On the Nonlinear Stability and the Existence of Selective Decay States of 3D Quasi-Geostrophic Potential Vorticity Equation


In this article, we study the dynamics of large-scale motion in atmosphere and ocean governed by the 3D quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible-irreversible coupling.


Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research


The work of Daozhi Han was supported by National Science Foundation (NSF) grant DMS-1912715.

Keywords and Phrases

Nonlinear Stability; Quasi-Geostrophic Equation; Selective Decay States

International Standard Serial Number (ISSN)

0170-4214; 1099-1476

Document Type

Article - Journal

Document Version


File Type





© 2020 John Wiley & Sons Ltd, All rights reserved.

Publication Date

01 Jan 2020