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

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, $\textrm {QG}^{+1}$ , 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 shallow water quasi-geostrophic equation ( $\textrm {SWQG}^{+1}$ ) model is formulated for the first time. The $\textrm {SWQG}^{+1}$ 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 modelling 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 $\textrm {SWQG}^{+1}$ captures the vorticity asymmetry in the shallow water system. Simulations of freely decaying turbulence in one layer show that $\textrm {SWQG}^{+1}$ 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.