# On the Grain-Modified Magnetic Diffusivity in Protoplanetary Disks

Rui Xu11affiliationmark:  22affiliationmark:  , Xue-Ning Bai33affiliationmark:
1 Department of Astrophysical Sciences, Princeton University, Princetion, NJ
08544<EMAIL_ADDRESS>2 Yuanpei College, Peking University, Beijing,
100871, China  3 Institute for Theory and Computation, Harvard-Smithsonian
Center for Astrophysics, 60 Garden St., MS-51, Cambridge, MA, 02138;
<EMAIL_ADDRESS>
# On the Grain-Modified Magnetic Diffusivities in Protoplanetary Disks

Rui Xu11affiliationmark:  22affiliationmark:  , Xue-Ning Bai33affiliationmark:
1 Department of Astrophysical Sciences, Princeton University, Princetion, NJ
08544<EMAIL_ADDRESS>2 Yuanpei College, Peking University, Beijing,
100871, China  3 Institute for Theory and Computation, Harvard-Smithsonian
Center for Astrophysics, 60 Garden St., MS-51, Cambridge, MA, 02138;
<EMAIL_ADDRESS>
###### Abstract

Weakly ionized protoplanetary disks (PPDs) are subject to non-ideal-
magnetohydrodynamic (MHD) effects including Ohmic resistivity, the Hall effect
and ambipolar diffusion (AD), and the resulting magnetic diffusivities
($\eta_{O},\eta_{H}$ and $\eta_{A}$) largely control the disk gas dynamics.
The presence of grains not only strongly reduces disk ionization fraction, but
also modify the scalings of $\eta_{H}$ and $\eta_{A}$ with magnetic field
strength. We derive analytically asymptotic expressions of $\eta_{H}$ and
$\eta_{A}$ in both strong and weak field limits and show that towards strong
field, $\eta_{H}$ can change sign (at a threshold field strength $B_{\rm
th}$), mimicking a flip of field polarity, and AD is substantially reduced.
Applying to PPDs, we find that when small $\sim 0.1$ ($0.01$)$\mu$m grains are
sufficiently abundant [mass ratio $\sim 0.01$ ($10^{-4}$)], $\eta_{H}$ can
change sign up to $\sim 2-3$ scale heights above midplane at modest field
strength (plasma $\beta\sim 100$) over a wide range of disk radii. Reduction
of AD is also substantial towards the AD dominated outer disk and may activate
the magneto-rotational instability. We further perform local non-ideal MHD
simulations of the inner disk (within 10 AU) and show that with sufficiently
abundant small grains, magnetic field amplification due to the Hall-shear
instability saturates at very low level near the threshold field strength
$B_{\rm th}$. Together with previous studies, we conclude by discussing the
grain-abundance-dependent phenomenology of PPD gas dynamics.

###### Subject headings:

accretion, accretion disks — magnetohydrodynamics — planetary systems:
protoplanetary disks

##  1\. Introduction

Gas dynamics of the weakly ionized protoplanetary disks (PPDs) is to a large
extent controlled by its coupling with magnetic fields, described by non-ideal
magnetohydrodynamic (MHD) effects, including Ohmic resistivity, the Hall
effect and ambipolar diffusion (AD). The three effects dominate in different
regions of PPDs, and affect the gas dynamics in different ways. A constant
theme of PPD research has been to estimate the strength of magnetic
diffusivities, and study their consequences to gas dynamics, especially on the
level of turbulence and efficiency of angular momentum transport (see Turner
et al., 2014 for recent review).

The strength of magnetic diffusivities largely depends on gas density $\rho$,
level of ionization, and magnetic field strength $B$. In the absence of
grains, the Ohmic, Hall and ambipolar diffusivities can be expressed in
particularly simple form (Salmeron & Wardle, 2003)

$\eta_{O}=\eta_{e}\ ,\quad\eta_{H}=\frac{cB}{4\pi en_{e}}\
,\quad\eta_{A}=\frac{B^{2}}{4\pi\gamma_{i}\rho\rho_{i}}\ ,$ (1)

where $\eta_{e}\propto(n/n_{e})$ is resistivity due to electrons, $n_{e}$, $n$
are the electron and neutral number densities, and $\rho_{i}$ is the ion
density.

These expressions are complicated by the presence of dust grains. On the one
hand, grains can substantially reduce the level of ionization, and hence
boosting all three magnetic diffusivities proportionally. On the other hand,
grains carry charge and result in much more complex dependence of
diffusivities on magnetic field strength (e.g., see Wardle, 2007 and Bai,
2014a), yet such complex dependence has not been systematically explored.

As an initial effort, Bai (2011b) (hereafter B11) considered the limit when
grains are tiny ($\sim$nm sized PAHs) so that the conductivity of charged
grains is similar to ions. He showed that when tiny grains are the dominant
charge carrier, Hall diffusivity diminishes and AD is substantially reduced in
the strong field limit, which may be important in the outer region of PPDs.

In this paper, we extend the work of B11 to more general situations with more
typical grain sizes (sub-micron to micron) relevant to PPDs. We focus on the
disk interior (rather than surface) where grains are likely the dominant
charge carrier, and are mostly singly charged, and systematically explore how
magnetic diffusivities depend on field strength. We hope to provide
researchers a useful guidance on modeling non-ideal MHD effects from
ionization chemistry, which has been fruitful in the recent years (e.g.,
Turner et al., 2007; Bai & Stone, 2013; Bai, 2013, 2014b, 2015; Lesur et al.,
2014; Gressel et al., 2015; Simon et al., 2015).

This work is also motivated by the recent advances in understanding the gas
dynamics of the Hall effect, where the Hall-shear instability may
substantially amplify the midplane field strength in the inner region of PPDs
(Kunz, 2008; Lesur et al., 2014; Bai, 2014b) when the background field is
aligned with disk rotation. With strong field amplification, we discuss
whether grain-modified diffusivities can feedback to PPD gas dynamics.

We present our simplified model in Section 2, with application to PPDs
discussed in Section 3. We perform MHD simulations of the inner disk and
discuss the results in Section 4. In Section 5, we summarize and conclude.

##  2\. A Simplified Model for Grain-modified Diffusivities

In weakly ionized gas, the general expressions for the three magnetic
diffusivities are (Wardle, 2007; Bai, 2011a)

$\begin{split}&\eta_{O}=\frac{c^{2}}{4\pi}\frac{1}{\sigma_{O}}\
,\quad\eta_{H}=\frac{c^{2}}{4\pi}\frac{\sigma_{H}}{\sigma_{H}^{2}+\sigma_{P}^{2}}\
,\\\
&\eta_{A}=\frac{c^{2}}{4\pi}\frac{\sigma_{P}}{\sigma_{H}^{2}+\sigma_{P}^{2}}-\eta_{O}\
,\end{split}$ (2)

where $\sigma_{O},\sigma_{H},\sigma_{P}$ are Ohmic, Hall and Pederson
conductivities, defined as

$\begin{split}&\sigma_{O}=\frac{ec}{B}\sum_{j}n_{j}|Z_{j}|\beta_{j}\ ,\\\
&\sigma_{H}=\frac{ec}{B}\sum_{j}\frac{n_{j}Z_{j}}{1+\beta_{j}^{2}}\ ,\\\
&\sigma_{P}=\frac{ec}{B}\sum_{j}\frac{n_{j}|Z_{j}|\beta_{j}}{1+\beta_{j}^{2}}\
,\end{split}$ (3)

where summation goes over all charged species $j$, with $n_{j}$, $Z_{j}e$
being the number density and charge of individual charged species. The Hall
parameter $\beta_{j}$ is given by

