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Kinetic Effects on Turbulence Driven by the Magnetorotational Instability

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Kinetic Effects on Turbulence Driven by the Magnetorotational Instability Kinetic Effects on Turbulence Driven by the Magnetorotational Instability in Black Hole Accretion Prateek Sharma A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the department of Astrophysical sciences September 2006 c Copyright by Prateek Sharma, 2006. All Rights Reserved Abstract Many astrophysical objects (e.g., spiral galaxies, the solar system, Saturn's rings, and luminous disks around compact objects) occur in the form of a disk. One of the important astrophysical problems is to understand how rotationally supported disks lose angular momentum, and accrete towards the bottom of the gravitational potential, converting gravitational energy into thermal (and radiation) energy. The magnetorotational instability (MRI), an instability causing turbulent trans- port in ionized accretion disks, is studied in the kinetic regime. Kinetic effects are important because radiatively inefficient accretion flows (RIAFs), like the one around the supermassive black hole in the center of our Galaxy, are collisionless. The ion Larmor radius is tiny compared to the scale of MHD turbulence so that the drift kinetic equation (DKE), obtained by averaging the Vlasov equation over the fast gy- romotion, is appropriate for evolving the distribution function. The kinetic MHD formalism, based on the moments of the DKE, is used for linear and nonlinear stud- ies. A Landau fluid closure for parallel heat flux, which models kinetic effects like collisionless damping, is used to close the moment hierarchy. We show that the kinetic MHD and drift kinetic formalisms give the same set of linear modes for a Keplerian disk. The BGK collision operator is used to study the transition of the MRI from kinetic to the MHD regime. The ZEUS MHD code iii is modified to include the key kinetic MHD terms: anisotropic pressure tensor and anisotropic thermal conduction. The modified code is used to simulate the collisionless MRI in a local shearing box. As magnetic field is amplified by the MRI, pressure anisotropy (p > p ) is created because of the adiabatic invariance (µ p =B). ? k / ? Larmor radius scale instabilities|mirror, ion-cyclotron, and firehose—are excited even at small pressure anisotropies (∆p=p 1/β). Pressure isotropization due to ∼ pitch angle scattering by these instabilities is included as a subgrid model. A key result of the kinetic MHD simulations is that the anisotropic stress can be as large as the Maxwell stress. It is shown, with the help of simple tests, that the centered differencing of anisotropic thermal conduction can cause the heat to flow from lower to higher temperatures, giving negative temperatures in regions with large temperature gradients. A new method, based on limiting the transverse temperature gradient, allows heat to flow only from higher to lower temperatures. Several tests and convergence studies are presented to compare the different methods. iv Acknowledgements I foremost thank my adviser Greg Hammett, whose guidance made this thesis possible. His insight, quest for perfection, and passion for science has always inspired me. He was always patient, and ensured that I understood every subtle point. Thanks to Eliot Quataert, who is an inspiring mentor, and initiated me into the fascinating field of astrophysics. He was so accessible that I never felt that he was not in Princeton. I am thankful to Jim Stone for his constant encouragement, and help with numerical methods, especially ZEUS. Finally, thanks to my loving and supporting family, and wonderful friends. Special thanks to my brother Rohit, sister Chubi, and cousins, who have always brought joy in my life. My wife Asha has been a loving and caring companion; her suggestion to keep it simple has significantly improved the thesis. The thesis work was supported by U.S. DOE contract # DE-AC02-76CH03073, and NASA grants NAG5-12043 and NNH06AD01I. Many thanks to the NASA web- sites for amazing pictures, some of which I have used in this thesis. v To my parents and grandparents vi Contents Abstract . iii Acknowledgements . v 1 Introduction 5 1.1 Accretion as an energy source . 8 1.1.1 The Eddington limit . 9 1.1.2 The emitted spectrum . 10 1.2 Accretion disk phenomenology . 11 1.2.1 Governing equations . 13 1.2.2 Fluctuations . 15 1.2.3 α disk models . 17 1.3 MRI: the source of disk turbulence . 