Energy-based analysis and anisotropic spectral distribution of internal gravity waves in strongly stratified turbulence

Stratified turbulence shows scale- and direction-dependent anisotropy and the coexistence of weak turbulence of internal gravity waves and strong turbulence of eddies. Straightforward application of standard analyses developed in isotropic turbulence sometimes masks important aspects of the anisotropic turbulence. To capture detailed structures of the energy distribution in the wave-number space, it is indispensable to examine the energy distribution with nonintegrated spectra by fixing the codimensional wave-number component or in the two-dimensional domain spanned by both the horizontal and the vertical wave numbers. Indices which separate the range of the anisotropic weak-wave turbulence in the wave-number space are proposed based on the decomposed energies. In addition, the dominance of the waves in the range is also verified by the small frequency deviation from the linear dispersion relation. In the wave-dominant range, the linear wave periods given by the linear dispersion relation are smaller than approximately one third of the eddy-turnover time. The linear wave periods reflect the anisotropy of the system, while the isotropic Brunt-Väisälä period is used to evaluate the Ozmidov wave number, which is necessarily isotropic. It is found that the time scales in consideration of the anisotropy of the flow field must be appropriately selected to obtain the critical wave number separating the weak-wave turbulence.

Figure 1

Integrated energy spectra: total kinetic energy, horizontal kinetic energy, vertical kinetic energy, and potential energy. (a) As functions of horizontal wave numbers integrated over the vertical wave numbers and (b) as functions of vertical wave numbers integrated over the horizontal wave numbers. Green, blue, and red dashed vertical lines, respectively, show the forced wave number $k_f$, the buoyancy wave number $k_b$, and the Ozmidov wave number $k_O$.

Figure 2

(a) Horizontal wave-number spectra and (b) vertical wave-number spectra of vortical kinetic energy, wave kinetic energy, and shear energy. The abscissa is scaled linearly for $k_{\perp },k_{\parallel }\le 1$ and logarithmically for $k_{\perp },k_{\parallel }\ge 1$. See also the caption to Fig. 1 for the vertical lines.

Figure 3

Kinetic energy spectra (a) for each $k_{\parallel }$ as a function of $k_{\perp }$ and (b) for each $k_{\perp }$ as a function of $k_{\parallel }$. See also the caption to Fig. 1 for the vertical lines.

Figure 4

Two-dimensional spectra of (a) total kinetic energy, (b) vortical energy, (c) wave kinetic energy, and (d) potential energy. Contours are drawn for ${10}^{-12},{10}^{-10},\cdots ,{10}^{-4}$. The critical wave number at which ${\chi }_{\mathit{k}}=1/3$ and that at which ${\chi }_{\mathit{k}}=1$ are represented by the thick and thin green curves, respectively. The 2D critical wave number at which ${\chi }_{2\mathrm{D}\mathit{k}}=1/3$ and that at which ${\chi }_{2\mathrm{D}\mathit{k}}=1$ are represented by the thick and thin yellow curves, respectively. The buoyancy wave number and the Ozmidov wave number are, respectively, represented by the blue and the magenta curves.

Figure 5

(a) Ratio of the wave kinetic energy to the total kinetic energy $K_{\mathrm{w}\mathit{k}}/K_{\mathit{k}}$, (b) relative difference between the wave kinetic energy and the potential energy $(K_{\mathrm{w}\mathit{k}}-V_{\mathit{k}})/(K_{\mathrm{w}\mathit{k}}+V_{\mathit{k}})$, and (c) ratio of the polarization anisotropic part to the kinetic energy $K_{z\mathrm{PA}\mathit{k}}/K_{\mathit{k}}$. Contours are drawn for every 0.2 in (a) and (c) and for every 0.1 in (b). The critical wave number at which ${\chi }_{\mathit{k}}=1/3$, the 2D critical wave number at which ${\chi }_{2\mathrm{D}\mathit{k}}=1/3$, the buoyancy wave number, and the Ozmidov wave number are represented by the green, yellow, blue, and magenta curves, respectively.

Figure 6

Ratio of the wave kinetic energy to the total kinetic energy $K_{\mathrm{w}\mathit{k}}/K_{\mathit{k}}$. (a) ${\gamma }_{\mathit{k}}=0.1$, (b) ${\gamma }_{\mathit{k}}=0.2$, (c) ${\gamma }_{\mathit{k}}=0.5$, (d) ${\gamma }_{\mathit{k}}=1$, (e) ${\gamma }_{\mathit{k}}=2$, and (f) ${\gamma }_{\mathit{k}}=5$. See also the caption to Fig. 5 for the curves.

Figure 7

Relative frequency deviation $\delta {\sigma }_{\mathit{k}}/{\sigma }_{\mathit{k}}$ for $\mathit{k}=(k_x,k_y,k_z)=(0,2^p,2^q)$, where $p,q=0,1,2,\cdots$. Contours are drawn for 1 and 10. See also the caption to Fig. 5 for the curves.