Universal Statistics of Inviscid Burgers Turbulence in Arbitrary Dimensions

We investigate the non-perturbative results of multi-dimensional forced Burgers equation coupled to the continuity equation. In the inviscid limit, we derive the exact exponents of two-point density correlation functions in the universal region in arbitrary dimensions. We then find the universal generating function and the tails of the probability density function (PDF) for the longitudinal velocity difference. Our results exhibit that in the inviscid limit, density fluctuations affect the master equation of the generating function in such a way that we can get a positive PDF with the well-known exponential tail. The exponent of the algebraic tail is derived to be −5/2 in any dimension. Finally we observe that various forcing spectrums do not alter the power law behaviour of the algebraic tail in these dimensions, due to a relation between forcing correlator exponent and the exponent of the two-point density correlation function. PACS numbers 47.27.Gs and 47.40.Ki The interest in solving the randomly driven Burgers equation with a large scale driving force is motivated by the hope that it can provide us with the first solvable model of turbulence. Consequently tremendous activity emerged on the nonperturbative understanding of Burgers turbulence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. As an original attempt, Polyakov offered a field theoretical method to derive the probability distribution or density function (PDF) of velocity difference in the problem of randomly driven Burgers equation in one dimension [1]. The problem of computation of correlation functions in the inertial range is reduced to the solution of a closed partial differential equation [1, 2]. However, Polyakov’s approach was based on the conjecture of the existence of the operator product expansion (OPE). This method was then extended to the forced Burgers equation coupled to the continuity equation and some results were derived for the behaviour of the probability density function tails and for the value of intermittency and density-density correlators exponents [3, 12]. On the other hand, some extensive numerical simulations show that the