2,992 research outputs found
One-armed spirals in locally isothermal, radially structured self-gravitating discs
We describe a new mechanism that leads to the destabilisation of
non-axisymmetric waves in astrophysical discs with an imposed radial
temperature gradient. This might apply, for example, to the outer parts of
protoplanetary discs. We use linear density wave theory to show that
non-axisymmetric perturbations generally do not conserve their angular momentum
in the presence of a forced temperature gradient. This implies an exchange of
angular momentum between linear perturbations and the background disc. In
particular, when the disturbance is a low-frequency trailing wave and the disc
temperature decreases outwards, this interaction is unstable and leads to the
growth of the wave. We demonstrate this phenomenon through numerical
hydrodynamic simulations of locally isothermal discs in 2D using the FARGO code
and in 3D with the ZEUS-MP and PLUTO codes. We consider radially structured
discs with a self-gravitating region which remains stable in the absence of a
temperature gradient. However, when a temperature gradient is imposed we
observe exponential growth of a one-armed spiral mode (azimuthal wavenumber
) with co-rotation radius outside the bulk of the spiral arm, resulting in
a nearly-stationary one-armed spiral pattern. The development of this one-armed
spiral does not require the movement of the central star, as found in previous
studies. Because destabilisation by a forced temperature gradient does not
explicitly require disc self-gravity, we suggest this mechanism may also affect
low-frequency one-armed oscillations in non-self-gravitating discs.Comment: 15 pages, accepted by MNRA
Effects of upper disc boundary conditions on the linear Rossby wave instability
The linear Rossby wave instability (RWI) in global, 3D polytropic discs is
revisited with a much simpler numerical method than that previously employed by
the author. The governing partial differential equation is solved with finite
differences in the radial direction and spectral collocation in the vertical
direction. RWI modes are calculated subject to different upper disc boundary
conditions. These include free surface, solid boundaries and variable vertical
domain size. Boundary conditions that oppose vertical motion increase the
instability growth rate by a few per cent. The magnitude of vertical flow
throughout the fluid column can be affected but the overall flow pattern is
qualitatively unchanged. Numerical results support the notion that the RWI is
intrinsically two dimensional. This implies that inconsistent upper disc
boundary conditions, such as vanishing enthalpy perturbation, may inhibit the
RWI in 3D.Comment: 5 pages, published in MNRAS; this article precedes arXiv:1209.0470
and details the numerical scheme employed therei
Testing large-scale vortex formation against viscous layers in three-dimensional discs
Vortex formation through the Rossby wave instability (RWI) in protoplanetary
discs has been invoked to play a role in planet formation theory, and suggested
to explain the observation of large dust asymmetries in several transitional
discs. However, whether or not vortex formation operates in layered accretion
discs, i.e. models of protoplanetary discs including dead zones near the disc
midplane --- regions that are magnetically inactive and the effective viscosity
greatly reduced --- has not been verified. As a first step toward testing the
robustness of vortex formation in layered discs, we present non-linear
hydrodynamical simulations of global 3D protoplanetary discs with an imposed
kinematic viscosity that increases away from the disc midplane. Two sets of
numerical experiments are performed: (i) non-axisymmetric instability of
artificial radial density bumps in viscous discs; (ii) vortex-formation at
planetary gap edges in layered discs. Experiment (i) shows that the linear
instability is largely unaffected by viscosity and remains dynamical. The
disc-planet simulations also show the initial development of vortices at gap
edges, but in layered discs the vortices are transient structures which
disappear well into the non-linear regime. We suggest that the long term
survival of columnar vortices, such as those formed via the RWI, requires low
effective viscosity throughout the vertical extent of the disc, so such
vortices do not persist in layered discs.Comment: 14 pages, accepted by MNRA
Linear stability of magnetized massive protoplanetary disks
Magneto-rotational instability (MRI) and gravitational instability (GI) are
the two principle routes to turbulent angular momentum transport in accretion
disks. Protoplanetary disks may develop both. This paper aims to reinvigorate
interest in the study of magnetized massive protoplanetary disks, starting from
the basic issue of stability. The local linear stability of a self-gravitating,
uniformly magnetized, differentially rotating, three-dimensional stratified
disk subject to axisymmetric perturbations is calculated numerically. The
formulation includes resistivity. It is found that the reduction in the disk
thickness by self-gravity can decrease MRI growth rates; the MRI becomes global
in the vertical direction, and MRI modes with small radial length scales are
stabilized. The maximum vertical field strength that permits the MRI in a
strongly self-gravitating polytropic disk with polytropic index is
estimated to be ,
where is the midplane sound speed and is the angular
velocity. In massive disks with layered resistivity, the MRI is not
well-localized to regions where the Elsasser number exceeds unity. For MRI
modes with radial length scales on the order of the disk thickness,
self-gravity can enhance density perturbations, an effect that becomes
significant in the presence of a strong toroidal field, and which depends on
the symmetry of the underlying MRI mode. In gravitationally unstable disks
where GI and MRI growth rates are comparable, the character of unstable modes
can transition smoothly between MRI and GI. Implications for non-linear
simulations are discussed briefly.Comment: Accepted by ApJ; project source code available at
https://github.com/minkailin/sgmr
Gravitational instability of planetary gaps and its effect on orbital migration
Gap formation by giant planets in self-gravitating disks may lead to a
gravitational edge instability (GEI). We demonstrate this GEI with global 3D
and 2D self-gravitating disk-planet simulations using the ZEUS, PLUTO and FARGO
hydrodynamic codes. High resolution 2D simulations show that an unstable outer
gap edge can lead to outwards orbital migration. Our results have important
implications for theories of giant planet formation in massive disks.Comment: Published in the proceedings of IAUS 299. Associated paper is
arXiv:1306.2514. Poster can be found at
http://cita.utoronto.ca/~mklin924/IAUposter.pd
The effect of self-gravity on vortex instabilities in disc-planet interactions
We study the effect of disc self-gravity on vortex-forming instabilities
associated with gaps opened by a Saturn mass planet in a protoplanetary disc.
