Compatibility Conditions, Complex Frequency, and Complex Vertical Wave Number for Models of Gravity Waves in the Thermosphere

Gravity wave propagation from the troposphere into the thermosphere has often been modeled by assuming a complex vertical wave number m , with the imaginary part accounting for viscous and thermal dissipation. An alternative is to assume a complex wave frequency ω , which leads to a simpler form of the dispersion relation and different wave solutions described by complex‐ ω ray theory. Here the solutions of complex ω and complex m are analyzed in terms of the compatibility conditions ∇× k =0 and k t =−∇ ω , where k is the wave number vector. Estimates derived here suggest that the complex‐ ω ray solutions can violate compatibility conditions above altitudes of weak viscosity. A possible consequence of this violation is an overestimation of the viscous damping of thermospheric gravity waves. The present results are based on a numerical solution of the viscous gravity‐wave dispersion relation in the anelastic approximation with a Prandtl number of unity and using conventional specifications of the dynamic viscosity of the thermosphere. Theoretical solutions for complex  m are also derived in the weak and strong viscosity limits.