The semi-implicit time-stepping scheme is often applied to the terms responsible for fast waves in large-scale global weather prediction and general circulation models to remove the time step restrictions associated with these waves. Both the phase and amplitude of fast gravity waves are distorted in such models. Because gravity waves carry very little energy, this distortion does not significantly impact the large-scale flow. At mesoscale resolutions the semi-implicit scheme can also be applied, but it has been generally assumed that the treatment of gravity waves is inaccurate at these scales as well. In this paper mesoscale convective systems in the midlatitudes driven by diabatic heating are studied. According to a recently developed mathematical theory, only gravity waves with wavelengths larger than the characteristic length scale of the heat source contain a significant amount of energy, and here it is shown that these gravity waves are accurately reproduced by a semiimplicit discretization of the 3D compressible governing equations 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.
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