410 research outputs found
Global MHD simulations of stratified and turbulent protoplanetary discs. I. Model properties
We present the results of global 3-D MHD simulations of stratified and
turbulent protoplanetary disc models. The aim of this work is to develop thin
disc models capable of sustaining turbulence for long run times, which can be
used for on-going studies of planet formation in turbulent discs. The results
are obtained using two codes written in spherical coordinates: GLOBAL and
NIRVANA. Both are time--explicit and use finite differences along with the
Constrained Transport algorithm to evolve the equations of MHD. In the presence
of a weak toroidal magnetic field, a thin protoplanetary disc in hydrostatic
equilibrium is destabilised by the magnetorotational instability (MRI). When
the resolution is large enough (25 vertical grid cells per scale height), the
entire disc settles into a turbulent quasi steady-state after about 300 orbits.
Angular momentum is transported outward such that the standard alpha parameter
is roughly 4-6*10^{-3}. We find that the initial toroidal flux is expelled from
the disc midplane and that the disc behaves essentially as a quasi-zero net
flux disc for the remainder of the simulation. As in previous studies, the disc
develops a dual structure composed of an MRI--driven turbulent core around its
midplane, and a magnetised corona stable to the MRI near its surface. By
varying disc parameters and boundary conditions, we show that these basic
properties of the models are robust. The high resolution disc models we present
in this paper achieve a quasi--steady state and sustain turbulence for hundreds
of orbits. As such, they are ideally suited to the study of outstanding
problems in planet formation such as disc--planet interactions and dust
dynamics.Comment: 19 pages, 29 figures, accepted in Astronomy & Astrophysic
The interaction of a giant planet with a disc with MHD turbulence I: The initial turbulent disc models
This is the first of a series of papers aimed at developing and interpreting
simulations of protoplanets interacting with turbulent accretion discs. Here we
study the disc models prior to the introduction of a protoplanet.We study
models in which a Keplerian domain is unstable to the magnetorotational
instability (MRI). Various models with B-fields having zero net flux are
considered.We relate the properties of the models to classical viscous disc
theory.All models attain a turbulent state with volume averaged stress
parameter alpha ~ 0.005. At any particular time the vertically and azimuthally
averaged value exhibited large fluctuations in radius. Time averaging over
periods exceeding 3 orbital periods at the outer boundary of the disc resulted
in a smoother quantity with radial variations within a factor of two or so. The
vertically and azimuthally averaged radial velocity showed much larger spatial
and temporal fluctuations, requiring additional time averaging for 7-8 orbital
periods at the outer boundary to limit them. Comparison with the value derived
from the averaged stress using viscous disc theory yielded schematic agreement
for feasible averaging times but with some indication that the effects of
residual fluctuations remained. The behaviour described above must be borne in
mind when considering laminar disc simulations with anomalous Navier--Stokes
viscosity. This is because the operation of a viscosity as in classical viscous
disc theory with anomalous viscosity coefficient cannot apply to a turbulent
disc undergoing rapid changes due to external perturbation. The classical
theory can only be used to describe the time averaged behaviour of the parts of
the disc that are in a statistically steady condition for long enough for
appropriate averaging to be carried out.Comment: 10 pages, 23 figures, accepted for publication in MNRAS. A gzipped
postscript version including high resolution figures is available at
http://www.maths.qmul.ac.uk/~rp
On the Ionisation Fraction in Protoplanetary Disks I: Comparing Different Reaction Networks
We calculate the ionisation fraction in protostellar disk models using a
number of different chemical reaction networks, including gas-phase and
gas-grain reaction schemes. The disk models we consider are conventional
alpha-disks, which include viscous heating and radiative cooling. The primary
source of ionisation is assumed to be X-ray irradiation from the central star.
