Weakly Nonlinear Response of a Stably Stratified Atmosphere to Diabatic Forcing in a Uniform Flow

Abstract

The weakly nonlinear response of a two-dimensional stably stratified atmosphere to prescribed diabatic heating in a uniform flow is investigated analytically using perturbation expansion in a small value of the nonlinearity factor for the thermally induced waves. The diabatic heating is assumed to have only a zeroth-order term specified to be vertically homogeneous between the surface and a certain height and bell shaped in the horizontal. The first-order (weakly nonlinear) solutions are obtained using the FFT algorithm after solutions in wavenumber space are obtained analytically. The forcing (F) to the first-order perturbation equation induced by the Jacobian of the zeroth-order (linear) solutions always represents cooling in the lower layer regardless of specified forcing type (cooling or heating). The vertical structure of F is related to the nondimensional heating depth (d) or the inverse Froude number. The first-order solutions are valid for relatively small values (<3) of d. The main nonlinear effect is to produce a strong convergence region near the surface associated with the zeroth-order perturbations regardless of the value of d. This convergence is responsible for producing upward motion in the center of the forcing region that extends upstream. As a result, the zeroth-order downward motion becomes weaker according to a degree of nonlinearity. The relative magnitude of the zeroth-order downward motion and the first-order upward motion upstream of the forcing can be determined by d. The source of the first-order wave energy is found to come mainly from the horizontal advection of the zeroth-order total wave energy by the first-order perturbation horizontal wind.