Algebraic Rossby Solitary Waves Excited by Non-Stationary External Source

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1 Commun. Theor. Phys. 58 (2012) Vol. 58, No. 3, September 15, 2012 Algebraic Rossby Solitary Waves Excited by Non-Stationary External Source YANG Hong-Wei ( ), 1 YIN Bao-Shu ( ), 2,3, DONG Huan-He (þ ), 1 and SHI Yun-Long (î ) 1 1 Information School, Shandong University of Science and Technology, Qingdao , China 2 Institute of Oceanology, Chinese Academy of Sciences, Qingdao , China 3 Key Laboratory of Ocean Circulation and Wave, Chinese Academy of Sciences, Qingdao , China (Received April 27, 2012) Abstract The paper deals with the effects of non-stationary external source forcing and dissipation on algebraic Rossby solitary waves. From quasi-geostrophic potential vorticity equation, basing on the multiple-scale method, an inhomogeneous Korteweg-de Vries Benjamin Ono Burgers (KdV-B-O-Burgers) equation is obtained. This equation has not been previously derived for Rossby waves. By analysis and calculation, four conservation laws associated with the above equation are first obtained. With the help of pseudo-spectral method, the waterfall plots are obtained and the evolutional characters of algebraic Rossby solitary waves are studied. The results show that non-stationary external source and dissipation have great effect on the generation and evolution of algebraic solitary Rossby waves. PACS numbers: Fg, Jr, Hm Key words: inhomogeneous Korteweg-de Vries Benjamin Ono Burgers equation, algebraic Rossby solitary waves, non-stationary external source, conservation laws 1 Introduction Rossby waves are one of the very important and meaningful waves in the ocean and atmosphere and are related to the rotation of earth and globe effect. It is regarded as one of the key dynamics of ocean response to large-scale atmospheric forcing. [1] A number of theoretical studies about Rossby waves have been carried out in the last decade or so, these studies could roughly be divided into two categories: one is the classical solitary waves, which the behavior of Rossby waves are governed by the well-known Korteweg-de Vries (KdV) type equation, [2 5] the outstanding feature of this type solitary waves is that they are very stable; the other is the algebraic solitary waves, [6 10] unlike the classical solitary waves, the evolution of weak nonlinear Rossby waves obeys an integrodifferential equation i.e. the Benjamin Ono (B-O) equation, furthermore the waveform of the algebraic solitary waves vanishes algebraically as x. It is known that the ocean and atmosphere are driven by external forcing. External forcing, nonlinear advection, rotating force field and gravitational field constitute essential characteristic of the oceanic and atmospheric motion. [11] So these factors must be considered in the research of the oceanic and atmospheric motion. The motion of the ocean and atmosphere must be taken as a forced nonlinear system. Here the external forcing includes topographic forcing, external source forcing and so on. In the past, many people had paid much attention to the topographic forcing effect and obtained a series of inspirable results. [12 16] External source general exists in the ocean and atmosphere, such as heat source, cold source, vorticity source and so on. The effect of external source on the oceanic and atmospheric motion is a very important and meaningful research subject. Observations show that external source, which supplies energy to the oceanic and atmospheric motion varies with time more slowly than the motion field varies with time. So, most researches involved stationary external source in the past, [17 19] little attention