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Plasma DynamosUCLA January 5th 2009
Steve Cowley, UKAEA Culham and Imperial
Thanks to Alex Schekochihin, Russell Kulsrud, Greg Hammett and Mark Rosin.
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Early magnetic fields -- what, when and how.
• After re-ionization the universe was probably a reasonably collisionless turbulent high plasma.
• Many large scale plasmas are quite collisionless. • I will argue that (random) magnetic fields grow rapidly
in such a plasma. • I will also argue that we need to know a lot more
about the small scale dynamics of high plasmas. We need an experiment at >> 1!
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Cluster Turbulence
The Coma Cluster: pressure map[Schuecker et al. 2004, A&A 426, 387]
L ~ 102…103 kpcU ~ 102…103 km/s (subsonic)L/U ~ 108…109yr
• Mergers• AGNs• Wakes
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Cluster Turbulence
Note: it is not obvious that there isturbulence! [A. Fabian 2003,MNRAS 344, L48]
• Mergers• AGNs• Wakes
L ~ 102…103 kpcU ~ 102…103 km/s (subsonic)L/U ~ 108…109yrmfp ~ 0.1…10 kpc
Re ~ 1…102
The Coma Cluster[Schuecker et al. 2004, A&A 426, 387]
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Cluster Magnetic Fields
Abell 400 cluster [Eilek & Owen 2002, ApJ 567, 202]
900 kpc
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Cluster MHD Turbulence
Turbulence scale is around here
TURBULENCEComa cluster
[Schuecker et al. 2004, A&A 426, 387]
MAGNETIC FIELDSHydra A Cluster
[Vogt & Enßlin 2005, A&A 434, 67]
•Magnetic Reynolds #, Rm ~ 10Magnetic Reynolds #, Rm ~ 102929..
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The Large Prandtl Number Case: Galaxies, Clusters etc.
• Magnetic Prandtl number = Pr = /.
• On the turnover time of the viscous eddies the “seed field” grows. The field develops structure below the viscous scale down to the resistive scale l= Pr -1/2 l
l
l = 10 -30kpcViscous scale
t
= 10 8 yearsViscous eddy Turnover.
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Isotropic Homogeneous Dynamo Folded Structure at Resistive Scale
Grayscale is |B|.
ScalarViscosity
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Plasma not Fluid
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Magnetized Viscosity --Anisotropic Pressure
Anisotropic pressure tensor in magnetized plasma. Because of fast motion around the field the tensor must be of the form:
DEFINITION OF PRESSURETENSOR.
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Magnetized Viscosity.
B
Collisionless particle motion restricted tobeing close to field line and conserving .
Compressing Field
Collisionless.Relaxed by Collisions.
P
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Incompressible Braginskii MHD.
Coefficients worked out by Braginskii Reviews of Plasma Physics Vol. 2.
Collisional limit
Unit vector along B
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Equilibrium -- Decreasing B.
V0 V0
B0
B0
Stretching rate
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Firehose Instability.
Look at instabilities that are smaller scale than the field and growing faster than the stretching rate. Treat B0 as quasi-constant during the growth.
We take perturbed velocity to go as:
The condition that the growth rate is faster than stretching rate is:
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Firehose Instability.
Linearized:
The x component becomes:
Perturbed field lineCurvature.
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Firehose Instability.
Putting this into force equation we get.
Alfven wave when no anisotropy
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More Firehose.
Parallel pressure forcessqueeze tube out.
Rosenbluth 1956Southwood and Kivelson 1993P|| P||
Tighter bend grows faster.
Unstable when
Growth rate at negligible B
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So What!? -- Nonlinear Firehose.
Nonlinear kinetic theory gives:
Rate of changeof B2 averaged along B.
Instability tries to keep average B constant by bending the field.
DivergingFlow.
Makes finite wiggles
Schekochihin et. al. Phys. Rev. Lett.
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Nonlinear Mirror Mode.
When the field increasing the plasma is unstable to the mirrorMode which creates little traps in the plasma.
Converging
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Stretching and compressing
Field decreasingP||>P FirehoseUnstable.
Field increasingP||< P Mirror modeUnstable.
Stretched at the turnover rate of theviscous eddies.
Using Braginskii’sExpression we getP||-P ~ Re-1/2P ~ P/6
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Scales
ll00~~1-3Mpc1-3Mpc
Viscous Scale l ~l0Re-3/4~10 - 30 kpc
l0 /u0~109
yearsl/ u~108
years
Mean-FreePath. mfp
~ l0Re-1~1-10kpc
kResistive Scale ~ l0Rm-1/2 ~ 104km
Ion LarmorRadiusScale @B = 1G~105km.
EB ?EV
Maximum growth rate
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ll00~~1-3Mpc1-3Mpc
Viscous Scale l ~l0Re-3/4~10 - 30 kpc
l0 /u0~109
yearsl/ u~108
years
Mean-FreePath. mfp
~ l0Re-1~1-10kpc
kResistive Scale ~ l0Rm-1/2 ~ 104km
Ion LarmorRadiusScale @B = 1G~105km.
EB ?EV
?
?
Scales
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What does small scale field do?
Enhanced particle scattering? Effective collisions increase --i? If so viscosity decreases -- Re gets large and turbulence has faster motions.
Dynamo Growth Time: ~ 0( i/L)(1/2) ~ 1000 years!
MAGNETIC FIELD CAN GROW ON TRIVIAL TIMESCALES.
Sharma, Hammett, Schekochihin, Kulsrud etc.
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Conclusions.
• Small scale fast growing instabilities to be expected in weak field magnetized fully ionized plasmas. Make finite wiggles on the scale almost of the ion larmor radius.
• May enhance collisions, dissipation and change the transport properties.