Plasma Stability in Alternate Confinement Concepts

E. Bickford Hooper

Lawrence Livermore National Laboratory Livermore, CA 94526

Global Climate & Energy Project Workshop Princeton May 1-2, 2006

Work performed under the auspices of the U. S. Department of Energy by University of California Lawrence Livermore National Laboratory under UCRL-PRES-220891 contract No. W-7405-Eng-48. I am pleased to thank many colleagues, including:

Per Brunsell Rick Ellis Adil Hassam Dan Den Hartog Alan Hoffman Jay Kesner Harry McLean Dick Post Dmitri Ryutov John Sarff Uri Shumlak

Any misinterpretations or errors are my responsibility A wide range of Innovative Confinement Concepts (“ICCs”) contribute to the physics of plasmas and fusion-energy

• Macroinstabilities and microinstabilities may be present – Macroinstabilities are large scale and usually described by fluid models, e.g. Magnetohydrodynamics (MHD) – Microinstabilities are fine scale (typically with wavelengths comparable to the ion Larmor radius) and usually require kinetic descriptions

This presentation focuses on macroinstabilities in ICCs

• These studies of stability complement those in tokamaks and stellarators • Control of instabilities will be essential for any ICC reactor Magnetic confinement devices have either toroidal or open magnetic field lines with the plasma weakly or highly constrained

Toroidal Self-Organized Highly Constrained Reversed-Field Pinch (RFP) Tokamak* Spheromak Spherical Torus (ST)* Field-Reversed Configuration Stellarator* (FRC) Levitated Dipole

*The stability of the tokamak, ST, and stellarator are not discussed here Open Weakly Constrained Highly Constrained Z-pinch Tandem Mirror Centrifugal Confinement Gas Dynamic Trap TOROIDAL SYSTEMS RFP –– Low toroidal field makes safety factor q << 1

m=1, n 6 ~ 0.2 resonances c

o

n

1,6 d

u q(r) c 1,7 t

i n

1,8 g shell m=0, all n

0 minor radius, r a

Susceptibility to m=1 tearing is the primary MHD stability issue.

MST:

R = 1.5 m, a = 0.5 m, Ip ≤ 0.55 MA J. Sarff, seminar at MIT (Feb. 4, 2005) Standard RFP operation: A spectrum of tearing modes develops through instability and nonlinear coupling.

Nonlinear coupling between the modes The spectrum spans a wide range leads to “sawteeth” –– dynamo events in of toroidal mode numbers (n) which poloidal flux injected by the ohmic transformer is converted into toroidal flux inner-most resonant m = 1, n = 6 B ÷ / B~1% 1% ÷ B φ (a)/B

0 0102030 Toroidal Mode, n Electron heat transport in standard RFP agrees well with stochastic magnetic expectations.

Field line “puncture plot” Fluctuations generate stochasticity Field line tracing:

– use measured equilibrium B(r) φ ~ – fluctuation B(r) from nonlinear MHD computation using measured η(r) and Lundquist number (S ≈ 106) Toroidal, ~ ~ – normalize B(r) to measured B(a) (< 2X correction required) directly from field line tracing χR-R

χe measured (m2/s) power-balance 2 magnetic Dm =〈∆r 〉/∆L diffusivity

predicted “Rechester-Rosenbluth”

χR−R = vTe Dm r/a Pulsed Poloidal (inductive) Current Drive (PPCD) targeted to outer-plasma region reduces MHD tearing instability.

