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 Min-Kai Lin Physics , 2012, DOI: 10.1111/j.1365-2966.2012.21955.x Abstract: Numerical simulations of global three-dimensional (3D), self-gravitating discs with a gap opened by an embedded planet are presented. The simulations are customised to examine planetary gap stability. Previous results, obtained by Lin & Papaloizou from two-dimensional (2D) disc models, are reproduced in 3D. These include (i) the development of vortices associated with local vortensity minima at gap edges and their merging on dynamical timescales in weakly self-gravitating discs, (ii) the increased number of vortices as the strength of self-gravity is increased and their resisted merging, and (iii) suppression of the vortex instability and development of global spiral arms associated with local vortensity maxima in massive discs. The vertical structure of these disturbances are examined. In terms of the relative density perturbation, the vortex disturbance has weak vertical dependence when self-gravity is neglected. Vortices become more vertically stratified with increasing self-gravity. This effect is seen even when the unperturbed region around the planet's orbital radius has a Toomre stability parameter ~10. The spiral modes display significant vertical structure at the gap edge, with the midplane density enhancement being several times larger than that near the upper disc boundary. However, for both instabilities the vertical Mach number is typically a few per cent,and on average vertical motions near the gap edge do not dominate horizontal motions.
 Physics , 2015, DOI: 10.1051/0004-6361/201424528 Abstract: The astonishing diversity in the observed planetary population requires theoretical efforts and advances in planet formation theories. Numerical approaches provide a method to tackle the weaknesses of current planet formation models and are an important tool to close gaps in poorly constrained areas. We present a global disk setup to model the first stages of giant planet formation via gravitational instabilities (GI) in 3D with the block-structured adaptive mesh refinement (AMR) hydrodynamics code ENZO. With this setup, we explore the impact of AMR techniques on the fragmentation and clumping due to large-scale instabilities using different AMR configurations. Additionally, we seek to derive general resolution criteria for global simulations of self-gravitating disks of variable extent. We run a grid of simulations with varying AMR settings, including runs with a static grid for comparison, and study the effects of varying the disk radius. Adopting a marginally stable disk profile (Q_init=1), we validate the numerical robustness of our model for different spatial extensions, from compact to larger, extended disks (R_disk = 10, 100 and 300 AU, M_disk ~ 0.05 M_Sun, M_star = 0.646 M_Sun). By combining our findings from the resolution and parameter studies we find a lower limit of the resolution to be able to resolve GI induced fragmentation features and distinct, turbulence inducing clumps. Irrespective of the physical extension of the disk, topologically disconnected clump features are only resolved if the fragmentation-active zone of the disk is resolved with at least 100 cells, which holds as a minimum requirement for all global disk setups. Our simulations illustrate the capabilities of AMR-based modeling techniques for planet formation simulations and underline the importance of balanced refinement settings to reproduce fragmenting structures.
 Physics , 1999, DOI: 10.1086/307465 Abstract: The stability of a rotating fluid disk to the formation of spiral arms is studied in the tightwinding approximation in the linear regime. The dispersion relation for spirals that was derived by Bertin et al. is shown to contain a new, acoustic instability beyond the Lindblad resonances that depends only on pressure and rotation. In this regime, pressure and gravity exchange roles as drivers and inhibitors of spiral wave structures. Other instabilities that are enhanced by pressure are also found in the general dispersion relation by including higher order terms in the small parameter 1/kr for wavenumber k and radius r. These instabilities are present even for large values of Toomre's parameter Q. Unstable growth rates are determined in four cases: a self-gravitating disk with a flat rotation curve, a self-gravitating disk with solid body rotation, a non-self-gravitating disk with solid body rotation, and a non-self-gravitating disk with Keplerian rotation. The most important application appears to be as a source of spiral structure, possibly leading to accretion in non-self-gravitating disks, such as some galactic nuclear disks, disks around black holes, and proto-planetary disks. All of these examples have short orbital times so the unstable growth time can be small.
