Plasma DynamosUCLA January 5th 2009

Steve Cowley, UKAEA Culham and Imperial

Thanks to Alex Schekochihin, Russell Kulsrud, Greg Hammett and Mark Rosin.

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!

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

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]

Cluster Magnetic Fields

Abell 400 cluster [Eilek & Owen 2002, ApJ 567, 202]

900 kpc

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..

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.

Isotropic Homogeneous Dynamo Folded Structure at Resistive Scale

Grayscale is |B|.

ScalarViscosity

Plasma not Fluid

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.

Magnetized Viscosity.

B

Collisionless particle motion restricted tobeing close to field line and conserving .

Compressing Field

Collisionless.Relaxed by Collisions.

P

Incompressible Braginskii MHD.

Coefficients worked out by Braginskii Reviews of Plasma Physics Vol. 2.

Collisional limit

Unit vector along B

Equilibrium -- Decreasing B.

V0 V0

B0

B0

Stretching rate

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:

Firehose Instability.

Linearized:

The x component becomes:

Perturbed field lineCurvature.

Firehose Instability.

Putting this into force equation we get.

Alfven wave when no anisotropy

More Firehose.

Parallel pressure forcessqueeze tube out.

Rosenbluth 1956Southwood and Kivelson 1993P|| P||

Tighter bend grows faster.

Unstable when

Growth rate at negligible B

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.

Nonlinear Mirror Mode.

When the field increasing the plasma is unstable to the mirrorMode which creates little traps in the plasma.

Converging

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

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

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

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.

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.