On how a joint interaction of two innocent partners (smooth advection & linear damping) produces a strong intermittency

Forced advection of passive scalar by a smooth $d$-dimensional incompressible velocity in the presence of a linear damping is studied. Acting separately advection and dumping do not lead to an essential intermittency of the steady scalar statistics, while being mixed together produce a very strong non-Gaussianity in the convective range: $q$-th (positive) moment of the absolute value of scalar difference, $$ is proportional to $r^{\xi_{q}}$, $\xi _{q}=\sqrt{d^{2}/4+\alpha dq/[ (d-1)D]}-d/2$, where $\alpha /D$ measures the rate of the damping in the units of the stretching rate. Probability density function (PDF) of the scalar difference is also found.Comment: 4 pages, RevTex, Submitted to Phys. Fluid

Dynamics of Fluctuation Dominated Phase Ordering: Hard-core Passive Sliders on a Fluctuating Surface

We study the dynamics of a system of hard-core particles sliding downwards on a one dimensional fluctuating interface, which in a special case can be mapped to the problem of a passive scalar advected by a Burgers fluid. Driven by the surface fluctuations, the particles show a tendency to cluster, but the hard-core interaction prevents collapse. We use numerical simulations to measure the auto-correlation function in steady state and in the aging regime, and space-time correlation functions in steady state. We have also calculated these quantities analytically in a related surface model. The steady state auto-correlation is a scaling function of t/L^z, where L is the system size and z the dynamic exponent. Starting from a finite intercept, the scaling function decays with a cusp, in the small argument limit. The finite value of the intercept indicates the existence of long range order in the system. The space-time correlation, which is a function of r/L and t/L^z, is non-monotonic in t for fixed r. The aging auto-correlation is a scaling function of t_1 and t_2 where t_1 is the waiting time and t_2 the time difference. This scaling function decays as a power law for t_2 \gg t_1; for t_1 \gg t_2, it decays with a cusp as in steady state. To reconcile the occurrence of strong fluctuations in the steady state with the fact of an ordered state, we measured the distribution function of the length of the largest cluster. This shows that fluctuations never destroy ordering, but rather the system meanders from one ordered configuration to another on a relatively rapid time scale

Statistical geometry in scalar turbulence

A general link between geometry and intermittency in passive scalar turbulence is established. Intermittency is qualitatively traced back to events where tracer particles stay for anomalousy long times in degenerate geometries characterized by strong clustering. The quantitative counterpart is the existence of special functions of particle configurations which are statistically invariant under the flow. These are the statistical integrals of motion controlling the scalar statistics at small scales and responsible for the breaking of scale invariance associated to intermittency.Comment: 4 pages, 5 figure

Non-universality of the scaling exponents of a passive scalar convected by a random flow

We consider passive scalar convected by multi-scale random velocity field with short yet finite temporal correlations. Taking Kraichnan's limit of a white Gaussian velocity as a zero approximation we develop perturbation theory with respect to a small correlation time and small non-Gaussianity of the velocity. We derive the renormalization (due to temporal correlations and non-Gaussianity) of the operator of turbulent diffusion. That allows us to calculate the respective corrections to the anomalous scaling exponents of the scalar field and show that they continuously depend on velocity correlation time and the degree of non-Gaussianity. The scalar exponents are thus non universal as was predicted by Shraiman and Siggia on a phenomenological ground (CRAS {\bf 321}, 279, 1995).Comment: 4 pages, RevTex 3.0, Submitted to Phys.Rev.Let

Statistical conservation laws in turbulent transport

We address the statistical theory of fields that are transported by a turbulent velocity field, both in forced and in unforced (decaying) experiments. We propose that with very few provisos on the transporting velocity field, correlation functions of the transported field in the forced case are dominated by statistically preserved structures. In decaying experiments (without forcing the transported fields) we identify infinitely many statistical constants of the motion, which are obtained by projecting the decaying correlation functions on the statistically preserved functions. We exemplify these ideas and provide numerical evidence using a simple model of turbulent transport. This example is chosen for its lack of Lagrangian structure, to stress the generality of the ideas

