the magnetothermal instability and its application to clusters of galaxies
DESCRIPTION
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. 50 kpc. 4x10 6 M Black Hole. Motivation. Sgr A * T ~ 1 keV, n ~ 10 cm -3 - PowerPoint PPT PresentationTRANSCRIPT
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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
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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.
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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)
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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
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Algorithm: Performance
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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
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•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
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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
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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
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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
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))(( 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
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AVk
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Adiabatic Multimode Example
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Conducting Boundaries
Magnetic Field Lines
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Conducting Boundary
Magnetic Energy Density
Kinetic Energy Density
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Conducting BoundaryTemperature Profiles
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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