Generation of field-reversed configurations via neutral beam injection

Machine configurations

Extensive details of the original configuration of Norman are given elsewhere^13,14.

Both the original and present forms are illustrated in Fig. 1. In the Norm configuration, plasmas primarily reside in the CV bounded at each end by divertors. Between the CV and divertors is a fueling region located between the M1 magnetic coil (closest to the CV) and the mirror plug (MP) (closest to the divertor). The M2 coil adjusts the field in the fueling region to facilitate neutral-gas ionization and refueling on field lines closer to the separatrix. Current is driven in the plasma by 8 neutral beams with 15 keV energy, that can deliver up to 13 MW of neutral power³⁷. Beams are injected 20° from normal to the CV with an impact parameter of 20 cm such that ionized particles will orbit in the ion-diamagnetic direction; the resulting ion current produces a field antiparallel to the external field produced by EQ magnet coils. Typically about 8 MW of NB power is attenuated by the plasma after accounting for duct losses and shine-through. NBI on its own tends to cause the plasma to rotate in the ion-diamagnetic direction. If the injection of angular momentum is left unchecked magneto hydrodynamic instabilities (typically n = 2 deformation) can arise. To counter this rotation concentric edge biasing electrodes in the divertors drive a radial current through the plasma that generates a torque in the electron diamagnetic direction³⁸. This balance of forces stabilizes the FRC core enabling a steady-state equilibrium to persist³⁹.

Contour lines are an example of typical magnetic flux surfaces and contour color is representative of typical plasma density profile.

Field reversal sequence

Seed plasmas are generated in Norman by an \(\overrightarrow{E}\times \overrightarrow{B}\) discharge of axial plasma guns and edge biasing electrodes fueled by gas injection in an open magnetic mirror configuration; plasma can be generated by the electrodes alone but this method is less reliable. Within 1 to 4 ms this generates a plasma target sufficient to begin trapping an injected neutral beam thereby providing a substantial fast-ion current. Over the next several ms plasma current grows leading to a robust FRC equilibrium in the CV. This is much longer than formation sequences in other formation methods, such as RMF⁴⁰. This is likely due to the slow process of fast ion accumulation and resulting current drive. This elegant procedure stands in stark contrast to the startup sequence in the original Norman design and its predecessors: gas injection, pre-ionization, theta-pinch, translation, acceleration, and collision-merging. Direct measurements of field reversal were obtained via an internal magnetic field probe in the C-2 device using this formation method⁴¹. The FRC equilibria created by both methods are very similar, as demonstrated in Fig. 2 where core plasma emission is confined within the excluded flux radius⁴², defined as

where plasma and vac subscripts refer to the quantities with and without plasma present.

(left) Images from a fast framing camera of O⁴⁺ emission at 650 nm with overlay of excluded flux radius and estimated X-points. Emission strongest in core where n_e and T_e are elevated. The magnetic axis points north, the camera’s view is mostly radial but pointed toward the midplane from  −82 cm along the axis. This perspective accounts for the apparent lack of symmetry of the X-points. (right) Top figure depicts plasma excluded flux radius (proxy for separatrix) for both generation cases and an additional case with theta-pinch generation and NBI sustainment (Norman configuration); the theta-pinch generated FRC is established quickly while the NBI generated FRC forms gradually. Lower figures compare electron density (n_e) and temperature (T_e) profiles for each case at times indicated by dashed lines on upper plot. Theta-pinch profiles are taken before fast ions accumulate enough to dominate the equilibrium. NBI profiles are taken in the middle of the equilibrium phase when thermal and fast ion currents are similar. Density and temperature have both increased significantly due to NBI heating and configuration optimization: magnetic field shaping, edge biasing, and fueling.

The characteristics of the formation phase, i.e., the rise of plasma current, can be controlled by adjusting settings for magnet coil currents, neutral beam settings, electrode voltage, plasma-gun discharge, and gas fueling. During the formation the external magnetic field must be strong enough to balance the pressure of the diamagnetic object and weak enough to allow adequate field reversal. Section “Direct field reversal by neutral beam injection” will describe the criteria for these boundaries. Norman has been operated with both hydrogen and deuterium for both background plasma and beam species, but we typically use hydrogen and our analysis here will focus on those discharges for simplicity. After formation the FRC equilibrium parameters and shape can be manipulated.

