The Magnetothermal Instability and its Application to Clusters of Galaxies Ian Parrish Advisor: James Stone Dept. of Astrophysical Sciences Princeton University/ UC Berkeley October 10, 2007 1 Motivation 50 kpc 4x106 M Black Hole mfp ~ 0.1RV Hydra A Cluster (Chandra) mfp ~ 0.1RV Sgr A* T ~ 4.5 keV n ~ 10-3-10-4 T ~ 1 keV, n ~ 10 cm-3 mfp ~ 0.05RV Rs ~ 1012 cm Collisionless Transport mfp ~ 1017 cm RS Motivating example suggested by E. Quataert 2 Talk Outline •Idea: Stability, Instability, and “Backward” Transport in Stratified Fluids, Steve Balbus, 2000. Physics of the Magnetothermal Instability (MTI). •Algorithm: Athena: State of the art, massively parallel MHD solver. Anisotropic thermal conduction module. •Verification and Exploration Verification of linear growth rates. Exploration of nonlinear consequences. •Application to Galaxy Clusters 3 Idea: Magnetothermal Instability Qualitative Mechanism Anisotropic heat flux given by Braginskii conductivity. Q Q dS 0 dz Q T Convective Stability in a Gravitational Field •Clasically: Schwarzschild Criterion dS 0 dz •Long MFP: Balbus Criterion Magnetic Field Lines dT 0 dz New Stability Criterion! 4 Algorithm: MHD with Athena @(½) + r ¢ (½v) @t · µ ¶ ¸ 2 @(½v) B BB + r ¢ ½vv + p + I¡ @t 8¼ 4¼ · µ ¶ ¸ @E B2 B(B ¢ v) +r¢ v E+p+ ¡ @t 8¼ 4¼ @B + r £ (v £ B) @t = 0; (1) = ½g; (2) = ½g ¢ v ¡ r ¢ Q; (3) = 0: (4) Athena: Higher order Godunov Scheme •Constrained Transport for preserving divergence free. •Unsplit CTU integrator 5 Algorithm: Heat Conduction h ³ ´i 3 P d ln P ½¡5=3 2 dt = ¡r ¢ Q = r ¢ ^b ÂC ^b ¢ rT ³ ´ ^bx^by @T Qx = ¡ÂC ^b2x @T + @x @y Verification (i-1,j-1) (i+1,j+1) (i,j) Qx (i-1,j-1) @T @y (i+1,j-1) •Gaussian Diffusion: 2nd order accurate. •Circular Field Lines. Â? Âk · 10¡4 Implemented through sub-cycling diffusion routine. 6 Algorithm: Performance 7 2 Dimensionless Growth Rate Dispersion Relation Dispersion Relation 2 4 6 8 10 12 14 'Tk2 Dimensionless Wavenumber 2 4 6 8 10 Weak Field Limit Instability Criterion 2 k 2 C P T 2 2 Tk N 0 z z 3 C P ln T P ln T limk 0 k v 0 0 z z z z 2 2 A 8 Dimensionless Growth Rate Linear Regime: Verification Linear Regime 2 'Tk 2 2 4 6 8 10 12 14 Dimensionless Wavenumber 2 4 6 8 10 ~3% error 9 Exploration: 3D Nonlinear Behavior •Subsonic convective turbulence, Mach ~ 1.5 x 10-3. •Magnetic dynamo leads to equipartition with kinetic energy. g •Efficient heat conduction. Steady state heat flux is 1/3 to 1/2 of Spitzer value. Magnetic Energy Density = B2/2 10 Exploration: 3D Nonlinear Behavior •Subsonic convective turbulence, Mach ~ 1.5 x 10-3. •Magnetic dynamo leads to equipartition with kinetic energy. •Efficient heat conduction. Steady state heat flux is 1/3 to 1/2 of Spitzer value. B2 V 4 2 A RMS Mach 11 Exploration: 3D Nonlinear Behavior MTI-Unstable Region •Subsonic convective turbulence, Mach ~ 1.5 x 10-3. Temperature •Magnetic dynamo leads to equipartition with kinetic energy. •Efficient heat conduction. Steady state heat flux is 1/3 to 1/2 of Spitzer value. •Temperature profile can be suppressed significantly. 