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

This paper presents a novel theoretical and numerical framework to demonstrate that non-rotating quasi-2D viscous flows over topographies settle into steady states where large-scale vortices occupy topographic valleys, a behavior distinct from rotating systems and significant for understanding slow planetary flows.

The Gist

Imagine the Earth's atmosphere and oceans as a giant, swirling bathtub. Usually, scientists study how water moves in this bathtub when the whole planet is spinning rapidly (like a figure skater). In that spinning world, the water tends to get "stuck" on bumps (mountains) and flow smoothly around them, creating predictable patterns.

But what happens if the bathtub stops spinning? That is the question this paper asks.

The authors, Pierpaolo Bilotto and Roberto Verzicco, wanted to understand how fluids behave over mountains and valleys when there is no spin to guide them. They built a digital "virtual bathtub" to simulate this. Here is what they found, explained through simple analogies:

1. The "Anti-Mountain" Rule

In a spinning world, a whirlpool (vortex) likes to sit right on top of a mountain. It's like a child sitting on a swing; the spin keeps them in place.

However, in their non-spinning simulation, the whirlpools act like shy cats. They actively avoid the mountains!

• The Analogy: Imagine a crowd of people trying to dance in a room with a giant, bumpy rock in the middle. If the room is spinning, people might get pinned against the rock. But if the room is still, the dancers will naturally spread out and stay in the flat, open spaces (the "valleys") to avoid tripping over the rock.
• The Result: The fluid creates two giant, opposing whirlpools that sit as far away from the mountain as possible, hugging the deepest parts of the "valley."

2. The "Energy Trap" (Metastability)

The researchers noticed something fascinating about how the fluid settles down.

• Low Energy (Slow Flow): If the water is moving slowly, it behaves like a tired hiker. It just walks straight to the most comfortable spot (the "ground state") and sits down.
• High Energy (Fast Flow): If the water is moving fast (turbulent), it's like a hyperactive dog in a park. It doesn't just go to the best spot; it gets stuck in "excited states."
  • The Analogy: Think of a marble rolling down a hill with lots of bumps. If you roll it slowly, it finds the very bottom. If you throw it hard, it might get stuck in a small dip halfway down the hill. It's not at the lowest point, but it's stuck there for a long time, bouncing around.
  • The Finding: In fast, turbulent flows, the system gets "trapped" in these temporary, sub-optimal patterns for a long time before finally settling.

3. The "Chaotic DJ" (Random Forcing)

To make the simulation more realistic, they added random "kicks" to the fluid, like a DJ randomly bumping the turntable.

• The Analogy: Imagine trying to balance a broomstick on your hand while someone keeps tapping your elbow randomly. You can never find a perfect, still balance.
• The Result: The fluid never settles into one perfect pattern. Instead, it constantly jumps between different "excited" patterns. However, even in this chaos, the rule holds: the whirlpools always avoid the mountain. They dance around the obstacle, never landing on it.

Why Does This Matter?

Most of our current weather and climate models assume the Earth spins fast enough to simplify the math. But this paper shows that in places where the spin is weak (like the equator of Earth, or the atmosphere of Venus), the rules change completely.

• The Takeaway: In a non-spinning world, the shape of the land (topography) acts like a giant magnet that repels storms and currents, pushing them into the deep valleys. This changes how we might predict weather on other planets or understand Earth's equatorial oceans.

In summary: When the world stops spinning, the fluid stops hugging the mountains and starts hiding in the valleys, getting stuck in temporary dance moves before finally finding a resting spot. It's a reminder that without the "spin" to organize things, nature finds its own, often surprising, ways to settle down.

Technical Summary

Here is a detailed technical summary of the paper "Topographic Effects on Steady-States of Non-Rotating Shallow Flows" by Bilotto and Verzicco.

1. Problem Statement

The paper addresses the long-time behavior of non-rotating, quasi-two-dimensional (2D) viscous flows over variable topography. While geophysical fluid dynamics (GFD) extensively studies rotating systems (where the Rossby number $Ro \ll 1$), many planetary environments (e.g., Venus, Titan, Earth's equatorial oceans) operate in regimes where rotation is negligible or absent ($Ro \gg 1$).

In rotating systems, the Coriolis force linearizes flow-topography interactions, often leading to flow alignment with topographic contours (Taylor columns). However, in non-rotating systems, the interaction is fully nonlinear. The authors investigate whether such systems can reach a stationary state, how topography influences the formation of coherent structures (vortices), and whether the "minimum enstrophy" principle (common in rotating/flat-bottom 2D turbulence) applies.

2. Methodology

Theoretical Framework

The authors derive a Non-Rotating Shallow Flow (NRSF) model from the 3D incompressible Navier-Stokes equations under the limit of infinite Rossby number ($Ro \to \infty$).

