tablenum

# Magnetorotational Instability in a Swirling Partially Ionized Gas

Amy Secunda1, Peter Donnel2, Hantao Ji1,3, and Jeremy Goodman1
1Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton, NJ 08544, USA
2ICFP undergraduate program, Physics department, Ecole Normale Supérieure,
Paris, 75005, France
3Princeton Plasma Physics Laboratory, Princeton, NJ 08540, USA

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The magnetorotational instability (MRI) has been proposed as the method of
angular momentum transport that enables accretion in astrophysical discs.
However, for weakly-ionized discs, such as protoplanetary discs, it remains
unclear whether the combined non-ideal magnetohydrodynamic (MHD) effects of
Ohmic resistivity, ambipolar diffusion, and the Hall effect make these discs
MRI-stable. While much effort has been made to simulate non-ideal MHD MRI,
these simulations make simplifying assumptions and are not always in agreement
with each other. Furthermore, it is difficult to directly observe the MRI
astrophysically because it occurs on small scales. Here, we propose the
concept of a swirling gas experiment of weakly-ionized argon gas between two
concentric cylinders threaded with an axial magnetic field that can be used to
study non-ideal MHD MRI. For our proposed experiment, we derive the
hydrodynamic equilibrium flow and a dispersion relation for MRI that includes
the three non-ideal effects. We solve this dispersion relation numerically for
the parameters of our proposed experiment. We find it should be possible to
produce non-ideal MRI in such an experiment because of the Hall effect, which
increases the MRI growth rate when the vertical magnetic field is anti-aligned
with the rotation axis. As a proof of concept, we also present experimental
results for a hydrodynamic flow in an unmagnetized prototype. We find that our
prototype has a small, but non-negligible, $\alpha$-parameter that could serve
as a baseline for comparison to our proposed magnetized experiment, which
could be subject to additional turbulence from the MRI.

###### keywords:

accretion discs – protoplanetary discs – instabilities – MHD – plasmas –
turbulence

††pubyear: 2022††pagerange: Magnetorotational Instability in a Swirling
Partially Ionized Gas–C.2

##  1 Introduction

Astrophysical accretion discs require a mechanism of outward angular momentum
transport in order for accretion to occur. For sufficiently well-ionized
accretion discs, such as an active galactic nucleus disc, the
magnetorotational instability (MRI, Balbus & Hawley, 1991) is a robust
mechanism for angular momentum transport. However, for weakly-ionized
accretion discs, such as protoplanetary discs, it is still heavily debated
whether the MRI is sufficient to account for observed accretion rates due to
non-ideal magnetohydrodynamic (MHD) effects, such as Ohmic resistivity and
ambipolar diffusion, which decouple the gas disc and magnetic field
stabilizing the disc (e.g. Gammie, 1996; Fleming et al., 2000; Sano & Stone,
2002; Bai, 2011; Bai & Stone, 2013; Gressel et al., 2015).

In protoplanetary discs, Ohmic resistivity dominates the high-density, weakly
magnetized, inner midplane. Ambipolar diffusion dominates the lower-density,
more strongly magnetized, outer regions and surface layers of the disc. A
third non-ideal MHD effect, the Hall effect, dominates in a regime somewhere
in between the other two effects at moderate densities and magnetic field
strengths. Unlike the diffusive non-ideal effects, the Hall effect has been
shown analytically and in simulations to moderately enhance (suppress) the MRI
when the magnetic field is anti-aligned (aligned) with the axis of rotation
and (Wardle, 1999; Bai et al., 2015a).

While the MRI has been studied extensively analytically and numerically, it is
often necessary to make simplifying assumptions in order to make the problem
tractable. For example, numerical simulations of protoplanetary disks are
often two-dimensional (e.g., Bai, 2017; Yang & Bai, 2021), local (shearing
box) approximations (e.g., Fleming & Stone, 2003; Bai & Stone, 2013; Simon et
al., 2013b, a; Lesur et al., 2014; Bai & Stone, 2014; Simon et al., 2015; Bai
et al., 2015a), of limited extent in one dimension (e.g., Cui & Bai, 2021),
and/or subject to numerical dispersion (e.g., Béthune et al., 2017; Bai,
2017). In addition, very few simulations of non-ideal MHD MRI include the Hall
effect, and due to the computationally expensive nature of these simulations,
wider parameter studies are not always feasible.

All simulations require some simplification of the physics and of the numerics
and disentangling the effects of these simplifications on the results can be
subtle and difficult. It is also difficult to directly observe the MRI
astrophysically because it occurs on small scales. Therefore, to better
understand the MRI, especially the standard version of it, or SMRI, when a
magnetic field is applied along the rotation axis, several laboratory
experiments have been proposed or attempted to generate the MRI that arises in
a (quasi-)Keplerian flow or to generate its analogues (e.g. Ji et al., 2001;
Sisan et al., 2004; Boldyrev et al., 2009; Nornberg et al., 2010; Roach et
al., 2012; Collins et al., 2012; Vasil, 2015; Bai et al., 2015b; Caspary et
al., 2018; Hung et al., 2019; Flanagan et al., 2020, see also Ji & Goodman
(2023) for a recent review). Wang et al. (2022) recently successfully produced
the SMRI in a Taylor-Couette cell with rotating magnetized liquid metal. This
experiment was able for the first time to experimentally confirm the existence
of an instability which up until this point had never been directly observed
in nature, only derived theoretically. However, the conditions of these
experiments are still far removed from the conditions of any astrophysical
disc.

In this paper, we propose the concept of a swirling gas experiment of
partially ionized argon gas between two concentric cylinders and threaded with
an axial magnetic field. Our proposed experiment will have a neutral number
density and temperature that falls within the values for a minimum mass solar
nebula (Hayashi, 1981), although a protoplanetary disc is primarily composed
of hydrogen and helium. Another key difference is that our experimental disc
will have a much higher ionization fraction of $\chi_{\rm i}\approx 10^{-3}$
as opposed to the ionization fraction of $\chi_{\rm i}\approx 10^{-13}$ in
protoplanetary discs (Lesur et al., 2022). Nonetheless, the advantage of our
proposed experiment is that we should be able to probe the ambipolar-, Hall-,
and Ohmic-dominated regimes in order to study how non-ideal effects suppress
or enhance the MRI. The MRI has never been directly observed astrophysically
or experimentally in a poorly-ionized gas. Doing so would allow for comparison
with analytic predictions and numerical simulations and provide insight on
what physics is most crucial to include in simulations that can be very
computationally expensive.

We first derive the hydrodynamic equilibrium conditions of our experiment in
Section 2. Next, we derive a dispersion relation for non-ideal MHD MRI,
describe our numerical solution, and present our parameter-dependent
predictions for producing the MRI in our experiment in Section 3. In Section 4
we describe the setup of and experimental results from a hydrodynamic
prototype containing air instead of argon to validate the concept. Finally, we
summarize our results in Section 5.

##  2 Hydrodynamic equilibrium flow

Figure 1: This cartoon of our proposed swirling gas experiment shows the top
view of the swirling gas between two concentric cylinders. The gas enters the
apparatus at the top of the diagram through a thin opening at an azimuthal
velocity $u_{\rm\theta}(r_{2})$. It spirals inward from the outer cylinder
wall at $r_{2}$ towards the inner cylinder at $r_{1}$ with a radial velocity
$u_{\rm r}(r)$ due to a pressure gradient imposed by a fan in the inner
cylinder. The gas passes out of the apparatus through holes in the inner
cylinder.

We show a cartoon version of a top view of our proposed experiment in Figure
1. In the steady-state setup of our cylindrical experiment, gas is injected at
the outer radius, $r_{2}$, with a large velocity of $u_{\rm\theta}(r_{2})$
that is mostly tangential to the outer cylinder. The gas then swirls radially
inwards with a radial velocity $u_{\rm r}<0$ due to a pressure gradient
imposed by a fan in the inner cylinder. Finally, the gas reaches the inner
cylinder with radius $r_{1}$ where it gets pumped out.

Note that gas is injected into the experiment at a prescribed rate. In a
steady state, the gas is also pumped out at this same rate. However, the
radial structures of the swirling gas are determined by the internal dynamics
of the flow. A large effective viscosity would allow rapid angular momentum
transport leading to a short residual time for gas to stay in the experiment.
On the other hand, a small effective viscosity would hinder angular momentum
transport leading to a long gas residual time. Therefore, just as in accretion
discs, an effective viscosity can be inferred from the radial profiles of the
flow. Studying this effective viscosity, including that due to MRI, is the
goal of our experiment.

In this section we derive the radial pressure and velocity profiles of the
steady state hydrodynamic equilibrium flow in our proposed experiment. We use
cylindrical coordinates ($r$, $\theta$, $z$), and assume any $\theta$ or $z$
dependencies are negligible ($\partial/\partial\theta=\partial/\partial z=0$),
which we expect to hold true far from the edges of the apparatus. We also
assume the vertical component of the velocity, $u_{\rm z}$, is negligible.
While the mean $u_{\rm z}$ should be roughly zero, there could be sizeable
fluctuations in $u_{\rm z}$ due to turbulence, especially in the Ekman layers.
However, we will assume this turbulent velocity is negligible relative to the
bulk velocity. In addition we assume that the gas temperature in the device is
homogeneous. In the magnetized case discussed in Section 3, the electrons and
ions will be hotter than the neutrals, but our plasma will be so poorly-
ionized the electrons and ions will be unable to sufficiently heat the
neutrals, which justifies our assumption of a homogeneous temperature.

For our equation of state we use, $p=\rho C^{2}$, where $p$ is the pressure,
$\rho$ is the gas density, and $C$ is the sound speed divided by the square
root of the adiabatic index ($\gamma_{\rm a}=7/5$). Unless otherwise noted, we
take $C$ to be constant. Finally, we assume there is an azimuthal force
density $F_{\rm\theta}<0$ acting on the azimuthal velocity. In Appendix B we
show that $F_{\rm\theta}$ represents a viscous force against the vertical
boundaries, including Ekman circulation.

