A Comparison of Time Discretization Schemes for Two-Timescale Problems in Geophysical Fluid Dynamics

In many geophysical fluid modeling applications there exist two very different time scales, essentially fast waves and slow vortices. At the largest planetary scales inertial-gravity waves propagate through the fluid at phase speeds much faster than particle velocities, while at small scales fast acoustic modes coexist with much slower vortex motions. If numerical models are to be affordable in this context, schemes which do not explicitly resolve the fast motion must be devised. Although this has traditionally been done for strictly numerical reasons, the result is physically equivalent to a temporal sub-grid-scale model. Examples of such schemes examined here are the semi-implicit, split explicit, and linear-propagator methods. All of these methods fail to resolve the fast motion accurately, but in very different ways. Recent progress in the field of rotating stratified turbulence is employed on a simplified two-time-scale version of the Boussinesq equations in order to determine how well these schemes represent the evolution of the slow variables. It is found that the split explicit and linear-propagator methods exhibit spurious interactions between the slow modes and their numerically retarded fast modes in certain conditions when ωΔt is large. The semi-implicit method is able to maintain a frequency separation, but it mimics the true evolution best when time filtering is applied.