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M. Chertkov

Publications and source records attributed to M. Chertkov.

33 records · Page 2Linked to original sources
arXiv2002

Decay of scalar turbulence revisited

We demonstrate that at long times the rate of passive scalar decay in a turbulent, or simply chaotic, flow is dominated by regions (in real space or in inverse space) where mixing is less efficient. We examine two situations. The first is of a spatially homogeneous stationary turbulent flow with both viscous and inertial scales present. It is shown that at large times scalar fluctuations decay algebraically in time at all spatial scales (particularly in the viscous range, where the velocity is smooth). The second example explains chaotic stationary flow in a disk/pipe. The boundary region of the flow controls the long-time decay, which is algebraic at some transient times, but becomes exponential, with the decay rate dependent on the scalar diffusion coefficient, at longer times.

nlin.CD↗
arXiv2000

Pulse confinement in optical fibers with random dispersion

Short range correlated uniform noise in the dispersion coefficient, inherent in many types of optical fibers, broadens and eventually destroys all initially ultra-short pulses. However, under the constraint that the integral of the random component of the dispersion coefficient is set to zero, or pinned, periodically or quasi-periodically along the fiber, the nature of the pulse propagation changes dramatically. For the case that randomness is added to constant positive dispersion, the pinning restriction significantly reduces pulse broadening. If the randomness is added to piecewise constant periodic dispersion, the pinning may even provide probability distributions of pulse parameters that are numerically indistinguishable from the statistically steady case. The pinning method can be used to both manufacture better fibers and upgrade existing fiber links.

nlin.CD↗
arXiv1999

Small-scale turbulent dynamo

Kinematic dynamo theory is presented here for turbulent conductive fluids. We describe how inhomogeneous magnetic fluctuations are generated below the viscous scale of turbulence where the spatial smoothness of the velocity permits a systematic analysis of the Lagrangian path dynamics. We find analytically the moments and multi-point correlation functions of the magnetic field at small yet finite magnetic diffusivity. We show that the field is concentrated in long narrow strips and describe anomalous scalings and angular singularities of the multi-point correlation functions which are manifestations of field's intermittency. The growth rate of the magnetic field in a typical realization is found to be half the difference of two Lyapunov exponents of the same sign.

chao-dyn↗
arXiv1997

Propagation of a Huygens front through turbulent medium

The dynamics of a thin Huygens front propagating through turbulent medium is considered. A rigorous asymptotic expression for the effective velocity $v_{F}$ proportional to the front area is derived. The small-scale fluctuations of the front position are shown to be strongly intermittent. This intermittency plays a crucial role in establishing a steady state magnitude of the front velocity. The results are compared with experimental data.

chao-dyn↗
arXiv1997

Intermittent dissipation of a passive scalar in turbulence

Probability density function (PDF) of passive scalar dissipation ${\cal P} (ε)$ is found analytically in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime) in two dimensions. The tail of PDF at $ε\gg<ε>$ is shown to be stretched exponent $\ln{\cal P}(ε)\proptoε^{1/3}$. At $ε\ll<ε>$, ${\cal P}\propto 1/\sqrtε$.

chao-dyn↗
arXiv1997

Inverse cascade and intermittency of passive scalar in 1d smooth flow

Random advection of Lagrangian tracer scalar field $θ(t,x)$ by a one-dimensional, spatially smooth and short-correlated in time velocity field is considered. Scalar fluctuations are maintained by a source concentrated at the integral scale $L$. The statistical properties of both scalar differences and the dissipation field are analytically determined, exploiting the dynamical formulation of the model. The Gaussianity known to be present at small scales for incompressible velocity fields emerges here at large scales ($x\gg L$). These scales are shown to be excited by an inverse cascade of $θ^{2}$ and the probability distribution function (PDF) of the corresponding scalar differences to approach the Gaussian form, as larger and larger scales are considered. Small scales ($x\ll L$) statistics is shown to be strongly non-Gaussian. Collapse of scaling exponents for scalar structure functions\thinspace takes place: moments of order $p\ge 1$ scale all linearly, independently of the order $p$. Smooth scaling $x^{p}$ is found for $-1<p<1$. Tails of scalar differences PDF are exponential while, at the center, a cusped shape tends to develop when smaller and smaller ratios $x/L$ are considered. The same tendency is present for scalar gradients PDF with respect to the inverse of the Péclet number (the pumping-to-diffusion scale ratio). The tails of the latter PDF are however much more extended, decaying as a stretched exponential of exponent 2/3, smaller than unity. This slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale.

