Energy Decay in Burgers Turbulence and Interface Growth. the Problem of Random Initial Conditions - II

Abstract

We present a study of the Burgers equation in one and two dimensions $d=1,2$ following the analytic approach indicated in the previous paper I. For the problem of the initial conditions decay we consider two classes of initial condition distributions $Q_{1,2}\sim \exp [-(1/4D)\int (\nabla h{)}^2$d{\bf x}] where $h$-field is unbounded ($Q_1$) or bounded ($Q_2,|h|\le H$). Avoiding the replica trick and using an integral representation of the logarithm we study the tractable field theory which has $d=2$ as a critical dimension. It is shown that the degenerate one-dimensional case has three stages of decay, when the kinetic energy density diminishes with time as $t^{-2/3}$, $t^{-2}$, $t^{-3/2}$ contrary to the predictions of the similarity hypothesis based on the second-order correlator of the distribution. In two dimensions we find the kinetic energy decay which is proportional to $t^{-1}{\ln }^{-1/2}(t)$. It is shown that the pure diffusion equation with the $Q_2$-type initial condition has non-trivial energy decay exponents indicating connection with the $O(2)$ non-linear $\sigma$-model.

39 pages, PHYZZX Tex, figures by request

Cited by in corpus (10)

• SCALING AND INTERMITTENCY IN BURGERS' TURBULENCE
• Exact solutions for the statistics of extrema of some random 1D landscapes, Application to the equilibrium and the dynamics of the toy model
• Canonical phase space approach to the noisy Burgers equation: Probability distributions
• Velocity fluctuations in forced Burgers turbulence
• Energy decay in three-dimensional freely cooling granular gas
• Solitons and diffusive modes in the noiseless Burgers equation: Stability analysis
• Universality classes for self-similarity of noiseless multi-dimensional Burgers turbulence and interface growth
• Dynamical Scaling in Dissipative Burgers Turbulence
• Low-dimensional modelling of a generalized Burgers equation
• The decay of plane wave pulses with complex structure in a nonlinear dissipative medium