The relation between energetic particles and the radial electric field in stellarators

Ethan Green, Wataru Hayashi, Xishuo Wei, Zhihong Lin, Hiroyuki Yamaguchi, Masaki Osakabe, Hideo Nuga, Kunihiro Ogawa, Ryosuke Seki, Mitsutaka Isobe, Yasuko Kawamoto, Akihiro Shimuzu, Takeishi Ido, and Masaki Nishiura

Email: ethanmg@uci.edu

1

Background and Motivation

Fusion reactions produce energetic particles (EP) which must be confined long enough to transfer their energy to the thermal plasma population, which is a particular challenge for stellarators.

Er may be used to shed fusion byproducts.

The radial Electric field (Er) can affect the EP population and the EP population can in turn affect Er. My goal is to understand and optimize stellarator design to make use of this synergy.

​

2

The radial electric field (Er)

• Non-ambipolar (different) radial diffusion rates lead to a radial charge gradient
• Electron root or ion root named by whichever species diffuses more
• As such electron root results in a positive Er and ion root a negative Er
• This talk will focus on the Electron root

​

3

Thesis Plan

1. Study the effects of Er on energetic particles (EP) [Green submitted to NF]
2. Study the effects of EP on Er
1. Can we inject NBI to particular locations in real and phase space in order to generate a desired Er
2. Will fast Alphas have any effect on Er
3. Study the cumulative effect of EP and Er on thermal confinement
4. Optimize stellarator design to make use of NBI control of Er

4

4

Outline:

1. Energetic particle transport under the effects of the radial electric field in LHD
1. Experiment Parameters
2. Simulation Parameters
3. Effects of constant radial electric field (Er) on orbit topologies
4. Effects of realistic experimental Er profile
1. Trapped and barely trapped Particles
2. Overall effects on particle distribution
5. Summary
2. Experiment
3. Future plans

5

LHD Experiment Parameters

• All experimental data and parameters come from shot #183281
• R_ax= 3.75 m
• B_ax = 2.64 T
• Clockwise configurations
• HIBP is used to measure electrostatic potential
• EP sourced by NBI 3 and 4
• ECH increases electron temperature enforcing an electron root

6

The Gyrokinetic Toroidal Code (GTC)

GTC is normally used to study low frequency micro turbulence, mesoscale AE and macroscopic MHD modes in the core of magnetic confinement devices

• Tokamak, Stellarators, FRC
• PIC code using gyrokinetic or fully kinetic equations to update particle positions in Boozer Coordinates and Cylindrical coordinates
• Cross scale turbulence
• Delta-f and full-f capabilities
• Fokker-Planck Collisions
• Electrons can be treated efficiently under fluid or drift kinetic models

7

GTC Simulation Parameters

• Magnetic equilibrium from Variational Moments Equilibrium Code (VMEC) run of shot #183281
• While GTC has electromagnetic and collisional capabilities, we chose to run these simulations electrostatic and collisionless to understand the basic effects Er on EP particle orbits
• Particles are considered lost when they reach the separatrix
• Particles are loaded uniformly and isotropically with kinetic energies between 20 and 60 keV (to match PNBI on LHD)
• Er is imposed on the simulations
• First we will examine the effects of a constant Er of -5, 0, 5 kV/m
• Then we will look at the effects of Er based on HIBP data of shot #183281

8

Effects of constant radial electric field

9

Trajectories of Passing Particles under different Er

• Little effect from Er
• Low pitch λ = 𝛍B₀/E

Er values in kV/m

a) Poloidal flux function 𝛙 normalized to separatrix as a function of time

b) Poloidal angle 𝚹 as a function of time.

c) Projection of particle trajectory in real space on plane moving toroidally with the particle

d) Particle trajectory on plane rotating with the minimum B field.

