Stability of Vertical Discretization Schemes for Semi-Implicit Primitive Equation Models: Theory and Application

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Abstract
The linear stability of vertical discretization schemes for semi-implicit primitive-equation models is thoroughly investigated. The equations are linearized about a stationary rotating basic state atmosphere that has a vertically shearing zonal wind. The amplification matrix of the finite-element model is constructed and its eigenvalues examined for possible instability. Investigating, the small time step limit of that matrix, we identify two operators whose eigenvectors are the “physical” and “computational” modes of the semi-implicit method, respectively, and whose eigenvalues are their frequencies. It is further shown that if the frequencies are real then the respective modes are stable. Switching off the rotation and horizontal advection in the above operators, we are able to state conditions on the implicit and explicit temperature profiles such that the unconditional instability is avoided (e.g., the so-called semi-implicit instability). These stability criteria may be easily extended to any type of vertical discretization.

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