Turbulent transport with intermittency: Expectation of a scalar concentration

Abstract

Scalar transport by turbulent flows is best described in terms of Lagrangian parcel motions. Here we measure the Eulerian distance travel along Lagrangian trajectories in a simple point vortex flow to determine the probabilistic impulse response function for scalar transport in the absence of molecular diffusion. As expected, the mean squared Eulerian displacement scales ballistically at very short times and diffusively for very long times, with the displacement distribution at any given time approximating that of a random walk. However, significant deviations in the displacement distributions from Rayleigh are found. The probability of long distance transport is reduced over inertial range time scales due to spatial and temporal intermittency. This can be modeled as a series of trapping events with durations uniformly distributed below the Eulerian integral time scale. The probability of long distance transport is, on the other hand, enhanced beyond that of the random walk for both times shorter than the Lagrangian integral time and times longer than the Eulerian integral time. The very short-time enhancement reflects the underlying Lagrangian velocity distribution, while that at very long-times results from the spatial and temporal variation of the flow at the largest scales. The probabilistic impulse response function, and with it the expectation value of the scalar concentration at any point in space and time, can thus be modeled using only the evolution of the lowest spatial wavenumber modes (the mean and the lowest harmonic) and an eddy based constrained random walk that captures the essential velocity phase relations associated with advection by vortex motions. Preliminary examination of Lagrangian tracers in three-dimensional homogeneous isotropic turbulence suggests that transport in that setting can be similarly modeled.

Vertical dispersion of Lagrangian tracers in fully developed stably stratified turbulence

“…To study the vertical dispersion of single-particles observed in the DNSs of SST in section III, we now present a stochastic model that combines a random wave model (to consider the effect of internal gravity waves) with a CTRW [35] (to capture the effect of overturning by turbulent or large-scale eddies).…”

“…We also present a model for the vertical single-and two-particle vertical dispersion that is in good agreement with the DNS results. The model consist of a continuous-time random walk (CTRW) based on a previous model for HIT [34,35] (to model trapping of tracers by turbulent eddies, and the effect of local overturning instabilities), and a random superposition of waves, and can capture the vertical dispersion of particles at all times and in all SST regimes considered. The superposition of linear and turbulent effects in the model allows us to identify the leading physical effects resulting in vertical dispersion at early and at late times in the flow (compared with the period of the internal gravity waves).…”

Single-particle dispersion in stably stratified turbulence

“…Since vertical dispersion is small, particle motions in planes perpendicular to the mean stratification can be approximated as two dimensional, and prediction of dispersion in this direction is relevant for the stably stratified atmosphere and for other geophysical flows. Thus, a model for the probability distribution P (x, t; x , t ) of finding a particle at (x, t) given a previous location (x , t ) has multiple applications, and would allow probabilistic prediction of the concentration of quantities transported by the flow without resorting to ensembles of deterministic simulations with small differences in the initial concentrations [27]. In the following we derive a model for this distribution resorting only to general statistical properties of the turbulent flow.…”

“…In the latter (transport perpendicular to the stratification), dispersion differs from HIT as it is strongly influenced by the large scale shearing flow generated by the stratification, and which plays an important role in the atmosphere [18][19][20]. The model used in this case is then a continuous-time eddy-constrained (CTEC) random walk (which accounts for particle trapping observed in HIT [27]), with a superposed drift caused by the vertically sheared horizontal winds (VSHW) in stably stratified turbulence.…”

Intermittency of three-dimensional perturbations in a point-vortex model

“…In the 1990s, there was a significant activity devoted to vortex gas modelling of (particularly decaying) 2-D turbulence [20][21][22][23][24], where merging rules for point vortices were prescribed, yielding 2-D turbulence-like behavior at reduced numerical cost. Point vortex models have also been used to investigate stirring by chaotic advection [25], as well as Lagrangian intermittency, pair dispersion and transport in turbulence [26][27][28]. Recently, vortex gas scaling arguments were leveraged to find a highly accurate local closure in baroclinic turbulence [29].…”