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

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 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

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,

FRCs have Potential Advantages as Fusion Reactors FRC  toroidal plasma configuration, with toroidal current, but minimal toroidal field. Problem: Bad curvature everywhere! H. Guo, Phys. Rev. Lett. 92, Disruptive Internal Tilt From 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, Psi Contours R.D. Milroy, et al, Phys. Fluids B 1, 1225 Disruptive Internal Tilt From 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 Tilt stabilized for very low elongation Radial shifting mode may become destabilized 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 modes Low n co-interchange modes (1<n<9) computed to be destructive to oblate FRCs (Belova 2001). For E<1, tilt becomes an external mode Can be stabilized by nearby conducting structures Tilt stabilized for very low elongation Radial shifting mode may become destabilized Pressure isosurface for n=2 co-interchange, calculated by HYM code Internal Tilt Mode in Prolate FRC (n=1)

MRX is a Flexible Facility for Oblate FRC Studies Spheromak Merging Scheme for FRC formation. FRC shape control via flexible external field set. Describe EF by Mirror Ratio Extensive Internal Magnetic Diagnostics. Passive stabilization with a conducting center column (sometimes). First experiments in Spring

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 B Z, B R, B T T i through Doppler Spectroscopy (He 468.6nm) Copper Center Column for Passive Stabilization 10cm radius,.5 cm thick, axial cut

FRC Formation By Spheromak Merging 1: Plasma Breakdown 2: Spheromak Formation 3: Spheromak Merging 4: Final FRC

Introduction to 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) B R n=1 (T) B R n=2 (T) Black Traces Merging End Time

Passive Stabilizer and Shape Control Determine the Plasma Lifetime Poloidal Flux (Wb) B R n=1 (T) B R n=2 (T) Black Traces Red Traces Merging End Time

Passive Stabilizer and Shape Control Determine the Plasma Lifetime Poloidal Flux (Wb) B R n=1 (T) B R n=2 (T) Black Traces Red Traces Blue Traces Large Non-Axisymmetries Before Merging Completion

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 B R Z Z For n=1  Tilt mode Consider Tilting to Predict Magnetics Structure n=2 Axial Co-Interchange

Axial Polarized Mode Appears Strongly in B R B R, HYM Z Z n=2 Axial Co-Interchange B R, Data Pressure isosurfaces at p=0.7p o Calculated for MRX equilibria from HYM code

Radial Polarized Mode Appears Strongly in B Z Z Z B Z Perturbation at Midplane Pressure isosurfaces at p=0.7p o Calculated for MRX equilibria from HYM code n=3 Radial Co-interchange B Z, HYM B Z, Data

n=1 Tilt Observed Without Center Conductor No center conductor n=1 (tilt) dominates B R spectrum, especially for MR<2.5 X X<index<X B R, n=1B R, n=2B R, n=3 No Center Conductor Helium

Center Column Reduces Tilt Signature B R, n=1B R, n=2B R, n=3 n=1 (tilt) reduced with center column n=2&3 axial modes reduced at large mirror ratio No Center Conductor Center Conductor Helium

n=1 Shifting Signature Observed Without Center Conductor B Z, n=1B Z, n=2B Z, n=3 Helium No center conductor n=1 (shift) increases with mirror ratio

n=1 Shifting Signature Largely Suppressed with Center Column B Z, n=1B Z, n=2B Z, n=3 n=1 reduced by center column n=2 & 3 not changed by the passive stabilizer No Center Conductor Center Conductor Helium

Lifetime is Strongly Correlated with B R Perturbations With Center- Column B R, n=1 B R, n=2 Large Mirror Ratio Small Mirror Ratio

Lifetime is Strongly Correlated with B R Perturbations With & Without Center-Column Large Mirror Ratio Small Mirror Ratio B R, n=1 B R, n=2 Helium

Lifetime is Strongly Correlated with B R Perturbations B Z, n=1 B Z, n=2 With & Without Center-Column No Correlation with BZ Perturbation. Large Mirror Ratio Small Mirror Ratio B R, n=1 B R, n=2 Helium

Lifetime is Strongly Correlated with B R Perturbations With & Without Center-Column Helium Large Mirror Ratio Small Mirror Ratio B R, n=1 Helium B R, n=2 Helium With & Without Center-Column Neon B R, n=1 Neon B R, 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.

Modify forms of p’(  ) and FF’(  ), and use  calculate from magnetics data. Using p(  ) and F(  ), calculate new J  =2  Rp’+2  0 FF’/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  2 old. Store  2 as  2 old. 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  distribution 2 and p’(  ) and FF’(  ). MRXFIT 1 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 Coils 2 Turns Per Coil Flux Core PF Windings 4 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 Coils 2 Turns Per Coil Flux Core PF Windings 4 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.

Equilibrium Properties Respond to the External Field (-Z T,R T ) (Z T,R T ) (0,R i ) (0,R o ) No Center Conductor Center Conductor *S.B. Zheng, A.J. Wooten, & E. R. Solano, Phys. Plasmas 3,1176 (1996) Definitions*: No Center Conductor Center Conductor

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 mirror ratio>2.5 in the tilt- stable regime. Plasmas are never strongly tilt stable. Simple model for center-column m=1 eddy currents used.

MRX Plasmas Transition to the Tilt Stable Regime Correlation between positive net torque and tilting Plasmas with mirror ratio>2.5 in the tilt- stable regime. Plasmas are never strongly tilt stable. Simple model for center-column m=1 eddy currents used.

Rigid Body Shift Often Present, But May Not Be So Deadly Midplane B Z contours at time of large shift 270  s 280  s290  s350  s 340  s 330  s320  s310  s300  s

Rigid Body Shift Often Present, But May Not Be a Deadly 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

HYM Calculations Indicate Reduced Growth Rates at Larger Mirror Ratio Increasing mirror ratio Consider MR=2.5 case N  /  o, Measured  /  o, HYM 2, axial10.3 3, axial N  /  o, Measured  /  o, HYM 2, radial , radial20.8 Consider MR=3.0 case

Local Mode Stability Improves At Low Elongation Ishida, Shibata, & Steinhauer J.R. Cary, Phys. Fluids 24, 2239 (1981) A. Ishida et al, Phys. Plasmas 3, 4278 (1996) HYM Ishida Approximation Growth Rates for Local Modes Reduced at Higher Mirror Ratio.

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>1 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 modes often remain. A regime with small elongation demonstrates improved stability to n>1 modes and extended lifetime. Equilibirium reconstruction technique has been demonstrated, illustrating FRC boundary control. The improved stability in the large elongation regimes 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 Lifetime Longest At Large Mirror Ratio 2 Trends Lifetime increases with larger mirror ratio. Center column does not 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:

Neon Tilting Suppressed With Center Column B R, n=1 B R, n=2 B R, n=3 Neon No Center Conductor Center Conductor

Center Column Reduces Rigid Body Shift Signature B Z, n=1 B Z, n=2B Z, n=3 Neon No Center Conductor Center Conductor

Analytic Equilibrium Model by Zheng Provides Approximation to Current Profile (-Z T,R T )(Z T,R T ) (0,R i ) (0,R o ) 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 B R, n=1 B R, n=2 B R, 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