层结流体中在热外源和β效应地形效应作用下的非线性Rossby孤立波和非齐次Schrödinger方程

在层结流体中, 从带有地形、热外源耗散的下边界条件以及带有热外源的准地转位涡方程开始, 使用小参数展开方法和多尺度时空伸长变换推导出了具有热外源、β效应和地形效应的强迫Rossby孤立波方程, 得到孤立Rossby振幅满足的带有地形与热外源的非齐次非线性的Schrödinger方程. 通过分析Rossby孤立波振幅的变化, 指出了热外源、β效应和地形效应都是诱导Rossby孤立波产生的重要因素, 给出了切变基本流下地形、热外源和层结流体中Rossby的相互作用.

Nonlinear solitary Rossby waves with external heating source and β effect topographic effect in stratified flows described by the inhomogeneous Schrödinger equation

Rossby waves are intrinsic in the large-scale systems of fluids, so they are the most important waves in the atmosphere and ocean. Theory and observation show that their basic characteristic is to satisfy the quasi-geostrophic and quasi-static equilibrium approximations. In stratified fluids, we discuss the long waves in a homogenous atmosphere and obtain the KdV equation, but the analysis is limited to the case that the velocity shear is small compared with a basic uniform zonal motion, and it gives no insight pertaining to the kinds of stream-line-flow patterns accompanying these waves. Here, the β-plane approximation f= f₀+ β₀ y (β₀ is a constant) is extended into f= f₀+ β(y) y, which includes a nonlinear function β(y) taking the place of β in the β-plane approximation. Such an approximation can depict more precisely the motion of the atmosphere and ocean, especially in the middle and high latitude regions. It generalizes the theory developed by Helfrich and Pedlosky for time-dependent coherent structures in a marginally stable zonal flow by including forcing. Such forcing could be due to topography or external source. We take the basic flow to be a shear and the Väisälä-Brunt frequency N a function of variable z. For the stratified fluids, based on the lower boundary with external heating source and topography, as well as the quasi-geostrophic potential vorticity equation with external heating source, an inhomogeneous nonlinear Schrödinger equation (including topographic forcing and an external heating source) is derived by using the perturbation method and stretching transforms of time and space. It is found that the external heating source, β effect and topography effect are the important factors that could induce the nonlinear solitary Rossby by inspection of the evolution of the amplitude of Rossby waves. On the assumption that nonlinear topographic effects and the dissipation of external heating source are balanced, an inhomogenous equation in which the coefficients depend on β(y), u(y,z) and N(z) is derived. Results show that the topography, external heating source and Rossby waves will interact with a basic stream function that has a shear. In stratified fluids, the inhomogeneous nonlinear Schrödinger equation is obtained for describing the evolution of the amplitude of solitary Rossby envelop solitary waves as the change of Rossby parameter β(y) with latitude y, topographic forcing and the external heating source.