Numerical models for the prediction of atmospheric motions are described by a finite number of coupled ordinary differential equations. We formally solve the initial-value problem for small-amplitude perturbations on some basic state as described by the prediction system. The solution and hence initial conditions are expressed as a sum over the normal modes of oscillation of the perturbation equations. The question as to which modes describe the evolution of meteorologically significant information may be answered for models which are used not only for prediction but also for climate simulation. Those modes which have a much larger amplitude in noisy real data than in climate simulation studies can be filtered from the initial data. The expansion of grid-point data into the normal modes of a model thus allows filtering in a more selective and rational fashion than has been possible using classical initialization procedures. Such an expansion also allows comparison of numerical simulation studies with spectral studies of actual free modes in the atmosphere. As an example of the model expansion procedure, we describe the application of a finite-difference approximation to the dynamics of a two-layer ocean model on a rotating sphere. In the limit of infinitesimal grid interval, the expansion of initial data is given by the Hough functions of tidal theory. For a finite grid interval, it is necessary to consider not only modes related to the Hough modes but also computational modes specific to the finite-difference equations employed. Examples of the eigenfrequencies and eigenfunctions for a basic state at rest are compared with those obtained assuming a basic state with a latitudinally varying zonal wind.
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Dickinson et al. (1972) studied this question.