Equilibrium and Stability of Oblate FreeBoundary FRCs in MRX

1
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/s
gerhard/seminar.html Linked from PPPL Employee
Homepage
2
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
• n1 tilt/shift instabilities without passive
  stabilization
• n?2 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
3
FRCs have Potential Advantages as Fusion Reactors
FRC?toroidal plasma configuration, with toroidal
current, but minimal toroidal field.
4
FRCs have Potential Advantages as Fusion Reactors
FRC?toroidal 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
5
FRCs have Potential Advantages as Fusion Reactors
Problem Often Predicted to Be Quite MHD Unstable
FRC?toroidal plasma configuration, with toroidal
current, but minimal toroidal field.
Pressure Contours, Disruptive Internal Tilt
Belova et al, Phys. Plasmas 2000
H. Guo, Phys. Rev. Lett. 92, 245001
6
FRC Stability is an Unresolved Issue
Internal Tilt Mode in Prolate FRC (n1)

• 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)

Pressure Contours, Disruptive Internal Tilt
Belova et al, Phys. Plasmas 2000
MHD Internal Tilt, Field Lines R.D. Milroy, et
al, Phys. Fluids B 1, 1225
7
FRC Stability is an Unresolved Issue
Internal Tilt Mode in Prolate FRC (n1)

• 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 Tilt?External Tilt (n1)

• For Elt1, 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.

MHD External Tilt Mode, Pressure
Contours Horiuchi Sato, Phys. Fluids B 1, 581
8
FRC Stability is an Unresolved Issue
Internal Tilt Mode in Prolate FRC (n1)

• 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 Tilt?External Tilt (n1)

• For Elt1, 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.

Co-Interchange Modes

• n?2 cousins of tilt/shift modes
• For n??, these are ballooning-like modes
• Low n co-interchange modes (1ltnlt9) computed to be
  destructive to oblate FRCs (Belova, 2001).
• Never experimentally studied.

Pressure isosurface for n2 axial co-interchange,
calculated by HYM code
9
MRX is a Flexible Facility for Oblate FRC Studies
MR2.0

• 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
MR2.4
52169
MR3.0
52191
MR4.0
51891
10
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 n0,1,2,3 in BZ, BR, BT
• Ti through Doppler Spectroscopy (He1 _at_ 468.6nm)
• Copper Center Column for Passive Stabilization
• 10cm radius, .5 cm thick, axial cut

11
FRC Formation By Spheromak Merging
Figure Courtesy of H. Ji.
Technique developed at TS-3, utilized on TS-4 and
SSX
12
Overview of MRX Results
13
Long-Lived FRC with Large Mirror Ratio Passive
Stabilizer
Movie of Shot 52259
14
Passive Stabilizer and Shape Control Extend the
Plasma Lifetime
Black Traces
53904, MR2.1
Poloidal Flux (Wb)
BR n1 (T)
BR n2 (T)
Merging End Time
15
Passive Stabilizer and Shape Control Extend the
Plasma Lifetime
Black Traces
53904, MR2.1
Poloidal Flux (Wb)
Red Traces
BR n1 (T)
52181, MR2.5
BR n2 (T)
Merging End Time
16
Passive Stabilizer and Shape Control Extend the
Plasma Lifetime
Black Traces
53904, MR2.1
Poloidal Flux (Wb)
Red Traces
BR n1 (T)
52181, MR2.5
BR n2 (T)
Blue Traces
52259, MR3.4
Large Non-Axisymmetries Before Merging Completion
17
Systematic Instability Studies Have Been Performed

• The instabilities have the characteristic of
  tilt/shift and co-interchange modes.
• The center column reduces the n1 tilt/shift
  amplitude.
• Co-interchange (n?2) modes reduced by shaping
  more than by the center column.
• Co-interchange modes can be as deadly as tilting.

