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 $m=1$) 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 $\Gamma=1$ is estimated to be $B_{z,\mathrm{max}} \simeq c_{s0}\Omega\sqrt{\mu_0/16\pi G}$, where $c_{s0}$ is the midplane sound speed and $\Omega$ 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 $,m,$ 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 $m.$ 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 $t_c=\beta\Omega_k^{-1}$, where $\Omega_k$ is the Keplerian frequency, and examine the effect of $\beta$ on the stability of gap edges and vortex lifetimes. We find increasing $\beta$ lowers the growth rate of non-axisymmetric perturbations, and the dominant azimuthal wavenumber $m$ decreases. We find a quasi-steady state consisting of one large-scale, over-dense vortex circulating the outer gap edge, typically lasting $O(10^3)$ 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 $\epsilon$-approximate of an unknown quantum process $c^*$ from a known finite concept class $C$ with probability $1-\delta$ using samples $\{(x_1,c^*(x_1)),(x_2,c^*(x_2)),\dots\}$, where $\{x_1,x_2, \dots\}$ are computational basis states sampled from an unknown distribution $D$ and $\{c^*(x_1),c^*(x_2),\dots\}$ are the (possibly mixed) quantum states outputted by $c^*$. 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 $c^*$ from an known finite set $C$, and we want to learn with probability $1-\delta$ an $\epsilon$-approximate of $c^*$ with as few copies of $c^*$ as possible. We show that the problem of PAC learning quantum process can be solved with $$O\left(\frac{\log|C| + \log(1/ \delta)} { \epsilon^2}\right)$$ samples when the outputs are pure states and $$O\left(\frac{\log^3 |C|(\log |C|+\log(1/ \delta))} { \epsilon^2}\right)$$ 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 $|C|$ 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