Normal Modes of a Semi-implicit Model

Abstract

The normal modes of a linearized discrete-time model have been determined. The model is spectral and global. Although the model actually uses the primitive equations, only the equivalent systems of shallow water equations are considered. The discrete-time scheme is semi-implicit. Modes of the corresponding continuous-time model are also determined. Structures of the corresponding modes of the two models are slightly different, especially those of the mixed Rossby-gravity or gravitational modes. The set of modes which is linearly independent in the continuous-time model is not, in general, linearly independent in the discrete-time model. For the nonlinear versions of the models, nonlinear normal mode initialization for one is not equivalent to that for the other. Errors due to ignoring thew differences when initializing or investigating model balances are described. It is shown that the errors are typically small and may be neglected in most instances.