Internal gravity waves in stratified flows with and without vortical modes

The comprehension of stratified flows is important for geophysical and astrophysical applications. The weak wave turbulence theory aims to provide a statistical description of internal gravity waves propagating in the bulk of such flows. However, internal gravity waves are usually perturbed by other structures present in stratified flow, namely the shear modes and the vortical modes. In order to check whether a weak internal gravity wave turbulence regime can occur, we perform direct numerical simulations of stratified turbulence without shear modes and with or without vortical modes at various Froude and buoyancy Reynolds numbers. We observe that removing vortical modes naturally helps to have a better overall balance between poloidal kinetic energy, involved in internal gravity waves, and potential energy. However, conversion between kinetic energy and potential energy does not necessarily show fluctuations around zero in our simulations, as we would expect for a system of weak waves. A spatiotemporal analysis reveals that removing vortical modes helps to concentrate the energy around the wave frequency, but it is not enough to observe a weak wave turbulence regime. Yet we observe that internal gravity waves whose frequency are large compared to the eddy turnover time are present, and we also find evidences for slow internal gravity waves interacting by triadic resonance instabilities in our strongly stratified flows simulations. Finally, we propose conditions that should be fulfilled in order to observe a weak internal gravity wave turbulence regime in real flows.

Figure 1

Illustration of the poloidal-toroidal basis $({\mathit{e}}_{\mathit{k}},{\mathit{e}}_{p\mathit{k}},{\mathit{e}}_{t\mathit{k}})$ defined by Eqs. (12). ${\theta }_{\mathit{k}}$ is the angle between ${\mathit{e}}_z$ and ${\mathit{e}}_{\mathit{k}}$. ${\phi }_{\mathit{k}}$ is the angle between the horizontal projection of $\mathit{k}$ and ${\mathit{e}}_x$.

Figure 2

Classification of regimes in our simulations using the isotropy coefficients. The large-scale isotropy coefficient $I_{\mathrm{kin}}$ is given by the color scale, and the small-scale isotropy coefficient $I_{\text{diss}}$ by the size of the symbols. For simulations with vortical modes $\left(\mathrm{a}\right)$, definitions (21) and (22) are used. For simulations without vortical modes $\left(\mathrm{b}\right)$, definitions (22) and (23) are used. The dotted blue lines correspond to $\mathcal{R}=10, F_h=0.14$, and $F_h=1$.

Figure 3

Vortical modes to total energy ratio (a) and relative difference between poloidal and potential energies $\tilde{\mathcal{D}}=(E_{\mathrm{polo}}-E_{\mathrm{pot}})/(E_{\mathrm{polo}}+E_{\mathrm{pot}})$ (b) vs $F_h$ for simulations with or without vortical modes. The red boxes indicate the simulations with $(N,{\mathcal{R}}_i)=(40,20)$ investigated in the next subsections.

Figure 4

Snapshots of the buoyancy fields for simulations $(N,{\mathcal{R}}_i)=(40,20)$ with $\left(\mathrm{a}\right)$ and without $\left(\mathrm{b}\right)$ vortical modes.

Figure 5

Temporal energy spectra for simulations $(N,{\mathcal{R}}_i)=(40,20)$ with $\left(\mathrm{a}\right)$ and without $\left(\mathrm{b}\right)$ vortical modes. The orange dotted lines correspond to the minimal and maximal frequencies of the linear internal gravity waves in the forcing region.

Figure 6

Compensated 1D spatial energy spectra for simulations $(N,{\mathcal{R}}_i)=(40,20)$ with $\left(\mathrm{a}\right)$ and without $\left(\mathrm{b}\right)$ vortical modes. Normalized $k_h$ energy fluxes for the same simulations with $\left(\mathrm{c}\right)$ and without $\left(\mathrm{d}\right)$ vortical modes. The orange dotted line corresponds to the maximal wave-vector modulus of the forcing region, the black dotted line to $k_b$, and the black dashed line to $k_O$.

Figure 7

Slices of the kinetic energy spectrum $E_{\mathrm{kin}}(k_h,k_z)$ for simulations $(N,{\mathcal{R}}_i)=(40,20)$ with and without vortical modes. The orange dotted line corresponds to the maximal wave-vector modulus of the forcing region, the black dotted line to $k_b$, the black dashed line to $k_O$, and the green dashed line to the dissipative wave vector. (a) $E_{\mathrm{kin}}$ vs $k_h$ with vortical modes. (b) $E_{\mathrm{kin}}$ vs $k_h$ without vortical modes. (c) $E_{\mathrm{kin}}$ vs $k_z$ with vortical modes. (d) $E_{\mathrm{kin}}$ vs $k_z$ without vortical modes.

Figure 8

$(k_h,k_z)$ spectra for the simulations at $(N,{\mathcal{R}}_i)=(40,20)$ with and without vortical modes. The cyan dotted lines correspond to ${\chi }_{\mathit{k}}=1/3$ and ${\chi }_{\mathit{k}}=3$, the cyan dashed line to $k=k_b$, the magenta dotted line to ${\gamma }_{\mathit{k}}=1$, and the green dotted line to the dissipative scale. The orange box corresponds to the forcing region. (a) $E_{\mathrm{toro}}$ with vortical modes. $\left(\mathrm{b}\right){\mathrm{E}}_{\mathrm{toro}}$ without vortical modes. (c) $E_{\mathrm{polo}}$ with vortical modes. $\left(\mathrm{d}\right){\mathrm{E}}_{\mathrm{polo}}$ without vortical modes. (e) $E_{\mathrm{pot}}$ with vortical modes. $\left(\mathrm{f}\right){\mathrm{E}}_{\mathrm{pot}}$ without vortical modes.

