Viscous Instanton for Burgers' Turbulence

We consider the tails of probability density functions (PDF) for different characteristics of velocity that satisfies Burgers equation driven by a large-scale force. The saddle-point approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special field-force configuration (instanton) that realizes the extremum of probability. We calculate high moments of the velocity gradient ${\mathrm{\partial }}_{x}u$ and find out that they correspond to the PDF with $\ln[{\cal P}(\partial_xu)]\propto-(-\partial_xu/{\rm Re})^{3/2}$where${\rm Re}$ is the Reynolds number. That stretched exponential form is valid for negative ${\mathrm{\partial }}_{x}u$ with the modulus much larger than its root-mean-square (rms) value. The respective tail of PDF for negative velocity differences $w$ is steeper than Gaussian, $\ln \mathcal{P}(w)\sim -(w\mathrm{/}{u}_{\mathrm{r}\mathrm{m}\mathrm{s}}{)}^3$, as well as single-point velocity PDF $\ln \mathcal{P}(u)\sim -(\mathrm{\mid }u\mathrm{\mid }\mathrm{/}{u}_{\mathrm{r}\mathrm{m}\mathrm{s}}{)}^3$. For high velocity derivatives $u^{(k)}={\mathrm{\partial }}_x^{k}u$, the general formula is found: $\ln \mathcal{P}(\mathrm{\mid }{u}^{(k)}\mathrm{\mid })\propto -(\mathrm{\mid }{u}^{(k)}\mathrm{\mid }\mathrm{/}{\mathrm{R}\mathrm{e}}^k{)}^{3\mathrm{/}(k+1)}$.