chao-dyn↗
arXiv1997

Inverse versus direct cascades in turbulent advection

A model of scalar turbulent advection in compressible flow is analytically investigated. It is shown that, depending on the dimensionality $d$ of space and the degree of compressibility of the smooth advecting velocity field, the cascade of the scalar is direct or inverse. If $d>4$, the cascade is always direct. For small enough degree of compressibility, the cascade is direct again. Otherwise it is inverse, i.e. very large scales are excited. The dynamical hint for the direction of the cascade is the sign of the Lyapunov exponent for particles separation. Positive Lyapunov exponents are associated to direct cascade and Gaussianity at small scales. Negative Lyapunov exponents lead to inverse cascade, Gaussianity at large scales and strong intermittency at small scales.

chao-dyn↗
arXiv1996

Instanton for random advection

A path integral over trajectories of $2n$ fluid particles is identified with a $2n$-th order correlation function of a passive scalar convected by $d$-dimensional short-correlated multi-scale incompressible random velocity flow. Strong intermittency of the scalar is described by means of an instanton calculus (saddle point $+$ fluctuations about it) in the path integral at $n\gg d$. Anomalous scaling exponent of the $2n$-th scalar's structural function is found analytically.

chao-dyn↗
arXiv1996

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).

chao-dyn↗
arXiv1996

Anomalous Scaling Exponents of a White-Advected Passive Scalar

For Kraichnan's problem of passive scalar advection by a velocity field delta-correlated in time, the limit of large space dimensionality $d\gg1$ is considered. Scaling exponents of the scalar field are analytically found to be $ζ_{2n}=nζ_2-2(2-ζ_2)n(n-1)/d$, while those of the dissipation field are $μ_{n}=-2(2-ζ_2)n(n-1)/d$ for orders $n\ll d$. The refined similarity hypothesis $ζ_{2n}=nζ_2+μ_{n}$ is thus established by a straightforward calculation for the case considered.

chao-dyn↗
arXiv1995

The Fourth-Order Correlation Function of a Randomly Advected Passive Scalar

Advection of a passive scalar $θ$ in $d=2$ by a large-scale velocity field rapidly changing in time is considered. The Gaussian feature of the passive scalar statistics in the convective interval was discovered in \cite{95CFKLa}. Here we examine deviations from the Gaussianity: we obtain analytically the simultaneous fourth-order correlation function of $θ$. Explicit expressions for fourth-order objects, like $\langle(θ_1-θ_2)^4\rangle$ are derived.

chao-dyn↗
arXiv1995

Normal and Anomalous Scaling of the Fourth-Order Correlation Function of a Randomly Advected Passive Scalar

For a delta-correlated velocity field, simultaneous correlation functions of a passive scalar satisfy closed equations. We analyze the equation for the four-point function. To describe a solution completely, one has to solve the matching problems at the scale of the source and at the diffusion scale. We solve both the matching problems and thus find the dependence of the four-point correlation function on the diffusion and pumping scale for large space dimensionality $d$. It is shown that anomalous scaling appears in the first order of $1/d$ perturbation theory. Anomalous dimensions are found analytically both for the scalar field and for it's derivatives, in particular, for the dissipation field.

chao-dyn↗
arXiv1995

Statistics of a Passive Scalar Advected by a Large-Scale 2D Velocity Field: Analytic Solution

Steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale. That corresponds to the so-called Batchelor regime where the velocity is replaced by its large-scale gradient. The probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate. The description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant. We have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics. The proof is valid for any finite correlation time of the velocity field. Whatever the statistics of the velocity field, the statistics of the passive scalar (averaged over time locally in space) is shown to approach gaussianity with increase in the Peclet number $Pe$ (the pumping-to-diffusion scale ratio). The first $n<\ln Pe$ simultaneous correlation functions are expressed via the flux of the square of the scalar and only one factor depending on the velocity field: the mean stretching rate, which can be calculated analytically in limiting cases. Non-Gaussian tails of the probability distributions at finite $Pe$ are found to be exponential.

cond-mat↗
arXiv1994

Dynamics of N-spins Quantum Ferromagnet with Uniform Exchange: Exact Solution

The transverse spin autocorrelation function at an arbitrary temperature are calculated rigorously for the system of $N$ uniformly exchanged quantum spins. At large $N$ the correlator in the para-phase is found to have a Gaussian bump at small times and a non vanishing tail at large times. A possible application of those exact results as forming a starting dynamical mean-field approximation for long-range quantum magnets and an accompanied physical picture are discussed.

cond-mat↗