Equilibrium and Stability of Oblate Free-Boundary FRCs

in MRX

S. P. Gerhardt, E. Belova, M. Inomoto*, M. Yamada, H. Ji, Y. Ren

Princeton Plasma Physics Laboratory

* Osaka University

Talk and Movies Available at http://w3.pppl.gov/~sgerhard/seminar.htmlLinked from PPPL Employee Homepage

Outline• Introduction

– Brief overview of relevant FRC issues– Introduction to the MRX facility– FRC formation by spheromak merging

• Overview of FRC Stability Results in MRX

• Systematic Studies of FRC Stability– n=1 tilt/shift instabilities without passive stabilization– n2 modes often limit lifetime after passive stabilizer is installed

• Modeling of Equilibrium and Stability Properties– Equilibrium reconstruction with new Grad-Shafranov code– Oblate Plasmas are Tilt Stable from a Rigid-Body Model– HYM calculations of Improved Stability Regime at Very Low Elongation

• Conclusions-E. Belova

FRCs have Potential Advantages as Fusion Reactors

FRCtoroidal plasma configuration, with toroidal current, but minimal toroidal field.

FRCs have Potential Advantages as Fusion Reactors

FRCtoroidal plasma configuration, with toroidal current, but minimal toroidal field.

• Intrinsically high (~1)

• Natural divertor structure

• Only circular axisymmetric coils

• No material objects linking plasma column (ideally)

• Translatable (formation and fusion in different places)

H. Guo, Phys. Rev. Lett. 92, 245001

FRCs have Potential Advantages as Fusion Reactors

FRCtoroidal plasma configuration, with toroidal current, but minimal toroidal field.

Problem:Often Predicted to Be Quite MHD Unstable

H. Guo, Phys. Rev. Lett. 92, 245001 Pressure Contours, Disruptive Internal Tilt Belova et al, Phys. Plasmas 2000

FRC Stability is an Unresolved IssueInternal Tilt Mode in Prolate FRC (n=1)

• Plasma current ring tilts to align its magnetic moment with the external field.• Growth rate is the Alfven transit time.• Essentially always unstable in MHD.• Never conclusively identified in experiments.• FLR/non-linear saturation effects almost certainly important. (Belova, 2004)

MHD Internal Tilt, Field LinesR.D. Milroy, et al, Phys. Fluids B 1, 1225

Pressure Contours, Disruptive Internal Tilt Belova et al, Phys. Plasmas 2000

FRC Stability is an Unresolved Issue

• Plasma current ring tilts to align its magnetic moment with the external field.• Growth rate is the Alfven transit time.• Essentially always unstable in MHD.• Never conclusively identified in experiments• FLR/non-linear saturation effects almost certainly important. (Belova, 2004)

Oblate FRC: Internal TiltExternal Tilt (n=1)

MHD External Tilt Mode, Pressure ContoursHoriuchi & Sato, Phys. Fluids B 1, 581

• For E<1, tilt becomes an external mode• Can be stabilized by nearby conducting structures, or by very low elongation.• Radial shifting mode may become destabilized.• Observed in oblate FRC experiments, avoided with passive stabilizers.

Internal Tilt Mode in Prolate FRC (n=1)

FRC Stability is an Unresolved Issue

• Plasma current ring tilts to align its magnetic moment with the external field.• Growth rate is the Alfven transit time.• Essentially always unstable in MHD.• Never conclusively identified in experiments• FLR/non-linear saturation effects almost certainly important. (Belova, 2004)

Oblate FRC: Internal TiltExternal Tilt (n=1)

Co-Interchange Modes• n2 cousins of tilt/shift modes• For n, these are ballooning-like modes• Low n co-interchange modes (1<n<9)

computed to be destructive to oblate FRCs (Belova, 2001).

• Never experimentally studied.Pressure isosurface for n=2 axialco-interchange, calculated by HYM code

Internal Tilt Mode in Prolate FRC (n=1)

• For E<1, tilt becomes an external mode• Can be stabilized by nearby conducting structures, or by very low elongation.• Radial shifting mode may become destabilized.• Observed in oblate FRC experiments, avoided with passive stabilizers.

MRX is a Flexible Facility for Oblate

FRC Studies• Spheromak merging scheme for FRC formation.

• FRC shape control via flexible external field (EF) set.

•Describe EF by Mirror Ratio (MR)

• Extensive internal magnetic diagnostics.

• Passive stabilization via a conducting center column (sometimes).

• First experiments in spring 2005.