Starting point of the particle in c and d marked with a black dot

𝝉₀ is the toroidal transit time of a 20 keV passing particle

10

𝜒 = 2θ-10𝜁

Trajectory of Helically trapped particles under different Er

• +Er reduces radial drift
• -Er increases radial drift
• -Er pushes towards barely trapped orbit

a) 𝛙 as a function of time

b) 𝚹 as a function of time.

c) Projection of particle trajectory in real space on plane moving toroidally with the particle

d) Particle orbit on flux surface over contours of magnetic field strength at normalized to B_ax. Trajectory color represents radial coordinate 𝛙. The arrow shows the direction of precession

11

Trajectory of Barely trapped particle

Two orbit types, both affected by B(m=1,n=0) toroidal mode

​

Helical boundary particles can pass through the inside of the torus by transitioning to a helically trapped state

Banana-like particles are in passing orbits on the outer side of the torus and bounce when they reach a strong enough B field

​

12

Banana-like

Helical boundary

Overall effects of confinement

• Loss fraction as a function of initial radial location under different constant Er
• Negative Er leads to increased losses due to affect on trapped and barely trapped particles
• Positive Er improves confinement

13

The main physical effect of Er on particle orbits comes from the relation between ExB drift and the curvature and grad B drift

• Radial drift is caused by the radial component of curvature and grad B drift, the sign of which only depends on the B field, and thus position
• Er>0 => v_ExB reinforces curvature and grad B drift on a flux surface, increasing helical precession of trapped particles.
• In clockwise configuration of LHD,
• Er>0 increases precession rate of helically trapped particles
• Reducing time spent drifting in one direction radially
• Reducing overall radial excursion
• improving confinement.
• Er<0 does the opposite

14

Effects of Experimental Radial electric field

15

Experimental Er profiles

Case - potential goes to 0 at separatrix

Er changes from + to - at 𝛙 ~ 0.6

Case + Er goes to 0 at separatrix

16

Potential data from HIBP shot #183281

Trapped particle in experimental Er

Orbit width widens as the particle in case- reaches negative Er

Reduced helical precession lets the particle in Case - escape.

c) Particle motion in 𝛙 and toroidal 𝛇 direction

17

Barely trapped

The particle in Er = 0 is in a helical boundary orbit until 15𝝉₀

In Case -, shifts to a banana- like orbit that slowly drifts out to a region of greater |B| variation, until transitioning to a helically trapped state and is lost

Case +, bounces between orbits but remains well confined over this time period

18

Fraction of particles lost as a function of initial location in phase space

Er affects low energy particles more strongly than high energy

Most lost particles are helically trapped (higher λ), although many come from barely trapped orbits as well ~0.7<λ<~0.95

19

λ = 𝛍B₀/E

Er =0 Case + Case -

Fraction of particles lost as a function of initial phase space location on different flux surfaces

​

• Only barely trapped particles can escape from deep in the core
• Negative Er makes confinement worse
• Positive Er makes confinement better

20

Fraction of particles lost as a function of initial location in real space

Lost particles come from low B region swept by helical orbits that touch the separatrix

21

Overall confinement with experimental Er

22

a) Fraction of lost particles as a function of time

b) fraction of lost particles at the end of simulation as a function of initial position.

Particles lost on first orbit in the Er=0 case are removed from both plots

a

b

Summary of Er effects on particle orbits

• Er has little effect on passing particles
• In LHD, positive Er reinforces helical precession of trapped particles, reducing radial excursion and improving confinement
• Negative Er does the opposite, increasing radial excursion and degrading confinement
• Increasing radial excursion also causes particles to feel greater variation in B, allowing more particles to reach barely trapped states, further reducing confinement
• The position of a particle in phase space as well as real space is important for analyzing the effects of Er on its trajectory
• Er due to potential variation <5 keV has been found to affect particles of energies up to at least 60 keV

23

Experiments On LHD

The data analyzed in the paper is from shot #183281, which was part of an experiment that occurred before I began work on this project. Since then we have had two experimental campaigns to analyze the effects of NBI on Er.

