Dynamo Effects in Laboratory Plasmas

S.C. PragerUniversity of Wisconsin

October, 2003

The lab plasma dynamo does• Generate current locally • Increase toroidal magnetic flux• Conserve magnetic helicity• Act through alpha and other effects• Arise from fluctuations superposed on the mean field• Achieve a nonlinearly saturated steady state

(with full backreaction)

The lab plasma dynamo does NOT• Generate magnetic field from a small seed field• Increase magnetic energy (it redistributes magnetic

field)

€

q =rBT

RBP

The toroidal magnetic field is measured by the safety factor

10 q

weak field,large fluctuations

self-organized

strong field,small fluctuations

externally controlled

Dynamo and self-organization occurs in laboratory plasmas with weak toroidal magnetic field

Examples: reversed field pinch (RFP) spheromak

The RFP: toroidal plasma with helical magnetic field

apply toroidal electric field

ET --> jT --> BP --> JP

The RFP

Today, approximate as cylinder

The MST Experiment(Madison Symmetric Torus)

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

T ~ 1 keV; n ~1013 cm-3; I ~ 0.5 MA, S ~ 106

The Spheromak

a compact torus

Outline

• Evidence for field generation

• The standard MHD model

• The backreaction

• Measurements of the MHD dynamo

• Dynamo effects beyond MHD (measurements)

• Open issues and relation to astrophysics

Evidence of field generation

• Cowling’s Theorem

• Toroidal flux generation

• Ohm’s law

Cowling’s theorem applied to the RFP

A time-independent, cylindrically symmetric plasma cannot contain a reversed magnetic field

Proof: assume Bz is reversed.

at the radius where Bz = 0

€

rE • dl = E ||rdϑ∫∫

€

=ηJθ r2π = ηdBz

drr2π ≠ 0

Thus, magnetic flux decays within reversal surface, in constrast to experiment

Bz

r

in experiment

-0.5

0.5

1.0

1.5

2.0

V/m

0.0

0.0 0.2 0.4 0.6 0.8 1.0ρ/a

E||

ηneo J||(Zeff = 2)

€

E ≠ η j

E||

ηj||

radius

additional current drive mechanism (dynamo)

The Standard MHD model• Mean field ohm’s law

€

⟨E⟩+ ⟨˜ v × ˜ B ⟩= η⟨ j⟩dynamo effect

€

˜ v , ˜ B

For high conductivity,

€

˜ v ≈˜ E × ⟨B⟩⟨B⟩2

€

˜ v × ˜ B ≈˜ E • ˜ B

B

€

˜ v , ˜ B Lab: from tearing instability (reconnection) Astrophysics: from convection, rotation…

The nonlinear dynamo

€

⟨E⟩

€

⟨ j⟩

⟨B⟩

€

⟨ ˜ v ⟩,⟨ ˜ B ⟩energysource

instability

dynamo

Quasilinear theory:

€

⟨ ˜ v × ˜ B ⟩ ~ ∇ • D∇⟨j⟩⟨B⟩

current diffusion

Nonlinear MHD computation: a complete description

(Bhattacharjee, Hamieri; Strauss;Boozer…..)

€

D ~ ˜ B 2

yields a collection of spatial Fourier modes (~R/a)

z

r

Flow vectors

In poloidal plane: 2 counter-rotating vortices, in toroidal plane: more complicated magnetic field: stochastic

Nonlinear MHD Computation

radius

The Lab Dynamo and the Backreaction

The lab dynamo is strong, with the backreaction,

self-induced

€

B >> ˜ B

Compare with backreaction theories predicting dynamo suppression (Cattaneo/Vainshtein, Kulsrud/Hahm, Gruzinov/Diamond, Bhattacharjee)

€

α =˜ v × ˜ B

B2 =

−η ˜ j • ˜ B + ˜ E • ˜ B

B2

€

α =αo −τ

3ρ˜ j • ˜ B

From Pouquet et al.,

for isotropic, homogenous turbulence

backreaction

Combining two equations,

€

α =αo +

τ

3ρη˜ E • ˜ B

1+τ

3ρηB

2

€

α =αo

1+τ

3ρηB

2

€

α =˜ E • ˜ B

B2

large resistivity

α-suppression with <B>

small resistivity

No obvious suppression, laboratory regime,Astrophysical regime???

