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.

中文翻译：

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热圈重力波模型的相容性条件，复数频率和复数垂直波数

从对流层到热层的重力波传播通常是通过假设垂直波数为m来建模的，其中虚部考虑了粘性和热耗散。另一种选择是假设复波频率ω，这导致色散关系的简化形式以及复ω射线理论所描述的不同波解。这里复杂的解决方案，ω和复杂米中的相容性条件方面进行了分析∇× ķ = 0和ķ_吨=-∇ ω，其中ķ是波数矢量。此处得出的估计值表明，ω射线解决方案可能会在弱粘度高度以上违反相容性条件。这种违反的可能后果是高估了热层重力波的粘性阻尼。目前的结果是基于粘性重力波色散关系在非弹性近似中的数值解，其普朗特数为1，并使用了热圈动态粘度的常规规范。复数m的理论解 也可以从弱和强粘度极限中得出。