A variety of motions associated with gravity waves and Kelvin-Helmholtz instabilities have recently been detected by high-power VHF and UHF radars at tropospheric, stratospheric, an mesospheric heights. The simultaneously observed mean temperature and wind profiles differ considerably from those used particularly in theoretical studies of wave-mean flow interactions. Therefore a linear operational model has been developed describing acoustic-gravity waves and Kelvin-Helmholtz instabilities in compressible atmospheres with arbitrary vertical profiles of the mean temperature, wind, and viscosity coefficient. The linearized hydrodynamic equations are solved by the nonstandard multilayer method and a Gauss-Seidel group iteration. Results from test computations of wave--critical layer interactions and a Kelvin-Helmholtz instability show good quantitative agreement with known theoretical results from simple incompressible models. Wave-induced instabilities are parameterized by mean eddy viscosity and thermal conductivity, assuming that the wave-perturbed gradient Richardson number is nowhere smaller than some critical value. A new iterative method is used to obtain the height profiles of the eddy transport coefficients. At mesospheric heights, wave-induced dissipation limits the amplitudes of internal gravity waves so that no significant nonlinearities are introduced in the hydrodynamic equations.