Functional integrals in Navier–Stokes incompressible fluid turbulence

Abstract

A variational principle is formulated for the dynamical evolution of the Hopf characteristic functional Φ=Φ[y(x),t] by employing an appropriate functional integral over all parameter fields y(x). It follows that the ratio of functional integrals Γ≡∫Φ*ΦD(y)/∫‖Φ‖2D(y) is an exact constant of the motion during the decay of boundary‐free Navier–Stokes incompressible fluid turbulence. Bearing the physical dimensions of inverse time, the constant of the motion Γ is a scalar function of the multipoint velocity correlation tensors embodied in Φ. For statistical situations such that the probability measure over the velocity‐field ensemble is semi‐Gaussian (i.e., the real part of ln Φ is a quadratic functional of y), Γ is evaluated explicitly in terms of the two‐point velocity correlation tensor.