To start, we assume that the gas is compressible. In steady state, mass
conservation gives,

$\frac{1}{r}\frac{\partial(r\rho u_{\rm r})}{\partial r}=0.$ (2.0.1)

Thus,

$r\rho u_{\rm r}=-K,$ (2.0.2)

where K is a positive constant. We estimate the radial mass flux, $\rho D$, at
radius $r$ as $\rho D=-2\pi r\rho u_{\rm r}H$, where $H$ is the height of the
cylinder and $D>0$ for inward flow. Using this and equation (2.0.2) we find,

$K=\frac{D\rho}{2\pi H}.$ (2.0.3)

The azimuthal component of the steady-state Navier-Stokes equation is,

$\rho\left[u_{\rm r}\frac{\partial u_{\rm\theta}}{\partial r}+\frac{u_{\rm
r}u_{\rm\theta}}{r}\right]=F_{\rm\theta}+\mu\left[\frac{1}{r}\frac{\partial}{\partial
r}\left(r\frac{\partial u_{\rm\theta}}{\partial
r}\right)-\frac{u_{\rm\theta}}{r^{2}}\right],$ (2.0.4)

where $\mu$ is the dynamic viscosity. If we divide this equation by $-K/r=\rho
u_{\rm r}$ we have,

$\frac{\partial u_{\rm\theta}}{\partial
r}+\frac{u_{\rm\theta}}{r}=\Gamma-\frac{\mu}{K}\left[\frac{\partial}{\partial
r}\left(r\frac{\partial u_{\rm\theta}}{\partial
r}\right)-\frac{u_{\rm\theta}}{r}\right],$ (2.0.5)

where $\Gamma=-rF_{\rm\theta}/K$. This equation has the general solution,

$u_{\rm\theta}=\frac{J}{r}+\frac{r\Gamma}{2}+\frac{a}{2-\frac{K}{\mu}}r^{1-\frac{K}{\mu}},$
(2.0.6)

where $J$, $\Gamma$, and $a$ are all constants. We provide the derivation of
this solution in Appendix A. We also show in Appendix A that in both limiting
cases, $K\ll\mu$ and $\mu\ll K$, we have

$u_{\rm\theta}=\frac{J}{r}+\frac{r\Gamma}{2},$ (2.0.7)

where the definition of $\Gamma$ can be adjusted if needed. If $K/\mu\approx
2$, the final term of (2.0.6) should be replaced by $ar^{-1}\ln(r/r_{1})$.

The radial component of the steady-state Navier-Stokes equation is,

$u_{\rm r}\frac{\partial u_{\rm r}}{\partial
r}-\frac{u_{\rm\theta}^{2}}{r}=-\frac{1}{\rho}\frac{\partial p}{\partial
r}-\frac{\mu u_{\rm r}}{K}\left[\frac{\partial}{\partial
r}\left(r\frac{\partial u_{\rm r}}{\partial r}\right)-\frac{u_{\rm
r}}{r}\right].$ (2.0.8)

Using the radial derivative of our equation of state,

$\frac{1}{\rho}\frac{\partial p}{\partial
r}=C^{2}\frac{\partial\ln(\rho)}{\partial r}.$ (2.0.9)

and the logarithmic derivative of equation (2.0.2),

$\frac{\partial\ln(\rho)}{\partial r}=-\frac{1}{r}-\frac{\partial\ln(u_{\rm
r})}{\partial r},$ (2.0.10)

in equation (2.0.8) we derive,

$\begin{split}\frac{\partial}{\partial r}\left(\frac{1}{2}u_{\rm
r}^{2}-C^{2}\ln(u_{\rm r})\right)+\frac{\mu u_{\rm
r}}{K}\left[\frac{\partial}{\partial r}\left(r\frac{\partial u_{\rm
r}}{\partial r}\right)-\frac{u_{\rm r}}{r}\right]\\\
=\frac{1}{r}\left(u_{\rm\theta}^{2}+C^{2}\right).\end{split}$ (2.0.11)

From equation (2.0.3) we have,

$\frac{K}{\mu}=\frac{D}{2\pi H\nu},$ (2.0.12)

where the kinematic viscosity, $\nu=\mu/\rho$. Anticipating the radial flow
will be larger than the viscosity for parameters of interest, we now take the
limit $\mu\ll K$ in equations (2.0.6) and (2.0.11) and,

$\frac{\partial}{\partial r}\left(\frac{1}{2}u_{\rm r}^{2}-C^{2}\ln(u_{\rm
r})\right)=\frac{1}{r}\left[\left(\frac{J}{r}+\frac{r\Gamma}{2}\right)^{2}+C^{2}\right].$
(2.0.13)

Defining the dimensionless variables, $U\equiv u_{\rm r}/C$, $V\equiv
u_{\rm\theta}/C$, $j\equiv J/r_{\rm 2}C$, $g\equiv\Gamma r_{\rm 2}/C$ and
$R\equiv r/r_{\rm 2}$, where $r_{\rm 2}$ is the outer cylinder radius,
equation (2.0.7) becomes,

$V=\frac{j}{R}+\frac{gR}{2}$ (2.0.14)

and equation (2.0.13) becomes,

$\left(1-\frac{1}{U^{2}}\right)\frac{\partial U^{2}}{\partial
R}=\frac{2}{R}\left[\left(\frac{j}{R}+\frac{gR}{2}\right)^{2}+1\right].$
(2.0.15)

If we define $X\equiv U^{2}$, then equation (2.0.15) can be integrated over
$X$ as,

$X-\ln X=-\left(\frac{j}{R}\right)^{2}+2(jg+1)\ln
R+\left(\frac{gR}{2}\right)^{2}+b,$ (2.0.16)

where $b$ is a constant that can be estimated at $R=1$ (i.e. $r=r_{\rm 2}$)
as,

$b=X(1)-\ln X(1)+j^{2}-\left(\frac{g}{2}\right)^{2}.$ (2.0.17)

We can compute $X(1)$ using equations (2.0.2) and (2.0.3) giving,

$X(1)=\left(\frac{D}{2\pi r_{\rm 2}HC}\right)^{2},$ (2.0.18)

where $D$ is the flux at $r_{2}$.

To compare the effect of the viscosity to the effect of the geometry of the
device it is useful to define the Shakura & Sunyaev (1973) disc viscosity
parameter, $\alpha$, as

$rF_{\rm\theta}=-\alpha\rho_{\rm 1}C^{2}.$ (2.0.19)

With this definition it is obvious that the bigger $\alpha$ is the bigger the
effect of $F_{\rm\theta}$ is. If we divide this definition by $K$ and make
everything dimensionless the result is,

$\alpha=-\frac{gU(r_{\rm 1})r_{\rm 1}}{r_{\rm 2}}.$ (2.0.20)

It is also useful to define the Reynolds number, which for a rotating flow
between two concentric cylinders is defined as,

$Re=\frac{(r_{\rm 2}^{2}-r_{\rm 1}^{2})(\Omega_{\rm 1}-\Omega_{\rm
2})}{2\nu}.$ (2.0.21)

Here $\Omega\equiv u_{\theta}/r$ is the angular velocity, and $\Omega_{1}$ and
$\Omega_{2}$ are the angular velocities near $r_{1}$ and $r_{2}$ (but outside
the boundary layers). Equation (2.0.7) gives us

$\Omega(r)=\Gamma/2+J/r^{2}\,,$ (2.0.22)

so for our experiment,

$Re=\frac{J(r_{\rm 2}^{2}-r_{\rm 1}^{2})^{2}}{2\nu r_{\rm 1}^{2}r_{\rm
2}^{2}}.$ (2.0.23)

The left panel of Figure 2 shows the radial profiles of the gas pressure,
radial velocity, and azimuthal velocity for our proposed experiment,
calculated using the equations in this section. The radial profiles shown in
brown are for $B_{\rm z}=0$. The solid and dashed lines correspond to gas
pressures, $p(r_{1})=10$ mTorr and $p(r_{1})=100$ mTorr, respectively. We set
$\Gamma=-0.01$ s-1, $u_{\rm r}(r_{1})=-250$ m s-1, and
$u_{\rm\theta}(r=1~{}{\rm m})=400$ m s-1. First we determine
$u_{\rm\theta}(r)$ using equation (2.0.7). Next, we use equation (2.0.16) to
determine $u_{\rm r}(r)$. Because the gas temperature is constant in our
calculations, once we have calculated $u_{\rm r}(r)$ we determine the gas
pressure from the gas density using equation (2.0.2), where we take $K=-u_{\rm
r}(r_{1})r_{1}p(r_{1})/(k_{\rm B}T_{\rm n})$, where $k_{\rm B}$ is Boltzmann’s
constant and $T_{\rm n}$ is the gas temperature.

For the chosen parameters, we calculate that the gas pressure increases by
roughly two orders of magnitude from the inner to outer radius. Therefore the
gas density is not constant and the assumption of incompressibility in Section
3.1 has to be taken as a first approximation. The magnitude of the radial
velocity decreases by two orders of magnitude from the inner to outer radius,
because it is inversely proportional to density (see Equation (2.0.2)). For
the same reason, the magnitude of the radial velocity decreases for higher
$p(r_{1})$. The azimuthal velocity does not depend on the pressure and
consistently goes from 1000 m s-1 at the inner radius to 250 m s-1 at the
outer radius.

Figure 2: From top to bottom, the left panel shows the radial profiles for the
gas pressure, radial velocity, and azimuthal velocity, and the right panel
shows the radial profiles for the electron temperature, electron number
density, and ionization fraction. The profiles for $B_{\rm z}=0$ ($B_{\rm
z}=-0.01$ T) are shown in brown (pink) and the profiles for $p(r_{1})=10$
mTorr ($p(r_{1})=100$ mTorr) are shown as solid (dashed lines).

