Issue |
A&A
Volume 515, June 2010
|
|
---|---|---|
Article Number | A70 | |
Number of page(s) | 17 | |
Section | Interstellar and circumstellar matter | |
DOI | https://doi.org/10.1051/0004-6361/200912834 | |
Published online | 10 June 2010 |
Trapping solids at the inner edge of the dead zone: 3-D global MHD simulations
N. Dzyurkevich - M. Flock - N. J. Turner -
H. Klahr - Th. Henning
Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
Received 6 July 2009 / Accepted 16 November 2009
Abstract
Context. The poorly-ionized interior of the protoplanetary
disk or ``dead zone'' is the location where dust coagulation processes
may be most efficient. However even here, planetesimal formation may be
limited by the loss of solid material through radial drift, and by
collisional fragmentation of the particles. Both depend on the
turbulent properties of the gas.
Aims. Our aim here is to investigate the possibility that solid
particles are trapped at local pressure maxima in the dynamically
evolving disk. We perform the first 3-D global non-ideal
magnetohydrodynamical (MHD) calculations of a section of the disk
treating the turbulence driven by the magneto-rotational instability
(MRI).
Methods. We use the ZeusMP code with a fixed Ohmic resistivity
distribution. The domain contains an inner MRI-active region near the
young star and an outer midplane dead zone, with the transition between
the two modeled by a sharp increase in the magnetic diffusivity.
Results. The azimuthal magnetic fields generated in the active
zone oscillate over time, changing sign about every 150 years. We
thus observe the radial structure of the ``butterfly pattern'' seen
previously in local shearing-box simulations. The mean magnetic field
diffuses from the active zone into the dead zone, where the Reynolds
stress nevertheless dominates, giving a residual
between 10-4 and 10-3.
The greater total accretion stress in the active zone leads to a net
reduction in the surface density, so that after 800 years an
approximate steady state is reached in which a local radial maximum in
the midplane pressure lies near the transition radius. We also observe
the formation of density ridges within the active zone.
Conclusions. The dead zone in our models possesses a mean
magnetic field, significant Reynolds stresses and a steady local
pressure maximum at the inner edge, where the outward migration of
planetary embryos and the efficient trapping of solid material are
possible.
Key words: planets and satellites: formation - magnetohydrodynamics - methods: numerical - instabilities - accretion, accretion disks - turbulence
1 Introduction
Forming planets in a protoplanetary disk with a power-law surface density profile is difficult for several reasons. First, solid material on accumulating into meter-sized boulders quickly spirals to the star, transferring its orbital angular momentum to the gas (Youdin & Chiang 2004; Weidenschilling 1977; Brauer et al. 2007; Takeuchi & Lin 2002; Nakagawa et al. 1986). Second, collisions between the constituents lead to disruption rather than growth when rather low speed thresholds are reached (Blum & Wurm 2000; Poppe et al. 1999; Blum et al. 1998). Bodies in the meter size range are destroyed in impacts as slow as some cm/s (Benz 2000). Turbulence in the gas readily yields collisions fast enough to terminate growth (Brauer et al. 2008a). Third, Earth-mass protoplanetary cores are prone to radial migration resulting from the tidal interaction with the gas, and in the classical type I picture quickly migrate all the way to the host star (Ward 1986; Goldreich & Tremaine 1980).
All three of these problems could be solved by the presence of local radial gas pressure maxima, which trap the drifting particles (Haghighipour & Boss 2003), leading to locally enhanced number densities and high rates of low-speed collision (Lyra et al. 2008; Brauer et al. 2008b; Kretke et al. 2009). With sufficient local enhancement, one can envision the direct formation of planetesimals via collapse under the self-gravity of the particle cloud, bypassing the size regime most susceptible to the radial drift. Furthermore, the radial migration of Earth-mass protoplanets can be slowed or stopped by varying the surface density and temperature gradients (Masset et al. 2006). Migration substantially slower than in the classical picture appears to be required to explain the observed exoplanet population under the sequential planet formation scenario (Schlaufman et al. 2009).
The formation of local pressure maxima is governed by the radial transport of gas within the disk. The magneto-rotational instability or MRI (Balbus & Hawley 1998,1991) is currently the best studied candidate to drive such flows. Local shearing-box calculations show that the instability leads to long-lasting turbulence and to angular momentum transfer by magnetic forces, provided the magnetic fields are well-coupled to the gas (Hawley et al. 1995; Sano et al. 2004; Brandenburg et al. 1995; Johansen et al. 2009). Global ideal MHD calculations have been performed in various astrophysical contexts: protoplanetary disks (Arlt & Rüdiger 2001; Fromang & Nelson 2009; Fromang 2005; Steinacker & Henning 2001; Fromang & Nelson 2006), black hole accretion toruses (Hawley 2000), and galactic disks (Dziourkevitch et al. 2004). All the global simulations included neither a physical magnetic diffusivity nor a physical viscosity. Meanwhile, from local shearing-box studies it is known that the strength of the saturated MRI turbulence depends critically on the resistivity and viscosity (Fromang & Papaloizou 2007; Fromang et al. 2007; Lesur & Longaretti 2007). In particular, whereas the molecular viscosity is small in protostellar disks, the gas is so weakly ionized in the cold disk interior (Gammie 1996; Semenov et al. 2004; Igea & Glassgold 1999) that the Ohmic resistivity shuts down the linear MRI and prevents the development of turbulence (Sano & Stone 2002b,a; Turner et al. 2007; Sano et al. 1998; Fleming & Stone 2003).
