Two-fluid Plasma-Neutral Model Development and Application

E.T. Meier*, V.S. Lukin**,U. Shumlak*

* University of Washington, PSI-Center, Seattle, WA** Naval Research Laboratory, Washington D.C.

Innovative Confinement Concepts 2010 WorkshopPrinceton, NJ

February 15-19, 2010

Resources used:• PSI-Center SGI Altix 350 cluster• DURIP SGI ICE Altix 8200 cluster (funded by an Air Force grant)This research is funded by DOE.

2

Motivation

In many ICC devices, neutral gas can dramatically affect the plasma behavior. Important effects include line radiation losses, charge exchange neutral penetration and associated confinement degradation, and momentum transfer between plasma species. By carrying a neutral fluid as a separate species along with the plasma fluid of a typical MHD model, these effects can be captured.

3

Outline

• Introduction to the plasma-neutral model• Applications

– Plasma acceleration with parallel-plate electrodes

• Are “critical ionization velocity” (CIV) effects captured?

– Plasma acceleration with coaxial electrodes• What is the effect on snowplow angle?

– Charge exchange drag on ion rotation in RMF-driven FRCs

• Summary

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Introduction to plasma-neutral model

5

Model is derived by generalizing work of Braginskii*.

* S.I. Braginskii, Transport Properties in a Plasma, in Review of Plasma Physics, M.A. Leontovich(ed.), Consultants Bureau, vol. 1, pp. 205-311 (1965)

Braginskii’s derivation assumes that .

0coll

f dtα∂

=∂∫ v .

This assumption is dropped, allowing species conversion. Ionization, recombination and charge exchange physics are captured. A Z=1 neutral gas is assumed (hydrogen/deuterium/tritium). The following reactions are considered:

electron impact ionization, 2e n i e− + −+ → + ,

radiative recombination, *e i n n hν− ++ → → + ,

and charge exchange, i n n i+ ++ → + .

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( )( )ii i ion recit

mρ ρ∂+∇ = Γ −Γ

∂vi

( )( )nn i rec ionnt

mρ ρ∂+∇ = Γ −Γ

∂vi

( ) ( )( ) reci i i i i i i ion i n cx i n i im m m mptρ ρ∂

+∇ + = × −Γ +Γ +Γ −∂

v v v I j B v v v vi

( ) ( )( ) recn n n n n i i ion i n in cx i i npt

m m m mρ ρ Γ −Γ +Γ −∂

+∇ + =∂

v v vv v v I v

( )( ) ( )11 1 cx ion n i i i ni i ion n ionp p p KE KE KEt

mγγ γ

φ+ Γ +Γ + − +Γ −

∂+∇ = ∇ − ∂ −

v vv vi i i

( ) ( )

1

rec i n i n i i cx i n i n

nn n n n

iKE

p p

KE m KE KE m

pt

γγ

φ

γ

+ Γ + −

∂ +∇ = ∇ −1 ∂ −1

− +Γ −

+

v v v

v v

vi i

i i

Neutral effects are captured in continuity, momentum and energy equations.

7

Atomic reaction source rates are empirical (Cx.) and semi-empirical (Ion. and Recomb.).

In general:

Charge exchange*:

Ionization**:

Recombination**:

Ion-neutral relative velocity:

* ORNL-6086/V1, Atomic Data for Fusion Vol. 1, C.F. Barnett** R. J. Goldston, P. H. Rutherford, Introduction to Plasma Physics, IOP Publishing Ltd., 1995

2v nσΓ =

1/22 m

sin cmi

v Em

=

, ( )2

2i

cm i n imE kT= − +v v

-193

1 3*10 *m scx in n iv n nΓ =

1/213-eV

3

T2*10 1e 6 / m s

ion eVTion i n

eV ion ion

n nT

φ

φ φ

− Γ = +

1/220 2

3eV

17*10 T m sion

rec inφ−

Γ =

8

Charge exchange is dominant at low temperatures ( < 10 eV).

