arXiv subjects

Chi-kwan Chan

Publications and source records attributed to Chi-kwan Chan.

64 records · Page 4Linked to original sources

arXiv2008

Oscillations of the Inner Regions of Viscous Accretion Disks

Although quasi-periodic oscillations (QPOs) have been discovered in different X-ray sources, their origin is still a matter of debate. Analytical studies of hydrodynamic accretion disks have shown three types of trapped global modes with properties that appear to agree with the observations. However, these studies take only linear effects into account and do not address the issues of mode excitation and decay. Moreover, observations suggest that resonances between modes play a crucial role. A systematic, numerical study of this problem is therefore needed. In this paper, we use a pseudo-spectral algorithm to perform a parameter study of the inner regions of hydrodynamic disks. By assuming alpha-viscosity, we show that steady state solutions rarely exist. The inner edges of the disks oscillate and excite axisymmetric waves. In addition, the flows inside the inner edges are sometimes unstable to non-axisymmetric perturbations. One-armed, or even two-armed, spirals are developed, which provides a plausible explanation for the high-frequency QPOs observed from accreting black holes. When the Reynolds numbers are above certain critical values, the inner disks go through some transient turbulent states characterized by strong trailing spirals; while large-scale leading spirals developed in the outer disks. We compared our numerical results with standard thin disk oscillation models. Although the non-axisymmetric features have their analytical counterparts, more careful study is needed to explain the axisymmetric oscillations.

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arXiv2008

The fundamental difference between shear alpha viscosity and turbulent magnetorotational stresses

Numerical simulations of turbulent, magnetized, differentially rotating flows driven by the magnetorotational instability are often used to calculate the effective values of alpha viscosity that is invoked in analytical models of accretion discs. In this paper we use various dynamical models of turbulent magnetohydrodynamic stresses, as well as numerical simulations of shearing boxes, to show that angular momentum transport in MRI-driven accretion discs cannot be described by the standard model for shear viscosity. In particular, we demonstrate that turbulent magnetorotational stresses are not linearly proportional to the local shear and vanish identically for angular velocity profiles that increase outwards.

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arXiv2008

Viscous, Resistive Magnetorotational Modes

We carry out a comprehensive analysis of the behavior of the magnetorotational instability (MRI) in viscous, resistive plasmas. We find exact, non-linear solutions of the non-ideal magnetohydrodynamic (MHD) equations describing the local dynamics of an incompressible, differentially rotating background threaded by a vertical magnetic field when disturbances with wavenumbers perpendicular to the shear are considered. We provide a geometrical description of these viscous, resistive MRI modes and show how their physical structure is modified as a function of the Reynolds and magnetic Reynolds numbers. We demonstrate that when finite dissipative effects are considered, velocity and magnetic field disturbances are no longer orthogonal (as it is the case in the ideal MHD limit) unless the magnetic Prandtl number is unity. We generalize previous results found in the ideal limit and show that a series of key properties of the mean Reynolds and Maxwell stresses also hold for the viscous, resistive MRI. In particular, we show that the Reynolds stress is always positive and the Maxwell stress is always negative. Therefore, even in the presence of viscosity and resistivity, the total mean angular momentum transport is always directed outwards. We also find that, for any combination of the Reynolds and magnetic Reynolds numbers, magnetic disturbances dominate both the energetics and the transport of angular momentum and that the total mean energy density is an upper bound for the total mean stress responsible for angular momentum transport. The ratios between the Maxwell and Reynolds stresses and between magnetic and kinetic energy densities increase with decreasing Reynolds numbers for any magnetic Reynolds number; the lowest limit of both ratios is reached in the ideal MHD regime.

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arXiv2007

Angular Momentum Transport in Accretion Disks: Scaling Laws in MRI-driven Turbulence

We present a scaling law that predicts the values of the stresses obtained in numerical simulations of saturated MRI-driven turbulence in non-stratified shearing boxes. It relates the turbulent stresses to the strength of the vertical magnetic field, the sound speed, the vertical size of the box, and the numerical resolution and predicts accurately the results of 35 numerical simulations performed for a wide variety of physical conditions. We use our result to show that the saturated stresses in simulations with zero net magnetic flux depend linearly on the numerical resolution and would become negligible if the resolution were set equal to the natural dissipation scale in astrophysical disks. We conclude that, in order for MRI-driven turbulent angular momentum transport to be able to account for the large value of the effective alpha viscosity inferred observationally, the disk must be threaded by a significant vertical magnetic field and the turbulent magnetic energy must be in near equipartition with the thermal energy. This result has important implications for the spectra of accretion disks and their stability.

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arXiv2006

Spectral Methods for Time-Dependent Studies of Accretion Flows. III. Three-Dimensional, Self-gravitating, Magnetohydrodynamic Disks

Accretion disks are three-dimensional, turbulent, often self-gravitating, magnetohydrodynamic flows, which can be modeled in detail with numerical simulations. In this paper, we present a new algorithm that is based on a spectral decomposition method to simulate such flows. Because of the high order of the method, we can solve the induction equation in terms of the magnetic potential and, therefore, ensure trivially that the magnetic fields in the numerical solution are divergence free. The spectral method also suffers minimally from numerical dissipation and allows for an easy implementation of models for sub-grid physics. Both properties make our method ideal for studying MHD turbulent flows such as those found in accretion disks around compact objects. We verify our algorithm with a series of standard tests and use it to show the development of MHD turbulnce in a simulation of an accretion disk. Finally, we study the evolution and saturation of the power spectrum of MHD turbulence driven by the magnetorotational instability.