In this paper we present the first global resistive MHD calculations to include the ``dead zone'' where the rapid diffusion of the magnetic fields prevents magnetorotational turbulence. Local pressure maxima form in the calculations in two ways: at dead zone edges and in zonal flows. The dead zone edges yield long-lived rings of enhanced surface density near locations where a gradient in the ionization fraction leads to a jump in the accretion stress (Kretke & Lin 2007). The zonal flows on the other hand result from local fluctuations in the Maxwell stress in the turbulence, and lead to pressure maxima with lifetimes of a few orbits (Johansen et al. 2009).
In the next section we describe our disk model and the choice of magnetic resistivity profiles. The third section presents the results, starting from the global properties of the magnetic field, followed by the effects of the resistivity jump on the surface density. In the fourth section we discuss the interaction of the magnetic fields with the density rings and with radial minima appearing in the turbulent activity. Our main results are summarized in Sect. 5.
2 Model description
The initial setup for the models is very similar to those studied by Fromang & Nelson (2006) for the ideal MHD case. In our models, MRI-driven turbulence is operating in locally-isothermal disk, with a fixed spatial distribution for the temperature. To describe the dead zone in the protoplanetary disk, we include the Ohmic dissipation in our models. The Ohmic dissipation is the largest non-ideal term in the induction equation under typical dusty conditions (Wardle 2007). Ambipolar diffusion and Hall effect are not considered for the sake of simplicity. The vertical profile of magnetic diffusivity (see Sect. 2.1) is adopted from separate chemistry calcullations and is fixed in space and time.
We solve the set of MHD equations using 3D global simulations on a
spherical grid
,
![]() |
(1) |
![]() |
(2) |
![]() |
(3) |
The notation is the usual one.




where a=3/2.
![]() |
Figure 1:
Vertical profiles of magnetic diffusivity. Black lines
show the profiles adopted for simulations of the dead zone, with
|
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We consider the inner part of the protoplanetary disk, which is
heavier and warmer compared to a minimum solar-nebula. In order to
mimic a ``dead zone'' and the ionization thresholds, we adopt fixed
magnetic diffusivity profiles
.
We use the estimations
from dynamical disk chemistry simulations for the adopted disk
parameters
(Turner et al. 2007). The surface density is
,
and the
temperature is
,
where Ris cylinder radius. The units are
km s-1 for velocity and
for magnetic diffusivity.
The models are listed in Table 1.
Our models include the disk part from 2 to 10 AU. A purely azimuthal
magnetic field is chosen as seed field for MRI turbulence, which is
everywhere in the disk. The Alfvén
limiter of
is applied. We use reflective
radial boundary condition, with buffer zones applied at radii
and
.
We apply the
magnetic diffusivity within the buffer zones:
is 10-5 at the radial boundary and decreases linearly towards the
physical domain. The buffer zones have both damping of radial
velocity towards zero at the border, and diffusing away the magnetic
eddies approaching the radial boundary.
Periodic boundary conditions are applied both for azimuthal and
vertical (i.e. )
domain borders.
Table 1:
Model properties and midplane -stresses inside
(
)
and outside (
)
of the
ionization threshold radius
.
2.1 Ionization thresholds and influence of dust grains
An estimate of the magnetic diffusivity vs. height at 4.5 AU is shown
in Fig. 1. The midplane diffusivity with dust grains
appears to be substantially higher then it is possible to include in
the MHD simulations. The four blue curves from top to bottom are
demonstrating the magnetic diffusivity in code units for the gas and
dust grains of 0.1, 1 and 10 microns, and no grains. We have used the
simple gas-phase reaction set of Oppenheimer & Dalgarno (1974) together with the grain
surface chemistry of Ilgner & Nelson (2006) for the classical dust to gas
ratio. Ionization by stellar X-rays, cosmic rays and long-lived
radionuclides is included. The penetration depths are assumed
for the X-rays and 96 for the cosmic rays.
The exact calculations of chemistry and dust behavior in the thermally
evolving global disk is a hard task with many free parameters. After
planetesimals form, the dust mass fraction will be lower than the
interstellar value. We shall bear in mind that CRP stopping depth can
be as low as
(Glassgold et al. 2009).
Here we simplify the situation and adopt the following
time-independent vertical profile of magnetic diffusivity,
where











2.2 Set of simulations
Models in Table 1 have an ionization threshold posed either
at
or
.
Midplane values for magnetic
diffusivity are noted in Table 1 as
and
(``Active'' and ``Dead'') for gas states inside and outside
the threshold radius.
The time duration of each model is
given in years, and the mark * is given when the steady-state
has not been reached.
Vertical profiles for magnetic diffusivity
follow Eq. (6) and are demonstrated in Fig. 1 with
black lines. Each model combines two diffusivity profiles, except the
run
.
In Table 1, notations are ``I'' for quasi-ideal MHD state with
(Fig. 1, solid black line),
``R''
for the gas disk with
-sized dust grains
(
,
dot-dashed black line), ``D'' for the case
of
grains (
,
dashed black line). Our
adopted magnetic diffusivity profiles for the disk with
and
-sized dust grains will allow the turbulent MRI
layers beyond 3H and 2H, correspondingly. The peak values of blue
curves in Fig. 1 are leading to unacceptable short time
steps in resistive MHD simulations. We have observed that it is not
convenient to compute the regions with
with
standard MHD codes, because of the dramatic shortening of the time
step. For this numerical reason, we take the magnetic diffusivity
slightly different as the chemistry models predict. Our adopted
profiles of magnetic diffusivity (black curves, Fig. 1)
allow to match the values of chemical models at 2 AU and remain above
the numerical dissipation for region between 2H and 3H. Reducing of
the magnetic diffusivity in the dead zone may influence how fast the
global magnetic fields are diffused into the dead zone, whereas the
MRI modes are damped all the same.