$\beta_{j}=\frac{|Z_{j}|eB}{m_{j}c}\frac{1}{\gamma_{j}\rho}\ ,$ (4)

which describes the ratio between the gyro-frequency of the charged species
and its collision frequency with the neutrals, where
$\gamma_{j}\equiv\langle\sigma v\rangle_{j}/(\mu_{n}+m_{j})$, with
$\langle\sigma v\rangle_{j}$ being rate coefficient for collisional momentum
transfer with the neutrals. The charged species $j$ is strongly coupled to the
neutrals if $\beta_{j}\ll 1$ (weak field), and it is strongly coupled to
magnetic fields when $\beta_{j}\gg 1$ (strong field). The values of
$\langle\sigma v\rangle_{j}$ for electron, ion and grain can be found in
Equation (14)-(16) of (Bai, 2011a, 2014a), giving Hall parameter

$\begin{split}\beta_{e}&\approx 2.1(B/n_{15})\min\ [1,\ T_{100}^{-1/2}]\ ,\\\
\beta_{i}&\approx 3.3\times 10^{-3}(B/n_{15})\ ,\\\
\beta_{g}&\approx(B/n_{15})\min\ [3.2\times 10^{-3}|Z_{g}|^{-1},\
10^{-9}a_{1}^{-2}T_{100}^{-1/2}]\end{split}$ (5)

where $B$ is magnetic field measured in Gauss,
$n_{15}=n_{H}/10^{15}\rm{cm^{3}}$, $Z_{g}$ is grain charge, $a_{1}$ is grain
size in $\mu m$, disk temperature $T_{100}=T//100{\rm K}$. Clearly,
$\beta_{e}\gg\beta_{i}$, and unless grain size is tiny ($a\lesssim$nm),
$\beta_{i}\gg\beta_{g}$.

As long as ion mass is much larger than neutral mass, different ion species
have their Hall parameters very close to $\beta_{i}$ quoted above, and can be
treated as a single species. In the interior of PPDs where ionization level is
low, chemistry calculations show that (small) grains are largely singly
charged until the ionization fraction exceeds grain abundance towards disk
surface (e.g., Figure 6 of Wardle, 2007 and Figure 1 of Bai, 2011a). We also
assume single-sized grains for simplicity, which is typically adopted in PPD
chemistry calculations. While a grain size distribution is expected in
reality, we expect a single-size treatment to capture and better clarify the
essence of their effects. Therefore, we consider only four types of charged
species: electrons, ions, and positive/negative charged grains, whose number
densities are represented by $n_{e}$, $n_{i}$, $n_{\rm gr}^{+}$ and $n_{\rm
gr}^{-}$.

Under these conditions, the Hall and Pederson conductivities read

$\begin{split}\sigma_{H}=&\frac{-1}{1+\beta_{e}^{2}}+\frac{n_{i}}{n_{e}}\frac{1}{1+\beta_{i}^{2}}+\frac{n_{gr}^{+}-n_{gr}^{-}}{n_{e}}\frac{1}{1+\beta_{g}^{2}}\
,\\\
\sigma_{P}=&\frac{\beta_{e}}{1+\beta_{e}^{2}}+\frac{n_{i}}{n_{e}}\frac{\beta_{i}}{1+\beta_{i}^{2}}+\frac{n_{\rm
gr}^{+}+n_{\rm gr}^{-}}{n_{e}}\frac{\beta_{g}}{1+\beta_{g}^{2}}\ .\end{split}$
(6)

Hereafter, we omit the prefactor $en_{e}c/B$ in conductivities and $cB/4\pi
en_{e}$ in diffusivities so as to make them dimensionless. The grain-free
diffusivities (1) corresponds to $\eta_{O}=1/\beta_{e}$, $\eta_{H}=1$ and
$\eta_{A}=\beta_{i}$, and hence the conventional Ohmic regime applies in weak
field with $\beta_{e}<1$, AD regime applies in strong field $\beta_{i}>1$, and
the Hall dominated regime lies in between. For brevity, we further define
$n_{\rm gr}^{\pm}\equiv n_{\rm gr}^{+}+n_{\rm gr}^{-}$.

Charge neutrality gurantees that only three of the four number densities are
independent. With additional normalization by $n_{e}$, only two of them serve
as independent model parameters. We find it useful to choose the two ratios
$n_{i}/n_{e}$, $n_{\rm gr}^{\pm}/n_{i}$ as model variables. Without grains,
their values are obviously $1$ and $0$, while when grains become important
charge carriers, $n_{i}\gg n_{e}$, and $n_{\rm gr}^{\pm}>n_{i}$.

The Hall parameters $\beta_{e,i,g}\propto B$ can be considered as a proxy for
magnetic field strength. According to (5), we fix $\beta_{i}=10^{-3}\beta_{e}$
in our calculations. The ratio $\beta_{g}/\beta_{i}\propto a^{-2}$ reflects
grain size and defines our last (third) model parameter. For $0.1\mu$m sized
grains, $\beta_{g}\sim 10^{-4}\beta_{i}$, and only for $a\sim 10^{-3}\mu$m we
have $\beta_{i}\sim\beta_{g}$.

Figure 1.— Dimensionless Ohmic (black), Hall (blue), and the ambipolar (red)
diffusivities as a function of magnetic field strength (manifested as the
electron Hall parameter $\beta_{e}$) under various representative parameters.
We set $n_{i}/n_{e}=1.2$ and $10^{4}$ in the top and bottom panels,
respectively. On the left two panels, we have $\beta_{g}/\beta_{i}=10^{-6}$
($\mu$m sized grain), while in right two panels we have
$\beta_{g}/\beta_{i}=10^{-2}$ ($\sim 0.01\mu$m sized grain). In each panel,
solid lines correspond to $(n_{gr}^{+}+n_{gr}^{-})/n_{i}=1$ ,while the dashed
lines correspond to $(n_{gr}^{+}+n_{gr}^{-})/n_{i}=10$.

###  2.1. Asymptotic Relations

We are mainly interested in reducing (6) to sufficiently simple and intuitive
expressions. Ohmic resistivity is the most straightforward and can be written
as

$\eta_{O}=\frac{1}{\beta_{e}(1+\theta_{1}+\theta_{2})}\ ,$ (7)

where we have defined

$\theta_{1}=\frac{n_{i}}{n_{e}}\frac{\beta_{i}}{\beta_{e}}\
,\quad\theta_{2}=\frac{n_{\rm gr}^{\pm}}{n_{e}}\frac{\beta_{g}}{\beta_{e}}\ .$
(8)

In general, Ohmic resistivity results from electron conductivity.
Contributions from the ions and grains become significant
($\theta_{1,2}\gtrsim 1$) only when $n_{i}/n_{e}>\beta_{e}/\beta_{i}\approx
1000$ and $n_{\rm gr}^{\pm}/n_{e}\gtrsim\beta_{e}/\beta_{g}$, both requiring
grain charge density to be overwhelmingly dominant.