##  3 Non-ideal magnetorotational instability

In the previous section we outlined the hydrodynamic equilibrium flow for our
proposed swirling gas experiment. In this section, we add a magnetic field to
this hydrodynamic flow that may or may not act to destabilize the flow. We
derive a dispersion relation for MRI in the presence of Ohmic, ambipolar, and
Hall effects. Then we describe our procedures for evaluating the physical
parameters of the dispersion relation numerically and solving for MRI growth
rates. Using these numerical solutions, we make predictions for MRI for a wide
range of parameters that are possible for a future swirling-argon-gas
experiment.

###  3.1 Deriving the Dispersion Relation

The equations of non-ideal MHD are,

  1. 1.

Continuity equation:

$\frac{\partial\rho}{\partial
t}+\overrightarrow{\nabla}\mn@boldsymbol{\cdot}(\rho\mn@boldsymbol{u})=0$
(3.1.1)

  2. 2.

Gauss’ law for magnetism:

$\overrightarrow{\nabla}\mn@boldsymbol{\cdot}\mn@boldsymbol{B}=0$ (3.1.2)

  3. 3.

Momentum equation:

$\rho\left(\frac{\partial\mn@boldsymbol{u}}{\partial
t}+\left(\mn@boldsymbol{u}\mn@boldsymbol{\cdot}\overrightarrow{\nabla}\right)\mn@boldsymbol{u}\right)$
$\hskip 36.98857pt=-\overrightarrow{\nabla}\left(p+\frac{B^{2}}{2\mu_{\rm
0}}\right)+\frac{\left(\mn@boldsymbol{B}\mn@boldsymbol{\cdot}\overrightarrow{\nabla}\right)\mn@boldsymbol{B}}{\mu_{\rm
0}}-\mu\left(\overrightarrow{\nabla}\times\left(\overrightarrow{\nabla}\times\mn@boldsymbol{u}\right)\right)$
(3.1.3)

  4. 4.

The induction equation:

$\frac{\partial\mn@boldsymbol{B}}{\partial
t}=\overrightarrow{\nabla}\mn@boldsymbol{\times}\left[\mn@boldsymbol{u}\mn@boldsymbol{\times
B}-\frac{\left(\overrightarrow{\nabla}\mn@boldsymbol{\times
B}\right)\mn@boldsymbol{\times B}}{\mu_{\rm 0}en_{\rm e}}\right.$ $\hskip
91.04881pt\left.+\frac{\left(\mn@boldsymbol{j}\mn@boldsymbol{\times
B}\right)\mn@boldsymbol{\times B}}{\gamma_{\rm in}\rho_{\rm
i}\rho}\right]+\eta_{\rm o}\overrightarrow{\nabla}^{2}\mn@boldsymbol{B},$
(3.1.4)

where $\mn@boldsymbol{B}$ is the magnetic field, $\mn@boldsymbol{u}$ is the
velocity vector, $\mn@boldsymbol{j}$ is the current, $\mu_{0}$ is the vacuum
permeability, $e$ is the charge of an electron, $n_{\rm e}$ is the electron
number density, $\gamma_{\rm in}$ is the ion-neutral collision rate, and
$\eta_{\rm o}$ is the Ohmic diffusivity. In the induction equation, the first
term in brackets corresponds to ideal MHD, the second term is the Hall term,
which accounts for the ions having greater inertia than the electrons, and the
third term is the ambipolar diffusion term, which accounts for the ion-neutral
drift. The final term in the induction equation is the resistive term.

Next we introduce axisymmetric velocity, $v_{\rm r},v_{\rm\theta},v_{\rm z}$,
magnetic field, $b_{\rm r},b_{\rm\theta},b_{\rm z}$, and pressure, $p_{1}$,
perturbations of the form, $X\equiv xe^{\gamma t-ik_{\rm r}r-ik_{\rm z}z}$. We
assume that the background flow is steady and incompressible, which means
equation 3.1.1 becomes $\overrightarrow{\nabla}\mn@boldsymbol{\cdot u}=0$. We
define

$\mn@boldsymbol{u}=\left(\begin{array}[]{c}\frac{-K}{\rho r}+v_{\rm r}\\\
\frac{\Gamma r}{2}+\frac{J}{r}+v_{\rm\theta}\\\ v_{\rm
z}\end{array}\right),\;\;\;\mn@boldsymbol{B}=\left(\begin{array}[]{c}b_{\rm
r}\\\ b_{\rm\theta}\\\ B_{\rm z}+b_{\rm z}\end{array}\right),\;\;\;p=p_{\rm
0}+p_{\rm 1}.$

Here the background flow has the radial velocity profile given by equation
(2.0.2), the azimuthal velocity profile given equation (2.0.7), and pressure
profile $p_{0}(r)$. The background magnetic field is purely vertical and
uniform, so that there is no background current.

If we then assume $k_{\rm r},k_{\rm z}\gg 1/r$ and linearize the MHD
equations, we have

$k_{\rm r}v_{\rm r}+k_{\rm z}v_{\rm z}=0,$ (3.1.5) $k_{\rm r}b_{\rm r}+k_{\rm
z}{b}_{\rm z}=0,$ (3.1.6) $\left(\gamma+\eta_{\rm o}k^{2}+\eta_{\rm
A}k^{2}\right)b_{\rm r}=-ik_{\rm z}B_{\rm z}v_{\rm r}-k_{\rm z}^{2}\eta_{\rm
H}b_{\rm\theta},$ (3.1.7) $\left(\gamma+\eta_{\rm o}k^{2}+\eta_{\rm A}k_{\rm
z}^{2}\right)b_{\rm\theta}=-ik_{\rm z}B_{\rm z}v_{\rm\theta}+k^{2}\eta_{\rm
H}b_{\rm r}+\frac{\partial\Omega}{\partial\ln r}b_{\rm r},$ (3.1.8)
$\left(\gamma+\nu k^{2}\right)v_{\rm r}=2\Omega v_{\rm\theta}-\frac{ik_{\rm
z}B_{\rm z}b_{\rm r}}{\mu_{\rm 0}\rho}+\frac{ik_{\rm r}p_{\rm
1}}{\rho}+\frac{ik_{\rm r}B_{\rm z}{b}_{\rm z}}{\mu_{\rm 0}\rho},$ (3.1.9)
$\left(\gamma+\nu k^{2}\right)v_{\rm\theta}=-\Gamma v_{\rm r}-\frac{ik_{\rm
z}B_{\rm z}b_{\rm\theta}}{\mu_{\rm 0}\rho},$ (3.1.10) $\left(\gamma+\nu
k^{2}\right)v_{\rm z}=\frac{ik_{\rm z}p_{\rm 1}}{\rho},$ (3.1.11)

where ambipolar diffusivity, $\eta_{\rm A}\equiv B_{\rm z}^{2}/({\mu_{\rm
0}\gamma\rho_{\rm i}\rho})$, and the Hall term, $\eta_{\rm H}\equiv B_{\rm
z}/(e\mu_{0}n_{\rm e})$. From these equations we can derive the fourth order
dispersion relation,

$\gamma^{4}+\Big{[}2(\nu+\eta_{\rm o})k^{2}+\eta_{\rm A}\left(k^{2}+k_{\rm
z}^{2}\right)\Big{]}\gamma^{3}$
$+\bigg{\\{}k^{2}\Big{[}\nu^{2}k^{2}+2\nu\left(2\eta_{\rm o}k^{2}+\eta_{\rm
A}(k^{2}+k_{\rm z}^{2})\right)+\left(\eta_{\rm o}+\eta_{\rm
A}\right)\left(\eta_{\rm o}k^{2}+\eta_{\rm A}k_{\rm z}^{2}\right)\Big{]}$
$\hskip 14.22636pt+2\Omega\Gamma\frac{k^{2}_{\rm z}}{k^{2}}+\eta_{\rm
H}k^{2}_{\rm z}\Big{[}(\Gamma-2\Omega)+k^{2}\eta_{\rm H}\Big{]}+2k_{\rm
z}^{2}V_{\rm A}^{2}\bigg{\\}}\gamma^{2}$ $+\bigg{\\{}\nu
k^{4}\Big{[}\nu\left(2\eta_{\rm o}k^{2}+\eta_{\rm A}\left(k^{2}+k_{\rm
z}^{2}\right)\right)+2\left(\eta_{\rm o}+\eta_{\rm A}\right)\left(\eta_{\rm
o}k^{2}+\eta_{\rm A}k_{\rm z}^{2}\right)\Big{]}$ $\hskip
14.22636pt+2\Omega\Gamma\frac{k^{2}_{\rm z}}{k^{2}}\left(2\eta_{\rm
o}k^{2}+\eta_{\rm A}\left(k^{2}+k_{\rm z}^{2}\right)\right)$ $\hskip
28.45274pt+2\nu\eta_{\rm H}k^{2}k^{2}_{\rm z}\Big{[}(\Gamma-2\Omega)+\eta_{\rm
H}k^{2}\Big{]}$ $\hskip 42.67912pt+k_{\rm z}^{2}V_{\rm
A}^{2}\Big{[}2(\nu+\eta_{\rm o})k^{2}+\eta_{\rm A}\left(k^{2}+k_{\rm
z}^{2}\right)\Big{]}\bigg{\\}}\gamma$ $+\left(k^{6}\nu^{2}+2\Omega\Gamma
k_{\rm z}^{2}\right)(\eta_{\rm o}+\eta_{\rm A})\left(\eta_{\rm
o}k^{2}+\eta_{\rm A}k_{\rm z}^{2}\right)$ $\hskip 14.22636pt+\nu^{2}\eta_{\rm
H}k^{4}k_{\rm z}^{2}\Big{[}(\Gamma-2\Omega)+\eta_{\rm H}k^{2}\Big{]}$ $\hskip
28.45274pt+\nu k^{2}k_{\rm z}^{2}V_{\rm A}^{2}\Big{[}2\eta_{\rm
o}k^{2}+\eta_{\rm A}(k^{2}+k_{\rm z}^{2})\Big{]}$ $+\frac{k_{\rm
z}^{4}}{k^{2}}\left(2\Omega\Gamma\eta_{\rm H}+2\Omega V_{\rm
A}^{2}\right)\Big{[}\frac{V_{\rm
A}^{2}k^{2}}{2\Omega}+(\Gamma-2\Omega)+\eta_{\rm H}k^{2}\Big{]}=0,$ (3.1.12)

where $V_{\rm A}=B_{\rm z}/\sqrt{\mu_{0}\rho}$ is the Alfvén velocity, and
$k^{2}\equiv k_{\rm r}^{2}+k_{\theta}^{2}$. For a more general angular-
velocity profile, $\Omega(r)$, the combinations $\Gamma$ and $\Gamma-2\Omega$
would be replaced by $r^{-1}d(r^{2}\Omega)/dr$ and $rd\Omega/dr$,
respectively. Notice that the latter always occurs added to $\eta_{\rm
H}k^{2}$, hinting at the importance of the Hall effect in promoting or
inhibiting instability, depending on the sign of $\eta_{\rm H}$. Kunz & Balbus
(2004) give a dispersion relation equivalent to equation (3.1.12), except for
the omission of the viscous terms, but allowing for an azimuthal as well as
vertical component of the background field.

