Final states of two-dimensional turbulence above large-scale topography: stationary vortex solutions and barotropic stability

TL;DR

This paper investigates the final states of 2D topographic turbulence, proposing a vortex model with Gaussian profiles and analyzing their stability, revealing energy-dependent vortex-topography correlations.

Contribution

It introduces an empirical vortex model with Gaussian profiles that accurately reproduces quasi-stationary states and provides stability analysis explaining vortex-topography correlations.

Findings

Vortices exhibit a 'sinh'-like PV-streamfunction relation after subtracting background flow.

The proposed Gaussian vortex model reproduces observed vortex structures.

Stability depends on background energy, with different vortex configurations stable at low and high energies.

Abstract

The final states of freely decaying two-dimensional (2D) topographic turbulence consist of a background flow and localized vortices. While the background flow satisfies a linear potential vorticity (PV)-streamfunction relation, the vortex structures remain poorly understood. To address this gap and ensure oceanic relevance, we examine quasi-stationary final states of 2D turbulence over a sinusoidal topography featuring a bump and a dip, where two oppositely signed vortices are locked to the topographic extrema. After subtracting the background flow, the vortices exhibit a "sinh"-like PV-streamfunction relation, as observed in flat-bottom turbulence. Motivated by Gaussian vortex profiles in flat-bottom turbulence, we propose an empirical model combining the background flow with Gaussian vortices centered at the topographic extrema. This model accurately reproduces quasi-stationary states and yields locally stationary solutions to the inviscid governing equation. We further test the model under complex topography and high-energy conditions, confirming that the "sinh"-like trend and Gaussian profiles are robust features of localized vortices. Linear stability analyses of these stationary vortex solutions reveal background flow-dependent stability: cyclone/elevation and anticyclone/depression configurations are stable at low background energy, while anticyclone/elevation and cyclone/depression configurations are stable at high background energy. These findings align with vortex-topography correlations observed in simulations across energy regimes. Our results provide explicit vortex solutions for quasi-stationary final states of 2D topographic turbulence and elucidate the mechanism underlying vortex-topography correlations through stability analyses of vortices embedded in topographic background flows.