Verification of field-reversal

A non-destructive direct measurement of the magnetic field inside the FRC is beyond current state-of-the-art diagnostic techniques. We are actively pursuing several methods, such as, Faraday rotation of far-infrared radiation, terahertz pulsed polarimetry, circular differential polarimetry, among others⁴³, to make this measurement, but as of now we rely on model-based inferences. Now we describe those models in detail, from comparison with early methodology to more sophisticated Monte Carlo and Bayesian techniques.

Past methodology

The results from Norman will be contrasted with the efforts on 2XIIB¹⁰. That effort was in support of the concept then labeled a field-reversed mirror. While approaching field reversal, it was not achieved. Based on our findings this was due to inadequate neutral current and impact parameter given beam energy and external magnetic field. In the meantime, research on field-reversed configurations (FRC) continued using other formation methods. One of these employed the traditional theta-pinch (some including translation). The other applied rotating magnetic fields. Pursuit of these methods, which continue to this day, were surveyed in two review papers^1,2. Since the 2XIIB experiments, FRC formation by NBI has not been attempted, until now.

This is the first experimental demonstration of successful NBI-based field reversal. To verify successful NBI-based field reversal, two non-direct approaches are taken here. The first adapts the field-reversal index method used in the 1979 paper¹⁰. The second approach builds on several fully two-dimensional (2D) reconstructions of experimental discharges to infer the trapped magnetic flux arising in true FRCs.

The original method defines the field reversal index ζ ≡ ∣ΔB∣/B_vac: here ΔB is the field increment on axis at the plasma mid-plane which is produced by plasma current, and B_vac is the on-axis vacuum field at the mid-plane. Field reversal requires ζ ≥ 1. The ΔB for an idealized thin current sheet is found by an integral of the fields from a ring current,

where I is the total sheet current, R_p and L_p are the sheet radius and half-length respectively. Complementing this is an equation for the field at the wall B_w (also a measurable). In the far-field approximation the field due to the sheet current at the wall is

where r_w is the wall radius. The current inferred from Eq. (3) is substituted into Eq. (2) thence yielding

A maximum field reversal of ζ = 0.9 was claimed for 2XIIB¹⁰. The foregoing method applied to Norman is shown in Fig. 3, clearly exhibiting the achievement of field reversal according to this simple model.

a Distribution of maximum ζ across 4671 shots with plasma radius (r_ΔΦ) greater than 40 cm which persisted longer than 30 ms and average ζ during the reversal process while 20 < r_ΔΦ < 35 cm. Dashed line indicates best result on 2XIIB. b Time history of a typical field-reversed configuration’s (shot #147783) magnetic field (B_z) at the wall (measured) and r = 0 cm (inferred by model described in Eq. (4)), as well as a calculation of ζ(t) itself. Shaded region indicates the pre-reversal averaging time window for the blue distribution in (a).

The model rests on assumptions about plasma current distribution. Using Norm’s large data set, more sophisticated techniques will be explored to strengthen the case.

Current Tomography equilibria reconstruction

Measurements verify the common assumption of azimuthal symmetry of the FRC. By exploiting this fact models can be developed to describe the equilibrium in 2D. In the following sections we examine two methods that rely on experimental data to produce plasma parameter profiles and magnetic field structures. Reconstructions of hundreds of shots showed field reversal; we will illustrate and compare results of both methods in Fig. 4 taking shot #147783 as an example. Neither of these reconstruction methods make any assumption about the presence of a field-reversed configuration; both methods can and do yield solutions where the field lines are open in some cases.

a–c Depiction of the FRC’s magnetic topology as it is generated. Contours represent the current tomography reconstructions of 2D flux surfaces from magnet currents, magnetic probes, and eddy current model inputs, vertical lines in (d) indicate time of each reconstruction. d Time evolution of amount of trapped flux as determined by current tomography (red), total plasma current (green), and the probability that field-reversal is present at that time (blue), error bars represent the Bayesian inference uncertainty in the calculations. e–h Profiles of magnetic field, electron density, electron temperature, and fast ion and total plasma current density, respectively. They are generated by the SEQUOIIA code from edge magnetic field measurements averaged over the steady-state equilibrium period (21–30 ms) and compared to experimental data and the current tomography method.

