Title: Local and non-local energy spectra of superfluid $^3$He turbulence,
Abstract: Below the phase transition temperature $Tc \simeq 10^{-3}$K He-3B has a
mixture of normal and superfluid components. Turbulence in this material is
carried predominantly by the superfluid component. We explore the statistical
properties of this quantum turbulence, stressing the differences from the
better known classical counterpart. To this aim we study the time-honored
Hall-Vinen-Bekarevich-Khalatnikov coarse-grained equations of superfluid
turbulence. We combine pseudo-spectral direct numerical simulations with
analytic considerations based on an integral closure for the energy flux. We
avoid the assumption of locality of the energy transfer which was used
previously in both analytic and numerical studies of the superfluid He-3B
turbulence. For T<0.37 Tc, with relatively weak mutual friction, we confirm the
previously found "subcritical" energy spectrum E(k), given by a superposition
of two power laws that can be approximated as $E(k)~ k^{-x}$ with an apparent
scaling exponent 5/3 <x(k)< 3. For T>0.37 Tc and with strong mutual friction,
we observed numerically and confirmed analytically the scale-invariant spectrum
$E(k)~ k^{-x}$ with a (k-independent) exponent x > 3 that gradually increases
with the temperature and reaches a value $x\simeq 9$ for $T\approx 0.72 Tc$. In
the near-critical regimes we discover a strong enhancement of intermittency
which exceeds by an order of magnitude the corresponding level in classical
hydrodynamic turbulence.