53904

52169

52191

51891

MR=2.0

MR=2.4

MR=3.0

MR=4.0

Comprehensive Diagnostics For Stability Studies

• 90 Channel Probe: 6x5 Array of Coil Triplets, 4cm Resolution, Scannable

• 105 Channel Toroidal Array: 7 Probes 5 coil triplets

Toroidal Mode Number n=0,1,2,3 in BZ, BR, BT

• Ti through Doppler Spectroscopy (He+1 @ 468.6nm)

• Copper Center Column for Passive Stabilization

•10cm radius, .5 cm thick, axial cut

FRC Formation By Spheromak Merging( a ) ( b )

( c ) ( d )

1: Plasma Breakdown 2: Spheromak Formation

3: Spheromak Merging 4: Final FRC

Technique developed at TS-3, utilized on TS-4 and SSX

Figure Courtesy of H. Ji.

Overview of MRX Results

Long-Lived FRC with Large Mirror Ratio & Passive Stabilizer

Movie of Shot 52259

Passive Stabilizer and Shape Control Extend the Plasma Lifetime

Poloidal Flux (Wb)

BR n=1 (T)

BR n=2 (T)

BlackTraces

Merging End Time

53904, MR=2.1

Passive Stabilizer and Shape Control Extend the Plasma Lifetime

Poloidal Flux (Wb)

BR n=1 (T)

BR n=2 (T)

BlackTraces

RedTraces

Merging End Time

53904, MR=2.1

52181, MR=2.5

Passive Stabilizer and Shape Control Extend the Plasma Lifetime

Poloidal Flux (Wb)

BR n=1 (T)

BR n=2 (T)

BlackTraces

RedTraces

BlueTraces

Large Non-Axisymmetries Before Merging Completion

53904, MR=2.1

52181, MR=2.5

52259, MR=3.4

Systematic Instability Studies Have Been Performed

• The instabilities have the characteristic of tilt/shift and

co-interchange modes.

• The center column reduces the n=1 tilt/shift amplitude.

• Co-interchange (n2) modes reduced by shaping

more than by the center column.

• Co-interchange modes can be as deadly as tilting.

Axial Polarized Mode Appears Strongly in BR

Z

Z

For n=1Tilt mode

Consider Tilting to Predict Magnetics Structure

n=2 Axial Co-Interchange

Axial Polarized Mode Appears Strongly in BR

BR,HYM

Z

Z

n=2 Axial Co-Interchange

52182

BR,Data

Pressure isosurfaces at p=0.7po

Calculated for MRX equilibria from HYM code

Radial Polarized Mode Appears Strongly in BZ

Z

Z

BZ Perturbation at Midplane

Pressure isosurfaces at p=0.7po

Calculated for MRX equilibria from HYM code

n=3 RadialCo-interchange

BZ,HYM

BZ,Data

n=1 Tilt Observed Without Center Conductor

• No center conductor• n=1 (tilt) dominates BR spectrum, especially for MR<2.5•1.8<MR<4.5 0.3<index<3

BR, n=1 BR, n=2 BR, n=3

No Center Conductor No Center ConductorNo Center Conductor

Helium

Center Column Reduces Tilt Signature

BR, n=1 BR, n=2 BR, n=3

• n=1 (tilt) reduced with center column• n=2&3 axial modes reduced at large mirror ratio

No Center Conductor No Center ConductorNo Center ConductorCenter ConductorCenter ConductorCenter Conductor

Helium

n=1 Shifting Signature Observed Without Center Conductor

BZ, n=1 BZ, n=2 BZ, n=3

Helium

• No center conductor• n=1 (shift) increases with mirror ratio

No Center Conductor No Center ConductorNo Center ConductorCenter ConductorCenter ConductorCenter Conductor

n=1 Shifting Signature Largely Suppressed with Center Column

BZ, n=1 BZ, n=2 BZ, n=3

• n=1 reduced by center column• n=2 & 3 not changed by the passive stabilizer

No Center Conductor No Center ConductorNo Center ConductorCenter ConductorCenter ConductorCenter Conductor

Helium

No Center Conductor No Center ConductorNo Center ConductorCenter ConductorCenter ConductorCenter Conductor

Lifetime is Strongly Correlated with BR Perturbations

With Center-Column

BR, n=1 BR, n=2Large Mirror Ratio

Large Mirror Ratio

Small Mirror RatioSmall Mirror Ratio

Helium

Lifetime is Strongly Correlated with BR Perturbations

With & Without Center-Column

Large Mirror Ratio

Large Mirror Ratio

Small Mirror Ratio

Small Mirror Ratio

BR, n=1 BR, n=2

Helium

Lifetime is Strongly Correlated with BR Perturbations

BZ, n=1 BZ, n=2

With & Without Center-Column

No Correlation with BZ Perturbation.