The poloidal current (and toroidal field) are transiently reduced 0 Standard Bφ(a) Field line puncture calculations (T) –0.04 PPCD show large areas of good surfaces –0.08 1.0 ~ Bθ rms (%) 0.5

0 1021027066 0102030 1000 Time (ms)

The peak electron temperature power balance increases significantly 100 1.0 (m2/s) PPCD-Improved 0.8 10

Te 0.6 χR-R (KeV) 1 0.4 r/a 0.2 Standard

0 0.2 0.4 0.6 0.8 1 r/a On the resistive-time of the conducting shell, external kink modes become resistive-wall modes Feedback control has been demonstrated on EXTRAP-T2R RFP: Summary

• m = 1 resistive tearing modes develop at mode-rational surfaces, q=1/n – These modes grow to sufficiently large amplitudes that their islands overlap and the magnetic field becomes stochastic – The electron thermal conductivity is large, the electron

temperature profile flat and peak Te low • PPCD changes the current and electric field profiles – Mode amplitudes are significantly reduced and good flux surfaces are calculated in much of the volume

– The electron-temperature profile becomes peaked and Te is increased by ~ 3 • On the resistive time of the wall, feedback stabilization reduces the RWM amplitudes significantly – The discharge duration is lengthened by a factor of 3 MHD stability in the gun-injected spheromak

SSPX (Sustained Spheromak Physics Experiment) –– a coaxial helicity-injected confinement experiment

• A large current is driven from the inner electrode to the flux conserver – Following formation, the current flows through the “donut-hole”, forming a column which pinches as shown

• The spheromak lies inside the separatrix, shown in red – Good energy confinement is found when magnetic surfaces are closed – Closed surfaces require low magnetic fluctuation levels

1 m MHD stability in the gun-injected spheromak

Experiment: low fluctuations Magnetic fluctuations occur due to MHD with low edge λ, no low- modes: order rational surfaces • On the column (outside the separatrix) where the current profile is similar to a z- pinch – The n=1 column mode drives current in the spheromak – The column mode is stabilized for

λ gun ≤ αλ fc where 1 < α < 1.5 and ∇× B = λ fcB • In the spheromak (inside the separatrix) – Internal modes occur on low-order rational surfaces, q = m/n – Generally, 0.5 ≤ q ≤ 1 – Experimentally, best stability occurs when the q-profile lies between 1/2– 2/3

λ = µ0 j/B The n=1 column mode reaches large amplitude (δB/B~10%). Nonlinear processes drive magnetic reconnection events which converts injected toroidal flux into poloidal flux

A strong n=1 mode develops Modeling and experiment show a consistent picture of during spheromak buildup and the physics processes during spheromak formation and sustainment at high gun current sustainment

The n=1 column mode The poloidal magnetic field and dominates the spectrum flux increase with each event

The reconnection events generate voltage spikes on the Te(experiment) is low during strong n=1 activity – gun, seen both in experiment – modeling (above) shows the magnetic surfaces and in resistive MHD simulations opening in each event, dropping Te E. B. Hooper, et al., Phys. Plasmas 12, 092503 (2005). Internal modes in the experiment are found to occur when the q-profile crosses low-order rational surfaces

The q-profile is sensitive to the ratio of gun current to gun flux Magnetic fluctuations correlate with the reconstructed q-profile Shown is the observed spectrum together with the maximum and Safety-factor minimum in the q-profile scaling with

λedge = µ0Igun/Ψgun

Good energy confinement is found when the q- profile has no low- order rational surfaces

H. S. McLean, Phys. Plasmas (to be published). Spheromak: Summary

• The n=1 column mode drives current via a dynamo – Injected toroidal flux is converted into poloidal flux by a reconnection event – The reconnection event opens magnetic flux surfaces allowing a large thermal conductivity to the walls This large heat leak will require separation of the current drive phase from a reactor burn phase –– a pulsed or refluxed spheromak is probably required

• Internal modes amplitudes are small when the q-profile does not span the 1/2 or 2/3 surface – Simulations find a similar effect, with poor confinement resulting when magnetic fluctuations generate islands or stochastic field lines Good energy confinement in a reactor may require current-profile control to shape the q-profile and maintain mode amplitudes < 1% FRC –– Macrostability stability

