Recently, Rossby solitary waves which determine the ocean’ s response to atmosphere and climate changes to a large extent have attracted increasing attention from meteorologists and oceanologists. As Russell observed, solitary waves are steady nonlinear waves.^[1] Two major theories to describe solitary waves have been generally formed, which are: classical solitary waves theory^{[2– 6]} and algebraic solitary waves theory.^{[7– 11]} The former was established by a KdV (Kortewegde Vries) type model to explain the evolution of classical solitary waves. The outstanding character of this type of solitary wave is that the waveform and the speed remain invariant during propagation and interaction. Meanwhile, algebraic solitary waves indicate the behaviors of solitary waves governed by the BO (Benjamin– Ono) model. In spite of its invariant of total mass, momentum and energy, its waveform and speed may vary because fission phenomenon may take place. These two kinds of models have allowed the remarkable development of solitary waves in a wide variety of physical contexts. However, most of the previous research about Rossby solitary waves was established in the zonal area, and we cannot apply them directly to the spherical earth; that is to say, solitary waves in a rotating fluid might be the next direction for our studies.

The main purpose of this paper is to construct a new model to describe algebraic Rossby solitary waves in a rotating fluid. Starting from rotational potential vorticity conserved equation with dissipation effect, by employing perturbation method and time-space stretching transformations, the model is established and is greatly different from the previous algebraic Rossby solitary wave models. Because of its similarity to the Boussinesq equation, we call it the generalized Boussinesq equation. As we are aware, conservation laws are very useful for studying behaviors of wave propagation; therefore, several conservation laws related to the present problem will be deduced in Section 3. The analytic solution of the generalized Boussinesq equation will be given in Section 4. Based on the analytical solution, the dissipation effect will be studied. Section 5 is focused on the solitary waves fission phenomenon from theoretical analysis and numerical simulation. Finally, some conclusions will be placed in Section 6.

According to Ref. [12], if we take the plane polar coordinate (r, θ ), then the rotational potential vorticity conserved equation including turbulent dissipation is, in non-dimensional form, given by

where β = β ₀(L²/U) and β ₀ = (ω ₀/R₀)cos ϕ ₀, in which R₀ is the Earth’ s radius, ω ₀ is the angular frequency of the Earth’ s rotation, ϕ ₀ is the latitude;

denotes the vorticity dissipation which is caused by the Ekman boundary layer, and λ ₀ is a dissipative coefficient; Q is the external source, the form of Q will be given in the latter; for the other characters, see Ref. [12].

Assume that the basic field is the rotating fluid with angular velocity Ω = Ω (r), then the stream function can be written as

where c₀ is a constant, which is regarded as a phase angle velocity; ε ψ is the disturbance part of stream-function, ε ≪ 1 is a small parameter; and, O(α ) = 1 is called detuning parameter and is used to denote the proximity of the system to a resonate state.

For the sake of studying the role of nonlinearity, the rotational angular velocity is taken as in the following form:

where ω ₀ is constant and Ω (r) is assumed to be smooth across r = r₂.

In the domain of [r₁, r₂], in order to balance nonlinearity and turbulent dissipation, and remove the dissipation caused by basic stream function, we take

By introducing Eqs. (2)– (4) into Eq. (1), we can obtain the following equation about perturbation stream function ψ :

For the domain [r₂, ∞ ), assuming β = 0 and neglecting dissipation effect, then we have

Because the waves in the fluid have multiple time-space scale features, we adopt the following time-space stretching transformations in Eq. (5):

and the perturbation stream function is taken as

Substituting Eqs. (7) and (8) into Eq. (5) yields

where the operator Γ is defined as

Assume that there is no perturbation at the boundary r = r₁, i.e., the boundary conditions are taken as

For ε ⁰, by assuming ψ ₀ = A(Θ , T)ϕ ₀(r), then ϕ ₀(r) satisfies

which is a variable coefficient eigenvalue problem, and A(Θ , T) is the unknown amplitude in the order O(ε ⁰), which needs higher-order equations to determine.

For ε ^1/2, by analysis, we deduce that

where ϕ ₁(r) satisfies the following equation:

Let us proceed to consider the equation of order ε . By multiplying the both sides of the order ε equation by

and integrating it with respect to r from r₁ to r₂, we obtain

in which we only solve the boundary conditions on ϕ and ψ ₂ to determine the equation governing the amplitude A.

