Mathematics Colloquia and Seminars

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Description

New intermediate models for the rotating shallow water equations (RSW) are derived by considering the nonlinear interactions between sub- sets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional (2D) quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vor- tical 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 simplest such model adds nonlinear interactions between one IG mode and two vortical modes. In sharp con- trast 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 numbers Ro in the range 0.1 < Ro < 1 (away from the QG parameter regime Ro → 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. Possi- ble advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting them a priori to any particular parameter regime), and (ii) insight into the physical and mathematical consequences of vortical-wave interactions.