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.

Ballistic aggregation for one-sided Brownian initial velocity

We study the one-dimensional ballistic aggregation process in the continuum limit for one-sided Brownian initial velocity (i.e. particles merge when they collide and move freely between collisions, and in the continuum limit the initial velocity on the right side is a Brownian motion that starts from the origin $x=0$). We consider the cases where the left side is either at rest or empty at $t=0$. We derive explicit expressions for the velocity distribution and the mean density and current profiles built by this out-of-equilibrium system. We find that on the right side the mean density remains constant whereas the mean current is uniform and grows linearly with time. All quantities show an exponential decay on the far left. We also obtain the properties of the leftmost cluster that travels towards the left. We find that in both cases relevant lengths and masses scale as $t^2$ and the evolution is self-similar.Comment: 18 pages, published in Physica

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 dynamics of a self-gravitating medium with random and non-random initial conditions

The dynamics of a one-dimensional self-gravitating medium, with initial density almost uniform is studied. Numerical experiments are performed with ordered and with Gaussian random initial conditions. The phase space portraits are shown to be qualitatively similar to shock waves, in particular with initial conditions of Brownian type. The PDF of the mass distribution is investigated.Comment: Latex, figures in eps, 23 pages, 11 figures. Revised versio

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