Calculations of the Hall and ambipolar diffusivities involve much more complex
algebra, and sufficiently simple expressions can only be obtained in
asymptotic regimes. In the weak field regime (equivalently, Ohmic dominated
regime) where $\beta_{g}\lesssim\beta_{i}\ll\beta_{e}\ll 1$, we Taylor expand
the expressions for $\sigma_{H}$, $\sigma_{P}$ in equation (6) and keep the
leading order terms111We have also assumed $n_{i}/n_{e}\ll 10^{6}$, or
$\beta_{i}\theta_{1}\ll\beta_{e}$, which is almost always the case from
chemistry calculations.. The results are

$\sigma_{H}\approx\beta_{e}^{2}\
,\quad\sigma_{P}\approx\beta_{e}[(1+\theta_{1}+\theta_{2})-\beta_{e}^{2}]\
.\\\ $ (9)

Substituting them to Equation (2), we obtain

$\begin{split}\eta_{H}&\approx\frac{1}{(1+\theta_{1}+\theta_{2})^{2}}\ .\\\
\eta_{A}&\approx\frac{\beta_{e}(\theta_{1}+\theta_{2})}{(1+\theta_{1}+\theta_{2})^{3}}=\beta_{i}\frac{n_{i}+n_{\rm
gr}^{\pm}(\beta_{g}/\beta_{i})}{n_{e}(1+\theta_{1}+\theta_{2})^{3}}\
.\end{split}$ (10)

These expressions reduces exactly to Equation (15) of B11
($\beta_{g}=\beta_{i}$) by replacing $\theta_{1}+\theta_{2}$ by his $\theta$,
and replacing $n_{i}+n_{\rm gr}^{\pm}(\beta_{g}/\beta_{i})$ by his
$\bar{n}\equiv n_{i}+n_{\rm gr}^{\pm}$. As already discussed there, ambipolar
diffusivity is enhanced by the presence of charged grains, and the Hall
diffusivity can be modestly reduced when $\theta_{1,2}\gtrsim 1$.

In the strong field limit where $1\ll\beta_{g}\lesssim\beta_{i}\ll\beta_{e}$,
we can reduce the Hall and Pederson conductivities to

$\begin{split}\sigma_{H}&\approx\frac{\theta_{1}\beta_{e}}{\beta_{i}}\bigg{[}\frac{1}{\beta_{i}^{2}}-\frac{1}{\beta_{g}^{2}}\bigg{]}+\frac{1}{\beta_{g}^{2}}\
,\\\
\sigma_{P}&\approx\beta_{e}\bigg{[}\frac{\theta_{1}}{\beta_{i}^{2}}+\frac{\theta_{2}}{\beta_{g}^{2}}\bigg{]}\
.\end{split}$ (11)

The corresponding Hall and ambipolar diffusivities are

$\begin{split}\eta_{H}&\approx\frac{\left[\theta_{1}\beta_{e}(\beta_{g}^{2}-\beta_{i}^{2})+\beta_{i}^{3}\right]\beta_{i}\beta_{g}^{2}}{\beta_{e}^{2}(\beta_{g}^{2}\theta_{1}+\beta_{i}^{2}\theta_{2})^{2}}\
,\\\
\eta_{A}&\approx\frac{\beta_{i}^{2}\beta_{g}^{2}}{\beta_{e}(\theta_{1}\beta_{g}^{2}+\theta_{2}\beta_{i}^{2})}\
.\end{split}$ (12)

In the tiny grain limit of $\beta_{g}=\beta_{i}$, these relatively complex
expressions can be further reduced to

$\begin{split}\eta_{H}&\approx\frac{\beta_{i}^{2}}{\beta_{e}^{2}\theta^{2}}=\left(\frac{n_{e}}{\overline{n}}\right)^{2}\
,\\\
\eta_{A}&\approx\frac{(\theta_{1}+\theta_{2})\beta_{i}^{6}}{\beta_{e}\beta_{i}^{4}(\theta_{1}+\theta_{2})^{2}}=\frac{\beta_{i}n_{e}}{\overline{n}}\
,\end{split}$ (13)

which agree exactly with equation (16) of B11.

From Equation (12), we see that in the strong field regime, $\eta_{H}$ becomes
negative when

$\theta_{1}\beta_{e}(\beta_{g}^{2}-\beta_{i}^{2})+\beta_{i}^{3}<0\ ,\ {\rm
or}\ \ \beta_{g}^{2}<\beta_{i}^{2}(1-\frac{n_{e}}{n_{i}})\ .$ (14)

In general, the sign of $\eta_{H}$ depends on the relative mass (mobility)
between the positive and negative charge carrier. In the absence of grains,
with electrons much more mobile than ions, $\eta_{H}$ is positive. In the
presence of tiny grains with $\beta_{g}=\beta_{i}$ as considered by B11,
charged grains are as mobile as normal ions. Therefore, negative charge
carriers are still effectively more mobile due to the presence of electrons,
and hence $\eta_{H}\geq 0$. Relation (14) states that, as long as grains are
not too small (i.e. well above nanometer size so that
$\beta_{g}\ll\beta_{i}$), $\eta_{H}$ can become negative whenever there are
more ions than electrons. In other words, there are more negatively charged
grains than positively ones, and hence negative charge carriers are
effectively less mobile. We will see in Section 3 that $n_{i}>n_{e}$ almost
always holds from ionization chemistry calculations in PPDs. The main
practical question then is whether disk magnetic field can reach the level
such that $\eta_{H}$ changes sign. We therefore define $B_{\rm th}$ as the
threshold field strength beyond which $\eta_{H}$ changes sign, and this is a
main quantity of interest that we study in the rest of the paper.

###  2.2. Representative Cases

In Figure 1, we show the dimensionless Ohmic, Hall and the ambipolar
diffusivities as a function of magnetic field strength (characterized by the
electron Hall parameter), covering most of the parameter space relevant for
grain size $a\gtrsim 0.01\mu$m (see figure caption). We first confirm that
$\eta_{H}\propto B$ and $\eta_{A}\propto B^{2}$ holds both in weak
($\beta_{e}\ll 1$) and strong ($\beta_{g}\gg 1$) field regimes as expected. In
between, we see that both $\eta_{H}$ and $\eta_{A}$ exhibit complex variations
with $B$, where $\eta_{H}$ changes sign in all cases, and $\eta_{A}$ is
reduced in steps compared with the $B^{2}$ scaling.

Top two panels of Figure 1 correspond to cases where grains are modestly
important charge carrier, giving $n_{i}/n_{e}$ not far from order unity. We
see that for field strength with $\beta_{i}<1$, $\eta_{H}$ and $\eta_{A}$
behave normally as in conventional Ohmic and Hall regimes. Sign change of
$\eta_{H}$ and reduction of $\eta_{A}$ are both achieved towards stronger
field (in the conventional AD regime) with $\beta_{i}>1$ (and $\beta_{g}<1$).

On the other hand, in the bottom two panels, assuming grains are the dominant
charge carriers such that $n_{i}/n_{e}\gg 1$, we see that magnetic
diffusivities behaves qualitatively differently. First, the threshold field
strength $B_{\rm th}$ for $\eta_{H}$ to change sign is much weaker, close to
$\beta_{e}=1$. Similarly, reduction of $\eta_{A}$ starts right from
$\beta_{e}=1$. Secondly, Ohmic resistivity dominates all the way through the
conventional Hall-regime ($\beta_{e}>1$, $\beta_{i}<1$), while the Hall effect
dominates only in much stronger field ($\beta_{i}>1$ and $\beta_{g}<1$), which
is in the conventional AD-dominated regime. AD dominates only when the field
is much stronger with $\beta_{g}>1$.