In the domain of [r₂, + ∞ ), the following transformations are adopted in Eq. (6):

and notes the perturbation function, and then introducing Eq. (15) into Eq. (6), we have the ε ⁰ order equation as follows:

From Eq. (16), we deduce that

where the integral constant is taken as zero. The solution of Eq. (17) can be written as

Taking the derivative with respect to r for both sides of Eq. (18) leads to

Because of the smoothness condition of the solution at r = r₂, we obtain

From Eq. (20), we can judge

By substituting Eq. (22) into Eq. (19), we have

where

From Eqs. (21) and (23), we have

Substituting Eqs. (22) and (24) into Eq. (14) yields

where

Equation (25) is a new integro-differential equation including the dissipation effect and is first derived to describe the evolution of Rossby solitary waves in a rotating fluid. According to the definition of algebraic Rossby solitary waves, ^[10] the Rossby solitary waves which are denoted as Eq. (25) are also called algebraic Rossby solitary waves. However, we note that the model is greatly different from the BO model, it has the ∂ ²A/∂ T² term instead of ∂ A/∂ T term. In addition, the integral term ∂ ⁴J(A(Θ , T))/∂ Θ ⁴ is used to express dispersion effect, and its form is different from the Hilbert transformation term in BO equation. Neglecting dissipation effect and replacing the term ∂ ⁴J(A(Θ , T))/∂ Θ ⁴ into ∂ ⁴A/∂ Θ ⁴, equation (25) deduces to Boussinesq equation. For convenience, we call Eq. (25) the generalized Boussinesq equation.

Conservation laws provide strong insights into how flows are constrained and play an increasingly important role in fluid dynamics. The purpose of this section is to present some conservation laws related to generalized Boussinesq equation and, furthermore, to express some conserved quantities associated with algebraic Rossby solitary waves. Because the generalized Boussinesq equation is new, the study about conserved quantities of algebraic Rossby solitary waves is original, even though lots of research about conservation laws of the soliton equation has been carried out.^{[13– 16]} Because the dissipation effect can cause the conserved quantities to be destroyed, the dissipation effect is ignored in this section, i.e., λ = 0.

Because of particularity of generalized Boussinesq equation, we need to rewrite Eq. (25) as the following system of evolutionary equations:

According to the definition of conservation law, if there are functions E and F to satisfy the following equation:

then equation (27) is called the conservation law of system (26), we call E conservation density and F conservation flux.

Because of periodicity condition, we can assume that the values of A, A_Θ , A_{Θ Θ} , A_{Θ Θ Θ} , B, B_Θ at Θ = 0 equal to those at Θ = 2π . First, it is not difficult to find the first conservation law, as follows:

The analogue to the KdV equation, is used to express the mass of algebraic Rossby solitary waves. Equation (28) shows that the mass of algebraic Rossby solitary waves do not vary with time; i.e., it is a conserved quantity.

Next, by direct calculation, we can also obtain the following equation:

where

In a similar way, by analogue the KdV equation, E₂ is used to express the momentum of algebraic Rossby solitary waves. Equation (29) shows that the momentum of algebraic Rossby solitary waves is conserved.

In what follows, we can prove that system (26) also satisfies the following conservation law:

where

and

Here, E₃ is used to express the energy of algebraic Rossby solitary waves. Equation (30) shows the energy of algebraic Rossby solitary waves is conserved. Please note that in the derivation process of Eqs. (29) and (30), the following relations are used:

Finally, let us seek the fourth conservation law of system (26). Multiplying Θ /E₁ for both sides of the first equation of system (26) and integrating with respect to Θ from 0 to 2π yields

where . According to Ono, ^[7]E₄ denotes the velocity of the center of gravity for the ensemble of solitary waves. Equation (32) shows that the velocity of the center of gravity of algebraic solitary waves is conserved.

In this section, four conservation relations of the generalized Boussinesq equation without dissipation effect are derived and we draw the conclusion that the mass, the momentum, the energy as well as the velocity of the center of gravity of algebraic Rossby solitary waves are conserved. In fact, besides the above four conservation laws that we have obtained, we need to ask whether there are other conservation laws of generalized Boussinesq equation? We also need to ask whether there is no limit to the KdV equation? These questions remain to be answered in the future.

