Signature and energetics of internal gravity waves in stratified turbulence

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.

Figure 1

Illustration of conical shells and their corresponding Fourier mode sums, $\mathcal{U}\left(\theta \right)$, shown on a cut at $k_x=0$. Note that modes on the $k_z$ axis corresponding to the shear modes are put into a separate individual shell. The same applies to modes on the horizontal $k_x\text{−}{k}_y$ plane.

Figure 2

Illustration of cylindrical forcing. The force $\widehat{\mathbf{f}}$ is applied in Fourier space and is active only on the cylinder areas, $A^{\prime }$ and $A^{\prime \prime }$, which in a discrete formulation become cylindrical shells. The forcing is active in an interval of $\theta$; the minimum value of $\theta$ that is forced is shown.

Figure 3

Realization of cylindrical forcing with $k_{h,f}=3, k_{v,f\min }=8, k_{v,f\max }=12$ on a ${256}^3$ grid. Visualization of $f_x$ on an $x\text{−}z$ plane (the units of $f_x$ are arbitrary).

Figure 4

Visualization of $x$ component of velocity on the central $x\text{−}z$ plane in run R9.9 at the last time instant of the simulation: (a) $u$ and (b) $u$ minus its horizontal average over the whole domain, i.e., keeping only the contributions of modes with $k_h\ge 1$ and removing the shear modes at $k_h=0$.

Figure 5

Visualization of ${\rho }^{\prime }g/{\rho }_0$ in run R9.9 (same plane and time instant as in Fig. 4).

Figure 6

Time evolution of volume-averaged quantities in run R3.8. (a) Kinetic energy evolution: $E_K, E_K^{\prime }$, and $u_h^2$; (b) evolution of $E_K^{\prime }, E_P$, and wave and vortex kinetic energies $E_w$ and $E_v$; (c) enlargement of panel (a), showing the initial transient period; (d) evolution of ${\text{Fr}}_h$ and ${\text{Re}}_b$ over time and vertical Froude number ${\text{Fr}}_v$ shown at the last time step of the run. The dashed lines in panels (a), (b), and (d) show the time at which the steady state is considered to begin.

Figure 7

Color maps of $E_K(\theta ,\omega )$ and $E_P(\theta ,\omega )$. Left column [(a), (c), (e)] $E_K(\theta ,\omega )$. Right column: [(b), (d)] $E_P(\theta ,\omega )$ and (f) $E_P^{\prime }(\theta ,\omega )$. [(a), (b)] Run R1.6; [(c), (d)] run R3.8; [(e), (f)] run R9.9. In panels (a) and (b), the black dashed curves show the dispersion relation of internal gravity waves, $\omega /N=\pm cos\theta$, while the red dashed lines show the minimum and maximum forcing angles, $\theta =\pm {\theta }_{f,\min }$ and $\theta =\pm {\theta }_{f,\max }$. The shear modes at $\theta =\pi /2$ are not plotted because their high intensity concentrated in a single point ($\omega =0$) makes all other points undistinguishable. Note that the color maps have a central symmetry with respect to $(\theta ,\omega )=(0,0)$ because they are constructed from real data.

Figure 8

Spatiotemporal analysis of wave and vortex components in run R9.9: (a) $E_w(\theta ,\omega )$; (b) $E_v(\theta ,\omega )$.

Figure 9

Kinetic and potential energy at large and small scales in $\theta \text{−}\omega$ space for run R9.9: (a) $E_K^{<}(\theta ,\omega )$, (b) $E_P^{<}(\theta ,\omega )$, (c) $E_K^{>}(\theta ,\omega )$, (d) $E_P^{>}(\theta ,\omega )$. The vertical dashed lines show the minimum forcing angle, $\theta =\pm {\theta }_{f,\min }$.

