Minimum-enstrophy solutions in topographic quasi-geostrophic flow on the rotating sphere

TL;DR

This paper extends the minimum-enstrophy theory to rotating spherical quasi-geostrophic flows, accounting for topography and Coriolis effects, and demonstrates existence, stability, and characteristic flow patterns through analytical and numerical methods.

Contribution

It introduces a new framework for minimum-enstrophy solutions on the sphere with topography, proving their stability and computing them numerically for planetary atmospheres.

Findings

Latitude-dependent flow structures with topographical trapping near poles.

Zonal flows are prominent near the equator depending on parameters.

Numerical stability of solutions is confirmed through structure-preserving simulations.

Abstract

The minimum-enstrophy theory of Bretherton and Haidvogel postulates that two-dimensional turbulent systems evolve to a state that minimises enstrophy at a fixed energy level. We extend this to the rotating spherical quasi-geostrophic setting, accounting for bottom topography and the fully nonlinear Coriolis effect, resulting in latitude-dependent effects not present in planar approximations. We prove existence and nonlinear stability of minimum-enstrophy solutions and describe analytically asymptotic regimes for certain rates of rotation, topography scales, and energy values. We compute the minimum-enstrophy solutions by a structure-preserving method for the quasi-geostrophic equations on the sphere. We apply the method to a range of parameter values, including those describing Jupiter's atmosphere. The results reveal a distinct latitude dependence of the flow, with a tendency for topographical trapping near the poles and zonal flow near the equator, depending on the chosen parameters. The predicted nonlinear stability is confirmed numerically by integrating perturbed solutions using a structure-preserving time discretisation.