented in this case are new and interesting.

Case 1: f1 (t) = −l/k ψ (x, y, t) =

( (

g1 (t) + ψ0 exp −ν k2 + l2 t × exp i

(

(

) )

kx + ly +

β kt k2 +l2

))

.

where k and l as the zonal and meridional wave number, Case 2: f1 = s1 t + s2 , where s1 , s2 ∈ R ψ (x, y, t ) = g2 (t) − 12 s1 y2 +

(

⎧ ⎪ ⎪ ⎨

ψ0

exp −k2 ν

(

1 2 3 s t 3 1

( ) )) + s1 s2 t 2 + 1 + s22 t

⎫ ⎪ ⎪ ⎬

1 + (s1 t + s2 ) [ ( )] . ⎪ ⎪ ⎪ ⎩× exp i k (x − (s1 t + s2 ) y) + β tan−1 (s1 t + s2 ) ⎪ ⎭ 2 2

s1 k

This result may be interpreted as a damped wave solution with a time-dependent meridional wave number.

( ) Γ3 + Γ4 = f1 (t) ∂∂x + f2 (t) − y f1′ (t)

∫ ( ) ∫ −ν f12 (t)dt ( Ae d0 sin f1 (t)y − f2 (t)dt ψ (x, y, t ) = f12 (t) ( )) ∫ + d1 cos f1 (t)y − f2 (t)dt

∂ ∂ψ

This case leads to new solution.

−

1 6

β y3 + h0 (t)y + h1 (t) −

f1′ (t) f1 (t)

xy +

f2 (t) f1 (t)

x

where h (t), h (t), f1 (t) and f2 (t) are arbitrary smooth functions of t. 0

Γ1 + Γ2 =

∂ ∂t

+

1

− 2δx

∂ ∂y

1. ψ = m1 + e

This case leads to three different solutions. The second solution is the only new solution.

m

[

m2 cos

(√ ) 3x 2δm

+ m3 sin

( √ )] 3x 2δm

.

2. ψ = m1 + m2 (y − c t ) + m3 (y − c t )2 + m4 exp − νc (y − c t ) . 3. ψ (x, y, t) = χ (γ ), where γ = a1 (x + b1 (y − c t)) and χ (γ ) = χ must satisfy χ (iv) + (cb1 2 ) χ ′′′ − ( β )2 χ ′ = 0.

(

ν a1 1+b1

)

ν a31 1+b21

where C1 , C2 , C3 , C4 and C5 are arbitrary constants. Therefore, (4.1) admits the five-dimensional Lie symmetry algebra aβ generated by the operators

( )( ) βt ∂ ∂ β ∂ Γ5 ≡ −y + x + + . (4.4) ∂x F ∂y F ∂ψ The symmetry obtained in (4.3) is not singular in β and consequently it also includes the case β = 0. Therefore, computing the symmetries for the case β = 0, we obtain the same symmetry of (4.3) with β = 0. Hence, the generators of the Lie ∂ ∂ ∂ ∂ Γ1 ≡ , Γ2 ≡ , Γ3 ≡ , Γ4 ≡ , ∂x ∂y ∂t ∂ψ

symmetry algebra are

∂ ∂ ∂ ∂ , Γ2 ≡ , Γ3 ≡ , Γ4 ≡ , ∂x ∂y ∂t ∂ψ

∂ ∂ +x . (4.5) ∂x ∂y The physical importance of these generators is as follows: Γ1 , Γ2 , Γ3 and Γ4 are the infinitesimal generators of translations in x, y, t and ψ , respectively. Γ5 is the rotation operator in the xy-plane. Γ1 ≡

Γ5 ≡ −y

It is easy to show that the Lie algebra (4.4) maps to the Lie algebra (4.5) under the transformation given by: t˜ = t , x˜ = x +

β

˜ =ψ− t , y˜ = y, ψ

β

y. (4.6) F F This transformation maps (4.1) to the same equation with β = 0. Therefore, (4.6) is an equivalence transformation for the class of equations of the form (4.1) with F ̸ = 0. The β−term can be neglected from (4.1) under symmetry analysis. Hence, every solution of (4.1) with β = 0 can be extended to a solution with β ̸ = 0 by means of the transformation (4.5). Please cite this article in press as: M.B. Abd-el-Malek, A.M. Amin, Lie group method for solving viscous barotropic vorticity equation in ocean climate models, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.11.016.

