# Presentation: 567457

## 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.html](http://w3.pppl.gov/~sgerhard/seminar.html)_
- Linked 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
- 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

## FRCs have Potential Advantages as Fusion Reactors

- FRCtoroidal plasma configuration, with toroidal current, but minimal toroidal field.

## 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

## FRCs have Potential Advantages as Fusion Reactors

- FRCtoroidal 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 Issue

**Internal 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 Lines
- R.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 Tilt********External Tilt (n=1)**

- MHD External Tilt Mode, Pressure Contours
- Horiuchi & 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 Tilt********External Tilt (n=1)**

**Co-Interchange Modes**

- n2 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 axial
- co-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

- 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)

**Black**
**Traces**

- 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)

**Black**
**Traces**

**Red**
**Traces**

- 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)

**Black**
**Traces**

**Red**
**Traces**

**Blue**
**Traces**

- 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 (n2) 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

- For n=1Tilt mode

- Consider Tilting to Predict Magnetics Structure

- n=2 Axial
- Co-Interchange

## Axial Polarized Mode Appears Strongly in BR

- BR,
- HYM

- 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

- BZ Perturbation at Midplane

- Pressure isosurfaces at p=0.7po
- Calculated for MRX equilibria from HYM code

- n=3 Radial
- Co-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 Conductor

- No 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 Conductor

- No Center Conductor

- Center Conductor

- Center Conductor

- Center 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 Conductor

- No Center Conductor

- Center Conductor

- Center Conductor

- Center 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 Conductor

- No Center Conductor

- Center Conductor

- Center Conductor

- Center Conductor

- Helium

- No Center Conductor

- No Center Conductor

- No Center Conductor

- Center Conductor

- Center Conductor

- Center Conductor

## Lifetime is Strongly Correlated with BR Perturbations

- Lifetime is Strongly Correlated with BR Perturbations

- With Center-Column

- BR, n=1

- BR, n=2

- Large Mirror Ratio

- Large Mirror Ratio

- Small Mirror Ratio

- Small 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

- 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-Column
**Helium**

- Large Mirror Ratio

- Large Mirror Ratio

- Small Mirror Ratio

- Small Mirror Ratio

- BR, n=1
- Helium

- BR, n=2
- Helium

- With & Without Center-Column
**Neon**

- BR, n=1
- Neon

- BR, n=2
- Neon

- 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 Z0.
- Equilibrium reconstruction can solve these problems.
- First ever application of this technique to an FRC plasma.
- Large non-axisymmetries present...calculate “nearby” equilibrium.

## MRXFIT1 Solves G-S Eqn. Subject to Magnetic Constraints

- Modify forms of p’() and FF’(), and use  calculate from magnetics data.

- Using p() and F(), calculate new
-  J=2Rp’+20FF’/R

- Use new J & Coil Currents to calculate new 

- Store  as old

- Compare  to old

- Converged

- Didn’t
- Converge

- 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 2old.

- Store 2 as 2old.

- Didn’t
- Converge

- Predict diagnostic signals based on equilibria. Compute 2

- Plotting and post-processing.

- Create input based on MRX data:
**1: 90 Channel Probe Scan**
**2: n=0 Component of N-Probes**
**3: 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)

## Fields Calculated From Axisymmetric Model With Flux Conserving Vessel

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

## 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.

## Equilibrium Properties Respond to the External Field

- 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

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

- Torque

**Tilting**

**Shifting**

- Force

- Tilting Stable: n>1
- Radial 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.

## Rigid Body Shift Often Present, But May Be Benign

- 270s

- 280s

- 290s

- 350s

- 340s

- 330s

- 320s

- 310s

- 300s

## Rigid Body Shift Often Present, But May Be Benign

- Midplane BZ contours at time of large shift

- 270s

- 280s

- 290s

- 350s

- 340s

- 330s

- 320s

- 310s

- 300s

## Rigid Body Shift Often Present, But May Be Benign

- Hypothesis:
- Strong field at large radius prevents *destructive* shift

## 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

## HYM Calculations Indicate Reduced Growth Rates at Larger Mirror Ratio

- Increasing mirror ratio

- Increasing mirror ratio

- Consider MR=2.5 case

| N | / o, Measured | / o, HYM |
| --- | --- | --- |
| 2, axial | 1 | 0.3 |
| 3, axial | 1.4 | 0.7 |

| N | / o, Measured | / o, 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

- Ishida, Shibata, & Steinhauer
- Rigid Displacement, n>>1 limit

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

- HYM

- HYM

- Ishida Estimate

- Growth Rates for Local Modes Reduced at Higher Mirror Ratio.

- Ishida Estimate

- Radially Polarized n=4

- Axially Polarized n=4

## 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 n2 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 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.

## Derivation of Formula For Local Mode Growth Rate

- 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)

- 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

## Plasma Parameters

|  | D2 | 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 |
| cii | 1 | 1 | 0.5 |
|  |  | 3-1 | 1.5-1 |
| E |  | .6-.3 | .6-.3 |
|  |  | 7-4 | 2-3 |

## 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

- 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:

## Neon Tilting Suppressed With Center Column

- BR, n=1

- BR, n=2

- BR, n=3

- Neon

- No Center Conductor

- No Center Conductor

- No 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 Conductor

- No Center Conductor

-     Center Conductor

-     Center Conductor

-     Center Conductor

## 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)

## 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