Overall, we see that the ratio $n_{i}/n_{e}$ largely controls the dependence
of $\eta_{H}$ and $\eta_{A}$ on magnetic field strength, especially towards
weaker field ($\beta_{i}\lesssim 1$). Other parameters (grain size and $n_{\rm
gr}^{\pm}/n_{i}$) mainly affect diffusivity behaviors towards strong field
with $\beta_{i}>1$, leading to complex dependence before reaching the
asymptotic regime (12) at $\beta_{g}>1$, and trend can be identified by
comparing the left and right panels of Figure 1.

Figure 2.— Results from ionization chemistry calculations at 1 AU (left) and 5
AU (right). Top panels: abundance of all charged species as a function of
height $z$ above disk midplane (normalized to disk scale height $H$), where
gr$n$(+/-) denote positively/negatively charged grains with grain charge $n$,
and the red/blue lines represent abundances of electrons/all ions. Bottom
panels: solid lines show threshold field strength $B_{\rm th}$ (beyond which
$\eta_{H}$ changes sign, in Gauss, see legend) as a function of $z$ based on
the chemistry calculations in the top panels. Also shown are the field
strengths that correspond to plasma $\beta=1$, 100 and $10^{4}$ (grey), as
well as field strengths that give electron/ion Hall parameters $\beta_{e}=1$
and $\beta_{i}=1$ (pink). Three combinations of grain size $a$ and dust-to-gas
mass ratios $f$ are considered in the bottom panels, as marked in legend,
while the top panels correspond to the $a=0.1\mu$m and $f=10^{-2}$ case.

##  3\. Application to Protoplanetary Disks: Ionization Chemistry

To test our analytical results, we conduct ionization-recombination chemistry
calculation using a complex reaction network developed in Bai & Goodman (2009)
and Bai (2011a) that followed from Ilgner & Nelson (2006). There are 175 gas-
phase species and over two thousands gas phase reactions extracted from the
latest version of the UMIST database (McElroy et al., 2013). The rate
coefficients in the data base are given as a function of temperature. One
change from our previous chemistry calculations is that when gas temperature
lies out of the stated range of validity, we still use the given formula as if
it remained valid (whereas originally we computed the coefficients using upper
or lower bound of valid temperature range). For inner disk temperatures ($\sim
100-300$K), about $10\%$ of the reactions are affected, majority of which are
neutral-neutral reactions, and/or reactions with large activation energy. In
the latter case, our original approach tends to significantly overestimate the
reaction rates at lower temperatures. Also, for many of these reactions, we
note that the rate coefficients listed in the KIDA chemical database generally
have different valid temperature ranges with similar coefficients. This change
does not affect the level of ionization in the presence of grains (but leads
to higher ionization fraction in the grain-free case), while the composition
of ion species is affected.

We include a single grain population with maximum grain charge of $\pm 2$,
with fiducial choices of grain size $a=0.1\mu$m and dust-to-gas mass ratio of
$f=10^{-2}$. Although such grain size and abundance are likely exaggerated
considering that substantial grain growth must have occurred in PPDs (e.g.,
Birnstiel et al., 2010), they are the best to illustrate the effects that we
discuss in this paper. Given that large uncertainties remain in our
understandings of grain size distribution in PPDs (e.g., D’Alessio et al.,
2006), we also consider a few other combinations of grain sizes and abundances
to study parameter depdendnce. Note that for chemical purposes, we are only
concerned with the abundance of sub-micron sized grains, which is the most
relevant to ionization and charging. We adopt the minimum-mass solar nebular
(MMSN) disk model that is vertically isothermal with standard cosmic-ray and
X-ray ionization prescriptions as described in Bai (2011a), together with a
constant ionization rate of $7\times 10^{-19}$ s-1 as a proxy for radioactive
decay (the exact value matters little).

The relative importance of non-ideal MHD effects in PPDs is conveniently
described by the dimensionless Elsasser numbers, defined as

$\Lambda\equiv\frac{v_{A}^{2}}{\eta_{O}\Omega_{K}}\
,\quad\chi\equiv\frac{v_{A}^{2}}{\eta_{H}\Omega_{K}}\ ,\quad
Am\equiv\frac{v_{A}^{2}}{\eta_{A}\Omega_{K}}\ ,$ (15)

where $v_{A}^{2}=B^{2}/4\pi\rho$ is the Alfvén speed, $\Omega_{K}$ is disk
angular frequency. Non-ideal MHD effects become dominant when the (absolute
value of) respective Elsasser numbers becomes smaller than order unity. Note
that in our definition, $\chi$ will change sign when $\eta_{H}$ becomes
negative.

Figure 3.— Threshold field strength $B_{\rm th}$ (in Gauss) beyond which the
Hall coefficient changes sign, plotted as a function of disk radius $R$ from
ionization chemistry calculations at disk midplane (left) and two scale
heights above midplane (right). Five combinations of grain sizes $a$ and dust-
to-gas mass ratios $f$ are shown, as marked in the legend. To guide the eye,
we also show field strengths that correspond to plasma $\beta=1$ and 100
(black solid), as well as field strengths that give electron/ion Hall
parameters $\beta_{e}=1$ and $\beta_{i}=1$ (grey dashed).

###  3.1. Threshold Field Strength in PPDs

In Figure 2, we show the vertical profile of charged species abundance at 1 AU
and 5 AU using fiducial grain size ($a=0.1\mu$m) and mass ratio ($f=10^{-2}$)
in the top two panels. We see that grains are the dominant charge carriers
within $\sim 2-3$ disk scale heights ($H$) about disk midplane, leading to
large ratios of $n_{i}/n_{e}$. This situation is similar to the solid curves
in the two bottom panels of Figure 1, and we have confirmed that our simple
four-species model well captures the $\eta_{H}$ and $\eta_{A}$ behaviors
computed from the full chemical abundances using (LABEL:eq:sigfull). as long
as most grains are singly charged.

In the bottom panels of Figure 2, we further show the threshold field strength
$B_{\rm th}$ (beyond which $\eta_{H}$ changes sign) as a function of height
above midplane. To guide eye, we further plot field strengths that correspond
to $\beta_{e}=1$ and $\beta_{i}=1$ (pink), as well as field strengths that
give plasma $\beta$, ratio of gas pressure to magnetic pressure, to be $1$,
$100$ and $10^{4}$.

We first focus on the fiducial case with $a=0.1\mu$m and $f=0.01$ (blue
lines). We see that within $\sim 2H$, $B_{\rm th}$ is well below the
$\beta_{i}=1$ line, consistent with our expectations given $n_{i}\gg n_{e}$.
Towards the surface, as ionization fraction approaches and then exceeds grain
abundance, we see while $\eta_{H}$ can still change sign, but at field
strength well beyond $\beta_{i}=1$, analogous to the top panels of Figure 1.
We also note that the threshold field strength $B_{\rm th}$ corresponds to
only modest level of magnetization, with $\beta\sim 100$ within $\pm 2H$ about
midplane, which is very likely accessible in real PPDs (see next subsection).