had been focused on the effect of the non-stationary external source on the Rossby solitary waves, especially on the algebraic Rossby solitary waves. But as the ancient Greece philosopher Heraclitus said that there is nothing permanent except change. It indicates that everything in the world is constant moving, developing and changing. The external source in the ocean and atmosphere is also. In [20], the author thought the external source varying with time i.e. non-stationary external source is more suitable for the realistic ocean and atmosphere system and discussed the solution s properties of the large scale atmospheric simultaneous equations under the influence of non-stationary external source. The research about the regularity of the oceanic and atmospheric motion under the influence of non-stationary external source has basic significance for realizing and forecasting the large-scale oceanic and atmospheric motion. [21] The aim of the paper is to study the generation and evolution of algebraic Rossby solitary waves excited by non-stationary external source. Starting from the quasi- Supported by Innovation Group Project of Chinese Academy of Sciences under Grant No. KZCX2-YW-Q07-01, National Sciences key Foundation of China under Grant No , Special Funding of Marine Science Study, State Ocean Administration under Grant No baoshuyin@126.com

2 426 Communications in Theoretical Physics Vol. 58 geostrophic potential vorticity equation including external source and dissipation, we will derive a new equation (inhomogeneous KdV-B-O-Burgers) in Sec. 2 which is first obtained and is suitable for describing the large-scale motion with external source and dissipation. In Sec. 3, the conservation laws associated with the equation are discussed. In Sec. 4, the numerical solutions of the inhomogeneous KdV-B-O-Burgers equation are obtained by using the pseudo-spectral method. Basing on the waterfall plots, we discuss the effect of dissipation and non-stationary external source on the algebraic Rossby solitary waves respectively. Finally, some conclusions are given in Sec Barotropic Model and KdV-B-O-Burgers Equation The non-dimensional barotrophic and quasigeostrophic potential vorticity equation including external source and turbulent dissipation on a β-plane channel can be written in the form [22] t 2 Ψ + J(Ψ, 2 Ψ) + β Ψ = µ 2 Ψ + Q, (1) where Ψ is the dimensionless stream function; 2 = 2 / / 2 denotes the two-dimensional Laplace operator; β = β 0 (L 2 /U) and β 0 = (ω 0 /R 0 )cosφ 0, in which R 0 is the Earth s radius, ω 0 is the angular frequency of the Earth s rotation, φ 0 is the latitude, L and U are the characteristic horizontal length and velocity scales; 2 Ψ expresses the vorticity dissipation which is caused by the Ekman boundary layer; µ is a dissipative coefficient and 0 µ 1. Especially, the distribution function of the external source is taken as Q(x, t), i.e the external source is non-stationary. According to Ono, [7] the region can be divided into three parts: (, ), [, L], and (L, + ). In order to consider the role of nonlinearity, we assume the following type of shear flows: u 1, y <, U = u(y), y L, (2) u 2, y > L, here u 1, u 2 are constants, u(y) is a function of y. For simplicity, u(y) is assumed to be smooth across y = and y = L. In order to consider weakly nonlinear perturbations on a zonal flow, we assume Ψ = y (U(s) c 0 + εα)ds + εψ, (3) where ε 1 is a small parameter characterizing the smallness of terms and measures the weakness of the nonlinearity; O(α) = 1 is a detuning parameter and reflects the proximity of the system to a resonate state; c 0 is a constant, which is regarded as a Rossby waves phase speed; ψ denotes