Transforming to a vertical coordinate $r(z,x,y,t)$, with $\dot{r} = \frac{\partial r}{\partial t}$ ...

The Boussinesq hydrostatic equations of motion in general-\/coordinate $r$ are\+:

\textbackslash{}f\{eqnarray\} \textbackslash{}label\{html\+:r-\/equations\}\textbackslash{}notag \textbackslash{} \textbackslash{}rho\+\_\+0 \textbackslash{}left( \textbackslash{}frac\{\textbackslash{}partial \textbackslash{}mathbf\{u\}\}\{\textbackslash{}partial t\} + ( f + \textbackslash{}zeta ) \textbackslash{}, \textbackslash{}hat\{\textbackslash{}mathbf\{z\}\} \textbackslash{}times \textbackslash{}mathbf\{u\} +
\begin{DoxyImage}
\includegraphics[width=\textwidth,height=\textheight/2,keepaspectratio=true]{dot_inline_dotgraph_2}
\doxyfigcaption{\{r\} \textbackslash{}, \textbackslash{}frac\{\textbackslash{}partial \textbackslash{}mathbf\{u\}\}\{\textbackslash{}partial r\} + \textbackslash{}nabla\+\_\+r \textbackslash{}, K \textbackslash{}right) \&= -\/\textbackslash{}nabla\+\_\+r \textbackslash{}, p -\/ \textbackslash{}rho \textbackslash{}nabla\+\_\+r \textbackslash{}, \textbackslash{}\+Phi + \textbackslash{}boldsymbol\{\textbackslash{}mathcal\{F\}\}}
\end{DoxyImage}