Single-particle dispersion in stably stratified turbulence

We present models for single-particle dispersion in vertical and horizontal directions of stably stratified flows. The model in the vertical direction is based on the observed Lagrangian spectrum of the vertical velocity, while the model in the horizontal direction is a combination of a continuous-time eddy-constrained random walk process with a contribution to transport from horizontal winds. Transport at times larger than the Lagrangian turnover time is not universal and dependent on these winds. The models yield results in good agreement with direct numerical simulations of stratified turbulence, for which single-particle dispersion differs from the well-studied case of homogeneous and isotropic turbulence.

Figure 1

(Left) Vertical (top) and horizontal (bottom) displacements for a few particles (indicated by different colors) in the TG4 run (TG forcing with $N=4)$. (Right) Same for the RND4 run (random forcing with $N=4)$.

Figure 2

(Left) Lagrangian power spectrum of the vertical velocity in all simulations. Power laws are indicated as a reference (see Table 2). All spectra display a peak at the Brunt-Väisälä frequency (indicated by the arrow), and for frequencies $\omega /N<1$ the shallow spectra are reminiscent of observations of oceanic internal gravity waves. (Right) Lagrangian power spectrum of the horizontal velocity. In the horizontal case there is no clear peak near the Brunt-Väisälä frequency.

Figure 3

PDF of vertical displacement waiting times for the simulations, and waiting times generated by the model consisting of a superposition of random waves (solid green in all cases). For TG runs the PDFs have been shifted vertically for better visualization. Power laws are shown as a reference (see Table 2), and vertical arrows indicate $t=1/N$.

Figure 4

PDFs of the local gradient Richardson number ${\text{Ri}}_g$ for all runs. A vertical solid line at ${\text{Ri}}_g=0$ and a vertical dashed line at ${\text{Ri}}_g=1/4$ are shown as references; for ${\text{Ri}}_g<1/4$ local shear instabilities can develop, while for ${\text{Ri}}_g<0$ overturning can occur.

Figure 5

Mean vertical quadratic dispersion $\delta z^2$ for all runs, normalized by the ratio $w^2/N^2$ of the mean squared vertical Lagrangian velocity to the squared Brunt-Väisälä frequency. Time is normalized by the Brunt-Väisälä period. All simulations collapse at early times, but the TG runs depart after $tN/2\pi \approx 1$. The thick (red) curve shows the model for RND4, which is in good agreement with the simulation until at late times molecular diffusion results in a slow vertical dispersion of the particles.

Figure 6

Isocontours of $P(r,t)$ for the simulations (dashed blue lines) and for the model (dotted red lines), for (a) TG4, (b) TG8, (c) RND4, and (d) RND8. The inset show the mean horizontal displacement with the same line labels (power laws are shown as references, see text for details). Vertical lines (from left to right in all panels) indicate, respectively, the Eulerian $T_e$ and Lagrangian $T_l$ turnover times.

Figure 7

Left: Mean horizontal quadratic dispersion for all runs (thick black curves), normalized by the product of the mean squared horizontal velocity $u_{\perp }^2$ and the squared Lagrangian turnover time $T_l^2$, as a function of time normalized by $T_l$. The same quantity obtained from the model is shown by the thin (red) curves, using the same line labels as their corresponding simulation. The mean quadratic dispersion obtained from the model is in good agreement with all simulations. Right: Mean horizontal displacements normalized by the time for all runs (thick black curves), such that curves are flat when $\langle r\rangle \sim t$. Again, $\langle r\rangle /t$ obtained from the model is shown in thin (red) curves.

Figure 8

Horizontal cuts (in the $x-y$ plane, and at $z=0)$ of the horizontal velocity at time $t=20$ for (left) the TG4 run, and (right) the RND4 run. Note the clear mean wind in the latter case, pointing approximately in the $\widehat{x}$ direction. Other layers (i.e., for other values of $z)$ have mean horizontal winds pointing in other directions.

Figure 9

PDFs of the mean horizontal winds for the TG4 and RND4 runs. (Left) PDF of ${\langle u_x\rangle }_{xy}$, i.e., of the velocity in the $\widehat{x}$ direction, $u_x$, averaged over the $x$ and $y$ coordinates, or equivalently, over planes at constant height. (Right) Same for the r.m.s. horizontal velocity $u_{\perp }={(u_x^2+u_y^2)}^{1/2}$, also averaged over planes at constant height.

Figure 10

$P(r,t)$ at different times for the simulations (solid lines) and for the model (dashed lines). Thick lines correspond to flows with $N=4$, and thin lines to $N=8$. PDFs peaked at small values of $r$ are for TG forcing, while PDFs peaked at large values of $r$ are for RND forcing (as labeled). (Left) $t=1.5\left(t<T_e\right)$; (right) $t=35\left(t>T_l\right)$.

Figure 11

(Left) Mean horizontal dispersion as a function of time for a run with Kolmogorov forcing and $N=8$ (thick black line). Dispersion in the random walk model is shown by the thin (red) line. (Right) Isocontours of $P(r,t)$ for the simulation (dashed blue curves) and for the model (dotted red curves). Vertical lines (from left to right) indicate, respectively, the Eulerian $T_e$ and Lagrangian $T_l$ turnover times.