An analytic model for steady state turbulence is employed to obtain the inertial range power spectrum of compressible turbulence. We assume that for homogeneous turbulence, the timescales controlling the energy injected at a given wavenumber from all smaller wavenumbers are equal for each spatial component. However, the longitudinal component energy is diverted into compression, so the rate controlling the energy that is transferred to all larger wavenumbers by the turbulent viscosity is reduced. The resulting inertial range is a power law with an index of −2. Indeed, such power spectra were observed in various astrophysical settings and also in numerical simulations.

中文翻译：

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可压缩湍流惯性范围的解析推导

利用稳态湍流的解析模型获得可压缩湍流的惯性范围功率谱。我们假设对于均匀湍流，控制在给定波数从所有较小波数注入的能量的时间尺度对于每个空间分量都是相等的。然而，纵向分量能量被转移到压缩中，因此控制通过湍流粘度转移到所有较大波数的能量的速率降低。由此产生的惯性范围是指数为 -2 的幂律。事实上，在各种天体物理环境和数值模拟中都观察到了这种功率谱。