The Effect of Critical Levels on 3D Orographic Flows: Linear Regime

Abstract

The effect of a critical level on airflow past an isolated axially symmetric obstacle is investigated in the small-amplitude hydrostatic limit for mean flows with linear negative shear. Only flows with mean Richardson numbers (Ri) greater or equal to ¼ are considered. The authors examine the problem using the linear, steady-state, inviscid, dynamic equations, which are well known to exhibit a singular behavior at critical levels, as well as a numerical model that has the capability of capturing both nonlinear and dissipative effects where these are significant. Linear theory predicts the 3D wave pattern with individual waves that are confined to paraboloidal envelopes below the critical level and strongly attenuated and directionally filtered above it. Asymptotic solutions for the wave field far from the mountain and below the critical level show large shear-induced modifications in the proximity of the critical level, where wave envelopes quickly widen with height. Above the critical level, the perturbation field consists mainly of waves with wavefronts perpendicular to the mean flow direction. A closed-form analytic formula for the mountain-wave drag, which is equally valid for mean flows with positive and negative shear, predicts a drag that is smaller than in the uniform wind case. In the limit of Ri d? ¼, in which linear theory predicts zero drag for an infinite ridge, drag on an axisymmetric mountain is nonzero. Numerical simulations with an anelastic, nonhydrostatic model confirm and qualify the analytic results. They indicate that the linear regime, in which analytic solutions are valid everywhere except in the vicinity of the critical level, exists for a range of mountain heights given Ri > 1. For Ri d? ¼ this same regime is difficult to achieve, as the flow is extremely sensitive to nonlinearities introduced through the lower boundary forcing that induce strong nonlinear effects near the critical level. Even well within the linear regime, flow in the vicinity of a critical level is dissipative in nature as evidenced by the development of a potential vorticity doublet.