Stability depends on the geometry r r Be c s [prolate (shown) or oblate], B Bo i external conducting wall, external magnetic mirror ratio, and other features

s • The ideal FRC has no current along B and thus has no current-driven, MHD modes (pressure-driven only) • Local ideal modes (n>>1) – interchange, co-interchange (ballooning) are predicted to be unstable but usually not observed – Stabilized by conducting shell, external magnetic mirror, etc. • Global ideal modes in absence of rotation (n=0, 1, >1) – axial shift, sideway shift, tilt may be theoretically unstable but have not been observed • Global ideal modes driven by rotation have been observed • Resistive tearing modes have been observed during formation

Refs.: M. Tuszewski, Nucl. Fusion 28, 2033 (1988); H. Ji et al., Phys. Plasmas 5, 3685 (1998). FRC –– Dominant global instability is usually the n=2, rotating interchange mode

• n=2 rotating interchange – Driven by centrifugal force due to plasma rotation. – Observed experimentally Side View in most FRCs End View

40

35

• Usually stabilized by external static multipole fields in θ- 30 (cm) r pinch formed FRCs. 25

20 • Instability not seen in translated FRCs due to 1.5 0.05-m gap 0.35-m gap (#13709) (#13863) ) development of moderate toroidal field and high shear.* -2 1.0 m • Instability is stabilized by Rotating Magnetic Fields 19 0.5 (RMF).** 0 hg2004.sta.2 0 1.0 2.0 3.0 • Finite Larmor-radius effects are usually stabilizing.*** Time (ms)

*H. Guo, et al., Phys. Rev. Letters 95, 175001 (2005). Use of separated RMF antennas, which produce same ** H. Guo, et al., PRL 94, 185001 (2005). ion spin-up, to illustrate ***E. Belova, et al., Phys. Plasmas 11, 2361 (2003). instability development when central RMF is not present RMF Can Stabilize Interchange Instabilities

• The radial inward force 1.0 produced by a partially penetrated dipole RMF with an elliptical distortion ξ is 0.8 Wall ) 0

given by, µ 2 / 2 e 0.6 B /( r Fr (r,θ) = − jz Bθ ⎛ r −r ⎞ 0.4 2 −2⎜ s ⎟ 2B r ⎜ δ* ⎟ ⎧ ξ⎫ = − ω s e ⎝ ⎠ ⎨1+ ⎬ µoδ* r ⎩ r ⎭ 0.2 hg2004.sta.6a 0 90 180 270 360

The condition for interchange stability is that the radial force response to 2 the perturbed ‘interchange’, Fr1, exceed the centrifugal term ρΩ , which 2 2 2 amounts to requiring Bω /µo > 1.3〈ρ〉Ω rs . (From A. L. Hoffman, presentation at ICC2006, http://icc2006.ph.utexas.edu/index.php) FRC Summary

• FRCs are potentially subject to ideal and resistive MHD modes • Experiments are usually more stable than theoretical predictions • Physics not included in most theories may explain this – Quadrupole fields – Kinetic effects, including FLR – Effects of the edge plasma, including line-tying – Profiles (beta, etc.) influence stability and may evolve to stable configurations • Techniques have been developed to stabilize residual modes, e.g. at U. Washington, in the MRX experiment at Princeton and in computational work by E. Belova – Conducting walls – External magnetic mirror ratio – Sheared velocity flows – Energetic ions from neutral beam injection, included in Belova’s models, can be stabilizing Levitated Dipole (LDX) –– MIT/Columbia

• Superconducting dipole

magnet, Ic>1 MA • 5 m diameter vacuum vessel • Dipole presently supported by three thin spokes; floating ring planned • Initial plasma studies use ECRH at two frequencies to generate a high β plasma

(Ee=50 keV) MHD stability of a dipole derives from plasma compressibility

Experiment

At low density the plasma is strongly unstable to an interchange mode

• A gas puff raises the background density – The plasma becomes quiescent and the β increases (Peak ~ 20%) –The instability creates Note that the Hot Electron Interchange hysteresis in β as the (HEI) stability is enhanced by ion gas fueling is increased diamagnetic currents (and thus high and decreased background density) Levitated dipole –– Summary