1. 5/28/24 We had 2 experimental scans in CW configuration to prioritize accurate Er data
1. Scan of perpendicular injection energy and parallel direction
2. Scan of ratio of perpendicular to parallel injection power
2. 10/2/25 Repeated injection energy scan in CCW configuration to prioritize FIDA and FILD data

24

Motivation: Manipulation of radial electric field through neutral beam injection

Two sets of scans

1. Scan 3 injection energies of NB5 by 3 different parallel beam arrangements (copassing, counterpassing, mixed) for a total of 9 shots.
2. Scan the fraction of beam power going from perpendicular to parallel. All shots in this scan will have a total injection power of 6 MW.

All shots will have ECH on for half of the shot to measure the effects in both electron root and ion root

25

3.6 m, 2.75 T, clockwise magnetic field from above, HIBP potential measurements,

~half of the shots were quiescent

Future work: Perform simulations to compare Er growth with experiment

Injection Energy Scan

26

Copassing particles -> increasing Er on edge, decreasing electron temperature

in electron root, and increasing electron temperature in ion root

Increasing energy -> decreasing Er on edge and decreasing electron

temperature and density in electron root

Counterpassing Passing

Increasing Perpendicular Injection Energy

Counterpassing Passing

27

Diagnosis of fast ion transport under the effects of the radial electric field

Motivation and method:

• Continuation of previous experiment to see the effects of Er on fast ion losses
• Scan of PNBI injection energy while holding injected power ~constant
• Scan of NNBI direction

Experimental conditions:

(R_ax, Polarity, B_t, γ, B_q) =

(3.9 m, CCW, -1.375 T, 1.254, 100.0%)

#194234 – #194244

E. Green, W. Hayashi (UC Irvine)

Results:

• By changing the PNBI injection energy we were able to detect differences in fast ion losses for same parallel injection (purely copassing shown on right)
• Moderate MHD activity was also observed, which will lead to difficulties in distinguishing effect of MHD and Er on EP losses
• ECH 77GHz 2 did not work so we did not get CTS measurements

Current Work–Self consistent Er simulations

​

• Coding in progress
• Works well for thermal species
• Trying to incorporate fast ions and subcycling
• NBI distribution loaded from GNET with help from PCMS

​

28

Future Work

• Nonlinear interactions between EP and Er
• Effects of Er on drift surfaces
• Relationship between Er, plasma flows, and plasma currents
• Stellarator optimization for NBI induced Er control

29

References

[1] L. Spitzer Jr., “The Stellarator Concept,” Phys. Fluids, vol. 1, no. 4, pp. 253–264, Jul. 1958, doi: 10.1063/1.1705883.

[2] A. H. Boozer, “Stellarators as a fast path to fusion,” Nucl. Fusion, vol. 61, no. 9, p. 096024, Aug. 2021, doi: 10.1088/1741-4326/ac170f.

[3] A. A. Galeev, R. Z. Sagdeev, H. P. Furth, and M. N. Rosenbluth, “Plasma Diffusion in a Toroidal Stellarator,” Phys. Rev. Lett., vol. 22, no. 11, pp. 511–514, Mar. 1969, doi: 10.1103/PhysRevLett.22.511.

[4] H. Nuga et al., “Degradation of fast-ion confinement depending on the neutral beam power in MHD quiescent LHD plasmas,” Nucl. Fusion, vol. 64, no. 6, p. 066001, Apr. 2024, doi: 10.1088/1741-4326/ad3971.

[5] E. J. Paul, A. Bhattacharjee, M. Landreman, D. Alex, J. L. Velasco, and R. Nies, “Energetic particle loss mechanisms in reactor-scale equilibria close to quasisymmetry,” Nucl. Fusion, vol. 62, no. 12, p. 126054, Nov. 2022, doi: 10.1088/1741-4326/ac9b07.

[6] P. J. Bonofiglo, D. W. Dudt, and C. P. S. Swanson, “Fast ion confinement in quasi-axisymmetric stellarator equilibria,” Nucl. Fusion, vol. 65, no. 2, p. 026050, Jan. 2025, doi: 10.1088/1741-4326/ada56d.

