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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?
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