###  3.2 Numerical Solution

Figure 3: The Ohmic ($\eta_{\rm o}$) and ambipolar ($\eta_{\rm A}$)
diffusivities, as well as the Hall term ($\eta_{\rm H}$) and total diffusivity
($\eta_{\rm tot}$), where $\eta_{\rm tot}=\eta_{\rm o}+\eta_{\rm A}$, as a
function of $p(r_{2})$ for our fiducial setup. In the left panel $B_{\rm
z}=0.001$ T and in the right panel $B_{\rm z}=0.01$ T. Figure 4: Blue dots
show pressures and magnetic field strengths for which the solution to Equation
(3.1.12) is positive, i.e. there is MRI growth with the Hall effect. Orange
dots show where there is MRI growth without the Hall effect (positive
solutions to Equation (3.1.12) with $\eta_{\rm H}=0$). Black dots show all
magnetic field strengths and pressures used in our calculation regardless of
whether MRI growth is found. Figure 5: The logarithm of the number of
e-folding times for MRI growth for the residual time of the gas in the
apparatus with the Hall effect (left panel) and without the Hall effect (right
panel) for a range of pressures at $r_{2}$ and vertical magnetic field
strengths. Figure 6: The logarithm of the Reynolds number (left panel), the
magnetic Reynolds number (center panel), and the absolute value of the Hall
parameter (right panel) as a function of pressure and magnetic field strength
for parameters with MRI growth for calculations with the Hall effect. Figure
7: The distribution of Reynolds numbers (top panel), magnetic Reynolds numbers
(center panel), and Hall parameters (bottom panel), for which there is MRI
growth for calculations with the Hall effect (in blue), MRI growth for
calculations without the Hall effect (in orange), and no MRI growth (dashed
black lines in black).

Figure 8: Same as Figure 4 but for two different powers: 10 kW (top panel) and
100 kW (bottom panel).

We solve for the real roots of this dispersion relation numerically. A
positive real root corresponds to MRI growth. To solve this dispersion
relation we need to determine the Alvfén velocity, viscosity, Ohmic and
ambipolar diffusivities, and Hall term for our proposed ionized swirling argon
experiment. To find these parameters we first determine the temperature and
density profiles of the experiment.

We set the ion and gas temperatures to a constant $T_{\rm i}=T_{\rm n}=500$ K.
As in Section 2, we calculate the radial density profile using Equation
(2.0.2), where we take $K=-u_{\rm r}(r_{1})r_{1}\rho(r_{1})$. However, when
$B\neq 0$, to calculate $u_{\rm r}(r)$ we need to add $j_{\theta}\times B_{\rm
z}$, where $j_{\theta}=(v_{\rm r}\times B_{\rm z})/(\eta_{\rm o}+\eta_{\rm
a})$ is the azimuthal component of the current, to the radial force balance
equation, Equation (2.0.8). Carrying this term through to Equation (2.0.15),
we get,

$\left(1-\frac{1}{U^{2}}\right)\frac{\partial U^{2}}{\partial
R}=\frac{2}{R}\left[\left(\frac{j}{R}+\frac{gR}{2}\right)^{2}+1\right]+\frac{2B_{\rm
z}^{2}r_{2}^{2}}{(\eta_{\rm o}+\eta_{\rm a})K}V^{2}R,$ (3.2.1)

which we then integrate for the initial radial velocity profile.

The electron density and temperature depend on the geometry of the device and
the input power, $P$. We use the model from Lieberman & Lichtenberg (2005),
outlined in Appendix C, to determine the electron temperature and density
profile of our experiment.

We show the radial profiles of the gas pressure, radial velocity, azimuthal
velocity, electron temperature, electron number density, and ionization
fraction in Figure 2 calculated for $p(r_{1})=10$ mTorr ($p(r_{1})=100$ mTorr)
and $B_{\rm z}=0$ and $B_{\rm z}=-0.01$ T as brown and pink solid (dashed)
lines, respectively. Compared to when $B_{\rm z}=0$, for $B_{\rm z}=-0.01$ T
there is an increase in both the pressure and radial velocity gradients. This
increase is due to the last term in Equation (3.2.1) which is zero for $B_{\rm
z}=0$. When this term is non-zero, both the pressure and radial velocity
gradients must increase to compensate. When the pressure gradient increases,
the neutral number density increases, which in turn increases the electron
number density. However, without increasing the power, for a higher density
the ionization fraction will decrease slightly, as will the electron
temperature.

As the initial pressure increases the relative impact of the added
$j_{\theta}\times B_{\rm z}$ term decreases. For $p(r_{1})=100$ mTorr (dashed
lines in Figure 2) the various radial profiles for the two magnetic field
strengths are nearly identical. Increasing the gas pressure increases the
electron number density, while decreasing the electron temperature and
ionization fraction, because the power is kept constant. For $p(r_{1})=100$
mTorr the radial velocity is similar to its value for $p(r_{1})=10$ mTorr and
$B_{\rm z}=0$. This similarity is because only increasing the gradient of the
pressure has an affect on the radial velocity, and at higher pressures, the
magnetic field is not significant enough to affect the radial velocity. Over
the range of parameters we use in our calculations, the electron temperature
tends to fall around $T_{\rm e}=1$ eV and the electron number density is
around $n_{\rm e}=10^{19}$ m-3.

As a simplification, we use the radially averaged values of these temperature
and density profiles to calculate $v_{\rm A}$, $\nu$, $\eta_{\rm o}$,
$\eta_{\rm a}$, and $\eta_{\rm H}$. We assume the viscosity is dominated by
neutral viscosity,

$\nu_{\rm nn}=\lambda_{\rm nn}\gamma_{\rm nn}=\frac{v_{\rm n}^{2}}{\gamma_{\rm
nn}}=\frac{1}{n_{\rm n}\sigma_{\rm n}}\sqrt{\frac{k_{\rm B}T_{\rm n}}{m_{\rm
n}}},$ (3.2.2)

where $\lambda_{\rm nn}=v_{\rm n}/\gamma_{\rm nn}$ is the neutral-neutral mean
free path, $\gamma_{\rm nn}=n_{\rm n}\sigma_{\rm n}\sqrt{4k_{\rm B}T_{\rm
n}/m_{\rm n}}$ is the neutral-neutral collision rate, $m_{\rm n}$ is the mass
of neutral argon, $n_{\rm n}$ is the neutral number density, $\sigma_{\rm
n}\approx 10^{-19}$ m2 is the neutral collision cross-section, and $v_{\rm
n}=\sqrt{2k_{\rm B}T_{\rm n}/m_{\rm n}}$ is the neutral velocity.

We calculate Ohmic resistivity as,

$\eta_{\rm o}=\frac{m_{\rm e}n_{\rm n}}{e^{2}n_{\rm e}}\langle\sigma
v\rangle_{\rm en},$ (3.2.3)

where $m_{\rm e}$ is the electron mass. $\langle\sigma v\rangle_{\rm en}$ is
the collision rate between electrons and neutrals which depends on
$\sigma_{\rm en}\approx 3.5\times 10^{-20}$ m2, the electron-neutral collision
cross-section at $T_{\rm e}\approx 4$ eV (Pitchford et al., 2013), and the
electron velocity, $v_{\rm e}=\sqrt{2k_{\rm B}T_{\rm e}/m_{\rm e}}$.

Parameter | Range
---|---
$p(r_{1})$ |  $10^{-3}$ – $10^{4}$ mTorr
$|B_{z}|$ |  $10^{-3}$ – 0.2 T
$P$ | 1 – 200 kW
$H/(r_{2}-r_{1})$ | 0.5 – 2
Table 1: The parameter ranges over which we numerically solve equation
(3.1.12) for the MRI growth rate.

$\eta_{\rm A}$ depends on the ion-neutral collision rate,

$\gamma_{\rm in}=\frac{\langle\sigma\nu\rangle_{\rm in}}{2m_{\rm i}},$ (3.2.4)

where $m_{\rm i}$ is the ion mass and $\langle\sigma\nu\rangle_{\rm in}$ is
the rate of ion-neutral interactions. At $T_{\rm i}=T_{\rm n}=500$ K ion-
neutral collisions are dominated by both polarization scattering and charge
exchange (Lieberman & Lichtenberg, 2005). We calculate the cross section for
resonant charge exchange for argon using Equation (14) in Rapp & Francis
(1962) and get $\sigma_{\rm cx}=$1.4\text{\times}{10}^{-18}$$ m2. We calculate
the cross section for polarization scattering as,

$\sigma_{\rm p}=2\pi Ze\left(\frac{\alpha_{\rm N}}{m_{\rm
r}}\right)^{1/2}\frac{1}{v_{\rm i}},$ (3.2.5)

where $Ze$ is the charge of the ion, $m_{\rm r}$ is the reduced mass,
$\alpha_{\rm N}=11.08/a^{3}$ is the polarizability of argon, $a$ is the Bohr
radius, and $v_{\rm i}=\sqrt{2k_{\rm B}T_{\rm i}/m_{\rm i}}$ is the ion
velocity (Lieberman & Lichtenberg, 2005). Note that the cross section is
inversely proportional to $v$, and therefore $\langle\sigma v\rangle_{\rm
p}=$6.96\text{\times}{10}^{-16}$$ m3 s-1 will be independent of velocity and
therefore temperature. To determine the ion-neutral collision rate we take the
sum of the polarization scattering rate and the rate of charge exchange.

