Vertical dispersion of Lagrangian tracers in fully developed stably stratified turbulence

We study the effect of different forcing functions and of the local gradient Richardson number ${\mathrm{Ri}}_g$ on the vertical transport of Lagrangian tracers in stably stratified turbulence under the Boussinesq approximation and present a wave and continuous-time random-walk model for single- and two-particle vertical dispersion. The model consists of a random superposition of linear waves with their amplitude, based on the observed Lagrangian spectrum of vertical velocity, and a random-walk process to capture overturning that depends on the statistics of ${\mathrm{Ri}}_g$ among other Eulerian quantities. The model is in good agreement with direct numerical simulations of stratified turbulence, where single- and two-particle dispersion differ from the homogeneous and isotropic case. Moreover, the model gives insight into the mixture of linear and nonlinear physics in the problem, as well as on the different processes responsible for vertical turbulent dispersion.

Figure 1

Mean vertical dispersion $\delta z^2$ for (a) runs in cubic domains (boxes with aspect ratio 1:1) and (b) runs in elongated domains with aspect ratio 1:4, in both figures with TG and RND forcing and with different Brunt-Väisälä frequencies. The dispersion is normalized by $U_z^2/N^2$, the ratio of the squared mean vertical velocity to the Brunt-Väisälä frequency, and time is normalized by the Brunt-Väisälä period. Power laws are indicated as references. In each figure, the Lagrangian time of each run is indicated by a vertical arrow with the same line style as the corresponding run, while the insets show the mean vertical dispersion of some of the simulations, not normalized, and compensated by the power laws indicated in the main panels for intermediate times.

Figure 2

Mean-square vertical displacement $\delta z^2$ for RND runs in domains with aspect ratio 1:4 and with $N=8$ (run ${\mathrm{RND}}_{4}8$ and runs ${\mathrm{RND}}_{4}8\mathrm{B}$ to ${\mathrm{RND}}_{4}8\mathrm{D}$, from higher to lower $\mathrm{Re}$ and $\mathrm{Rb})$: (a) $\delta z^2\left(t\right)$ without normalization and (b) same as (a), normalized by the ratio $U_z^2/N^2$ and with time normalized by the Brunt-Väisälä period.

Figure 3

The PDFs of the Eulerian local gradient Richardson number ${\mathrm{Ri}}_g$, for all runs in cubic domains (runs ${\mathrm{TG}}_{1}4$ and ${\mathrm{TG}}_{1}8$ with TG forcing and $N=4$ and 8, respectively, and runs ${\mathrm{RND}}_{1}4$ and ${\mathrm{RND}}_{1}8$ with RND forcing and $N=4$ and 8, respectively). Vertical solid lines at ${\mathrm{Ri}}_g=0$ and $1/4$ are shown as references.

Figure 4

Vertically averaged absolute value of the Eulerian vertical velocity $\langle |u_z{|\rangle }_z$ for run ${\mathrm{TG}}_{4}4$ (TG forcing, $N=4$, and aspect ratio 1:4). Bright regions correspond to large vertical velocities in absolute value. As the domain is periodic in both the $x$ and $y$ directions, the regions with large $\langle |u_z{|\rangle }_z$ can be enclosed by four circles (cylinders when extended in the $z$ direction), indicated as a reference by the black solid lines.

Figure 5

Thick black lines show the Lagrangian PDFs of the gradient Richardson number ${\mathrm{Ri}}_g$ for TG runs in boxes with 1:4 aspect ratio. This PDF is compared with (a) the PDFs of ${\mathrm{Ri}}_g$ restricted to Lagrangian velocities $|v_z|>\langle |v_z|\rangle +2{\sigma }_{v_z}$ (where ${\sigma }_{v_z}$ is the dispersion in $v_z)$, shown as thin blue lines, and (b) the PDFs of ${\mathrm{Ri}}_g$ restricted to particles in the circular regions indicated in Fig. 4. Vertical lines at ${\mathrm{Ri}}_g=0$ and ${\text{Ri}}_g=1/4$ are shown as references.

Figure 6

Thick black lines show the PDFs of the Lagrangian vertical velocity $v_z$ for TG runs in domains with 1:4 aspect ratio and varying $N$. The PDFs restricted to particles in instants with (a) ${\mathrm{Ri}}_g<1/4$ or (b) ${\mathrm{Ri}}_g<0$, for the same runs, are shown as thin red lines.