18 1.3.1 Insufficiency of hydrodynamics . 18 1.3.2 MHD accretion disks: Linear analysis . 20 1.3.3 MHD accretion disks: Nonlinear simulations . 25 1.4 Radiatively inefficient accretion flows . 29 1.4.1 RIAF models . 33 1.4.2 The Galactic center . 36 1.5 Motivation . 38 1.6 Overview . 40 vii 2 Description of collisionless plasmas 44 2.1 The Vlasov equation . 45 2.2 The drift kinetic equation . 46 2.3 Kinetic MHD equations . 48 2.4 Landau fluid closure . 49 2.4.1 The moment hierarchy . 50 2.4.2 The 3+1 Landau closure . 53 2.5 Collisional effects . 55 2.5.1 The high collisionality limit . 55 2.5.2 3+1 closure with collisions . 56 2.6 Nonlinear implementation of closure . 57 2.6.1 The effects of small-scale anisotropy-driven instabilities . 61 3 Transition from collisionless to collisional MRI 67 3.1 Introduction . 68 3.2 Linearized kinetic MHD equations . 70 3.3 Kinetic closure including collisions . 73 3.4 Comparison with Landau fluid closure . 77 3.5 Collisionality dependence of the MRI growth rate . 80 3.6 Summary and Discussion . 85 4 Nonlinear Simulations of kinetic MRI 88 4.1 Introduction . 89 4.2 Governing equations . 92 4.2.1 Linear modes . 98 4.2.2 Isotropization of the pressure tensor in collisionless plasmas . 100 4.2.3 Pressure anisotropy limits . 103 4.3 Kinetic MHD simulations in shearing box . 105 viii 4.3.1 Shearing box . 105 4.3.2 Numerical methods . 108 4.3.3 Shearing box and kinetic MHD . 109 4.3.4 Shearing box parameters and initial conditions . 110 4.4 Results . 111 4.4.1 Fiducial run . 112 4.4.2 The double adiabatic limit . 119 4.4.3 Varying conductivity . 121 4.4.4 Different pitch angle scattering models . 122 4.5 Additional simulations . 125 4.5.1 High β simulations . 125 4.5.2 Runs with β = 400 . 131 4.6 Summary and Discussion . 134 5 Anisotropic conduction with large temperature gradients 144 5.1 Introduction . 147 5.2 Anisotropic thermal conduction . 150 5.2.1 Centered asymmetric scheme . 153 5.2.2 Centered symmetric scheme . 154 5.3 Negative temperature with centered differencing . 157 5.3.1 Asymmetric method . 157 5.3.2 Symmetric method . 157 5.4 Slope limited fluxes . 160 5.4.1 Limiting the asymmetric method . 161 5.4.2 Limiting the symmetric method . 162 5.5 Limiting using the entropy-like source function . 164 5.6 Mathematical properties . 166 5.6.1 Behavior at temperature extrema . 166 ix 5.6.2 The entropy-like condition, s_∗ = q T 0 . 167 − · r ≥ 5.7 Further tests . 170 5.7.1 Circular diffusion of hot patch . 171 5.7.2 Convergence studies: measuring χ?;num . 177 5.8 Conclusions . 181 6 Conclusions 183 6.1 Summary . 184 6.2 Future directions . 187 A Accretion models 191 A.1 Efficiency of black hole accretion . 191 A.2 Bondi accretion . 193 B Linear closure for high and low collisionality 197 B.1 Closure for high collisionality: ζ 1 . 197 j j B.2 Closure for low collisionality: ζ 1 . 199 j j C Kinetic MHD simulations: modifications to ZEUS 201 C.1 Grid and variables . 201 C.1.1 Determination of δt: Stability and positivity . 203 C.2 Implementation of the pressure anisotropy \hard wall" . 204 C.2.1 Implementation of the advective part of q . 206 r · ? C.3 Numerical tests . 207 C.3.1 Tests for anisotropic conduction . 207 C.3.2 Collisionless damping of fast mode in 1-D . 207 C.3.3 Mirror instability in 1-D . 209 C.3.4 Shear generated pressure anisotropy: Firehose instability in 2-D 210 D Error analysis 213 x E Entropy condition for an ideal gas 215 xi List of Tables 1.1 Dim SMBHs in the Galactic center and nearby galaxies . 32 1.2 Plasma parameters for Sgr A∗ . 37 4.1 Vertical field simulations with β = 400 . 138 4.2 Statistics for Model Zl4 . 139 4.3 Simulations with an explicit collision term . 140 6 4.4 Bφ = Bz, β = 10 simulations . 141 6 4.5 Only Bz, β = 10 simulations . 142 4.6 β = 400 simulations with different field orientations . 143 5.1 Diffusion in circular field lines: 50 50 grid . 171 × 5.2 Diffusion in circular field lines: 100 100 grid . 171 × 5.3 Diffusion in circular field lines: 200 200 grid . 173 × 5.4 Diffusion in circular field lines: 400 400 grid . 173 × 5.5 Asymptotic slopes for convergence of error χ = T −1(0; 0) ?;num j − T −1(0; 0) . 179 iso j 5.6 Asymptotic slopes for convergence of χ?;num in the ring diffusion test 181 1 List of Figures 1.1 An artist's impression of a binary accretion disk . 6 1.2 Disk and jet associated with a supermassive black hole . 6 1.3 Density-temperature diagram for hydrogen . 12 1.4 Spring model of the MRI . 23 1.5 Magnetic energy and stresses for an MHD simulation with net vertical flux . ..
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