Random Advection Research Articles

An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Peclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Peclet), to two algebraic singularities at both unit and zero scalar values (large Peclet).

Advection-mediated coexistence of competing species

We study a Lotka–Volterra reaction–diffusion–advection model for two competing species in a heterogeneous environment. The species are assumed to be identical except for their dispersal strategies: one disperses by random diffusion only, the other by both random diffusion and advection along an environmental gradient. When the two competitors have the same diffusion rates and the strength of the advection is relatively weak in comparison to that of the random dispersal, we show that the competitor that moves towards more favourable environments has the competitive advantage, provided that the underlying spatial domain is convex, and the competitive advantage can be reversed for certain non-convex habitats. When the advection is strong relative to the dispersal, we show that both species can invade when they are rare, and the two competitors can coexist stably. The biological explanation is that, for sufficiently strong advection, the ‘smarter' competitor will move towards more favourable environments and is concentrated at the place with maximum resources. This leaves enough room for the other species to survive, since it can live upon regions with finer quality resources.

Random advection in a fractal medium with finite correlation length

We investigate tracer advection in fractal media with finite correlation length. The process is superdiffusive at early times and transforms to a classical diffusion regime later. At large time the spatial tail of the concentration has a two-stage structure with a long-distant part corresponding to superdiffusive regime.

An idealised model of turbulent dispersion: one rectangular pulse initial condition

AbstractWe examine an idealised model of turbulent dispersion which was introduced by Zimmerman and Chatwin. It involves deterministic diffusion in a periodic one‐dimensional domain, for a specified initial concentration field. A stochastic element is introduced by sampling at random across the domain, equivalent to random advection by a velocity which does not vary spatially. We consider initial conditions consisting of a single rectangular pulse, displaced a distance X0 from the centre of the domain. We present numerical results for the dependence of the variance, skewness, kurtosis and probability density function (pdf) of concentration on X0 and on time, and we derive the large time asymptotic forms for these quantities. At large time, the pdf is inevitably bimodal, with the peaks at the smallest and largest concentrations. To allow for a range of possible distances between strands of high concentration, as would be expected in a real turbulent flow, we suggest a new model pdf for concentration, obtained by averaging over different values of X0. For a uniform distribution of X0 values, we show that this gives a unimodal concentration pdf at large time, with the peak at the mean concentration. Copyright © 2007 John Wiley & Sons, Ltd.

Dynamics of spatial Fourier modes in turbulence

We report an experimental investigation of the statistical time dynamic of spatial Fourier modes in a fully developped turbulent jet flow. Measurements rely on an original acoustic scattering technique, allowing the direct and continuous probing in a time of spatial Fourier modes of the turbulent vorticity field at well defined spatial wavevectors. From the recordings of the amplitude of the spatial Fourier modes corresponding to different wavevectors in the inertial range, we compute the time correlation functions of the modal amplitude. The modal vorticity dynamics exhibits two well separated time scales: a short one which is scale dependant and related to the well known sweeping effect (random advection), and a much longer one, of order the integral time scale, which is not scale dependent. By computing the cross-correlation of the amplitude of two different wavevectors, we evidence a clear statistical dependance between scales, whatever their separation, at variance with the original Kolmogorov 41 theory. We ascribe our experimental results to a manifestation of a new type of statistical intermittency. We call this new statistical feature a large scale time intermittency effect, in contrast with the usual small scale spatial intermittency reported in Eulerian velocity measurements. We propose to attribute this time intermittency, to the fluctuations in time of the injected energy at large scale, in the spirit of the objection formulated by Landau against the K41 theory.

Clustering of randomly advected low-inertia particles: A solvable model

Measurements and simulations indicate that the particle-pair radial distribution function in isotropic turbulence is a power law in a range of length scales below the Kolmogorov scale for Stokes number St<<1. In this range, the exponent is proportional to St1St2 for unlike particles (1 and 2) in a bidispersion, hence St2 for a monodispersion. Here, this result is derived from a model of particle response to random advection. The analysis generalizes a geometrical interpretation of clustering to polydispersions and suggests an economical Monte Carlo simulation method.

Anomalous diffusion in a self-similar random advection field

The problem on passive scalar advection in random (statistically homogeneous) self-similar media is solved. Even in superdiffusive mode the solution is found not to possess "heavy" power tails. Instead, they are exponential (not Gaussian). The derivation is conducted by means of the scaling analysis.

Composition-Space Behavior of Diffusion-Reaction Systems

Species concentrations in homogeneous well-stirred reacting (WSR) systems with multiple time scales mostly access only a small portion (slow manifold) of the permitted composition space. This offers a mathematically sound basis for reducing complex kinetics. Computational investigation of such slow manifolds in inhomogeneous reaction systems undergoing diffusion in addition to chemical reaction is the subject of this paper. We perform simulations of a one-dimensional diffusion–reaction system from random non-premixed initial conditions to study the accessed composition space for a variety of diffusion to reaction time scale ratios. A detailed hydrogen–oxygen chemical kinetic set (6 species, 14 reactions) is used. Our results show that, to first order, the effect of random advection and diffusion is merely to increase the accessed state-space dimensionality over that of the WSR by the number of relevant mixture fractions. Circumstances under which the above simplification is not accurate are also identified.

