4.3.2 Surface kinematic boundary condition

In order to construct the boundary condition to be placed at the free surface $z=\eta$ (see Figure 4.1), consider the algebraic equation which defines the position of the free surface

$$\eta - z = 0.$$ (4.27)

In the special case when there is zero water penetrating the free surface, $D(\eta-z)/Dt = 0$ since the free surface in this case is a material boundary for which a particle initially on the boundary will remain on the boundary. The same reasoning was used previously to derive the bottom kinematic boundary condition. The advent of a fresh water flux means that the free surface is generally not an impermeable material boundary. Rather, the material time derivative of $(\eta-z)$has a source term determined by the fresh water flux

$$\frac{D(\eta - z)}{Dt} = q_{w} \qquad z = \eta,$$ (4.28)

where q_w is the volume per unit time per unit area (dimensions of a velocity) of fresh water entering the ocean through the free surface (q_w > 0 for water entering the ocean across the free surface). This result leads to the surface kinematic boundary condition

 

$$(\partial_{t} + {\bf u}_{h} \cdot \nabla_{h}) \, \eta = w + q_{w} \qquad z = \eta.$$ (4.29)

As such, the surface height $z=\eta$ has a time tendency $\partial_{t} \, \eta$ determined by an advective flux of height $-{\bf u}_{h} \cdot \nabla_{h} \, \eta$, the Eulerian vertical velocity $w(\eta)$, and the fresh water velocity q_w.