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 $\Omega_{m}(t)$ ($1 \leq m < \infty$) 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 $\{D_{1},\,D_{m}\}$ 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 $Ro^{-1},\,Re$ and $Fr^{-1}$.