Experiments with a supported ring verify the predicted role of plasma compressibility in the stability of the hot-electron mode

In future levitated, high density experiments thermal species will dominate

both β and ne • Experiments are planned to study the properties of the predicted modes

• The background MHD mode is predicted to generate convective cells which transport particles but not energy, and thus can remove fusion ash: OPEN SYSTEMS Z-pinch imbedded in a flowing plasma

Theory predicts that a Z-pinch imbedded Simulations confirm the stabilizing in a sheared axial flow can be stabilized impact of flow

Sheared flow Static pinch vz’ = 0.2 kvA

Ref.: U. Shumlak and C. W. Hartman, Phys. Rev. Lett. 75, 3285 (1995) T. D. Arber and D. F. Howell, Phys. Plasmas 3, 554 (1996) U. Shumlak, Phys. Plasmas 10, 1683 (2003) and presentation at ICC2006, http://icc2006.ph.utexas.edu/index.php The ZAP experiment finds low fluctuations in the presence of sheared axial flow

The pinch is formed by a gun discharge –– inertia and gun currents maintain the flow until the accelerator plasma empties

Measured magnetic fluctuation levels The external plasma flow shear is high at the beginning of the quiescent period (t ≈ 42 ms) • The plasma has low fluctuations for ~ 2000 x the mode growth time for a static pinch At the end of the quiescent phase the velocity profile is measured to be uniform Z-pinch imbedded in sheared plasma flow –– Summary

• Experiments and theory show that a sheared axial plasma flow is stabilizing for the MHD sausage and kink modes

• Detailed measurements have been made on the ZaP experiment at U. Washington – The results strongly suggest that the sheared flow is responsible for the relatively quiescent period in the pinch – The experimentally-measured profiles of density, magnetic field, etc. provide guidance for future experiments

Experiments continue with the purpose of better understanding the stabilization mechanism, obtaining a hotter plasma, etc. Rotating mirror –– centrifugal confinement

MCX

BACKGROUND • MCX is a simple mirror with plasma rotating azimuthally • This provides centrifugal confinement but the system is expected to be flute unstable • However, velocity shear should suppress the interchanges if the speed is supersonic (assuming parabolic type profiles) • V’ shear stabilization has been confirmed by analysis and 3D MHD numerical simulation

EXPERIMENT • MCX rotates supersonically with Mach numbers 1-3 • MCX discharges are relatively steady over 1000’s of instability growth times • Velocity profile has been measured, it is parabolic type, and the velocity shear exceeds the growth rate by up to x 5. Velocity shear is large enough for mode stabilization

• Not rigid rotor rotation

C3+ • Similar profiles for different C lines Ù C2+ E×B drift is the dominant motion. C+ H

R int R out

• Velocity shear from C+ Doppler shifts shows stability threshold is exceeded! during HR mode

during O mode

1/ 2 RΩ' > γ ln( R1/ 3 ) ⇔ stable int []µ Voltage across plasma remains steady for 1000’s of MHD instability times

Experiment:

• MHD instability growth time τMHD ~ 2 - 20µs τmom τMHD • Measured momentum confinement time

τmom ~200µs

ms • No “major disruptions” => MHD Stable?

Simulation:

• First picture: Rotating mirror – no velocity shear – Interchange unstable • Velocity shear turned on and the plasma evolves – At equilibrium the interchange mode is stabilized

Ref.: R. F. Ellis, et al., Phys. Plasmas 8, 2057 (2001); Y.-M. Huang and A. B. Hassam, Phys. Rev. Lett. 87, 23500021 (2001). Centrifugal-mirror confinement –– Summary

Simulation demonstrates that rotational velocity shear can stabilize the interchange mode

The experiment “finds” an operating mode with high voltage and improved confinement which correlates with the measured velocity shear

• The shear is calculated to be sufficient for stability

• In agreement with this result, the measured momentum confinement time is found to be ~5 times the instability growth rate in the absence of velocity shear Gas Dynamic Trap (GDT)