<sigma*v> (m3/s) vs. T (eV)

1E-20

1E-19

1E-18

1E-17

1E-16

1E-15

1E-14

1E-13

1E-12

0 10 20 30 40 50 60

Te, eV

<sig

ma*

v>, m

^3/s

sigma*v, charge exchange

sigma*v, ionization

sigma*v, recombination

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Higher moments of collision terms require additional attention.

Ionization 0th moment

. . .

i e n

ion ion ion

e n ionion

f f fd d dt t tn n vσ

∂ ∂ ∂= = −

∂ ∂ ∂

= = Γ

∫ ∫ ∫v v v

R and Q terms are not yet included for ionization, recombination, or charge exchange.

Ionization 1st moment

( ). .

.

i ii i i

ion ion

ii i ion i

ionion

i i ion i

f fm d m dt t

fm m dt

m

∂ ∂= +

∂ ∂

∂= Γ +

∂

= Γ +

∫ ∫

∫

v v v w v

v w v

v R

Ionization 2nd moment

( )2 2 2

. .

2 2

. .

2

1 1 22 2

1 1 2 21 2

i ii i i i

ion ion

i ii i ion i i i

ion ion

ion ioni i ion i i i i

f fm d m dt t

f fm m d m dt t

m m Q

∂ ∂= + +

∂ ∂

∂ ∂= Γ + +

∂ ∂

= Γ + +

∫ ∫

∫ ∫

v v v v w w v

v v w v w v

v v R

i

i

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• Alfvén conceived of CIV as a mechanism for coalescence of planets from interstellar neutral gas [1].– When neutral gas particle kinetic energy exceeds ionization

energy, ionization occurs:

• Fundamental energy transfer process (neutral kinetic energy electron kinetic energy ionization) is not known with certainty.

• For a discussion of CIV from a theoretical perspective, see [2].

• CIV effects have been shown repeatedly in laboratory experiments. For a review of experimental work, see [3].

Critical ionization velocity (CIV) was originally conceived by Hannes Alfvén.

2crit ion nv mφ=

[1] Alfvén, H., On the Origin of the Solar System, 194 pp., Oxford Univ. Press, New York, 1954[2] Lai, S.T., A review of critical ionization velocity, Reviews of Geophysics 39, 4, 2001[3] Brenning, N., Review of the CIV Phenomenon, Space Science Reviews 59: 200-314, 1992

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• Ionization can be thought of as the sum of electron impact ionization and ionization due to CIV.

Critical ionization velocity can be included in the plasma-neutral model.

ion e impact CIV−Γ = Γ + Γ

• ΓCIV is probably caused by a kinetic instability (perhaps a modifiedtwo-stream instability).

• ΓCIV is assumed to be somewhere near the peak of Γe-impact which is a maximum at 100 eV.

( )

( )

1/213-

3

2*10 1e 6 / m s

100

ion eVTeVe impact eV i n

eV ion ion

CIV CIV e impact

TT n nT

f eV

φ

φ φ

−

−

−

Γ = + Γ = Γ

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Plasma acceleration withparallel-plate electrodes

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MHD simulation of parallel-plate electrode acceleration is a 1D problem.

V-

+

Slug of neutral gas Low-density background of plasma and neutral gas

+x

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When peak density ≈ 10-8 kg/m3, CIV restricts the peak plasma speed.

With CIV

WithoutCIV

Ion speed (m/s) Ion density (kg/m3) Neutral density (kg/m3)

Vcrit = 36 km/s

Vcrit

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At higher densities, ionization energy investment limits kinetic energy.

With CIV

WithoutCIV

Ion speed (m/s) Ion density (kg/m3) Neutral density (kg/m3)

Vcrit

Vcrit

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With enough power, rapid ionization allows v > vCIV.

