Vertical evolution of gravity wave spectra and the parameterization of associated wave drag

Abstract

Extensions of Weinstock's theory of nonlinear gravity waves and a parameterization of the related momentum deposition are developed. Our approach, which combines aspects of Hines' Doppler spreading theory with Weinstock's theory of nonlinear wave diffusion, treats the low‐frequency part of the gravity wave spectrum as an additional background flow for higher‐frequency waves. This technique allows one to calculate frequency shifting and wave amplitude damping produced by the interaction with this additional background wind. For a nearly monochromatic spectrum the parameterization formulae for wave drag coincide with those of Lindzen. It is shown that two processes should be distinguished: wave breaking due to instabilities and saturation due to nonlinear diffusionlike processes. The criteria for wave breaking and wave saturation in terms of wave spectra are derived. For a saturated spectrum the power spectral density's (PSD) dependence S(m) = AN2/m3 is obtained, where m is the vertical wavenumber and N is the Brunt‐Väisäla frequency. Unlike Weinstock's original formulation, our coefficient of proportionality A is a slowly varying function of m and mean wind. For vertical wavelengths ranging from 10 km to 100 m and for typical wind shears, A varies from one half to one ninth. Calculations of spectral evolution with height as well as related profiles of wave drag are shown. These results reproduce vertical wavenumber spectral tail slopes which vary near the −3 value reported by observations. An explanation of these variations is given.