  We develop Ladyzhenskaya-Prodi-Serrin type spectral regularity criteria for
3D incompressible Navier-Stokes equations in a torus. Concretely, for any
$N>0$, let $w_N$ be the sum of all spectral components of the velocity fields
whose all three wave numbers are greater than $N$ absolutely. Then, we show
that for any $N>0$, the finiteness of the Serrin type norm of $w_N$ implies the
regularity of the flow. It implies that if the flow breaks down in a finite
time, the energy of the velocity fields cascades down to the arbitrarily large
spectral components of $w_N$ and corresponding energy spectrum, in some sense,
roughly decays slower than $\kappa^{-2}$