disturbance stream-function. In the domain [, L], in order to achieve a balance among external source, turbulent dissipation and nonlinearity, we take Q(x, t) = ε 5/2 q(x, t), µ = ε 3/2 µ 0. (4) Substituting Eq. (2), Eq. (3) and Eq. (4) into Eq. (1), yields: t + (u c 0 + εα) ] 2 ψ + (β u ) ψ + εj(ψ, 2 ψ) = ε 3/2 µ 0 2 ψ + ε 3/2 q. (5) In the domain (, ) and (L, ), the parameter β is smaller than that in the domain [, L], here we assume β(y) = 0 for y > L. Furthermore, in these areas, the external source and turbulent dissipation is absent and we only consider the features of disturbances generated. Then the governing equations in these areas are t + (u 1 c 0 + εα) ] 2 ψ + εj(ψ, 2 ψ) = 0, y <, t + (u 2 c 0 + εα) ] 2 ψ + εj(ψ, 2 ψ) = 0, y > L. (6) For convenience, we can rewrite Eq. (6) as the following synthetic form t + (u 1,2 c 0 + εα) ] 2 ψ + εj(ψ, 2 ψ) = 0. (7) In order to consider the influence of the external source on Rossby waves and the effect of nonlinear terms, we introduce the stretching transformation and the perturbation expansion of ψ in in the domain [, L] as follows: X = ε 1/2 x, T = ε 3/2 t, (8) ψ in = ψ 1 (X, y, T) + εψ 2 (X, y, T) + (9) Substituting Eq. (8) and Eq. (9) into Eq. (5), we get O(ε 1/2 ): ( 2 ψ 1 X 2 + β ) u ψ 1 = 0. (10) u c 0 Separating ψ 1 as ψ 1 = A(X, T)φ(y), then we have from Eq. (10) ( d 2 dy 2 + β ) u φ(y) = 0. (11) u c 0 Equation (11) is a eigenvalue problem and describes the space structure of the wave along direction. Provided that the boundary conditions for φ are given, c 0 will be determined as the eigen-value. A(X, T) is the unknown amplitude in the order O(ε 1/2 ), which needs higher order equations to determine. Proceeding to O(ε 3/2 ), we obtain

3 No. 3 Communications in Theoretical Physics 427 ( 2 ψ 2 X 2 + β ) u ψ 2 = 1 [( A u c 0 u c 0 T + α A ) β u X + µ 0A φ (u c 0 ) 3 A ( β u ) φ u c 0 X 3φ + 2 A A ] u c 0 X + q. (12) Multiplying the both sides of Eq. (12) by φ and integrating it with respect to y from to L, we get [ φ ψ 2 X dφ ] L ( A dy ψ 2 = T + α A ) X + µ β u 0A (u c 0 ) 2 φ2 dy 3 A L X 3 φ 2 dy + A A ( β u ) φ 3 qφ dy + dy. (13) X u c 0 u c 0 In Eq. (13), if the boundary conditions on φ and ψ 2 are known, the equation governing the amplitude A will be determined. So in the following we will consider the boundary conditions on φ and ψ 2 by using the Eq. (7). For the two external regions of y > L, we adopt the transformations in the forms: and the external stream function ψ ex is set ξ = x, T = ε 3/2 t, (14) ψ ex = ψ(ξ, y, T, ε). (15) Substituting Eq. (14) and Eq. (15) into Eq. (7), we get the lowest-order equation for the external region (u 1,2 c 0 ) ( 2 ) ξ ξ ψ = 0. (16) 2 Obviously, Eq. (16) reduces to ( 2 ) ξ ψ = 0. (17) 2 According to Ono, [7] the solutions of Eq. (17), which satisfy boundary conditions ψ 0 as y, are in the following form: ψ(ξ, y, T, ε) = ±P ψ(ξ y L, ±L, T, ε) π (y L) 2 + (ξ ξ ) 2 dξ, (18) where the upper and the lower signs denote that for y > L and for y < respectively, and P stands for the principal value of the integration. Taking the derivative with respect to y for both sides of Eq. (18), then we have ψ(ξ, y, T, ε) = ±P π ψ(ξ, ±L, T, ε) (ξ ξ ) 2 (y L) 2 [(y L) 2 + (ξ ξ ) 2 ] 2 dξ. (19) Assuming that the inner solution matches smoothly with the outer solutions at y = ±L, then from Eq. (18) and Eq. (19), we obtain From Eq. (20), we get ψ 1 (X, ±L, T) + εψ 2 (X, ±L, T) = ψ(ξ, ±L, T, ε) + O(ε 2 ), (20) ψ 1 (X, ±L, T) By combining Eq. (22) with Eq. (19), we obtain ψ(ξ, ±L, T, ε) + ε ψ 2(X, ±L, T) = ψ(ξ, ±L, T, ε) + O(ε 2 ). (21) A(X, T)φ(±L) = ψ(ξ, ±L, T, ε), ψ 2 (X, ±L, T) = 0. (22) = ε 1/2 H(A(X, T)) φ(±l) = εφ(±l) 2 J (A(X, T)) X X 2, (23) where H(A(X, T)) (P/π) A(X, T)/(X X )dx is the well-known Hilbert transform and J (A(X, T)) (P/π) A(X, T)ln X X dx. Then we obtain from Eq. (21) and Eq. (23) φ ψ 2 (X, ±L, T) (±L) = 0, Substituting Eq. (22) and Eq. (24) into Eq. (13) yields = φ(±l) 2 J (A(X, T)) X 2. (24) A T + α A X + a 1A A X + a 3 A 2 X 3 + a 3 X 3 J (A(X, T)) + µ 0A = a 4 q, (25) the Eq. (25) can be rewritten as follows: 3 A T + α A X + a 1A A X + a 3 A 2 X 3 + a 3 X 2 H(A(X, T)) + µ 0A = a 4 q, (26) 2