With CIV

WithoutCIV

Ion speed (m/s) Ion density (kg/m3) Neutral density (kg/m3)

Vcrit

Vcrit

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Plasma acceleration withcoaxial electrodes

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Neutral gas effects are often important in coaxial acceleration.

V-

+

Slug of neutral gas

Low-density background of plasma and neutral gas

z

r

r = 0.1 m

r = 0.2 m

z = 1

m

CL

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Snowplow angle is changed significantly when neutral physics are included.

17o

24o

Initial slug has density of 6e-5 kg/m3. As seen in parallel-plate acceleration, the role of CIV is limited at this high density. At lower densities, CIV could further reduce snowplow effect.

Parameters (densities, voltage, spatial dimensions) are similar to the U. Washington ZaPexperiment.

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Charge exchange drag on FRC ion rotation

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Anomalous drag favorably limits ion rotation in FRCs… neutral effects are likely

important.

• Current can be driven in FRCs by Rotating Magnetic Field (RMF) current drive. Formation and sustainment can be accomplished with RMF.

• Electron rotation is desired. Ion rotation can result in rotational instabilities and is not desired.

• Past and present researchers [2,3] at RPPL on the TCSU experiment [1] have studied on ion rotation damping mechanisms, but uncertainty persists.

1. Translation Confinement and Sustainment Upgrade experiment at the University of Washington Redmond Plasma Physics Laboratory.2. C. Deards, ongoing work.3. A.M. Peters, PhD thesis, 2003.

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The two-fluid plasma-neutral model is applicable to the FRC ion rotation problem.

• The two-fluid plasma neutral model can capture the ionization, recombination and charge exchange effects.

• Preliminary computations have been done, and neutral drag effects are seen.

• Preliminary simulation details are listed below.

• Like the TCS device, maximum radius is 40 cm. Total axial length is 3 m.

• Domain is periodic axially.• Initial condition is a computed equilibrium

(using a code by G. Marklin).• Maximum initial density is 1e19 m-3.• Maximum initial total temperature (2Ti =

2Te = T) is 80 eV.• Radius-dependent azimuthal body force =

75r N/m3 such that, if density is uniform, rotational acceleration is uniform.

• Total simulated time is 20 µsec.

• Neutral gas initial density is one tenth the initial maximum plasma density.

• Gas is injected uniformly from the radial wall at a rate of 1.5 µg/µsec. (Compare to total initial plasma density of 10 µg.)

• Neutral products of charge exchange are allowed to leave the system (with associated mass, momentum, and energy).

– CX neutral m.f.p. is longer than machine size.

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Uniform azimuthal force is applied to the FRC plasma.

• Color shows azimuthal velocity.• Black contours are of flux; dark black line is the separatrix.• In the case which includes neutral effects, maximum azimuthal velolcityis 26% lower.

Without neutral effects With neutral effects

Initial condition

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Azimuthal velocity evolves to steady state in 20 µsec.

Azimuthal velocity (m/s) Plasma density (kg/m3) Neutral density (kg/m3)

Without neutral effects

With neutral effects, azimuthal velocity is damped.

(Variables are plotted at a slice along the axial midplane.)r/rmax r/rmax r/rmax

r/rmax r/rmax r/rmax

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Summary

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Summary

• Critical ionization velocity (CIV) effects can be captured in the fluid model presented.– Additional research needed to properly specify the

magnitude of CIV ionization.• A 6 degree change in the snowplow angle is seen when

neutral effects are included.• Charge exchange drag on ion rotation in FRCs can be

captured as demonstrated. Additional work remains if high-fidelity (useful) results are desired.

• Viscosity is thought to be critical [Deards, Peters] so the viscosity model used for these simulations should be improved. (Presently, a uniform viscosity is used.)

• Radial wall BCs on neutral and plasma density and pressure are not physically meaningful.

• Neutral gas inflow should be changed to approximate an experimental scenario.

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Two-fluid Plasma-Neutral Model Development and Application

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