Forced Burgers Turbulence in 3-Dimensions

Abstract

We investigate non-perturbative results of inviscid forced Burgers equation supplemented to continuity equation in three-dimensions. The exact two-point correlation function of density is calculated in three-dimensions. The two-point correlator $<ρ(\bf x_1) ρ(\bf x_2)>$ behaves as $ |{\bf {x_1 - x_2}}|^{-α_3}$ and in the universal region $α_3 = 7/2$ while in the non-universal region $α_3 = 3$. In the non-universal region we drive a Kramers-Moyal equation governing the evolution of the probability density function (PDF) of longitudinal velocity increments for three dimensional Burgers turbulence. In this region we prove Yakhot's conjecture {[Phys. Rev. E {\bf 57}, 1737 (1998)]} for the equation of PDF for three dimensional Burgers turbulence. We also derive the intermittency exponents for the longitudinal structure functions and show that in the inertial regime one point $U_{rms}$ enters in the PDF of velocity difference.