2.3 Calculation of turbulent stresses
An important outcome of our simulations is the magnitude of the
Reynolds and Maxwell stresses. To calculate the latter, we use the
approach described in Fromang & Nelson (2006) for curvi-linear coordinates. The
turbulent viscosity can be described as
,
where the main component of the
stress tensor
is
or

The mean pressure for azimuthal domain

3 Results
In this section we describe our results and focus on two main issues. First, we study the time evolution and radial dependence of magnetic fields in our models. Secondly we study the formation of long-lived density rings which may or may not be able to trap solids in the disk and thus trigger the onset of planet formation. Table 1 represents the set of models. In Sect. 3.1 we discuss the issue of resolution. In Sect. 3.2 we describe the properties of global models with the emphasis on the evolution of the magnetic fields. In Sect. 3.3, the radial behavior of resulting Maxwell and Reynolds stresses is described. In Sect. 3.4 we explore the evolution of the pressure rings in time. In Sect. 3.5 we demonstrate the change of rotation and the turbulent properties of the gas in the rings and in the pressure bump at the inner edge of the dead zone.
3.1 Azimuthal MRI and the issue of resolution
The MRI from a purely azimuthal magnetic field (AMRI) leads to
non-axisymmetric perturbations. The radial displacements of the
initial azimuthal field are enhanced due to the differential rotation.
This leads to the appearance of field components Br and turbulent .
The excess of magnetic pressure and the buoyancy lead to
the generation of the vertical magnetic field component. The linear
analysis has been done in Balbus & Hawley (1992). The critical
wavelength for AMRI in units of the azimuthal grid size is
which follows from Eq. (15) in Hawley et al. (1995). When


All our runs are made with the initial uniform plasma beta of 25.
Following Eq. (11), we have
everywhere in the MRI-active disk for the
models with resolution of
[256:128:64]. Model
with
halved resolution has
.
Note, that
the Ohmic dissipation poses an additional limitation for the excited
MRI wavelength.
![]() |
Figure 2:
Inverse plasma |
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The effective resolution of 10.47 is holding during the linear stage
of AMRI. The azimuthal magnetic field is breaking into filaments of
opposite signs with
in
MRI-active zones. The inverse plasma beta grows to 100 at the
midplane and the decreases below unity in the upper disk layers. The
effect of expelling the magnetic field into the corona becomes visible
after 30 local orbits (Sect. 3.2, Fig. 6). Note, that
in low-resolution tests with
(
with resolution of
,
excluded from
Table 1), there is no MRI exited and the azimuthal magnetic field
remains in its initial shape for few hundred years.
Figure 2 demonstrates how the inverse plasma beta,
,
changes due to MRI from
(red dotted line) to the convex shape (black
lines). The solid line stands for inverse plasma beta averaged within
the active zone (
AU). The dashed line stands for the
averaged over the patch of the dead zone (
AU). In
the midplane we find the minimum of magnetic pressure, with
.
The plasma beta is reaching 1 at 2.8H both
in model
and in the low-resolution run
(Fig. 2, left). The resulting vertical profile of the
magnetic pressure is very similar to those shown in Fromang & Nelson
(2006). It is remarkable, that the dead zone builds up the same
vertical distribution of magnetic pressure as the active zone,
predominantly due to the smooth azimuthal magnetic field component.
Radial dependence of the inverse plasma beta (Fig. 2, right)
in the normal resolution run
shows that upper layers
possess the constant
,
whereas the midplane
layers, i.e. from midplane to 2H, are oscillating and slightly
decrease towards the inner radius within the active zone. The dotted
line in Fig. 2 (top right, model
)
shows that
the magnetic pressure falls to zero at
at the
midplane. The reason is a diffusion of the mean azimuthal magnetic
field from the active zone into the dead zone. This diffused field has
the opposite sign to the primordial field. Its time propagation into
the dead zone is shown in Fig. 4 (Sect. 3.2). The
low-resolution model
shows that the expelling of the
azimuthal field into the corona is not reaching the same extend as in
the normal resolution model for radii r>6 AU.
This vertical re-distribution of the azimuthal magnetic field affects
the effective resolution. In the left panels of Fig. 2, we
adopt solid lines for active and dashed lines for dead zone values.
Green vertical bars in Fig. 2 mark the midplane region with
(Eq. (11)). The blue vertical
bars show the disk height where Elsässer number
drops below unity. The criterion for MRI
instability
has been introduced in Sano & Stone (2001) for the
case of non-ideal MHD with Ohmic dissipation. After 300 years, the
vertical profile is changed so much that the midplane
layers are resolved only with
,
for example in the active zone (2.5 AU to 4.5 AU) in normal resolution
model (green bars, Fig. 2). On the other side, model
becomes well-resolved in the layers |z/H|> 2. Effective
resolution of
is reached in the active
layers above the dead zone, where vertical MRI is launched outside of
the
line. Interesting to note, that the Elsässer number
drops below unity roughly at the
same height, when we compare normal and low resolution models
(Fig. 2, top left and bottom left). When looking for
numerical values of
get at the location
of blue bars in the active zone, we find
for
and 4 for
.
These
numbers are only approximate values, because it is difficult to
calculate them accurately at the height at which
.
All in
all, there are surprisingly small differences between
and
models. The instability occurs in both cases and leads
to similar physics, though the speed of the total field amplification
is slower for
(Table 1).
![]() |
Figure 3:
Elsässer number
|
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Figure 4:
Temporal evolution of azimuthal magnetic field as a function
of radius for
|
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Figure 3 demonstrates the Elsässer number and criterion in the
plane of azimuthally averaged disk. The
solid black line of
marks the locations where the
regeneration of vertical magnetic field cannot be efficient enough.