Using different grain size and abundances yield different threshold field
strengths $B_{\rm th}$. Reducing dust-to-gas mass ratio by a factor of 100 to
$f=10^{-4}$ increases $B_{\rm th}$ by about an order of magnitude, and we see
that at both 1 AU and 5 AU, field strength near or above equipartition
($\beta=1$) is needed to have $\eta_{H}$ change sign. This is much less likely
to be achieved in PPDs, at least under standard disk models where field
strength well below equipartition is sufficient to drive disk accretion at
expected accretion rate (e.g., Bai, 2013). On the other hand, for the same
dust-to-gas mass ratio $f=10^{-4}$, reducing the grain size further to
$0.01\mu$m yields results similar to the fiducial case. This is because
smaller grains have higher total surface area/abundance at given mass, both
are favorable to make charged grains dominant charge carriers.

###  3.2. An Empirical Criterion on Grain Size and Abundance

Extending the discussion to a broad range of disk radii, we show in Figure 3
the threshold field strength $B_{\rm th}$ as a function of orbital radius $R$,
and present the results at disk midplane ($z=0$) and intermediate height
($z=2H$). Moving towards larger disk radii, ionization fraction increases
systematically (due to reduced density and deeper penetration of external
ionization), as can be traced from Figure 2. This makes $B_{\rm th}$ shift up
relative to the $\beta_{i}=1$ line as discussed before. In the mean time, as
gas density/pressure drops, field strength at fixed plasma $\beta$ also shifts
up relative to the $\beta_{i}=1$ line, as is straightforward to show from
their definitions ($B\propto\sqrt{\rho}$ v.s. $B\propto\rho$). As a result, we
see from Figure 3 that line showing $B_{\rm th}$ as a function of $R$ is
approximately parallel the constant plasma $\beta$ line over a very wide range
of radii.

In Figure 3, we have considered a broad collection of grain sizes and
abundances. We are particularly interested in the question of for which
combinations of grain sizes and abundances can $\eta_{H}$ change sign in the
expected range of PPD magnetic field strength. We consider plasma $\beta=100$
as a target field strength, which is easily achieved in either pure
magnetorotational instability (MRI, Balbus & Hawley, 1991) turbulence (e.g.,
Davis et al., 2010), or in more realistic studies of PPD gas dynamics (e.g.,
Bai & Stone, 2013; Lesur et al., 2014; Bai, 2014b, 2015). On the other hand,
field strength above equipartition is unlikely in the disk interior. For
reference, we also mark the possible field strength of the solar nebular
inferred from the Semekona meteorites (Fu et al., 2014), which likely
corresponds to midplane field strength at $2-3$ AU during the epoch of
chondrule formation. Although large uncertainties remain, field strength of up
to $\beta=100$ in an MMSN disk is reasonably consistent with the inferred
nebular field strength.

From Figure 3, it is clear that small grains in large abundance is essential
to make $B_{\rm th}$ sufficiently weak (plasma $\beta\gtrsim 100$) that can
potentially be achieved in PPDs. In particular, the results from our fiducial
choice of $a=0.1\mu$m, $f=10^{-2}$ are very similar to those obtained with the
$a=0.01\mu$m, $f=10^{-4}$ case, at both $z=0$ and $z=2H$. This might suggest
an empirical criterion where $f/a^{2}$ should be at least order unity (with
$a$ measured in $\mu$m) in order to achieve a sufficiently weak threshold
field $B_{\rm th}$. Note that a constant $f/a^{2}$ corresponds to a constant
$na$ where $n$ is grain number density. Therefore, the controlling parameter
is not the total grain surface area ($\propto na^{2}$), but lies in between
total surface area and number density. We might further generalize this
consideration into a distribution of grain sizes, and define

$G\equiv\int\frac{df/da}{(a/{\mu}{\rm m})^{2}}da\ .$ (16)

Our discussions above suggest $G\gtrsim 1$ to achieve sufficiently weak
$B_{\rm th}$.

Figure 4.— The ambipolar Elsasser number $Am$ (solid) and Hall Elsasser number
$\chi$ (dashed) as a function of field strength (in Gauss) from ionization
chemistry calculations at 30 AU (left) and 100 AU (right). Top and bottom
panels show the results from calculations at disk midplane, and at two scale
heights above midplane, respectively. Three combinations of grain sizes $a$
and dust-to-gas mass ratios $f$ are shown, as marked in the legends. We also
mark the field strength corresponding to plasma $\beta=1$, as well as field
strengths that give ion Hall parameters $\beta_{i}=1$, in vertical dash-dotted
lines.

We note that when assuming the standard MRN size distribution for interstellar
grains (Mathis et al., 1977), with $dn/da\propto a^{-3.5}$ between $a_{\rm
min}\approx 0.005\mu$m and $a_{\rm max}\approx 0.25\mu$m, we obtain $G\approx
22\gg 1$. Therefore, while grain growth in PPDs must substantially reduces $G$
from interstellar value, we do expect that grain-modified diffusivities is
likely relevant at least in some stages of PPD evolution.

###  3.3. Diffusivities in the Outer Disk

Although $\eta_{H}$ can change sign essentially at all distances, it is mainly
the inner disk ($\lesssim 10$ or at most a few tens of 10 AU) that is the most
relevant. Conventionally, $\beta_{i}>1$ marks the dominance of AD over the
Hall effect, which is achieved at lower density regions towards the outer
disk. The complex dependence of $\eta_{A}$ and $\eta_{H}$ on field strength in
the presence of abundant small grains complicates the situation. In Figure 4,
we show the AD and Hall Elsasser numbers $Am$ and $\chi$ as a function of
magnetic field strength at 30 AU and 100 AU, at both disk midplane and
intermediate height ($z=2H$). We consider three combination of grain sizes and
abundances as before and discuss the results below.

We first see that $Am$ is constant at both weak and strong field regimes, as
expected. In the presence of abundant small grains ($G\gtrsim 1$, black and
red lines), the difference in $Am$ between the two regimes can reach several
orders of magnitude. In between at intermediate field strength, we have
$\eta_{A}\approx$constant or $Am\propto B^{2}$ (see also Figure 1). Therefore,
AD behaves the same way as Ohmic resistivity (except for its anisotropy).
These trends are the same as those discussed in B11 in the presence of tiny
($a\sim$nm) grains. If we expect field strengths in the outer PPDs correspond
to plasma $\beta\sim 1-10^{4}$, then roughly speaking, the midplane region up
to $\sim 100$ AU, and upper layer ($z=2H$) up to $\sim 30$ AU would lie in
this intermediate regime when $G\gtrsim 1$.

With less amount of small grains ($G\ll 1$, blue lines in Figure 4), on the
other hand, reduction of AD occurs at much stronger field. For both 30 AU and
100 AU, both at midplane and a few scale heights above, such reduction
requires field strength above equipartition level, thus is unlikely to be
relevant. In fact, for our choice of $a=0.1\mu$m and $f=10^{-4}$, the
ionization fraction at disk midplane already exceeds grain abundance, and
hence the role of grains is only minor. In this case, we see that $Am\sim 1$
is a good proxy over a wide range of disk radii and vertical locations,
consistent with previous findings (Bai, 2011a, b; Perez-Becker & Chiang,
2011).