In this section, we will derive the analytical solution of generalized Boussinesq equation and analyze the dissipation effect. First, we will look for the exact solution of generalized Boussinesq equation without the dissipation effect. Based on the exact solution, the approximate analytical solution of generalized Boussinesq equation with dissipation effect will be derived. By employing the approximate analytical solution, the dissipation effect will be considered.

Without the dissipation effect, equation (25) can be rewritten as

where

By assuming that A is the function of η = Θ − μ T, then equation (33) can be simplified into

By integrating Eq. (34) twice with respect to the variable η and taking the integration constants as zero, we have

We can prove that equation (35) has the following solution:

where and Here, we note that μ shows the propagation speed of algebraic Rossby solitary waves and can be taken as positive or negative. When μ is positive, the solitary waves propagate towards the right; when it is negative, they propagate towards the left. This is a special feature of the new algebraic Rossby solitary waves and is different from the common algebraic Rossby solitary waves, which propagate towards one direction.

In the following, we consider the dissipation effect and look for the approximate analytical solution of Eq. (25). Taking a new space coordinate

and assuming A₀ = A₀(λ T), then we obtain

By taking two time scales as follows:

and expanding the solution as follows:

the following approximate equations can be obtained:

Setting

and introducing it into Eq. (41) gives

The solution of Eq. (42) has been given in Eq. (36) in the following form:

By employing the definition of ζ as well as Eq. (37), it is easy to find the approximate analytical solution of Eq. (25), as follows:

Next, we need to determine the expression of A₀(λ T), equation (43) will be discussed.

Suppose

and substitute it to Eq. (43), equation (43) can be rewritten as

where

Only when the following condition is satisfied:

then equation (47) can be solved. Here, K(γ ) is the solution of the following equation:

The solution of Eq. (49) is

By employing Eq. (48) and the expression of K(γ ), then we have

where is the initial amplitude. Therefore, the approximate analytical solutions of generalized Boussinesq equation is

Figures 1– 3 give the evolutions of amplitude and vertex of algebraic solitary wave with and without the dissipation effect, respectively. From Fig. 1, we can find that the amplitude of algebraic solitary waves remains invariant during propagation process without the dissipation effect. While in Fig. 2, due to dissipation effect, the amplitude decreases with the time evolution. By comparing the two conditions (λ = 0 and λ = 1.2) in Fig. 3, we can conclude that dissipation effect also has an important impact on the propagation speed of algebraic solitary waves because it can cause the propagation speed to decrease.

Fig. 1.

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	Fig. 1. Solitary waves evolution in the absence of dissipation (λ = 0).

Fig. 2.

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	Fig. 2. Solitary waves evolution in the presence of dissipation (λ = 1.2).

Fig. 3.

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	Fig. 3. The trace of the vertex of solitary waves.

In this section, we adopt an integrable ODE method^[17] to obtain traveling type solution of Eq. (33). Furthermore, by employing the perturbation method, ^[18] the approximate analytical solution of Eq. (25) is also derived. With the help of the solution, the evolution of algebraic solitary waves under the influence of dissipation effect is analyzed. In fact, there are other methods to solve the above equation, such as multiple exp-function method, ^{[19, 20]} bilinear method, ^[21] Darboux transformation method, ^{[22, 23]} Riccati equation expansion method, ^[24] similarity reduction, ^[25] and so on. They are of interest in analytical geophysical fluid dynamics and deserve to be studied in the future.

In Section 4, we study the steady progressing wave solution of Eq. (25) due to the equilibrium between nonlinearity and dispersion. In the present section, we investigate unsteady behaviors of algebraic Rossby solitary waves. In fact, the algebraic Rossby solitary waves that we derive in this paper cannot be called the common “ soliton” , because in the nonlinear process, an algebraic Rossby solitary wave at initial time can split into a series of algebraic Rossby solitary waves with the passage of time, which is greatly different from the classic solitary waves. Next, we will describe the fission process of algebraic Rossby solitary waves and discuss the impact of the detuning parameter α on the fission process.

Here, we take the initial value of algebraic solitary wave as follows:

Without loss of generality, suppose that the initial algebraic solitary wave asymptotically splits into two algebraic Rossby solitary waves with different amplitudes and speeds

where , , .