We consider a number of gas-phase chemical networks. In general we find that
the simple models predict higher fractional ionisation levels and more
extensive active zones than the more complex models. When heavy metal atoms are
included the simple models predict that the disk is magnetically active
throughout. The complex models predict that extensive regions of the disk
remain magnetically uncoupled even with a fractional abundance of magnesium of
10(-8). The addition of submicron sized grains with a concentration of 10(-12)
causes the size of the dead zone to increase dramatically for all kinetic
models considered. We find that the simple and complex gas-grain reaction
schemes agree on the size and structure of the resulting dead zone. We examine
the effects of depleting the concentration of small grains as a crude means of
modeling the growth of grains during planet formation. We find that a depletion
factor of 10(-4) causes the gas-grain chemistry to converge to the gas-phase
chemistry when heavy metals are absent. 10(-8) is required when magnesium is
included. This suggests that efficient grain growth and settling will be
required in protoplanetary disks, before a substantial fraction of the disk
mass in the planet forming zone between 1 - 10 AU becomes magnetically active
and turbulent.Comment: 21 pages, 23 figures, accepted for publication in A & A Includes
correction to our implementation of the Umebayashi-Nakano reaction networ
On the Orbital Evolution of Low Mass Protoplanets in Turbulent, Magnetised Disks
(Abridged).We present the results of MHD simulations of low mass protoplanets
interacting with turbulent disks. We calculate the orbital evolution of
`planetesimals' and protoplanets with masses in the range 0 < m_p < 30 M_Earth.
Planetesimals and protoplanets undergo stochastic migration due to interaction
with turbulent density fluctuations. Over run times of ~ 150 planet orbits,
stochastic migration dominates over type I migration for many models. Fourier
analysis of the torques experienced by planets indicates that the torque
fluctuations contain components with significant power whose time scales of
variation are similar to the simulation run times. These low frequency
fluctuations partly explain the dominance of stochastic torques, and may
provide a powerful means of counteracting the type I migration of some planets
in turbulent disks. Turbulence is a source of eccentricity driving.
Planetesimals attained eccentricities in the range 0.02 < e < 0.14, m_p=1
M_Earth planets attained eccentricities 0.02 < e < 0.08, and m_p=10 M_Earth
protoplanets reached 0.02 < e < 0.03. This is in basic agreement with a model
in which turbulence drives e-growth, and interaction with disk material at
coorbital Lindblad resonances causes e-damping. These results are significant
for planet formation. Stochastic migration may prevent some planet cores
migrating into their star via type I before becoming gas giants. The growth of
planetary cores may be enhanced by preventing isolation. Eccentricity
excitation by turbulence, however, may reduce growth rates of planetary cores
during the runaway and oligarchic growth stages, and cause collisions between
planetesimals to become destructive.Comment: 21 pages, 16 figures. Accepted for publication in Astronomy &
Astrophysics. A version with full resolution, colour figures is available
from: http://www.maths.qmul.ac.uk/~rpn/preprint
The interaction of planets with a disc with MHD turbulence III: Flow morphology and conditions for gap formation in local and global simulations
We present the results of both global cylindrical disc simulations and local
shearing box simulations of protoplanets interacting with a disc undergoing MHD
turbulence with zero net flux magnetic fields. We investigate the nature of the
disc response and conditions for gap formation. This issue is an important one
for determining the type and nature of the migration of the protoplanet, with
the presence of a deep gap being believed to enable slower migration. For both
types of simulation we find a common pattern of behaviour for which the main
parameter determining the nature of the response is , with
, , , and being the protoplanet mass, the central mass, the
orbital radius and the disc semi-thickness respectively. We find that as this
parameter is increased towards 0.1, the presence of the protoplanet is first
indicated by the appearance of the well known trailing wake which, although it
may appear erratic on account of the turbulence, appears to be well defined.