The current tomography method uses Bayesian inversion to find the most likely plasma current distribution given the information from a variety of magnetic sensors, external magnet currents, and a physics model. Two advantages of the Bayesian inversion are its ability to combine sensor information from heterogeneous sources and its flexibility in incorporating physics models of variable complexity and certainty. This flexibility makes it ideally suited to study the time evolution of plasma during formation. The method has been validated extensively in axisymmetric plasmas in Tokamaks (JET, TCV) and FRCs (C-2U, Norman)^44,45,46,47

The inference process takes place in two steps. First, the evolution of magnet currents are used to estimate the currents induced in the CV using an eddy current model. This gives an initial guess for the CV currents prior mean. In the second step Bayesian inference is used to calculate a posterior distribution of plasma currents and CV currents given the magnetic measurements and the magnet currents.

These parameters are obtained directly from the data by running an optimization algorithm that maximizes the evidence (empirical Bayes’), making the inference completely data driven. From the inferred currents, it is straightforward to obtain magnetic flux. Figure 4a–d shows the plasma flux evolution resulting from the inference process including a time history of trapped flux and total plasma current, which is determined to be  ~300 kA. The inference process creates a distribution of possible states, and by interrogating that distribution we can arrive at the probability that the state is an FRC. This is written as P(FRC) in Fig. 4d.

Monte Carlo equilibria reconstruction

To gain further insight into the internal magnetic field structure of the FRC, TAE developed a 2D axisymmetric equilibrium reconstruction tool to solve Ampere’s law (\(\overrightarrow{\nabla }\times \overrightarrow{B}={\mu }_{0}\overrightarrow{J}\)) called SEQUOIIA (Synthetic Equilibrium from Observational Input Interpretative Algorithm)⁴⁸. It uses experimental measurements of density and temperature along with machine inputs, such as coil currents and neutral beam parameters, in a realistic Norm geometry.

The magnetic field is calculated from diamagnetic currents in the thermal background plasma, fast ion currents, and vacuum field with coil and eddy currents. Neutral beam fast ion density and current are calculated using a 2D full-orbit Monte Carlo code with realistic neutral beam sources and sinks.

The simulation begins with the vacuum magnetic field and thermal plasma. Profiles and currents are calculated in the initial field. Fast ions are introduced and parameters are recalculated using the quasi-neutrality condition n_i = n_e − n_f, where n_f is the fast ion density. In the modified magnetic field, the thermal diamagnetic and fast ion currents are calculated in an iterative manner until a steady state solution is achieved. The difference between experimental magnetic flux and simulated magnetic flux is minimized by adjusting n_f by varying fast ion losses. The final steady state solution is also compared with the other experimental measurements.

Radial profiles from reconstructions of shot #147783 during its steady-state equilibrium (21–30 ms) are shown in Fig. 4e–h. The resulting equilibrium state is a field-reversed configuration with a separatrix radius of 0.4 m, axial length of 2 m, and trapped poloidal flux of  ~ 6 mWb. A total plasma current, produced by thermal plasma and fast ions, of 350 kA is required to create the resulting FRC. Most of the thermal plasma current is near the boundary of the closed field region resulting in a peaking of total current near the separatrix as shown in Fig. 4h. The total energy of the plasma is 9 kJ with nearly equal contributions from the thermal plasma and fast ions.

Mirror-to-FRC transition

During the transition from a high-beta mirror plasma to a field-reversed configuration, the structures of certain fast-ion-driven wave modes observed in C-2W evolve consistent with the magnetic topology change. In C-2W, magnetic fluctuations are monitored using an array of 3-axis Mirnov probes mounted on the interior walls of the CV. These probes are arranged in arrays of eight rings of eight probes each that are spaced at nearly regular intervals azimuthally, and an additional linear array covering most of the length of the CV⁴⁹. The ring arrays can be used to extract the azimuthal mode number, n, of the fluctuations and the linear array can inform on the axial structure of the modes. Here, we discuss two distinct observations: first, magnetic oscillations related to the azimuthal structure and precession of the fast-ion orbits; and second, an energetic-particle mode corresponding to the fast-ion axial bounce motion and axial turning points.

Precession motion

The evolution of fast-ion driven fluctuations from an n = 1 toroidal mode structure to n = 2 follows changes in the excluded flux radius. These fluctuations are attributed to the fast ion precession mode, a finite Larmor radius mode first described in the context of relativistic electron layers in the Astron device^50,51. This mode arises from energetic particle orbit precession phase coherence induced by the self-field generated by the injected energetic particles (electrons in Astron, ions in Norman). The precession phase coherence leads to macroscopic perturbations to the magnetic field and density profiles observable in magnetic field measurements at the vessel wall⁵² and line-integrated density measurements through the plasma, respectively.

