Steady-state bifurcation analysis of a strong nonlinear atmospheric vorticity equation

TL;DR

This paper analyzes the bifurcation behavior of a nonlinear atmospheric vorticity equation, revealing conditions for multiple steady states and employing a Lagrangian approach to overcome compactness challenges.

Contribution

It introduces a novel bifurcation analysis for a strong nonlinear quasi-geostrophic equation using Lagrangian formulation to establish steady-state solutions.

Findings

Unique steady state for 
for 
>1

Multiple steady states exist for 
<
_{crit}<1

The analysis employs a Lagrangian formulation to address compactness issues in the nonlinear equation.

Abstract

The quasi-geostrophic equation or the Euler equation with dissipation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205-1216] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by $\kappa>0$. It is proved that the steady-state solution is unique for $\kappa >1$ while multiple steady-state solutions exist for $\kappa<\kappa_{crit}$ with respect to critical value $\kappa_{crit}<1$. Without involvement of viscosity, the equation has strong nonlinearity as its nonlinear part contains the highest order derivative term. Steady-state bifurcation analysis is essentially based on the compactness, which can be simply obtained for semi-linear equations such as the Navier-Stokes equations but is not available for the quasi-geostrophic equation in the Euler formulation. Therefore the Lagrangian formulation of the equation is employed to gain the required compactness.