  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.
