Weakly non-linear response of a stably stratified shear flow to thermal forcing

Weakly non-linear response of a stably-stratified shear flow without a critical level of specified heating is investigated analytically using the perturbation method. The heating is prescribed to be bell-shaped in the horizontal and uniform from the surface to the non-dimensional heating depth of one, in the vertical. The non-dimensional vertical wind shear varies from 0.1 to 0.6. As the vertical wind shear increases, the magnitude of the zeroth-order (linear) perturbation vertical velocity above the forcing region increases by extracting energy from the mean shear flow. On the other hand, the magnitude of the perturbation horizontal velocity in the forcing region decreases with increasing wind shear because part of the heating is used to compensate for the positive vorticity induced by the positive wind shear. The forcing to the 1st-order equation exists not only in the specified heating region but also above the heating region because of the wind shear. The magnitude of the 1st-order (weakly non-linear) perturbation vertical velocity above the specified forcing region, varies irregularly with vertical wind shear, because the forcing to the 1st-order equation depends non-linearly on the wind shear. In the forcing region, the magnitude of the perturbation horizontal velocity decreases with increasing wind shear due to both the reduced effective forcing results from the compensation of the vorticity by the wind shear and the forcing to the 1st-order equation which decreases with increasing wind shear. The magnitudes of the momentum flux for the zeroth-order and 1st-order perturbations decrease monotonously with increasing wind shear, and the vertical convergence of the momentum flux exists below the forcing top. Above the forcing region, the vertical convergence/divergence of the momentum flux exists for the 1st-order perturbation. It is suggested that vertical convergence/divergence of the momentum flux can influence large-scale mean motion in terms of gravity wave drag; hence, gravity wave drag induced by the thermal forcing should be included in large-scale models.