Vertical normal modes in a hybrid-coordinate system

Vertical normal modes are eigensolutions of a vertical structure equation derived from a set of Primitive Equations (PE), traditionally formulated in a z-, p-, or modified p-coordinate (e.g., sigma-coordinate) system. In recent years, atmospheric PE models using a hybrid-coordinate system, in which a terrain-following sigma coordinate is used in the lower levels and isentropic coordinate is used in the upper levels, become more and more popular (e.g., Zhu et al. 1992; Bleck and Benjamin 1993; Konor and Arakawa 1997). The advantage of using such a coordinate system is obvious. Since a theta-surface is a material surface, the use of it as a coordinate can effectively provide a much needed spatial resolution for upper-level disturbances such as tropopause folding, jet-stream meandering, and mid-to-upper-level frontogenesis, while at lower levels the sigma surfaces are not affected by topography.

However, because of the complexity in their vertical coordinates, hybrid-coordinate models generally have difficulty to obtain their vertical structure functions. Such a problem prevents these models from an application of the normal-model-initialization procedure and a baroclinic coupling of horizontal winds in their variational data assimilation system. In this study, vertical normal modes in a hybrid-coordinate system are derived. It is hoped that the analyses of these vertical normal modes will shed some light on the hybrid-coordinate model's initialization and data assimilation problems.