Figure 7

Isocontours of the joint probability distribution function of ${\mathrm{Ri}}_g$ and $|v_z|,P({\mathrm{Ri}}_g,|v_z\left|\right)$, for runs (a) ${\mathrm{TG}}_{4}4$, (b) ${\mathrm{TG}}_{4}8$, and (c) ${\mathrm{TG}}_{4}12$.

Figure 8

The PDFs of $\theta$ as seen by the Lagrangian particles (thick black curves) and the same PDFs restricted (thin red curves) to (a) particles at times with ${\mathrm{Ri}}_g<1/4$ and (b) particles at times with ${\mathrm{Ri}}_g<0$. Except for the run ${\mathrm{TG}}_{4}12$, all PDFs are compatible with Gaussian statistics for $\theta$.

Figure 9

The PDFs of the Lagrangian vertical temperature gradients ${\partial }_z\theta$ (thick black curves) and the same PDFs restricted (thin red curves) to (a) ${\mathrm{Ri}}_g<1/4$ and (b) ${\mathrm{Ri}}_g<0$.

Figure 10

Overturning probability normalized by the forcing wave number and unit length, $R/k_f{L}_0$, as a function of the Reynolds buoyancy number $\mathrm{Rb}$ for all simulations in Table 1.

Figure 11

Power spectrum of the Lagrangian vertical velocity for (a) runs in cubic domains and (b) runs in elongated domains. Frequencies have been normalized by the Brunt-Väisälä frequency. The solid vertical lines indicate (from left to right) $\omega =N/2$ and $\omega =N$.

Figure 12

Mean dispersion obtained from a random superposition of waves as in Eq. (15) (thick black lines) and from Eq. (16) (thin red lines), with the parameter $A_0$ to match the Eulerian vertical rms velocity and $N$ the Brunt-Väisälä frequency for some of the TG runs (see the labels in the legend).

Figure 13

Mean vertical dispersion $\delta z^2$ for (a) runs in elongated domains with amplitudes rescaled for better visualization (see the legend) and (b) runs in cubic domains, for RND and TG forcing and with different values of $N$. Normalizations of $\delta z^2$ and of time are the same as in Fig. 1. In both (a) and (b) the thick black lines show the results from the DNSs and the thin red lines show the results obtained from the single-particle dispersion model. The inset in (b) shows a close-up of $\delta z^2$ for run ${\mathrm{RND}}_{1}8$ (thick dash-dotted curve) together with the mean-square vertical displacement predicted by the full model and by the superposition of random waves alone.

Figure 14

Mean-square vertical displacement $\delta z^2$ in linear scale for four cases: (a) ${\mathrm{TG}}_{4}8$, (b) ${\mathrm{TG}}_{1}4$, (c) ${\mathrm{RND}}_{4}8$, and (d) ${\mathrm{RND}}_{1}8$. The four cases are illustrative of runs with TG or RND forcing, with different levels of stratification and with different domain aspect ratios. The thick black curves show the results from the DNSs, while the thin red lines show the results obtained from the model. In spite of the randomness in the DNSs and in the CTRW model, all cases display reasonable agreement.

Figure 15

(a) Sketch of the two contributions to vertical dispersion in the model. The random superposition of waves dominates at early times (red dashed curve), while the effect of turbulent eddies and overturning events become relevant at late times (blue dotted curve). The total vertical dispersion (black solid curve) results from the superposition of both effects. (b) A few particle trajectories from the ${\mathrm{TG}}_{4}4$ run projected on the $x\text{−}z$ plane. Note the small and wavy particles' displacements in the vertical directions, interrupted by sudden and close to circular trajectories that result in a larger vertical displacement, which we associate by trapping by overturning eddies (see Sec. 5 for details).

Figure 16

(a) Two-particle vertical dispersion for particles with initial vertical separation $\delta z_0\ll \eta$ and two initial horizontal separations $\delta r_0=\eta$ (black lines) $\delta r_0=2\eta$ (gray lines). Results from three simulations with TG forcing are shown. The vertical dispersion was normalized by ${\left(w{K}_{\parallel }\delta r_0\right)}^2/N^2$ and time by $1/N$. With this normalization all curves collapse during the ballistic regime. A power law at later times is indicated as a reference. (b) Two-particle vertical dispersion from DNSs with RND or TG forcing (thick lines) and from the model (thin lines). The amplitudes of the curves have been rescaled for better visualization (see the rescaling factor in the legend).