The stability of a thermally radiating stratified shear layer

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Abstract
A linear stability analysis is applied to a stratified, thermally radiating, un-bounded shear layer. Temperature disturbances are assumed to be optically thin. Both viscous and inviscid neutral stability boundaries are determined numerically for hyperbolic-tangent mean velocity and potential-temperature profiles. For these profiles, long-wavelength disturbances are completely destabilized (the critical Richardson number Ric → ∞ as the wavenumber k → 0) in the inviscid limit. A similar situation is found for the case of discontinuous step-function profiles. However, in contrast to the non-radiating problem, the functional form of the neutral stability boundary is not the same for both the continuous and discontinuous profiles in the limit k → 0. Application of the viscous results to the atmospheres of the earth and Venus yield critical Richardson numbers in excess of ¼.