Large Mirror Ratio

Large Mirror Ratio

Small Mirror Ratio

Small Mirror Ratio

BR, n=1 BR, n=2

Helium

Lifetime is Strongly Correlated with BR Perturbations

With & Without Center-ColumnHelium

Large Mirror Ratio

Large Mirror Ratio

Small Mirror Ratio

Small Mirror Ratio

BR, n=1Helium

BR, n=2Helium

With & Without Center-ColumnNeon

BR, n=1Neon

BR, n=2Neon

Helium & Neon

Multiple Tools Used to Model Oblate FRCs

• MHD equilibria computed using new free-boundary Grad-Shafranov solver.

• Simple rigid-body model used to estimate rigid body shifting/tilting.

• MHD computations with the HYM code.• Calculation by E. Belova

Multiple Tools Used to Model Oblate FRCs

• MHD equilibria computed using new free-boundary Grad-Shafranov solver.

• Plasmas shape modifications due to external field are important to understand.

• Stability modeling requires knowledge of current profile, pressure profile, external field,…

• 90-channel probe scan lacks coverage area and axisymmetry.

• N-Probes lack information Z0.

• Equilibrium reconstruction can solve these problems.

• First ever application of this technique to an FRC plasma.

• Large non-axisymmetries present…calculate “nearby” equilibrium.

Modify forms of p’() and FF’(), and use

calculate from magnetics data.

Using p() and F(), calculate new

J=2Rp’+20FF’/R

Use new J & Coil Currents to

calculate new

Store as old

Compare to old

Converged

Didn’tConverge

Reevaluate P and F

with new

G-S Solver Loop

Find separatix flux (sep) using contour following

algorithm

If not Iteration 1:Compare 2 to 2

old.

Store 2 as 2old.

Didn’tConverge

Predict diagnostic signals based on

equilibria. Compute 2

Plotting and post-

processing.

Create input based on MRX data:1: 90 Channel Probe Scan2: n=0 Component of N-Probes3: Coil Currents

Create guesses to the distribution2 and p’() and FF’().

MRXFIT1 Solves G-S Eqn. Subject to Magnetic Constraints

1) J.K. Anderson et.al. Nuclear Fusion 44, 162 (2004)2) S.B. Zheng, A.J. Wooten, & E. R. Solano, Phys. Plasmas 3,1176 (1996)

Converged

Fields Calculated From Axisymmetric Model With Flux Conserving Vessel

*J.K. Anderson et.al. Nuclear Fusion 44, 162 (2004)

Equilibrium Field Coils

Vacuum Vessel is Treated as a Flux Conserver

Shaping Field Coils2 Turns Per Coil

Flux Core PF Windings4 Turns Per Coil

Windings of Future Ohmic Solenoid

Fields Calculated From Axisymmetric Model With Flux Conserving Vessel

*J.K. Anderson et.al. Nuclear Fusion 44, 162 (2004)

Equilibrium Field Coils

Vacuum Vessel is Treated as a Flux Conserver

Shaping Field Coils2 Turns Per Coil

Flux Core PF Windings4 Turns Per Coil

Windings of Future Ohmic Solenoid

Coils Only

Coils & Flux Conserver

Measurement

MRXFIT Code Finds MHD Equilibria Consistent with Magnetics Data

• Iterative free-boundary Grad-

Shafranov solver.• Flexible Plasma Boundary

• Center Column Limited• SF Coil Limited• X-points

• P’() & FF’() optimized for solution

matching measured magnetics.

• Equilibria interfaced to HYM stability

code.

€

p' ψ( ) = Ci

) ψ α i∑

€

FF' ψ( ) = CF

) ψ γF

Equilibrium Properties Respond to the External Field

€

R0 =Ro + R i( )

2

€

κ =Z ta ~ 2E

€

δ = R0 − R t( )a

€

a =Ro − R i( )

2

(-ZT,RT)

(ZT,RT)

(0,Ri)

(0,Ro)

Center Conductor

*S.B. Zheng, A.J. Wooten, & E. R. Solano, Phys. Plasmas 3,1176 (1996)

Definitions*:

Center Conductor

No Center Conductor

No Center Conductor

Pink region: Caution! Large Non-axisymmetries!

