7.3.1 The barotropic system

For the barotropic system, split the vertically integrated transport into two terms

$${\bf U} = {\bf U}_{0} + \int_{0}^{\eta} dz \, {\bf u}_{h},$$ (7.26)

where

$${\bf U}_{0} = \int_{-H}^{0} dz \, {\bf u}_{h}$$ (7.27)

is the vertically integrated transport contained within $-H \le z \le 0$, and $\int_{0}^{\eta} dz \, {\bf u}_{h}$ is that transport maintained within the free surface layer. Consequently,

$$\partial_{t} {\bf U} = \partial_{t} {\bf U}_{0} + \eta_{t} \, {\bf u}_{h}(\eta) + \int_{0}^{\eta} dz \, \partial_{t} {\bf u}_{h}.$$ (7.28)

In a similar manner, split the vertically integrated forcing into two terms

$$X_{0} + \int_{0}^{\eta} dz \left( - \nabla \cdot (u {\bf u}) + \f... ...\kappa_mu_z)_z - \frac{(p_{b})_{\lambda} }{a \rho_\circ \cos\phi} + F^u \right)$$ X =

$$Y_{0} + \int_{0}^{\eta} dz \left( -\nabla \cdot (v {\bf u}) - \fr... ...phi}{a} + (\kappa_mv_z)_z -\frac{(p_{b})_{\phi} }{a \rho_\circ } + F^v \right),$$ Y = (7.29)

where X₀ and Y₀ are the contributions from $-H \le z \le 0$

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

$$\int_{-H}^{0} dz \left( -\nabla \cdot (v {\bf u}) - \frac{u^2\tan\phi}{a} + (\kappa_mv_z)_z -\frac{(p_{b})_{\phi} }{a \rho_\circ } + F^v \right).$$ Y₀ = (7.30)

Using these expressions in equation (7.22) for the zonal transport yields

$$\partial_{t} \, U_{0} - f V_{0} + \frac{H \, (p_{a} + p_{s})_{\la... ... X_{0} = - \frac{\eta \, (p_{a} + p_{s})_{\lambda} }{a \, \rho_{o} \, \cos\phi}$$

$$+ \int_{0}^{\eta} dz \left(-u_{t} + f v - \nabla \cdot (u {\bf u}... ...\kappa_mu_z)_z -\frac{(p_{b})_{\lambda} }{a \rho_\circ \cos\phi} + F^u \right).$$ (7.31)

Note the cancelation of the $\eta_{t} \, u(\eta)$ term. Use of the zonal momentum equation (7.20) provides for a convenient cancellation of the remaining terms, hence revealing that the vertically integrated transport ${\bf U}_0$ satisfies the equation

$$\partial_{t} \, U_{0} - f V_{0} = - \frac{H \, (p_{a} + p_{s})_{\lambda}}{a \, \rho_{o} \, \cos\phi} + X_{0}.$$ (7.32)

Similarly, the meridional transport V₀ satisfies

$$\partial_{t} \, V_{0} + f U_{0} = - \frac{H \, (p_{a} + p_{s})_{\phi} }{a \, \rho_{o}} + Y_{0}.$$ (7.33)