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4.1. Reductions barotropic vorticity equation for F ̸ = 0 and β = 0 Here we give the list of reduced equations of (4.1) with the parameters F ̸ = 0 and β = 0 based on the invariant symmetry (4.4). Moreover, the differential equations obtained by reduction could be investigated again by means of symmetry techniques. ∂ , leads to C1 ̸= 0, C5 ̸= 0 and all others are zeros in Case (1): The reduction of (4.1) under Γ4 + Γ5 = −y ∂∂x + x ∂∂y + ∂ψ (4.3), the auxiliary equation will be dx

−y

=

dy

dt

=

x

=

0

dψ

.

C5

(4.7)

Solving (4.7) we get

√

q1 = x2 + y2 , q2 = t ,

⎫ ⎪ ⎬

x ⎭ χ (q1 , q2 ) = ψ + C5 tan−1 ⎪

.

(4.8)

y

By substituting (4.8) in (4.1), we get

δq2 − F χq2 −

C5 q1

( ) 1 δq1 q1 + δq1 ,

δq 1 = ν

where δ (q1 , q2 ) = χq1 q1 +

1 q1

(4.9)

q1

χ q1 .

Case (2): The reduction of (4.1) under Γ1 + Γ3 + Γ5 = zeros in (4.3), the auxiliary equation will be dx C4

=

dy 0

dt

=

C3

=

dψ C5

∂ ∂x

+

∂ ∂t

+

∂ , ∂ψ

leads to C3 ̸ = 0, C4 ̸ = 0, C5 ̸ = 0 and all others are

.

(4.10)

Solving (4.10) we get q1 = x − m1 t , q2 = y, χ (q1 , q2 ) = ψ − m2 t

} ,

(4.11)

where m1 , m2 are arbitrary constants. By substituting (4.11) in (4.1), we get m 1 δq 1 − F

m1 χq2 − m2 − χq1 δq2 + χq2 δq1 + ν

)

(

where δ (q1 , q2 ) = χq1 q1 + χq2 q2 . Case (3): The reduction of (4.1) under Γ3 + Γ5 = auxiliary equation will be dx 0

=

dy 0

=

dt C3

=

dψ C5

∂ ∂t

(

) δq1 q1 + δq2 q2 = 0,

+

∂ , ∂ψ

(4.12)

leads to C3 ̸ = 0, C5 ̸ = 0 and all others are zeros in (4.3), the

.

(4.13)

,

(4.14)

Solving (4.10) we get q 1 = x, q2 = y, χ (q1 , q2 ) = ψ − c t

}

where c is arbitrary constant. By substituting (4.14) in (4.1), we get

( ) − F c + χq1 δq2 − χq2 δq1 = ν δq1 q1 + δq2 q2 , where δ (q1 , q2 ) = χq1 q1 + χq2 q2 . Case (4): The reduction of (4.1) under Γ3 + Γ4 + Γ5 = −y ∂∂x + x ∂∂y + others are zeros in (4.3), the auxiliary equation will be dx

−y

=

dy x

=

dt C3

=

dψ C5

.

(4.15) ∂ ∂t

+

∂ , ∂ψ

leads to C1 ̸ = 0, C3 ̸ = 0, C5 ̸ = 0 and all

(4.16)

Solving (4.16) we get q1 = x cos(ε t) + y sin(ε t), q2 = −x sin(ε t) + y cos(ε t), , χ (q1 , q2 ) = ψ − ε c t

}

(4.17)

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where ε = ±1 and c = C5 /C3 . By substituting (4.17) in (4.1), we get

ε

(

q2 δq1 − q1 δq2 − ε F

)

q2 χq1 − q1 χq2 − c + χq1 δq2 − χq2 δq1 = ν

(

)

