21.4 Time Stepping Schemes

Basically, MOM uses a ceneterd leapfrog time stepping scheme to integrate prognostic equations forward in time. To handle the computational mode characteristic of the leapfrog scheme, there are three choices: periodic application of a forward timestep, periodic application of an Euler backward timestep, or a Robert time filter applied every time step.

If option robert_time_filter is enabled, a Robert time filter is applied every time step. If option robert_time_filter is not enabled, then the type of mixing is determined by logical variable eb which if true indicates an Euler backward mixing scheme, otherwise a forward mixing scheme is used. Intervals between mixing timesteps are set by logical switch mixts. In the past, mixing time steps were applied every 17 time steps. This interval seems to work satisfactorily and has been retained. Of the two mixing schemes (forward and Euler backward), the Euler backward is more dissipative. It also damps spatial scales whereas the forward scheme does not (Haltiner and Williams 1980).

The focus now will be to explain how these schemes are implemented within MOM. The following description assumes that the idea of a memory window as outlined in Section 11.3.1 is understood. Refer to Figure 21.1 and note the four columns: Time Step,  Type of Time Step, a partially opened memory window with two disk areas indicated bu column jmw < jmt, and a fully opened memory window with no disk area indicated by column jmw=jmt.

Two sets of indices are required to act as pointers to specific areas on disk and in the memory window: one set points to $\tau-1$, $\tau$, and $\tau+1$ locations on disk and these indices are named taum1disk, taudisk, and taup1disk; the other set points to $\tau-1$, $\tau$, and $\tau+1$ locations within the memory window and these indices are named taum1, tau, and taup1.

At the beginning of each time step, memory window indices are updated as shown schematically in the Type of Time Step column. The whole idea is to get data positioned properly inside the memory window so that the equations can be solved for various types of time steps with only minimal changes^21.4.

To actually see how disk indices are cycled for various types of time steps, enable option trace_indices as described in Chapter 38.