Topographic Effects on Steady-States of Non-Rotating Shallow Flows

TL;DR

This paper investigates how topography influences the long-term behavior of non-rotating shallow viscous flows, revealing that large-scale vortices tend to settle in valleys and that attractor states depend on Reynolds number.

Contribution

It introduces a new theoretical and numerical framework for analyzing non-rotating shallow flows over topography, extending understanding of steady states in turbulent regimes.

Findings

Large-scale vortices settle in topographic valleys.

Attractors depend on Reynolds number and are delocalized under turbulence.

Steady states differ from rotating flow phenomenology.

Abstract

In this work, we discuss the long-time behavior of non-rotating quasi-2D viscous flows over topographies. We develop a novel theoretical and numerical framework for the analysis of these flows, derived as a dimensional reduction of the 3D Navier-Stokes equations in the limit of infinite Rossby number $\mathit{Ro}$. We numerically determine dynamical attractors for fixed kinetic energy, focusing on the dependence of the final state on the Reynolds number. Under turbulent conditions, the attractor is no longer unique but delocalized, spanning the lowest excited states of the deterministic system. Regardless of the realized stationary configuration, large-scale vortices settle within topographic valleys, in contrast with the phenomenology of the rotating case. These findings have significant implications for understanding steady turbulent regimes in slowly rotating ($\mathit{Ro} \gg 1$) planetary environments.