Shear and convective instabilities, Doppler spreading and non-linear 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 altitude in response to decreasing atmospheric density until they are removed by diffusive damping. A wave of intrinsic frequency ω 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)1/2 are permitted to grow in amplitude as they propagate upward in the atmosphere. If the gravity wave temporal spectrum varies as ω−P, we show that the vertical wave number spectrum must vary as m−2p+l and the zonal (or meridional) wave number spectrum must vary as k−(2p+ 1)/3. For p near 2, the diffusion theory predicts that the spectra are proportional to ω−2, m−3 and k−5/3. Because the joint (m, ω) intrinsic spectrum for scale-independent diffusive filtering is not separable, the theory predicts that the m-spectrum of vertical winds is proportional to m5-2p. The model spectra compare favorably with recent lidar and radar observations of middle atmosphere density, temperature and horizontal wind fluctuations.