dynamo effects in laboratory plasmas s.c. prager university of wisconsin october, 2003
TRANSCRIPT
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?