We also see that in the presence of abundant small grains, substantial
reduction of AD towards strong field ($\beta_{i}>1$) is accompanied by sign
change of $\eta_{H}$, and the Hall effect can dominate over AD (in the
conventional AD dominated regime). This is consistent features seen in Figure
1 (e.g., bottom panels). Therefore, the Hall effect may play a more important
role than commonly assumed.

Table 1List of all quasi-1D simulations.

$R$ (AU) | $\beta_{0}$ | $f$ | Diffusivity | $\beta_{\rm mid}$ | $\alpha^{\rm Max}$ | $T_{z\phi}^{z_{b}}$ | $\dot{M}_{w}$ | $z_{b}$
---|---|---|---|---|---|---|---|---
1 | $10^{5}$ | $10^{-2}$ | full | $9.50\times 10^{3}$ | $4.00\times 10^{-4}$ | $1.76\times 10^{-4}$ | $2.02\times 10^{-5}$ | 3.94
1 | $10^{5}$ | $10^{-2}$ | simple | $8.54\times 10^{3}$ | $5.69\times 10^{-4}$ | $1.95\times 10^{-4}$ | $3.15\times 10^{-5}$ | 3.86
1 | $10^{5}$ | $10^{-4}$ | any | $8.10\times 10^{2}$ | $5.96\times 10^{-3}$ | $2.67\times 10^{-4}$ | $6.27\times 10^{-5}$ | 4.05
1 | $10^{4}$ | $10^{-2}$ | full | $5.38\times 10^{3}$ | $1.25\times 10^{-3}$ | $7.46\times 10^{-4}$ | $4.57\times 10^{-5}$ | 3.83
1 | $10^{4}$ | $10^{-2}$ | simple | $4.62\times 10^{3}$ | $2.09\times 10^{-3}$ | $7.84\times 10^{-4}$ | $7.22\times 10^{-5}$ | 4.00
1 | $10^{4}$ | $10^{-4}$ | any | $4.81\times 10^{2}$ | $1.80\times 10^{-2}$ | $1.24\times 10^{-3}$ | $1.30\times 10^{-4}$ | 3.95
5 | $10^{5}$ | $10^{-2}$ | full | $5.58\times 10^{3}$ | $2.04\times 10^{-4}$ | $2.70\times 10^{-4}$ | $4.97\times 10^{-5}$ | 3.94
5 | $10^{5}$ | $10^{-2}$ | simple | $3.41\times 10^{3}$ | $4.62\times 10^{-4}$ | $3.32\times 10^{-4}$ | $6.70\times 10^{-5}$ | 3.91
5 | $10^{5}$ | $10^{-4}$ | any | $2.77\times 10^{2}$ | $5.20\times 10^{-3}$ | $4.99\times 10^{-4}$ | $1.43\times 10^{-4}$ | 3.91
5 | $10^{4}$ | $10^{-2}$ | full | $2.61\times 10^{3}$ | $1.13\times 10^{-3}$ | $1.22\times 10^{-3}$ | $1.61\times 10^{-4}$ | 3.83
5 | $10^{4}$ | $10^{-2}$ | simple | $2.08\times 10^{3}$ | $1.88\times 10^{-3}$ | $1.30\times 10^{-3}$ | $1.86\times 10^{-4}$ | 3.91
5 | $10^{4}$ | $10^{-4}$ | any | $4.61\times 10^{2}$ | $1.75\times 10^{-2}$ | $1.83\times 10^{-3}$ | $2.29\times 10^{-4}$ | 3.71

Note: we use grain size $a=0.1\mu$m for all runs, and “any” means “full” or
“simple” diffusivity treatments yield same results. See Section 4.1 for more
details.

The AD Elsasser number $Am$ directly controls the level of the MRI turbulence,
and $Am\sim 1$ is generally considered to be sufficiently strong to
substantially damp the MRI (Bai & Stone, 2011; Simon et al., 2013b, a).
Further introducing a modestly strong Hall term would enhance the level of
turbulence if net vertical field is aligned with disk rotation
${\boldsymbol{B}}_{0}\cdot{\boldsymbol{\Omega}}>0$, while reduce it for
opposite polarity (Sano & Stone, 2002; Kunz & Lesur, 2013; Bai, 2015; Simon et
al., 2015). The grain-modified magnetic diffusivities in the outer disk
studied here may lead to two interesting effects when $G\gtrsim 1$. First, the
fact that AD behaves similarly as Ohmic resistivity at intermediate field
strength may lead to cycles of growth and decay of the MRI as observed in
resistive MRI simulations (Simon et al., 2011). Second, the sign change of
$\eta_{H}$ means that the enhancement of the MRI by the Hall effect would
occur at anti-aligned field polarity, rather than aligned polarity. These
effects deserve further investigation via three-dimensional MHD simulations.

##  4\. MHD simulations of the inner PPD

###  4.1. Simulation Setup and Parameters

To further study the effect of grain-modified magnetic diffusivities on PPD
gas dynamics, we follow Bai (2014b) (hereafter B14) and conduct local
stratified shearing-box MHD simulations that include all three non-ideal MHD
terms. Note that B14 simply assumed $\eta_{H}\propto B$, $\eta_{A}\propto
B^{2}$ in the calculations, and used a 2D diffusivity table showing the
proportional coefficients as a function of density and ionization rate to
compute the diffusivities. To incorporate the complex dependence of $\eta_{H}$
and $\eta_{A}$ on magnetic field strength, we extend the diffusivity table to
3D that explicitly accounts for this dependence.

Here we mainly focus on the inner disk region ($R<10$ AU), where it suffices
to conduct quasi-1D simulations222It means that the simulation box is largely
1D in vertical, but allows for $4$ cells in radial and azimuthal dimensions,
which facilitates the simulations to relax to a steady state. (as verified in
Bai, 2015) because the MRI is largely suppressed. The simulations use
isothermal equation of state, and the simulation box extends from $-8H$ to
$8H$ in $z$, with a resolution of 18 cells per $H$. We consider examples at
two radii $R=1$ AU and $5$ AU, using diffusivity tables with fixed grain size
$a=0.1\mu$m but two different abundances $f=10^{-2}$ and $10^{-4}$. An
artificial boost of ionization rate beyond column densities of $\sim 0.03$ g
cm-2 is included to mimic the effect of FUV ionization that brings the gas to
ideal MHD regime. All simulations are conducted with net vertical magnetic
field $B_{z0}$ threading the disk, and the field strength is parameterized by
plasma $\beta_{0}$ defined as ratio of midplane gas pressure to magnetic
pressure of the net vertical field. We consider $\beta_{0}=10^{4}$ and
$10^{5}$, as adopted in most previous works chosen to yield accretion rate in
the expected range for the MMSN disk model. In code units, we have midplane
density $\rho_{\rm mid}=1$, isothermal sound speed $c_{s}=1$, Keplerian
frequency $\Omega_{K}=1$, and magnetic permeability $\mu_{m}=1$.