It is shown analytically and confirmed through linear calculations that vortex
modes with low azimuthal mode number are stabilised by increasing
self-gravity if the basic state is fixed. However, linear calculations show
that the combined effect of self-gravity through the background and through the
linear response shifts the most unstable vortex mode to higher Nonlinear
hydrodynamic simulations of planetary gaps show more vortices develop with
increasing strength of self-gravity. For sufficiently large disc mass the
vortex instability is suppressed and replaced by a new global instability,
consistent with analytical expectations. In the nonlinear regime, vortex
merging is increasingly delayed as the disc mass increases and multiple
vortices may persist until the end of simulations. With self-gravity, the
post-merger vortex is localised in azimuth and has similar structure to a
Kida-like vortex. This is unlike the case without self-gravity where vortices
merge to form a single vortex extended in azimuth. We also performed a series
of supplementary simulations of co-orbital Kida-like vortices and found that
self-gravity enables such vortices to execute horseshoe turns upon encountering
each other. As a result vortex merging is avoided on time-scales where it would
occur without self-gravity. Thus we suggest that mutual repulsion of
self-gravitating vortices in a rotating flow is responsible for the delayed
vortex merging above. The effect of self-gravity on vortex-induced migration is
briefly discussed. We found that when self-gravity is included, the
vortex-induced type III migration of Lin & Papaloizou (2010) is delayed but the
extent of migration is unchanged.Comment: 21 pages, 19 figures. Accepted by MNRAS. Displayed abstract is
shortene
Gap formation and stability in non-isothermal protoplanetary discs
Several observations of transition discs show lopsided dust-distributions. A
potential explanation is the formation of a large-scale vortex acting as a
dust-trap at the edge of a gap opened by a giant planet. Numerical models of
gap-edge vortices have thus far employed locally isothermal discs, but the
theory of this vortex-forming or `Rossby wave' instability was originally
developed for adiabatic discs. We generalise the study of planetary gap
stability to non-isothermal discs using customised numerical simulations of
disc-planet systems where the planet opens an unstable gap. We include in the
energy equation a simple cooling function with cooling timescale
, where is the Keplerian frequency, and
examine the effect of on the stability of gap edges and vortex
lifetimes. We find increasing lowers the growth rate of
non-axisymmetric perturbations, and the dominant azimuthal wavenumber
decreases. We find a quasi-steady state consisting of one large-scale,
over-dense vortex circulating the outer gap edge, typically lasting
orbits. Vortex lifetimes were found to generally increase with cooling times up
to an optimal value, beyond which vortex lifetimes decrease. This non-monotonic
dependence is qualitatively consistent with recent studies using strictly
isothermal discs that vary the disc aspect ratio.Comment: 12 pages, 13 figures, 1 table, accepted by MNRA
Steady state of dust distributions in disk vortices: Observational predictions and applications to transitional disks
The Atacama Large Millimeter Array (ALMA) has been returning images of
transitional disks in which large asymmetries are seen in the distribution of
mm-sized dust in the outer disk. The explanation in vogue borrows from the
vortex literature by suggesting that these asymmetries are the result of dust
trapping in giant vortices, excited via Rossby wave instability (RWI) at
planetary gap edges. Due to the drag force, dust trapped in vortices will
accumulate in the center, and diffusion is needed to maintain a steady state
over the lifetime of the disk. While previous work derived semi-analytical
models of the process, in this paper we provide analytical steady-state
solutions. Exact solutions exist for certain vortex models. The solution is
determined by the vortex rotation profile, the gas scale height, the vortex
aspect ratio, and the ratio of dust diffusion to gas-dust friction. In
principle, all these quantities can be derived from observations, which would
give validation of the model, also giving constrains on the strength of the
turbulence inside the vortex core. Based on our solution, we derive quantities
such as the gas-dust contrast, the trapped dust mass, and the dust contrast at
the same orbital location. We apply our model to the recently imaged Oph IRS 48
system, finding values within the range of the observational uncertainties.Comment: 11 pages, 3 figures, accepted by Ap
Sample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem
We generalize the PAC (probably approximately correct) learning model to the
quantum world by generalizing the concepts from classical functions to quantum
processes, defining the problem of \emph{PAC learning quantum process}, and
study its sample complexity. In the problem of PAC learning quantum process, we
want to learn an -approximate of an unknown quantum process
from a known finite concept class with probability using samples
, where are
computational basis states sampled from an unknown distribution and
are the (possibly mixed) quantum states outputted
by . The special case of PAC-learning quantum process under constant input
reduces to a natural problem which we named as approximate state
discrimination, where we are given copies of an unknown quantum state
from an known finite set , and we want to learn with probability
an -approximate of with as few copies of as possible. We
show that the problem of PAC learning quantum process can be solved with
samples when
the outputs are pure states and samples if the outputs can be mixed. Some
implications of our results are that we can PAC-learn a polynomial sized
quantum circuit in polynomial samples and approximate state discrimination can
be solved in polynomial samples even when concept class size is
exponential in the number of qubits, an exponentially improvement over a full
state tomography
Vertical shear instability in the solar nebula
We quantify the thermodynamic requirement for the Vertical Shear Instability
and evaluate its relevance to realistic protoplanetary disks as a potential
route to hydrodynamic turbulence.Comment: IAUS 314 proceedings, main paper arXiv:1505.0216
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