Weixing Ding

University of California, Los Angeles,USA

collaborators: D.L. Brower, W. Bergerson, D. Craig, D. Demers, G.Fiksel, D.J. Den Hartog, J. Reusch, S. C. Prager, J. Sarff, T.Yates and MST team

(Oct. 14, 2009 Princeton)

Particle Transport due to Stochastic Magnetic Field in a High-Temperature Plasma

Introduction

Free energy

instabilities

Fluctuations

Particle (charge)

Momentum

Energy Transport

J// (r)

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δ r

B ,δr J ,δn

Tearing

Electron Density Relaxation Due to Magnetic Field Perturbation

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ne (r) [1019 m−3]

Magnetic Field Fluctuations

Driven by Tearing Instabilities in MST

MST

L.Zeng (UCLA) et al DIII-D by reflectometry

Driven by I-coil externally in DIII-D

DIII-D

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

MST Reversed-Field Pinch (RFP) is toroidal configuration with relatively weak toroidal magnetic field BT ( i.e., BT ~ Bp)

Madison Symmetric Torus

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q(r) =r

R

BT

BP

<1 For plasma w/o current profile control

R0 = 1.5 m, a = 0.51 m, Ip ~400 kA

BT ~ 3-4 kG, ne ~ 1019 m-3 ,Te~Ti~300eV~7%

Fluctuations and Transport in the MST

Generation of magnetic flux (dynamo)

Particle and energytransport

Ion heating

Momentum transport

Magnetic reconnection events

Outline

(1) Density Relaxation and particle transport during reconnection;

(2) Measured magnetic fluctuation-induced particle flux accounts for particle transport;

GOAL: Identify the role of stochastic magnetic field in particle transport during reconnection

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Γre =

< δΓ//,eδbr >

B (=

< δj//,eδbr >

eB)

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∂ne

∂t

Particle Diffusion Rate in a Stochastic Magnetic Field in MSTT

oroi

dal A

ngle

/

r/a

Puncture Plot of B Field in MST Magnetic diffusivity coefficient from field line tracing

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Dm =< (Δr)2 >

2Δl

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~ 1.0 ×10−4 m

(Hudson and Gennady, 2006)

Rechester & Rosenbluth (1978) derived a quasi-linear coefficient

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Dm = πR0 q(δbr

m,n

B0

)2

m,n

∑ ~ 2.0 ×10−4 m

Harvey (1981) and Finn (1990) suggest particle diffusion rate Di ~ Dm cs (or Vi,th)

Plugging in MST parameters: p ~ 1-2 ms (quasi-linear estimate) in ambient case p ~ 0.1-0.2 ms (experiment) during relaxation event

Measured Total Particle Flux during Reconnection

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∂n

∂t+ •∇ < nVr >= Se (Se <<

∂n

∂t)

Γr × 2πrL = −∂

∂tn(r)dV ≅ −

∂n

∂t∫ πr2L

Γr = −∂n

∂t

r

2r

L

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// B

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⊥ B

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Γr =< nVr >

Total particle flux surges to 1.5*1021 m-2s-1 during reconnection

Fluctuation-Induced Particle Flux Contributes to Total Flux

Particle transport is determined by perpendicular momentum balance

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nr E ⊥+ n(

r V ×

r B )⊥ = 0

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n = n0 + δnr E ⊥ =

r E ⊥0 + δ

r E ⊥

r V =

r V 0 + δ

r V

r B =

r B 0 + δ

r B

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< nVr >= n0

< E⊥0 >

B0

+< δnδE⊥ >

B0

+ V//

< δnδbr >

B0

+ n0

< δV//δbr >

B0

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<δΓ//δbr >

B0

Particle transport arises from particle streaming along stochastic field lines

total flux pinch electrostatic magnetic fluctuation

Pinch and electrostatic fluctuation-induced particle flux are negligible

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Γr =< nVr >= n0

< E⊥0 >

B0

+< δnδE⊥ >

B0

+ V//

< δnδbr >

B0

+ n0

< δV//δbr >

B0

Lei, et al. PRL 2002

total flux pinch electrostatic magnetic fluctuation

electrostatic

Magnetic Fluctuation-Induced Particle Flux

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ΓrM = V//

< δnδbr >

B0

+ n< δV//δbr >

B0

<B0>

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δbr

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δbr€

δbr

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δbr€

δn(+)

