The accuracy and efficiency of semi-implicit time stepping for mesoscale storm dynamics

The semi-implicit time-stepping scheme is often applied to the terms respon sible for fast waves in large-scale global weather prediction and general c irculation models to remove the time step restrictions associated with thes e waves. Both the phase and amplitude of fast gravity waves are distorted i n such models. Because gravity waves carry very little energy, this distort ion does not significantly impact the large-scale flow. At mesoscale resolu tions the semi-implicit scheme can also be applied, but it has been general ly assumed that the treatment of gravity waves is inaccurate at these scale s as well. In this paper mesoscale convective systems in the midlatitudes d riven by diabatic heating are studied. According to a recently developed ma thematical theory, only gravity waves with wavelengths larger than the char acteristic length scale of the heat source contain a significant amount of energy, and here it is shown that these gravity waves are accurately reprod uced by a semiimplicit discretization of the 3D compressible governing equa tions with a time step appropriate for the dominant solution component. It will also be shown that the structure equation reduces to the gravity wave equation of the mathematical theory when the appropriate scaling arguments are applied.