On the Asymptotic Regimes and the Strongly Strati ed Limit of Rotating Boussinesq Equations

Asymptotic regimes of geophysical dynamics are described for di erent Burger number limits. Rotating Boussinesq equations are analyzed in the asymptotic limit of strong strati cation in the Burger number of order one situation as well as in the asymptotic regime of strong strati cation and weak rotation. It is shown that in both regimes horizontally averaged buoyancy variable is an adiabatic invariant for the full Boussinesq system. Spectral phase shift corrections to the buoyancy time scale associated with vertical shearing of this invariant are deduced. Statistical dephasing e ects induced by turbulent processes on inertial-gravity waves are evidenced. The `split' of the energy transfer of the vortical and the wave components is established in the Craya-Herring cyclic basis. As the Burger number increases from zero to in nity, we demonstrate gradual unfreezing of energy cascades for ageostrophic dynamics. The energy spectrum and the anisotropic spectral eddy viscosity are deduced with an explicit dependence on the anisotropic rotation/strati cation time scale which depends on the vertical aspect ratio parameter. Intermediate asymptotic regime corresponding to strong strati cation and weak rotation is analyzed where the e ects of weak rotation are accounted for by an asymptotic expansion with full control (saturation) of vertical shearing. The regularizing e ect of weak rotation di ers from regularizations based on vertical viscosity. Two scalar prognostic equations for ageostrophic components (divergent velocity potential and geostrophic departure) are obtained. This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS119480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001