I thank Alan Plumb and Noboru Nakamura for valuable discussions throughout this work. I am also grateful for Olivier Pauluis and anonymous reviewers whose comments led to substantial improvements of the manuscript. The author is supported by the National Science Foundation (NSF) Climate and Large-Scale Dynamics program under Grants AGS-1042787 and AGS-1064079.
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More generally, the continuity equation in the η (a monotonic function of pressure p) vertical coordinate on the sphere can be written as