ON THE ASYMPTOTIC REGIMES AND THE STRONGLY STRATIFIED LIMIT OF ROTATING BOUSSINESQ EQUATIONS

Asymptotic regimes of geophysical dynamics are described for different Burger number limits. Rotating Boussinesq equations are analyzed in t he asymptotic limit of strong stratification in the Burger number of o rder one situation as well as in the asymptotic regime of strong strat ification and weak rotation. It is shown that in both regimes the hori zontally averaged buoyancy variable is an adiabatic invariant (approxi mate conservation law) 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 effects induced by turbulent processes on inertial-gravity waves are evidenced . The ''split'' of the energy transfer of the vortical and the wave co mponents is established in the Craya-Herring cyclic basis. As the Burg er number increases from zero to infinity, we demonstrate gradual unfr eezing of energy cascades for ageostrophic dynamics. This property is related to the nonlinear geostrophic adjustment mechanism which is the capacity of ageostrophic dynamics to transfer energy to small scales. The energy spectrum and the anisotropic spectral eddy viscosity are d educed with an explicit dependence on the anisotropic rotation/stratif ication time scale which depends on the vertical aspect ratio paramete r. Intermediate asymptotic regime corresponding to strong stratificati on and weak rotation is analyzed where the effects of weak rotation ar e accounted for by an asymptotic expansion with full control (saturati on) of vertical shearing. The regularizing effect of weak rotation dif fers from regularizations based on vertical viscosity. Two scalar prog nostic equations for ageostrophic components (divergent velocity poten tial and geostrophic departure) are obtained.