  The turbulence of superfluid helium is investigated numerically at finite
temperature. Direct numerical simulations are performed with a "truncated HVBK"
model, which combines the continuous description of the
Hall-Vinen-Bekeravich-Khalatnikov equations with the additional constraint that
this continuous description cannot extend beyond a quantum length scale
associated with the mean spacing between individual superfluid vortices. A good
agreement is found with experimental measurements of the vortex density.
Besides, by varying the turbulence intensity only, it is observed that the
inter-vortex spacing varies with the Reynolds number as $Re^{-3/4}$, like the
viscous length scale in classical turbulence. In the high temperature limit,
Kolmogorov's inertial cascade is recovered, as expected from previous numerical
and experimental studies. As the temperature decreases, the inertial cascade
remains present at large scales while, at small scales, the system evolves
towards a statistical equipartition of kinetic energy among spectral modes,
with a characteristic $k^2$ velocity spectrum. The accumulation of superfluid
excitations on a range of mesoscales enables the superfluid to keep dissipating
kinetic energy through mutual friction with the residual normal fluid, although
the later becomes rare at low temperature. It is found that most of the
superfluid vorticity can concentrate on these mesoscales at low temperature,
while it is concentrated in the inertial range at higher temperature. This
observation should have consequences on the interpretation of decaying
turbulence experiments, which are often based on vortex line density
measurements.