It is well known that the effect of the Hall term depends on the polarity of
background vertical field $B_{z0}$ threading the disk (e.g., Wardle, 1999;
Balbus & Terquem, 2001; Wardle & Salmeron, 2012). Of particular interest is
the case where ${\boldsymbol{B}}_{0}\cdot{\boldsymbol{\Omega}}>0$, or
background field being aligned with rotation axis. Linear analysis as well as
numerical simulations have shown that in this case, horizontal components of
the magnetic field can be strongly amplified due to the Hall shear instability
(HSI, Kunz, 2008; Lesur et al., 2014; Bai, 2014b). Because both radial and
azimuthal components are amplified, Maxwell stress ($-B_{R}B_{\phi}$) is
strongly enhanced which likely plays a non-negligible role in disk angular
momentum transport. On the other hand, when
${\boldsymbol{B}}_{0}\cdot{\boldsymbol{\Omega}}<0$, the opposite occurs, and
the horizontal field is reduced to close to zero near the midplane region
(Bai, 2014b, 2015). Because the interesting effects we have discussed in this
paper operate when magnetic field gets amplified, we consider only aligned
polarity ${\boldsymbol{B}}_{0}\cdot{\boldsymbol{\Omega}}>0$ in our
simulations.

Figure 5.— Top: vertical profiles for three components of magnetic field
strength (in code units) from simulations at 1AU (left) and 5 AU (right) in
steady state, where equipartition field strength at midplane corresponds to
$B=\sqrt{2}$. We have chosen grain size $a=0.1\mu$m and dust-to-gas mass ratio
$f=0.01$, with weak net vertical field $\beta_{0}=10^{5}$. Dashed lines
correspond to simulations assuming $\eta_{H}\propto B$ and $\eta_{A}\propto
B^{2}$ in the weak field limit, while solid lines (except the cyan line)
correspond to simulations using full dependence of $\eta_{H}$ and $\eta_{A}$
on $B$. Cyan solid lines mark the threshold field strength above which
$\eta_{H}$ changes sign. Bottom: vertical profiles of Ohmic, Hall and AD
Elsasser numbers from simulations using full dependence of $\eta_{H}$ and
$\eta_{A}$ on $B$ at 1 AU (left) and 5 AU (right).

We run the simulations following the procedures of Bai (2014b). We first turn
off the Hall effect during initial evolution to time $t=120\Omega^{-1}$. Then
we turn on the Hall effect and run the simulations to $t=240\Omega^{-1}$,
where steady state magnetic configuration fully saturates. In all cases, the
saturated field geometry obeys “odd-z” symmetry (e.g., Figure 9 of Bai &
Stone, 2013) where horizontal components of magnetic field remain the same
sign across the disk, and reach maximum around disk midplane. While this
symmetry is unphysical for disk wind launching, it is likely an artifact of
shearing-box and here we set this issue aside333Whether a physical “even-z”
symmetry can be obtained in shearing-box also depends on the diffusivity
profiles. With $a=0.1\mu$m and $f=10^{-4}$, Bai (2014b, 2015) obtained a
physical wind geometry at $R=5$ AU with $\beta_{0}=10^{5}$. We can reproduce
this result but we fail to obtain physical symmetry with $f=10^{-2}$.. Because
magnetic diffusivities in the disk midplane region are always excessively
large at the inner disk, a diffusivity cap $\eta_{\rm cap}$ for each non-ideal
MHD term is implemented to prevent prohibitively small numerical timestep.
Most previous works used $\eta_{\rm cap}\lesssim 10H^{2}\Omega$. We use
$\eta_{\rm cap}=10H^{2}\Omega$ in the runs up to $t=240\Omega^{-1}$ as
described before, but then enlarge $\eta_{\rm cap}$ to $100H^{2}\Omega$ and
run for another $\Delta t=60\Omega^{-1}$, which appears sufficient for the
system to relax to a new state. We do observe changes in $B_{R}$ at midplane
by up to $30\%$ after enlarging the diffusivity cap, while typically the
changes are within $10\%$, and is much smaller for midplane $B_{\phi}$.

Figure 6.— Vertical profiles of Maxwell stress from simulations at 1 AU (left)
and 5 AU (right). We have chosen grain size $a=0.1\mu$m and dust-to-gas mass
ratio $f=10^{-4}$ (blue) and $f=0.01$ (black and red), with weak net vertical
field $\beta_{0}=10^{5}$. Black lines correspond to simulations assuming
$\eta_{H}\propto B$ and $\eta_{A}\propto B^{2}$ in the weak field limit, while
red lines correspond to simulations using full dependence of $\eta_{H}$ and
$\eta_{A}$ on $B$.

In parallel to simulations that take into account the full dependence of
$\eta_{H}$, $\eta_{A}$ on $B$ (referred as “full”), we also conduct
simulations assuming $\eta_{H}\propto B$ and $\eta_{A}\propto B^{2}$, taking
the proportional constants from the weak field limit of the full dependence
(referred to as “simple”), and compare the difference between the two sets of
runs.

Table 1 lists all our quasi-1D simulations. We also show four main simulation
diagnostics at saturation: $\beta_{\rm mid}$ is the plasma $\beta$ at disk
midplane (based on total field strength), $\alpha^{\rm Max}$ is the vertically
integrated dimensionless Shakura-Sunyaev $\alpha$ based on Maxwell stress
$\int_{-z_{b}}^{z_{b}}(-B_{R}B_{\phi})dz/\int_{-z_{b}}^{z_{b}}\rho
c_{s}^{2}dz$, $T^{z\phi}|_{z_{b}}$ is proportional to the torque exerted at
the wind base $z_{b}$, and $\dot{M}_{w}=\rho v_{z}$ is the mass loss rate
measured at the top of the simulation box (single-sided). The wind base
$z_{b}$ is defined as the location where rotation transitions from sub-
Keplerian to super-Keplerian (in practice, we choose it to be the first
minimum of $|v_{\phi}-v_{K}|$ as one varies $z$ from boundary towards
midplane, as discussed in B14). All aspects of these diagnostic quantities
have been discussed in the literature, and this table mainly serves as a
reference.

###  4.2. Simulation Results

In Figure 5, we show the simulation results with $a=0.1\mu$m, $f=0.01$ and
$\beta_{0}=10^{5}$ at 1 AU and 5 AU, respectively. The top panels show the
vertical profiles of magnetic field strength in code units, where plasma
$\beta=1$ at midplane corresponds to $B=\sqrt{2}$. As usual, toroidal field is
the dominant component, and it maintains a relatively flat profile across the
disk midplane because of excessively low ionization level there (lack of
charge carriers can not support current). We also mark the threshold field
strength $B_{\rm th}$ at which $\eta_{H}$ changes sign, which is computed
based on the diffusivity table directly. We have verified that the line shapes
follow exactly the corresponding curves shown in Figure 2 in logarithmic
scale. In linear scale, the minimum of $B_{\rm th}$ at $z=2-3H$ is the most
prominent feature, and we see that the saturated field strength from our
simulations (dominated by toroidal field) matches closely to the minimum value
of $B_{\rm th}$. On the other hand, simulations assuming $\eta_{H}\propto B$
yield stronger field around disk midplane as a result of the HSI. This
comparison clearly shows that $B_{\rm th}$ provides an approximate upper limit
that magnetic field strength can grow, resulting in pre-matured saturation of
the HSI.

The bottom panels of Figure 5 show the Elsasser number profiles for the three
non-ideal MHD effects. We first see that the locations where $B<B_{\rm th}$ in
the top panels have negative Hall Elsasser number $\chi$ seen in the bottom
panels. In other regions where $\eta_{H}$ is positive, we also see that both
$\eta_{H}$ and $\eta_{A}$ are reduced compared to the “simple” case assuming
$\eta_{H}\propto B$ and $\eta_{A}\propto B^{2}$, which is due to a). weaker
magnetic field strength and b). reduction of $|\eta_{H}|$ and $\eta_{A}$
towards stronger field.

