The stability of a thermally radiating stratified shear layer, including self-absorption

Abstract

A linear stability analysis is applied to a stably stratified, thermally radiating shear layer. The grey Milne–Eddington approximation is employed as a radiation model. In contrast to a previously reported optically thin analysis, no inviscid instability exists, in the limit of vanishing horizontal wavenumber, for this selfabsorbing model. The inviscid neutral-stability boundary (Richardson number us. dimensionless wavenumber) for the Milne–Eddington approximation converges to the optically thick limit as the optical depth of the shear layer is increased. As the optical depth of the shear layer is decreased, the inviscid Milne–Eddington neutral-stability boundary approaches the optically thin limit, although not uniformly in the wavenumber. For fixed mean velocity gradient and fluid properties, the inviscid critical Richardson number approaches infinity as the optical depth of the shear layer approaches zero. Viscous effects neutralize this radiative destabilization, and the critical Richardson number eventually returns to zero as the optical depth continues to decrease. A shearlayer thickness exists for which the viscous critical Richardson number is a maximum. For shear depths greater than this thickness, self-absorption effects increase the stability; and for shear depths less than this thickness, viscous effects increase the stability. Results of the analysis are applied to the atmospheres of Venus and the earth. A critical Richardson number somewhat above the non-radiating value of 3 (although below the previously reported optically thin value) is found for the lower troposphere of the earth. No substantial effect is found for the earth's lower stratosphere or for the 100 km level above Venus.