Anelastic internal wave reflection and transmission in uniform retrograde shear

We perform fully nonlinear simulations in two dimensions of a horizontally periodic, vertically localized, anelastic internal wavepacket in order to examine the effects of weak and strong nonlinearity upon wavepackets approaching a reflection level in uniform retrograde shear. Transmission, reflection, and momentum deposition are measured in terms of the horizontal momentum associated with the wave-induced mean flow. These are determined in part as they depend upon the initial wavenumber vector, k = (k,m), which determines the modulational stability (if |m/k| 0.7) or instability (if |m/k| 0.7) of moderately large amplitude quasi-monochromatic internal wavepackets. Whether modulationally stable or unstable, the evolution of the wavepacket is determined by the height of the reflection level predicted by linear theory, zr, relative to the height, z , at which weak nonlinearity becomes significant, and the height, zb > z , at which linear theory predicts anelastic waves first overturn in the absence of shear. If zr < z , the amplitude remains sufficiently small and the waves reflect as predicted by linear theory. If zr is moderately larger than z , a fraction of the momentum associated with the wavepackets transmits past the reflection level. This is because the positive shear associated with the wave-induced mean flow can partially shield the wavepacket from the influence of the negative background shear enhancing its transmission. The effect is enhanced for weakly nonlinear modulational unstable wavepackets that narrow and grow in amplitude faster than the anelastic growth rate. However, as nonlinear effects become more pronounced, a significant fraction of the momentum associated with the wavepacket is irreversibly deposited to the background below the reflection level. This is particularly the case for modulationally unstable wavepackets, whose enhanced amplitude growth leads to overturning below the predicted breaking level. Because the growth in the amplitude envelope of modulationally stable wavepackets is retarded by weakly nonlinear effects, reflection is enhanced and transmission retarded relative to their modulationally unstable counterparts. Applications to mountain wave propagation through the stratosphere in the winter hemisphere are discussed. C © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4864104]