14.4.4 Time steps

Namelist /tsteps/

Historically, ocean time is measured in terms of tracer time steps. It should be noted that the equations in MOM can be solved asynchronously (Bryan 1984) using one timestep for the internal mode dtuv, another for the external mode dtsf, and a third for the tracers dtts. Each are input through namelist.

Basically asynchronous timestepping can be done because the three processes have different time scales. Since the timescale for adjustment of density is much greater than that of velocity, it is reasoned that integrations to equilibrium can be speeded up by taking a large time step on the tracer equations (within CFL restrictions) and letting the velocities come into geostrophic adjustment with the density. If the problem is linear and only the equilibrium solution is sought, the equilibrium solution is unique and it doesn't matter how the integration gets there. However, if the solution is non-linear enough to have multiple equilibria or the transient response is of interest, all three time steps should be synchronous with the rigid lid, and dtts=dtuv for the explicit free surface.

A similar argument can be made for the adjustment time scale of the deep layers being much greater than that of the surface layers. An acceleration with depth factor dtxcel_k, initialized to 1.0 for all levels, is used to increase the length of the tracer time step with depth to reach equilibrium sooner (Bryan 1984).

Given the resolution defined by module grids, a time step can be estimated from the linear CFL condition (see Haltiner and Williams 1980)

$$\Delta t \leq \frac{\Delta_{min}}{2\cdot c \cdot \sin(2\pi\Delta_{min}/L)}$$ (14.2)

where $\Delta_{min}$ is the minimum grid cell width, c is the wavespeed, and L is the scale of the wave. The most restrictive scale is $L=4\Delta_{min}$. The external gravity wave is the fastest wave with $c=\sqrt{grav\cdot H}$ but this is filtered out of the equations by the rigid lid approximation. If the explicit free surface option is enabled, then this wave speed must be resolved and this accounts for the relatively short time step on the external mode. Next fastest are the low frequency external mode barotropic Rossby basin-scale waves with $c=-\frac{\beta}{k^2 + l^2}$ where $k=\frac{2\pi}{L_x}$ and $l=\frac{2\pi}{L_y}$ are zonal and meridional wavenumbers. These waves limit the barotropic time step when using the rigid lid stream function method. Eastward traveling Rossby waves have small scales and their speed is too slow to be a limiting factor on the time step. Small scale internal mode gravity waves and the non-dispersive Kelvin waves travel with maximum speed $c \approx 3$ m/sec. In general, these are the waves that restrict the baroclinic time step. The trace time step is limited by advective velocity which can easily reach 1 or 2 m/sec in boundary currents.

In some models, wavespeed is not the limiting factor for determining timestep length. For instance, when vertical resolution is approximately 10 meters thick, the time step may be limited by vertical velocity near the surface in regions such as the equator. Regardless of what limits the time step, it is recommended that diagnostic option stability_tests be enabled to show how close the model is to the local CFL condition and where that position is located. Large vertical diffusion coefficients can also limit the timestep length and when this is the case, option implicitvmix should be enabled to solve the vertical diffusion components implicitly.

Here are some rough examples from models run at GFDL.

• For a $2.4^\circ$ by $2.4^\circ$ mid latitude thermocline model, the following settings were used: dtsf=dtuv=3000.0 sec; dtts = 30000.0 with acceleration of dtts with depth using dtxcel₁=1.0to dtxcel_km=5.0.

• For a $\Delta_\lambda=1^\circ$ by $\Delta_\phi=1/3^\circ$equatorial basin the following settings were used: dtsf = dtuv = dtts = 3000.0 sec.

The following variables set the time steps in seconds.

•  dtts = time step length for tracers

•  dtuv = time step length for internal mode velocities

•  dtsf = time step length for external mode velocities