6.6 The baroclinic mode

The time tendency for the zonal baroclinic velocity is given by

$$\partial_{t} \widehat{u} \partial_{t} (u - \overline{u})$$ =

$$\left(1 - \frac{1}{H} \int^{0}_{-H} dz \right) \left(G^{u} - \lef... ...{a \, \rho_\circ \, \cos\phi} \right) (p_{b} + p_{l} + p_{a})_{\lambda} \right)$$ = (6.50)

where the vector ${\bf G}$ was introduced in equation (6.36). Since the atmospheric and lid pressures are depth independent, these pressures have zero deviation from their depth average. Hence, the tendency for the baroclinic velocity is independent of the atmospheric and lid pressures. In determining the baroclinic velocity, it is therefore sufficient to consider

 

$$\partial_{t} \widehat{u} \partial_{t} (u' - \overline{u}')$$ = (6.51)

where u' and $\overline{u}'$ satisfy the full zonal velocity equation and vertically averaged zonal velocity equation, respectively, but without any atmospheric or lid pressure contributing to the forcing. Without the atmospheric and lid pressure contributions, all forcing terms in the two ``primed'' equations are known. Consequently, the primed velocities can be time stepped in a straightforward manner without using any tricks such as those needed to eliminate the atmospheric and lid pressures from the streamfunction equation. Time stepping the primed velocities then yields the updated baroclinic velocity through equation (6.51). After obtaining the updated baroclinic velocity, the full horizontal velocity field

$$(u,v) = (\overline{u},\overline{v}) + (\widehat{u},\widehat{v})$$ (6.52)

is known at the new time step. Formulation of the rigid lid streamfunction method is now complete.