Viscous Instanton for Burgers' Turbulence

E. Balkovsky; G. Falkovich; I. Kolokolov; V. Lebedev

doi:10.1103/PhysRevLett.78.1452

paper

Viscous Instanton for Burgers' Turbulence

arXiv:chao-dyn/9603015 · doi:10.1103/PhysRevLett.78.1452

Abstract

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 $\partial_{x} u$ and find out that they correspond to the PDF with $ln [P (\partial_{x} u)] \propto - (- \partial_{x} u / Re)^{3/2}$ where $Re$ is the Reynolds number. That stretched exponential form is valid for negative $\partial_{x} u$ 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, $ln P (w) \sim - (w / u_{rms})^{3}$ , as well as single-point velocity PDF $ln P (u) \sim - (∣ u ∣/ u_{rms})^{3}$ . For high velocity derivatives $u^{(k)} = \partial_{x}^{k} u$ , the general formula is found: $ln P (∣ u^{(k)} ∣) \propto - (∣ u^{(k)} ∣/ Re^{k})^{3/ (k + 1)}$ .

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Viscous Instanton for Burgers' Turbulence

Abstract

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