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We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of $\textit{interval collapse times}$ $\tau_{\rm col}$, the time taken for an interval of length $\ell$, demarcated by a pair of Lagrangian tracers, to collapse at a shock. By calculating the dynamic scaling exponent of the order-$p$ moment of $\tau_{\rm col}$, we show that (a) there is $\textit{not one but an infinity of characteristic time scales}$ and (b) the probability distribution function of $\tau_{\rm col}$ is non-Gaussian and has a power-law tail. Our study is based on (a) a theoretical framework that allows us to obtain dynamic-multiscaling exponents analytically, (b) extensive direct numerical simulations, and (c) a careful comparison of the results of (a) and (b). We discuss possible generalizations of our work to dimensions $d >1 $, for the stochastically forced Burgers equation, and to other compressible flows that exhibit turbulence with shocks.

The rate of collision and the relative velocities of the colliding particles in turbulent flows is a crucial part of several natural phenomena, e.g., rain formation in warm clouds and planetesimal formation in a protoplanetary disks. The particles are often modeled as passive, but heavy and inertial. Within this model, large relative velocities emerge due to formation of singularities (caustics) of in the gradient matrix of the velocities of the particles. Using extensive direct numerical simulations of heavy particles in both two (direct and inverse cascade) and three dimensional turbulent flows we calculate the rate of formation of caustics, $J$ as a function of the Stokes number (${\rm St}$).The best approximation to our data is $J \sim \exp(-C/{\rm St})$, in the limit ${\rm St} \to 0 $ where $C$ is a non-universal constant., Comment: Consistent with the version accepted for publication in Philosophical Transactions of the Royal Society A

We study the spreading of viruses, such as SARS-CoV-2, by airborne aerosols, via a new first-passage-time problem for Lagrangian tracers that are advected by a turbulent flow: By direct numerical simulations of the three-dimensional (3D) incompressible, Navier-Stokes equation, we obtain the time $t_R$ at which a tracer, initially at the origin of a sphere of radius $R$, crosses the surface of the sphere \textit{for the first time}. We obtain the probability distribution function $\mathcal{P}(R,t_R)$ and show that it displays two qualitatively different behaviors: (a) for $R \ll L_{\rm I}$, $\mathcal{P}(R,t_R)$ has a power-law tail $\sim t_R^{-\alpha}$, with the exponent $\alpha = 4$ and $L_{\rm I}$ the integral scale of the turbulent flow; (b) for $l_{\rm I} \lesssim R $, the tail of $\mathcal{P}(R,t_R)$ decays exponentially. We develop models that allow us to obtain these asymptotic behaviors analytically. We show how to use $\mathcal{P}(R,t_R)$ to develop social-distancing guidelines for the mitigation of the spreading of airborne aerosols with viruses such as SARS-CoV-2.

We consider the kinematic fluctuation dynamo problem in a flow that is random, white-in-time, with both solenoidal and potential components. This model is a generalization of the well-studied Kazantsev model. If both the solenoidal and potential parts have the same scaling exponent, then, as the compressibility of the flow increases, the growth rate decreases but remains positive. If the scaling exponents for the solenoidal and potential parts differ, in particular if they correspond to typical Kolmogorov and Burgers values, we again find that an increase in compressibility slows down the growth rate but does not turn it off. The slow down is, however, weaker and the critical magnetic Reynolds number is lower than when both the solenoidal and potential components display the Kolmogorov scaling. Intriguingly, we find that there exist cases, when the potential part is smoother than the solenoidal part, for which an increase in compressibility increases the growth rate. We also find that the critical value of the scaling exponent above which a dynamo is seen is unity irrespective of the compressibility. Finally, we realize that the dimension $d = 3$ is special, since for all other values of $d$ the critical exponent is higher and depends on the compressibility., Comment: 12 pages, 6 figures