• Assumptions: Shallow domain ($D \ll L$), hydrostatic balance, rigid lid approximation (eliminating gravity waves), and stress-free bottom topography.
• Governing Equations: The system is reduced to a scalar potential vorticity ($q$) equation coupled with a mass-transport stream function ($\psi$):
  $$h \partial_t q + J(q, \psi) = \nu \Delta(hq) + f$$
  where $q = \zeta/h$ (relative vorticity $\zeta$ divided by fluid depth $h$), $J$ is the Jacobian, and $h(x,y)$ is the variable fluid depth determined by the bottom topography $b(x,y)$.
• Key Challenge: Unlike standard 2D Navier-Stokes, the relationship between $q$ and $\psi$ is non-local and implicit due to the variable depth $h$. The operator $L$ relating them ($q = L[\psi]$) cannot be efficiently inverted in Fourier space.

Numerical Approach

• Discretization: A second-order finite difference method on a $512 \times 512$ grid.
• Time Integration: A Crank-Nicolson implicit scheme for linear terms embedded within a third-order explicit Runge-Kutta scheme for nonlinear terms.
• Handling the Stream Function: To solve the implicit relation $q = L[\psi]$, the authors employ a recursive iterative approach at each time step to converge on $\psi$ until a tolerance of $10^{-15}$ is met.
• Symmetry Preservation: The numerical scheme (using Arakawa's Jacobian and specific gradient calculations on diagonal grid points) is designed to preserve the discrete symmetries (rotations and reflections) of the continuous equations, which is crucial for correctly capturing the final attractors.
• Forcing Strategies:
  1. Deterministic (Fixed Energy): To study intrinsic attractors, the authors enforce constant kinetic energy by adding a non-local feedback term that balances viscous dissipation. This transforms the decaying flow into a forced-dissipative system on a constant-energy manifold.
  2. Stochastic: A random, spatially homogeneous forcing is applied to simulate turbulent regimes, allowing the system to explore a broader phase space.

3. Key Contributions

1. Novel Numerical Framework: Development of a robust numerical scheme specifically tailored for the NRSF model, handling the complex inversion of the mass-transport stream function operator while preserving symmetries.
2. Rejection of Rotating Paradigms: The study demonstrates that non-rotating flows exhibit fundamentally different phenomenology compared to rotating geostrophic flows. Specifically, vortices avoid topographic hills and settle in valleys, contrary to the alignment seen in rotating systems.
3. Reynolds Number Dependence: The authors show that stationary solutions in non-rotating shallow flows are not unique and depend on the Reynolds number ($Re_L$). This challenges the "selective decay" principle, which predicts that stationary states are independent of viscosity and determined solely by energy minimization.
4. Metastability and Excited States: The discovery that at high Reynolds numbers, the system can become temporally trapped in "excited states" (metastable configurations) rather than relaxing immediately to the global ground state.

4. Results

Deterministic Regime (Fixed Energy)

• Ground State: Regardless of the Reynolds number, the system eventually settles into a configuration of two large-scale, opposing vortices (a dipole) that are maximally spaced from each other and from the topographic hill.
• Mechanism: This behavior is driven by the conservation of potential vorticity ($q = \zeta/h$). As fluid moves over a hill ($h$ decreases), relative vorticity $\zeta$ must decrease to conserve $q$. Conversely, in valleys ($h$ increases), $\zeta$ increases. Consequently, vortices are repelled from hills and attracted to valleys.
• Reynolds Number Effects:
  • At low $Re_L$, the flow relaxes directly to the ground state.
  • At high $Re_L$ (e.g., $Re_L = 10^4$), the flow exhibits metastability. It can get trapped in intermediate "excited states" (e.g., anti-symmetric configurations) for long periods before eventually relaxing to the ground state.
• Symmetry Breaking: The final state depends on the symmetry class of the initial conditions. The system converges to one of six possible eigenstate configurations (related to the lowest eigenfunctions of the operator $L$), determined by spontaneous symmetry breaking.

Stochastic Regime (Turbulent Forcing)

• Inverse Cascade: The system exhibits a classic 2D turbulence inverse energy cascade, building large-scale structures.
• Delocalized Attractor: Unlike the deterministic case, the stochastic forcing prevents the system from settling into a single unique ground state. Instead, the system explores an ensemble of configurations.
• Persistent Behavior: Despite the randomness, the system remains confined to a region of phase space characterized by a dipole of vortices that consistently avoid the topographic hill. The flow continuously transitions between different excited states.

5. Significance and Implications

• Planetary Science: The findings are critical for understanding the atmospheric and oceanic dynamics of slowly rotating or non-rotating planetary bodies (e.g., Venus, Titan, and Earth's equatorial regions). It suggests that topography in these environments acts as a repulsive barrier for vortices rather than a guide for alignment.
• Theoretical Fluid Dynamics: The work challenges the universality of the "minimum enstrophy" principle. It proves that in the absence of strong rotation, the nonlinear coupling between flow and topography creates a landscape where stationary states are viscosity-dependent and the system can be trapped in metastable excited states.
• Turbulence Modeling: The study highlights that in non-rotating shallow flows, the concept of a unique "steady state" is often an oversimplification; the system may exist in a delocalized, fluctuating state spanning multiple metastable configurations.

In summary, Bilotto and Verzicco provide a rigorous theoretical and numerical demonstration that non-rotating shallow flows over topography organize into vortex dipoles that avoid hills, a behavior governed by potential vorticity conservation and distinct from the alignment seen in rotating systems. This behavior is highly sensitive to the Reynolds number, leading to complex metastable dynamics in turbulent regimes.