12 Application: Clusters of Galaxies Expectations from Structure Formation Hydro Simulation: ΛCDM Cosmology, Eulerian Expect: steep temperature profile Rv ~ 1-3 Mpc M ~ 1014 – 1015 solar masses (84% dark matter, 13% ICM, 3% stars) T ~ 1-15 keV LX ~ 1043 – 1046 erg/s B ~ 1.0 µG Loken, Norman, et al (2002) Anisotropic Thermal Conduction Dominates 13 Application: Clusters of Galaxies Observational Data Plot from DeGrandi and Molendi 2002 ICM unstable to the MTI on scales greater than 1/ 2 crit T 4.6 kpc 5 keV 1/ 2 2000 14 Simulation: Clusters of Galaxies Temperature Profile becomes Isothermal 15 Simulation: Clusters of Galaxies Magnetic Dynamo: B2 amplified by ~ 60 Vigorous Convection: Mean Mach: ~ 0.1 Peak Mach: > 0.6 16 Summary Future Work •Verification and validation of MHD + anisotropic thermal conduction. •Galaxy cluster heating/cooling mechanisms: jets, bubbles, cosmic rays, etc. •Nonlinear behavior of the MTI. •Application to neutron stars. •Application to the thermal structure of clusters of galaxies. •Mergers of galaxy clusters with dark matter. •Physics of the MTI. Acknowledgements •DOE CSGF Fellowship, Chandra Fellowship •Many calculations performed on Princeton’s Orangena Supercomputer 17 Questions? 18 Adiabatic Single Mode Example 19 Single Mode Evolution Magnetic Energy Density Kinetic Energy Density Single Mode Perturbation Tk 2 1.04, 1.09 N N 20 Single Mode Evolution •Saturated State should be new isothermal temperature profile •Analogous to MRI Saturated State where angular velocity profile is flat. 21 Dependence on Magnetic Field Instability Criterion: C P ln T k v 0 z z 2 2 A 22 Conducting Boundaries Temperature Fluctuations 23 Models with Convectively Stable Layers MTI Stable MTI Unstable MTI Stable •Heat flux primarily due to Advective component. •Very efficient total heat flow 24 Future Work & Applications Full 3-D Calculations •Potential for a dynamo in three-dimensions (early evidence) •Convection is intrinsically 3D •Application-Specific Simulations •Clusters of Galaxies •Atmospheres of Neutron Stars Acknowledgements: Aristotle Socrates, Prateek Sharma, Steve Balbus, Ben Chandran, Elliot Quataert, Nadia Zakamska, Greg Hammett •Funding: Department of Energy Computational Science Graduate Fellowship (CSGF) 25 SUPPLEMENTARY MATERIAL 26 Analogy with MRI Magneto-Rotational Magneto-Thermal •Keplerian Profile •Convectively Stable Profile •Conserved Quantity: •Conserved Quantity: Angular Momentum •Free Energy Source: Angular Velocity Gradient •Weak Field Required Unstable When: d k v 0 d ln R 2 2 A Entropy •Free Energy Source: Temperature Gradient •Weak Field Required Unstable When: C P ln T k v 0 z z 2 2 A 27 Heat Conduction Algorithm 3 Pd ln P Q bˆ bˆ T 2 dt T T T T R.H.S. bx bx by by by bx y y x y x x (i-1,j-1) By (i+1,j+1) 1. Magnetic Fields Defined at Faces Bx (i,j) By Bx 2. Interpolate Fields 3. Calculate Unit Vectors (i-1,j-1) (i+1,j-1) Ti 1, j 1 Ti 1, j 1 ˆ Ti 1, j 1 Ti 1, j 1 ˆ ˆ ˆ bx ,i 1, j by ,i 1, j bx ,i 1, j by ,ii1, j 2 y 2y ˆ ˆ T +Symmetric b b x y Term x y 2x 28 Heat Flux with Stable Layers 29 Outline & Motivation Goal: Numerical simulation of plasma physics with MHD in astrophysics. Solar Corona •Verification of algorithms •Application to Astrophysical Problems Outline: •Physics of the Magnetothermal Instability (MTI) •Verification of Growth Rates Around 2 R☼: •Nonlinear Consequences n ~ 3 x 1015, T ~ few 106 K •Application to Galaxy Clusters λmfp > distance from the sun 30 Cooling Flows? •Problem: Cooling time at