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δn(−)

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δn(−)

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δn(+)

Density-fluctuation dependent flux

Velocity-fluctuation dependent flux

<B0>

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δV//(+)

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δV//(−)

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δbr

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δbr

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δbr

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δbr

Measurement of Magnetic Fluctuation-Induced Particle Flux

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V//,e =J

neLaser Faraday rotation B

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J =∇ × B

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δbr(r) Laser Faraday rotation

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δne Laser interferometer ( )

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∇δne

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Γr = V//

< δnδbr >

B0

+ n< δu//δbr >

B0

Motion Stark effect

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V//,i , δV//,i CHERS, Mode Rotation, Ion Doppler Spectroscopy

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B0

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δV//,e Not measured, inferred from quasi-neutrality

7 independent quantities are needed to determine electron and ion flux

FIR Polarimeter-Interferometer System

SignalBeam

25mWEachλ≈432μm125WContinuousλ=μm

Grating Tuned CO Pump Laser2

Waveguides

Plasma VacuumDuct

SchottkyDetectors

BeamSplittersWireMesh

9ChordHαArray

25mWEachλ≈432μm

Translating Mirror

9.27

3 Wave FIR Laser Local Oscillator Beam

MST Far-Infrared Differential Interferometer System

5o=Ro1.5m=a.52m

P43

P36

P28

P21

P13

P06N02N09

N17N24

N32

MST

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φ ~ ndl∫ + ˜ n dl∫

Ψ ~ nr B • d

r l + n ˜ b • d

r l + ˜ n

r B • d

r l ∫∫∫

Interferometer density fluctuation

Faraday rotation magnetic field

11 chords, separation 8 cm,phase resolution 0.05 degree, time

response up to 1μs

Ding, Brower, et al

PRL(2003),(2004)

RSI(2004),(2008)

See poster: P03-23(Bergerson)

Measurement of Equilibrium Magnetic Field and Current Denisty

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V//,e =J0

ene 0

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V//,e

< δnδbr >

B0

1.2

1.0

0.8

0.6

0.4-2 -1 0 1 2

Time [ms]

1.5

1.0

-2 -1 0 1 2Time [ms]

1.2

1.0

0.8

0.6

0.4-2 -1 0 1 2

Time [ms]

2.5

2.0

1.5-2 -1 0 1 2

Time [ms]1.5

1.0

-2 -1 0 1 2Time [ms]Time [ms]

0.44

0.40

0.36

0.32

B0

[ T ]

1.51.00.50.0-0.5-1.0Time [ms]

By Motion Stark Effect

By Faraday rotation

Measurement of density and magnetic fluctuation

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V//,e

< δnδbr >

B0 Density

fluctuations

Radial Magnetic fluctuations

1.5

1.0

0.5

0.0

[1017m

-3]

1.00.80.60.40.20.0r/a

80

60

40

20

0

[Gauss]

1.00.80.60.40.20.0r/a

radial magnetic field br

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δn(r)

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δbr(r)

m/n=1/6

Magnetic fluctuation-induced electron particle flux

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Γr =< nVr >= V//,e

< δnδbr >

B0

+ n0

< δV//,eδbr >

B0

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< nVr >=< nVr,e >=< nVr,i >

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V//,e

< δnδbr >

B0

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V//,i

< δnδbr >

B0

(Total Flux)

