Sensitivity of Internal Gravity Waves Solutions to the Time Step of a Semi-Implicit Semi-Lagrangian Nonhydrostatic Model

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Abstract
The combination of semi-implicit and semi-Lagrangian marching algorithms leads to stable integration of the meteorological equations with long time steps even for large advecting velocities and fast-moving free waves. In recent years, however, attention has been drawn to the fact that forced stationary waves may not be handled properly when the time step exceeds some limit dictated by the advecting velocity and the mesh size. The MC2 (Mesoscale Compressible Community) model is used to investigate numerically the behavior of internal gravity waves forced by stable flow over topography as a function of time step. Although the semi-implicit semi-Lagrangian scheme employed in the MC2 model allows larger time steps than classic Eulerian schemes, it leads to an inaccurate solution when the time step is increased beyond some limit. A stability analysis of the numerical response is conducted in order to explain model results. The response of two off-centered semi-implicit schemes proposed by Rivest et al. and Tanguay et al. to solve this problem for large-scale models is studied. The analysis of the response of forced mesoscale disturbances shows that these alternative schemes may allow the time step limit of forced mesoscale models to extend somewhat but that solutions still diverge for sufficiently large Courant number.

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