7.3.4 Summary of the linear barotropic system

The linearized barotropic system consists of the vertically integrated momentum and continuity equations, where the vertical integral extends between $-H \le z \le 0$. The results are the dynamical equations

$$\partial_{t} \, {\bf U}_{0} -f\hat{z} \wedge {\bf U}_{0} - H \, \nabla_{h} \, (p_{a} + p_{s})/\rho_{o} + {\bf X}_{0}$$ = (7.42)

$$\partial_{t} \eta^{0} -\nabla_{h} \cdot {\bf U}_{0} + q_{w},$$ = (7.43)

where the vertically integrated forcing is given by

$${\bf X}_{0} = \int_{-H}^{0} dz \biggl[ -\nabla \cdot ({\bf... ...b}/\rho_{o}) + (\kappa_m {\bf u}_z)_z + F^{\bf u} \biggr].$$ (7.44)

The shallow water limit consists of setting ${\bf X}_{0} = 0$. The result is the linear shallow water equations. In general, the barotropic system for ${\bf U}_0$ and $\eta^{0}$ is identical to the linearized version of the barotropic equations for ${\bf U}$ and $\eta$, where the ${\bf U}, \eta$ system results from vertically integrating the momentum and continuity equations between $-H \le z \le \eta$.

Note that the barotropic system has been formulated for the vertically integrated transport. After solving for this transport, the vertically averaged velocity can simply be diagnosed and then the full barotropic plus baroclinic velocity field can be constructed. An alternative approach is to directly time step the prognostic equations for the vertically averaged velocity and hence save some computational time by avoiding the extra diagnostic step. However, undocumented experience with shallow water models (Pacanowski) has shown that time stepping the equations for $\overline{{\bf u}}_{h}$ provides for less numerical stability and introduces more noise. The formulation of MOM's free surface is motivated by this experience.

In the shallow water system, linear waves with speeds $\sqrt{g \, H}$ are admitted. In the deep ocean, these waves can reach over 200 m/sec, and they are generally much faster than the first baroclinic wave motions. An explicit time stepping algorithm is therefore constrained by the need to resolve these fast shallow water waves. When ${\bf X}_{0}$ is not neglected, there are additional nonlinear couplings between the shallow water system and the depth dependent baroclinic system. Due to the complexity of this nonlinear coupling, and due to the large contribution from the slower baroclinic modes to the temporal changes of ${\bf X}_{0}$, it is reasonable to hold ${\bf X}_{0}$ constant when integrating over the small barotropic time steps, and to only update them at the long baroclinic time steps. This is the approach used by Blumberg and Mellor (1987), Killworth et al. (1991), Hallberg (1995), and in the MOM implementation of the explicit free surface (Section 29.5). It is useful to compare the linearized barotropic system with the rigid lid method. Indeed, the comparison is trivial: the only difference is that the rigid lid equations set $\eta$ to zero and introduce a streamfunction to describe the divergence-free transport ${\bf U}_0$. The vertically integrated forcing ${\bf X}_{0}$ is formally the same in both the rigid lid and free surface. The essential difference is that ${\bf X}_{0}$ in the free surface contains contributions from a generally nonzero vertical velocity w(0). But note that in the steady state with fresh water input set to zero, the free surface w(0) vanishes. Hence, the steady state solutions realized with the linearized free surface reduce to the rigid lid solutions. Such connection between the two solution methods provides an essential point of departure for developing the free surface.