18
Axial Polarized Mode Appears Strongly in BR
n2 Axial Co-Interchange
Z
Consider Tilting to Predict Magnetics Structure
For n1?Tilt mode
19
Axial Polarized Mode Appears Strongly in BR
n2 Axial Co-Interchange
Z
BR, HYM
BR, Data
Pressure isosurfaces at p0.7po Calculated for
MRX equilibria from HYM code
52182
20
Radial Polarized Mode Appears Strongly in BZ
BZ Perturbation at Midplane
n3 Radial Co-interchange
BZ, HYM
BZ, Data
Pressure isosurfaces at p0.7po Calculated for
MRX equilibria from HYM code
21
n1 Tilt Observed Without Center Conductor
BR, n1
BR, n2
BR, n3
No Center Conductor
No Center Conductor
No Center Conductor

• No center conductor
• n1 (tilt) dominates BR spectrum, especially for
  MRlt2.5
• 1.8ltMRlt4.5 ? 0.3ltindexlt3

Helium
22
Center Column Reduces Tilt Signature
BR, n1
BR, n2
BR, n3
No Center Conductor
No Center Conductor
No Center Conductor
Center Conductor
Center Conductor
Center Conductor

• n1 (tilt) reduced with center column
• n23 axial modes reduced at large mirror ratio

Helium
23
n1 Shifting Signature Observed Without Center
Conductor
BZ, n1
BZ, n2
BZ, n3
No Center Conductor
No Center Conductor
No Center Conductor
Center Conductor
Center Conductor
Center Conductor

• No center conductor
• n1 (shift) increases with mirror ratio

Helium
24
n1 Shifting Signature Largely Suppressed with
Center Column
BZ, n1
BZ, n2
BZ, n3
No Center Conductor
No Center Conductor
No Center Conductor
No Center Conductor
No Center Conductor
No Center Conductor
Center Conductor
Center Conductor
Center Conductor
Center Conductor
Center Conductor
Center Conductor

• n1 reduced by center column
• n2 3 not changed by the passive stabilizer

Helium
25
Lifetime is Strongly Correlated with BR
Perturbations
Large Mirror Ratio
Large Mirror Ratio
BR, n1
BR, n2
With Center-Column
Small Mirror Ratio
Small Mirror Ratio
Helium
26
Lifetime is Strongly Correlated with BR
Perturbations
Large Mirror Ratio
Large Mirror Ratio
BR, n1
BR, n2
With Without Center-Column
Small Mirror Ratio
Small Mirror Ratio
Helium
27
Lifetime is Strongly Correlated with BR
Perturbations
Large Mirror Ratio
Large Mirror Ratio
BR, n1
BR, n2
With Without Center-Column
Small Mirror Ratio
Small Mirror Ratio
BZ, n2
BZ, n1
No Correlation with BZ Perturbation.
Helium
28
Lifetime is Strongly Correlated with BR
Perturbations
Large Mirror Ratio
Large Mirror Ratio
BR, n1 Helium
BR, n2 Helium
With Without Center-Column Helium
Small Mirror Ratio
Small Mirror Ratio
BR, n1 Neon
BR, n2 Neon
With Without Center-Column Neon
Helium Neon
29
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

30
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 Z??0.
• Equilibrium reconstruction can solve these
  problems.
• First ever application of this technique to an
  FRC plasma.
• Large non-axisymmetries presentcalculate
  nearby equilibrium.

31
MRXFIT1 Solves G-S Eqn. Subject to Magnetic
Constraints
Create guesses to the ? distribution2 and p(?)
and FF(?).
Create input based on MRX data 1 90 Channel
Probe Scan 2 n0 Component of N-Probes 3 Coil
Currents
Find separatix flux (?sep) using contour
following algorithm
Modify forms of p(?) and FF(?), and use ?
calculate from magnetics data.
Store ? as ?old
Reevaluate P and F with new ?
Using p(?) and F(?), calculate new
J?2?Rp2??0FF/R
Store ?2 as ?2old.
Didnt Converge
Didnt Converge
Use new J? Coil Currents to calculate new ?
Compare ? to ?old
Converged
G-S Solver Loop
If not Iteration 1 Compare ?2 to ?2old.
Plotting and post-processing.
Predict diagnostic signals based on equilibria.
Compute ?2
Converged
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)
32
Fields Calculated From Axisymmetric Model With
Flux Conserving Vessel
J.K. Anderson et.al. Nuclear Fusion 44, 162
(2004)
33
Fields Calculated From Axisymmetric Model With
Flux Conserving Vessel
J.K. Anderson et.al. Nuclear Fusion 44, 162
(2004)
34
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.

