  In this paper, we will prove a new, scale critical regularity criterion for
solutions of the Navier--Stokes equation that are sufficiently close to being
eigenfunctions of the Laplacian. This estimate improves previous regularity
criteria requiring control on the $\dot{H}^\alpha$ norm of $u,$ with $2\leq
\alpha<\frac{5}{2},$ to a regularity criterion requiring control on the
$\dot{H}^\alpha$ norm multiplied by the deficit in the interpolation inequality
for the embedding of $\dot{H}^{\alpha-2}\cap\dot{H}^{\alpha} \hookrightarrow
\dot{H}^{\alpha-1}.$ This regularity criterion suggests, at least
heuristically, the possibility of some relationship between potential blowup
solutions of the Navier--Stokes equation and the Kolmogorov-Obhukov spectrum in
the theory of turbulence.