Measurements of MHD dynamo

€

E + ˜ v × ˜ B = η jMeasure each term in Ohm’s law

In the hot core

€

˜ v passive spectroscopy,active spectroscopy (under development)(charge exchange recombination spectroscopy)(den Hartog, Craig, Ennis)

Laser Faraday rotation (Ding, Brower, UCLA)

Motional Stark effect (Craig, den Hartog, under development

€

˜ B

In the cool edge

Insertable probes: magnetic, Langmuir (E), spectroscopic

Active Spectroscopy

30 keV H Beam

Beam CurrentMonitor

Perpendicular Viewing Chords

22.5° ViewingChordMST Vessel

3-Wave Polarimeter-Interferometer System

MST R0 = 1.50 ma = 0.52 mIp = 400 kAne ~ 1019 m-3

B0 ~ 4 kG

Faraday rotation/interferometer system

Spectroscopic probe

Measure quantities during discrete dynamo event

ToroidalMagneticFlux(Wb)

MST

time (ms)

Flow velocity fluctuations

time (ms)

r/a = 0.9

€

⟨ ˜ v × ˜ B ⟩

€

η⟨ j⟩− ⟨E⟩

MHD dynamo dominant at some radii, not everywhere

r/a = 0.8

Measurement of MHD dynamo

0

-10

-20

0

-20

-10

Volts m

Volts m

-0.5 0 0.5time (ms)

r/a = 0.9

r/a = 0.8

Dynamo Effects Beyond MHD

• Hall dynamo

• Diamagnetic dynamo

• Kinetic dynamo (current transport)

Hall dynamo: a two-fluid effect

€

η j = ˜ v × ˜ B −˜ j × ˜ B

neMHD

dynamoHall

dynamo

Two fluid effects also alter the <v x B> dynamo

Quasilinear Theory of Hall Dynamo

Three layer analysis

Ideal MHD

ve ~ vi

Ideal two-fluid

ve ~ vi

distance from reconnection layer

0 dR, de ρs

Resistive two-fluid

vi ~ 0

V. Mirnov

For experimental parameters

-1

0

1

2

3

4

5

6

0.001 electron skin depth 0.05 ion Larmor radius 1 3

DISTANCE FROM RESONANCE SURFACE X/L

€

⟨˜ j ט B ⟩||

ne

€

⟨˜ v ט B ⟩||×100

distance from resonant surface

de ρs

Faraday rotation angle

time (ms)24 26

80

60

40

20

0

P(f) [Gs

2

/kHz]

806040200f [kHz]

standard 400ka ppcd 400ka

magnetic turbulence

Tearing Modes

Magnetic fluctuations

€

˜ B ( f )

Measuring fluctuations with Faraday rotation

100

80

60

40

20

0-2 -1 0 1 2

Time [ms]

Time Evolution of Current Density Fluctuation

100

80

60

40

20

01.00.80.60.40.20.0

r/a

w=8cm rs=17 cm

(b)

m = 1, n = 6

The reconnection “current sheet”

Hall Dynamo Measurements

€

ηJ0 ≈ 0.5V /m60

40

20

0

-2 -1 0 1 2Time [ms]

E// <δJxδ >B // /ne

1.7 /V m 0.50 /V m

W. Ding et al

Hall dynamo localized in radius

30

20

10

00.80.60.40.2

r/a

< δJxδ >B // /nee

The diamagnetic dynamo

€

η j||− E

||= ˜ v × ˜ B

||−

˜ j × ˜ B ||

ne€

ηj − E = v × B +j × B

ne−∇pe

ne

€

η j − E =˜ E • ˜ B

B+∇˜ p e • ˜ B

B

parallel component of mean-fields,

or, writing yields

€

˜ v +˜ j

ne= ˜ E −∇˜ p e( ) ×

B

B2

MHD dynamo

diamagnetic dynamo

Measurement of diamagnetic dynamo

Ji et al

TPE-1RM20 RFP

Different dynamo mechanisms dominate in different parameter regimes

Kinetic Dynamo

• Radial transport of parallel current (electron momentum) by particle motion along stochastic magnetic field

• Can show,

radial flux of parallel current ~

€

˜ p || ˜ B r

not yet measured

Open questions(and relation to astrophysics)

Nonlinear aspects of MHD dynamo• Is nonlinear physics of growing field similar to that of steady state

dynamo

• Does strong dynamo effect in lab have implication for astrophysical dynamo saturation?

• What is the role of reconnection in astrophysical dynamos?

• Does current (magnetic field) transport play a role in astrophysics?

• What is role of nonlinear coupling in altering wave functions near reconnection surface?(Need a nonlinear theory)

Non-MHD effects

• What are the relative contributions of the various mechanisms? Dependence on parameters?

• Does the detailed mechanism matter?

• Are non-MHD mechanisms active in astrophysics?