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arXiv2006

MHD Simulations of Accretion onto Sgr A*: Quiescent Fluctuations, Outbursts, and Quasi-Periodicity

High resolution observations of Sgr A* have revealed a wide variety of phenomena, ranging from intense rapid flares to quasi-periodic oscillations, making this object an ideal system to study the properties of low luminosity accreting black holes. In this paper, we use a pseudo-spectral algorithm to construct and evolve a three-dimensional magnetohydrodynamic model of the accretion disk in Sgr A*. Assuming a hybrid thermal-nonthermal emission scheme, we show that the MHD turbulence can by itself only produce factor of two fluctuations in luminosity. These amplitudes in variation cannot explain the magnitude of flares observed in this system. However, we also demonstrate that density perturbations in the disk do produce outbursts qualitatively similar to those observed by XMM-Newton in X-rays and ground-based facilities in the near infrared. Quasi-periodic oscillations emerge naturally in the simulated lightcurves. We attribute these to non-axisymmetric density perturbations that emerge as the disk evolves back toward its quiescent state.

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arXiv2006

A Local Model for Angular Momentum Transport in Accretion Disks Driven by the Magnetorotational Instability

We develop a local model for the exponential growth and saturation of the Reynolds and Maxwell stresses in turbulent flows driven by the magnetorotational instability. We first derive equations that describe the effects of the instability on the growth and pumping of the stresses. We highlight the relevance of a new type of correlations that couples the dynamical evolution of the Reynolds and Maxwell stresses and plays a key role in developing and sustaining the magnetorotational turbulence. We then supplement these equations with a phenomenological description of the triple correlations that lead to a saturated turbulent state. We show that the steady-state limit of the model describes successfully the correlations among stresses found in numerical simulations of shearing boxes.

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arXiv2006

The signature of the magnetorotational instability in the Reynolds and Maxwell stress tensors in accretion discs

The magnetorotational instability is thought to be responsible for the generation of magnetohydrodynamic turbulence that leads to enhanced outward angular momentum transport in accretion discs. Here, we present the first formal analytical proof showing that, during the exponential growth of the instability, the mean (averaged over the disc scale-height) Reynolds stress is always positive, the mean Maxwell stress is always negative, and hence the mean total stress is positive and leads to a net outward flux of angular momentum. More importantly, we show that the ratio of the Maxwell to the Reynolds stresses during the late times of the exponential growth of the instability is determined only by the local shear and does not depend on the initial spectrum of perturbations or the strength of the seed magnetic. Even though we derived these properties of the stress tensors for the exponential growth of the instability in incompressible flows, numerical simulations of shearing boxes show that this characteristic is qualitatively preserved under more general conditions, even during the saturated turbulent state generated by the instability.

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arXiv2005

Spectral Methods for Time-Dependent Studies of Accretion Flows. II. Two-Dimensional Hydrodynamic Disks with Self-Gravity

Spectral methods are well suited for solving hydrodynamic problems in which the self-gravity of the flow needs to be considered. Because Poisson's equation is linear, the numerical solution for the gravitational potential for each individual mode of the density can be pre-computed, thus reducing substantially the computational cost of the method. In this second paper, we describe two different approaches to computing the gravitational field of a two-dimensional flow with pseudo-spectral methods. For situations in which the density profile is independent of the third coordinate (i.e., an infinite cylinder), we use a standard Poisson solver in spectral space. On the other hand, for situations in which the density profile is a delta function along the third coordinate (i.e., an infinitesimally thin disk), or any other function known a priori, we perform a direct integration of Poisson's equation using a Green's functions approach. We devise a number of test problems to verify the implementations of these two methods. Finally, we use our method to study the stability of polytropic, self-gravitating disks. We find that, when the polytropic index Gamma is <= 4/3, Toomre's criterion correctly describes the stability of the disk. However, when Gamma > 4/3 and for large values of the polytropic constant K, the numerical solutions are always stable, even when the linear criterion predicts the contrary. We show that, in the latter case, the minimum wavelength of the unstable modes is larger than the extent of the unstable region and hence the local linear analysis is inapplicable.

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arXiv2005

Spectral Methods for Time-Dependent Studies of Accretion Flows. I. Two-dimensional, Viscous, Hydrodynamic Disks

We present a numerical method for studying the normal modes of accretion flows around black holes. In this first paper, we focus on two-dimensional, viscous, hydrodynamic disks, for which the linear modes have been calculated analytically in previous investigations. We use pseudo-spectral methods and low storage Runge-Kutta methods to solve the continuity equation, the Navier-Stokes equation, and the energy equation. We devise a number of test problems to verify the implementation. These tests demonstrate the ability of spectral methods to handle accurately advection problems and to reproduce correctly the stability criteria for differentially rotating hydrodynamic flows. They also show that our implementation is able to handle sound wave correctly with non-reflective boundary conditions, to recover the standard solution for a viscous spreading ring, and produce correctly the Shakura-Sunyaev steady disk solution. Finally, we have applied our algorithm to the problem of a non-axisymmetric viscous spreading ring and verify that such configuration is unstable to non-antisymmetric perturbations.

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