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

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.