Models
and
have their most active MRI
layers at 2.5H above the ``dead'' midplane. Comparison of the contour
plots of the Elsässer number for
,
and
shows, that MRI-active zones have very similar
appearance. At the midplane of the active zone, there are
yellow-orange areas of low Elsasser number at
AU
(
,
),
AU (
), and
AU
(
), which are corresponding to the enhanced gas
pressure. In Sect. 3.4 we describe the formation of the density
rings at these locations in more details. Condition
is fulfilled in the whole disk in every model
(Turner & Sano 2008) and the magnetic fields may be pumped into the dead zone
from the active layers.
3.2 Oscillating magnetic fields and saturation
![]() |
Figure 5:
Temporal evolution of azimuthal magnetic field as a function
of radius for
|
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After most of the initial azimuthal magnetic flux has been shifted to
the upper disk layers, the two scale heights to the adjacent midplane
develop the oscillating axisymmetric magnetic field. For model
,
the oscillations last for over 400 years and then decay. In
the MRI-active zone, i.e. between 2.5 and 4.5 AU, the
sign-switching occurs within about every 120 to 150 years. In
Fig. 4 we demonstrate the time evolution of the azimuthally
averaged
as a function of the radial distance for the
model. The fluctuating part of the azimuthal magnetic field is
roughly ten times stronger than the mean part,
.
This property appears to be typical for
MRI, as the observations of the magnetic fields in the galactic disks
show (see Beck (2000) for review). When the active zone is
stretched up to 6.5 AU, it is not affecting the time period of the
sign reversals. Model
demonstrates that sign
reversals occur not only in time, but also along the radius, as shown
in Fig. 5. The first positive
-stripe is
starting at 60 years at 3 AU and progresses to 6 AU in 160 years
(Fig. 5). The stripes of
in (r,t)-plane are
stretched along the line of local orbit, which is plotted with black
dashed line for
in Fig. 5. At later
times, the mixing and interaction of the waves is breaking radial
reversals into less regular oscillations both in
Fig. 4 and in Fig. 5. The field diffusion in
the dead zone outside of
AU (model
)
and
outside of
AU (model
)
does not follow
the line of local orbit,
r3/2, but propagates with
diffusion time
r2. For the
run, the region
with
develops the negative azimuthal magnetic fields
due to diffusion of the magnetic field. After 600 years there is a
weaker wave of positive
,
which has a shorter time period.
Oscillations in the sign of
at the
H appear less
clearly, compared to the azimuthal magnetic field at the midplane
(Figs. 4, 5). A comparison with
shows why. The reason is the interaction of MRI waves in the
upper layers between active and dead zones. For the case of a very
thick dead zone (
), the sign reversals in
at
show most pronounced intervals. This is due to the fact, that
MRI can be best exited between 1H and 2H layers of the active zone.
In the model
,
the layers above and below the dead zone
are turbulent and interacting with MRI modes at same height in the
active zone, what leads to a more irregular picture in oscillations of
at
H. The color-coded presentation of
as a function of (z,t) reveals a butterfly diagram, if
the azimuthal magnetic field is averaged within the disk region
between 2 and 4 AU (Fig. 6, top). The low plasma
does not prevent the upper disk layers from being very turbulent, as
one can see from left and middle panels for Br and
.
Comparing the contours of dominating
and other two
turbulent field components
Br(z,t) and
demonstrates again that the MRI turbulence and vertical redistribution
of azimuthal magnetic field are connected. The period of
oscillations is about 150 years, corresponding well to the radial
changes of
sign shown in Figs. 4 and 5. When averaging over the whole radial extent of the
disk, or at least within the dead zone, then the butterfly picture
disappears (Fig. 6, bottom).
Volume-averaging of the magnetic energy shows, that its total value is
oscillating in time, with a period correlated to the sign-switch in
azimuthal magnetic field within
relative to the midplane.
Figure 7 shows the magnetic energy for each model in Table 1
and the corresponding total alpha stresses. The oscillations of
energies in time can be clearly correlated with the butterfly diagram.
Oscillations are weakly visible in the total alpha stress
(Fig. 7, right). The magnetic energy curves reach a
constant value in model
,
which has the smallest active
zone (from 2.5 to 4.5 AU). Model
looses the total
magnetic energy continously during 400 years due to higher magnetic
dissipation. Model
reaches a steady-state when stresses
and magnetic energy remain unchanged from 400 to over 600 years. The
longer is the MRI-active domain, the longer it takes for the
simulation to reach the steady-state. Models
and
hold oscillatory (non-stationary) magnetic fields for 900 years.
The closed boundary conditions enforce the conservation of the total flux in the domain. Fluxes of vertical magnetic field remain zero through the whole simulation. The effect of the boundary choice on the butterfly diagram remains to be investigated in future work. The local box simulations, made for open vertical boundaries, show the butterfly picture as well (Turner & Sano 2008).
3.3 Maxwell and Reynolds stresses
Radial inhomogeneity in turbulent viscosity has been suggested as the
mechanism to produce the pressure maximum, which is efficient in dust
trapping, and therefor important in the planet formation theory. In
our models, the turbulent viscosity is driven by MRI and the
inhomogeneity in turbulent stresses appears naturally as the result of
simulations, when we include the sharp gas ionization threshold.
Indeed, we find a density bump forming behind the ionization threshold
in our simulations (Sect. 3.5), and a corresponding jump in
turbulent stress.
![]() |
Figure 6:
Horizontally averaged magnetic fields components
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The time evolution of the Maxwell stress
(Eq. (9)) is demonstrated in Fig. 9. There is a weak
Maxwell stress of about 10-5 in the dead zone, which can
periodically become negative. One can see the sharp border in
between the active and the dead zones in slices for z=0,
and
.
The traces of sign reversals in the azimuthal
magnetic field are also visible in the Maxwell stress.
Figure 9 shows that exactly at 150 years the Maxwell stress
reaches its maximum of 10-1, when calculated in units of initial
pressure
(Eq. (9)). Later on, the
saturation of MRI sets in and the total stress is between 10-3and 10-2 (see Table 1). The dark-orange filaments of very weak
Maxwell stress in the active zone,
10-7, correspond to the
location where the reversals of axisymmetric azimuthal field happen.
