Acoustic-gravity waves in quasi-isothermal atmospheres with a random vertical temperature profile

We study acoustic-gravity waves in a quasi-isothermal atmosphere in the presence of a weak random addition to the vertical temperature profile, which simulates the real atmosphere of the Earth at altitudes greater than $\sim 200$ km. The resulting stochastic equation is closed in the Bourret approximation. The poles of the obtained mean Green's function determine the generalized dispersion relation for acoustic-gravity waves. Two particular cases are considered: random inhomogeneities in the form of white noise ($\delta$-correlated in space) and the opposite case of a $\delta$-shaped noise spectrum. In both cases, instability of acoustic-gravity waves is predicted and the corresponding instability growth rates are determined.