6.4.1 Tendencies for the vertically averaged velocities

The first step is to integrate the horizontal velocity equations (4.1) and (4.2) vertically over the full depth of the rigid lid ocean

$$\partial_{t} U - f \, V - \left( \frac{H}{a \, \rho_{o} \, \cos\phi} \right) \, (p_{a}+p_{l})_{\lambda} + X_{0}$$ = (6.15)

$$\partial_{t} V + f \, U - \left( \frac{H}{a \, \rho_{o} } \right) \, (p_{a}+p_{l})_{\phi} + Y_{0}.$$ = (6.16)

The vertical integral of depth dependent quantities (mod the Coriolis force) has been lumped into the terms

  

$$\int_{-H}^{0} dz \left( - \nabla \cdot (u {\bf u}) + \frac{uv\tan... ...kappa_mu_z)_z + F^u - \frac{ (p_{b})_{\lambda} }{a \rho_\circ \cos\phi} \right)$$ X₀ = (6.17)

$$\int_{-H}^{0} dz \left( -\nabla \cdot (v {\bf u}) - \frac{u^2\tan\phi}{a} + (\kappa_mv_z)_z + F^v -\frac{(p_{b})_{\phi}}{a \rho_\circ } \right).$$ Y₀ = (6.18)

These equations are implemented in MOM through a straightforward vertical integration of the forcing terms. To facilitate physical interpretation of the surface terms forcing the vertically integrated momentum, it is useful to perform some manipulations on X₀ and Y₀. First, use the bottom kinematic boundary condition (4.24) and the surface kinematic condition w(0) = 0 in order to bring the vertical integral of the convergence of the advective flux to the form

$$- \int^{0}_{-H} dz \; \biggl( \nabla_{h} \cdot (\alpha \, {\bf u}_{h}) + (\alpha \, w)_{z} \biggr) - \nabla_{h} \cdot \int^{0}_{-H} dz \; \alpha \, {\bf u}_{h}$$ =

$$\alpha(-H) \biggl[ \nabla_{h} H \cdot {\bf u}_{h}(-H) + w(-H) \biggr] - w(0) \, \alpha(0)$$ +

$$-\nabla_{h} \cdot \int^{0}_{-H} dz \; \alpha \, {\bf u}_{h},$$ = (6.19)

where $\alpha = u,v$. Second, recall from equations (9.187) and (9.193) that the horizontal momentum friction can be written as a Laplacian acting on the horizontal velocity, plus an extra ``metric'' term

$${\bf F} = \nabla_{h} \cdot (A_{m} \, \nabla_{h} {\bf u}) + {\bf F}_{metric}.$$ (6.20)

As such, the depth integral of momentum friction can be written

$${ \int_{-H}^{0} dz \biggl( (\kappa_{m} \, {\bf u}_{z})_{z} + {\bf F} \biggr) }$$

$$\left( \frac{{\bf\tau}_{surf} - {\bf\tau}_{bottom}}{\rho_{o}} \ri... ...F}_{metric} + \nabla_{h} \cdot \int_{-H}^{0} dz \, A_{m} \, \nabla_{h} {\bf u},$$ = (6.21)

where

$${\bf\tau}_{surf} \rho_{o} \, \kappa_{m} \, {\bf u}_{z}\vert _{z=0}$$ = (6.22)

$${\bf\tau}_{bottom} \rho_{o} \, \biggl( \kappa_{m} \, {\bf u}_{z} + A_{m} \, (\nabla_{h}H) \cdot (\nabla_{h} {\bf u}) \biggr)_{z=-H}$$ = (6.23)

are the surface and bottom stresses ( $\mbox{dyn}\,\mbox{cm}^{-2}$) due to the winds at the ocean surface and friction and topography at the bottom. The above results render

$${ X_{0} = \frac{ \Delta(\tau^{\lambda})}{\rho_{o}} - \int^{0}_{-H} dz \; \frac{(p_{b})_{\lambda}}{a \rho_\circ \cos\phi} }$$

$$\int_{-H}^{0} dz \left( \frac{uv\tan\phi}{a} + F^u_{metric} \righ... ...nt^{0}_{-H} dz \; \left(u \, {\bf u}_{h} - A_{m} \, \nabla_{h} u\right) \right)$$ + (6.24)

$${ X_{0} = \frac{ \Delta(\tau^{\phi})}{\rho_{o}} - \int^{0}_{-H} dz \; \frac{(p_{b})_{\phi}}{a \rho_\circ} }$$

$$\int_{-H}^{0} dz \left( - \frac{u^2\tan\phi}{a} + F^v_{metric} \r... ...nt^{0}_{-H} dz \;\left( v \, {\bf u}_{h}- A_{m} \, \nabla_{h} v\right) \right).$$ + (6.25)

Using an overline to denote a vertical column average, and using the expression (6.14) for the horizontal pressure gradient, yields the vertically averaged velocity equations

$$\partial_{t} \overline{u} f \, \overline{v} - \left( \frac{1}{a \, \rho_{o} \, \cos\phi} \r... ...{b})_{\lambda}} \biggr] + (\Delta \tau^{\lambda} / \rho_{o} \, H) + \Gamma^{u},$$ = (6.26)

$$\partial_{t} \overline{v} -f \, \overline{u} - \left( \frac{1}{a \, \rho_{o}} \right) \bigg... ...e{ (p_{b})_{\phi}} \biggr] + (\Delta \tau^{\phi} / \rho_{o} \, H) + \Gamma^{v},$$ = (6.27)

where

$$\Gamma^{u} \overline{F^{u}_{metric} } + \overline{u \, v} \, (\tan\phi/a) + ... ...t \biggl[ H \, \overline{ (A_{m} \, \nabla_{h} - {\bf u}_{h}) \, u } \, \biggr]$$ = (6.28)

$$\Gamma^{v} \overline{F^{v}_{metric}} - \overline{u \, u} \, (\tan\phi/a) + H... ... \biggl[ H \, \overline{ (A_{m} \, \nabla_{h} - {\bf u}_{h}) \, v } \, \biggr].$$ = (6.29)

represent the vertical average of the horizontal friction metric terms, advection metric terms, divergence of the horizontal viscous fluxes, and convergence of the horizontal advective fluxes.