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

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.

## 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.