The macroscopic perturbations are expected to exhibit a toroidal mode structure matching that of the fast ion orbits themselves⁵³. The mode frequencies evolve over the course of each discharge, but typically fall within the range of 50–400 kHz, consistent with expected precession frequencies of the fast ion orbits. The mode frequencies begin higher at the start of the shot and decrease as the field is reversed. The toroidal mode shape evolution is experimentally consistent with the expected evolution of the fast ion orbit types as the field reverses, in the absence of direct measurements of fast ion orbit types. Thus, as the field reversal parameter increases and fast ion orbit shapes transition from predominantly n = 1 (cyclotron orbits) to n = 2 (betatron orbits), the strength of fluctuations with the latter toroidal mode shape is expected to increase⁵⁴.

Indeed, this transition is consistently observed during FRC formation. Figure 5 depicts this transition in the dominant fast ion mode structure during FRC formation for several shots with similar excluded flux evolutions taken from across the Norman operational campaign without the use of theta-pinch formation sections. The amplitude of these fluctuations are small in both cases due to the stabilizing effect of edge biasing electrodes⁵⁵.

Amplitudes of both of these fluctuations are small due to electrode biasing stabilization. Data from 25 neutral beam injection generated field-reversed configuration shots across the Norm experimental campaign used to generate this plot. Dark traces are the medians of all shots in the series.

Axial bounce motion

Energetic-particle modes (EPMs) are plasma wave modes that correspond to resonances of the characteristic periodic motions of fast ions⁵⁶. In open field-line mirror-trapped plasmas, fast ions will bounce axially between turning points with a frequency ω_z determined by the external magnetic field profiles. In FRC plasmas, fast ions will bounce axially between the ends of the FRC, a shorter distance leading to a higher ω_z. During the mirror-to-FRC transition, the fast-ion turning points are expected to move toward the mid-plane and the axial bounce frequency will increase accordingly. Recently, an EPM was identified in C-2W that corresponds to a resonance of the axial bounce motion and its harmonics⁵⁷. We have called this mode the Axial Bounce Mode (ABM) and will adopt this language herein. The results of several experiments on the ABM have been reported⁵⁷, and we expect to publish a more detailed article soon.

The ABM in C-2W manifests as an axisymmetric magnetic fluctuation (n = 0 azimuthal mode structure) primarily in the B_z and B_r components of the magnetic field; eddy currents in the wall artificially reduce the observed amplitudes of the B_r fluctuations and we restrict ourselves to the B_z component in diagnosing this mode. In dedicated mirror experiments, the turning points of fast ions sourced from NBI were dynamically scanned using field shaping to establish that the ABM is a resonance of their axial bounce motion; the mode was shown to oscillate with frequency ω = ℓω_z, where ℓ is a positive integer and ω_z is the axial bounce frequency of the fast ions in the mirror field. The mode typically has a standing wave structure with odd parity about the mid-plane in B_z, the magnitude peaks near the fast ion turning points, and the amplitude increases with the total NBI power. We have established robust control of the mode using beam species, energy, and injection angle, and suppression of the ABM shows it is not detrimental to plasma confinement. The mode is understood to arise from the axial bunching of fast ions in phase space. A reduced model of ion rings in an axial potential well has been constructed to reproduce many experimentally observed features of the ABM⁵⁸.

Since the mode is observed in both mirror and FRC plasmas, it can be used to assess changes to the magnetic topology after the mirror-to-FRC transition. Figure 6 shows an example of the transition from a mirror plasma to an FRC and how the features of the ABM evolve. The plasma is initially kept as a mirror plasma with a small r_Δϕ using a lower level of NBI power, Fig. 6a. Applying additional NBI power at 10 ms increases the trapped flux until FRC formation at approximately 16 ms. Figure 6b shows a spectrogram of the Mirnov probe signals at z = ± 0.45 m decomposed into n = 0 fluctuations with the fundamental and two harmonics of the ABM mode indicated. In the mirror region, we can see the mode has a low amplitude due to the lower beam power and has a nearly constant frequency that matches well with the bounce frequency determined from the vacuum magnetic field; typically, the third harmonic has the largest amplitude in the mirror regime, as is the case here. When the beam power is increased, we see the amplitude of the mode increase rapidly, and the frequency begins to rise; this increase in frequency is due to a constriction of the fast-ion turning points towards the mid-plane.