Natural variability of atmospheric temperatures and geomagnetic intensity over a wide range of time scales.

The majority of numerical models in climatology and geomagnetism rely on deterministic finite-difference techniques and attempt to include as many empirical constraints on the many processes and boundary conditions applicable to their very complex systems. Despite their sophistication, many of these models are unable to reproduce basic aspects of climatic or geomagnetic dynamics. We show that a simple stochastic model, which treats the flux of heat energy in the atmosphere by convective instabilities with random advection and diffusive mixing, does a remarkable job at matching the observed power spectrum of historical and proxy records for atmospheric temperatures from time scales of one day to one million years (Myr). With this approach distinct changes in the power-spectral form can be associated with characteristic time scales of ocean mixing and radiative damping. Similarly, a simple model of the diffusion of magnetic intensity in Earth's core coupled with amplification and destruction of the local intensity can reproduce the observed 1/f noise behavior of Earth's geomagnetic intensity from time scales of 1 (Myr) to 100 yr. In addition, the statistics of the fluctuations in the polarity reversal rate from time scales of 1 Myr to 100 Myr are consistent with the hypothesis that reversals are the result of variations in 1/f noise geomagnetic intensity above a certain threshold, suggesting that reversals may be associated with internal fluctuations rather than changes in mantle thermal or magnetic boundary conditions.

On the FKPP equation with Gaussian shear advection

A boundary value problem for the Fisher–Kolmogorov–Petrovskii–Piskunov (FKPP) equation with shear random advection is investigated. It is demonstrated that the upper bounds for front propagation in the FKPP equation with Gaussian advection obtained by an analysis of its linear ensemble-averaged upper solution can give too rough predictions. It is shown that the unboundedness of the Gaussian advection affects ensemble-averaged solutions of linear and nonlinear problems in a different way. This analytical prediction is confirmed by direct numerical simulations. The phenomenon of propagation failure due to zero boundary conditions is studied and the critical conditions are found. Numerical procedure for the analysis of random wave propagation is suggested by introduction of mean front position and its variance. Some numerical experiments are presented.

Interactions of disparate scales in drift-wave turbulence

Renormalized statistical theory is used to calculate the interactions between short scales (wave vector k) and long scales (wave vector q<<k) in the Hasegawa-Mima model of drift-wave turbulence (generalized to include proper nonadiabatic response for k( ||)=0 fluctuations). The calculations include the zonal-flow growth rate as a special case, but also describe long-wavelength fluctuations with q oriented at an arbitrary angle to the background gradient. The results are fully renormalized. They are subtly different from those of previous authors, in both mathematical form and physical interpretation. A term arising in previous treatments that is related to the propagation of short-scale wave packets is shown to be a higher-order effect that must consistently be neglected to lowest order in a systematic expansion in q/k. Rigorous functional methods are used to show that the long-wavelength growth rate gamma(q) is related to second-order functional variations of the short-wavelength energy and to derive a heuristic algorithm. The principal results are recovered from simple estimates involving the first-order wave-number distension rate gamma;((1))(k)equals, single dot abovek small middle dotnablaOmega;(k)/k(2), where Omega;(k) is a nonlinear random advection frequency. Fokker-Planck analysis involving gamma;((1))(k) is used to heuristically recover the evolution equation for the small scales, and a random-walk flux argument that relates gamma;((1))(k) to an effective autocorrelation time is used to give an independent calculation of gamma(q). Both the rigorous and heuristic derivations demonstrate that the results do not depend on, and cannot be derived from, properties of linear normal modes; they are intrinsically nonlinear. The importance of random-Galilean-invariant renormalization is stressed.

Geometric properties of passive random advection

Geometric properties of a random Gaussian short-time correlated velocity field are studied by considering the statistics of a passively advected metric tensor. That describes the universal properties of the fluctuations of tensor objects frozen into the fluid and passively advected by it. The problem of the one-point statistics of covariant and contravariant tensors is solved exactly, provided that the advected fields do not reach diffusive scales, which would break the symmetry of the problem.

A closure method for random advection of a passive scalar

A novel functional method is applied to calculate the statistics of a passive scalar in an isotropic turbulent velocity field. The method yields asymptotic series expansions for small velocity correlation time from which approximate closure equations are derived. The closure method admits a diagram expansion, and is implemented as a Mathematica program. Padé summation of the asymptotic series yields accurate values for the effective diffusivity and gives formulas expressing the Lagrangian correlation of the velocity in terms of the Eulerian correlation. The approximations compare very favorably with numerical simulations of advection by a Gaussian velocity field.

