at fixed and moving r surfaces we set (see :numref:`zandp-vert-coord`):
\dot{r}=0 \text{ at } r=R_{\rm fixed}(x, y)\text{ (ocean bottom, top of the atmosphere)}
\dot{r}=\frac{Dr}{Dt} \text{ at } r=R_{\rm moving}(x, y)\text{ (ocean surface, bottom of the atmosphere)}
Here
R_{\rm moving}=R_{o} + \eta
where R_{o}(x,y) is the ‘r-value’ (height or pressure, depending on whether we are in the atmosphere or ocean) of the ‘moving surface’ in the resting fluid and \eta is the departure from R_{o}(x,y) in the presence of motion.
\vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0
where \vec{\mathbf{n}} is the normal to a solid boundary.