The radial profiles for $p$, $n_{\rm e}$, $\chi_{\rm i}$, $T_{\rm e}$, and
$u_{\rm r}$ depend on $\eta_{\rm o}$ and $\eta_{\rm a}$, which in turn depend
on $p$, $n_{\rm e}$, $\chi_{\rm i}$, and $T_{\rm e}$. Therefore we use an
iterative approach. First, we calculate the radial profiles using an initial
guess for the diffusivities. Next, we use the average values from the radial
profiles to calculate $\eta_{\rm o}$ and $\eta_{\rm a}$, and then we
recalculate the radial profiles using the updated diffusivities. We continue
to iterate until the change in $\eta_{\rm o}+\eta_{\rm a}$ is less than 20 m2
s-1, which should be sufficient because even at the lowest pressures and
magnetic field strengths $\eta_{\rm o}+\eta_{\rm a}>400$ m2 s-1.

Finally, we define $k_{\rm r}\equiv\pi/(2(r_{\rm 2}-r_{\rm 1}))$ and $k_{\rm
z}\equiv\pi/2H$, where $H$ is the height of the apparatus. The instability
will appear first at these longer wavelengths, since flows characterized by
these wavelengths are less affected by the viscosity which tends to stabilize
the flow. We use an averaged orbital frequency,
$\Omega\equiv\sqrt{\Omega(r_{\rm 1})\Omega(r_{\rm 2})}$.

###  3.3 Predictions for MRI

This section presents the numerical solutions to Equation (3.1.12) using the
methods described in Section 3.2 for the range of parameters outlined in Table
1. Over a range of magnetic field strengths and pressures, we use a fiducial
power, $P=200$ kW, and aspect ratio, $H/(r_{2}-r_{1})=2$ ($H=2$ m, $r_{1}=0.4$
m, $r_{2}=1.4$ m). We use this fiducial aspect ratio because if
$H/(r_{2}-r_{1})<2$ the MRI cells will be squashed in the axial direction
making it difficult to produce. The aspect ratio in an actual protoplanetary
disc is $\ll 1$. However, our dispersion relation is a local approximation, so
our experiment should be thought of as a localized part of a disc that overall
has an aspect ratio $\ll 1$. We use $P=200$ kW because the power needs to be
high to reach sufficient ionization fractions (see Appendix C). The magnetic
fields probed here are several orders of magnitude larger than what is
predicted for a protoplanetary disc (Lesur et al., 2022). However, we are
limited on the lower end of our parameter range by the strength of Earth’s
magnetic field unless a magnetic shield is used.

Figure 3 shows the magnitudes of the diffusivities as a function of pressure
at $r_{2}$ with an initial magnetic field strength of $B_{\rm z}=10^{-3}$ T
and $B_{\rm z}=0.01$ T in the left and right panels, respectively. At both
magnetic field strengths ambipolar diffusivity is only significant at low
pressures, the Hall term is dominant at low to intermediate pressures, and
Ohmic diffusivity dominates at higher pressures. Ohmic diffusivity is
independent of magnetic field strength, while $\eta_{\rm A}\propto B^{2}$ and
$\eta_{\rm H}\propto B$. As a result, for $B_{\rm z}=10^{-3}$ T ambipolar
diffusion and the Hall effect are weaker and Ohmic diffusivity becomes
dominant at roughly $10^{2}$ mTorr versus 104 mTorr for $B_{\rm z}=0.01$ T.
Therefore, for lower magnetic field strengths, the Hall effect is only
dominant at the lowest pressures ($\lesssim 50$ mTorr).

For $B_{\rm z}=0.01$ T, ambipolar diffusion can be significant at the lowest
pressures, leading to the Hall effect being most dominant at intermediate
pressures around 100 mTorr. The Hall effect in this case remains dominant up
to around 5 Torr, at which point Ohmic diffusivity becomes dominant. Overall,
as magnetic field strength increases, the region of Hall dominance shifts to
higher pressures.

Which non-ideal effect is dominant at different pressures and magnetic field
strengths dictates at which pressures and magnetic field strengths there is
MRI growth, as Figure 4 shows. The black dots in Figure 4 show all the values
of $-B_{\rm z}$ and $p(r_{2})$ used in our calculation regardless of whether
MRI growth was found. Although we input an evenly logarithmically spaced grid
of $p(r_{1})$ from $10^{-3}$ – $10^{4}$ mTorr, when we solve Equation (2.0.2)
for $p(r)$, because $u_{\rm r}$ decreases by over 2 orders of magnitude from
$r_{1}$ to $r_{2}$, the pressure increases with radius (see Figure 2). As a
result, the minimum $p(r_{2})\sim 1$ mTorr. The results we show in this figure
are all calculated for negative values of $B_{\rm z}$, or in other words a
magnetic field that is anti-aligned with the axis of rotation.

The blue dots in Figure 4 show values of $-B_{\rm z}$ and $p(r_{2})$ at which
we get MRI growth with the Hall effect. The orange dots shows values at which
we get growth without the Hall effect, that is when we remove the Hall effect
from our dispersion relation by setting $\eta_{\rm H}=0$. While doing so does
not represent a physical case, it is informative to compare our results with
and without the Hall effect to study the impact the Hall effect has on the
stability of our proposed experiment.

At low magnetic field strengths, there is only MRI growth when $p(r_{2})$ is
less than a few hundred mTorr, because Ohmic diffusion quickly dominates over
the other non-ideal effects. Without the Hall term, there is only MRI growth
at these low to intermediate magnetic field strengths, because when the
magnetic field is weak, ambipolar diffusion is weak enough to allow MRI
growth. MRI growth extends to even lower pressures when the Hall term is
included suggesting Hall MRI growth can destabilize a flow otherwise
stabilized due to ambipolar diffusivity.

At $-B_{\rm z}\gtrsim 10^{-2}$ T ambipolar diffusion will be very strong,
especially at lower pressures. As a result, there will be no MRI growth
without the Hall effect. However, the Hall term is also larger for stronger
magnetic fields, and so it still becomes dominant over ambipolar diffusion at
intermediate pressures and acts to destabilize a flow stabilized due to
ambipolar diffusion. Furthermore, because Ohmic resistivity is independent of
magnetic field strength, the Hall effect remains dominant up to higher
pressures. For $-B_{\rm z}\gtrsim 0.1$ T, MRI growth becomes possible at
pressures up to $10-100$ Torr.

We show the logarithm of the number of e-folding times within the residual
time of the gas in the experiment in Figure 5. We show calculations with the
Hall effect in the left panel and without the Hall effect in the right panel.
The number of e-folding times ranges from around 4 – 200 for calculations with
the Hall effect and from around 6 – 40 for calculations without. When the
vertical magnetic field is anti-aligned with the axis of rotation, the Hall
effect not only increases the parameter-space for MRI instability, but also
increases the growth rate of the MRI leading to higher numbers of e-folding
times overall.

Figure 6 shows the logarithm of the Reynolds number (left panel), the magnetic
Reynolds number, $R_{\rm m}\equiv\Omega/k^{2}\eta_{\rm tot}$ (center panel),
and the hall parameter $h\equiv|\eta_{\rm H}|/\eta_{\rm tot}$ (right panel),
as a function of pressure and magnetic field strength for parameters that lead
to MRI growth for calculations with the Hall effect. Figure 7 shows the
distributions of $R_{\rm e}$, $R_{\rm m}$, and $h$ for parameters that have
MRI growth for calculations with the Hall effect (in blue) and calculations
without the Hall effect (in orange). The dashed black lines show the
distributions of $R_{\rm e}$, $R_{\rm m}$, and $h$ for parameters that have no
MRI growth. The Reynolds number for our range of parameters goes from roughly
$R_{\rm e}=1-10^{4}$ and increases as a function of pressure. MRI growth is
most common for intermediate Reynolds numbers.

The magnetic Reynolds number is much smaller for our range of parameters,
ranging from around $R_{\rm m}=10^{-3}-1$. The magnetic Reynolds number
decreases as the magnetic field strength increases, because $\eta_{\rm
A}\propto B^{2}$. Without the Hall effect, MRI growth is only possible for the
largest values of $R_{\rm m}$ in our calculations. MRI growth is possible for
a wider range of $R_{\rm m}$ if the Hall effect is included.

As Figure 3 suggests, the Hall parameter is strongest at intermediate
pressures, where ambipolar diffusion is no longer as strong. At these
intermediate pressures the Hall parameter increases as magnetic field strength
increases, because $\eta_{\rm H}\propto B$. Because the Hall effect needs to
be strong enough to overcome Ohmic resistivity and ambipolar diffusion, MRI
growth only occurs when $h>1$.

Figure 8 is the same as Figure 4, but for calculations with $P=$10 kW (top
panel), and $P=$100 kW (bottom panel). Decreasing the power reduces the range
of parameters where MRI growth is possible. Only at $P\gtrsim 100$ kW is it
possible to get MRI growth without the Hall effect. At $P=10$ kW it is only
possible to get MRI growth with strong magnetic fields. Our calculation with
$P=$1 kW had no MRI growth for $H=2$ m, $r_{2}-r_{1}=1$ m. However, for a
smaller apparatus MRI growth does occur at the largest magnetic field
strengths because the power per area increases, which increases the ionization
rate.

