1.3.1. Kinematic Boundary conditions Vertical

at fixed and moving \(r\) surfaces we set (see Figure 1.18):

(1.7)\[\dot{r}=0 \text{ at } r=R_{\rm fixed}(x, y)\text{ (ocean bottom, top of the atmosphere)}\]
(1.8)\[\dot{r}=\frac{Dr}{Dt} \text{ at } r=R_{\rm moving}(x, y)\text{ (ocean surface, bottom of the atmosphere)}\]


\[R_{\rm moving}=R_{o} + \eta\]

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. Horizontal

(1.9)\[\vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0\]

where \(\vec{\mathbf{n}}\) is the normal to a solid boundary.