4 428 Communications in Theoretical Physics Vol. 58 where a = a 1 = a 2 = φ 2 (β u )/(u c 0 )dy, φ 3 [(β u )/(u c 0 )] dy/a, φ 2 dy/a, a 3 = [φ 2 (L) + φ 2 ()]/a, a 4 = φ/(u c 0 )dy/a. Equation (26) is an inhomogeneous integro-differential equation including dissipation and external source terms. In the absence of the dissipation and the external source, when a 3 = 0, Eq. (26) degenerates to the standard KdV equation; when a 2 = 0, Eq. (26) degenerates to the B-O equation. Because the term µ 0 A expresses the dissipation effect and has the same physical meaning with the term 2 A/X 2 in Burgers equation, so we call Eq. (26) forced KdV-B-O-Burgers equation. As we know that the socalled KdV-B-O-Burgers equation is first obtained here. The properties of the equation will be discussed in the next section. 3 Conservation Laws Related to KdV-B-O- Burgers Equation Conservation laws are a common feature of mathematical physics, where they describe the conservation of fundamental physical quantities and play various important roles in the analysis of unsteady problems of wave propagation. As is known, some famous soliton equations such as the KdV equation, mkdv equation and so on have an infinite number of conserved quantities; [23 24] it is also proved that the B-O equation has four conservation laws in [6]. Now, if we can conclude that the KdV-B-O-Burgers equation also has conservation laws in the absence of dissipation effect? In the presence of dissipation effect, what is the conservation relations of the algebraic Rossby solitary waves from the KdV-B-O-Burgers equation? In this section, we will discuss these problems. In what follows, we assume that A, A X, A XX, A XXX vanish as X and neglect the external source effect, i.e. a 4 = 0. First, let us integrate Eq. (26) itself with respect to X from to +, then we get Q 1 = AdX = exp( µ 0 T) A(X, 0)dX, (27) which shows that Q 1 is a time-invariant quantity as the dissipation effect is neglected i.e. µ 0 = 0. Here Q 1 is regarded as the mass of the algebraic solitary waves, so we conclude that the mass of the algebraic solitary waves is conserved without dissipation. From Eq. (27), we can also get that the mass of the waves decreases exponentially with the increasing of time T and the dissipative coefficient µ 0 in the presence of dissipation effect. Next, Eq. (26) has another simple conservation law, which becomes clear if we multiply Eq. (26) by A(X, T) and carry the integration, by using the property of the Hilbert operator H : f(x)h(f(x))dx = 0, in which f(x) 0 as X, then we get Q 2 = A 2 dx = exp( 2µ 0 T) A 2 (X, 0)dX. (28) Similar to the mass Q 1, Q 2 is regarded as the momentum of the waves and is conserved without dissipation. The momentum of the waves also decreases exponentially with the increasing of time T and the dissipative coefficient µ 0 in the presence of dissipation effect. Further Q 1 describes the conservation of mass the solitary waves, Q 2 describes the conservation of momentum of the solitary waves, then we think if there should be a corresponding conserved density, which can be associated with the energy of the solitary waves. To confirm this, we assume µ 0 = 0 and construct d dt Q 3 = d dt [ 1 3 A A2 X + a 3 A ] a 1 X H(A) dx, (29) by