  Turbulence governed by the Navier-Stokes equations shows a tendency to evolve
towards a state in which the nonlinearity is diminished. In fully developed
turbulence this tendency can be measured by comparing the variance of the
nonlinear term to the variance of the same quantity measured in a Gaussian
field with the same energy distribution. In order to study this phenomenon at
high Reynolds numbers, a version of the Direct Interaction Approximation is
used to obtain a closed expression for the statistical average of the
mean-square nonlinearity. The wavenumber spectrum of the mean-square nonlinear
term is evaluated and its scaling in the inertial range is investigated as a
function of the Reynolds number. Its scaling is dominated by the sweeping by
the energetic scales, but this sweeping is weaker than predicted by a random
sweeping estimate. At inertial range scales, the depletion of nonlinearity as a
function of the wavenumber is observed to be constant. At large it is observed
that the mean-square nonlinearity is larger than its Gaussian estimate, which
is shown to be related to the non-Gaussianity of the Reynolds-stress
fluctuations at these scales.
