Hayashi, Y., 1982: Interpretations of space-time spectral energy equations. Journal of the Atmospheric Sciences, 39 (3), 685-688.

Abstract: Interpretations are given of two different formulations of space-time spectral energy equations derived by Kao (1968) and Hayashi (1980).
Contrary to Kao's interpretation, it is argued that his formulation does not describe how spectral energy is maintained, since his equation corresponds to the imaginary part of the energy equation of space-time Fourier components which governs the frequency (time change of phase).
On the other hand, Hayashi's formulation is consistent with Saltzman's (1957) wavenumber spectral energy equation, since his formulation corresponds to the real part which governs the growth rate (time change of amplitude).