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

Coupling large eddies and waves in turbulence: Case study of magnetic helicity at the ion inertial scale