  The very small scales of isotropic, Navier-Stokes turbulence at Reynolds
number ${\cal R}_\lambda \approx 15$ are studied by high-resolution direct
numerical simulation (DNS) and by integration of the direct-interaction (DIA)
equations. The DNS follows the tail of the energy spectrum over more than
thirty decades of magnitude. The energy spectrum in the far-dissipation range
$5k_d < k < 10k_d$ is well-fitted by $k^\alpha\exp(-ck/k_d)$, where $k_d$ is
the Kolmogorov dissipation wavenumber, $\alpha \approx 3.3$ and $c\approx 7.1$.
For values of $m$ that emphasize the far-dissipation range, the fields
$(-\nabla^2)^m{\bf u}$ exhibit strong spatial intermittency, associated with
gentle spatial variations of the lower-$k$ part of the velocity field. DIA
analysis gives a prefactor $k^3$ and an exponential decay more rapid than DNS.
Averaging over an ensemble of DIA solutions, suggested by the observed
intermittency, removes some of the discrepancy.

Chaotic Dynamics;Chaotic Dynamics