Density fluctuation contributes to electron particle flux

Magnetic fluctuation-induced ion particle flux

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Γr =< nVr >= V//,i

< δnδbr >

B0

+ n0

< δV//,iδbr >

B0

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< nVr >

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E⊥ =<δvφ ×δbr >

Vr,i = E⊥× B0 /B02

1.50

0.75

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Γr,i = n0Vr,i = n0

< δV//,iδbr >

B0

Den Hartog et. Al ,PoP,1999

(Ion Flux)

Velocity fluctuation contributes to ion flux

Magnetic Fluctuation Induced Flux Balances the Change of Density

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∂ < ne >

∂t

−∇ • Γre

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∂ < ne >

∂t+∇ • Γr

e ≈ 0

2.0

1.5

1.0

0.5

0.0

ne

(0) [10

13

cm-3]

210-1-2Time [ms]

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< ne >

− ∇ • Γre∫ dt

Integrating over time

Magnetic fluctuation induced particle transport drives density change

-4

-3

-2

-1

0

1

2

[1022m

-3s

-1]

-2 -1 0 1 2Time [ms]

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r /a = 0.04

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< nVr >=V//

< δnδbr >

B0

+ n0

< δV//δbr >

B0

1.50

0.75

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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< nVr,e >= V//,e

< δnδbr >

B0

+ n0

< δV//,eδbr >

B0

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< nVr,i >= V//

< δnδbr >

B0

+ n0

< δV//,iδbr >

B0

Electron and ion flux arise from different mechanism

Electron flux

Ion flux

Total flux <nVr>

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Γq = Γr,i − Γr,e =

=< ˜ j // ˜ b r >

eB≈

R

nB2(m

rBp +

n

RBT )

1

eμ0

<1

r˜ b r

∂

∂rr ˜ b θ >

≈R

neB2(r k •

r B ) | ˜ b r || ˜ j φ | cos(Δ)

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∇ ×δr B = μ0δ

r J

| r − rs |

rs

<<1

(m,n) are poloidal and toroidal mode number

Ambipolarity of Particle Transport (difference between electron and ion particle flux)

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Γr

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Γq

2.0

1.5

1.0

0.5

0.0

-0.5

1021[m-2s

-1]

-1.0 -0.5 0.0 0.5 1.0Time [ms]

Total Particle FluxCharge Flux

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Γq ≈ 2%Γr

Ding, et al. PRL 2007

Particle diffusivity is (30X) larger than QL prediction

Rechester&Rosenbluth(1978)

(without ambipolarity )

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30 × DmVi,th

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0.5 × DmVe,th~ ~

(30 times) Predicted Particle Transport

~Predicted Heat Transport

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χeRR (De ) ~ DmVe,th

Experimentally, particle diffusion rate is approximately electron diffusion rate in a stochastic magnetic field.

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DExp = 730 m2 /s

Harvey (1981)

(with ambipolariy)

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Di ~ DmVi,th

Summary

(1)Rapid particle transport is observed in MST associated with the stochastic field;

(2) Electron particle flux is comparable to ion flux ( ), accounting for global density change during a reconnection event;

(3) Particle transport is much larger than the expected from quasi-linear theory.

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V//,e

< δnδbr >

B

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n< δV//,iδbr >

B

Magnetic Fluctuation Measurement Method

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Ampere's Law : δr B • d

r l = μ0L

∫ δI

Faraday Rotation Fluctuation :

δΨ = cF n0∫ δr B • d

r l ≈ cF n 0 δ

r B • d

r l ∫

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δ r

B • dr l ≈

L

∫ δBz∫ dz[ ]x1

− δBz∫ dz[ ]x 2

≈ μ0δIφ =δΨ1 −δΨ2

cF n 0

Plasma

X1 X2

Ding, et al. PRL (2003)

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⊗ δI

-15

-10

-5

0

5

10

15

δjϕ

[ /A cm

2 ]

-40 -20 0 20 40 [ ]r cm

Localization of Density Fluctuations

Physically

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δn ~ −δr v r • ∇n0

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δvr

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δvr

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δvr

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m =1