Numerical Solution of the Vertical Structure Equation in the Normal Mode Method

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  • 1 Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, OK 73019
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Abstract

In a diagnostic study by expanding global data in normal mode functions, Kasahara and Puri found that for zonal wavenumber one, even the seventh vertical mode (the highest mode they presented) contains about 50% of the energy of the external mode. The vertical normal modes are eigensolutions of the vertical structure equation, and each mode is associated with well‐defined physical significance. Consequently, it is of interest to look into the accuracy of representation of, say, the first ten vertical modes in a discretized model because seriously misrepresented normal mode functions may not be able to honestly express the physics embedded in the data to be expanded. Along this line, a systematic method of obtaining matching eigensolutions of the vertical structure equation of a multilayered stratified atmosphere was developed. The resultant eigensolutions were used to investigate the influence of the upper boundary condition, the judicious method of the vertical grid levels and the relative accuracy of a finite‐difference and a finite‐element method in obtaining the discretized vertical normal mode functions. An important conclusion of this study is that in a discretized model, an inadequate grid resolution in the upper domain may result in considerable misrepresentation of the vertical structure functions even in the lower part of the domain for vertical modes higher than mode 5.

Abstract

In a diagnostic study by expanding global data in normal mode functions, Kasahara and Puri found that for zonal wavenumber one, even the seventh vertical mode (the highest mode they presented) contains about 50% of the energy of the external mode. The vertical normal modes are eigensolutions of the vertical structure equation, and each mode is associated with well‐defined physical significance. Consequently, it is of interest to look into the accuracy of representation of, say, the first ten vertical modes in a discretized model because seriously misrepresented normal mode functions may not be able to honestly express the physics embedded in the data to be expanded. Along this line, a systematic method of obtaining matching eigensolutions of the vertical structure equation of a multilayered stratified atmosphere was developed. The resultant eigensolutions were used to investigate the influence of the upper boundary condition, the judicious method of the vertical grid levels and the relative accuracy of a finite‐difference and a finite‐element method in obtaining the discretized vertical normal mode functions. An important conclusion of this study is that in a discretized model, an inadequate grid resolution in the upper domain may result in considerable misrepresentation of the vertical structure functions even in the lower part of the domain for vertical modes higher than mode 5.

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