Internal gravity waves propagating within homogeneous stratified turbulence are the subject of the present study. A spatiotemporal analysis is carried out on the results of direct numerical simulations including a forcing term, with the aim of showing the energy content of the simulations as a function of frequency, $\omega$, and wave-vector inclination to the horizontal, $\theta$. Clear signatures of the dispersion relation of internal gravity waves, $\omega =\pm Ncos\theta$, where $N$ is the Brunt-Väisälä frequency, are observed in all our simulations, which have low Froude number, ${\text{Fr}}_h\ll 1$, and increasing buoyancy Reynolds number up to ${\text{Re}}_b\approx 10$. Interestingly, we observe the presence of high-frequency waves with $\omega \sim N$ and a corresponding low-frequency vortex mode, both containing a non-negligible amount of energy. These waves are large-scale waves, their energy signature being found at scales larger than the forcing scales. We also observe the growth of energy in the shear modes, constituting a horizontal mean flow, and we show that their continuous growth is due to an upscale energy transfer, from the forcing scales to larger horizontal as well as vertical scales. These shear modes are found to be responsible for Doppler shifting the frequency of the large-scale waves. When considering the wave energy across the simulations at varying ${\text{Re}}_b$, such energy is seen to reduce as ${\text{Re}}_b$ is increased and the flow enters the strongly stratified turbulence regime. The classical wave-vortex decomposition, based on a purely spatial decomposition of instantaneous snapshots of the flow, is analyzed within the current framework and is seen to correspond relatively well to the “true” wave signal identified by the spatiotemporal analysis, at least for the large-scale waves with $\omega \sim N$. Distinct energy peaks in $\theta \text{−}\omega$ space highlight that the waves have preferential directions of propagation, specifically $\theta ={45}^{\circ }$ and $\theta \approx {55}^{\circ }$, similar to observations in studies of wave radiation from localized regions of turbulence. This suggests that the same wave-generation mechanisms may be relevant for homogeneous and inhomogeneous stratified turbulent flows.

中文翻译：

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内部重力波在分层湍流中的特征和能量学

在均匀分层湍流中传播的内部重力波是本研究的主题。对包含强迫项的直接数值模拟的结果进行时空分析，目的是显示模拟的能量含量与频率的关系，$\omega$，并且波矢向水平方向倾斜， $\theta$。内部引力波的色散关系的清晰特征，$\omega =\pm ñcos\theta$，在哪里 $ñ$ 在我们所有的模拟中都观察到Brunt-Väisälä频率是低的弗洛德数， ${\text{r}}_H\ll \mathrm{1个}$，并增加雷诺数的浮力 ${\text{回覆}}_b\approx 10$。有趣的是，我们观察到高频波的存在$\omega 〜ñ$以及相应的低频涡旋模式，两者都包含不可忽略的能量。这些波是大规模的波，其能量特征在大于强迫尺度的尺度上被发现。我们还观察到剪切模式下能量的增长，构成水平平均流，并且我们表明它们的持续增长归因于从强迫尺度到较大的水平尺度和垂直尺度的高端能量传递。发现这些剪切模式是造成多普勒频移大尺度波频率的原因。当考虑整个模拟的波能变化时${\text{回覆}}_b$，这种能量会减少为 ${\text{回覆}}_b$流量增加，气流进入强分层湍流区。基于流的瞬时快照的纯空间分解的经典波涡旋分解在当前框架内进行了分析，并且至少相对于时空分析所识别的“真实”波信号相对较好。大波浪$\omega 〜ñ$。在不同的能量峰值$\theta \text{-}\omega$ 空间凸显出波浪具有优先的传播方向，特别是 $\theta ={45}^{\circ }$ 和 $\theta \approx {55}^{\circ }$，类似于研究湍流局部区域的波辐射的观察结果。这表明相同的波浪发生机理可能与均匀和不均匀的分层湍流有关。