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

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.