The results described above are all for cases in which the vertical magnetic
field is anti-aligned with the axis of rotation ($B_{\rm z}<0$). We also
performed calculations for which the magnetic field is aligned with the axis
of rotation ($B_{\rm z}>0$). For cases when $B_{\rm z}>0$, if we do not
include the Hall effect in our dispersion relation the results are identical
to when $B_{\rm z}<0$. This symmetry is expected because ambipolar diffusion
and Ohmic resistivity are diffusive effects which depend on even powers of
$B_{\rm z}$, and hence are independent of sign. On the other hand, when we do
include the Hall effect it acts to suppress MRI growth, leading to no MRI
growth for our range of parameters. Therefore, to achieve MRI growth it is
important to have $B_{\rm z}<0$.

##  4 Prototype experimental results

Figure 9: A diagram showing the overall setup of our prototype. The air
expelled from the fog machine goes through the top tube and enters the gap
between the concentric cylinders through a small opening along the edge of the
outer cylinder. The fog then rotates in between the cylinders and eventually
leaves through holes in the inner cylinder, pulled in by the fan. The exit
tube has an anemometer to measure the flux of the air. Figure 10: Photograph
of the apparatus from the top. The inner cylinder with holes is visible, as
well as the opening through which the light shines. During operation, we cover
all but a small section of the side panel so that light only enters at a
specific height.

###  4.1 Description of the apparatus

To test our experimental design we built a small non-magnetized prototype. The
prototype is composed of two concentric cylinders and uses air instead of
argon. The outer cylinder of our prototype has a radius of 29 cm and an
opening to let air emitted from a fog machine (DYNO-FOG II by American DJ)
enter. On another part of the outer cylinder is a small window for the
external light, which is provided by a laser sheet. We initially used a
construction lamp for the lighting, but found that the light did not penetrate
close enough to the inner cylinder. The inner cylinder has a radius of 7.5 cm
and it has 80 identical holes (10 ranks of 8 holes) to allow the gas to flow
out of the apparatus. The height of the prototype is 60 cm and the upper
boundary is plexiglass to allow filming of the flow with a camera. The entire
apparatus is airtight.

A gradient of pressure is imposed by a fan placed at the exit pipe. An
anemometer is also placed at the exit pipe to measure the flux of the gas. The
apparatus scheme is shown in Figure 9, and Figure 10 shows a photograph of the
prototype.

To run the experiment, we turn on the laser sheet, start the camera, start the
fog machine, and then turn the fan on. As the fog is rapidly sucked into the
apparatus, the camera records the fog that is illuminated by the laser sheet.
We describe our method of processing this recording in the following section.

###  4.2 Description of the Analysis Methods

To record our experiment, we use a phantom ir300 camera, which is able to
record 900 images per second. We optimized the height of the camera
($\sim$22.5cm above the apparatus), the camera lens, and the exposure time
(700 $\mu$s). To initially process the video, we use a program specifically
for this camera brand: Phantom Camera Control Software. Each frame of the
video is processed as a separate image.

The initial images produced are too low contrast to be analyzed. We improve
the contrast in two main steps. First, we compute a local averaged image using
10 images (no more so that we have approximately the same amount of fog in
every image). We then divide each image by this local averaged image to
normalize the images, improving the contrast. Unfortunately the noise is also
increased by this process. Therefore, our second step is to reduce the noise
using a local filter. We create this filter by replacing each pixel by a
median pixel of itself and its eight neighbor pixels. This process
significantly reduces the noise, as the noise is often localized around one
pixel. Additionally, it does not significantly affect the actual signal, which
is typically spread out over significantly more than eight pixels.

To track the motion of the fog we compute the correlations between different
regions of each image for 11 evenly spaced bins in radius and 100 evenly
spaced bins in azimuth. For example, if we have areas A and B, the image in
which area B most closely resembles area A is selected and then the time
difference between the two images and the angular displacement is used to
calculate an azimuthal velocity. To calculate the azimuthal velocity we assume
the radial velocity can be neglected. This assumption should be valid because
for this prototype we expect the $u_{\rm\theta}\gg|u_{\rm r}|$. However, over
longer timescales we could also track the radial velocity by looking at the
radial displacement instead. We also assume $u_{\rm z}$ is negligible, as the
bulk $u_{\rm z}$ should be near zero in our apparatus. However, there could be
velocity fluctuations due to turbulence, in particular in the Ekman layers.
Here, we avoid the Ekman layers where turbulence should be largest and only
measure $u_{\rm\theta}$ at intermediate heights where it should be much larger
than the other velocity components. In our apparatus the camera is located at
a specific height and only records movement in the azimuthal direction. Future
experiments could rotate the camera to determine $u_{\rm z}$, which could be
used to verify that $u_{\rm z}$ is negligible.

###  4.3 Experimental Results

Figure 11 shows the azimuthal velocity as a function of radius in our
experimental device. We show the mean of the distribution of azimuthal
velocities for each radius bin in blue. The errors are calculated as the
standard deviation of the azimuthal velocities in each radial bin divided by
the number of measurements in that bin. The errors are largest at radii closer
to the inner cylinder, because the inner radii are farthest from the light
(see Figure 9). Insufficient illumination appears to be the largest source of
error in our data. The flux measured by the anemometer in the exit pipe is
$D=0.11$ m3 s-1. Using Equation (2.0.12) and the kinematic viscosity of air at
room temperature and normal pressure, $\nu\approx 1.5\times 10^{-5}$ m2 s-1,
we find $K/\mu=$1.95\text{\times}{10}^{3}$$. Therefore, $K\gg\mu$ and we can
use Equation (2.0.7) to find a fit to the azimuthal velocity.

We first make the assumption that $F_{\rm\theta}=0$, which means equation
(2.0.7) becomes $u_{\rm\theta}=J/r$. The orange line in Figure 11 shows this
function for $u_{\rm\theta}$ fit to our data. We find $J=0.86\pm 0.02$ m2 s-1
with a large reduced-$\chi^{2}=70$. If we assume instead $F_{\rm\theta}<0$,
equation (2.0.7) becomes $u_{\rm\theta}=J/r+\Gamma r/2$ . The green line in
Figure 11 shows this function fit to our data. We find $J=0.66\pm 0.02$ m2 s-1
and $\Gamma=8.54\pm 0.66$ s-1. With these values reduced-$\chi^{2}=3.5$, which
is a factor of 20 smaller than for our fit with $F_{\rm\theta}=0$, suggesting
$F_{\rm\theta}<0$ in our experiment.

We show the radial velocity as a function of radius in Figure 12. We derive
this radial velocity profile using our fitted values of $J$ and $\Gamma$ and
the flux measured by the anemometer, $D$, to integrate equation (2.0.15). We
find that in our prototype the radial velocity is at least an order of
magnitude smaller than the azimuthal velocity, which validates our assumption
that the radial velocity can be neglected in order to determine the azimuthal
velocity, as described in Section 4.2.

Figure 11: The azimuthal velocity as a function of radius measured at a height
of 38 cm in our prototype experiment. We fit a model $u_{\rm\theta}=J/r$ to
our data, shown in orange, and find $J=0.86\pm 0.02$ m2 s-1 with
reduced-$\chi^{2}=70$. We also fit a model $u_{\rm\theta}=J/r+\Gamma r/2$ to
our data, shown in green, and find $J=0.66\pm 0.02$ m2 s-1 and $\Gamma=8.54\pm
0.66$ s-1 with reduced-$\chi^{2}=3.6$. Figure 12: The radial velocity as a
function of radius in our prototype. To find $u_{\rm r}(r)$ we integrate
equation (2.0.15) using the $J$ and $\Gamma$ from our fit in Figure 11.

Figure 13 again shows the azimuthal velocity as a function of radius, but now
for several different heights inside the apparatus. The radial profiles for
the various heights are mostly consistent with each other. The profiles for
heights of 28 cm and 48 cm differ the most from the other profiles at the
largest radii. However, they are still consistent to within around 20%.

Figure 13: Radial profiles of the azimuthal velocity taken at multiple
heights. The profiles for the top and bottom height differ the most from the
other profiles at the largest radii, but are still consistent to within $\sim
20\%$.

We use Equation (2.0.23) to get a Reynolds number for our system,
$Re=4.36\times 10^{5}$. We also use Equation (2.0.20) to find
$\alpha=2.85\times 10^{-6}$ in our prototype, which is many orders of
magnitude smaller than the $\alpha$ estimated from the accretion rates in
protoplanetary discs, $\alpha_{\rm ppd}\approx 10^{-2}$ (Hartmann et al.,
1998). Converting the kinematic viscosity of air at room temperature and
normal pressure to $\alpha$ we get, $\alpha_{\rm
air}=$4.2\text{\times}{10}^{-9}$$. Therefore for our prototype we have
$\alpha_{\rm air}\ll\alpha\ll\alpha_{\rm ppd}$, which means the viscosity is
small, but non-negligible.

This small viscosity could be due to either hydrodynamic turbulence or Ekman
effects. The residual time of the gas in the device, $\sim 1.4$ s, is around
an order of magnitude shorter than the Ekman time (Kageyama et al., 2004),

$\tau_{\rm E}\approx\frac{H}{2\sqrt{\nu\Omega}}\approx 12\rm{~{}s}.$ (4.3.1)

Therefore, hydrodynamic turbulence must play some role in driving the
viscosity. From Figure 11 we can see that at inner radii the azimuthal
velocity is lower than it would be if angular momentum is conserved (i.e.
$F_{\rm\theta}=0$), and at outer radii the azimuthal velocity is somewhat
higher, suggesting that this low level of turbulence is leading to angular
momentum being transferred outwards, which in turn leads to accretion.