The intermittent behavior and hierarchical clustering of the cosmic mass field

The hierarchical clustering model of the cosmic mass field is examined in the context of intermittency. We show that the mass field satisfying the correlation hierarchy $\xi_n\simeq Q_n(\xi_2)^{n-1}$ is intermittent if $\kappa < d$, where $d$ is the dimension of the field, and $\kappa$ is the power-law index of the non-linear power spectrum in the discrete wavelet transform (DWT) representation. We also find that a field with singular clustering can be described by hierarchical clustering models with scale-dependent coefficients $Q_n$ and that this scale-dependence is completely determined by the intermittent exponent and $\kappa$. Moreover, the singular exponents of a field can be calculated by the asymptotic behavior of $Q_n$ when $n$ is large. Applying this result to the transmitted flux of HS1700 Ly$\alpha$ forests, we find that the underlying mass field of the Ly$\alpha$ forests is significantly intermittent. On physical scales less than about 2.0 h$^{-1}$ Mpc, the observed intermittent behavior is qualitatively different from the prediction of the hierarchical clustering with constant $Q_n$. The observations, however, do show the existence of an asymptotic value for the singular exponents. Therefore, the mass field can be described by the hierarchical clustering model with scale-dependent $Q_n$. The singular exponent indicates that the cosmic mass field at redshift $\sim 2$ is weakly singular at least on physical scales as small as 10 h$^{-1}$ kpc.Comment: AAS Latex file, 33 pages,5 figures included, accepted for publication in Ap

Dynamics of a passive sliding particle on a randomly fluctuating surface

We study the motion of a particle sliding under the action of an external field on a stochastically fluctuating one-dimensional Edwards-Wilkinson surface. Numerical simulations using the single-step model shows that the mean-square displacement of the sliding particle shows distinct dynamic scaling behavior, depending on whether the surface fluctuates faster or slower than the motion of the particle. When the surface fluctuations occur on a time scale much smaller than the particle motion, we find that the characteristic length scale shows anomalous diffusion with $\xi(t)\sim t^{2\phi}$, where $\phi\approx 0.67$ from numerical data. On the other hand, when the particle moves faster than the surface, its dynamics is controlled by the surface fluctuations and $\xi(t)\sim t^{{1/2}}$. A self-consistent approximation predicts that the anomalous diffusion exponent is $\phi={2/3}$, in good agreement with simulation results. We also discuss the possibility of a slow cross-over towards asymptotic diffusive behavior. The probability distribution of the displacement has a Gaussian form in both the cases.Comment: 6 pages, 4 figures, error in reference corrected and new reference added, submitted to Phys. Rev.

Passive Sliders on Growing Surfaces and (anti-)Advection in Burger's Flows

We study the fluctuations of particles sliding on a stochastically growing surface. This problem can be mapped to motion of passive scalars in a randomly stirred Burger's flow. Renormalization group studies, simulations, and scaling arguments in one dimension, suggest a rich set of phenomena: If particles slide with the avalanche of growth sites (advection with the fluid), they tend to cluster and follow the surface dynamics. However, for particles sliding against the avalanche (anti-advection), we find slower diffusion dynamics, and density fluctuations with no simple relation to the underlying fluid, possibly with continuously varying exponents.Comment: 4 pages revtex

Fronts in passive scalar turbulence

The evolution of scalar fields transported by turbulent flow is characterized by the presence of fronts, which rule the small-scale statistics of scalar fluctuations. With the aid of numerical simulations, it is shown that: isotropy is not recovered, in the classical sense, at small scales; scaling exponents are universal with respect to the scalar injection mechanisms; high-order exponents saturate to a constant value; non-mature fronts dominate the statistics of intense fluctuations. Results on the statistics inside the plateaux, where fluctuations are weak, are also presented. Finally, we analyze the statistics of scalar dissipation and scalar fluxes.Comment: 18 pages, 27 figure