Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact solutions for nonlinear trapped lee waves in the $β$-plane approximation

Lili Fan, Ruonan Liu, Heyang Li

Abstract

In this paper, we construct exact solutions that character three-dimensional, nonlinear trapped lee waves propagation superimposed on longitudinal atmospheric currents in the $β$-plane approximation. The solutions obtained are presented in Lagrangian coordinates, and are Gerstner-like solutions. In the process, we also derive the dispersion relation and analyze the density, pressure and the vorticity qualitatively.

Exact solutions for nonlinear trapped lee waves in the $β$-plane approximation

Abstract

In this paper, we construct exact solutions that character three-dimensional, nonlinear trapped lee waves propagation superimposed on longitudinal atmospheric currents in the -plane approximation. The solutions obtained are presented in Lagrangian coordinates, and are Gerstner-like solutions. In the process, we also derive the dispersion relation and analyze the density, pressure and the vorticity qualitatively.
Paper Structure (6 sections, 1 theorem, 35 equations, 1 figure)

This paper contains 6 sections, 1 theorem, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The governing equations in the $\beta$-plane approximation
  3. Exact solutions for the trapped lee waves
  4. Qualitative Analysis
  5. Temperature, density and pressure
  6. The vorticity

Key Result

Theorem 3.1

Given satisfying $r_0-m(s)< 0$ and a density function where $F:(0,\infty)\rightarrow (0, \infty)$ is continuously differentiable and monotone increasing, then for a given arbitrary wavenumber $k > 0$, there is a wave speed and an associated pressure distribution with $\mathcal{F}'=F$, such that the velocity field $(u, v, w)$ determined by 3.1 solves the system 2.1-2.4.

Figures (1)

  • Figure 1: (a) Typical graph of the map $\Psi'(X)=-3X^2+2AX+3$ with $X=e^\xi$ for $A=3+\frac{8}{m_s^2}$ and $m_s=3\times10^{-4}$. The zero of the map is $X_1=\frac{A-\sqrt{A^2-9}}{3}\sim 1.5\times 10^{-8}$. (b) Typical graph of the map $\Psi (X)=1+X(AX-X^2-3)$ with $X=e^\xi$ for $A=3+\frac{8}{m_s^2}$ and $m_s=3\times10^{-4}$, where $\Psi (X_1)_{\min}=\frac{2A^3+(18-2A^2)\sqrt{A^2-9}-27A}{27}+1>0$.

Theorems & Definitions (3)

  • Theorem 3.1
  • proof
  • Remark 3.1