The propagation of gravity waves through a critical layer for conditions of moderate wind shear

Solutions of the linearised hydrodynamic equations for a viscous atmosphere using (i) a full-wave integration procedure and (ii) a simplified analytical approach are used to examine the attenuation of gravity waves passing through a critical layer, where the horizontal phase velocity is equal to that of the mean wind. Particular attenuation is paid to the variation of this attenuation with values of Richardson number, Ri, greater than unity. The two sets of results are in good agreement with the predictions of Booker and Bretherton (1967) for an inviscid fluid for values of Ri up to about 4. However, a marked discrepancy from these predicted values is found for larger values of Ri, the present results indicating substantially smaller attenuation. Further calculations suggest that the wave-amplitude attenuation factor predicted by the inviscid model is approached asymptotically in the limit of vanishingly small viscosity and thermal conductivity coefficients.The inclusion of viscosity and thermal conduction gives rise to three characteristic modes of propagation for each direction of energy flow, in place of the single mode occurring in the inviscid case. The importance of energy exchange between these modes in the propagation through the critical layer is demonstrated.