The weak Maxwell stress is located at r=3.5 AU for many years, what
appears as a systematic stripe when looking at z=0 and
horizontal slices of
in Fig. 9. This is
a location where the density ring is created (more in Sect. 3.4) and
also most of
reversals take place during the time period
from 200 to 700 years. Negative values of Maxwell stress appear at
3H above the midplane. This is the region of low plasma beta, and the
turbulence at this height is no more MRI-driven.
Table 2:
-stresses inside (A) and outside (D) of the
ionization threshold radius for four layers above the
midplane.
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Figure 7:
Top: total magnetic energy evolution;
Bottom: total alpha
stress evolution for models
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In the dead zone, the value
is due
to Reynolds stress contribution. In order to provide the
understanding of how the turbulence there looks like, we present a
snap-shot of turbulent
stress for the
model
(Fig. 8). The dead zone is filled with vertical pillars of
the
stress of opposite sign. In the
plane, those
pillars look like tightly-wrapped spirals. The spiral waves are
launched from the dead-zone edge (
AU), where the
non-axisymmetric fluctuations in all velocity components are
MRI-generated. The weak spiral structures can be found in the gas
density as well.
We have calculated the turbulent stresses as the function of radius.
The vertical averaging has been done separately for four
midplane-symmetric layers:
,
,
and
,
where
c0=H/R=0.07. The resulting
stresses for these layers are given in Table 2 as
correspondingly and plotted in
Figs. 10 and 11 with solid, dotted, dashed and
dot-dashed black lines.
In Table 2, the mark * indicates that the steady-state has not yet
been reached.
Solid blue lines in Fig. 10 and 11 represent the
integrated over the whole
disk thickness. The strongest total stress alpha of all models is
obtained in model
,
which remains of the same order of
magnitude with radius.
The turbulent stresses have r-2 slope during the
``butterfly'' evolution stage in models
and
.
For the
model, the time averaging of the stresses
between 250 and 600 years results in a
.
Afterwards,
the turbulence in
reaches a ``butterfly''-free state
(t>600 years). When the temporal averaging is made between 400 and
800 years, instead of from 250 to 600 years, we obtain
which is constant with radius (Fig. 11).
In the
model, the Maxwell stress is the main contribution
to
only for layers
outside the
threshold
(Fig. 11).
The gas density perturbations look like spiral waves, and the Maxwell
stress is strongly reduced outside of the ionization threshold.
Model
shows a steeper fall in Reynolds stress and in
total
in the dead zone, compared to the
model. The
active layers above and below the dead zone are very thin and only
marginally unstable to MRI. Model
has
and
partly
layers which are deactivated in the dead zone. The
pumping of the waves happens mostly from the MRI-active zone, and not
from both active zone and adjacent layers as in
.
Comparing models
and
,
we conclude that the
pumping of the hydro-dynamical waves into the dead zone is coming both
from ``sandwich'' active layers and from the active zone within
.
The summary of turbulent
stress values for each layer above
the midplane is presented in Table 2. The radial averaging
has been done separately for active (A) and dead (D) zones. The
turbulent
stress is increasing from midplane to
,
and drops in the fourth layer. The
most prominent decrease in
stress from active to dead zone
happens within adjacent to the midplane
layers
for models
and
.
In the case of very thick
dead zone (
), the decrease in turbulent stress at the dead
zone edge is significant in all three layers,
.
3.4 Development of pressure maxima and trapping of solids
Due to active secretion through MRI-active zone, the pressure bump at the inner edge of the dead zone is formed within a hundred of inner orbits.The change in the radial density gradient appears in the disk layers
starting from the midplane and up to 3H. The piling up of the density
behind the
is accompanied by a broad gap before the
threshold. In addition, we find the rings of enhanced density within
the MRI-active zone. The pressure bump at the inner edge of the dead
zone and the rings of enhanced density within the MRI-active
zone are formed due to different processes, which we discuss in
Sect. 4. Figure 12 (top) shows the changes of surface
density for models
and
.
In our
simulations, the number of density rings depends on the extension of
the active zone. In the model without a dead zone, there are three
rings of enhanced density appearing within the domain before the quasi
steady-state is reached (Fig. 12, bottom right). The thicker
dead zone (model
)
seems to be more efficient in piling up
the higher bump: the maximum of the surface density peak is larger,
when compared to the snap-shots of surface density in
at
the same time.
First, we consider the accumulation of the density at the inner edge
of the dead zone. The pressure bump is fixed in time at the location
behind the ionization threshold
,
as in Fig. 13
(left). There are three stages in the evolution of the pressure
maximum at the inner edge of the dead zone (Fig. 13, middle),
for example when considering model
.
One can recognize the
period of very fast mass excavation for the time from t=0 to 150 years. During
years the peak of surface density is
still growing at a roughly ten times slower speed. After t=600 years there is no further increase of surface density, what
corresponds to the steady-state in the presence of saturated
MRI-turbulence. The most straight-forward explanation for three
stages in density excavation gives a time-dependent evolution of the
Maxwell stress, since it governs the accretion in the MRI-active zone
and in the active layers. The quasi-steady state is reached for
models
and
after 600 years, and the maximum
of the pressure bump at the ionization threshold and the minimum of
the surface density in the gap remain unchanged. The density rings in
the active zone appear to be long-living but less stable features. We
observe the merging of two rings of enhanced density after 640 years
in model
(Fig. 13, bottom).
