  We say that the solution u to the Navier-Stokes equations converges to a
solution v to the Euler equations in the vanishing viscosity limit if u
converges to v in the energy norm uniformly over a finite time interval.
Working specifically in the unit disk, we show that a necessary and sufficient
condition for the vanishing viscosity limit to hold is the vanishing with the
viscosity of the time-space average of the energy of u in a boundary layer of
width proportional to the viscosity due to modes (eigenfunctions of the Stokes
operator) whose frequencies in the radial or the tangential direction lie
between L and M. Here, L must be of order less than 1/(viscosity) and M must be
of order greater than 1/(viscosity).