35
Equilibrium Properties Respond to the External
Field
Definitions
No Center Conductor
Center Conductor
No Center Conductor
Center Conductor
Pink region Caution! Large Non-axisymmetries!
S.B. Zheng, A.J. Wooten, E. R. Solano, Phys.
Plasmas 3,1176 (1996)
36
Rigid-Body Model Used to Estimate Tilt/Shift
Stability
Tilting
Shifting
Force
Torque
Tilting Stable ngt1 Radial Shifting Stable nlt0
H. Ji, et al, Phys. Plasmas 5, 3685 (1998)
37
MRX Plasmas Transition to the Tilt Stable Regime

• Plasmas with MRgt2.5 predicted to be in the
  tilt-stable regime.
• Simple model for center-column m1 eddy currents
  used.
• Marginal comparison Tilt often develops during
  merging phase.

38
Rigid Body Shift Often Present, But May Be Benign
270?s
340?s
280?s
290?s
350?s
330?s
320?s
310?s
300?s
39
Rigid Body Shift Often Present, But May Be Benign
Midplane BZ contours at time of large shift
270?s
340?s
280?s
290?s
350?s
330?s
320?s
310?s
300?s
40
Rigid Body Shift Often Present, But May Be Benign
Hypothesis Strong field at large radius prevents
destructive shift
41
HYM Calculations Indicate Reduced Growth Rates at
Larger Mirror Ratio
4 Configurations Considered So Far
Increasing mirror ratio
Increasing mirror ratio
Calculations By E. Belova
42
HYM Calculations Indicate Reduced Growth Rates at
Larger Mirror Ratio
Consider MR2.5 case
Increasing mirror ratio
Consider MR3.0 case
Increasing mirror ratio
43
Local Mode Stability Improves At High Mirror Ratio
Radially Polarized n4
Axially Polarized n4
HYM
Ishida, Shibata, Steinhauer Rigid Displacement,
ngtgt1 limit
HYM
Ishida Estimate
Ishida Estimate
Growth Rates for Local Modes Reduced at Higher
Mirror Ratio.
J.R. Cary, Phys. Fluids 24, 2239 (1981) A. Ishida
et al, Phys. Plasmas 3, 4278 (1996)
44
Results Supportive of Proposed SPIRIT Program
(Self-organized Plasma with Induction,
Reconnection, and Injection Techniques)

• Merging spheromaks for formation of oblate FRC.
• Process has been demonstrated in MRX.
• Shaping and passive conductors to stabilize n1
  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 n?2 modes.
• Need for beam is clearly demonstrated, especially
  at lower elongation.

45
Conclusions

• FRCs formed in MRX under a variety of conditions
• Large n1 tilt/shift instabilities observed in
  MRX plasmas without passive stabilization.
• Shaping passive stabilization substantially
  reduces n1 modes, but destructive n?2 axial
  modes often remain.
• A regime with small elongation demonstrates
  improved stability to n?2 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.

46
Derivation of Formula For Local Mode Growth Rate
P1
P2
P2 P3
P2

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

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)
47
Plasma Parameters
48
Plasma Lifetime Longest At Large Mirror Ratio
Helium
Neon
2 Trends

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

49
Condition For Kinetic Effects
Kinetic effects matter when
The Diamagnetic drift Frequency is
Note that for ?1
The wavenumber is related to the Major radius as
The separatrix radius is related to the null
radius by
Combine these as
50
Neon Tilting Suppressed With Center Column
BR, n1
BR, n3
BR, n2
No Center Conductor
No Center Conductor
No Center Conductor
Center Conductor
Center Conductor
Center Conductor
Neon
51
Center Column Reduces Rigid Body Shift Signature
BZ, n1
BZ, n2
BZ, n3
No Center Conductor
No Center Conductor
No Center Conductor
Center Conductor
Center Conductor
Center Conductor
Neon
52
Analytic Equilibrium Model by Zheng Provides
Approximation to Current Profile

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

Poloidal flux specified as
Magnetic Field
Used to Generate Initial Equilibrium for MRXFIT
S.B. Zheng, A.J. Wooten, E. R. Solano, Phys.
Plasmas 3,1176 (1996)
53
Spheromak Tilt is Dominated by n1
BR, n1
BR, n2
BR, n3
54
Strong n1 during Tilting Spheromak
55
Poor FRC with no Passive Stabilizer and Low
Mirror Ratio
Movie of Shot 54154
56
Lifetime Not Longer with the Stabilizer at Low
Mirror Ratio
Movie of Shot 52181