 Physics , 2014, DOI: 10.1007/s10509-014-1994-8 Abstract: We studied global nonaxisymmetric hydrodynamic instabilities in an extensive collection of hot, self-gravitating polytropic disk systems, systems that covered a wide expanse of the parameter space relevant to protostellar and protoplanetary systems. We examined equilibrium disk models varying three parameters: the ratio of the inner to outer equatorial radii, the ratio of star mass to disk mass, and the rotation law exponent $q$. We took the polytropic index $n$ = 1.5 and examined the exponents $q =$ 1.5 and 2, and the transitional one $q$ = 1.75. For each of these sets of parameters, we examined models with inner to outer radius ratios from 0.1 to 0.75, and star mass to disk mass ratios from 0 to 10$^3$. We numerically calculated the growth rates and oscillation frequencies of low-order nonaxisymmetric disk modes, modes with azimuthal dependence $\propto$ e$^{im\phi}$. Low-$m$ modes are found to dominate with the character and strength of instability strongly dependent on disk self-gravity. Representatives of each mode type are examined in detail, and torques and mass transport rates are calculated.
 Physics , 2001, DOI: 10.1051/0004-6361:20010224 Abstract: In the dynamics of accretion disks, the presence of collective effects associated with the self-gravity of the disk is expected to affect not only the momentum transport, but also the relevant energy balance equations, which could differ substantially from the non-self-gravitating case. Here we follow the picture that, when the disk is sufficiently cold, the stirring due to Jeans-related instabilities acts as a source of effective heating. The corresponding reformulation of the energy equations allows us to: ({\it i}) demonstrate how self-regulation can be established, so that the stability parameter $Q$ is maintained close to a threshold value, with weak dependence on radius; ({\it ii}) rediscuss the opacity properties in the self-gravitating regime. In particular, we show that, if cooling is dominated by {\it bremsstrahlung}, an optically thin stationary accretion solution is thermally stable, even in the non-advective case, provided the disk is self-gravitating. The details of the cooling function have little effect on the structure of such accretion disk, which is in any case brought by self-gravity to self-regulate. This condition of self-gravitating accretion is expected to be appropriate for the outer regions of many disks of astrophysical interest. With the reformulation of the energy equations described in this paper we have also secured: ({\it iii}) a starting point for the study of the emission properties of self-gravitating accretion disks; ({\it iv}) a tool to analyze the structure of the transition region, where the disk becomes self-gravitating.
 Physics , 2011, DOI: 10.1111/j.1365-2966.2011.18797.x Abstract: We study the stability of gaps opened by a giant planet in a self-gravitating protoplanetary disc. We find a linear instability associated with both the self-gravity of the disc and local vortensity maxima which coincide with gap edges. For our models, these edge modes develop and extend to twice the orbital radius of a Saturn mass planet in discs with disc-to-star mass ratio >0.06, corresponding to a Toomre Q < 1.5 at the outer disc boundary. Unlike the local vortex-forming instabilities associated with gap edges in weakly or non-self-gravitating low viscosity discs, the edge modes are global and exist only in sufficiently massive discs, but for the typical viscosity values adopted for protoplanetary discs. Analytic modelling and linear calculations show edge modes may be interpreted as a localised disturbance associated with a gap edge inducing activity in the extended disc, through the launching of density waves excited at Lindblad resonances. Nonlinear hydrodynamic simulations are performed to investigate the evolution of edge modes in disc-planet systems. The form and growth rates of unstable modes are consistent with linear theory. Their dependence on viscosity and gravitational softening is also explored. We also performed a first study of the effect of edge modes on planetary migration. We found that if edge modes develop, then the average disc-on-planet torque becomes more positive with increasing disc mass. In simulations where the planet was allowed to migrate, although a fast type III migration could be seen that was similar to that seen in non-self-gravitating discs, we found that it was possible for the planet to interact gravitationally with the spiral arms associated with an edge mode and that this could result in the planet being scattered outwards. Thus orbital migration is likely to be complex and non monotonic in massive discs of the type we consider.
 Mir Abbas Jalali Physics , 2013, DOI: 10.1088/0004-637X/772/1/75 Abstract: We use the Fokker-Planck equation and model the dispersive dynamics of solid particles in annular protoplanetary disks whose gas component is more massive than the particle phase. We model particle--gas interactions as hard sphere collisions, determine the functional form of diffusion coefficients, and show the existence of two global unstable modes in the particle phase. These modes have spiral patterns with the azimuthal wavenumber $m=1$ and rotate slowly. We show that in ring-shaped disks, the phase space density of solid particles increases linearly in time towards an accumulation point near the location of pressure maximum, while instabilities grow exponentially. Therefore, planetesimals and planetary cores can be efficiently produced near the peaks of unstable density waves. In this mechanism, particles migrating towards the accumulation point will not participate in the formation of planets, and should eventually form a debris ring like the main asteroid belt or classical Kuiper belt objects. We present the implications of global instabilities to the formation of ice giants and terrestrial planets in the solar system.