Plasma In the basic GDT an axisymmetric mirror is

absorber stabilized by the outflow of a collisional plasma End tank through the good-curvature region of the mirrors Mirror Bwall Confinement region throat The Rosenbluth-Longmire stability integral is: Bo a(z) Bmir d 2a a3 p + p + ρv 2 dz > 0 2 ()|| ⊥ ∫ dz

End-tank field line curvature The MHD-stabilizing effect of flow in good curvature stabilizes the plasma was demonstrated in a cold, gun-injected plasma on the Novosibirsk GDT • The plasma was stable when the field lines in the end-tank had sufficient good curvature. • The plasma was unstable when: – the field lines in the end tank were straight – the mirror ratio became large so there was insufficient outflow of plasma – the central cell curvature was increased

Ref. I. I. Ivanov, et al., Phys. Plasmas 1, 1529 (1994). GDT –– Neutral-beam injected plasma (Possible neutron source for fusion testing)

A hot, low-collisionallity plasma is formed in the Novosibirsk GDT by neutral beam injection B(midplane) = 0.22T B(mirrors) = 2.5-16T

Ebeam = 15-17 keV n = 0.1-2x1020 m–3

Te = 45-90 eV

βmax = 0.4 (hot ions at turning points)

Several stabilizing mechanisms were identified: • The hot-ion Larmor radius stabilized all modes with m > 1 • Stability for m = 1 resulted from: – Good field line curvature in end cells, including a cusp field – Non-paraxial effects – Fast-ion axial profile Measured stability limit in reasonable agreement with the energy principle Ref.: A. V. Anikeev, et al., Phys. Plasmas 4, 347 (1997). GDT –– Summary

• Outflow of plasma in the exit fan of the axisymmetric mirror is stabilizing due to good magnetic curvature • A hot-ion, neutral-beam injected GDT is stabilized by a mixture of good curvature and other effects – Also, the energetic ions were found to have the spatial distribution predicted from electron drag, with no indication of beam-driven instabilities This version of the GDT is a neutron source candidate for testing fusion materials and components • The stability measurements are encouraging for a GDT operating in the collisional regime – The stability limit with plasma outflow was in poor quantitative agreement with theory, probably due to the high resistivity of the cold plasma –– Further experiments are needed – Good energy balance in an open magnetic trap would require a long (>km) central cell –– The GDT is a candidate for a large, central power or hydrogen-producing plant – Proposed mechanisms to reduce electron energy flow to the end wall might reduce the required length for a GDT reactor An axisymmetric tandem mirror has been proposed –– injected energetic ions reflect in the good curvature outside the last mirror –– The Kinetic Stabilizer

Injected beams In the “conventional” tandem mirror, quadrupole magnetic fields provide stability but generate cross-field ion transport and other problems. The kinetic stabilizer has been proposed to overcome these problems Energetic ions +++ compress to a high Density ++++ ++++++ density in the good curvature region, Position providing MHD stability to the entire tandem mirror R. F. Post, Trans. of Fusion Technology, 39, 25, (2001); presentation at ICC2006, http://icc2006.ph.utexas.edu/index.php The kinetic-stabilized tandem mirror is predicted to be MHD stable

The Rosenbluth-Longmire “Expander” integral and a MHD stability code have been evaluated Central mirror cell (end) using models for the [fusion-plasma region] stabilizer injectors and plasmas in an plug and central cell When the field curvature throughout the tandem Kinetic Stabilizer mirror is optimized, the energetic- calculations predict MHD ion stability injectors

“Plug” mirror cell

Summary: Modeling is promising for stabilizing axisymmetric tandem mirrors. Experiments are needed to verify this result CONCLUSIONS

Many of the ICC experiments are successfully addressing macro (MHD) stability

This progress is a result (in large measure) of a good understanding of the driving forces of the modes

Computational simulations have contributed in the interpretation and guidance of experiments

The successful control of MHD stability is a necessary (but not sufficient) requirement for a reactor. The ICC experiments are highly promising in this regard