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An analysis of the free modes of one, two and four layer models based on sigma coordinates
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    "It is well-known that the computational stability of explicit schemes for numerical integration of the quasi-static equations is dependent upon the satisfaction of a criterion of the form, [omega] [delta] t < 1. The parameter, [omega], represents the frequency of oscillation of a wave motion admitted by the difference equations. It is seen that (1) requires the use of a smaller time step when a high frequency oscillation is admitted by the system of equations. In connection with our work on the development of semi-implicit integration methods (1), the need arose for a good estimate of the frequencies admitted by multi-level quasi-static models. In particular, we were led to enquire into possible differences in the modes when alternate formulations of the vertical coordinate were employed. In this paper, we shall-present the results of analyses of the free modes admitted in one, two and four layer models based upon the a-coordinate systems introduced by Phillips (2), and Shuman and Hovermale (3). The perturbation equations will be developed using an unspecified a-type coordinate. The specialization to specific coordinates and vertical resolution will then be made, and the analysis indicated. In the concluding section, the results will be summarized and their significance evaluated"--Introduction.

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