Once the above parameter exceeds a number around unity a gap starts to develop
inside which the magnetic energy density tends to be concentrated in the high
density wakes. This gap formation condition can be understood from simple
dimensional considerations of the conditions for nonlinearity, and the balance
of angular momentum transport due to Maxwell and Reynolds' stresses with that
due to tidal torques. An important result is that the basic flow morphology in
the vicinity of the protoplanet is very similar in both the local and global
simulations. This indicates that local shearing box simulations, which are
computationally less demanding, capture much of the physics of disc-planet
interaction. Thus they may provide a useful tool for studying the local
interaction between forming protoplanets and turbulent, protostellar discs.Comment: 20 pages, 28 figures (some colour), accepted for publication in
M.N.R.A.S. with minor modification. A pdf version containing high resolution
colour figures is available from
http://www.maths.qmul.ac.uk/~rpn/projects/mhd along with additional images
and movies. A companion paper accepted without change by M.N.R.A.S. is also
availabl
The interaction of a giant planet with a disc with MHD turbulence II: The interaction of the planet with the disc
We present a global MHD simulation of a turbulent accretion disc interacting
with a protoplanet of 5 Jupiter masses. The disc model had H/r=0.1,and a value
of the Shakura & Sunyaev alpha ~ 0.005. The protoplanet opened a gap in the
disc, with the interaction leading to inward migration on the expected time
scale. Spiral waves were launched by the protoplanet and although they were
diffused and dissipated through interaction with the turbulence, they produced
an outward angular momentum flow which compensated for a reduced flux
associated with the turbulence, so maintaining the gap. When compared with
laminar disc models with the same estimated alpha, the gap was found to be
deeper and wider indicating that the turbulent disc behaved as if it possessed
a smaller alpha. This may arise for two reasons. First, the turbulence does not
provide a source of constantly acting friction in the near vicinity of the
planet that leads to steady mass flow into the gap region. Instead the
turbulence is characterised by large fluctuations in the radial velocity, and
time averaging over significant time scales is required to recover the
underlying mass flow through the disc. Near the planet the disc material
experiences high amplitude perturbations on time scales that are short relative
to the time scale required for averaging. The disc response is thus likely to
be altered relative to a Navier--Stokes model. Second, the simulation indicates
that an ordered magnetic connection between the inner and outer disc can occur
enabling angular momentum to flow out across the gap, helping to maintain it
independently of the protoplanet's tide. This type of effect may assist gap
formation for smaller mass protoplanets which otherwise would not be able to
maintain them.Comment: 14 pages, 17 figures, accepted for publication in MNRAS. A gzipped
postscript version including high resolution figures is available at
http://www.maths.qmw.ac.uk/~rp
On the Ionisation Fraction in Protoplanetary Disks II: The Effect of Turbulent Mixing on Gas--phase Chemistry
We calculate the ionisation fraction in protostellar disk models using two
different gas-phase chemical networks, and examine the effect of turbulent
mixing by modelling the diffusion of chemical species vertically through the
disk. The aim is to determine in which regions of the disk gas can couple to a
magnetic field and sustain MHD turbulence. We find that the effect of diffusion
depends crucially on the elemental abundance of heavy metals (magnesium)
included in the chemical model. In the absence of heavy metals, diffusion has
essentially no effect on the ionisation structure of the disks, as the
recombination time scale is much shorter than the turbulent diffusion time
scale. When metals are included with an elemental abundance above a threshold
value, the diffusion can dramatically reduce the size of the magnetically
decoupled region, or even remove it altogther. For a complex chemistry the
elemental abundance of magnesium required to remove the dead zone is 10(-10) -
10(-8). We also find that diffusion can modify the reaction pathways, giving
rise to dominant species when diffusion is switched on that are minor species
when diffusion is absent. This suggests that there may be chemical signatures
of diffusive mixing that could be used to indirectly detect turbulent activity
in protoplanetary disks. We find examples of models in which the dead zone in
the outer disk region is rendered deeper when diffusion is switched on. Overall
these results suggest that global MHD turbulence in protoplanetary disks may be
self-sustaining under favourable circumstances, as turbulent mixing can help
maintain the ionisation fraction above that necessary to ensure good coupling
between the gas and magnetic field.Comment: 11 pages, 7 figures; accepted for publication in A &
The interaction of planets with a disc with MHD turbulence IV: Migration rates of embedded protoplanets
(Abridged) We present global disc and local shearing box simulations of
planets interacting with a MHD turbulent disc. We examine the torque exerted by
the disc on the embedded planets as a function of planet mass, and thus make a
