Conservation of energy for schemes applied to the propagation of shallow-water inertia-gravity waves in regions with varying depth

The linear equations governing the propagation of inertia-gravity waves in geophysical uid ows are dis- cretized on the Arakawa C-grid using centered di erences in space. In contrast to the constant depth case it is demonstrated that varying depth may give rise to increasing energy (and loss of stability) using the natural approximations for the Coriolis terms found in many well-known codes. This is true no matter which numerical method is used to propagate the equations. By a simple trick based on a modi ed weighting that ensures that the propagation matrices for the spatially discretized equations become similar to skew-symmetric matrices, this problem is removed and the energy is conserved in regions with varying depth too. We give a number of examples both of model problems and large-scale problems in order to illustrate this behaviour. In real applications di usion, explicit through frictional terms or implicit through numerical di usion, is intro- duced both for physical reasons, but often also in order to stabilize the numerical experiments. The growing modes associated with varying depth, the C-grid and equal weighting may force us to enhance the di usion more than we would like from physical considerations. The modi ed weighting o ers a simple solution to this problem.