Characteristics of drift effects arising from nonlinear symmetry of the quasi-geostrophic equation

TL;DR

This paper analyzes the large-time behavior of solutions to the quasi-geostrophic and convection-diffusion equations, highlighting how nonlinear symmetry influences drift effects and asymptotic profiles in meteorological models.

Contribution

It provides a comparative analysis of nonlinear drift effects in two related meteorological equations, emphasizing the role of spatial symmetry in their asymptotic behavior.

Findings

Nonlinear effects differ in direction between the two equations.

Asymptotic profiles are distorted by nonlinear effects.

Spatial symmetry influences the first approximation's behavior.

Abstract

This paper compares two similar diffusion equations that appear in meteorology. One is the quasi-geostrophic equation, and the other is the convection-diffusion equation. Both are two-dimensional bilinear equations, and the order of differentiation is the same. Naturally, their scales also coincide. However, the direction in which the nonlinear effects act differs: one acts along the isothermal surface, while the other acts along the temperature gradient in a specified direction. The main assertion quantifies this difference through the large-time behavior of their solutions. In particular, the nonlinear distortions in the asymptotic profiles of both equations are compared. In this context, the spatial symmetry of the first approximation plays a crucial role, but the solutions require no symmetry. As an appendix, the mixed problem of those models are studied.