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
2y


 ˆ ˆ T



 +Symmetric
b
b

 x y

Term
x 
y 
2x
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