A general, linearized vertical structure equation for the vertical velocity: Properties, scalings and special cases

A general, linear vertical structure equation for the vertical velocity com ponent, including explicit forcing terms in the momentum, thermodynamic and continuity equations, is derived for horizontally-homogeneous flows. The b asic flow is assumed to depend on height alone and is in geostrophic and hy drostatic balance. Scale analysis is used to show that this equation incorp orates a variety of familiar special cases including the lee-wave equation, Eady's equation and the quasi-geostrophic omega equation, the different fl ow regimes being identified in terms of the Rossby, Froude and Richardson n umbers. Using the vertical structure equation, a wave-stress conservation p rinciple is derived that is valid for basic flows whose magnitude and direc tion vary with height. In addition to providing some unification to the man y flavours of vertical velocity equation in the Literature, this derivation was motivated by the need to provide a starting point for a wide class of analytical problems in the study of baroclinic instability and inertia-grav ity wave dynamics.