Spatially correlated and inhomogeneous random advection

We study the equilibrium statistics of an idealized model of random advection. Theoretical results for large-scale homogeneous forcing—notably the shapes of the PDFs—are recovered; they are compared with results for various inhomogeneous and intermittent forcings. The correlation times of the forcings are varied. The PDFs are interpreted in terms of the underlying mixing time scales. Conditional statistics of the tracer field—the conditional diffusion in particular—are computed; they are used to elucidate the influence of small-scale tracer–field correlations. We contrast results for spatially smooth and spatially rough velocity fields.

Shell model for time-correlated random advection of passive scalars.

We study a minimal shell model for the advection of a passive scalar by a Gaussian time-correlated velocity field. The anomalous scaling properties of the white noise limit are studied analytically. The effect of the time correlations are investigated using perturbation theory around the white noise limit and nonperturbatively by numerical integration. The time correlation of the velocity field is seen to enhance the intermittency of the passive scalar.

Effects of a large-scale mean circulating flow on passive scalar statistics in a model of random advection

We study the effects of a large-scale mean circulating flow on passive scalar statistics in turbulent advection using a two-dimensional lattice model. The incompressible advecting velocity field consists of a large-scale circulation plus fluctuations. The latter are modeled by a random Gaussian field that has a finite correlation time but is statistically independent at different lattice points. In the presence of the large-scale flow, we find that the profiles of both the mean and the rms fluctuation of the passive scalar are modified significantly, and have close resemblance to those observed experimentally in turbulent convection. The one-point probability density functions (PDFs) of the passive scalar at the two sides are affected by the large-scale flow and become more skewed. There is, however, not much change for the PDFs at other locations. Furthermore, the shape of the PDFs of the scalar increments remains the same with or without the presence of the large-scale flow, supporting the idea that large-scale effects can be filtered out by studying increments.

Structure of the three-point correlation function of a passive scalar in the presence of a mean gradient

The three-point correlation function of a passive scalar advected by a random, incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological Hopf equations. Numerical solutions are provided in the case where the velocity is Gaussian, white in time, and with a power law in space, $〈[v(\stackrel{\ensuremath{\rightarrow}}{r})\ensuremath{-}v(\stackrel{\ensuremath{\rightarrow}}{0}){]}^{2}〉\ensuremath{\sim}{r}^{2\ensuremath{-}\ensuremath{\epsilon}},$ and in the model introduced by Shraiman and Siggia [C. R. Acad. Sci. 321, 279 (1995)]. Anomalous scaling exponents are found in both models. The numerics agrees with all the available analytic, perturbative results. In addition, the angular dependence of the correlation function is explicitly determined. In the Batchelor limit of random advection by a smooth velocity field, the exponent is found to remain very close to 1, as found experimentally. In this limit, the three-point correlation function is found to be very well represented, away from collinearity, by an explicit integral representation.

Inverse cascade and intermittency of passive scalar in one-dimensional smooth flow

Random advection of a Lagrangian tracer scalar field $\ensuremath{\theta}(t,x)$ 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 Gaussian statistics known to be present at small scales for incompressible velocity fields emerges here at large scales $(x\ensuremath{\gg}L).$ These scales are shown to be excited by an inverse cascade of ${\ensuremath{\theta}}^{2}$ and the probability distribution function (PDF) of the corresponding scalar differences to approach the Gaussian form, as larger and larger scales are considered. Small-scale $(x\ensuremath{\ll}L)$ statistics is shown to be strongly non-Gaussian. A collapse of scaling exponents for scalar structure functions takes place: Moments of order $p&gt;~1$ all scale linearly, independently of the order $p.$ Smooth scaling ${x}^{p}$ is found for $\ensuremath{-}1&lt;p&lt;1.$ Tails of the scalar difference 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 the scalar gradient PDF with respect to the inverse of the P\'eclet 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.

Passive scalar conditional statistics in a model of random advection

We study numerically a model of random advection of a passive scalar by an incompressible velocity field of different prescribed statistics. Our focus is on the conditional statistics of the passive scalar and specifically on two conditional averages: the averages of the time derivative squared and the second time derivative of the scalar when its fluctuation is at a given value. We find that these two conditional averages can be quite well approximated by polynomials whose coefficients can be expressed in terms of scalar moments and correlations of the scalar with its time derivatives. With the fitted polynomials for the conditional averages, analytical forms for the probability density function (pdf) of the scalar are obtained. The variation of the coefficients with the parameters of the model result in a change in the pdf. Three different kinds of velocity statistics, (i) Gaussian, (ii) exponential, and (iii) triangular, are studied, and the same qualitative results are found demonstrating that the one-point statistics of the velocity field do not affect the statistical properties of the passive scalar.

Instanton for random advection

A path integral over trajectories of 2n fluid particles is identified with a 2nth order correlation function of a passive scalar convected by d-dimensional short-correlated multiscale incompressible random velocity flow. Strong intermittency of the scalar is described by means of an instanton calculus (saddle point plus fluctuations about it) in the path integral at n\ensuremath{\gg}:d. The anomalous scaling exponent of the 2nth scalar's structural function is found analytically.