NIMROD Simulations of FRC Formation and Merging

R. D. Milroy and C.C. KimyRPPL / Ψ-Center – University of Washington

OutlineBegin study of FRC formation, translation and mergingConcentrate on “dynamic formation”Concentrate on dynamic formation– FRC is formed in a θ-pinch where the coils are fire sequentially from

one end.– FRC is formed and rapidly accelerated out of the formation chamber

at sonic velocities.

In Hall calculations, significant net toroidal field and helicityg yis generated during the “dynamic formation” process.Two FRCs can be formed in two separate θ-pinch devices and accelerated towards each other and merge to form aand accelerated towards each other – and merge to form a single FRC.The “merging chamber”, may be a smaller radius, equal g g , y , qradius, or larger radius than the formation sections.

NIMROD Equationsq

( ) =⋅∇+∂∂ untn 0

Continuity

⋅∇−∇−×=⎟⎠⎞

⎜⎝⎛ ∇⋅+∂∂

∂

ΠPBJuuut

t

ρMomentum

+⋅∇−⋅∇Π−⋅∇−=⎟⎠⎞

⎜⎝⎛ ∇⋅+∂∂

−uuu QqPT

tn

ssssssss :1γ

Temperature

×−∇=∂∂ EBtFaraday’s Law

( ) ( )⎥⎦⎤

⎢⎣⎡ +⋅∇+∂∂

+∇−×++×−= VJJVJPBJJBuE e tnem

nee2

1ηGeneralizedOhm’s Law

Chodura Resistivityy

“Chodura” resistivity is a semi-empirical formula to captureChodura resistivity is a semi-empirical formula to capture macroscopic effects of microinstabilities.– Has been used extensively in previous FRC formation studies

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−==

s

dpicch

chech v

vfcnem exp1 ,2 ωννη

– Has recently been added as an option in NIMROD, and used form most calculations in this study.

Initial ConditionsBegin with uniform pressure vacuum field solution– Flux interpolation between coils, is same as for Eθ.p , θ

Chodura resistivity annihilates flux at near experimentally y p yobserved rates during formation – slower in formed FRC.Viscosity: ν = 500 m2 / sec (Reynold’s no. ~ 200)Use a 16 x 96 cell grid (4th order polynomial)Hall term is included in formation calculations

Boundary Conditionsy

A set of external coils can be simulated by specifying anA set of external coils can be simulated by specifying an axially varying Eθ on the radial boundary.Allow an arbitrary number of coils.– Use a tanh function to interpolate between coils– Voltage time history ~ cos{(t-tf)/t1/4·(2/π)}

Velocity: ⎬⎫

⎨⎧ ×

= 0minv BEVelocity:Optionally set Bθ = 0 on radial boundary– This is found to be very important to the generation of net toroidal

⎭⎬

⎩⎨=⊥ 0,min 2B

v

magnetic flux and helicity during dynamic formation.

Use mass conserving boundary condition on density.

Calculation Parameters

n =2x1020 (3 mTorr fill)no 2x10 (3 mTorr fill)Initial temperature (To = 5 eV)Bbias = -0.1 T under θ-pinch: +0.2 T in translation regionbias p gApply voltage to θ-pinch coils: – Vo= 30 kV, t¼ = 13 µsec. (ΔB=0.5 T)– Fire coils sequentially at 0, 1, 2, 3, & 4 µsec.

Pressure evolution during formationg

θ-pinch coils are fired sequentially at 0,1,2,3,4 µsec.– The FRC is formed and accelerated simultaneously

Bθ evolution during formationθ g

Toroidal magnetic field is generated by strong Jθ near the g g y g θreversal layer. This is near (at) the wall during reversal.After field-lines are closed, little net helicity is generated.

Toroidal field generation during dynamic formation

Two measures of net toroidal flux:

( )ψ

∫=0

∫- Helicity= ; Net toroidal flux ( ) ψψχψ

ψψdK

o∫ =

−=0

2 ∫=Φ dzdrBθ

Solution is not sensitive to the assumed resistivity form, but it is very sensitive to the application of the Bθ=0 boundary condition!y pp θ y

Application to the PHD experimentpp p

Simulation geometryDensity ramps down in merging chamber

Single end of PHD experimentg pInitial PHD simulations are of a single end, followed by the ejection of the FRC into the translation / merging section.j g gParameters: Bbias = -0.05 T, V = 30 kV, t1/4=18.5 μsec, ΔB=0.7 T, ndens=2x1020 (2x1019 in translation region)Coils fired at t=0, 0.25, 0.5, 0.75, 1.0 μsec. – Reduced time between firing was required to compress FRC prior to

ejection, and leads to a significant reduction in net toroidal flux.

Formation in PHDFormation in PHDPressure and magnetic flux

Time history of key parameters during formation in PHD

Formation & merging in PHDg gPressure and magnetic flux

Formation & merging in PHDg gPressure and magnetic flux

Time history of key parameters during formation and merging in PHDg g

Formation and merging in an expanded chamber

ParametersR 0 3 i f ti ti 0 7 i i ti– Rwall = 0.3 m in formation section, 0.7 m in merging section

– Bbias = -0.05 T, V = 30 kV, t1/4=10.0 μsec, ΔB=0.675 T, ndens=2x1020

(2x1019 in translation region)– Bbias = 0.08 T in merging section– Coils fired at 1, 2, 3, … 9 μsec

Formation with elongated θ-pinchg pPressure and magnetic flux

Significant toroidal field is generatedg gBθ magnitude reaches values ~ 25% of poloidal field

FRC doesn’t survive merge in simulationgPressure and magnetic flux

Time history of key parameters during formation and “merging”g g

SummaryySuccesses– NIMROD simulations of the dynamic formation of FRCs has beguny g– The Hall term may explain the observed toroidal field generation when a Bθ=0 boundary condition is applied to n=0 on the radial boundary.

DifficultiesDifficulties– Have not yet successfully simulated FRC merging in an expanded

confinement chamber. (Perhaps more sophisticated viscosity is required.)

Plans– Comparison with observations on the PHD experiment– Study development of instabilities during formation by switching to 3D y p g y g

calculations and employing 5 and 11 toroidal modes.