We uncover universal statistical properties of the trajectories of heavy inertial particles in three-dimensional, statistically steady, homogeneous, and isotropic turbulent flows by extensive direct numerical simulations. We show that the probability distribution functions (PDFs) $P(\phi)$, of the angle $\phi$ between the Eulerian velocity ${\bf u}$ and the particle velocity ${\bf v}$, at this point and time, shows a power-law region in which $P(\phi) \sim \phi^{-\gamma}$, with a new universal exponent $\gamma \simeq 4$. Furthermore, the PDFs of the trajectory curvature $\kappa$ and modulus $\theta$ of the torsion $\vartheta$ have power-law tails that scale, respectively, as $P(\kappa) \sim \kappa^{-h_\kappa}$, as $\kappa \to \infty$, and $P(\theta) \sim \theta^{-h_\theta}$, as $\theta \to \infty$, with exponents $h_\kappa \simeq 2.5$ and $h_\theta \simeq 3$ that are universal to the extent that they do not depend on the Stokes number ${\rm St}$ (given our error bars). We also show that $\gamma$, $h_\kappa$ and $h_\theta$ can be obtained by using simple stochastic models. We characterize the complexity of heavy-particle trajectories by the number $N_{\rm I}(t,{\rm St})$ of points (up until time $t$) at which $\vartheta$ changes sign. We show that $n_{\rm I}({\rm St}) \equiv \lim_{t\to\infty} \frac{N_{\rm I}(t,{\rm St})}{t} \sim {\rm St}^{-\Delta}$, with $\Delta \simeq 0.4$ a universal exponent.

We solve for the motion of charged particles in a helical time-periodic ABC (Arnold-Beltrami-Childress) magnetic field. The magnetic field lines of a stationary ABC field with coefficients $A=B=C=1$ are chaotic, and we show that the motion of a charged particle in such a field is also chaotic at late times with positive Lyapunov exponent. We further show that in time-periodic ABC fields, the kinetic energy of a charged particle can increase indefinitely with time. At late times the mean kinetic energy grows as a power law in time with an exponent that approaches unity. For an initial distribution of particles, whose kinetic energy is uniformly distributed within some interval, the PDF of kinetic energy is, at late times, close to a Gaussian but with steeper tails., Comment: uses Revtex 4 instead of Revtex 4-1

We obtain, by extensive direct numerical simulations, time-dependent and equal-time structure functions for the vorticity, in both quasi-Lagrangian and Eulerian frames, for the direct-cascade regime in two-dimensional fluid turbulence with air-drag-induced friction. We show that differ- ent ways of extracting time scales from these time-dependent structure functions lead to different dynamic-multiscaling exponents, which are related to equal-time multiscaling exponents by different classes of bridge relations; for a representative value of the friction we verify that, given our error bars, these bridge relations hold., Comment: Published in Phys. Rev. Lett, Supplemental materials added as appendix

We present a natural framework for studying the persistence problem in two-dimensional fluid turbulence by using the Okubo-Weiss parameter $\Lambda$ to distinguish between vortical and extensional regions. We then use a direct numerical simulation (DNS) of the two-dimensional, incompressible Navier--Stokes equation with Ekman friction to study probability distribution functions (PDFs) of the persistence times of vortical and extensional regions by employing both Eulerian and Lagrangian measurements. We find that, in the Eulerian case, the persistence-time PDFs have exponential tails; by contrast, this PDF for Lagrangian particles, in vortical regions, has a power-law tail with an exponent $\theta=2.9\pm0.2$., Comment: consistent with the published version

We carry out a direct numerical simulation (DNS) study that reveals the effects of polymers on statistically steady, forced, homogeneous, isotropic fluid turbulence. We find clear manifestations of dissipation-reduction phenomena: On the addition of polymers to the turbulent fluid, we obtain a reduction in the energy dissipation rate, a significant modification of the fluid energy spectrum, especially in the deep-dissipation range, a suppression of small-scale intermittency, and a decrease in small-scale vorticity filaments. We also compare our results with recent experiments and earlier DNS studies of decaying fluid turbulence with polymer additives., Comment: consistent with the published version

We carry out systematic and high-resolution studies of dynamo action in a shell model for magnetohydrodynamic (MHD) turbulence over wide ranges of the magnetic Prandtl number $Pr_{\rm M}$ and the magnetic Reynolds number $Re_{\rm M}$. Our study suggests that it is natural to think of dynamo onset as a nonequilibrium, first-order phase transition between two different turbulent, but statistically steady, states. The ratio of the magnetic and kinetic energies is a convenient order parameter for this transition. By using this order parameter, we obtain the stability diagram (or nonequilibrium phase diagram) for dynamo formation in our MHD shell model in the $(Pr^{-1}_{\rm M}, Re_{\rm M})$ plane. The dynamo boundary, which separates dynamo and no-dynamo regions, appears to have a fractal character. We obtain hysteretic behavior of the order parameter across this boundary and suggestions of nucleation-type phenomena., Comment: 9 pages, 9 figures

