Long-Time Tails and the Large-Eddy Behavior of a Randomly Stirred Fluid

The long-wavelength and low-frequency behavior of an incompressible, randomly stirred fluid is determined in $d$ dimensions by renormalization-group arguments. A free-field fixed point, which describes conventional hydrodynamics, is stable for $d>~2$ although we find nontrivial corrections to the leading behavior. These corrections give rise to long-time tails in a fluid near thermal equilibrium. A nontrivial fixed point controls the behavior for $d<\mathrm{}2\mathrm{}$, which is determined to all orders in $\epsilon =2-d$.