Transport in a field aligned magnetized plasma and neutral gas boundary:

The end of the plasma…C.M. Cooper, W. Gekelman, P. Pribyl, J. Maggs, Z. Lucky

University of California, Los AngelesFebruary 15, 2013

Overview• Motivation– Relation to Auroral Physics, Divertor Physics

• Summary of Enormous Toroidal Plasma Device– Model of ETPD plasma regions

• Neutral Boundary Layer model: Plasma Termination1. Occurs at plasma/neutral gas pressure equilibrium 2. Plasma terminates on current-free ambipolar sheath

determined by a generalized Ohm’s law3. Heating, ionization occur in NBL

Measured scaling laws in ETPD

Motivation: Aurora

Schematic view of parallel currents j||, plasma motions, vperp, electric fields, and potential contours above auroral

arc, sketched in green [Haerendel 1996]

Aurora over Jokulsarlon Lake, IcelandStephane Vetter, “Nuits sacrees” Feb. 2011

The auroral-arc current loop associated with a perpendicular electric field imposed

in the magnetosphere [Borovsky 1993]

J. Borovsky, J. Geo Res 98, A4 6101-6138 (1993)G. Haerendel, J. Atmos. Terr. Phys. 58, 1-4 71-83 (1996)

Motivation: Gaseous Divertors

Gaseous divertors designed to dissipate plasma particles, momentum, and energy on neutral gas before touching wallGaseous divertor design for JET

Gaseous divertor design for DIII-D

R. W. Conn, Fusion and Eng. Design 14 81-97 (1991)

Diagram of the ETPD

Toroidal Magnets (red) provide a 250 G confining field

Vertical Magnets (blue) provide a 6 G vertical field to space out rings

ETPD Plasma (pink) follows the helical magnetic field up to 120 m

LaB6 source injects primaries to form then heat the plasma

R.J. Taylor et al, Nuclear Fusion 42 46-51 (2002)C.M. Cooper et al, Rev of Sci Inst. 81 083503 (2010)

Frame and Shielding

Heater

CathodeAnode

Picture of a LaB6 Cathode

Experimental Setup• Primary electrons boiled off LaB6 are accelerated along the field by anode.

• Primaries ionize neutral gas to create and heat plasma.

• Data taken where plasma ends, 90% around machine.

Probe

400 V discharge2 mTorr

300 V discharge2.5 mTorr

220 V discharge3.6 mTorr

Neutral Boundary Layer

2.2 kA

Energy Balance in ETPDe

ee

e ee e e e

e eeee

e ee ei

e

iiii

e ee ee

e

ee e

ee H

EAT

HEAT

HEA

T

HEAT

HEA

T

HEAT

HEA

T

HEAT

HEA

T

HEAT

HEA

T

HEAT

220 V

<100 kV

B

Energy Balance in Aurora

Atmo-sphere

Ionosphere

HEAT

HeliumGase

e ee

eHEAT

3 Zones in ETPD Plasma

Heat

Density

- Conduction/ ionization- Radial diffusion

Energy loss length Efield Term.Energy input length

+ Thermalization of primaries+ ionization > losses

mprimmfp 15~, mrrp 15~/ mvpin 5.1~/

- Ohmic Heating and cooling- Ionization and Radial diffusion

Take Data

Ambipolar Termination Sheath in NBL!

Plasma potential in neutral gas boundary indicative of large field-

aligned electric field

Neutral fill of 3.6 mTorr He, a plasma discharge of 194 A 220 V, Btor=250 G, Bver=6 G,

Relative x coord (cm) Relative x coord (cm) Relative x coord (cm) Relative x coord (cm)

Rela

tive

y co

ord

(cm

)

Fp (eV)-40 -20 0 20 40 60-60 -40 -20 0 20 40 60 -40 -20 0 20 40 -40 -20 0 20 40 60

40

20

0

-20

-40

1

0

-1

-2

-3

-4

GenerateRadial Profiles

Neutral fill of 3.6 mTorr He, a plasma discharge of 194 A 220 V, Btor=250 G, Bver=6 G,

Relative x coord (cm) Relative x coord (cm) Relative x coord (cm) Relative x coord (cm)

Rela

tive

y co

ord

(cm

)

Fp (eV)-40 -20 0 20 40 60-60 -40 -20 0 20 40 60 -40 -20 0 20 40 -40 -20 0 20 40 60

40

20

0

-20

-40

1

0

-1

-2

-3

-4

Used to study rate at which particles, momentum, and heat move across the field and out of the plasma

Neutral fill of 3.6 mTorr He, a plasma discharge of 194 A 220 V, Btor=250 G, Bver=6 G,

Generate Axial Profiles in NBLPlot data along the center of plasma coordinate “s” Three zones! A B C

