The invariants of the velocity gradient tensor, R and Q, and their
enstrophy and strain components are studied in the logarithmic layer of an
incompressible turbulent channel flow. The velocities are filtered in the three
spatial directions and the results analyzed at different scales. We show that
the R--Q plane does not capture the changes undergone by the flow as the
filter width increases, and that the enstrophy/enstrophy-production and
strain/strain-production planes represent better choices. We also show that the
conditional mean trajectories may differ significantly from the instantaneous
behavior of the flow since they are the result of an averaging process where
the mean is 3-5 times smaller than the corresponding standard deviation. The
orbital periods in the R--Q plane are shown to be independent of the
intensity of the events, and of the same order of magnitude than those in the
enstrophy/enstrophy-production and strain/strain-production planes. Our final
goal is to test whether the dynamics of the flow are self-similar in the
inertial range, and the answer turns out to be no. The mean shear is found to
be responsible for the absence of self-similarity and progressively controls
the dynamics of the eddies observed as the filter width increases. However, a
self-similar behavior emerges when the calculations are repeated for the
fluctuating velocity gradient tensor. Finally, the turbulent cascade in terms
of vortex stretching is considered by computing the alignment of the vorticity
at a given scale with the strain at a different one. These results generally
support a non-negligible role of the phenomenological energy-cascade model
formulated in terms of vortex stretching