[7] H. Biglari, P. H. Diamond, and P. W. Terry, “Influence of sheared poloidal rotation on edge turbulence,” Phys. Fluids B Plasma Phys., vol. 2, no. 1, pp. 1–4, Jan. 1990, doi: 10.1063/1.859529.

[8] Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and R. B. White, “Turbulent Transport Reduction by Zonal Flows: Massively Parallel Simulations,” Science, vol. 281, no. 5384, pp. 1835–1837, Sep. 1998, doi: 10.1126/science.281.5384.1835.

[9] K. Hanatani and F.-P. Penningsfeld, “Resonant superbanana and resonant banana losses of injected fast ions in Heliotron E and Wendelstein VII-A: effects of the radial electric field,” Nucl. Fusion, vol. 32, no. 10, p. 1769, Oct. 1992, doi: 10.1088/0029-5515/32/10/I06.

[10] K. Ida et al., “Control of the radial electric field shear by modification of the magnetic field configuration in LHD,” Nucl. Fusion, vol. 45, no. 5, p. 391, Apr. 2005, doi: 10.1088/0029-5515/45/5/010.

[11] P. Helander, A. G. Goodman, C. D. Beidler, M. Kuczyński, and H. M. Smith, “Optimised stellarators with a positive radial electric field,” May 29, 2024, arXiv: arXiv:2405.07085. doi: 10.48550/arXiv.2405.07085.

[12] K. Ida, “Experimental studies of the physical mechanism determining the radial electric field and its radial structure in a toroidal plasma,” Plasma Phys. Control. Fusion, vol. 40, no. 8, p. 1429, Aug. 1998, doi: 10.1088/0741-3335/40/8/002.

[13] M. Landreman and P. J. Catto, “Effects of the radial electric field in a quasisymmetric stellarator,” Plasma Phys. Control. Fusion, vol. 53, no. 1, p. 015004, Nov. 2010, doi: 10.1088/0741-3335/53/1/015004.

[14] H. E. Mynick, “Effect of collisionless detrapping on nonaxisymmetric transport in a stellarator with radial electric field,” Phys. Fluids, vol. 26, no. 9, pp. 2609–2615, Sep. 1983, doi: 10.1063/1.864452.

[15] H. E. Mynick and W. N. G. Hitchon, “Effect of the ambipolar potential on stellarator confinement,” Nucl. Fusion, vol. 23, no. 8, p. 1053, Aug. 1983, doi: 10.1088/0029-5515/23/8/006.

[16] X. D. Du et al., “Resistive Interchange Modes Destabilized by Helically Trapped Energetic Ions in a Helical Plasma,” Phys. Rev. Lett., vol. 114, no. 15, p. 155003, Apr. 2015, doi: 10.1103/PhysRevLett.114.155003.

30

References

[17] M. Nishiura et al., “Core density profile control by energetic ion anisotropy in LHD,” Phys. Plasmas, vol. 31, no. 6, p. 062505, Jun. 2024, doi: 10.1063/5.0201440.

[18] K. Ida et al., “Reduction of Ion Thermal Diffusivity Associated with the Transition of the Radial Electric Field in Neutral-Beam-Heated Plasmas in the Large Helical Device,” Phys. Rev. Lett., vol. 86, no. 23, pp. 5297–5300, Jun. 2001, doi: 10.1103/PhysRevLett.86.5297.

[19] T. Watanabe et al., “Magnetic field structure and confinement of energetic particles in the LHD,” Nucl. Fusion, vol. 46, no. 2, p. 291, Jan. 2006, doi: 10.1088/0029-5515/46/2/013.

[20] J. Y. Fu, J. H. Nicolau, P. F. Liu, X. S. Wei, Y. Xiao, and Z. Lin, “Global gyrokinetic simulation of neoclassical ambipolar electric field and its effects on microturbulence in W7-X stellarator,” Phys. Plasmas, vol. 28, no. 6, p. 062309, Jun. 2021, doi: 10.1063/5.0047291.

