Emergent vorticity asymmetry of one and two-layer shallow water system captured by a next-order balanced model

This paper introduces a next-order balanced model (SWQG$^{+1}$) for one- and two-layer shallow water systems that successfully captures emergent vorticity asymmetry and finite strain-driven front divergence, phenomena missed by the standard Quasi-Geostrophic approximation while still filtering out inertial-gravity waves.

The Gist

Imagine the Earth's atmosphere and oceans as a giant, chaotic dance floor. The dancers are swirling currents, storms, and eddies. For decades, scientists have used a simplified rulebook to predict how these dancers move. This rulebook is called Quasi-Geostrophic (QG) theory.

The QG rulebook is great at describing the "main dance": the big, slow, swirling patterns that dominate our weather and oceans. It assumes the dancers are perfectly balanced, moving in a smooth, predictable rhythm. However, there's a problem: real life isn't perfectly balanced.

In the real world, the dance floor has a secret asymmetry. Cyclones (spinning counter-clockwise in the Northern Hemisphere) and anticyclones (spinning clockwise) don't behave exactly the same way. They have different personalities, different strengths, and they interact differently with the floor. The old QG rulebook misses this entirely; it thinks a left-handed spin is identical to a right-handed spin, just mirrored.

This paper introduces a new, upgraded rulebook called SWQG+1. Think of it as taking the old rulebook and adding a "next-level" chapter that captures the messy, real-world quirks of the dance without getting bogged down in the chaotic noise of high-speed gravity waves (which are like the dancers tripping over their own feet).

Here is a breakdown of what the authors did, using simple analogies:

1. The Problem: The "Perfect Mirror" Fallacy

The old model (QG) treats the fluid like a perfect mirror. If you have a spinning vortex, the model thinks a clockwise spin and a counter-clockwise spin are mathematically identical twins.

• Reality: In the ocean and atmosphere, they are not twins. One might be a sturdy, long-lived giant, while the other is a fragile, short-lived wisp.
• The Consequence: Because the old model can't tell them apart, it fails to predict important things like why storms lean in a certain direction or how energy moves through the system.

2. The Solution: The "SWQG+1" Upgrade

The authors created a new model, SWQG+1.

• The Analogy: Imagine the old model was a black-and-white sketch of a dancer. It got the pose right but missed the muscle definition and the subtle sway of the body. The new model (SWQG+1) is a high-definition, color photograph. It keeps the same basic pose (the "balanced" motion) but adds the extra details (the "asymmetry") that make the dancer look real.
• How it works: It uses a mathematical trick called "Potentials." Instead of tracking every single drop of water, it tracks a single "master variable" called Potential Vorticity (PV). Think of PV as the "DNA" of the weather system. If you know the DNA, you can reconstruct the whole body (wind, pressure, height).
• The Upgrade: The new model calculates the DNA with slightly more precision. It accounts for the fact that the "height" of the water (or air) and the "spin" are slightly out of sync in a way that creates a bias.

3. The Experiments: Proving the Upgrade Works

The authors ran two major tests to see if their new rulebook works better than the old one.

Test A: The "Free-For-All" (One-Layer Simulation)

• The Setup: They created a digital ocean with random swirls and let them crash into each other until they settled down.
• The Result: In the real shallow water model, the swirls eventually became negatively skewed. This is a fancy way of saying: "The clockwise spins (anticyclones) became the dominant, stronger survivors, while the counter-clockwise ones faded away."
• The Old Model: Failed. It kept the spins perfectly balanced, like a coin toss.
• The New Model (SWQG+1): Succeeded. It correctly predicted that the clockwise spins would win out. It captured the "personality" difference between the two types of spins.

Test B: The "Jet Stream" (Two-Layer Simulation)

• The Setup: They simulated a jet stream (like the polar jet stream) that becomes unstable and breaks into waves. This is a two-layer system (like a top layer of air and a bottom layer).
• The Result: Here, the asymmetry flipped! The new model showed that in this specific two-layer dance, the counter-clockwise spins became the dominant survivors initially.
• Why? It's due to "vortex stretching." Imagine a figure skater pulling their arms in to spin faster. In the two-layer system, the interaction between the layers stretches the vortices, making the counter-clockwise ones grow faster.
• The New Model: Captured this complex, counter-intuitive behavior perfectly, whereas the old model would have missed it entirely.

