  We revisit the issue of whether thermal fluctuations are relevant for
incompressible fluid turbulence, and estimate the scale at which they become
important. As anticipated by Betchov in a prescient series of works more than
six decades ago, this scale is about equal to the Kolmogorov length, even
though that is several orders of magnitude above the mean free path. This
result implies that the deterministic version of the incompressible
Navier-Stokes equation is inadequate to describe the dissipation range of
turbulence in molecular fluids. Within this range, the fluctuating
hydrodynamics equation of Landau and Lifschitz is more appropriate. In
particular, our analysis implies that both the exponentially decaying energy
spectrum and the far-dissipation range intermittency predicted by Kraichnan for
deterministic Navier-Stokes will be generally replaced by Gaussian thermal
equipartition at scales just below the Kolmogorov length. Stochastic shell
model simulations at high Reynolds numbers verify our theoretical predictions
and reveal furthermore that inertial-range intermittency can propagate deep
into the dissipation range, leading to large fluctuations in the equipartition
length scale. We explain the failure of previous scaling arguments for the
validity of deterministic Navier-Stokes equations at any Reynolds number and we
provide a mathematical interpretation and physical justification of the
fluctuating Navier-Stokes equation as an ``effective field-theory'' valid below
some high-wavenumber cutoff $\Lambda$, rather than as a continuum stochastic
partial differential equation. At Reynolds number around a million the
strongest turbulent excitations observed in our simulation penetrate down to a
length-scale of microns. However, for longer observation times or higher
Reynolds numbers, more extreme turbulent events could lead to a local breakdown
of fluctuating hydrodynamics.
