Abstract
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 wave-number space. On the other hand, vertical wave-momentum flux, even when it is invariant for waves of fixed frequency, is not invariant in the wave-number 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 in two ways: from the gravity-wave energy equation and also by Fourier transforming the solution to the gravity-wave equations. The derived spectral transformation law should be a key point in spectral gravity-wave parametrizations. Copyright © 2005 Royal Meteorological Society
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