TY - JOUR

T1 - Spectral methods for time-dependent studies of accretion flows. II. Two-dimensional hydrodynamic disks with self-gravity

AU - Chan, Chi Kwan

AU - Psaltis, Dimitrios

AU - Özel, Feryal

N1 - Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2006/7/1

Y1 - 2006/7/1

N2 - 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 precomputed, 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 pseudospectral 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 δ-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 Γ is ≤4/3, Toomre's criterion correctly describes the stability of the disk. However, when Γ > 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.

AB - 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 precomputed, 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 pseudospectral 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 δ-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 Γ is ≤4/3, Toomre's criterion correctly describes the stability of the disk. However, when Γ > 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.

KW - Accretion, accretion disks

KW - Black hole physics

KW - Hydrodynamics

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U2 - 10.1086/500394

DO - 10.1086/500394

M3 - Article

AN - SCOPUS:33746897333

VL - 645

SP - 506

EP - 518

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 1 I

ER -