TTF Workshop

25-28 March, 2008

Boulder, CO

Parallel and Perpendicular Flowsdriven by Stochastic Magnetic Fields in MST

D.L. Brower

W.X. Ding, T. YatesUniversity of California at Los Angeles

G. Fiksel, J.S. Sarff, S.C. Prager and the MST GroupUniversity of Wisconsin-Madison

D. CraigWheaton College

GOAL: Identify the role of stochastic magnetic fields in momentumtransport and flow generation during reconnection

Results:

(1) transport of parallel momentum from stochastic fields is measuredand found to be significant in the plasma core……matching parallel velocity relaxation

(2) Stochastic field driven charge transport produces localized perpendicular flow and radial electric field (shear)

…….to be presented at poster

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

Madison Symmetric Torus - MST

R0 = 1.5 m, a = 0.51 m, Ip < 600 kA BT ~ 3-4 kG, ne ~ 1019 m-3, Te0 < 1.3 keV τE ~ 10 ms, β =

/B2(a)=15%

Field Line Puncture Plot

Toro

idal

Ang

le / π

r/a

Puncture Plot of B Field in MST

magnetic field in MST is typically stochastic andshould contribute to momentum transport

• J. Finn (1991?) suggests momentum diffusion with D ~ Dm cs• Plugging in MST parameters: τm ~ 1 ms in ambient case

τm ~ 100 µs during relaxation event

• Momentum density in z direction being transferred in xdirection is

• Assuming a strong B field with small fluctuations and taking zalong B, we can write the contribution from parallel streamingparticles using vz ≅ v|| and vx ≅ v|| (bx/B) as

• Averaging over flux surface (z and y directions) we get theaverage momentum flux.

momentum transport by stochastic B field

vdvvmf xz3

! v)(x,

!

f (x,v)" m v||2bx

Bd3v =

p||bx

B

B

bp x >< ||

• If = 0, then we are left with

• In order for the momentum at x to change, need more fluxcoming in from –x than that leaving toward +x.

• above relation provides a direct measure of the transport of momentum along the mean magnetic field direction due to stochastic magnetic field.

• Substituting , we get two terms

!

Flux : "rmom

=< ˜ p

||˜ b x >

B

!!

"

#

$$

%

& ><'(

)

>

B

&

' (

)

* + $ n%

< ˜ T ||

˜ b x

>

B

&

' (

)

* +

!

˜ p //

= T//

˜ n + n ˜ T //

Momentum transported from core to the edgeduring reconnection

During reconnection, parallel momentum redistributes within ~100 µs,much faster than classical collision time.

25

20

15

10

km

/s

-2 -1 0 1 2time [ms]

Toroidal flow r/a=0

Stochastic field-driven transport is directly measured(first term only)

• Measure T//,ion with Rutherford Scattering (or CHERS)• Measure br and B with FIR laser Faraday Rotation• Measure with differential FIR interferometer

• Measurement details provided at poster….!

˜ n and " ˜ n

!

"< #v||

>

"t~ $T

||%

< ˜ n ˜ b r

>

B

&

' (

)

* + $ n%

< ˜ T ||

˜ b r

>

B

&

' (

)

* +

Convection

FIR Polarimeter-Interferometer System

MST!

" ~ ndl# + ˜ n dl#

Interferometer

11 chords, Δx~ 8 cm, Δφ~ 0.05o , Δt~ 1µs

!

" ˜ #

"x"x ~ 1mm

!

" ~ nr B • d

r l + n ˜

r

b • dr l + ˜ n

r B • d

r l ###

Differential Interferometer

Faraday rotation

!

˜ n

!

" ˜ n

!

˜ b and ˜ j

!

˜

r

b • dr l

L

" = µ0 ˜ I # $˜ %

1& ˜ %

2

cF n 0' ˜ j #

At x=-2 cm

Momentum flux and parallel momentum change

!

"mom

= T//,i

< ˜ n ̃ b r

>

B

Magnetic fluctuation-induced (convective) flux drives parallel momentum drop!

r / a = 0.04 -T//,i"< ˜ n ̃ b

r>

B

# < $V// >

#t-8

-6

-4

-2

0

2

4

N/m

3

1.51.00.50.0-0.5-1.0Time [ms]

!

" < #V// >

"t+ T//,i$

< ˜ n ̃ b r

>

B% 0

m=1, n=6,7,8,9

800

600

400

200

0

Ti [e

V]

1.51.00.50.0-0.5-1.0Time [ms]

Momentum flux changes direction away from magnetic axis

!

r /a = 0.04 " < #V

//>

"t

-$ •%mom

!

r /a = 0.45

-8

-6

-4

-2

0

2

4

N/m

3

1.51.00.50.0-0.5-1.0Time [ms]

4

2

0

-2

-4

N/m

3

1.51.00.50.0-0.5-1.0time [ms]

Direction change consistent with momentum transport(not total momentum loss)

30

20

10

0

-10

[km

/s]

1.51.00.50.0-0.5-1.0Time [ms]

40

30

20

10

0

-10

V// [

km

/s]

1.51.00.50.0-0.5-1.0Time [ms]

!

< "V//

>

# $ •%mom& dt

Summary: Stochastic magnetic fields drive parallel and perpendicular flows

Measurements indicate :

Momentum transport(convective)

!

"rmom

=< ˜ p //

˜ b r >stochastic magnetic field

tearing mode coupling

Parallel Velocity Relaxation

Charge Transport

Perpendicular Zonal Flow

Radial Electric Field !

Er

= "Te#n

e

nee

Increasing Density Gradient

!

"q = "ri#"r

e

!

V"i

= #Er

B