Relative x coord (cm) Relative x coord (cm) Relative x coord (cm) Relative x coord (cm)

Rela

tive

y co

ord

(cm

)

Fp (eV)-40 -20 0 20 40 60-60 -40 -20 0 20 40 60 -40 -20 0 20 40 -40 -20 0 20 40 60

40

20

0

-20

-40

1

0

-1

-2

-3

-4

A B C

Neutral Boundary Layer Model• Three-fluid, current-free

,, ppspp Svns

ensnsppee

Bpne

s vvnms

Tknt

s

peE ,,71.0

pionzssppe

Bspp SEEvens

Tkvn

,,2

3

• Momentum equation: pressure equilibration

• Heat equation: ionization and Ohmic heating

• Momentum equation: generalized Ohms Law

• Continuity equation: ionization and losses

0,,,,

nsnpspinp vvmpps

,, npsnn Svns A

B

C

Zone A: Pressure Equilibration

• Diffusivity argument: Neutral density convected out along field by plasma, diffuses across field

𝐷𝑎,∨¿≫𝐷𝑛=𝑘𝐵𝑇 𝑛

𝜈𝑛

>𝐷𝑎 ,⊥

/

=Momentum gained from plasmaMomentum lost to stationary neutrals

𝑝𝑝≥𝑝𝑛For

• Drag force limited to cross-field neutral diffusion rate

• Axial neutral flow develops from force balance

Electron Temperature in Zone A

• Constant

• Ionization ~ 0

mm

men

e

iTe 001~

eVTe 2.2

eT

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

3

2(eV)

Te

0.5

0.4

0.3

0.2

0.1

0

Plasma m

ach at 2.2 eV

Plasma Velocity in Zone A

• Plasma velocity is flat• Momentum balance

• Sets a critical velocity

,,

,,

np

psseq cv

cxinsnspei vvnm 5.0,,

ann

ap

scr

D

D

scmv

,,

,,

5,

~

~

/102

rprspie vms

p,,

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

4

2

0

(105cm/s)

vp

Plasma Density in Zone A

• Plasma density is dropping from radial losses

cm

DDD Bohmar

10~2

1~4~ ,

2,,

p

rrprp

sp

nD

s

nv

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

1.2

0.8

0.4(1012/cm3)

nerD

,aD

BohmD

ee

e

i

i

e

eie i i i

i

iei

e

iei

ee

iei

e

ii

e

ei e i e

i

Current-free plasma sees neutral gas at end

Electrons stop, ion current penetrates

Electric field develops, drives electron current

Electric field maintains Current free term.

Zone B: Ambipolar Electric Field

Potential Structure in NBL

lei

len

lDColormap of plasma potential from 1,300 points interpolated onto toroidal coordinates

• Plasma iso-potentials form “nested V’s”

• White arrows show electric field, parallel field 10x

• Quasineutral

Electron Temperature in Zone B

• Electron temperature rises with potential• For

eVT pe 1~~

eni

een m

mL

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

3

2(eV)

Te

Fp

0.5

0.4

0.3

0.2

0.1

0

Plasma m

ach at 2.2 eV

Plasma Velocity in Zone B

• Plasma velocity drops due to drag on neutrals

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

4

2

0

(105cm/s)

vp

The difference between momentum and temperature loss

Electrons• Momentum loss: – faster• Energy loss: – slower

In electric field,

electrons HEAT.

en

eni

e

m

m

cxin 5.

Ions• Momentum loss: – slower• Energy loss: – faster

In electric field, ions SLOW.

cxin .5

Plasma Density in Zone B

• Plasma density is dropping from radial losses

• Balance between Flux conservation

Adiabatic heatingpsp nv ,

pe nT

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

1.2

0.8

0.4(1012/cm3)

ne

Plasma Termination

Use generalized Ohms law to solve for electric field

ensppee

pnpe vnm

s

TntEen

s

p ,)71.0(0

mVe

vmE enspe /2~

19.0,

0 eE

1.0ntfor

0,

toten

i

en

e JJJ

E

Zone C: Additional Ionization

eVTe 2.2

R. K. Janev, Elementary Processes in Hydrogen and Helium Plasmas (1987)

Electron Temperature in Zone C

• Prediction: Electron temperature continues to rise rapidly with potential

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

3

2(eV)

Te

Electron Temperature in Zone C

eVp 5.2~

1000~|

|

2.2

7.4

eV

eV

v

v

IONIZATION

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

3

2(eV)

Te

Plasma Density in Zone C

• Prediction: Plasma density continues to drop

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

1.2

0.8

0.4(1012/cm3)

ne

Plasma Density in Zone C

IONIZATION

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

1.2

0.8

0.4(1012/cm3)

ne

10%

Plasma Production in BoundaryIf all energy gained went into plasma production

how much could you make? For

What’s the mean free path of ionization?