![]() |
Figure 8:
Snap-shots of turbulent |
Open with DEXTER |
The radial pressure gradient is negative in the smooth unperturbed
disk, what leads to sub-Keplerian gas rotation. Dust grains undergo
an orbital decay (Adachi et al. 1976), because they experience the ``head''
wind. For example, the meter-size particle will migrate from 1 AU into
the Sun within few hundred years (MMSN model). When local positive
exponent of the disk midplane pressure appears, the dust grains may
experience ``tail'' wind and the hydrodynamical drag will lead to their
outward migration (Nakagawa et al. 1986). The criterion for outward migration
of the dust grain due to the gas drag is
where


The right panels in Fig. 13 show where criteria for outward migration are fulfilled in our models. Black lines stand for criterion given in Eq. (12) and the red dashed line is for Eq. (13). The outward migration of planetesimals is possible within the pressure bump at the inner edge of the dead zone (Fig. 13). Conditions for embryos are more difficult to satisfy. Not all density rings within the active zone provide sufficiently strong positive pressure gradients, so that planetary embryos cannot be stopped there. The exception is model

![]() |
Figure 9:
Model
|
Open with DEXTER |
![]() |
Figure 10:
Reynolds, Maxwell and total |
Open with DEXTER |
![]() |
Figure 11:
Reynolds, Maxwell and total |
Open with DEXTER |
![]() |
Figure 12:
Surface density after 600 years of evolutions. Red solid
line stands for the initial surface density profile, black solid
line represents the final surface density. Red dotted line indicates
the cosmic ray adsorbtion depth
|
Open with DEXTER |
![]() |
Figure 13:
Relative surface density
|
Open with DEXTER |
3.5 Super-Keplerian rotation and similarity to zonal flows
![]() |
Figure 14:
Radial and vertical turbulent properties in model
|
Open with DEXTER |
![]() |
Figure 15:
Radial and vertical turbulent properties in model
|
Open with DEXTER |
When a quasi-steady state is reached, all time derivatives can be
neglected and the Navier-Stokes equation for radial velocity gives:
![]() |
(14) |
In our locally-isothermal simulations, the temperature is constant on cylinders. The steady-state solution is then given as
![]() |
(15) |
It follows from the last equations, the gas may reach purely Keplerian rotation when the density profile is locally changed to

The vertical profiles of turbulent velocity dispersion and the mean
radial drift velocity
are plotted in
Fig. 14 (
)
and Fig. 15 (
).
The root-mean-square turbulent velocities can be described with the
same shape of the vertical profile both in the active and in the dead
zone. The turbulent velocity dispersion reaches 0.4 in the
corona. This agrees with the result of Fromang & Nelson (2006). The
top left panels in Figs. 14 and 15 show the
root-mean-square turbulent velocities averaged separately for active
(A) and for dead (D) zones. It is interesting to note, that the levels
of
,
at the
midplane are not very different for the two zones. Midplane values
vary from 0.03 for
to 0.16 for
.
As expected, the perturbations of
rotational profile are higher in the active zone. The radial
dependence of
,
shows that the turbulent dispersion is
significantly reduced within the pressure bump at the inner edge of
the dead zone (Figs. 14 and 15, bottom right).
The time-average of
shows that the super-Keplerian rotation
at the density rings is a long-lasting effect. Relatively large dust
particles (i.e. for Stokes number
)
will migrate outwards
with the velocity
(Klahr & Lin 2001). The areas of
outward migration are up to 0.5 AU broad and outward velocities reach
0.15 km s-1 (Figs. 14 and 15, top right).
There are certain similarities between the density bump and rings we
found and the zonal flows described in Johansen et al. (2009). The enhancement
in density has been reaching 10 pressure scale heights both in large
local-box simulations and in the global model of Lyra et al. (2008).
We
estimate the rings to be roughly 1.5 AU broad, what gives maximum
6 pressure scale heights. We show the radial correlations of the
following parameters:
for various
heights in the disk,
,
magnetic pressure
B2/B02, and Maxwell stress
(Figs. 14, 15, bottom left). The
gas around the rings of enhanced density within the active zone in
and
models shows similar turbulent
properties to the zonal flows and corresponding density maximum as in
Johansen et al. (2009): the magnetic pressure and Maxwell stress are strongest
where the density minima are excavated, the maximal
is
half-phase shifted. The maximal deviations from Kepler velocity have
same amplitude as in zonal flows in Johansen et al. (2009), but the relative
amplitude of density
is ten times larger in our models. The maxima of magnetic pressure
B2/B02 and Maxwell stress
are
located between the rings and are about ten times stronger compared to
the values in Johansen et al. (2009).
In addition, the pressure bump at the ionization thresholds has
significantly less turbulence; the
has a
value only about 0.05 and not 0.5 as outside of the density bump. In
the case of
,
the density ring in the active zone is a
``calm place'' as well. Model
keeps the butterfly pattern
of the azimuthal fields much longer. This is the reason why the
turbulent velocities decrease in the density rings weaker in the case
of
,
compared to
.
4 Synthesis: connection between pressure maxima and ``butterfly'' structures
4.1 Density rings in the MRI-active region
In order to provide the comparison with previous global simulations
(Fromang & Nelson 2006), we have done the fully MRI-active disk model. The
turbulent and magnetic properties of the fully MRI-active disk model
are very close to those presented in Fromang & Nelson (2006) for
run S4. The toroidal magnetic field is expelled from the midplane
into the upper disk layers within the linear stage of MRI turbulence.
The peak value of the volume-integrated turbulent
stress is
reaching 0.019 in our model
(Fig. 7) and 0.013 in the S4 model (Fromang & Nelson 2006). At the end of the simulation,
the turbulent
stress is 0.005 in gas pressure units.
