4.3.1 Bottom kinematic boundary condition

The bottom kinematic boundary condition is the no-normal flow condition

$$\hat{n}_{bottom} \cdot {\bf u}= 0.$$ (4.22)

Namely, with the ocean bottom defined by the algebraic expression $f(\lambda, \phi, z) = z + H(\lambda,\phi) = 0$, the unit vector normal to the ocean bottom is given by

$$\hat{n}_{bottom} \frac{\nabla f}{\vert\nabla f\vert}$$ =

$$\frac{(\nabla_{h}H, 1)}{\sqrt{1 + \vert\nabla_{h}H\vert^{2}}}.$$ = (4.23)

As such, the no-normal flow condition implies

 

$$- {\bf u}_{h} \cdot \nabla_{h} H \qquad z = -H(\lambda,\phi).$$ w = (4.24)

For the degenerate case of a steep sidewall, $\hat{n}_{bottom}$orients horizontally and the usual lateral boundary condition

$$\hat{n}_{wall}\cdot {\bf u}_h= 0$$ (4.25)

is retained, where $\hat{n}_{wall}$ is a horizontal unit vector normal to the sidewall.

A second derivation uses the fact that the bottom is a material surface at $z=-H(\lambda,\phi)$. This fact means that a particle initially on the bottom will remain so, and hence

$$\frac{D(z+H)}{Dt} = 0.$$ (4.26)

This result implies again equation (4.24).

For the rigid lid, a third derivation allows for a more ready implementation in MOM of the bottom condition. The bottom kinematic condition can be generated by integrating Equation (4.3) from the surface to the ocean bottom using Equations (4.1), (4.2), (6.4), and (6.5). The finite difference equivalent of this method is used to generate vertical velocities in the interior as well as at the bottom. A more complete discussion of the discrete vertical velocity at the ocean bottom is given in Section 22.3.3.