1

The Magnetothermal Instability and its Application to Clusters of

Galaxies

Ian Parrish

Advisor: James StoneDept. of Astrophysical Sciences

Princeton University/ UC Berkeley

October 10, 2007

2

Hydra A Cluster (Chandra)

Collisionless Transport

T ~ 4.5 keV n ~ 10-3-10-4

Motivation

Vmfp R1.0~ Vmfp R1.0~

Vmfp R05.0~

4x106 M

BlackHole

Sgr A*

T ~ 1 keV, n ~ 10 cm-3

Rs ~ 1012 cmSmfp Rcm10~ 17

50 kpc

Motivating example suggested by E. Quataert

3

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

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Convective Stability in a Gravitational Field

•Clasically: Schwarzschild Criterion

•Long MFP: Balbus Criterion

New Stability Criterion!

Idea: Magnetothermal Instability Qualitative Mechanism

0dz

dS

0dz

dS

0dz

dT

Q Q TQ

Magnetic Field Lines

Anisotropic heat flux given by Braginskii conductivity.

5

Algorithm: MHD with Athena

Athena: Higher order Godunov Scheme

•Constrained Transport for preserving divergence free.

•Unsplit CTU integrator

@(½)@t

+ r ¢(½v) = 0; (1)

@(½v)@t

+r ¢·½vv +

µp+

B2

8¼

¶I ¡

BB4¼

¸= ½g; (2)

@E@t

+r ¢·vµE +p+

B2

8¼

¶¡B(B ¢v)

4¼

¸= ½g¢v ¡ r ¢Q; (3)

@B@t

+r £ (v £ B) = 0: (4)

@(½)@t

+ r ¢(½v) = 0; (1)

@(½v)@t

+r ¢·½vv +

µp+

B2

8¼

¶I ¡

BB4¼

¸= ½g; (2)

@E@t

+r ¢·vµE +p+

B2

8¼

¶¡B(B ¢v)

4¼

¸= ½g¢v ¡ r ¢Q; (3)

@B@t

+r £ (v £ B) = 0: (4)

6

Algorithm: Heat Conduction

(i,j)

(i+1,j+1)

(i+1,j-1)(i-1,j-1)

(i-1,j-1)

B

Verification

•Gaussian Diffusion: 2nd order accurate.

•Circular Field Lines.

Implemented through sub-cycling diffusion routine.

32P d ln P ½¡ 5=3

dt = ¡ r ¢Q = r ¢hb̂³ÂC b̂¢r T

´i32P d ln P ½¡ 5=3

dt = ¡ r ¢Q = r ¢hb̂³ÂC b̂¢r T

´i

Qx = ¡ ÂC³b̂2x

@T@x + b̂xb̂y @T@y

´Qx = ¡ ÂC

³b̂2x

@T@x + b̂xb̂y @T@y

´

@T@y@T@yQxQx Â?

Âk· 10¡ 4Â?

Âk· 10¡ 4

7

Algorithm: Performance

8

Dispersion Relation2 4 6 8 10 12 14

'Tk2

10

8

6

4

2

2

2 23 2 2 0Ck P T

Tk Nz z

Weak Field Limit

Dimensionless Growth Rate

Dimensionless Wavenumber

Dispersion Relation

0ln

0ln 0lim22

z

T

z

P

z

T

z

Pvk kC

A Instability

Criterion

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Linear Regime2 4 6 8 10 12 14

'Tk2

10

8

6

4

2

2

Dimensionless Growth Rate

Dimensionless Wavenumber

~3% error

Linear Regime: Verification

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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.

Magnetic Energy Density = B2/2

g

11

•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.

RMS Mach

Exploration: 3D Nonlinear Behavior

4

22 B

VA

12

MTI-Unstable Region

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.

•Temperature profile can be suppressed significantly.

Tem

pera

ture

13

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

Anisotropic Thermal Conduction Dominates

Loken, Norman, et al (2002)

Application: Clusters of Galaxies

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Application: Clusters of Galaxies

Plot from DeGrandi and Molendi 2002

ICM unstable to the MTI on scales greater than2/12/1

crit

2000

keV 5 kpc 6.4

T

Observational Data

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Simulation: Clusters of Galaxies

Temperature Profile becomes Isothermal

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Simulation: Clusters of Galaxies

Magnetic Dynamo:

B2 amplified by ~ 60

Vigorous Convection:

Mean Mach: ~ 0.1

Peak Mach: > 0.6

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Summary•Physics of the MTI.

•Verification and validation of MHD + anisotropic thermal conduction.

•Nonlinear behavior of the MTI.

•Application to the thermal structure of clusters of galaxies.

Future Work•Galaxy cluster heating/cooling mechanisms: jets, bubbles, cosmic rays, etc.

