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 component, including explicit forcing terms in the momentum, thermodynamic and continuity equations, is derived for horizontally‐homogeneous flows. The basic flow is assumed to depend on height alone and is in geostrophic and hydrostatic balance. Scale analysis is used to show that this equation incorporates a variety of familiar special cases including the lee‐wave equation, Eady's equation and the quasi‐geostrophic omega equation, the different flow regimes being identified in terms of the Rossby, Froude and Richardson numbers. Using the vertical structure equation, a wave‐stress conservation principle is derived that is valid for basic flows whose magnitude and direction vary with height. In addition to providing some unification to the many 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‐gravity wave dynamics.