##  5 Discussion and Conclusions

In this paper we have examined the feasibility of obtaining non-ideal MHD MRI
growth in a swirling argon gas laboratory experiment meant to resemble a
protoplanetary disc. First, we derive the hydrodynamic equilibrium flow of our
proposed experiment. Next, we derive the dispersion relation for MRI growth
with Ohmic resistivity, ambipolar diffusion, and the Hall effect. We then
solve this dispersion relation numerically for a simulated experiment over a
range of parameters. We find that for $P=200$ kW MRI growth is possible for
$10$ mTorr $\lesssim p(r_{2})\lesssim 1$ Torr and $10^{-3}$ T $\lesssim B_{\rm
z}\lesssim$2\text{\times}{10}^{-2}$$ T without the Hall effect and for $1$
mTorr $\lesssim p(r_{2})\lesssim 100$ Torr and $10^{-3}$ T $\lesssim|B_{\rm
z}|\lesssim 0.2$ T with the Hall effect, when the magnetic field is anti-
aligned with the axis of rotation. In cases where there is MRI growth several
e-folding times, sometimes over 100, will occur within the residual timescale.

At lower powers it is difficult to get MRI without the Hall effect and the
range of parameters over which MRI growth is possible with the Hall effect,
decreases as power decreases. In general the range of parameters over which
MRI growth is possible is larger when we include the Hall effect in our
dispersion relation than when we do not.

We also describe our un-magnetized prototype. Using only an anemometer at the
exit pipe and the method described in Section 4.2 to fit a model to
$u_{\rm\theta}(r)$, we are able to determine the radial velocity, Reynolds
number, and $\alpha$-parameter of our experiment. We are also able to validate
our assumptions that $K\gg\mu$ and that our flow is constant as a function of
height to within 20%.

We find that in our prototype the azimuthal velocity depends on a frictional
force $F_{\rm\theta}<0$. While the corresponding $\alpha$-parameter is much
lower than that of a protoplanetary disc, it is still significant compared to
the viscosity of air at room temperature and normal pressure, and appears to
lead to outward angular momentum transport.

We can use this $\alpha$-parameter as a baseline to compare to a future
magnetized experiment, like the one proposed here. However, because it is a
prototype, the apparatus described in Section 4 has a small radial extent and
low azimuthal velocities compared to our proposed magnetized experiment. In
particular, the azimuthal velocity in our proposed experiment will be
supersonic at standard atmospheric pressure, while it is subsonic in our
prototype. As a result, there will be several differences between the two. For
example, the Reynolds number in our proposed experiment will be smaller than
in our prototype (see Figure 7), because even though the azimuthal velocity in
our proposed experiment is over 100 times larger, the density will be several
orders of magnitude smaller. Another example is that, as Figure 2 shows, the
pressure in our proposed experiment will increase by around 2 orders of
magnitude from the inner to the outer radius, as opposed to being nearly
constant in our prototype. Because the gas temperature is assumed to be
constant in our proposed experiment, this increase in pressure suggests the
steady-state flow may not be incompressible as we have assumed when deriving
the dispersion relation in Section 3.1. Future work should derive a dispersion
relation for compressible gas.

Our calculations show that the Hall effect enhances the range of parameters
for which we get MRI growth and the rate of MRI growth when the magnetic field
is anti-aligned with the rotation axis ($B_{\rm z}<0$). Our results are
consistent with Wardle (1999) and Bai et al. (2015a) who showed the Hall
effect can enhance the MRI when $B_{\rm z}<0$.

On the other hand, we found that all MRI growth is suppressed when the
magnetic field and rotation axis are aligned. MRI growth is suppressed because
the Hall effect tends to stabilize the MRI when $B_{\rm z}>0$, and also
because our proposed experiment is stable to the Hall-Shear Instability (HSI,
Kunz, 2008). Our proposed experiment is HSI stable because our shear rate is
very low. As a result, HSI would only be achievable at high pressures and low
magnetic field strengths, where in our case Ohmic resistivity is strong enough
to suppress the HSI.

While our proposed experiment will probe whether MRI is possible in a non-
ideal MHD gas disc with similar gas number densities and gas pressures to
those assumed for a protoplanetary disc, we are still limited in our ability
to fully replicate the environment of a protoplanetary disc. First, we are
using argon gas for our experiment, when a protoplanetary disc is primarily
composed of hydrogen and helium gas, as well as many species of dust. Second,
our magnetic field strength will be several orders of magnitude stronger than
the predicted magnetic field strengths for protoplanetary discs, because we
are limited by the Earth’s magnetic field, $B\sim 10^{-4}$ T. Also, while we
probe values of plasma $\beta=P_{\rm th}/P_{\rm B}\gtrsim 10^{4}$ as are
expected for a protoplanetary disc (Lesur et al., 2022), our calculations only
produce MRI in cases where $\beta\leq 10$.

In addition, we have used an aspect ratio, $H/r>1$, when a protoplanetary disc
will have $H/r\ll 1$. However, we have used a local approximation to derive
our dispersion relation and so our experiment can be thought of as a local
approximation for a global disc that may as a whole have $H/r\ll 1$. Finally,
the ionization fractions ($\chi_{\rm i}\approx 10^{-5}$ – $10^{-1}$) in our
proposed experiment are many orders of higher than those for a protoplanetary
disc ($\chi_{\rm i}\approx 10^{-13}$ – $10^{-6}$, Lesur et al., 2022).
However, these ionization fractions are similar to those in the interstellar
medium, which is also subject to non-ideal MHD MRI turbulence.

While there are key differences between our proposed experiment and a
protoplanetary disk, it will still be useful to show whether the MRI can grow
in weakly-ionized gas subject to non-ideal effects. In addition, with this
experiment we should be able to explore the physics of the MRI in a parameter
study complementary in some respects (e.g. Reynolds number) to parameter
studies performed with numerical simulations, and without several of the
simplifications made in these simulations. Furthermore, the results of this
proposed experiment can be used to compare to analytic and numerical solutions
and provide insight on the significant physics to include.

The next step should be to design, and if feasible to build, an experiment of
radius on the order of $r_{2}=1.4$ m with a power on the order of $P=200$ kW.
Our calculations suggest that setting the pressure at the outer radius of the
apparatus to around $p(r_{2})=100$ mTorr and generating a vertical magnetic
field with a magnitude of around $B_{\rm z}=0.01$ T that is anti-aligned with
the axis of rotation is ideal for trying to produce MRI in the presence of
non-ideal MHD effects. Doing so would enable us to directly observe the MRI
under conditions similar to those of protoplanetary discs, which could provide
insight to the origin of angular momentum transport in these discs.

## Acknowledgements

We thank the anonymous referee for their helpful comments which improved the
clarity of this paper. A.S. is supported by a fellowship from the NSF Graduate
Research Fellowship Program under Grant No. DGE-1656466. H.J. acknowledges
support by the Laboratory Directed Research and Development (LDRD) Program at
Princeton Plasma Physics Laboratory under the U.S. Department of Energy
Contract No. DE-AC02-09CH11466. J.G. acknowledges support from NASA grant
17-ATP17-0094 and NSF grant AST-2108871. H.J. thanks Jill Foley, Michael
Kennelly, Mark Nornberg, Stefan Gerhardt, Brandon Fetroe, Yevgeny Raitses,
Jenna Kefeli, Hans Rinderknecht, Enrique Merino, Igor Kaganovich, Joey
McDonald, Alex Gurak, Erdem Oz, Cami Collins, Courtney Kaita, Bob Cutler, Eric
Edlund, Austin Wang, Kyle Kremer, Owen Williams, Lex Smits, Tapash Sarkar,
Matthew Basile, Samuel Greess, Erik Gilson, and Daniel Gift for their
contributions over many years to this project.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A Solution for the Azimuthal Velocity

To find the azimuthal velocity we first note that since $\Gamma$ has no $r$
dependence, we can write the homogeneous form of equation (2.0.5) as,

$\frac{\partial{u_{\rm\theta}}}{\partial{r}}+\frac{u_{\theta}}{r}+\frac{\mu}{K}\left[\frac{\partial}{\partial{r}}\left(r\frac{\partial{u_{\rm\theta}}}{\partial{r}}\right)-\frac{u_{\rm\theta}}{r}\right]=0,$
(A.0.2)

where we have set $\Gamma=0$. This equation is an equidimensional linear (or
Cauchy-Euler) ordinary differential equation (ODE), which have power-law
solutions. If we define $f$ such that $u_{\rm\theta}=f/r$, equation (A.0.2)
can be written as,

$f^{\prime\prime}=\frac{f^{\prime}}{r}\left(1-\frac{K}{\mu}\right),$ (A.0.3)

which we can integrate to get,

$f^{\prime}=ar^{1-\frac{K}{\mu}},$ (A.0.4)

and integrate again to get,

$u_{\rm\theta}=\frac{J}{r}+\frac{a}{2-\frac{K}{\mu}}r^{1-\frac{K}{\mu}}.$
(A.0.5)

Thus we have a general solution to equation (A.0.2), if $K/\mu\neq 2$. For
$K/\mu\approx 2$, $J/r$ is the only linearly independent power-law solution.
In this case we find a second linearly independent solution in the form of a
logarithm times the first solution, which gives us,

$u_{\rm\theta}=\frac{J}{r}+\frac{a}{r}\ln(r/r_{1}).$ (A.0.6)

We now look for a particular solution to

$\frac{\partial{u_{\rm\theta}}}{\partial{r}}+\frac{u_{\rm\theta}}{r}+\frac{\mu}{K}\left[\frac{\partial}{\partial{r}}\left(r\frac{\partial{u_{\rm\theta}}}{\partial{r}}\right)-\frac{u_{\rm\theta}}{r}\right]=\Gamma.$
(A.0.7)

Solutions to Cauchy-Euler inhomogenous ODEs have particular solutions $\propto
r$. We take $r\Gamma/2$ as a particular solution to this equation. Therefore
our solution to equation (2.0.5) is,

$u_{\rm\theta}=\frac{J}{r}+\frac{r\Gamma}{2}+\frac{a}{2-\frac{K}{\mu}}r^{1-\frac{K}{\mu}},$
(A.0.8)

where the last term would be replaced by $ar^{-1}\ln(r/r_{1})$ if
$K/\mu\approx 2$.