•Application to neutron stars.

•Mergers of galaxy clusters with dark matter.

•DOE CSGF Fellowship, Chandra Fellowship

•Many calculations performed on Princeton’s Orangena Supercomputer

Acknowledgements

18

Questions?

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Adiabatic Single Mode Example

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Single Mode Evolution

2

1.04, 1.09Tk

N N

Magnetic Energy Density

Kinetic Energy Density

Single Mode Perturbation

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Single Mode Evolution

•Saturated State should be new isothermal temperature profile

•Analogous to MRI Saturated State where angular velocity profile is flat.

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Dependence on Magnetic Field

Instability Criterion:2 2 ln

0CA

P Tk v

z z

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Conducting Boundaries

Temperature Fluctuations

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Models with Convectively Stable Layers

•Heat flux primarily due to Advective component.

•Very efficient total heat flow

MTI Stable

MTI Stable

MTI Unstable

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Future Work & ApplicationsFull 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)

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SUPPLEMENTARY MATERIAL

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Analogy with MRIMagneto-Rotational Magneto-Thermal

2 2 ln0C

A

P Tk v

z z

2 2 0lnA

dk v

d R

•Keplerian Profile

•Conserved Quantity:

Angular Momentum

•Free Energy Source:

Angular Velocity Gradient

•Weak Field Required

•Convectively Stable Profile

•Conserved Quantity:

Entropy

•Free Energy Source:

Temperature Gradient

•Weak Field Required

Unstable When: Unstable When:

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Heat Conduction Algorithm

(i,j)

(i+1,j+1)

(i+1,j-1)(i-1,j-1)

(i-1,j-1)

Bx

By

1. Magnetic Fields Defined at FacesBx

By

2. Interpolate Fields

3. Calculate Unit Vectors

1, 1 1, 1 1, 1 1, 1, 1, , 1, , 1, , 1,

ˆ ˆ ˆ ˆ2 2ˆ ˆ

2

i j i j i j i jx i j y i j x i j y ii j

x y

T T T Tb b b b

y yTb b

x y x

y

Tb

x

Tbb

yy

Tb

x

Tbb

x

Tbbdt

PPd

yxyyxx

R.H.S.

ˆˆln

2

3Q

+Symmetric Term

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Heat Flux with Stable Layers

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Outline & MotivationGoal: Numerical simulation of plasma physics with MHD in astrophysics.

•Verification of algorithms

•Application to Astrophysical Problems

Outline:

•Physics of the Magnetothermal Instability (MTI)

•Verification of Growth Rates

•Nonlinear Consequences

•Application to Galaxy Clusters

Solar Corona

Around 2 R☼:

n ~ 3 x 1015, T ~ few 106 K

λmfp > distance from the sun

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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)

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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

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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:

•Softened NFW Potential:

•Initial Hydrostatic Equilibrium: Convectively Stable, MTI Unstable

•Magnetic Field: Smooth Azimuthal/Chaotic

•Resolution: Scales down to 5-10 kpc (the coherence length) within 200 kpc box requires roughly a 5123 domain

20

))(( scDM rrrr

M

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Application II: Neutron StarsNeutron Star Parameters

•R = 10 km (Manhattan)

•M = 1.4 solar masses

•B = 108 – 1015 G

Properties

•Semi-relativistic

•Semi-degenerate

•In ocean, not fully quasi-neutral

•In crust, Coulomb Crystal?

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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.

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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

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Non-Linear Evolution III

•Advective Heat Flux is dominant

•Settling of atmosphere to isothermal equilibrium

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Adiabatic Multimode Evolution

No Net Magnetic Flux leads to decay by Anti-Dynamo Theorem

39

Effect of Finite B on Temperature Profile

Stability Parameter:

2max

22

AVk

40

Adiabatic Multimode Example

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Conducting Boundaries

Magnetic Field Lines

42

Conducting Boundary

Magnetic Energy Density

Kinetic Energy Density

43

Conducting BoundaryTemperature Profiles

44

Extension to 3DHow to get there• ATHENA is already parallelized for 3D• Need to parallelize heat conduction algorithm• Parallel scalability up to 2,048 processorsWhat can be studied• Confirm linear and non-linear properties in 2D• Convection is intrinsically 3D—measure heat

conduction• Possibility of a dynamo?

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Initial ConditionsPressure Profile

Bx

0

0

20

30

( )

( ) (1 )

( ) (1 )

( ) (1 )

g z g

T z T z

z T z

P z T z

g p

•Convectively Stable Atmosphere

•Ideal MHD (ATHENA)

•Anisotropic Heat Conduction (Braginskii)

•BC’s: adiabatic or conducting at y-boundary, periodic in x