On the doppler shifting effect in an atmospheric gravity wave spectrum - Pulido, Manuel Arturo (2005)
The power spectrum evolution of a transient gravity wave disturbance propagating conservatively upwards in a shear flow is examined. It is proven that the wave action for a disturbance suffering Doppler shifting is invariant in the wavenumber space. On the other hand, vertical wave momentum flux, even when it is invariant for waves of fixed frequency, is not invariant in the wavenumber space for a transient disturbance.  The principle is used to derive a transformation law between a source spectrum and the resultant Doppler-shifted spectrum. The Doppler shifted spectrum has a $-3$ power law in the spectral tail, the asymptotic behaviour of the tail is shown to be independent of the gravity wave source. The result is obtained following two ways, from gravity wave energy equation and also Fourier transforming the solution to the gravity wave equations. The derived spectral transformation law should be a key point in spectral gravity wave parameterisations.