4. Why Should We Care?

This isn't just about math; it's about better predictions.

• Better Forecasts: By capturing these subtle asymmetries, we can better predict how ocean currents move heat, how storms form, and how energy is transported across the globe.
• Efficiency: The best part? The new model is still very fast. It doesn't require a supercomputer to run every single drop of water. It keeps the simplicity of the old model (one main variable to track) but adds the accuracy of the complex model.
• Real-World Application: This helps us understand everything from the Gulf Stream's path to the formation of massive storms on Earth and even the giant storms on Jupiter.

The Bottom Line

The authors took a classic, simplified model of the ocean and atmosphere and gave it a "turbo boost." They didn't throw away the old, reliable engine; they just added a new gear that allows the car to handle the bumps and curves of the real world much better.

SWQG+1 is the new "Goldilocks" model: it's not too simple (missing the asymmetry) and not too complex (too slow to run). It's just right for understanding the messy, asymmetric dance of our planet's fluids.

Technical Summary

1. Problem Statement

The shallow water (SW) system is a fundamental model for geophysical fluid dynamics (GFD), capturing essential features of atmospheric and oceanic flows. However, the standard Quasi-Geostrophic (QG) approximation (specifically the Shallow Water QG or SWQG), while computationally efficient and free of inertial-gravity waves, fails to capture a critical emergent phenomenon: vorticity asymmetry.

• The Limitation: In full SW simulations, cyclonic and anticyclonic vortices evolve differently. For instance, in freely decaying turbulence, vorticity becomes negatively skewed (anticyclones are more robust), and in baroclinic instability, complex asymmetries arise. SWQG, being symmetric in vorticity evolution, cannot reproduce these effects.
• The Gap: Existing balanced models (e.g., Frontal Geostrophic, Semi-Geostrophic, Balance Equation) either lack higher-order Rossby number accuracy, fail to allow for divergent flow, or are numerically intractable due to complex nonlinear elliptic inversions (e.g., Monge-Ampère equations).
• Objective: The authors aim to develop a next-order-in-Rossby-number balanced model (SWQG+1) that retains the computational advantages of QG (single prognostic variable, no gravity waves) while capturing the ageostrophic physics responsible for vorticity asymmetry and finite divergence.

2. Methodology

The paper presents a rigorous derivation and numerical validation of the SWQG+1 model for both single-layer and multi-layer (specifically two-layer) shallow water systems.

A. Theoretical Derivation

• Asymptotic Expansion: The model is derived by expanding the shallow water equations in the small Rossby number ($\varepsilon$) up to order $O(\varepsilon^2)$, while keeping the Burger number ($Bu$) and Charney-Green number ($\gamma$) of order $O(1)$.
• Potential Formulation: Unlike previous derivations (e.g., Warn et al., 1995), the authors introduce a new formulation using "potentials" inspired by Muraki et al. (1999).
  • Velocity and height are expressed via a streamfunction ($\Phi$) and two ageostrophic potentials ($F, G$).
  • This formulation ensures that all diagnostic inversions involve the same linear elliptic operator: the Screened Poisson operator (used in standard QG).
• Prognostic Variable: The model remains PV-based. Potential Vorticity ($q$) is the sole prognostic variable. All other physical variables (velocity, height, divergence) are diagnosed from $q$ via a set of linear elliptic inversions.
  • Inversion Steps:
    1. Invert $q$ to find the zeroth-order streamfunction $\Phi_0$ (Standard QG).
    2. Invert a nonlinear source term (dependent on $\Phi_0$) to find the first-order streamfunction correction $\Phi_1$.
    3. Invert source terms derived from thermal wind balance to find the ageostrophic potentials $F_1$ and $G_1$.
• Multi-Layer Extension: The authors generalize this derivation to a two-layer system, defining a coupled operator $\mathcal{L}$ for the inversions, allowing the model to handle baroclinic modes.

