Frequency shift of acoustic gravity waves in a stratified, isothermal, turbulent atmosphere

We study the modification of the frequency of small amplitude acoustic gravity waves which propagate in an isothermal turbulent atmosphere that is stratified by a homogeneous gravitational field. Using a Green's function method, the dispersion relation for the frequency of the waves is formulated as an integral eigenvalue equation and it is solved by perturbation techniques. We draw the following main conclusions: (a) for an arbitrary turbulent correlation spectrum the dispersion relation has a root with a negative imaginary part in the unphysical Riemann sheet of the dispersion function, leading to wave attenuation, much in the same way as it happens for Landau damping; (b) the real part of this root differs from the frequency of an acoustic gravity wave propagating in a nonturbulent medium and, for all forms of the turbulent correlation spectrum, the absolute value of this difference increases if gravity increases; (c) for a Gaussian turbulent correlation spectrum, this difference is always positive; (d) conversely if this frequency difference is known from observations, the auto-correlation function of the temperature fluctuations can be calculated through a simple inversion formula.