adding (A 2 (a 3 /a 1 )H(A X )) Eq. (26) to (/X) Eq. (26) [A X + (a 3 /a 1 )H(A)] and integrating it, by virtue of the relation 2 H(A) ( 2 A ) X 2 = H X 2, uhvdx = vhudx, (30) after tedious calculation, we find that dq 3 /dt = 0. Here Q 3 expresses the energy of the solitary waves. As we anticipated, the energy of the solitary waves is conserved without dissipation. Finally, beside the above three important conserved quantities, we will derive the fourth conserved quantity. Defining a quantity related to the phase of the solitary waves: Q 4 = d XAdX. (31) dt Then applying dq 2 /dt = 0 and the above assumption A, A X, A XX, A XXX vanish as X and µ 0 = 0, we are easy to deduce d Q 4 /dt = 0. While here we are interested into the quantity Q 4 = Q 4 /Q 1, which expresses the velocity of the center of gravity for the ensemble of such waves according to [6]. Then by using dq 1 /dt = 0 and d Q 4 /dt = 0, we obtain dq 4 /dt = 0, which shows that the velocity of the center of gravity is conserved without dissipation. In this section, we obtain four conservation relations of KdV-B-O-Burgers equation and draw the conclusion that the dissipation effect causes the mass, the momentum, the energy and the velocity of the center of gravity to vary. In fact, after the above four conservation laws are given, we can wonder whether there exist other conservation laws?

5 No. 3 Communications in Theoretical Physics 429 Whether there is no limit as KdV equation? Which remains to be studied in the future. 4 Numerical Solutions and Discussion In Sec. 3, we discuss the conservation relations of the homogeneous KdV-B-O-Burgers equation and investigate the dissipation effect. In this section, we will consider what happens to the algebraic Rossby solitary waves under the influence of external source and dissipation, especially what happens to the algebraic Rossby solitary waves under the influence of the non-stationary external source. In order to answer the above problems, we need to solve the inhomogeneous KdV-B-O-Burgers equation. But it is difficult to solve the equation analytically, so here we will present numerical solutions of Eq. (26) by using the pseudo-spectral method. [25] Once the zonal flow u(y) and external source function q(x, T) are given, it is not difficult to obtain the coefficients of Eq. (26) with the help of Eq. (11) and Eq. (24). For simplicity, in this paper we take α = 2, a 1 = 1, a 2 = 1, a 3 = 0.5, a 4 = 1. As an initial condition, we take A(X, T) = 0, T = 0. The external source, as a forcing for wave generation, is taken as follows q(x, T) = q 1 exp[ (X X 0 ) 2 /4 + M 1 T] + q 2 exp[ (X X 1 ) 2 /4 + M 2 T], where q 1, q 2 express external source intension, X 0, X 1 express external source center. Here we will focus attention on the difference of stationary and non-stationary external source, so we take X 0 = 0, X 1 = 1, q 1 = 1, q 2 = 1, and M 1, M 2 will be given in the following. 4.1 The Effect of Dissipation on Algebraic Rossby Solitary Waves In this subsection, we will discuss what role the dissipation effect plays in the generation and evolution of algebraic Rossby solitary waves. The simulated results are shown in Fig. 1 and Fig. 2. Figure 1(a) gives the time evolution of the disturbances excited by the stationary external source in the absence of dissipation in the region of 20 X 120 and the time of 0 T 30. Figure 1(b) gives the waveform at the end of calculation time, i.e. T = 30. Accordingly, Fig. 2 gives the condition in the presence of dissipation. It is easy to see from Fig. 1(a) that the stationary external source excites a stationary solitary wave in the forcing region and modulated cnoidal wavetrains in the downstream