Instantons in the theory of turbulence

Abstract

We describe a method for finding the non-Gaussian tails of probability distribution function (PDF) for solutions of stochastic differential equations, such as convection equation for a passive scalar, random driven Navier-Stokes equation etc. Existence of such tails is generally regarded as a manifestation of intermittency phenomenon. Our formalism is based on the WKB approximation in the functional integral for the conditional probability of a large fluctuation. Then the main contribution to the functional integral is given by a coupled field-force configuration — instanton. We argue that the tails of the single-point velocity probability distribution function (PDF) are generally non-Gaussian in developed turbulence. By using instanton formalism for the Navier-Stokes equation, we establish the relation between the PDF tails of the velocity and those of the external forcing. In particular, we show that a Gaussian random force having correlation scale L and correlation time τ produces velocity PDF tails ln $P(\upsilon )\alpha - {\upsilon ^4}at\upsilon \gg {\upsilon _{rms,}}L/\tau$. For a short-correlated forcing when $\tau \ll L/{\upsilon _{rms}}$ there is an intermediate asymptotics ln $P(\upsilon )\alpha - {\upsilon ^3}atL/\tau \gg \upsilon \gg {\upsilon _{rms}}$. We consider the tails of probability density function (PDF) for the velocity that satisfies Burgers equation driven by a Gaussian large-scale force. The instantonic calculations show that for the PDFs of velocity and its derivatives ${u^{(k)}} = \partial _x^ku$, the general formula is found: ln $P(|{u^{(k)}}|)\alpha - {(|{u^{(k)}}|/{{\mathop{\rm Re}\nolimits} ^k})^{3/(k + 1)}}$. We consider high-order correlation functions of the passive scalar in the Kraichnan model. Using the instanton formalism we find the exponents ζn of the structure functions S _n for $n \gg$ 1 at the condition $d{\zeta _2} \gg 1$ (where d is the dimensionality of space). At $n < {n_c}$ (where ${n_c} = d{\zeta _{_2}}/\left[ {2\left( {2 - {\zeta _2}} \right)} \right]$) the exponents are ${\zeta _n} = \left( {{\zeta _2}/4} \right)\left( {{2_n} - {n^2}/{n_c}} \right)$, while at n > n _c they are n-independent: ${\zeta _n} = {\zeta _2}{n_c}/4$. Besides ζ _n , we also estimate n-dependent factors in S _n and critical behavior of S _n at n close to n _c