Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations

We study the three-dimensional, incompressible, non-hydrostatic Boussinesq fluid equations, which are applicable to the dynamics of the oceans and atmosphere. These equations describe the interplay between velocity and buoyancy in a rotating frame. A hierarchy of dynamical variables is introduced whose members ${{ \Omega }_{m}}(t)$ ($1\leqslant m<\infty $ ) are made up from the respective sum of the L 2m -norms of vorticity and the density gradient. Each ${{ \Omega }_{m}}(t)$ has a lower bound in terms of the inverse Rossby number, Ro −1, that turns out to be crucial to the argument. For convenience, the ${{ \Omega }_{m}}$ are also scaled into a new set of variables D m (t). By assuming the existence and uniqueness of solutions, conditional upper bounds are found on the D m (t) in terms of Ro −1 and the Reynolds number Re. These upper bounds vary across bands in the $\left\{{{D}_{1}},\,{{D}_{m}}\right\}$ phase plane. The boundaries of these bands depend subtly upon Ro −1, Re, and the inverse Froude number Fr −1. For example, solutions in the lower band conditionally live in an absorbing ball in which the maximum value of ${{ \Omega }_{1}}$ deviates from Re 3/4 as a function of $R{{o}^{-1}},\,Re$ and Fr −1