New intermediate models for rotating shallow water and an investigation of the preference for anticyclones

Abstract

New intermediate models for the rotating shallow water (RSW) equations are derived by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vortical eigenmodes. Continuing past QG in a non-perturbative manner, the new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. The such simplest model adds nonlinear interactions between one IG mode and two vortical modes. In sharp contrast to QG, the latter model behaves similar to the full RSW equations for decay from balanced initial conditions as well as unbalanced random initial conditions with divergence-free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency and cyclone/anticyclone asymmetry. In particular, dominance of anticyclones is observed for Rossby numbersRoin the range 0.1 <Ro< 1 (away from the QG parameter regimeRo→ 0). A hierarchy of models is explored to determine the effects of wave-vortical and wave–wave interactions on statistics such as the skewness of vorticity in decaying turbulence. Possible advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting thema priorito any particular parameter regime) and (ii) insight into the physical and mathematical consequences of vortical–wave interactions.