[21] V. V. Nemov, S. V. Kasilov, W. Kernbichler, and G. O. Leitold, “Poloidal motion of trapped particle orbits in real-space coordinates,” Phys. Plasmas, vol. 15, no. 5, p. 052501, May 2008, doi: 10.1063/1.2912456.

[22] A. LeViness et al., “Energetic particle optimization of quasi-axisymmetric stellarator equilibria,” Nucl. Fusion, vol. 63, no. 1, p. 016018, Dec. 2022, doi: 10.1088/1741-4326/aca4e3.

[23] P. Helander and A. N. Simakov, “Intrinsic Ambipolarity and Rotation in Stellarators,” Phys. Rev. Lett., vol. 101, no. 14, p. 145003, Sep. 2008, doi: 10.1103/PhysRevLett.101.145003.

[24] X. Xu et al., “Active generation and control of radial electric field by local neutral beamlets injection in tokamaks,” Nucl. Fusion, vol. 64, no. 2, p. 026012, Jan. 2024, doi: 10.1088/1741-4326/ad169e.

[25] Y.-S. Na, T. S. Hahm, P. H. Diamond, A. Di Siena, J. Garcia, and Z. Lin, “How fast ions mitigate turbulence and enhance confinement in tokamak fusion plasmas,” Nat. Rev. Phys., vol. 7, no. 4, pp. 1–13, Apr. 2025, doi: 10.1038/s42254-025-00814-8.

[26] A. Iiyoshi et al., “Overview of the Large Helical Device project,” Nucl. Fusion, vol. 39, no. 9Y, p. 1245, Sep. 1999, doi: 10.1088/0029-5515/39/9Y/313.

[27] T. Ido et al., “6 MeV heavy ion beam probe on the Large Helical Device,” Rev. Sci. Instrum., vol. 77, no. 10, p. 10F523, Oct. 2006, doi: 10.1063/1.2338311.

[28] M. Nishiura et al., “Enhanced beam transport via space charge mitigation in a multistage accelerator for fusion plasma diagnostics,” Jul. 28, 2025, arXiv: arXiv:2507.20948. doi: 10.48550/arXiv.2507.20948.

[29] C. Suzuki, K. Ida, Y. Suzuki, M. Yoshida, M. Emoto, and M. Yokoyama, “Development and application of real-time magnetic coordinate mapping system in the Large Helical Device,” Plasma Phys. Control. Fusion, vol. 55, no. 1, p. 014016, Dec. 2012, doi: 10.1088/0741-3335/55/1/014016.

[30] S. P. Hirshman and J. C. Whitson, “Steepest‐descent moment method for three‐dimensional magnetohydrodynamic equilibria,” Phys. Fluids, vol. 26, no. 12, pp. 3553–3568, Dec. 1983, doi: 10.1063/1.864116.

[31] R. Sanchez, S. P. Hirshman, A. S. Ware, L. A. Berry, and D. A. Spong, “Ballooning stability optimization of low-aspect-ratio stellarators*,” Plasma Phys. Control. Fusion, vol. 42, no. 6, p. 641, Jun. 2000, doi: 10.1088/0741-3335/42/6/303.

[32] P. Liu et al., “Regulation of Alfvén Eigenmodes by Microturbulence in Fusion Plasmas,” Phys. Rev. Lett., vol. 128, no. 18, p. 185001, May 2022, doi: 10.1103/PhysRevLett.128.185001.

[33] J. W. Connor and R. J. Hastie, “Neoclassical diffusion in an l = 3 stellarator,” Phys. Fluids, vol. 17, no. 1, pp. 114–123, Jan. 1974, doi: 10.1063/1.1694573.

[34] T. Watanabe et al., “Magnetic field structure and confinement of energetic particles in the LHD,” Nucl. Fusion, vol. 46, no. 2, p. 291, Jan. 2006, doi: 10.1088/0029-5515/46/2/013.

31