Rigid-Body Model Used to Estimate Tilt/Shift Stability

€

NX = θX πR2JφBZ 1− ndecay( )∫∫ dA ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

ndecay R,Z( ) =

−R

BZ

∂BZ

∂R−

Z

R2

∂

∂R2RBR + ZBZ( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

Current Profile : Jφ = Jφ(R,Z)

Equilibrium Field : BZ = BZ (R,Z)

BR = BR (R,Z)

⎧ ⎨ ⎩

H. Ji, et al, Phys. Plasmas 5, 3685 (1998)

Torque

Tilting Shifting

€

FX = −ξX πR2JφBZnshift∫∫ dA ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥€

nshift R,Z( ) = −R

BZ

∂BZ

∂R

Force

Tilting Stable: n>1Radial Shifting Stable: n<0

MRX Plasmas Transition to the Tilt Stable Regime

• Plasmas with MR>2.5 predicted to be in the tilt-stable regime.• Simple model for center-column m=1 eddy currents used.• Marginal comparison: Tilt often develops during merging phase.

€

T = πR2JφBZ 1− ndecay( )∫∫ dA

Rigid Body Shift Often Present, But May Be Benign

€

γshift

γ0

270s 280s 290s 350s340s330s320s310s300s

Rigid Body Shift Often Present, But May Be Benign

Midplane BZ contours at time of large shift

€

γshift

γ0

270s 280s 290s 350s340s330s320s310s300s

Rigid Body Shift Often Present, But May Be Benign

Hypothesis:Strong field at large radius prevents destructive shift

€

γshift

γ0

HYM Calculations Indicate Reduced Growth Rates at Larger Mirror Ratio

4 Configurations Considered So Far

Increasing mirror ratio

Increasing mirror ratio

€

γγ0

€

γγ0

Calculations By E. Belova

HYM Calculations Indicate Reduced Growth Rates at Larger Mirror Ratio

Increasing mirror ratio

Increasing mirror ratio

€

γγ0

€

γγ0

Consider MR=2.5 case

N γ/ γo, Measured

γ/ γo, HYM

2, axial 1 0.3

3, axial 1.4 0.7

N γ/ γo, Measured

γ/ γo, HYM

2, radial 1.2 0.65

3, radial 2 0.8

Consider MR=3.0 case

Local Mode Stability Improves At High Mirror Ratio

€

ω2 =δW

T

Ishida, Shibata, & SteinhauerRigid Displacement, n>>1 limit

J.R. Cary, Phys. Fluids 24, 2239 (1981)A. Ishida et al, Phys. Plasmas 3, 4278 (1996)

HYMHYM

Ishida Estimate

€

ω2 =−12

∂∂

dlκRB2∫

2π ρdl

B∫

Growth Rates for Local Modes Reduced at Higher Mirror Ratio.

Ishida Estimate

Radially Polarized n=4Axially Polarized n=4

€

γγ0

€

γγ0

Results Supportive of Proposed SPIRIT* Program

• Merging spheromaks for formation of oblate FRC. Process has been demonstrated in MRX.

• Shaping and passive conductors to stabilize n=1 modes. Demonstrated to work with a center column.– SPIRIT program calls for conducting shells.

• Transformer to increase B and heat the plasma.– Prototype transformers under construction using laboratory funds.

• Neutral beam to stabilize dangerous n2 modes.– Need for beam is clearly demonstrated, especially at lower elongation.

(*Self-organized Plasma with Induction, Reconnection, and Injection Techniques)

Conclusions• FRCs formed in MRX under a variety of conditions

• Large n=1 tilt/shift instabilities observed in MRX plasmas without passive stabilization.

• Shaping + passive stabilization substantially reduces n=1 modes, but destructive n2 axial modes often remain.

• A regime with small elongation demonstrates improved stability to n2 axial modes and extended lifetime.

• Equilibrium reconstruction technique has been demonstrated, illustrating FRC boundary control.

• The improved stability in the small elongation regime is confirmed by a combination of modeling techniques.