We systematise the study of dynamic multiscaling of time-dependent structure functions in different models of passive-scalar and fluid turbulence. We show that, by suitably normalising these structure functions, we can eliminate their dependence on the origin of time at which we start our measurements and that these normalised structure functions yield the same linear bridge relations that relate the dynamic-multiscaling and equal-time exponents for statistically steady turbulence. We show analytically, for both the Kraichnan Model of passive-scalar turbulence and its shell model analogue, and numerically, for the GOY shell model of fluid turbulence and a shell model for passive-scalar turbulence, that these exponents and bridge relations are the same for statistically steady and decaying turbulence. Thus we provide strong evidence for dynamic universality, i.e., dynamic-multiscaling exponents do not depend on whether the turbulence decays or is statistically steady., Comment: 29 pages, 18 figures. For figures with better resolution, please download the preprint available at the website of the International Collaboration of Turbulence Research

We give an overview of the progress that has been made in recent years in understanding the dynamic multiscaling of homogeneous, isotropic turbulence and related problems. We emphasise the similarity of this problem with the dynamic scaling of time-dependent correlation functions in the vicinity of a critical point in, e.g., a spin system. The universality of dynamic-multiscaling exponents in fluid turbulence is explored by detailed simulations of the GOY shell model for fluid turbulence., Comment: 13 pages, 10 figures, Submitted for the Proceedings of the Statphys 23 Conference

The existence of drag reduction by polymer additives, well established for wall-bounded turbulent flows, is controversial in homogeneous, isotropic turbulence. To settle this controversy we carry out a high-resolution direct numerical simulation (DNS) of decaying, homogeneous, isotropic turbulence with polymer additives. Our study reveals clear manifestations of drag-reduction-type phenomena: On the addition of polymers to the turbulent fluid we obtain a reduction in the energy dissipation rate, a significant modification of the fluid energy spectrum especially in the deep-dissipation range, a suppression of small-scale intermittency, and a decrease in small-scale vorticity filaments., Comment: 4 pages, 3 figures

We present the first study of the dynamic scaling or multiscaling of passive-scalar and passive-vector turbulence. For the Kraichnan version of passive-scalar and passive-vector turbulence we show analytically, in both Eulerian and quasi-Lagrangian frameworks, that simple dynamic scaling is obtained but with different dynamic exponents. By developing the multifractal model we show that dynamic multiscaling occurs in passive-scalar turbulence only if the advecting velocity field is itself multifractal. We substantiate our results by detailed numerical simulations in shell models of passive-scalar advection., Comment: published version

We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum $\sim 1/k$, where $k$ is the wave number. From very-high-resolution numerical simulations, in the limit of vanishing viscosity, we find evidence for multiscaling of velocity structure functions which cannot be falsified by standard tests. We find a new artifact in which logarithmic corrections can appear disguised as anomalous scaling and conclude that bifractal scaling is likely., Comment: previous version replaced, with major changes

Active turbulent advection is considered in the context of magneto-hydrodynamics. In this case, an auxiliary passive field bears no apparent connection to the active field. The scaling properties of the two fields are different. In the framework of a shell model, we show that the two-point structure function of the passive field has a unique zero mode, characterizing the scaling of this field only. In other words, the existence of statistical invariants for the decaying passive field carries no information on the scaling properties of the active field., Comment: 4 pages, 2 figures

We show that different ways of extracting time scales from time-dependent velocity structure functions lead to different dynamic-multiscaling exponents in fluid turbulence. These exponents are related to equal-time multiscaling exponents by different classes of bridge relations which we derive. We check this explicitly by detailed numerical simulations of the GOY shell model for fluid turbulence. Our results can be generalized to any system in which both equal-time and time-dependent structure functions show multiscaling., Comment: 4 pages 4 figures