region, which are positive stationary disturbance. The amplitude of the solitary wave is almost invariable with time T. The waveform at the end of calculation time (T = 30) can be found in Fig. 1(b). Figure 1(b) shows the amplitude of the solitary wave is about 0.5 at the end of calculation time. In addition, there exists a stationary buffer region between the solitary wave and the modulated cnoidal wavetrains. Fig. 1 (a) Wavetrains excited by stationary external source (M 1 = M 2 = 0) without dissipation (µ 0 = 0). (b) Waveform at the end of calculation time (M 1 = M 2 = 0) without dissipation µ 0 = 0). Fig. 2 (a) Wavetrains excited by stationary external source (M 1 = M 2 = 0) with dissipation (µ 0 = 0.3). (b) Waveform at the end of calculation time (M 1 = M 2 = 0) with dissipation µ 0 = 0.3). From Fig. 2, we can find that there is still a stationary solitary wave in the forcing region and fewer modulated

6 430 Communications in Theoretical Physics Vol. 58 cnoidal wavetrains in the downstream region. But, obvious the solitary wave and modulated cnoidal wavetrains in Fig. 2 is different from them in Fig. 1. By comparing Fig. 1 and Fig. 2, we can obtain that the dissipation effect has great influence on the disturbances excited by the stationary external source. First, dissipation effect makes the amplitude of the solitary wave in the forcing region decrease. At the end of calculation time, the amplitude of solitary wave in Fig. 2(b) is less than it in Fig. 1(b). Second, the dissipation effect causes the modulated cnoidal wavetrains in the downstream region to be dissipated. In Fig. 2, the modulated cnoidal wavetrains almost disappear with time T increasing. Last, the dissipation effect makes the buffer region between the solitary wave and the modulated cnoidal wavetrains vanish. In a word, it can be concluded that the dissipation effect makes the amplitude of solitary wave in the forcing region decrease and the modulated cnoidal wavetrains in the downstream region disappear. 4.2 The Effect of Non-stationary External Source on Algebraic Rossby Solitary Waves In this subsection, the calculation cases M 1 = M 2 = and M 1 = M 2 = are designed to investigate the effect of non-stationary external source on algebraic Rossby solitary waves. The simulated results are shown in Fig. 3 and Fig. 4. Figure 3 shows that the external source increases with time T, while Fig. 4 shows that the external source decreases with time T. From Fig. 3 and Fig. 4, we will find that a nonstationary solitary wave is generated in the forcing region. Figure 3 shows that the amplitude of the solitary wave increases with time T. At the end of the calculation time, the amplitude of the solitary wave in Fig. 3(b) is about 1. Figure 4 shows that the amplitude of the solitary wave increases firstly and then decreases with time T. The amplitude of the solitary wave in Fig. 4(b) is the lest among Fig. 1(b), Fig. 3(b) and Fig. 4(b). Comparison of Fig. 1, Fig. 3 and Fig. 4, we can see that the modulated cnoidal wavetrains will be excited in the downstream region and the variation of external source has almost no effect on the amplitude, the propagate speed and quantity of the modulated cnoidal wavetrains. Similar with Fig. 1, there also exists a buffer region between the solitary wave and the modulated cnoidal wavetrains in Fig. 3 and Fig. 4. The buffer region is stationary and located in the horizontal plane in Fig. 1, while it is unsteady in Fig. 3 and Fig. 4. There is a concave below the horizontal plane in Fig. 3 and a convex above the horizontal plane in Fig. 4. Fig. 3 (a) Wavetrains excited by non-stationary external source (M 1 = M 2 = 0.065) without dissipation (µ 0 = 0). (b) Waveform at the end of calculation time (M 1 = M 2 = 0.065) without dissipation (µ 0 = 0). Fig. 4 (a) Wavetrains excited by non-stationary external source (M 1 = M 2 = 0.065) without dissipation (µ 0 = 0). (b) Waveform at the end of calculation time (M 1 = M 2 = 0.065) without dissipation (µ 0 = 0). In conclusion, by comparing Fig. 1, Fig. 3, and Fig. 4,