In our simulations, the MRI turbulence evolves through three stages:
(a) linear growth, (b) oscillatory saturation regime and (c)
non-oscillating steady state. The stage (b) is best to observe if the
dead zone is included. The oscillations are regular and can be
registered in total magnetic energy and in turbulent stress
time evolution. The dominating azimuthal magnetic field component in
the MRI-active zone switches its sign with a period of about 150 years
or 30 local orbits, and within the radial extent of 1.5 AU. The sign
reversals of the azimuthal magnetic field with respect to the
midplane, known as butterfly diagram, have also been observed in local
shearing-box simulations of the stratified disk (Turner et al. 2007; Johansen et al. 2009).
In case of the global simulation (model
), averaging
over the whole eight AU of the MRI-active domain brings a very
irregular butterfly picture. Numerous reversals of the azimuthal
magnetic field along the radial extent lead to seemingly irregular
peaks in
and in the magnetic energy.
We observe in our models, that the reversals in
come along
with the density rings. Thus, it is important to understand what
causes radial and temporal reversals in the magnetic field, and what
determines a period. We observe that the amplitude of the
oscillations in magnetic energy is higher, if we increase the radial
extent of the active zone (models
and
). In
addition, it is important to know how long the ``butterfly'' structure
can survive. The magnetic field is oscillating over the whole
duration of the local-box simulations. In our global runs, the
life-time of the oscillating stage (b) seems to depend on the extent
of the MRI-active zone (Fig. 7). The thickness of the dead
zone also influences the life-time of the oscillatory regime (b), as
we found from a comparison of magnetic energy curves for models
and
.
The MRI waves in active layers above and
below the dead zone are interacting with the turbulent magnetic field
inside the threshold radius. If the layers above and below the dead
zone are as thin as in our model
,
the turbulent
stress in the active zone appears to be lower, the same applies to the
total magnetic energy. The oscillations of the azimuthal magnetic
field have ceased after 400 years, and after 600 years in the case of
thinner dead zone (model
,
Fig. 7). The
turbulent
stress is roughly constant with radius during the
non-oscillating steady stage (c).
The inner radial boundary with its resistive buffer (
)
and the inner edge of the dead zone
enclose the
MRI-active zone in our models
,
and
.
We observe a butterfly diagram in the MRI-active zone of
our models, when plotting
.
There are no oscillations of
in or
around of the dead zone. We obtain the clearest oscillations of
at the midplane of active zone,
.
The
-sign reversal happens every 150 years and within
(models
,
)
and
(
), i.e. within every 1.5 AU. Those
reversals are stretched along the line of the local orbit in (r,t)space .
The rings of enhanced density appear already at t=150 years, roughly
at the time when the linear AMRI breaks into a nonlinear regime and
the ``butterfly'' is initiated. During the oscillatory stage (b) of the
MRI evolution, there is a radial dependence in turbulent stresses according to
.
At the inner radii,
the
stress is high and it leads to the effect of fast local
excavation of the density and accumulation of it at some outer radius.
The radial reversals of the azimuthal magnetic field are aligned with
the rings of enhanced density. The radial location of magnetic field
reversals remains constant over hundreds of years. At the same
location, we find the stripes of weakest Maxwell stress
10-7(Fig. 11). As soon as the ring of density is created at a certain
location, there is a corresponding change in the rotation and in the
shear. On the one hand, over-density leads to the local
relation which can be too low to excite AMRI. On
the other hand, the change in the rotation reduces the shear and it
leads to local stabilization of MRI within the density ring. The
consequence is that the rings of enhanced density are less turbulent
compared to the density minima between the rings. In contrast to the
local-box studies (Guan et al. 2009; Johansen et al. 2009), there is more then one density
ring forming in the case of a longish MRI-active zone. The merging
between rings is possible. In the
model, we observe
the appearance of three weak density rings.
4.2 Pressure maximum at the inner edge of the dead zone
The formation of the pressure bump at the dead zone edge is not
directly correlated with the ``butterfly'' magnetic structures within
the active zone. The strength of the turbulent viscosity in the
MRI-active zone determines the time scale to form such a pressure
bump. The extent of the MRI-active zone has an effect on the value of
the total turbulent stress in the active zone. As it follows
from Table 1, the turbulence in models
and
provide the largest stresses of
and
,
followed by
with
and
with
.
The radial extent of the
active zone was reduced from 8 AU to 4 AU and to 2 AU. This sequence is
consistent with the results of Guan et al. (2009); Bodo et al. (2008), where a similar
decrease of total alpha stress is demonstrated when the size of the
local box is reduced.
The jump in the turbulent viscosity is responsible for the formation
of a pressure trap at the dead zone edge. This jump is strongest at
the midplane. The intensity of MRI-driven turbulence grows strongly
with the disk height. The locally calculated stresses are different
for each pressure scale height layer: there is up to 1 order of
magnitude difference when comparing the values for midplane and top
layers of the active zone. As expected, the
midplane value of Maxwell stress decreases drastically in the dead
zone, but the Reynolds stress does not ``feel'' the presence of the dead
zone. We observe only a slight bump in the Reynolds stress at the
location of the density maximum. Within the dead zone, there are
spiral density waves propagating from the inner edge outwards. The
vertical velocity dispersion is non-zero as well. We conclude that the
resulting
of about 10-3 is due to the waves pumped
vertically from the active layers and radially from the active zone
through the threshold radius. For the disk evolution, it is important
to have a significant residual
in the dead zone in order to
reach the quasi-steady state without getting unstable (i.e., a
gravitational instability of the density ring).
5 Conclusions
We have presented the results of the first global non-ideal 3D MHD simulations of stratified protoplanetary disks. The domain spans the transition from the MRI-active region near the star to the dead zone at greater distances. The main results are as follows.
- -
- As suggested in Kretke et al. (2008), the height-averaged accretion
stress shows a smooth radial transition across the dead zone edge.
The stress peaks well off the midplane at
. Consequently, averaging over the full disk thickness yields only a mild jump, despite the steepness of the midplane radial stress gradient.