 Physics , 2004, Abstract: In two previous publications$^{1,2}$, we have demonstrated that stationary rotation of magnetized plasma about a compact central object permits an enormous number of different MHD instabilities, with the well-known magneto-rotational instability as just one of them. We here concentrate on the new instabilities found that are driven by transonic transitions of the poloidal flow. A particularly promising class of instabilities, from the point of view of MHD turbulence in accretion disks, is the class of {\em trans-slow Alfven continuum modes}, that occur when the poloidal flow exceeds a critical value of the slow magnetosonic speed. When this happens, virtually every magnetic/flow surface of the disk becomes unstable with respect to highly localized modes of the continuous spectrum. The mode structures rotate, in turn, about the rotating disk. These structure lock and become explosively unstable when the mass of the central object is increased beyond a certain critical value. Their growth rates then become huge, of the order of the Alfven transit time. These instabilities appear to have all requisite properties to facilitate accretion flows across magnetic surfaces and jet formation.[1] R. Keppens, F. Casse, J.P. Goedbloed, "Waves and instabilities in accretion disks: Magnetohydrodynamic spectroscopic analysis", Astrophys. J. {\bf 569}, L121--L126 (2002).[2] J.P. Goedbloed, A.J.C. Belien, B. van der Holst, R. Keppens, "Unstable continuous spectra of transonic axisymmetric plasmas", Phys. Plasmas {\bf 11}, 28--54 (2004).
 Physics , 1999, Abstract: A linear stability analysis has been performed onto a self-gravitating magnetized gas disk bounded by external pressure. The resulting dispersion relation is fully explained by three kinds of instability: a Parker-type instability driven by self-gravity, usual Jeans gravitational instability and convection. In the direction parallel to the magnetic fields, the magnetic tension completely suppresses the convection. If the adiabatic index $\gamma$ is less than a certain critical value, the perturbations trigger the Parker as well as the Jeans instability in the disk. Consequently, the growth rate curve has two maxima: one at small wavenumber due to a combination of the Parker and Jeans instabilities, and the other at somewhat larger wavenumber mostly due to the Parker instability. In the horizontal direction perpendicular to the fields, the convection makes the growth rate increase monotonically upto a limiting value as the perturbation wavenumber gets large. However, at small wavenumbers, the Jeans instability becomes effective and develops a peak in the growth rate curve. Depending on the system parameters, the maximum growth rate of the convection may or may not be higher than the peak due to the Jeans-Parker instability. Therefore, a cooperative action of the Jeans and Parker instabilities can have chances to over-ride the convection and may develop large scale structures of cylindrical shape in non-linear stage. In thick disks the cylinder is expected to align its axis perpendicular to the field, while in thin ones parallel to it.
 Physics , 2009, Abstract: Duschl et al. (2000) have shown that the standard model for geometrically thin accretion disks ($\alpha$-disks) leads to inconsistency if self-gravity play a role. This problem arise from parametrization of viscosity in terms of local sound velocity and vertical disks scale hight. The $\beta$-viscosity prescription was introduced by Duschl et al. (2000), which has been derived from rotating shear flow experiment ($\nu=\beta \Omega R^2$). Following the Duschl et al. (2000) suggestion for a $\beta$-prescription for viscosity, we apply this model for a thin self-gravitating disk around newborn stars. Our result is quite different with standard alpha disks in the outer part of the disks where the self-gravity becomes important. In the inner part of the disks, our solution converged to the standard $\alpha$ disks. It has been presented that for beta model, Toomre parameter is more than unity everywhere which means that gravitational fragmentation can be occur everywhere. We suggest that the kind of hydrodynamically driven viscosity, $\beta$-model, can be used for modeling of accretion disks from proto-stellar disks to AGN and galactic disks. It would be interest to investigate ADAF-type solution for follow any effects by $\beta$-viscosity model. An important property of the $\beta$-disk is that they are viscously stable.
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