Optimal Vertical Discretization for Atmospheric Models

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Abstract
The vertical levels used in atmospheric models are selected for a variety of reasons, but the selection process has not been systematized. The study presented herein represents an attempt to do so. An atmospheric model is linearized about a state of rest and vertical modes are determined for the vertical structure equation-an ordinary differential equation-by solving a difference form of that equation. Since the solutions of the differential equation do not correspond to the solutions of the difference equation, the distribution of points used for the difference equation (the vertical levels) is adjusted until both sets of solutions coalesce. This distribution is considered the optimum set of model levels. To test the impact of such a distribution on a numerical integration, the NCAR CCMOB is integrated using both its standard levels and the levels determined above. Comparisons of the integrations show that the solutions separate as time evolves, despite the fact that the initial conditions for the separate integrations are as similar as possible. The results suggest that care in selecting vertical levels is essential to successful integration and that perhaps a start on finding a systematic way of choosing levels has been made.

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