Derivation of Formula For Local Mode Growth Rate

€

T = Mn ξ2d3x∫∫∫ = 2π dψ∫ JdχMn

X2

R2B2+

Y2R2

n2+ B2Z2

⎛

⎝ ⎜

⎞

⎠ ⎟∫

€

rξ=ξ

re ψ +ξφ

r e φ +ξ χ

r e χ

€

X ψ,χ( ) = RBξψ

Y ψ,χ( ) =in

Rξφ

Z ψ,χ( ) =ξ χ

B

P1 P2

P1: I. B. Bernstein, E.A. Frieman, M.D. Kruskal, and R.M. Kulsrud, Proc. Royal Society A 244, 17 (1958)P2: J.R. Cary, Phys. Fluids 24, 2239 (1981)P3: A. Ishida, N. Shibata, and L.C. Steinhauer, Phys. Plasmas 3, 4278 (1996)P4: A. Ishida, N. Shibata, and L.C. Steinhauer, Phys. Plasmas 1, 4022 (1994)

€

X = rBcos θ − θ0( )

Z =1

Bsin θ − θ0( )

€

T'= 2π nMdl

B∫

P2 & P3

€

∂W = d3xr Q

2−

r J ⋅

r Q ×

r ξ − γp ∇ ⋅

r ξ ( )

2

+ ∇ ⋅r ξ ( )

r ξ ⋅∇p( )

⎧ ⎨ ⎩

⎫ ⎬ ⎭∫∫∫

€

W'< −1

2

∂

∂ψκRB2dl∫

P2

€

ω2 =W'

T'=

−1

2

∂

∂ψκRB2dl∫

2π nMdl

B∫

• First Written Explicitly in Ishida, Shibata, and Steinhauer, Phys. Plasmas 3, 4278 (1996)• Approximate agreement with Variational Analysis for Prolate FRCs in P4

Plasma ParametersD2 Helium Neon

Fill Pressure (mTorr, molecules/cm-3)

8-10, 3x1014 7-9.5, 2x1014 3.5-5, 1.3x1014

ne, Te 1x1014, 10 (1-2)x1014,10-14 (2-3)x1014, 10

BZ,Sep (Gauss) .03-.02 .03-.02 .03-.02

VA (m/s) 3-2x104 2-3x104 1x104

ZS (m)

A (s) 3-7 5-10 10-30

i,mfp (cm) 4 3 2

ωcii 1 1 0.5

3-1 1.5-1

E .6-.3 .6-.3

7-4 2-3

€

s

€

s /E

Plasma Lifetime Longest At Large Mirror Ratio

2 Trends• Lifetime increases with larger mirror ratio.• Center column does not substantially increase the lifetime.

Helium Neon

Condition For Kinetic Effects

€

γ=CVA

ZS

= CVA

ERS

< ω *

€

ω* =r k ⋅

r V D =

T

eBLp

k =kvth

2

2Lpω ci

Kinetic effects matter when:

The Diamagnetic drift Frequency is:

€

k =n

Rnull€

vth ~ VANote that for ~1:

The wavenumber is related to the Major radius as:

The separatrix radius is related to the null radius by:

€

R0 =1.4Rnull

Combine these as:

€

γ<ω*

CVA

ERS

<kvth

2

2L pω ci

C

1.4E<

nρ i

2L p

s

E<

1.4

2

⎛

⎝ ⎜

⎞

⎠ ⎟n

Neon Tilting Suppressed With Center Column

BR, n=1 BR, n=2 BR, n=3

Neon

No Center Conductor No Center ConductorNo Center Conductor Center Conductor Center Conductor Center Conductor

Center Column Reduces Rigid Body Shift Signature

BZ, n=1 BZ, n=2 BZ, n=3

Neon

No Center Conductor No Center ConductorNo Center Conductor

Center Conductor Center Conductor Center Conductor

Analytic Equilibrium Model by Zheng Provides Approximation to Current Profile

(-ZT,RT) (ZT,RT)

(0,Ri)

(0,Ro)

6 Fit parameters in Model:• 4 Parameters determine the Plasma shape• 2 Parameters determine Pressure and Toroidal field:

€

−2( )2 0

dp

dψ= A1

2πμ 0( )2F

dF

dψ= A2

Poloidal flux specified as:

Magnetic Field:

€

Ψ =c1 + c2 R2 + c3 (R4 − 4R2 Z2 )+ c4 R2 ln R( ) − Z2[ ]+

A1

8R4 −

A2

2Z2

€

B = F∇φ+1

2π∇ψ ×∇φ

Used to Generate Initial Equilibrium for MRXFIT

S.B. Zheng, A.J. Wooten, & E. R. Solano, Phys. Plasmas 3,1176 (1996)

Spheromak Tilt is Dominated by n=1

BR, n=1

BR, n=2

BR, n=3

Strong n=1 during Tilting Spheromak

Poor FRC with no Passive Stabilizer and Low Mirror Ratio

Movie of Shot 54154

Lifetime Not Longer with the Stabilizer at Low Mirror Ratio

Movie of Shot 52181