Extended Sawyer–Eliassen Equation for Frontal Circulations in the Presence of small Viscous Moist Symmetric Stability

Abstract

The Sawyer?Eliassen (S?E) equation for frontal circulations forced by a geostrophic stretching deformation is extended to include the effects of both negative moist potential vorticity (MPV) and eddy viscosity. Since the moist (precipitation) region depends on the vertical motion and thus needs to be solved together with the frontal circulation, the extended S?E equation is a nonlinear, elliptic, partial differential equation of sixth order. When MPV is positive and viscosity is negligible, this equation degenerates into the conventional S?E equation. The existence, uniqueness and stability of the solutions of the extended S?E equation in the presence of negative MPV (but still stable to viscous moist symmetric perturbations) are examined both analytically and numerically.