Hydrostatic Pressure Gradient -- semi-implicit scheme

Figure: Sketch of the leapfrog time stepping sequence in NEMO from Leclair and Madec [2009]. The use of a semi-implicit computation of the hydrostatic pressure gradient requires the tracer equation to be stepped forward prior to the momentum equation. The need for knowledge of the vertical scale factor (here denoted as $h$) requires the sea surface height and the continuity equation to be stepped forward prior to the computation of the tracer equation. Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here (see § 6.5).

The range of stability of the Leap-Frog scheme can be extended by a factor of two by introducing a semi-implicit computation of the hydrostatic pressure gradient term [Brown and Campana, 1978]. Instead of evaluating the pressure at $t$, a linear combination of values at $t-\rdt$, $t$ and $t+\rdt$ is used (see § 6.4.5). This technique, controlled by the nn_dynhpg_rst namelist parameter, does not introduce a significant additional computational cost when tracers and thus density is time stepped before the dynamics. This time step ordering is used in NEMO (Fig.3.1).

This technique, used in several GCMs (NEMO, POP or MOM for instance), makes the Leap-Frog scheme as efficient ^3.1 as the Forward-Backward scheme used in MOM [Griffies et al., 2005] and more efficient than the LF-AM3 scheme (leapfrog time stepping combined with a third order Adams-Moulthon interpolation for the predictor phase) used in ROMS [Shchepetkin and McWilliams, 2005].

In fact, this technique is efficient when the physical phenomenon that limits the time-step is internal gravity waves (IGWs). Indeed, it is equivalent to applying a time filter to the pressure gradient to eliminate high frequency IGWs. Obviously, the doubling of the time-step is achievable only if no other factors control the time-step, such as the stability limits associated with advection, diffusion or Coriolis terms. For example, it is inefficient in low resolution global ocean configurations, since inertial oscillations in the vicinity of the North Pole are the limiting factor for the time step. It is also often inefficient in very high resolution configurations where strong currents and small grid cells exert the strongest constraint on the time step.

Footnotes

... efficient^3.1

The efficiency is defined as the maximum allowed Courant number of the time stepping scheme divided by the number of computations of the right-hand side per time step.