7.4 Stresses at the ocean surface and bottom

In Section 4.3.3, a discussion was given of the dynamic boundary conditions. These conditions determine how the momentum field is forced at the ocean surface and bottom. That discussion is now extended by performing a few manipulations on the expressions for the vertically integrated forcing discussed in Section 7.2.4. The ideas in this section are also relevant for considering how the baroclinic system is forced at the ocean surface and bottom, and so they will be revisited in that context later in this chapter. The formulation here employs the barotropic equations as defined by vertically integrating between -H and $\eta$. The limit of these results for $\eta \rightarrow 0$recovers the expressions relevant for a linearized free surface method.

First, use the kinematic boundary conditions (7.15) and (7.16) to establish the identity

 

$$\alpha(\eta) \, \eta_{t} - \int_{-H}^{\eta} dz \; \nabla \cdot (\alpha \, {\bf u} ) \alpha(\eta) \, [\eta_{t} - w(\eta) ] + w(-H) \, \alpha(-H) - \int_{-H}^{\eta} dz \; \nabla_{h} \cdot (\alpha \, {\bf u}_{h})$$ =

$$\alpha(\eta)\left(\, q_{w} - \, {\bf u}_{h}(\eta) \cdot \nabla \eta \right) -\alpha(-H) \, {\bf u}_{h}(H) \cdot \nabla H$$ =

$$- \int_{-H}^{\eta} dz \; \nabla_{h} \cdot (\alpha \, {\bf u}_{h})$$

$$\alpha(\eta)\, q_{w} - \nabla_{h} \cdot \int_{-H}^{\eta} dz \; \alpha \, {\bf u}_{h}.$$ = (7.45)

Note that Killworth et al. (1991) performed similar manipulations, yet their equations (19) and (20) are missing a $\cos\phi$ factor in the $\partial / \partial \phi$ terms, which precludes them from exposing the convergence as done in equation (7.45). Additionally, they omit the freshwater forcing term.

Second, recall from the discussion of momentum friction in Chapter 9 (i.e., equations (9.187) and (9.193)) that the second order friction operator can be written as a Laplacian acting on the horizontal velocity, plus an extra metric term

$${\bf F} = \nabla_{h} \cdot (A \, \nabla_{h} {\bf u}_{h}) + {\bf F}_{metric}.$$ (7.46)

The biharmonic friction operator described in Section 9.5 has a similar form, and so no loss of generality arises from assuming such. Consequently, the vertical integral of momentum friction can be rewritten as

$$\int_{-H}^{\eta} dz {\bf F} - A \,(\nabla_{h}\eta) \cdot (\nabla_{h} {\bf u}_{h})\vert _{z=\eta} - A \, (\nabla_{h}H) \cdot (\nabla_{h} {\bf u}_{h})\vert _{z=-H}$$ =

$$+ \int_{-H}^{\eta} dz {\bf F}_{metric} + \nabla_{h} \cdot \int_{-H}^{\eta} dz A \, \nabla_{h} {\bf u}_{h}.$$ (7.47)

Combining these two results brings the vertically integrated forcing to the form

  

$$\frac{\tau_{surf}^{\lambda}}{\rho_{o}} - \frac{\tau_{bottom}^{\la... ... \cdot \int^{\eta}_{-H} dz \; \left(u \, {\bf u}_{h} - A \, \nabla_{h} u\right)$$ X =

$$+ \int_{-H}^{\eta} dz \left( \frac{uv\tan\phi}{a} - \frac{(p_{b})_{\lambda}}{a \rho_\circ \cos\phi} + F^u_{metric} \right),$$ (7.48)

$$\frac{\tau_{surf}^{\phi}}{\rho_{o}} - \frac{\tau_{bottom}^{\phi}}... ...} \cdot \int^{\eta}_{-H} dz \;\left( v \, {\bf u}_{h}- A \, \nabla_{h} v\right)$$ Y =

$$+ \int_{-H}^{\eta} dz \left( - \frac{u^2\tan\phi}{a} -\frac{(p_{b})_{\phi}}{a \rho_\circ } + F^v_{metric} \right).$$ (7.49)

In these expressions, the surface and bottom stresses have been introduced

 

$$\frac{{\bf\tau}_{surf}}{\rho_{o}} \kappa_m {\bf u}_z - A \,(\nabla_{h}\eta) \cdot (\nabla_{h} {\bf u}) + q_{w}\, {\bf u}_{h} \qquad z=\eta(\lambda,\phi,t)$$ = (7.50)

$$\frac{{\bf\tau}_{bottom}}{\rho_{o}} \kappa_m {\bf u}_z + A \, (\nabla_{h} H) \cdot (\nabla_{h} {\bf u}) \qquad z= -H(\lambda,\phi).$$ = (7.51)

Each term is now interpreted physically and their appearance in the numerical model indicated.