Uncovering the Varieties of Three-dimensional Hall-MHD Turbulence

We carry out extensive pseudospectral direct numerical simulations (DNSs) of decaying three-dimensional (3D) Hall magnetohydrodynamics (3D HMHD) plasma turbulence at three magnetic Prandtl numbers Prₘ=0.1, $1.0$ and $10.0$. Our DNSs have been designed to uncover the dependence of the statistical properties of 3D HMHD turbulence on Prₘ and to bring out the subtle interplay between three lengths, the kinetic and magnetic dissipation length scales ηᵤ, and ηb and the ion-inertial scale dᵢ, below which we see the manifestations of the Hall term. This interplay, qualitatively apparent from isosurface plots of the moduli of the vorticity and the current density, is exposed clearly by the kinetic-energy and magnetic-energy spectra, Eᵤ(k) and Eb(k), respectively. We find two different inertial regions, In the first inertial region k<kᵢ~1/dᵢ, both the kinetic-energy and magnetic-energy spectra, Eᵤ(k) and Eb(k), respectively, display power-law regions with an exponent that is consistent with Kolmogorov-type $-5/3$ scaling, at all values of Prₘ. In the second inertial region k > kᵢ, the scaling of Eb(k) depends upon PrM: At Prₘ=0.1, the spectral-scaling exponent is $-17/3$, but for Prₘ=1 and $10$ this exponent is $-11/3$. We then show theoretically that Eᵤ(k) ~ k² Eb(k) for Prₘ 1 and Eb(k) ~ k² Eᵤ(k) for Prₘ 1; our DNS results are consistent with our theoretical predictions. We examine, furthermore, left- and right-polarised fluctuations of the fields that lead, respectively, to the dominance of ion-cyclotron or whistler waves.