A Comparison of Two Cumulus Parameterization Schemes in a Linear Model of Wave-CSK

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Abstract
A linear model of wave-CISK consisting of the primitive equations formulated on an equatorial beta-plane with a resting basic state, is used to compare the relatively simple cumulus parameterization scheme developed by Stevens and Lindzen with the relatively complex one developed by Arakawa and Schubert. The comparison is based mainly on the instability characteristics and vertical structure of the wave. Discrete forms of the equations are developed and the inhomogeneous vertical structure equation is solved as a generalized matrix eigenvalue problem. The eigenvalues correspond to complex equivalent depths which determine the growth rates through the use of the dispersion relation for an equatorial beta-plane. The eigenvectors, which are the vertical velocities at discrete levels in the model, determine the vertical structure. Results indicate that the growth rates obtained with Arakawa and Schubert's parameterization are smaller than those obtained with Stevens and Lindzen's parameterization. Comparison of the vertical structure of the unstable modes shows that the cumulus heating profile for the Arakawa and Schubert parameterization, which is determined internally in the model, is remarkably similar to the specified profile used in the Stevens and Lindzen parameterization. Middle and upper tropospheric moisture is found to play only a small role in the Stevens and Lindzen parameterization. Differences from previous studies using the Arakawa and Schubert parameterization are found; specifically, a greater number of unstable modes and a vertical heating distribution for the most unstable mode that has a first internal mode structure, even though it is not associated with the first internal mode.

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