Figure 9

Ratios of energy for the simulations at $(N,{\mathcal{R}}_i)=(40,20)$ with and without vortical modes. The cyan dotted lines correspond to ${\chi }_{\mathit{k}}=1/3$ and ${\chi }_{\mathit{k}}=3$, the cyan dashed line to $k=k_b$, and the green dotted line to the dissipative scale. The orange box corresponds to the forcing region. (a) $E_{\mathrm{toro}}/E$ with vortical modes. (b) $E_{\mathrm{toro}}/E$ without vortical modes. (c) $\tilde{\mathcal{D}}=(E_{\mathrm{polo}}-E_{\mathrm{pot}})/(E_{\mathrm{polo}}+E_{\mathrm{pot}})$ with vortical modes. $\left(\mathrm{d}\right)\tilde{\mathcal{D}}$ without vortical modes. A wave-dominated region is expected for the simulation without vortical where ${\chi }_{\mathit{k}}<1/3$, for which $E_{\mathrm{toro}}=0$ and $\tilde{\mathcal{D}}\ll 1$.

Figure 10

Normalized conversion to potential energy $\tilde{\mathcal{B}}$ (31) for the simulations at $(N,{\mathcal{R}}_i)=(40,20)$ with $\left(\mathrm{a}\right)$ and without $\left(\mathrm{b}\right)$ vortical modes. The magenta dotted line corresponds to ${\gamma }_{\mathit{k}}=1$, and the green dotted line to the dissipative scale. The orange box corresponds to the forcing region.

Figure 11

Slices of $E_{\mathrm{equi}}(k_h,k_z,\omega )$ for the simulations at $(N,{\mathcal{R}}_i)=(40,20)$ with and without vortical modes. (a) $k_z=25.1$ with vortical modes. (b) $k_z=25.1$ without vortical modes. (c) $k_h=25.1$ with vortical modes. (d) $k_h=25.1$ without vortical modes. The linear dispersion relation ${\omega }_{\mathit{k}}=N{k}_h/k$ is plotted in black, the lines ${\omega }_{\mathit{k}}\pm \delta \omega /2$ (35) in blue, and the yellow dashed lines correspond to the lines ${\omega }_{\mathit{k}}\pm \delta {\omega }_{\text{doppler}}$ (34).

Figure 12

Spatiotemporal analysis of the simulations with $(N,{\mathcal{R}}_i)=(40,20)$. $E_{\mathrm{equi}}(k_h,k_z,\omega )/\underset{\omega }{max}E(k_h,k_z,\omega )$ for the simulation with vortical modes $\left(\mathrm{a}\right)$ and for the simulation without vortical modes $\left(\mathrm{b}\right)$. Only some couples $(k_h,k_z)$ are shown. Line colors corresponds to different values of $k/k_b$. The vertical black lines corresponds to $\omega ={\omega }_{\mathit{k}}$.

Figure 13

$\delta \omega /{\omega }_{\mathit{k}}$ for simulations with $(N,{\mathcal{R}}_i)=(40,20)$ with vortical modes (a) and without vortical modes (b). Dotted lines correspond to ${\chi }_{\mathit{k}}=1/3$ and ${\chi }_{\mathit{k}}=3$, the dashed line to $k=k_b$, and the orange box to the forcing region.

Figure 14

Integrated spatiotemporal spectra of the total energy as a function of ${\omega }_{\mathit{k}}/N=sin{\theta }_{\mathit{k}}$ and $\omega$ for different simulations at the same viscosity $\nu =1/\left(N^2{\mathcal{R}}_i\right)$. (a) $(N,{\mathcal{R}}_i)=(20,160)$ with vortical modes, (b) $(N,{\mathcal{R}}_i)=(20,160)$ without vortical modes, (c) $(N,{\mathcal{R}}_i)=(40,40)$ with vortical modes, (d) $(N,{\mathcal{R}}_i)=(40,40)$ without vortical modes, (e) $(N,{\mathcal{R}}_i)=(80,10)$ with vortical modes, and (f) $(N,{\mathcal{R}}_i)=(80,10)$ without vortical modes. The white dotted line represents $\omega ={\omega }_{\mathit{k}}$, i.e., the linear dispersion relation.

Figure 15

Wave energy ratio ${\tilde{E}}_{\mathrm{wave}}(k_h,k_z)$ (37) for simulations with $(N,{\mathcal{R}}_i)=(40,20)$ with vortical modes $\left(\mathrm{a}\right)$ and without vortical modes $\left(\mathrm{b}\right)$. The cyan dotted lines correspond to ${\chi }_{\mathit{k}}=1/3$, and the cyan dashed line to $k=k_b$. The orange box corresponds to the forcing region. The forcing angle ${\theta }_f$ is represented by a full orange line, while first harmonics (38, 39, 40) excited by the TRI are indicated by dashed orange lines.

Figure 16

${\tilde{E}}_{\mathrm{wave}}$ as a function of $F_h\left(\mathrm{a}\right)$ and $\mathcal{R}\left(\mathrm{b}\right)$ for all our simulations with or without vortical modes. The red boxes indicate the simulations with $(N,{\mathcal{R}}_i)=(40,20)$ investigated in the previous subsections.

Figure 17

Simulations of the present study and Refs. [31, 42, 46, 74], and the experiments [45] in the $(F_h,\mathcal{R})$ parameters space. The colored full lines corresponding to conditions (42), (45), and (46) (see legend). The blue dashed line corresponds to $\mathrm{Re}=500$. The colored region is where a weak wave turbulence regime is expected.