Scale-dependent anisotropy in forced stratified turbulence

In stratified turbulence, buoyancy forces inhibit vertical motion and lead to anisotropy over a wide range of length scales, which is characterized by layerwise pancake vortices, thin regions of strong shear, and patches of small-scale turbulence. It has long been known that stratified turbulence becomes increasingly isotropic as one moves to smaller length scales, as the eddy timescale decreases towards and below the buoyancy period. This paper investigates the anisotropy of stratified turbulence across scales and the transition towards isotropy at small scales, using a variety of techniques. Direct numerical simulations of strongly stratified turbulence, with buoyancy Reynolds numbers ${\text{Re}}_b$ up to 50, are analyzed. We examine the relative contributions of different components of the strain rate tensor to the kinetic energy dissipation, the invariants of the isotropy tensor, directional kinetic energy spectra, and the subfilter energy flux across different length scales. At small scales, the degree of isotropy is determined by ${\text{Re}}_b$, while at the Ozmidov and larger scales, the anisotropy also depends on the Froude number. The change in the anisotropy with scale and with the parameters is examined in detail. Interestingly, Ozmidov-scale eddies are found to become increasingly isotropic as ${\text{Re}}_b$ increases, as characterized by the isotropy tensor invariants and the subfilter energy flux. At larger scales, the energy spectra for near-vertical wave vectors have a spectral slope around $-3$, which shallow towards $-1$ for near-horizontal wave vectors. These spectra converge beyond the Ozmidov scale, increasingly so for large ${\text{Re}}_b$. These results suggest that ${\text{Re}}_b\gtrsim 500$ would be necessary to obtain the same degree of small-scale isotropy that is found in similarly sized simulations of unstratified turbulence.

Figure 1

Time series of (a) kinetic energy and (b) kinetic energy dissipation for simulations with $N=0.15$ and different $n$.

Figure 2

Vertical ($x,z$) slices of $u$ at $t=1000$ for $N=0.15$ and $n=\left(\mathrm{a}\right)384$, (b) 576, (c) 1024, and (d) 1536. Contour spacing is 0.009.

Figure 3

Vertical ($x,z$) slices of $u$ at $t=1000$ for $n=576$ and $N=\left(\mathrm{a}\right)0$, (b) 0.15, and (c) 0.3. Contour spacing is 0.009.

Figure 4

(a) Relative contribution of different components to the total dissipation $\overline{{\epsilon }_{ij}/\epsilon }$, and (b) mean square distance ${\chi }_{\epsilon }$ between the $\overline{{\epsilon }_{ij}/\epsilon }$ and their isotropic values. In panel (a) the isotropic values $1/10$ (for $i\ne j$) and $2/15$ (for $i=j$) are denoted by horizontal lines, and the points for (1,1) and (1,3) are not visible because, due to axisymmetry, they are directly behind the points for (2,2) and (2,3), respectively. In panels (a) and (b), all stratified simulations are included, and values are plotted against ${\text{Re}}_b$. The solid line shows the power law best fit.

Figure 5

Isotropy tensor invariants of all simulations. Panel (a) shows the full Lumley triangle, and panel (b) is a zoom on the bottom left side. Shading denotes $N=0$ (white), 0.15 (gray), and 0.3 (black), and symbols denote $n=384$ (square), 576 (diamond), 1024 (triangle), and 1536 (star). The triangle boundary and reference points for 1D, isotropic 2D, and isotropic 3D turbulence are also indicated for clarity [43]. Invariants are computed after filtering out the forcing scales $k<6$.

Figure 6

Invariants after high-pass filtering scales relative to the Ozmidov scale. Invariants are computed after removing (a) $k<0.25k_O$, (b) $k<0.5k_O$, (c) $k<k_O$, and (d) $k<1.4k_O$. Symbols and shading are as in Fig. 5. Note the increase in zoom from panels (a) to (d). In panel (d) the $n=384$ case with $N=0.3$ (black square) is not included because $1.4k_O$ is equal to the truncation wave number.

Figure 7

Directional kinetic energy spectra for $N=0$ for (a) $n=384$, (b) 576, (c) 1024, and (d) 1536. The reference lines have slopes of $-5/3$ and $-3$.

Figure 8

Directional kinetic energy spectra for $N=0.15$ for (a) $n=384$, (b) 576, (c) 1024, and (d) 1536. In all cases, $k_b\approx 9$ and $k_O\approx 43$ are shown with vertical lines. The reference lines have slopes of $-5/3$ and $-3$.

Figure 9

Directional kinetic energy spectra for $N=0.30$ for (a) $n=384$, (b) 576, (c) 1024, and (d) 1536. In all cases, $k_b\approx 17$ and $k_O\approx 120$ are shown with vertical lines. The reference lines have slopes of $-5/3$ and $-3$.

Figure 10

Standard deviation of the directional spectra $E(k,i)$ over $i$, normalized by the mean, for (a) $N=0$, (b) 0.15, and (c) 0.30. In the stratified cases (b) and (c), the buoyancy and Ozmidov wave numbers $k_b$ and $k_O$, with $k_O>k_b$, are shown with vertical lines.

Figure 11

Ratio of the Ozmidov-scale energy in $O_{k,6}$ to $O_{k,1}$.

Figure 12

The minimum value of $\sigma \left(k\right)/\mu \left(k\right)$ over $k$. All stratified simulations are included, and values are plotted vs ${\mathrm{Re}}_b$. The solid line shows the power law best fit.

Figure 13

Relative contribution of different components to the subfilter-scale dissipation ${\mathrm{\Pi }}_{ij}/\mathrm{\Pi }$ for filter cutoff wave number $k_c=\left(\mathrm{a}\right)0.25k_O$, (b) $0.5k_O$, (c) $k_O$, and (d) $1.4k_O$. The isotropic values $1/10$ (for $i\ne j$) and $2/15$ (for $i=j$) are denoted by horizontal lines. All stratified simulations are included and values are plotted against ${\mathrm{Re}}_b$.

Figure 14

Mean square distance ${\chi }_{\mathrm{\Pi }}$ between the ${\mathrm{\Pi }}_{ij}/\mathrm{\Pi }$ and their isotropic values.