1.3.1. Kinematic Boundary conditions¶
1.3.1.1. Vertical¶
at fixed and moving \(r\) surfaces we set (see Figure 1.18):
(1.7)¶\[\dot{r}=0 \text{ at } r=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}\]
(1.8)¶\[\dot{r}=\frac{Dr}{Dt} \text{ at } r=R_{moving}(x,y)\text{ (ocean surface, bottom of the atmosphere)}\]
Here
\[R_{moving}=R_{o}+\eta\]
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.
1.3.1.2. Horizontal¶
(1.9)¶\[\vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0\]
where \(\vec{\mathbf{n}}\) is the normal to a solid boundary.