( ) δq1 q1 + δq2 q2 ,

(4.18)

where δ (q1 , q2 ) = χq1 q1 + χq2 q2 . 5. Conclusion We know that it is useful and meaningful to investigate the models of atmospheric and oceanic dynamics. But due to their nonlinearity, it is hard to give out their exact solutions. The early study of numerical weather prediction was carried out by investing a series of models, the two dimensional barotropic model takes as a first object of study, the first publication dealt with the numerical properties of barotropic equation. In this paper the viscous barotropic non-divergent vorticity equation (β BVE) studied, first, making use of Lie group method, we analyze the symmetry of (β BVE) including its point Lie symmetries, Lie algebra, the corresponding Lie group and one-dimensional optimal system of subalgebra of symmetry group. For ignoring the restriction on group invariant solutions we cannot consider any boundary value problems of the (β BVE) during the Lie symmetries calculation. Inserting the boundary condition is easy and does not change the principle procedure for obtaining the exact solution [22], considering the boundary value problem during Lie symmetries calculation perhaps reduces the number of group invariant solution [22]. The obtained symmetries are used to reduce the viscous barotropic non-divergent vorticity equation to an ordinary differential equation and the explicit solutions are obtained for all cases. The invariant ∂ solution under Γ3 (f2 ) ≡ f2 ∂ψ , which obtained in optimal one dimensional Lie subalgebra does not seem to be very useful to be constructed alone. For this reason, we used Γ3 + Γ4 for constructing the invariant solution. In cases (1) and (2) we applied the boundary condition after obtaining the symmetries. In case (1) the forms of explicit solution depend on the value of Reynolds number. So, we studied the effects of Reynolds number on the behavior of the solution, and the solutions are graphically represented. In case (2) the time translational symmetries combined with moving coordinate symmetries make us create wave solutions that are more complicated than the linear damped Rossby waves. In case (3) specific choices f1 (t) = 0 which leads to one dimensional damped Rossby wave and f1 (t) = − sl leads to two dimensional damped Rossby wave. In case (4) we combined the moving coordinate symmetries which leads to complicated wave solution depending on the specific choices of f1 (t) and f2 (t). In case (5) the translation with time symmetries combined with translation with y spiral symmetries, the solution in this case is obtained by solving the linearized ordinary differential equation. The diffusion term υ ∇ 2 ξ in vorticity equation causes the vortex lines to diffuse through the moving fluid, and velocity component vanished after large value of time. For the viscous barotropic non-divergent vorticity equation with two parameters F and β , it is shown that in the case F ̸ = 0 there exists a well-defined point transformation to set β = 0. Hence, the β term can be neglected under symmetry analysis. Also, we note the Lie symmetry algebra in the cases F = 0, β = 0 and F = 0, β ̸ = 0 are neither isomorphic to each other nor isomorphic to the case F ̸ = 0. Consequently, it is not possible to find point transformations that relate the corresponding partial differential equations to each others. Also, there is no useful reduction that can be achieved using the ∂ subalgebra ∂ψ . Furthermore, as an extension to the present work, we intend to determine the invariant solutions of the viscous barotropic non-divergent vorticity equation with two parameters F and β by investigating the reduced differential equations again by the means of symmetry techniques. Finally, it is clear that the obtained solutions in Section 3 are new solutions and more general, and all solutions obtained by [18,22,32,33] are special cases. All the obtained solutions are verified that satisfy the viscous barotropic non-divergent vorticity equation on the f - and β -planes. Acknowledgments The authors would like to express their gratitude to The American University in Cairo for supporting this research by a research grant number 10100000-1050-13030-52111602. One of the authors (M.B. Abd-el-Malek) would like to express his thanks to the International Center of Mechanical Sciences, Udine, Italy, (CISM) for supporting him to attend a school on Ocean Climate Models. Also, the authors would like to express their gratitude to the potential reviewers of the journal for their valuable suggestions and critical review that improved the paper to the present form. References [1] [2] [3] [4] [5] [6] [7] [8]

G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1969. S. Pond, G.L. Pickard, Introductory Dynamical Oceanography, second ed, Butterworth-Heinemann, 1983, p. 329. J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, Berlin Heidelberg, 1992. J. Pedlosky, Ocean Circulation Theory, Springer-Verlag, Berlin Heidelberg, 1996. P.K. Kundu, L.M. Cohen, Fluid Mechanics, Academic Press, New York, 2004. j. Mashall, R.A. Plumb, Atmosphere, Ocean, and Climate Dynamics, Academic Press is an imprint of Elsevier, 2008. L.D. Talley, G.L. Pickard, W.J. Emery, J.H. Swift, Descriptive Physical Oceanography, sixth ed, Elsevier, 2011, p. 555. Geoffrey K. Vallis, Climate and the Oceans, Princeton University Press, 2011.

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