Algebraic Tails of Probability Density Functions in the Random-Force-Driven Burgers Turbulence

The dynamics of velocity fluctuations governed by the Burgers equation, driven by the white-in-time random forcing function with $\overline{\left[f\right(x+r,t)-f({x,t}^{\prime }){]}^2}\propto r^{\xi }\delta \left({t-t}^{\prime }\right)$ is considered on the interval $0<x<L$. The properties of the probability density function of velocity differences $P(\Delta u,r)$ are investigated for the three cases $\xi =\{0;1/2;2\}$. It is shown that the tail of the probability density function in the interval $\Delta u/r^z\ll -1\mathrm{}$; $|\Delta u|\ll u_{\mathrm{rms}}$ and $r\ll L$ is accurately described by the asymptotic algebraic relation $P(\Delta u,r)\propto r/(\Delta u{)}^{\gamma }$ with $\gamma =1+1\mathrm{}/z$, where $z=(\xi +1\mathrm{})/3$. A detailed numerical investigation, performed in this work, supports this result.