Turbulence, critical fluctuations, and intermittency

An explicit analogy is developed between the long-range fluctuations near a critical point and the long-range fluctuations in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ space in a turbulent fluid in the limit of large Reynolds number. Mathematically the analogy is between the probability functional of the spins on a lattice and the probability functional for the Fourier-transformed velocity fluctuations. The latter is given by the stationary solution of an exact Fokker-Planck equation originally derived by Edwards. This functional, which is not explicitly known, plays the role of a Hamiltonian for stationary turbulence. The limit of zero viscosity is equivalent to the temperature approaching the critical temperature from above. The order parameter is the vorticity-vorticity-correlation function in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ space, which in three dimensions is the same as the energy spectrum $E(k)$. The mechanism of energy dissipation constrains the critical exponent $\ensuremath{\gamma}$ to satisfy $\ensuremath{\gamma}=1$. The fluid is stirred at an external length scale ${k}_{0}^{\ensuremath{-}1}$. If effects due to the nonzero value of ${k}_{0}$ are ignored, the scaling properties of the full probability functional can be explicitly calculated, and lead to the original Kolmogorov theory. This can be characterized by the critical exponents $\ensuremath{\nu}=\frac{3}{4}$, $\ensuremath{\eta}=\frac{2}{3}$, indicating that the equivalent Hamiltonian is of long range. For ${k}_{0}\ensuremath{\ne}0$, the range of fluctuations in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$, space still diverges for zero viscosity. This limit no longer leads to a probability functional with the scale symmetry of a fixed point. We examine the cascade process under the assumption that $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$-space correlation functions go as some power of $r$. The original Kolmogorov theory is recovered by neglecting fluctuations in the cascade process. When these fluctuations are included, correlation functions of different order do not scale in the same way. This is in accord with phenomenological theories of intermittency and with experiment. An effective critical exponent ${\ensuremath{\eta}}^{\ensuremath{'}}=\frac{2}{3}+\ensuremath{\zeta}$ is still defined by the energy spectrum $E(k)$. The parameter $E(k)$ is one of a family of intermittency exponents which are intrinsic properties of the Navier-Stokes equations. Their calculation from the dynamical equations remains an objective for future work.