Abstract
The general shallow wave equation can be used to describe the fluid motion which own uniform density and nearly uniform speed. In fact, these problems we face in nature do not satisfy these conditions, for example, the motion system of nonlinear rotation fluid governed by the frictional dissipation as well as the atmosphere system with greater viscosity. In these conditions, we need to amend the general shallow wave equation and consider the dissipative and viscous effect. In this article, starting from the shallow wave equation with dissipative and viscous effects in the horizontal direction, by virtue of β plane approximation and quasi-geostrophic approximation of large-scale motion, we derive the dissipative Petviashvili equation to describe the two-dimensional Rossby waves. Based on the ansatz function method, we obtain the exact analytical solutions of dissipative Petviashvili equation and discuss the influence of dissipation on the two-dimensional Rossby waves.
Highlights
The basic equations describing the motion of two-layer barotropic fluid is the following nondimensional shallow water wave equations[1]
Equation (1) includes two kinds of waves, that is, inertial-gravity waves and Rossby waves. These two kinds of waves reflect the motion of fluid under the influence of gravity and rotation in the rotation earth, so they have important meaning for fluid dynamics
For equation (30), when l and m are both zero, that is, g and a are both zero, by balancing the highest order derivative term and nonlinear term, the authors have obtained the exact analytical solutions based on the extension Jacobi elliptic function method,[2] Backlund transformations,[15] Binary Bell polynomial,[16] Rational function,[17] and so on.[18,19,20]
Summary
The basic equations describing the motion of two-layer barotropic fluid is the following nondimensional shallow water wave equations[1]. In this article, starting from the shallow water wave equations with viscous and friction dissipation, the dissipative potential vorticity equation will be derived and conserved state will be analyzed; the dissipative Petviashvili equation will be derived to describe two-dimensional Rossby waves. For equation (30), when l and m are both zero, that is, g and a are both zero, by balancing the highest order derivative term and nonlinear term, the authors have obtained the exact analytical solutions based on the extension Jacobi elliptic function method,[2] Backlund transformations,[15] Binary Bell polynomial,[16] Rational function,[17] and so on.[18,19,20] But when the viscous and friction dissipation are present, the extension Jacobi elliptic function method is invalid, so here we adopt the ansatz function method.[21,22,23] Assuming. We can obtain the other one exact analytical solution of dissipative Petviashvili equation
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