7 No. 3 Communications in Theoretical Physics 431 we will find that the instability of external source has great effect on the solitary wave and the modulated wavetrains. 5 Conclusion In the paper, a new governing equation is derived by the multiple-scale method to describe the amplitude of algebraic Rossby solitary waves under the influence of non-stationary external source and dissipation. By analysis and calculation, we obtain four conserved quantities of KdV-B-O-Burgers equation without dissipation. It is convenient to regard the four conserved quantities as mass, momentum, energy, velocity of the center of gravity of algebraic Rossby solitary waves. In the last section, the numerical solutions of inhomogeneous KdV-B- O-Burgers equation are obtained by using the pseudospectral method. By analyzing the waterfall plots, we discuss the effect of dissipation and non-stationary external source on algebraic Rossby solitary waves respectively and obtain the following conclusions: (i) The dissipation effect makes the amplitude of the solitary wave in the forcing region decrease and the modulated cnoidal wavetrains in the downstream region disappear. (ii) The solitary wave in the forcing region of stationary external source is excited and its amplitude remains invariant with time T; While the solitary wave in the forcing region of non-stationary external source is also excited, but when M 1, M 2 > 0, its amplitude increases; when M 1, M 2 < 0, its amplitude increases firstly and then decreases with time T. (iii) The instability of external source has almost no effect on the amplitude, propagate speed and quantity of the modulated wavetrains, but makes the buffer region between the solitary wave and the modulated wavetrains unstable. References [1] S.G.H. Philander, Revi. Geop. Spac. Phys. 16 (1978) 15. [2] J.P. Body, J. Phys. Ocean. 13 (1983) 428. [3] O.G. Derzho and R.H.J. Grimshaw, Stud. in Appl. Math. 115 (2005) 387. [4] D. Lokenath, J. Math. Anal. Appl. 333 (2007) 164. [5] D.H. Luo, J. Atmos. Sci. 62 (2005) 22. [6] H. Ono, J. Phys. Soc. Jpn. 39 (1975) [7] H. Ono, J. Phys. Soc. Jpn. 50 (1981) [8] D.H. Luo, Sci. China (Ser. B) 32 (1989) [9] D.R. Christie, J. Atmos. Sci. 46 (1989) [10] L. Meng and K.L. Lv, Chin. J. Comp. Phys. 19 (2002) 159. [11] J.P. Li and J.F. Chou, Sci. China (Ser. D) 27 (1997) 89. [12] R.H.J. Grimshaw and N. Smyth, J. Flu. Mech. 169 (1986) 429. [13] B.K. Tan and J.P. Boyd, Wave Motion 26 (1997) 239. [14] P.D. Killworth and J.R. Blundell, J. Phys. Ocea. 29 (1999) [15] L.G. Yang, C.J. Da, et al., Chin. J. Ocean. Limn. 26 (2008) 334. [16] H.W. Yang, B.S. Yin, D.Z. Yang, and Z.H. Xu, Commun. Theor. Phys. 27 (2012) 473. [17] B.K. Tan and R.S. Wu, Sci. Atmos. Sin. 19 (1995) 289. [18] L. Meng and K.L. Lv, Chin. J. Comp. Phys. 19 (2002) 89. [19] J. Song and J.F. Lai, Acta Phys. Sin. 59 (2010) [20] J.P. Li and J.F. Chou, Chin. Sci. Bull. 40 (1995) [21] Q.C. Zeng, Mathematical Physical Basis of Numerical Weather Prediction, Science Press, Beijing (1979). [22] J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York (1979). [23] Z.H. Song, J.Q. Mei, and H.Q. Zhang, Commun. Theor. Phys. 53 (2010) 693. [24] X. Yu, Y.T. Gao, Z.Y. Sun, and Y. Lin, Commun. Theor. Phys. 55 (2011) 629. [25] B. Fornberg, A Practical Guide to Pseudo-Spectral Method, Cambridge University Press, Cambridge (1996).