a Temporal evolution of excluded flux radius, r_Δϕ, (left abscissa) and neutral-beam injection power (right abscissa); mirror and FRC plasma regimes are highlighted by orange and blue, respectively, with a transition zone between them. b Spectrogram of B_z fluctuations with n = 0 azimuthal structure and averaged over measurements at axial locations z = ± 0.48 m, the ABM and harmonics are indicated by arrows and the vacuum bounce frequency of fast ions sourced from NBI by the dashed white line. c Axial structure of the axial bounce mode in the Mirror and FRC regions (left abscissa) and vacuum field magnitude (right abscissa); the turning point regions identified by the peak amplitude of the mode (shaded regions) and the turning points of the vacuum field (black dashed lines) are indicated. Error bars represent the standard deviation of the mode amplitude over time.

In Fig. 6c, cross-correlation of the ring Mirnov array with the linear Mirnov array is used to extract the axial structure of the mode in the mirror and FRC regions of Fig. 6a. In the mirror, the mode is observed to peak around z = ±1.6 m, a good match for the vacuum field turning point at approximately z = ±1.5 m, calculated using ionization at the mid-plane of the vacuum field profile indicated and the nominal beam injection angle. Once the FRC has formed the mode peaks sharply at the probes located at z = ±0.48 m and rapidly decays indicating the reduced extent of the axial confinement of the fast ions. The evolution of this n = 0 ABM as presented for the shot here is consistently observed in the formation of FRC plasmas across all Norm campaigns and represents a clear signature of the changing magnetic topology induced by the NBI reversal process.

Direct field reversal by neutral beam injection

Now we construct a theoretical model to describe the generation of an FRC object. It is a zero-dimensional construct focusing on beam and plasma currents. It derives a minimum criterion for field reversal and is compared to experimental results.

A cylindrical ring of ion current creates a magnetic field,

where I_f is the total ion current and L is the length of the cylindrical ring of current. If the ion current is carried by fast ions injected by neutral beams then the ion current buildup is determined by the equation \({I}_{f}=\frac{{I}_{b}{\tau }_{\ell }}{{\tau }_{2\pi }}\left[1-\exp \left(-\frac{t}{{\tau }_{\ell }}\right)\right]\), and the steady-state fast ion current is \({I}_{f}=\frac{{I}_{b}{\tau }_{\ell }}{{\tau }_{2\pi }}\), where τ_ℓ is the lifetime of fast ions determined by loss mechanisms such as charge-exchange and collisions with background plasma, τ_2π is the average orbit time about the cylindrical axis, and I_b is the portion of injected neutral beam current that has been trapped in the plasma (i.e., that part which is not lost to beam shine-through S), I_b = (1 − S)I_inj, where I_inj is the injected current of neutral particles.

A background magnetic field, B, is reversed by the neutral beam injected fast ion current if B_f > B. The condition for fast ion current to reverse the field, B_f > B, is that \(\frac{{\mu }_{0}{I}_{b}{\tau }_{\ell }}{{\tau }_{2\pi }L} > B\), or in other words,

In the initial vacuum magnetic field, the most efficient way to impart current from the beams is if the impact parameter of beam injection is equal to the initial ion gyroradius so that the fast ion orbits are Larmor orbits encircling the cylindrical axis and the average period of gyration is equal to the gyroperiod, \({\tau }_{2\pi }=\frac{1}{f}=\frac{2\pi }{\Omega }=\frac{2\pi }{eB/m}\). The requirement to reverse the field becomes

This expression is approximate because the magnetic field changes as the fast ion current builds up, however it is a useful rule of thumb which has been validated empirically. Experiments which scanned over various values of magnetic field and neutral beam current have been conducted on Norman to explore the operational boundaries between reversed and non-reversed mirror plasmas. Figure 7 shows estimated values of the field produced by the fast ions versus the external field for discharges that achieved r_ΔΦ > 35 cm (shown as green dots) and those that did not (as red crosses). The quantities shown are window-averaged between 1 and 5 milliseconds. The apparent boundary for FRC formation aligns well with theoretical arguments laid out above and reflects what has long been observed in our NBI generated FRC campaigns.

Discharges that eventually achieved r_ΔΦ > 35 cm are shown as green dots, while those that did not are shown as red crosses. The boundary where B_f = B₀ is shown as the black dashed line. Only hydrogen plasma discharges with hydrogen beams are shown.