[Paper Review] Topographic Effects on Steady-States of Non-Rotating Shallow Flows

Pierpaolo Bilotto, Roberto Verzicco|arXiv (Cornell University)|Mar 7, 2026

Fluid Dynamics and Turbulent Flows0 citations

TL;DR

The paper derives a non-rotating shallow-flow model (NRSF) from 3D Navier–Stokes in the infinite Rossby-number limit, analyzes long-time behavior over topography, and shows vortices avoid hills with stationary states depending on Reynolds number; forcing can yield either a ground state or metastable excited states.

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.

Motivation & Objective

• Motivate understanding of long-time behavior of non-rotating quasi-2D viscous flows over topography.
• Derive a 2D effective model from 3D Navier–Stokes in the high-Rossby-number limit and with a rigid lid.
• Investigate stationarity notions and attractors under both deterministic and stochastic forcing.
• Characterize how topography couples nonlinearly to flow and influences vortex organization.

Proposed method

• Derive the non-rotating shallow flow (NRSF) model from 3D incompressible Navier–Stokes with a height-variable fluid column h(x,y).
• Express dynamics in terms of potential vorticity q = ζ/h and a mass-transport stream function ψ via q = L[ψ] with L defined by h.
• Close the system with q_t = (ν/h)Δζ + 𝔣/h and h u = ∇⊥ψ, leading to a single prognostic equation for q.
• Impose a rigid lid and periodic topography to study asymptotic states in a quasi-2D setting.
• Develop a numerical scheme that preserves symmetries using Crank–Nicolson, Runge–Kutta, Arakawa’s Jacobian, and an iterative inversion of L to obtain ψ from q.

Figure 1: An illustration of the fluid with the relevant variables

Experimental results

Research questions

• RQ1 Do non-rotating shallow flows over topography reach stationary states, and how do these depend on Reynolds number?
• RQ2 How does energy balance (with or without forcing) influence the long-time attractors and possible metastable states?
• RQ3 How does topography steer large-scale vortices and differ from rotating-geostrophic expectations?

Key findings

• Vortices consistently avoid topographic hills in non-rotating flows, with large-scale dipoles forming in valleys or away from hills.
• Under fixed energy and increasing Re_L, the system can trap in metastable excited states before relaxing to a final pattern.
• For deterministic dissipation-balanced dynamics, the final state depends on Re_L, and the ground state is not universal as in selective decay.
• Stochastic forcing sustains turbulence and yields an ensemble of dipole configurations that still avoid the hill, indicating non-uniqueness of ground states.
• The asymptotic patterns resemble lowest-eigenfunction structures of the perturbed Laplacian operator, linking stationary states to eigenmodes of the topography-coupled operator.

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. III.1 . 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)