- -
- Weak accretion flows within the dead zone are driven mainly
by Reynolds stresses. Spiral density waves propagating horizontally
produce non-zero velocities and angular momentum transport near the
midplane, apparently with little associated mixing. The dead zone
midplane vertical velocity dispersion
is 0.03 times the sound speed, two to three times less than the radial and azimuthal components (Fig. 15). The waves are pumped both by the active region near the star and by the active layers above and below the dead zone. In model
where the dead zone is very thick, the pumping from the surface layers is weak and the stress falls steeply with radius within the dead zone (Fig. 11).
- -
- The excavation of gas from the active region during the linear growth and after the saturation of the MRI leads to the creation of a steady local radial gas pressure maximum near the dead zone edge, and to the formation of dense rings within the active region, resembling the zonal flows described in Johansen et al. (2009) (Figs. 12, 13). Super-Keplerian rotation is observed where the radial pressure gradient is positive. The corresponding outward radial drift speeds for bodies of unit Stokes number can exceed 10% of the sound speed. The pressure maxima are thus likely locations for the concentration of solid particles.
- -
- The turbulent velocity dispersion, magnetic pressure and Maxwell stresses all are greatest in the density minima between the rings. The dense rings are ``quiet'' locations where the turbulence is substantially weaker. Such an environment may be helpful for the growth of larger bodies.
- -
- The rings within the active zone sometimes move about, leading to mergers. In contrast, the bump at the dead zone inner edge is stationary. Planetary embryos with masses in the type I migration regime can be retained at the dead zone inner edge as proposed by Schlaufman et al. (2009); Ida & Lin (2008).
6 Outlook
The causes of the magnetic field oscillations appearing in the butterfly diagram remain to be clarified. In particular, it is not known what processes drive the quasi-periodic reversals in the azimuthal magnetic fields shown in Fig. 4.
While we have used a fixed magnetic diffusivity distribution, the
degree of ionization will in fact change as the disk evolves. Annuli
of increasing surface density will absorb the ionizing stellar X-rays
and interstellar cosmic rays further from the midplane, and will have
higher recombination rates due to the greater densities. These
effects will likely mean contrasts in the strength of the turbulence
between the rings and inter-ring regions, even greater than those
observed in our calculations. Stronger magnetic fields and higher
mass flow rates in the inter-ring regions could lead in turn to growth
in the surface density contrast over time, analogous to the viscous
instability of the -model in the radiation-pressure dominated
regime (Piran 1978; Lightman & Eardley 1974).
The resistivity can show another kind of time variation near the boundary of the thermally-ionized region. Changes in the temperature can alter the strength of the turbulence and thus the heating rate. In this way, radial heat transport can activate previously dead gas (Wünsch et al. 2006,2005). Radial oscillations of the ionization front may be the consequence. Overall, due to the fixed magnetic diffusivity, it is probable that our models underestimate the evolution resulting from the ionization thresholds.
Owing to the effects of the radial boundary conditions, our global simulations have limitations for measuring quantities such as the accretion rate and the mean radial velocity. Fixing the rate of flow across the outer radial boundary is a promising avenue for further exploration in this direction.
AcknowledgementsN. Dzyurkevich acknowledges the support of the Deutsches Zentrum für Luft- und Raumfahrt (DLR). N. Dzyurkevich, M. Flock and H. Klahr were supported in part by the Deutsche Forschungsgemeinschaft (DFG) through Forschergruppe 759, ``The Formation of Planets: The Critical First Growth Phase''. The participation of N. J. Turner was made possible by the NASA Solar Systems Origins program under grant 07-SSO07-0044, and by the Alexander von Humboldt Foundation through a Fellowship for Experienced Researchers. The parallel computations were performed on the PIA cluster of the Max Planck Institute for Astronomy Heidelberg, located at the computing center of the Max Planck Society in Garching.
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Footnotes
- ... Turner
- Permanent address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA.
All Tables
Table 1:
Model properties and midplane -stresses inside
(
)
and outside (
)
of the
ionization threshold radius
.
Table 2:
-stresses inside (A) and outside (D) of the
ionization threshold radius for four layers above the
midplane.
All Figures
![]() |
Figure 1:
Vertical profiles of magnetic diffusivity. Black lines
show the profiles adopted for simulations of the dead zone, with
|
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Inverse plasma |
Open with DEXTER | |
In the text |
![]() |
Figure 3:
Elsässer number
|
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Temporal evolution of azimuthal magnetic field as a function
of radius for
|
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Temporal evolution of azimuthal magnetic field as a function
of radius for
|
Open with DEXTER | |
In the text |
![]() |
Figure 6:
Horizontally averaged magnetic fields components
|
Open with DEXTER | |
In the text |
![]() |
Figure 7:
Top: total magnetic energy evolution;
Bottom: total alpha
stress evolution for models
|
Open with DEXTER | |
In the text |
![]() |
Figure 8:
Snap-shots of turbulent |
Open with DEXTER | |
In the text |
![]() |
Figure 9:
Model
|
Open with DEXTER | |
In the text |
![]() |
Figure 10:
Reynolds, Maxwell and total |
Open with DEXTER | |
In the text |
![]() |
Figure 11:
Reynolds, Maxwell and total |
Open with DEXTER | |
In the text |
![]() |
Figure 12:
Surface density after 600 years of evolutions. Red solid
line stands for the initial surface density profile, black solid
line represents the final surface density. Red dotted line indicates
the cosmic ray adsorbtion depth
|
Open with DEXTER | |
In the text |
![]() |
Figure 13:
Relative surface density
|
Open with DEXTER | |
In the text |
![]() |
Figure 14:
Radial and vertical turbulent properties in model
|
Open with DEXTER | |
In the text |
![]() |
Figure 15:
Radial and vertical turbulent properties in model
|
Open with DEXTER | |
In the text |
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