Universality classes for self-similarity of noiseless multi-dimensional Burgers turbulence and interface growth

The present work is devoted to the evolution of random solutions of the unforced Burgers and KPZ equations in d-dimensions in the limit of vanishing viscosity. We consider a cellular model and as initial condition assign a value for the velocity potential chosen independently within each cell. We show that the asymptotic behavior of the turbulence at large times is determined by the tail of the initial potential probability distribution function. Three classes of initial distribution leading to self-similar evolution are identified: (a) distributions with a power-law tail, (b) compactly supported potential, (c) stretched exponential tails. In class (c) we find that the mean potential (mean height of the surface) increases logarithmically with time and the 'turbulence energy' E(t) (mean square gradient of the surface) decays as 1/t times a logarithmic correction. In classes (a) and (b) we find that the changes in the mean potential and energy have a power-law time dependence. In class (c) the roughness of the surface, measured by its mean--square gradient, may either decrease or increase with time. We discuss also the influence of finite viscosity and long range correlation on the late stage evolution of the Burgers turbulenceComment: 21 pages, no figures, LaTeX, submitted to Phys. Rev.

The global picture of self-similar and not self-similar decay in Burgers Turbulence

This paper continue earlier investigations on the decay of Burgers turbulence in one dimension from Gaussian random initial conditions of the power-law spectral type $E_0(k)\sim|k|^n$. Depending on the power $n$, different characteristic regions are distinguished. The main focus of this paper is to delineate the regions in wave-number $k$ and time $t$ in which self-similarity can (and cannot) be observed, taking into account small-$k$ and large-$k$ cutoffs. The evolution of the spectrum can be inferred using physical arguments describing the competition between the initial spectrum and the new frequencies generated by the dynamics. For large wavenumbers, we always have $k^{-2}$ region, associated to the shocks. When $n$ is less than one, the large-scale part of the spectrum is preserved in time and the global evolution is self-similar, so that scaling arguments perfectly predict the behavior in time of the energy and of the integral scale. If $n$ is larger than two, the spectrum tends for long times to a universal scaling form independent of the initial conditions, with universal behavior $k^2$ at small wavenumbers. In the interval $2<n$ the leading behaviour is self-similar, independent of $n$ and with universal behavior $k^2$ at small wavenumber. When $1<n<2$, the spectrum has three scaling regions : first, a $|k|^n$ region at very small $k$\ms1 with a time-independent constant, second, a $k^2$ region at intermediate wavenumbers, finally, the usual $k^{-2}$ region. In the remaining interval, $n<-3$ the small-$k$ cutoff dominates, and $n$ also plays no role. We find also (numerically) the subleading term $\sim k^2$ in the evolution of the spectrum in the interval $-3<n<1$. High-resolution numerical simulations have been performed confirming both scaling predictions and analytical asymptotic theory.Comment: 14 pages, 19 figure

On the decay of Burgers turbulence

This work is devoted to the decay ofrandom solutions of the unforced Burgers equation in one dimension in the limit of vanishing viscosity. The initial velocity is homogeneous and Gaussian with a spectrum proportional to $k^n$ at small wavenumbers $k$ and falling off quickly at large wavenumbers. In physical space, at sufficiently large distances, there is an ``outer region'', where the velocity correlation function preserves exactly its initial form (a power law) when $n$ is not an even integer. When $1<n<2$ the spectrum, at long times, has three scaling regions : first, a $|k|^n$ region at very small $k$\ms1 with a time-independent constant, stemming from this outer region, in which the initial conditions are essentially frozen; second, a $k^2$ region at intermediate wavenumbers, related to a self-similarly evolving ``inner region'' in physical space and, finally, the usual $k^{-2}$ region, associated to the shocks. The switching from the $|k|^n$ to the $k^2$ region occurs around a wave number $k_s(t) \propto t^{-1/[2(2-n)]}$, while the switching from $k^2$ to $k^{-2}$ occurs around $k_L(t)\propto t^{-1/2}$ (ignoring logarithmic corrections in both instances). The key element in the derivation of the results is an extension of the Kida (1979) log-corrected $1/t$ law for the energy decay when $n=2$ to the case of arbitrary integer or non-integer $n>1$. A systematic derivation is given in which both the leading term and estimates of higher order corrections can be obtained. High-resolution numerical simulations are presented which support our findings.Comment: In LaTeX with 11 PostScript figures. 56 pages. One figure contributed by Alain Noullez (Observatoire de Nice, France

Instanton Theory of Burgers Shocks and Intermittency

A lagrangian approach to Burgers turbulence is carried out along the lines of the field theoretical Martin-Siggia-Rose formalism of stochastic hydrodynamics. We derive, from an analysis based on the hypothesis of unbroken galilean invariance, the asymptotic form of the probability distribution function of negative velocity-differences. The origin of Burgers intermittency is found to rely on the dynamical coupling between shocks, identified to instantons, and non-coherent background fluctuations, which, then, cannot be discarded in a consistent statistical description of the flow.Comment: 7 pages; LaTe

Is the cosmic UV background fluctuating at redshift z ~ 6 ?

We study the Gunn-Peterson effect of the photo-ionized intergalactic medium(IGM) in the redshift range 5< z <6.4 using semi-analytic simulations based on the lognormal model. Assuming a rapidly evolved and spatially uniform ionizing background, the simulation can produce all the observed abnormal statistical features near redshift z ~ 6. They include: 1) rapidly increase of absorption depths; 2) large scatter in the optical depths; 3) long-tailed distributions of transmitted flux and 4) long dark gaps in spectra. These abnormal features are mainly due to rare events, which correspond to the long-tailed probability distribution of the IGM density field, and therefore, they may not imply significantly spatial fluctuations in the UV ionizing background at z ~ 6.Comment: 12 pages, 4 figs, accepted by ApJ

Statistical properties of the Burgers equation with Brownian initial velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its $n$-point correlations. In the same limit, we derive the $n-$point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.Comment: 42 pages, published in J. Stat. Phy

Models of Passive and Reactive Tracer Motion: an Application of Ito Calculus

By means of Ito calculus it is possible to find, in a straight-forward way, the analytical solution to some equations related to the passive tracer transport problem in a velocity field that obeys the multidimensional Burgers equation and to a simple model of reactive tracer motion.Comment: revised version 7 pages, Latex, to appear as a letter to J. of Physics

Spatiotemporal complexity of the universe at subhorizon scales

This is a short note on the spatiotemporal complexity of the dynamical state(s) of the universe at subhorizon scales (up to 300 Mpc). There are reasons, based mainly on infrared radiative divergences, to believe that one can encounter a flicker noise in the time domain, while in the space domain, the scaling laws are reflected in the (multi)fractal distribution of galaxies and their clusters. There exist recent suggestions on a unifying treatment of these two aspects within the concept of spatiotemporal complexity of dynamical systems driven out of equilibrium. Spatiotemporal complexity of the subhorizon dynamical state(s) of the universe is a conceptually nice idea and may lead to progress in our understanding of the material structures at large scalesComment: references update