Pseudomomentum Diagnostics for Two-Dimensional Stratified Compressible Flow

Abstract

Expressions are derived for the local pseudomomentum density in two-dimensional compressible stratified flow and are compared with the expressions for pseudomomentum in two-dimensional Boussinesq and anelastic flow derived by Shepherd and by Scinocca and Shepherd. To facilitate this comparison, algebraically simpler expressions for the anelastic and Boussinesq pseudomomentum are also obtained. When the vertical wind shear in the reference-state flow is constant with height, the Boussinesq pseudomomentum is shown to reduce to a particularly simple form in which the pseudomomentum is proportional to the perturbation vorticity times the fluid-parcel displacement. The extension of these compressible pseudomomentum diagnostics to viscous flow and to three-dimensional flows with zero potential vorticity is also discussed. An expression is derived for the pseudomomentum flux in stratified compressible flow. This flux is shown to simultaneously satisfy the group-velocity condition for both sound waves and gravity waves in an isothermal atmosphere with a constant basic-state wind speed. Consistent with the earlier results of Andrews and McIntyre, it is shown that for inviscid flow over a topographic barrier, the pseudomomentum flux through the lower boundary is identical to the cross-mountain pressure drag?provided that the flow is steady and that the elevation of the topography returns to its upstream value on the downstream side of the mountain.