5.2 Methods for solving the separated equations

In symbols, the horizontal velocity ${\bf u}_{h}$ is separated into two parts. The vertically averaged velocity representing the approximate barotropic or external part is given by

 

$$\overline{{\bf u}}_{h} = \frac{1}{H+\eta} \, \int^{\eta}_{-H} dz \; {\bf u}_{h},$$ (5.1)

where $H(\lambda, \phi)$ is the distance from the resting ocean surface z=0 to the bottom, and $\eta(\lambda,\phi,t)$ is the departure of the ocean surface height from z=0. Typically, $\vert\eta\vert \le 200\mbox{cm}$, but may be much larger, if tides are taken into consideration. In general, fields which are averaged over the vertical coordinate will be denoted with the overbar. The residual

$$\widehat{{\bf u}}_{h} = {\bf u}_{h} - \overline{{\bf u}}_{h}$$ (5.2)

is a depth dependent velocity, which embodies the approximate baroclinic or internal mode flow. Often, it will be convenient to introduce the vertically integrated horizontal velocity field

 

$${\bf U} = (H+\eta) \, \overline{\bf {u}}_{h} = \int^{\eta}_{-H} dz \; {\bf u}_{h}.$$ (5.3)

Additionally, the following vertically integrated velocity

 

$${\bf U}_0 = \int^{0}_{-H} dz \; {\bf u}_{h}$$ (5.4)

will prove useful. In the fixed surface / rigid lid method (see below), there is no distinction between ${\bf U}$ and ${\bf U}_0$ in the baroclinic model part, since $\eta =0$ is assumed. Additionally, with $\eta =0$ and w(z=0)=0, then $\nabla_{h} \cdot {\bf U} = 0$. This result is exploited in the rigid lid formulation, as seen in Section 5.2.1.

The dynamical equations for the vertically averaged velocity are generally more complicated than the unaveraged equations. Two means for handling these equations are implemented in MOM:

• The fixed surface / rigid lid method. This method fixes the upper surface to $\eta =0$, and closes the upper boundary with w(z=0)=0. There are two flavors of this method: the streamfunction and the surface pressure methods.
• The free surface / non-rigid lid method. This method allows for a freely evolving surface $\eta \ne 0$, and it uses open boundary conditions at z=0 with $w(0) \ne 0$ for the baroclinic and tracer equations. There are two flavors of this method: the explicit and implicit free surface methods.

In short, these two methods differ fundamentally in how they handle the upper ocean boundary conditions.