We now look at the two limiting cases, $K\ll\mu$ and $K\gg\mu$. The first
case, $K\ll\mu$, is straightforward. $K/\mu\rightarrow 0$ and we get

$u_{\rm\theta}=\frac{J}{r}+\frac{r\Gamma}{2},$ (A.0.9)

where $\Gamma\rightarrow\Gamma+a$.

For $K\gg\mu$, we define the function,

$f(x,y)=\frac{x^{(1-y)}}{2-y},$ (A.0.10)

where $y\equiv K/\mu$ and $x\equiv ra^{(1/(1-K/\mu))}$, which are both
dimensionless. The question is, what is the limit of $f(x,y)$ when
$y\rightarrow\infty$ at fixed $x$. If $|x|>1$, $f(x,y)\rightarrow 0$, when
$y\rightarrow\infty$. If $|x|<1$, $f(x,y)\rightarrow-\infty$, when
$y\rightarrow\infty$, which would mean a supersonic flow. Therefore, it is
necessary for $a=0$ to avoid a supersonic flow. This solution then gives us
Equation (A.0.9) for both limiting cases.

##  Appendix B A model to illustrate the physical meaning of $F_{\theta}$

To illustrate the physical meaning of $F_{\rm\theta}$, we relax our assumption
that $u_{\rm r}$ and $u_{\rm\theta}$ are independent of $z$. The azimuthal
component of the Navier-Stokes equation is now,

$\rho\left[u_{\rm r}\frac{\partial u_{\rm\theta}}{\partial r}+\frac{u_{\rm
r}u_{\rm\theta}}{r}\right]=\mu\left[\frac{1}{r}\frac{\partial}{\partial
r}\left(r\frac{\partial u_{\rm\theta}}{\partial
r}\right)-\frac{u_{\rm\theta}}{r^{2}}+\frac{\partial^{2}u_{\rm\theta}}{\partial
z^{2}}\right]$ (B.0.11)

We can then write the solution to this equation as,

$u_{\rm\theta}=\left(\frac{J}{r}+f(r)\right)g(z)$ (B.0.12)

Substituting equation (B.0.12) into equation (B.0.11), we can separate the two
variables and get,

$f^{\prime}+\frac{f}{r}+\frac{\mu}{K}\left(f^{\prime}+rf^{\prime\prime}-\frac{f}{r}\right)=-\frac{\mu
Jg^{\prime\prime}}{Kg}=\Gamma,$ (B.0.13)

where we have assumed $f(r)\ll J/r$. $\Gamma$ is still a constant because the
left part of the equation is only $r$-dependent and the middle part is only
$z$-dependent. The solution for the left hand side is the same as in Appendix
A. Solving for $g(z)$ yields,

$g(z)=a\cos\left(\sqrt{\frac{-\Gamma K}{\mu
J}}z\right)+b\sin\left(\sqrt{\frac{-\Gamma K}{\mu J}}z\right).$ (B.0.14)

If we take the coordinate origin at the center of the apparatus we have
$g(z)=g(-z)$ and $g(\frac{H}{2})=0$. These boundary conditions give us $b=0$
and $\Gamma=\frac{\mu J\pi^{2}}{KH^{2}}$. Thus we get,

$u_{\rm\theta}(r,z)=\left(\frac{J}{r}+\frac{J\pi^{3}\nu
r}{DH}+\frac{a}{2-\frac{D}{2\pi H\nu}}r^{1-\frac{D}{2\pi
H\nu}}\right)\cos\left(\frac{\pi z}{H}\right)$ (B.0.15)

after using $K=\frac{D\rho}{2\pi H}$. A numerical estimation with our
parameters confirms that the assumption $f(r)\ll\frac{J}{r}$ is valid. To
simplify we take the limiting case as in Equation (A.0.9),

$u_{\rm\theta}(r,z)=\left(\frac{J}{r}+\frac{J\pi^{3}\nu
r}{DH}\right)\cos\left(\frac{\pi z}{H}\right).$ (B.0.16)

This model is not fully correct because it does not take into account $u_{\rm
z}$, which has to exist because of the Ekman circulation. We use it here only
to illustrate a few intuitive points about the physical meaning of
$F_{\rm\theta}=-K\Gamma/r$. First, $\Gamma$ is proportional to $\nu$, which is
characteristic for a viscous drag force. Second, $\Gamma$ decreases as $H$
increases, meaning it is greatest at the center of the apparatus. Finally if
$D$ increases, the Reynolds number increases and thus $\Gamma/J$ decreases.
Therefore $F_{\rm\theta}$ represents viscous drag against the vertical
boundaries, including Ekman circulation.

##  Appendix C Model to compute electron density and temperature

In this section, we briefly summarize the model derived in Lieberman &
Lichtenberg (2005) for electron temperature and number density. We use this
model to compute these values in our numerical calculations. As a
simplification, we only take vertical loss into account, neglecting radial
loss to the axial magnetic field. In this Appendix all temperatures are given
in electronvolts.

###  C.1 Computation of $T_{\rm e}$

For a cylindrical discharge model in the low-to-intermediate ion mean free
path regime, we can determine the electron temperature from particle balance
by equating the total surface particle loss to the total volume ionization,

$n_{\rm e}u_{\rm B}A=K_{\rm iz}n_{\rm n}n_{\rm e}\pi r_{21}^{2}H,$ (C.1.17)

where $u_{\rm b}=\sqrt{T_{\rm e}/m_{\rm i}}$ is the characteristic velocity at
which ions flow out of the apparatus known as the Bohm velocity, $A=2\pi
r_{21}^{2}Hh_{\rm H}$ is the area for particle loss, $K_{\rm iz}$ is the
ionization rate constant, and $r_{21}=r_{2}-r_{1}$.

$h_{\rm H}=\frac{0.86}{\sqrt{3+\frac{H}{2\lambda_{\rm i}}+(0.86Hv_{\rm i}/(\pi
D_{\rm a}))^{2}}},$ (C.1.18)

is the edge to center density ratio (Lee & Lieberman, 1995), where
$\lambda_{\rm i}=(1/n_{\rm i}\sigma_{\rm i})$ is the ion mean free path,
$D_{\rm a}\approx D_{\rm i}T_{\rm e}/T_{\rm i}$ is the macroscopic ambipolar
diffusion coefficient, and $\sigma_{\rm i}$ is the ion collision cross-
section. The ambipolar diffusion coefficient is, $D_{\rm i}=\pi/8\lambda_{\rm
i}^{2}\nu_{\rm i}$, where $\nu_{\rm i}=v_{\rm i}/\lambda_{\rm i}$ is the ion
viscosity and $v_{\rm i}=\sqrt{8T_{\rm i}/\pi m_{\rm i}}$. Using these
definitions we can simplify the above equation to,

$h_{\rm H}=\frac{0.86}{\sqrt{3+\frac{H}{2\lambda_{\rm
i}}+0.19\left(\frac{H}{\lambda_{\rm i}}\right)^{2}\frac{T_{\rm i}}{T_{\rm
e}}}}.$ (C.1.19)

We can simplify equation C.1.17 to obtain,

$\frac{K_{\rm iz}(T_{\rm e})}{u_{\rm B}(T_{\rm e})}=\frac{1}{n_{\rm n}d_{\rm
eff}},$ (C.1.20)

where $d_{\rm eff}=H/2h_{\rm H}$ is an effective plasma size for particle
loss. Using the electron temperature dependence of $K_{\rm iz}$ and $u_{\rm
b}$ we get (see Figure 10.1 of Lieberman & Lichtenberg (2005)),

$T_{\rm e}=\frac{1\rm{~{}eV}}{0.167+0.1315\log_{\rm 10}(n_{\rm n}d_{\rm
eff})},$ (C.1.21)

Because $K_{\rm iz}$ varies exponentially with $T_{\rm e}$, $T_{\rm e}$ only
varies logarithmically with $n_{\rm n}$ and $d_{\rm eff}$. Figure 2 shows that
$T_{\rm e}$ varies by only a factor of 2 from the inner to the outer radius of
our proposed experiment, even though the gas density increases by around 2
orders of magnitude. In addition, the temperature depends only on particle
conservation, not on the electron density. Thus it is independent of the input
power.

To accurately compute $T_{\rm e}$ in our numerical calculation, we first make
an initial guess for $T_{\rm e}$ and use this value to compute $h_{\rm H}$. We
then use this value for $h_{\rm H}$ to re-compute $T_{\rm e}$. We iterate
through this process until $T_{\rm e}$ converges to within $10^{-3}$ eV.

###  C.2 Computation of $n_{\rm e}$

We use the electron temperature to compute the electron number density. We
assume Maxwellian electrons absorbing electrical power, $P$, and equate the
total power absorbed to the total power lost,

$P=en_{\rm e}u_{\rm B}AE_{\rm tot},$ (C.2.22)

where $E_{\rm tot}=E_{\rm i}+E_{\rm e}+E_{\rm c}$ is the total energy lost,
which depends on the collisional energy lost per electron-ion pair created,
$E_{\rm c}$, as well as the kinetic energy carried by electrons and ions to
the walls, $E_{\rm e}$ and $E_{\rm i}$, respectively.

For Maxwellian electrons $E_{\rm e}=2T_{\rm e}$. $E_{\rm i}$ is the sum of the
ion energy entering the sheath ($1/(2m_{\rm i})u_{\rm B}^{2}=T_{\rm e}/2$) and
the energy gained by the electron as it traverses the sheath ($\sim 4.7T_{\rm
e}$ for an insulating wall). We approximate $E_{\rm c}$ as,

$\log_{\rm 10}E_{\rm c}=2.18e^{-3\log_{\rm 10}T_{\rm e}}+1.28,$ (C.2.23)

which accounts for the loss of electron energy due to ionization, excitation,
and polarization scattering against neutral atoms (Lieberman & Lichtenberg,
2005; Gudmundsson, 2002).

We can solve equation (C.2.22) for $n_{\rm e}$ to get,

$n_{\rm e}=\frac{P}{eU_{\rm b}A_{\rm eff}E_{\rm tot}}.$ (C.2.24)