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

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.

Figures (6)

• Figure 1: An illustration of the fluid with the relevant variables
• Figure 2: Plots of the stream function (a, b, d, e) and of the rate of change (c, f) for two simulations in a rotating frame, as detailed in Sec. \ref{['sec:numerical_validation']}. In the top row, the topography is a dipole of a Gaussian hill and valley, while the bottom row employs a random topography. In panels (a, b, d, e), the colors show the stream function, black lines represent topographic contours (negative values are dashed, black lines represent the contours at one and two $\sigma$ of the Gaussian), and purple arrows point along streamlines of the velocity field. The insets in panels (c, f) zoom on the transient during which coherent vortices form.
• Figure 3: The initial (a) and final (b, c, d) state over a Gaussian hill, for three different configurations of parameters $E$ and $\nu$. Colors represent the rescaled potential vorticity $\tilde{q}$, purple arrows point along streamlines of the velocity field, and black lines mark the contours at one and two $\sigma$ of the Gaussian hill.
• Figure 4: The maximum values of the terms in Eq. \ref{['eqn:stationary_energy']} as a function of the advective time $\tau$, for the simulations shown in Fig. \ref{['fig:steadE_pv']}. Different dynamical regimes are labeled as: (a), vortex dipole formation; (b), relaxation to a metastable state; (c), final relaxation to the fixed point.
• Figure 5: Illustrations of the lowest eigenfunctions of the operator $\mathcal{L}$ with different symmetries. The respective eigenvalues are displayed on top of each panel. The omitted $\mathfrak{q}_3$ corresponds to a $\pi/2$ rotation of $\mathfrak{q}_2$, as they span the two-/dimensional eigenspace of the degenerate eigenvalue $\lambda_2=\lambda_3$. Black lines represent the contours at one and two $\sigma$ of the Gaussian hill. Purple arrows point along streamlines of the velocity field.