An analysis of the vertical structure equation in sigma coordinates |
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| Authors: | A Staniforth M Béland J Côté |
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| Institution: | Recherche en prévision numérique , Service de l'environnement atmosphérique , 2121 Route Trans‐canadienne, Dorval, Québec, H9P 1J3 |
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| Abstract: | Abstract An analysis of the vertical structure equation of sigma coordinate primitive equation models is given that brings together and extends the work of several authors. We derive the vertical structure equation, and obtain its solution for a two‐parameter family of vertical structure profiles that includes those of previous studies. For this family, it is shown that in the limiting case of an unbounded atmosphere the spectrum becomes partially continuous, rather than entirely discrete as in the bounded case. A criterion is obtained for the validity of the linearization used to derive the vertical structure equations, and it turns out that this criterion is satisfied by all but one of the previous studies. Asymptotic expansions are derived and used to explain two observations of Wiin‐Nielsen (1971a), viz. why the equivalent depths of the internal modes are relatively insensitive to the precise choice of lower boundary condition, and why one choice in particular leads to the elimination of the external mode; these asymptotic expansions also yield surprisingly accurate numerical values for the equivalent depths. Finally, the projection of atmospheric data onto modes found by direct numerical approximation of the vertical structure equation is shown, particularly for the least grave modes, to be very sensitive to resolution; consequently care must be exercised when interpreting the results of data projection studies that use this approach. |
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