Energy stability of the Charney-DeVore quasi-geostrophic equation for atmospheric blocking

TL;DR

This paper analyzes the energy stability of the Charney-DeVore quasi-geostrophic equation, proving stability conditions, the necessity of certain average flow constraints, and confirming multiple equilibrium states through numerical methods.

Contribution

It establishes conditions for the asymptotic stability of the basic flow and demonstrates the coexistence of multiple equilibrium states using numerical continuation.

Findings

Basic flow is asymptotically stable in certain parameter regions.

Multiple equilibrium states coexist depending on topographic amplitude.

Stability of these states is verified through high-resolution simulations.

Abstract

Charney and DeVore [J. Atmos. Sci. 36 (1979), 1205-1216] found multiple equilibrium states as a consequence of bottom topography in their pioneering work on the quasi-geostrophic barotropic flow over topography in a $\beta$-plane channel. In the present paper, we prove that the basic flow is asymptotically stable in a parameter region, including the flat topography situation, which excludes the existence of multiple equilibrium states therein. Moreover, we show that an additional condition on the average zonal force or the average zonal velocity is indispensable to the well-posedness of the Charney-DeVore quasi-geostrophic equation. Coexistence of at least three equilibrium states is confirmed by a pseudo-arclength continuation method for different topographic amplitudes. The stabilities of the equilibrium states are examined by high-resolution direct numerical simulations.