4.4.3 Surface kinematic boundary conditions revisited

The discussion in Section 4.3.2 provided one method to derive the surface kinematic boundary condition. This section provides another, which is based on the volume or mass conserving balance equations for the layer thickness.

For a volume conserving fluid, the thickness equation (4.40) can be written

 

$$(\partial_{t} + {\bf u}_{h} \cdot \nabla_{h} ) \, \eta q_{w} + w_{1} - \int_{z_1}^{\eta} dz \, \nabla_{h} \cdot {\bf u}_{h}.$$ = (4.49)

To recover the surface boundary condition (4.29), vertically integrate the incompressibility condition (4.3), $\nabla \cdot {\bf u} = 0$, to yield an expression for the vertical velocity at the bottom of the surface layer

$$w(\eta) + \int_\eta^{z_1} dz\, w_z$$ w₁ =

$$w(\eta) + \int_{z_1}^{\eta} dz\, \nabla_{h} \cdot {\bf u}_{h}.$$ = (4.50)

This expression in equation (4.49) then yields the surface kinematic boundary condition

$$(\partial_{t} + {\bf u}_{h} \cdot \nabla_{h} ) \, \eta q_{w} + w(\eta).$$ = (4.51)

For a mass conserving fluid, the thickness equation (4.45) can be written

$$\rho(\eta) (\partial_{t} + {\bf u}_{h} \cdot \nabla_{h} ) \, \eta w_{1} \, \rho_{1} + Q_{w} - \int_{z_1}^{\eta} dz \, (D_h\rho/Dt + \rho \, \nabla_{h} \cdot {\bf u}_{h})$$ = (4.52)

With the identity

$$w_{1}\rho_1 w(\eta)\rho(\eta) - \int_{z_1}^\eta dz\, \frac{\partial (w \rho)}{\partial z}$$ = (4.53)

the horizontal Langrangian derivative $D_h\rho/Dt$ can be completed to the full Lagrangian derivative

$$\rho(\eta) (\partial_{t} + {\bf u}_{h} \cdot \nabla_{h} ) \, \eta w(\eta) \, \rho(\eta) + Q_{w} - \int_{z_1}^{\eta} dz \, (D\rho/Dt + \rho \, \nabla \cdot {\bf u}).$$ = (4.54)

The term on the right hand side under the integral vanishes due to mass conservation

$$\frac{D\rho}{Dt} + \rho \, \nabla \cdot {\bf u}$$ = 0. (4.55)

As such, one recovers the surface kinematic boundary condition appropriate for a mass conserving fluid

$$(\partial_{t} + {\bf u}_{h} \cdot \nabla_{h} ) \, \eta w + \frac{Q_{w}}{\rho} \qquad z=\eta.$$ = (4.56)

The quantity

$$q_w = \frac{Q_{w}}{\rho(\eta)}$$ (4.57)

is the volume flux in a mass conserving model. Thus, the only apparent difference from the volume conserving kinematic condition (4.29) is the approximation used for the calculation of the volume flux through the sea surface. However, other differences as thermal expansion are hidden in the vertical velocity $w(\eta)$.