In Section 3, we have derive four conservation laws of algebraic Rossby solitary waves without dissipation, we will seek the two amplitudes of algebraic Rossby solitary waves after fission. The conserved quantities E_i(i = 1, 2) at T = 0 can be calculated from the initial value (52) with the help of system (26) and Eq. (35) to give

On the other hand, the conserved quantities can be considered as the sum of those associated with each emergent algebraic solitary wave after fission; it follows that:

By equating corresponding quantities in Eqs. (54) and (55), we obtain

Assuming that A₁ > A₂, q₀ > 0, and σ ₀ = 0.05, without loss of generality, we obtain

Once the amplitudes A₁ and A₂ are obtained, the corresponding propagation speeds μ ₁ and μ ₂ are easy to be obtained. The two algebraic solitary waves which generate due to fission can be described clearly. Here, we need emphasis that the conditions for the initial disturbance (52) with A₀ = 20q₀ and σ ₀ = 0.05 is necessary to split into two solitary waves alone.

The theoretical analysis for the fission phenomenon of algebraic Rossby solitary waves has been carried out, and we will verify the fission phenomenon of algebraic solitary waves by numerical simulation and discuss the effect of detuning parameter α on the fission process of solitary waves in the latter. Figs. 4– 6 describe the fission process of the algebraic solitary waves when α = 0, 10, − 1.3, respectively.

Fig. 4.

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	Fig. 4. α = 0, a1 = − 1, a2 = − 1, a3 = 1, r2 = 1, λ = 0, θ 0 = π /2. (a) T = 0, (b) T = 0.3, (c) T = 0.6.

Fig. 5.

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	Fig. 5. α = 10, a1 = − 1, a2 = − 1, a3 = 1, r2 = 1, λ = 0, θ 0 = π /2. (a) T = 0, (b) T = 0.3, (c) T = 0.6.

Fig. 6.

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	Fig. 6. α = − 1.3, a1 = − 1, a2 = − 1, a3 = 1, r2 = 1, λ = 0, θ 0 = π /2. (a) T = 0, (b) T = 0.3, (c) T = 0.6.

The three time points T = 0, T = 0.3, and T = 0.6 are taken in Figs. 4– 6 to show how the algebraic solitary waves evolve with time. By comparing the three states of algebraic solitary waves at different time points, we can draw the following conclusions: the algebraic solitary wave splits into two solitary waves with different amplitudes, they propagate toward the same direction, the solitary wave with large amplitude runs faster than one with a small amplitude, and the distance between the two solitary waves becomes larger and larger.

Meanwhile, we contrast Figs. 4– 6 which owe different detuning parameter α . It is easy to find that the decrease of detuning parameter α can accelerate the occurrence of solitary wave fission phenomenon.

In this paper, with the help of perturbation expansions and stretching transformations of time and space method, a new model for algebraic solitary waves in a rotating fluid is obtained in the context of a generalized Boussinesq equation with a dissipation effect. This is greatly different from the classic Rossby solitary waves models such as KdV model, mKdV model, and is also different from the common algebraic Rossby solitary waves models such as BO model and ILW model. By rewriting the generalized Boussinesq equation as a suitable system of evolutionary equations, the conservation laws related to generalized Boussinesq equation without dissipation are obtained, which show that the mass, the momentum, the energy, the velocity of the center of gravity of algebraic Rossby solitary waves are conserved without dissipation effect. The analytical solution of generalized Boussinesq equation without dissipation is also obtained. In particular, the approximate analytical solution of generalized Boussinesq equation with dissipation is derived and used to analyze the impact of dissipation effect on the amplitude and propagation speed of algebraic Rossby solitary waves. Finally, we study the fission feature of algebraic solitary waves from theoretical analysis and numerical simulation. The main results are as follows.

(i) The generalized Boussinesq equation with dissipation effect can be used to describe the evolution of Rossby solitary waves. Because it includes the term ∂ ²A/∂ T² instead of the term ∂ A/∂ T, which is new and worth being studied.

(ii) By discussing the approximate analytical solution of generalized Boussinesq equation, we can conclude that dissipation effect can cause the amplitude and the propagation speed of solitary waves decrease.

(iii) The algebraic solitary waves can split into several solitary waves with different amplitudes, they propagate toward the same direction, the solitary wave with large amplitude runs faster than the one with a small amplitude, and the distance between these solitary waves becomes larger and larger. Meanwhile, we also find that the decrease of detuning parameter α can accelerate the occurrence of solitary waves fission phenomenon.