first study of orbital migration of planets due to interaction with turbulent
discs. Global simulations were performed for a disc with H/R=0.07 and planet
masses M_p=3,10,30 Earth masses, and 3 Jupiter masses. Shearing box runs were
performed for values of (M_p/M_*)/(H/R)^3=0.1,0.3,1.0 and 2.0, M_* being the
central mass. These allow embedded and gap forming planets to be examined. In
all cases the instantaneous torque exerted on a planet showed strong
fluctuations. In the embedded cases it oscillated between negative and positive
values, and migration occurs as a random walk, unlike the usual type I
migration. Running time averages for embedded planets over 20-25 orbital
periods show that large fluctuations occur on longer time scales, preventing
convergence of the average torque to well defined values, or even to a well
defined sign. Fluctuations become relatively smaller for larger masses, giving
better convergence, due to the planet's perturbation of the disc becoming
larger than the turbulence in its vicinity. Eventually gap formation occurs,
with a transition to type II migration. The existence of significant
fluctuations occurring in turbulent discs on long time scales is important for
lower mass embedded protoplanets. If significant fluctuations occur on the
longest disc evolutionary time scales, convergence of torque running averages
for practical purposes will not occur, and the migration behaviour of low mass
protoplanets considered as an ensemble would be very different from predictions
of type I theory for laminar discs.Comment: 19 pages, 24 figures (some colour), submitted to M.N.R.A.S. A gzipped
postscript version containing high resolution colour figures is available
from http://www.maths.qmul.ac.uk/~rp
On the dynamics of planetesimals embedded in turbulent protoplanetary discs
(abridged) Angular momentum transport and accretion in protoplanetary discs
are generally believed to be driven by MHD turbulence via the
magneto-rotational instability (MRI). The dynamics of solid bodies embedded in
such discs (dust grains, boulders, planetesimals and planets) may be strongly
affected by the turbulence, such that the formation pathways for planetary
systems are determined in part by the strength and spatial distribution of the
turbulent flow.
We examine the dynamics of planetesimals, with radii between 1m \^a 10 km,
embedded in turbulent protoplanetary discs, using three dimensional MHD
simulations. The planetesimals experience gas drag and stochastic gravitational
forces due to the turbulent disc. We use, and compare the results from, local
shearing box simulations and global models in this study.
The main aims of this work are to examine: the growth, and possible
saturation, of the velocity dispersion of embedded planetesimals as a function
of their size and disc parameters; the rate of radial migration and diffusion
of planetesimals; the conditions under which the results from shearing box and
global simulations agree.
We find good agreement between local and global simulations when shearing
boxes of dimension 4H x 16H x 2H are used (H being the local scale height). The
magnitude of the density fluctuations obtained is sensitive to the box size,
due to the excitation and propagation of spiral density waves. This affects the
stochastic forcing experienced by planetesimals. [...]
Our models show that fully developed MHD turbulence in protoplanetary discs
would have a destructive effect on embedded planetesimals. Relatively low
levels of turbulence are required for traditional models of planetesimal
accretion to operate, this being consistent with the existence of a dead zone
in protoplanetary discs.Comment: 23 pages, 28 figures, 3 tables, accepted for publication in MNRA
On the migration of protoplanets embedded in circumbinary disks
We present the results of hydrodynamical simulations of low mass protoplanets
embedded in circumbinary accretion disks. The aim is to examine the migration
and long term orbital evolution of the protoplanets, in order to establish the
stability properties of planets that form in circumbinary disks. Simulations
were performed using a grid--based hydrodynamics code. First we present a set
of calculations that study how a binary interacts with a circumbinary disk. We
evolve the system for 10^5 binary orbits, which is the time needed for the
system to reach a quasi-equilibrium state. From this time onward the apsidal
lines of the disk and the binary are aligned, and the binary eccentricity
remains essentially unchanged with a value of e_b ~ 0.08. Once this stationary
state is obtained, we embed a low mass protoplanet in the disk and let it
evolve under the action of the binary and disk forces. We consider protoplanets
with masses of 5, 10 and 20 Earth masses. In each case, we find that inward
migration of the protoplanet is stopped at the edge of the tidally truncated
cavity formed by the binary. This effect is due to positve corotation torques,
which can counterbalance the net negative Lindblad torques in disk regions
where the surface density profile has a sufficiently large positive gradient.
Halting of migration occurs in a region of long-term stability, suggesting that
low mass circumbinary planets may be common, and that gas giant circumbinary
planets should be able to form in circumbinary disks.Comment: 10 pages, 10 figures, accepted for publication in A&
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