7.2.4 Vertically integrated momentum equations

From the definition of ${\bf U}$ and Leibnitz's Rule (4.65), the time tendency ${\bf U}_{t}$ takes the form

 

$${\bf U}_{t} \eta_{t} \; {\bf u}_{h}(\eta) + \int_{-H}^\eta dz \, \partial_{t} {\bf u}_{h}.$$ = (7.19)

In this expression, ${\bf u}_{h}(\eta)$ is the horizontal current on the ocean side of the free surface $z=\eta$, and the term $\eta_{t} \, {\bf u}_{h}(\eta)$ arises from changes in the free surface height. The following discussion amounts to an insertion of the terms from the momentum equations into $\int_{-H}^\eta dz \, \partial_{t} {\bf u}_{h}$. Recall the considerations of Section 7.1, which rendered the decomposition (7.9) of hydrostatic pressure into a depth dependent ``baroclinic pressure'' p<sub>b</sub> and a depth independent ``barotropic pressure'' p_a + p_s. As a result, the horizontal momentum equations take the form

  

$$f v - \frac{(p_{a} + p_{s} + p_{b})_{\lambda}}{a \, \rho_\circ \,... ...phi} - \nabla \cdot (u {\bf u}) + \frac{uv\tan\phi}{a} + (\kappa_m u_z)_z + F^u$$ u_t = (7.20)

$$-fu - \frac{(p_{a} + p_{s} + p_{b})_\phi}{a \, \rho_\circ} - \nabla \cdot (v {\bf u}) - \frac{u^2\tan\phi}{a} + (\kappa_m v_z)_z + F^v.$$ v_t = (7.21)

A vertical integral of these equations between $-H \le z \le \eta$ leads to the equations for the vertically integrated velocity

  

$$\eta_{t} \; u(\eta) - \left( \frac{H + \eta }{a \, \rho_\circ \, \cos\phi } \right) \; (p_{a} + p_{s})_{\lambda} + X$$ U_t - f V = (7.22)

$$\eta_{t} \; v(\eta) - \left(\frac{H + \eta}{a \, \rho_\circ }\right) \; (p_{a} + p_{s})_{\phi} + Y.$$ V_t + fU = (7.23)

The forcing terms

$$\int_{-H}^{\eta} dz \left( - \nabla \cdot (u {\bf u}) + \frac{uv\... ...\kappa_mu_z)_z - \frac{(p_{b})_{\lambda} }{a \rho_\circ \cos\phi} + F^u \right)$$ X = (7.24)

$$\int_{-H}^{\eta} dz \left( -\nabla \cdot (v {\bf u}) - \frac{u^2\tan\phi}{a} + (\kappa_mv_z)_z -\frac{(p_{b})_{\phi} }{a \rho_\circ } + F^v \right)$$ Y = (7.25)

contain vertically integrated baroclinic-baroclinic, barotropic-barotropic, and baroclinic-barotropic interactions as well as the vertically integrated baroclinic pressure gradient and friction. Note that the nonlinear term $\eta_{t} \; {\bf u}_{h}(\eta)$ follows from equation (7.19); it arises from the time dependence of the free surface $\eta$. Since the surface and atmospheric pressures are independent of depth, their horizontal gradients can be trivially integrated vertically, hence the $H+\eta$ factors multiplying these terms in equations (7.22) and (7.23). The equation (7.18) for the free surface height $\eta$, and the equations (7.22) and (7.23) for ${\bf U}$, form the dynamical equations for the barotropic system. Again, this is the barotropic system defined by vertically integrating the continuity and momentum equations between $-H \le z \le \eta$.