center of cluster much faster than age of cluster •Theory A: Cooling flow drops out of obs. To colder phase •Observation: No cool mass observed! •Theory B: Another source of heat…from a central AGN, from cosmic rays, Compton heating, thermal conduction (Graphic from Peterson & Fabian, astro-ph/0512549) 31 Thermal Conduction •Rechester & Rosenbluth/Chandran & Cowley: Effective conductivity for chaotic field lines…Spitzer/100 (too slow) •Narayan & Medvedev: Consider multiple correlation lengths…Spitzer/a few (fast enough?) •Zakamska & Narayan: Sometimes it works. •ZN: AGN heating models produce thermal instability! •Chandran: Generalization of MTI to include cosmic ray pressure 32 Clusters: Case for Simulation •Difficult to calculate effective conductivity in tangled field line structure analytically •Heat transport requires convective mixing length model •Convection modifies field structure….feedback loop Plan: M0 •Softened NFW Potential: (r rc )( r rs ) 2 •Initial Hydrostatic Equilibrium: Convectively Stable, MTI Unstable DM •Magnetic Field: Smooth Azimuthal/Chaotic •Resolution: Scales down to 5-10 kpc (the coherence length) within 200 kpc box requires roughly a 5123 domain 33 Application II: Neutron Stars Neutron Star Parameters •R = 10 km (Manhattan) •M = 1.4 solar masses •B = 108 – 1015 G Properties •Semi-relativistic •Semi-degenerate •In ocean, not fully quasineutral •In crust, Coulomb Crystal? 34 Neutron Star Atmosphere (from Ventura & Potekhin) Tout = 5 x 105 K (solid) Tout = 2 x 106 K (dashed) Various values of cosθ Construct an atmosphere •EOS: Paczynski, semi-degenerate, semi-relativistic •Opacity: Thompson scattering (dominant), free-free emission •Conduction: Degenerate, reduced Debye screening (Schatz, et al) •Integrate constant-flux atmosphere with shooting method. 35 Instability Analysis and Simulation •Potential for instability near equator where B-field lines are perpendicular to temperature gradient •MTI damped: •Outer parts due to radiative transport •Inner part due to stronger magnetic field and cross-field collisions •Check analytically if unstable •Simulate plane-parallel patch in 3D with Athena •Estimate heat transport properties, and new saturated T-profile 36 Non-Linear Evolution III •Advective Heat Flux is dominant •Settling of atmosphere to isothermal equilibrium 37 Adiabatic Multimode Evolution No Net Magnetic Flux leads to decay by Anti-Dynamo Theorem 38 Effect of Finite B on Temperature Profile Stability Parameter: 2 kV 2 A 2 max 39 Adiabatic Multimode Example 40 Conducting Boundaries Magnetic Field Lines 41 Conducting Boundary Magnetic Energy Density Kinetic Energy Density 42 Conducting Boundary Temperature Profiles 43 Extension to 3D How to get there • ATHENA is already parallelized for 3D • Need to parallelize heat conduction algorithm • Parallel scalability up to 2,048 processors What can be studied • Confirm linear and non-linear properties in 2D • Convection is intrinsically 3D—measure heat conduction • Possibility of a dynamo? 44 Initial Conditions Pressure Profile p g Bx g ( z ) g0 •Convectively Stable Atmosphere T ( z ) (1 T0 z ) •Ideal MHD (ATHENA) ( z ) (1 T0 z ) 2 •Anisotropic Heat Conduction (Braginskii) P ( z ) (1 T0 z )3 •BC’s: adiabatic or conducting at y-boundary, periodic in x 45