  In turbulence, for neutral or conducting fluids, a large ratio of scales is
excited because of the possible occurrence of inverse cascades to large, global
scales together with direct cascades to small, dissipative scales, as observed
in the atmosphere and oceans, or in the solar environment. In this context,
using direct numerical simulations with forcing, we analyze scale dynamics in
the presence of magnetic fields with a generalized Ohm's law including a Hall
current. The ion inertial length epsilon_H serves as the control parameter at
fixed Reynolds number. Both the magnetic and generalized helicity -- invariants
in the ideal case -- grow linearly with time, as expected from classical
arguments. The cross-correlation between the velocity and magnetic field grows
as well, more so in relative terms for a stronger Hall current. We find that
the helical growth rates vary exponentially with epsilon_H, provided the ion
inertial scale resides within the inverse cascade range. These exponential
variations are recovered phenomenologically using simple scaling arguments.
They are directly linked to the wavenumber power-law dependence of generalized
and magnetic helicity, k^(-2), in their inverse ranges. This illustrates and
confirms the important role of the interplay between large and small scales in
the dynamics of turbulent flows.
