Microphysical Regulation of Non-Ideal MHD in Weakly-Ionized Systems: Does the Hall Effect Matter?

TL;DR

The paper addresses the breakdown of standard non-ideal MHD in weakly ionized plasmas when magnetic-field gradients drive drift speeds $v_{ m drift}$ above the thermal speed $v_T$, which invalidates the usual derivation of Ohmic, Hall, and ambipolar coefficients. It develops a multi-species framework and a practical prescription to modify the diffusion coefficients into drift- and instability-informed effective forms, including an anomalous Ohmic term that activates in the superthermal regime. The authors show that Hall effects typically lose dynamical importance under these corrections, while strong diffusion suppresses small-scale magnetic structure and drives the system toward subthermal drift, constraining magnetic amplification. The proposed, computationally inexpensive correction can be implemented atop existing chemistry outputs, enabling more physically consistent simulations of weakly ionized environments such as circumstellar disks and star-forming regions, and guiding future PIC/MHD studies of the microphysics involved.

Abstract

The magnetohydrodynamics (MHD) equations plus 'non-ideal' (Ohmic, Hall, ambipolar) resistivities are widely used to model weakly-ionized astrophysical systems. We show that if gradients in the magnetic field become too steep, the implied charge drift speeds become much faster than microphysical signal speeds, invalidating the assumptions used to derive both the resistivities and MHD equations themselves. Generically this situation will excite microscale instabilities that suppress the drift and current. We show this could be relevant at low ionization fractions especially if Hall terms appear significant, external forces induce supersonic motions, or dust grains become a dominant charge carrier. Considering well-established treatments of super-thermal drifts in laboratory, terrestrial, and Solar plasmas as well as conduction and viscosity models, we generalize a simple prescription to rectify these issues, where the resistivities are multiplied by a correction factor that depends only on already-known macroscopic quantities. This is generalized for multi-species and weakly-ionized systems, and leaves the equations unchanged in the drift limits for which they are derived, but restores physical behavior (driving the system back towards slow drift by diffusing away small-scale gradients in the magnetic field) if the limits are violated. This has important consequences: restoring intuitive behaviors such as the system becoming hydrodynamic in the limit of zero ionization; suppressing magnetic structure on scales below a critical length which can comparable to circumstellar disk sizes; limiting the maximum magnetic amplification; and suppressing the effects of the Hall term in particular. This likely implies that the Hall term does not become dynamically important under most conditions of interest in these systems.

Figures (4)

