MRI turbulence in accretion discs at large magnetic Prandtl numbers

The effect of large magnetic Prandtl number Pm (the ratio of viscosity to
resistivity) on the turbulent transport and energetics of the magnetorotational
instability (MRI) is poorly understood, despite the realization of this regime
in astrophysical environments as disparate as discs from binary neutron star
mergers, the inner regions of low mass X-ray binaries and active galactic
nuclei, and the interiors of protoneutron stars. We investigate the MRI dynamo
and associated turbulence in the regime $\text{Pm}>1$ by carrying out fully
compressible, 3D MHD shearing box simulations using the finite-volume code
\textsc{PLUTO}, focusing mostly on the case of Keplerian shear relevant to
accretion discs. We find that when the magnetic Reynolds number is kept fixed,
the turbulent transport (as measured by the stress-to-thermal-pressure ratio
$\alpha$) scales with the magnetic Prandtl number as $\alpha \sim
\text{Pm}^{\delta}$, with $\delta \sim 0.5-0.7$ up to $\text{Pm} \sim 128$.
However, this scaling weakens as the magnetic Reynolds number is increased.
Importantly, compared to previous studies, we find a new effect at very large
Pm -- the turbulent energy and stress begin to plateau, no longer depending on
${\rm Pm}$. To understand these results we have carried out a detailed analysis
of the turbulent dynamics in Fourier space, focusing on the effect increasing
Pm has on the transverse cascade -- a key non-linear process induced by the
disc shear flow that is responsible for the sustenance of MRI turbulence.
Finally, we find that the scaling of turbulent transport with Pm is sensitive
to the box vertical-to-radial aspect ratio, as well as to the background shear:
tall boxes exhibit weaker scaling compared to cubic boxes, while MRI turbulence
in sub-Keplerian shear flows (characteristic of protoneutron stars) exhibits
stronger scaling than it does in Keplerian discs.