B. Numerical Simulations

The authors validate the model against full shallow water simulations using the spectral code Dedalus.

1. Freely Decaying Turbulence (One-Layer):
   • Setup: Random balanced initial conditions with zero initial skewness.
   • Goal: Test if SWQG+1 captures the emergence of negative vorticity skewness observed in full SW.
2. Baroclinic Instability (Two-Layer):
   • Setup: Evolution of a baroclinically unstable jet (simulating a polar jet stream).
   • Goal: Test if SWQG+1 captures the positive vorticity skewness and divergence patterns associated with baroclinic fronts.
3. Strain-Driven Fronts: A minimal model of a cold filament under barotropic strain to analyze the mechanism of vortex stretching and divergence.

3. Key Contributions

• New Formulation: Introduced a "potential" based derivation for SWQG+1 that unifies the elliptic inversions, making the multi-layer extension straightforward and numerically efficient.
• First Multi-Layer SWQG+1: Formulated the SWQG+1 model for multiple layers for the first time, enabling the study of baroclinic processes within a balanced framework.
• Diagnostic Relations: Provided explicit diagnostic relations for ageostrophic velocities and divergence, extending the utility of geostrophic balance to submesoscale regimes where geostrophy fails.
• Energy and Enstrophy Behavior: Demonstrated that while SWQG+1 does not strictly conserve total energy or potential enstrophy (due to the lack of explicit source terms in the PV equation), it mimics the evolution of these quantities in the full SW model with high fidelity.

4. Results

• Vorticity Asymmetry (One-Layer):
  • Full SW simulations show a transition to negative vorticity skewness (dominance of robust anticyclones).
  • SWQG+1 successfully captures this negative skewness, matching the ensemble mean of full SW simulations across various Rossby numbers ($\varepsilon = 0.01$ to $0.12$).
  • The mechanism is identified as the correlation between vorticity and height (nonlinear PV term), which scales with the Froude number squared.
• Vorticity Asymmetry (Two-Layer):
  • In baroclinic instability, the full SW model exhibits positive vorticity skewness (cyclonic bias) during the initial growth phase.
  • SWQG+1 captures this positive skewness, a feature absent in one-layer simulations.
  • The model correctly reproduces the divergence fields associated with baroclinic fronts (convergence in the lower layer correlating with cyclonic vorticity), driven by vortex stretching.
• Divergence and Fronts:
  • The model successfully diagnoses finite divergence in strain-driven fronts, showing that baroclinic cyclonic vorticity correlates with convergence. This confirms the model's ability to represent ageostrophic frontogenesis.
• Energy Conservation:
  • While SWQG+1 does not conserve energy exactly (unlike the full SW model), the time evolution of total energy, Eddy Kinetic Energy (EKE), and Available Potential Energy (APE) closely tracks the full SW model.

5. Significance and Outlook

• Bridging the Gap: SWQG+1 serves as a crucial intermediate model between the computationally cheap but inaccurate SWQG and the expensive, wave-filled full shallow water equations. It allows for the study of balanced transport and asymmetry without the noise of inertial-gravity waves.
• Applications:
  • Eddy Parameterization: Can improve the modeling of jet locations and eddy variability in ocean general circulation models (OGCMs), where standard QG fails.
  • Moisture and Storm Tracks: The framework is extendable to moist PV, offering a way to study large-scale storm track evolution without gravity wave complications.
  • Coherent Structures: Provides a more realistic theoretical basis for studying the formation and destruction of ocean eddies and planetary vortices (e.g., on Jupiter) than SWQG.
• Future Work: The authors suggest extending the model to spherical domains, non-constant Coriolis parameters ($f$), and topography, as well as integrating it into data assimilation systems for high-resolution satellite altimetry.

In conclusion, the paper establishes SWQG+1 as a robust, next-order balanced model that successfully captures emergent vorticity asymmetries and finite divergence in both one and two-layer shallow water systems, offering a powerful tool for geophysical fluid dynamics research.