Diffusive filtering theory of gravity wave spectra in the atmosphere

Abstract

Shear and convective instabilities, Doppler spreading, and nonlinear wave‐wave interactions are mechanisms which have been proposed to explain the form of the gravity wave vertical wave number spectrum of horizontal winds. In this paper we present an alternative explanation by assuming that the damping effects of molecular viscosity, turbulence, and off‐resonance wave‐wave interactions can all be characterized in terms of a scale‐independent diffusivity (D) which increases with altitude. The components of the gravity wave source spectrum are assumed to grow exponentially with increasing altitude in response to decreasing atmospheric density until they are removed by diffusive damping. A wave of intrinsic frequency Cu and vertical wave number m is assumed to be completely damped when the effective vertical velocity of momentum diffusion (mD) exceeds the vertical phase velocity of the wave (ω/m). Only waves satisfying mD < ω/m, or equivalently m2D < ω and m < (ω/D)½ are permitted to grow in amplitude as they propagate upward in the atmosphere. If the gravity wave temporal spectrum of horizontal winds varies as ω−p, we show that the vertical wave number spectrum must vary as m−2p + 1 and the zonal (or meridional) wave number spectrum must vary as ω−p we show that the vertical wave number spectrum must very as k−(2p + 1)/3. For p = 2, and ω m and ω spectra are and where N is the buoyancy frequency, ƒ is the inertial frequency, is a form of the Richardson number, and is the variance of the vertical shear of the horizontal wind. Similar models are developed for scale‐dependent diffusion. In this case the spectra are proportional to ω−p, m−3 and k(−p ‐ 1)/(p + 1). Because the joint (m, ω) intrinsic spectra for both scale‐independent and scale‐dependent diffusive filtering are not separable, the theory predicts that the m spectra of vertical winds are proportional to m5 ‐ 2p for scale‐independent diffusion and m(7 ‐ 3p)/(p ‐ 1) for scale‐dependent diffusion. The model spectra compare favorably with recent lidar and radar observations of middle atmosphere density, temperature, and wind fluctuations.