New models for the rotating shallow water and Boussinesq equations by subsets of mode interactions

Models for the rotating shallow water equations (RSW) are derived by considering the nonlinear interactions between subsets of linear eigenmodes. The quasi-geostrophic (QG) equation results when only vortical eigenmodes are involved. Continuing past QG, new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. One such model adds the interactions between one IG mode and two vortical modes. Unlike QG, this model behaves similar to RSW in both decay and forced scenarios for which the initial conditions or forcing is either balanced or unbalanced with divergence- free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency, and cyclone/anticyclone asymmetry. Dominance of anticyclones is observed observed away from the QG regime. A hierarchy of models investigates wave-vortical and wave-wave interaction effects.;The nonlinear dynamics of IG modes in rotating stratified fluids is investigated. From the rotating Boussinesq equations (RBE), a partial differential equation system is derived; the GGG model, consisting of only the interactions among the IG modes. This subsystem conserves energy and is not restricted to resonant interactions. Comparing GGG to RBE gauges the importance of wave-vortical-wave vs. wave-wave-wave interactions in determining the transfer and distribution of wave-mode energy. The numerical setting is a skewed domain in which the effects of rotation and stratification are equal. The focus is on the equilibration of wave-mode energy and its spectral scaling driven by random large-scale forcing. In a setting under strong rotation and stratification and as skewed as numerical resources would allow, the RBE wave-mode energy equilibrates and its spectrum scales between k-1 and k-5/3. However, when forcing is restricted to only wave modes the wave-mode energy fails to equilibrate in both RBE and GGG at achieved resolutions. This demonstrates the importance of the vortical mode (wave-vortical-wave interactions). However, in a less skewed domain under weaker rotation and stratification the energy in both systems equilibrates and becomes resolution independent; both systems yield identical power-law scaling of wave-mode energy spectra. These results highlight the difficulty in properly resolving wave-mode interactions.