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

TL;DR

The paper introduces SWQG$^{+1}$, a next-order balanced model for shallow-water dynamics that captures emergent vorticity asymmetry missed by SWQG. By employing a potential-based formulation with PV as the sole prognostic variable and a set of four Screened-Poisson inversions, the model extends naturally from one to multiple layers while filtering inertial-gravity waves. Nonlinear simulations in both one- and two-layer configurations show SWQG$^{+1}$ reproduces the negative vorticity skewness seen in freely decaying shallow-water turbulence and the positive skewness associated with baroclinic jets, respectively, while preserving energy-PV evolution trends similar to the full system. The results highlight SWQG$^{+1}$ as a practical, PV-based tool for studying balanced geophysical flows, offering improved representation of ageostrophic processes and frontogenesis while remaining computationally efficient and adaptable to moist or topographic extensions.

Abstract

The turbulent evolution of the shallow water system exhibits asymmetry in vorticity. This emergent phenomenon can be classified as "balanced", that is, it is not due to the inertial-gravity wave modes. The Quasi-Geostrophic (QG) system, the canonical model for balanced motion, has a symmetric evolution of vorticity, thus misses this phenomenon. Here we present a next-order-in-Rossby extension of QG, QG$^{+1}$, in the shallow water context. We recapitulate the derivation of the model in one-layer shallow water grounded in physical principles and provide a new formulation using "potentials". Then, the multi-layer extension of the SWQG$^{+1}$ model is formulated for the first time. The SWQG$^{+1}$ system is still balanced in the sense that there is only one prognostic variable, potential vorticity (PV), and all other variables are diagnosed from PV. It filters out inertial gravity waves by design. This feature is attractive for modeling the dynamics of balanced motions that dominate transport in geophysical systems. The diagnostic relations connect ageostrophic physical variables and extend the massively useful geostrophic balance. Simulations of these systems in classical set-ups provide evidence that SWQG$^{+1}$ captures the vorticity asymmetry in the shallow water system. Simulations of freely decaying turbulence in one-layer show that SWQG$^{+1}$ can capture the negatively skewed vorticity, and simulations of the nonlinear evolution of a baroclinically unstable jet show that it can capture vorticity asymmetry and finite divergence of strain-driven fronts.

Figures (10)

• Figure 1: Vorticity ($\{\varepsilon\}\zeta/f$) fields (top) and height ($\{\varepsilon/Bu\}h/H$) fields (bottom) for $\varepsilon=0.1$ at time $t/T=200$ from the shallow water simulation (left) and SWQG+1 (right).
• Figure 2: Left: time series of vorticity ($\zeta$) skewness for the $\varepsilon=0.1$ simulations from the shallow water model as well as SWQG+1. The lighter lines are the individual ensemble members. The darker lines are the ensemble mean, and the $1/\sqrt{10}$ of the ensemble standard deviation is the filled color around the mean. Right: the vorticity ($\zeta$) skewness at $t/T=200$ for $\varepsilon=0.01,0.03,0.05,0.07,0.1,0.12$. The error bar is $1/\sqrt{10}$ of the ensemble standard deviation.
• Figure 3: The same as Figure \ref{['fig:SW_vortskew_timeline']} but for PV ($q$).
• Figure 4: Left: the time series of the total energy, EKE, and APE \ref{['eq:SW_energies']} for the $\varepsilon=0.1$ simulations of the shallow water model and SWQG+1. Right: the time series of the potential enstrophy \ref{['eq:poten_ens']}.
• Figure 5: Time series of the total energy at the QG level \ref{['eq:energy_QGlev']} in the SWQG+1 simulations, compared to the mean PV ($\left\langle {q} \right\rangle$) of the shallow water model.