Figure 10

Evolution over the course of simulation R3.8 of 1D horizontal and vertical energy spectra: (a) horizontal spectra $E\left({\kappa }_h\right)$ and (b) vertical spectra $E\left(k_v\right)$. In both panels, the energy spectra obtained after filtering out the shear modes are also given. The time at which the spectra are taken is indicated by the line weight of the curves, with increasing line weight corresponding to increasing time. The first spectrum plotted in panels (a) and (b) corresponds to the initial conditions at $t=0$. In both plots, the wave number is shifted by half a wave-number increment to show the spectra at ${\kappa }_h=0$ and $k_v=0$. In panel (b), the forcing range in terms of $k_v$ is given, while in panel (a) the forcing at $k_h=3$ corresponds to the forcing being active from ${\kappa }_h=0$ to ${\kappa }_h=3$.

Figure 11

Color map of ${log}_{10}\left(E_P^{<}(\theta ,\omega )\right)$ for run R9.9 together with plots of ${\omega }_{\mathrm{Doppler}}/N$ (black dashed lines). The vertical bands in the plot are due to the fact that over this limited number of wave numbers several values of $\theta$ are not sampled. For clarity, values below ${10}^{-7}$ are filtered out.

Figure 12

Wave and vortex envelopes. The color maps are of data from run R9.9: (a) ${log}_{10}\left(E_K^{\prime }(\theta ,\omega )\right)$ and (b) ${log}_{10}\left(E_P^{\prime }(\theta ,\omega )\right)$. For clarity, values below ${10}^{-5}$ are filtered out in both color maps.

Figure 13

Wave and vortex envelopes for run R1.6 superimposed on color maps of (a) ${log}_{10}\left(E_K(\theta ,\omega )\right)$ and (b) ${log}_{10}\left(E_P(\theta ,\omega )\right)$. For clarity, values below ${10}^{-4.5}$ in panel (a) and below ${10}^{-5}$ in panel (b) are filtered out.

Figure 14

Wave and vortex energy across the three DNS runs shown as a function of ${\text{Re}}_b$. (a) Kinetic and potential energy of $\omega \sim N$ waves and kinetic energy of $\omega \approx 0$ vortices. The star symbols at ${\text{Re}}_b=9.9$ indicate the wave and vortex kinetic energy calculated from $E_K^{\prime }(\theta ,\omega )$ instead of $E_K(\theta ,\omega )$, and the wave potential energy calculated from $E_P^{\prime }(\theta ,\omega )$ instead of $E_P(\theta ,\omega )$. The color code for these quantities is maintained. (b) Wave and vortex kinetic energy $E_w$ and $E_v$ as obtained from the wave-vortex decomposition. In both panels, the energy is normalized by the overall kinetic energy not counting the shear modes, $E_K^{\prime }$.

Figure 15

Plots of $E_{P,\max }\left(\theta \right)$ and of $E_{P,\max }^{\prime }\left(\theta \right)$ (in case of run R9.9), the maxima of the potential energy within the wave envelope as a function of $\theta$. The corresponding run is indicated on the right of each plot.

Figure 16

Plots of $E_{K,\max }\left(\theta \right)$ and of $E_{K,\max }^{\prime }\left(\theta \right)$ (in case of run R9.9).

Figure 17

Potential energy $E_P(\theta ,\omega )$ for run R1.6 as obtained from (a) reduced analysis and (b) 4D Fourier transform. The color map is the same for both figures.

Figure 18

Comparison of $E_{P,\max }\left(\theta \right)$ obtained from the reduced and full analyses for run R1.6.

Figure 19

Kinetic energy content $E_K(k,\theta ,\omega )$ at the first eight wave numbers of run R9.9. The color map is the same across all color plots and is given on the plot corresponding to $k=2$; the upper limit of the color map has been chosen to make the wave energy visible in all plots, i.e., the energy in the vicinity of the dispersion relation. This results in the energy close to $\omega \approx 0$ (corresponding to vortex energy) saturating in a few cases as it is higher than this upper limit.