• Figure 1: Illustration of the regimes in density $\bar{n} = \bar{\rho}/m_{p}$ and magnetic field strength $\bar{B} \equiv | \bar{\bf B}|$, in a weakly-ionized gas ($n_{\pm} \ll n$), where different non-ideal MHD terms derived in § \ref{['sec:deriv']} (Ohmic resistivity $\eta_{O} {\bf J}$, Hall $\eta_{H} {\bf J} \times {\bf b}$, and ambipolar drift $\eta_{A} {\bf J} \times {\bf b} \times {\bf b}$) would each be relatively more important in the induction equation. Quantities like temperature, $v_{T}$, $\omega_{\rm coll}$, and ionization fractions at each $B$, $n$ are estimated as described in § \ref{['sec:multispecies']}-\ref{['sec:plots']}, for a multi-species (electrons, multiple ions, dust grains) system. Left: The case with "classical" non-ideal coefficients (§ \ref{['sec:deriv.nonideal.mhd']}, Eq. \ref{['eqn:nonideal.classical']}), which are valid only if the drift velocity between charge carriers $v_{\rm drift}\sim |{\bf J}|/ n_{-} |q_{-}| \sim c B/4\pi n_{-} |q_{-}| \ell_{B}$ ($\ell_{B} \sim |{\bf B}|/|\nabla {\bf B}|$) and "slip" velocity between neutrals and carriers (Eq. \ref{['eqn:vdrift']}) are much less than the thermal velocities $v_{T}$ ($v_{\rm drift} \ll v_{T}$). This in turn requires that the magnetic gradient scale length $\ell_{B}$ is much larger than some $\ell_{\rm crit}$ (Eq. \ref{['eqn:lcrit']}). The multi-species coefficients come from NICIL (§ \ref{['sec:plots']}) and follow the expressions in Appendix \ref{['sec:extra.multi']}. Right: The case if the implied drift becomes superthermal: $v_{\rm drift} \gtrsim v_{T}$. We use the corrected "effective" coefficients (§ \ref{['sec:superthermal']}; Eqs. \ref{['eqn:etas.general']}-\ref{['eqn:etas.general.specific']}), which account for superthermal drift and its effect both on direct particle collision rates and effective collisions via scattering (anomalous resistivity), assuming drift velocities are superthermal (here imposing $v_{\rm drift} = 10\,v_{T}$ for illustration), or $\ell_{B} \lesssim \ell_{\rm crit}$ (strong magnetic field gradients), with the effective $\eta_{O}^{\rm an}$ and $v_{T}$ for a multi-species dusty fluid following § \ref{['sec:multispecies']}, Eqs. \ref{['eqn:eta.an']}-\ref{['eqn:vT']}. When the drift becomes superthermal, the effective Ohmic term becomes larger than the Hall term, and increases greatly in relative importance even at low densities. Where it predominates, we label whether the anomalous ($\omega_{\rm an}$) or direct collision ($\omega_{\rm coll}$) term is most important.
• Figure 2: Ohmic, Hall, and ambipolar diffusivities $\eta_{O,H,A}$ computed for a multispecies, weakly-ionized gas with total (mostly neutral) density $n$. We compare values for slow drift+slip $v_{\rm drift},\,v_{\rm slip} \ll v_{T}$ ($\ell_{B} \gg \ell_{\rm crit}$; dotted) vs superthermal drift ($v_{\rm drift}\sim 10\,v_{T}$, $\ell_{B} \sim 0.1\,\ell_{\rm crit}^{\rm drift}$; solid) including the proposed corrections in Eq. \ref{['eqn:etas.general.specific']} to account for enhanced scattering. We also compare $\ell_{B}\sim0.1\,\ell_{\rm crit}$ ($v_{\rm drift,\,max}=|{\bf v}_{\rm drift}+{\bf v}_{\rm slip}| \sim 10\,v_{T}$; dashed), which has superthermal slip but not drift at low-$n$. For Hall, the lines overlap. We consider three models for $B$, $T$, ionization, etc. from NICIL (§ \ref{['sec:plots']}): a protostellar collapse model with non-ideal MHD ( top), a barytropic equation-of-state model with stronger $B$ ( middle), and a circumstellar disk midplane model with lower $T$ and weaker ionization ( bottom). The models change the value of $\eta_{O,H,A}$ (e.g. collapse models become hot at $n \gtrsim 10^{14} {\rm cm^{-3}}$, causing rapid ionization and decreasing $\eta$), but do not change the systematic offsets of interest here. With $\ell_{B} \gg \ell_{\rm crit}$ (sub-thermal drift), the system transitions from ambipolar to Hall to Ohmic with increasing $n$. Superthermal drift/slip greatly enhances $\eta_{O}$ and eliminates the Hall regime.
• Figure 3: Critical magnetic gradient scale-length $\ell_{\rm crit}$, below which the drift ($\ell_{\rm crit}^{\rm drift}$) or slip ($\ell_{\rm crit}^{\rm slip}$) speeds become superthermal, versus density $n$ for the same three model variants as Fig. \ref{['fig:etas']}. Strong gradients below this scale will be rapidly erased by enhanced resistivity (Fig. \ref{['fig:etas']}; § \ref{['sec:consequences']}). For comparison we plot the disk scale-height or thermal-pressure scale-length $h_{T}$ at each $n$ in the models. At circumstellar disk radii from $\sim 0.1-100\,$au (densities $n \sim 10^{10}-10^{18} {\rm cm^{-3}}$), and at densities where the Hall diffusion would naively dominate absent anomalous correction terms (for $v_{\rm drift} \ll v_{T}$), we have $\ell_{\rm crit} \gg h_{T}$, so magnetic structures and Hall effects on disk scales will be strongly suppressed.
• Figure 4: Ohmic, Hall, and ambipolar diffusivities $\eta_{O,H,A}$ as a function of density as in Fig. \ref{['fig:etas']} (for the "circumstellar disk"-like case), but for the specific numerical examples in § \ref{['sec:numerical.example']}. Left: Comparison of a generally-subthermal case ($\ell_{B} = 10\,\ell_{\rm crit}$; dotted) to a superthermal drift case ($\ell_{B} = 0.1\,\ell_{\rm crit}^{\rm drift}$; solid). Right: Same but for the superthermal slip case $\ell_{B}=0.1\,\ell_{\rm crit}^{\rm slip}$. For each, we compare (1) the prediction from the simple parameterized rescaling we propose in Eq. \ref{['eqn:etas.general.specific']} (for the multi-species scalings in § \ref{['sec:multispecies']}; "Approximate" in thin lines); to (2) the case where ignore charged dust grains as a current carrier (treat them as part of the neutral inertia taking $\tilde{\beta}_{\rm grain} \rightarrow 0$) and solve the full non-linear three-species problem with the exact coefficients (§ \ref{['sec:deriv']}; "Exact" 3-species in intermediate lines); and (2) the results of exactly solving the non-linear multi-species equations as described in § \ref{['sec:extra.multi:ov']} with all possible pairs of species featuring simple Epstein-type scalings and anomalous terms (§ \ref{['sec:numerical.example']}; "Exact" 5-species in thick lines). The 3 and 5-species methods give excellent agreement, indicating that grain current and differences between exact velocity-dependent scalings in the superthermal regime make little difference. The closed-form analytic approximate scaling from Eq. \ref{['eqn:etas.general.specific']} reproduces these reasonably well in all regimes of interest. Key qualitative results (e.g. vanishing of the Hall-dominated regime) are robust.