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 fluid flows are discretized on the Arakawa C-grid using centered differences in space. In contrast to the constant depth case it is demonstrated that varying depth may give rise to increasing energy land 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 modified 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 diffusion, explicit through frictional terms or implicit through numerical diffusion, is introduced 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 diffusion more than we would like from physical considerations. The modified weighting offers a simple solution to this problem. Copyright (C) 2000 John Wiley & Sons, Ltd.