Velocity correlations in a randomly stirred fluid: A variational principle for path-integral functionals

A variational principle is applied to the generating functional of velocity correlations in an incompressible fluid described by the Navier-Stokes equations with random Gaussian stirring forces. The generating functional of velocity correlations gives, in a field-theoretic language, a complete statistical description of this driven, stationary Markov process in terms of path integrals and can be viewed as a generalized free energy. The statistical properties of velocity fluctuations generated by white-noise stirring forces whose energy-injection rate into the fluid is wave-number power-law distributed are investigated. A boundary dimension $d^{*}$ depending upon the forcing spectrum is found above which long-wavelength velocity fluctuations are weakly coupled. In the strong-coupling regime $d<\mathrm{}{d}^{*}$, static and dynamic exponents characterizing the wave-number dependence of long-wavelength fluctuations are evaluated. For $d>\mathrm{}{d}^{*}$, corrections to the linear theory caused by the nonlinear mode-coupling terms are determined. The results are compared with renormalization-group calculations. Turbulent velocity fluctuations are discussed in the limit of vanishing viscosity. The relationship between forcing and energy spectrum in $d$ dimensions is investigated and the results are compared with second-order closure approximations of statistical turbulence theories. A stirring-force spectrum $\sim k^{-d}$ leads to a Kolmogorov distribution of energy over wave number.