Any energy gained by electric field is quickly absorbed by ionization

abab vvTT ,

ionzQQ

%10~2.26.24

5.2

eVeV

eV

Tn

n

eionze

e

cmvn

v

eVn

pz 10~| 7.4

0.5

0.4

0.3

0.2

0.1

0

Plasma m

ach at 2.2 eV

Plasma Velocity in Zone C

• Prediction: Plasma velocity will drop to 0

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

4

2

0

(105cm/s)

vp

Plasma Velocity in Zone C

• Additional flux is generated by the ionization at the end of the plasma• Still slowing down after ionization ends

IONIZATION

0.5

0.4

0.3

0.2

0.1

0

Plasma m

ach at 2.2 eV

2400 2600 2800 3000 3200Distance from anode (cm)

-1

-2

-3(eV)

Fp

4

2

0

(105cm/s)

vp

NBL Conserved QuatitiesAxial Plasma Flux:

2400 2600 2800 3000 3200Distance from anode (cm)

Normalized Plasma Pressure:• End of the plasma occurs where pp=pn

• Dropping pressure “interrupted” by termination E field

2400 2600 2800 3000 3200Distance from anode (cm)

Plasma flux:• Measure of axial particle flux• Only loss of axial flux is radial flux• Filling in of profile• Kinetic effects (trapped particles)

1.4

1.0

0.6

P p=(n

p(Ti+

T e)/(n

nTn)) 4

2

0

Gp,

s=n pv

p,s

Normalized Plasma Pressure:

49 kW

43 kW

39 kW

Scaling of Ambipolar Electric Field

The location and magnitude of Electric field terminating the plasma change as a function of input power

The potential marks a “footprint” for the boundary of the plasma

Measure plasmaparameters

Scaled Plasma Parameters pre-NBL

• Axial flow drops

• Extra power raises density

• Te is flat at 2.2 eV(a)

(b)

(c)

Scaling Study

e

vmE enspe

ts 19.0,

,

nn

pie

r

pspteq nT

snTT

D

Rvs

)()(ln 0

2,

,

Measured electric field, Es,m, scales like the theoretical ambipolar value

Measured termination point, seq,m, coincides with location of pressure equilibration

Es,m=Es,t

seq,m=seq,t

pp=pn

Modified Ambipolar Flow in ETPD

Ion current

Electron current

Ion currentIon current

Electron current

Electron current

Ambipolar Flow in NBL

• Magnetic field data from probe indicates axial current

• Negative current carried by electrons moving in positive direction

Data taken at s = 2870 cm, halfway through NBL

Non-Zero Current in NBL

•NBL not current free

•Current trends to zero

•Drift speed associated with current << bulk flow

measmeasT BJ ,

0.9𝑣𝑒 , 𝑠<𝑣𝑖 ,𝑠<𝑣𝑒 , 𝑠

Types of Current

Non Divergence-FreeDivergence-FreeTr

ansp

ort

Caus

ing

Non

Tra

nspo

rt

Caus

ing

Some currents diverge and need to be closed to maintain J=0

Some currents

are associated

with particle

drifts and transport

• Poloidal Diamagnetic Current (-)

• Poloidal Hall (i-n) Current (+)

• Parallel (e-n) Current• Radial Pederson (i-n) Current• Vertical/B Currents

Note: Steadystate so Jpolz = 0

Estimation of Jr

• Axial current tied to Radial current

Jr

JS

ArJSo+DS = JSo + JrAS

Kirchoff’s Junction Rule

inirie

rrrr vm

n

pEE

*

*, rinir EJ

Future Work on NBL

• Observational– Measure plasma potential and flows in Aurora

• Experimental– Study NBL for different pressures and/or gases

• Simulation– Model in DEGAS (neutral gas collisional code)*– Model in UEDGE (transport code)*– Runge Kutta solver for fluid model**

Future work: 1.5-D SimulationMeasured 5 things: Model neutral gasSolve 5 equations + neutral gas transport:

ppiee JvTn ,,,,

ene

ezez

e Sz

nv

z

vn

ine

iziz

e Sz

nv

z

vn

eveee

eez

eze zn

z

T

z

nT

z

vvn

eTez

eeee

eze Sz

vTn

z

q

z

Tvn

2

3

ivee

iiz

ize zn

z

nT

z

vvn

Generalized Source/Sink calibrated by data

Conclusions–Plasma terminates on ambipolar electric field– Electric field occurs where diminishing plasma

pressure and neutral pressure equilibrate–NBL dominated by termination field through

Ohmic heating and ionization–NBL has a small modified ambipolar diffusion

due to differences in directional particle motilities

Thanks!