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 ${\mathrm{\Omega }}_m(t)$ ($1\le m<\mathrm{\infty }$) are made up from the respective sum of the $L^{2m}$-norms of vorticity and the density gradient. Each ${\mathrm{\Omega }}_m(t)$ has a lower bound in terms of the inverse Rossby number, $R{o}^{-1}$, that turns out to be crucial to the argument. For convenience, the ${\mathrm{\Omega }}_m$ 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 $R{o}^{-1}$ 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 $R{o}^{-1}$, $Re$, and the inverse Froude number $F{r}^{-1}$. For example, solutions in the lower band conditionally live in an absorbing ball in which the maximum value of ${\mathrm{\Omega }}_1$ deviates from $R{e}^{3\mathrm{/}4}$ as a function of $R{o}^{-1},Re$ and $F{r}^{-1}$.