We also performed simulations with reduced grain abundance ($a=0.1\mu$m and
$f=10^{-4}$). In this case, however, the threshold field strength is much
stronger, and we find that simulations using full diffusivity dependence on
$B$ yield identical results as assuming $\eta_{H}\propto B$, $\eta_{A}\propto
B^{2}$ scalings. The latter was adopted in Bai (2014b), and hence his results
were unaffected. This is because with reduced grain abundance, the threshold
field strength is much stronger (as seen in Figure 2), and exceeds the
strength that can be attained from field amplification by the HSI.

Increasing the net vertical magnetic flux naturally leads to stronger wind,
with larger wind torque and higher mass outflow rate, as listed in Table 1.
Other than the wind being stronger, we find that the general phenomenologies
discussed above hold exactly the same way, where field amplification by the
HSI saturates at weaker field strength when full dependence of diffusivities
on $B$ is considered.

Field amplification by the HSI always weakens with increasing grain abundance
as a result of enhanced resistive dissipation. This is already evident by
comparing the effectively grain-free simulations of Lesur et al. (2014) and
the simulations of Bai (2014b) with $a=0.1\mu$m and $f=10^{-4}$, where the
former showed extreme level of field amplification around the disk midplane
that approach equipartition field strength, and the latter only found modest
level of amplification. The trend continues when increasing dust-to-gas ratio
to $f=10^{-2}$, and assuming simple $\eta_{H}\propto B$, $\eta_{A}\propto
B^{2}$ dependence, midplane field strength is already reduced by a factor of
$\sim 3$. Our simulations show that a further reduction is achieved when full
dependence of diffusivities on $B$ is considered. This further reduction in
field strength is better viewed when plotting the Maxwell stress
($-B_{R}B_{\phi}$) profiles, which depends quadratically on field strength.
This is shown in Figure 6, and the vertically integrated values $\alpha^{\rm
Max}$ are listed in Table 1.

Overall, our studies show that whether the full dependence on field strength
needs to be taken into account when small grains are abundant (or effectively,
$G\gtrsim 1$ defined in (16)). Reducing grain abundance leads to larger
threshold field strength $B_{\rm th}$, and and in the mean time higher level
of field amplification by the HSI. It turns out that the former increases much
faster, and the effect studied in this paper becomes essentially irrelevant
when $G\ll 1$.

##  5\. Summary and Discussion

We have conducted a comprehensive study on the dependence of magnetic
diffusivities on magnetic field strength in the presence of charged grains
with application to PPDs. Ohmic resistivity $\eta_{O}$ is always independent
of field strength. The Hall and ambipolar diffusivities $\eta_{H}$ and
$\eta_{A}$ obey simple asymptotic relations $\eta_{H}\propto B$ and
$\eta_{A}\propto B^{2}$ in both weak field regime where $\beta_{e}<1$, and
very strong field regime where $\beta_{g}>1$, while show complex dependence in
between (see Figure 1). In particular, when relation (14) is satisfied (easily
achieved when $n_{e}$ falls below $n_{i}$ due to the presence of charged
grains), $\eta_{H}$ changes sign towards strong field, accompanied by
reduction of AD. The threshold field strength $B_{\rm th}$ beyond which
$\eta_{H}$ changes sign largely depends the ratio $n_{e}/n_{i}$, and is weaker
when $n_{e}/n_{i}$ is smaller (e.g., charged grains more dominant). Similarly,
more abundant charged grains leads to stronger reduction of $\eta_{A}/B^{2}$
between the weak and strong field regimes.

Applying to standard MMSN model of PPDs with ionization chemistry, we find
that in order for the threshold field $B_{\rm th}$ to be well below
equipartition level, a large amount of small grains are required, as
summarized in an empirical condition $G\gtrsim 1$ [defined in Equation (16)],
which is applicable to a wide range of disk radii. When the same condition is
satisfied, AD is also reduced substantially and may promote the action of the
MRI in the outer PPDs.

We further conducted quasi-1D shearing-box simulations for inner region of
PPDs including all three non-ideal MHD effects with net vertical field aligned
with rotation. We show that when small grains are abundant, magnetic field
amplification due to the Hall-shear instability saturates at close to the
minimum value of $B_{\rm th}$. Therefore, besides the fact that abundant small
grains limit magnetic field growth by enhancing resistivity, they also affect
the gas dynamics in a more complex way by modifying the field-strength
dependence of magnetic diffusivities, especially by enabling $\eta_{H}$ to
change sign towards strong field. On the other hand, with substantial grain
growth (so that $G\ll 1$), the threshold field strength becomes too strong to
be accessible via internal MHD processes in PPDs, and hence it suffices to
simply adopt $\eta_{H}\propto B$ and $\eta_{A}\propto B^{2}$ in the weak field
limit.

Our work helps clarify how the presence of grains affects ionization chemistry
and magnetic diffusivities in PPDs, and hence the general disk dynamics. This
is particularly relevant to the Hall effect: inclusion of the Hall effect not
only makes PPD gas dynamics dependent on the polarity of net vertical magnetic
field, but also leads to more sensitive dependence on grain abundance, as
compared to the Hall-free case of Bai & Stone (2013). In combination with
other recent studies, we can identify three regimes on the gas dynamics of
inner PPDs ($\lesssim 10-15$ AU) depending on the grain abundance.

  * •

Grain-free ($G\sim 0$): very strong magnetic field amplification by the HSI
when ${\boldsymbol{\Omega}}\cdot{\boldsymbol{B}}>0$ (Lesur et al., 2014; Bai,
2014b), and bursty behavior at $\sim 5-10$ AU when
${\boldsymbol{\Omega}}\cdot{\boldsymbol{B}}<0$ (Simon et al., 2015).

  * •

Modest grain abundance ($G<1$): modest magnetic field amplification by the HSI
when ${\boldsymbol{\Omega}}\cdot{\boldsymbol{B}}>0$, while horizontal field is
reduced to close to zero when ${\boldsymbol{\Omega}}\cdot{\boldsymbol{B}}<0$
(Bai, 2014b, 2015).

  * •

High grain abundance ($G\gtrsim 1$): the HSI saturates due to sign change of
$\eta_{H}$ at threshold field strength $B_{\rm th}$ when
${\boldsymbol{\Omega}}\cdot{\boldsymbol{B}}>0$.

Exploration of the Hall effect in PPDs is still at an early stage. There are
still pressing issues especially concerning the symmetry of disk wind,
direction of magnetic flux transport, wind kinematics, etc. The complexity
introduced by grains further adds to the richness of the overall subject of
PPD gas dynamics. Conversely, it is well known that grain size and spatial
distribution in PPDs depends on level of turbulence and global structure of
the disk, and evolve with time (e.g., Birnstiel et al., 2010). The two
interrelated aspects are inherent in PPD dynamics, and further call for more
in-depth investigations in the future.

We thank K. Öberg for helpful advice on disk chemistry. XNB acknowledges
support from Institute for Theory and Computation (ITC) at Harvard-Smithsonian
Center for Astrophysics.

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