Hostname: page-component-76d6cb85b7-8p85h Total loading time: 0 Render date: 2026-07-14T04:18:24.838Z Has data issue: false hasContentIssue false
  • English
  • Français

Dependence of field-reversed configuration formation in collisional merging experiments on the mirror magnetic field strength

Part of: Open Magnetic Systems for Plasma Confinement

Published online by Cambridge University Press:  08 July 2026

Satsuki Ishiwata Open the ORCID record for Satsuki Ishiwata [Opens in a new window] ,
Loren Steinhauer  and
Tomohiko Asai Open the ORCID record for Tomohiko Asai [Opens in a new window]
Satsuki Ishiwata*
Affiliation:
College of Science and Technology, Nihon University , Tokyo, Japan
Loren Steinhauer
Affiliation:
TAE Technologies Inc., Foothill Ranch, CA, USA
Tomohiko Asai
Affiliation:
College of Science and Technology, Nihon University , Tokyo, Japan
*
Corresponding author: Satsuki Ishiwata, cssa25003@g.nihon-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Collisional merging experiments on field-reversed configurations (FRCs) in the FRC amplification via translation–collisional merging (FAT-CM) device have suggested that the mirror magnetic field strength influences the final merged plasma state. Analytical models do not account for mirror magnetic fields at the ends of the confinement section. To explore their effect, experiments were conducted at FAT-CM, where the strength of mirror magnetic fields were varied. Increasing the mirror field strength was found to promote relaxation to an FRC with larger radius and shorter axial length. Based on these observations, the dependence of equilibrium states on the mirror field strength and the resulting plasma parameters were investigated using a model that reconstructs fully two-dimensional equilibria primarily constrained by wall-mounted magnetic probe measurements. Consistent with the experimental results, computed equilibria with stronger mirror fields converged to plasma states with a larger radius and shorter length. Further, these results showed a bifurcation of topologies with a closed FRC core for higher mirror field and an open-field high-$\beta$ mirror for lower field. This result agreed with internal magnetic probe data which confirm the FRC-to-high-$\beta$ mirror transition. This demonstrates the importance of employing fully two-dimensional modelling of plasma equilibria.

Keywords

fusion plasmaplasma diagnostics

Information

Type
Research Article
Information
Journal of Plasma Physics , Volume 92 , Issue 4 , June 2026 , E93
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2026. Published by Cambridge University Press

1. Introduction

Field-reversed configurations (FRCs) are elongated and axisymmetric compact-toroid plasma structures comprising a closed-field core surrounded by an open-field periphery, as shown in figure 1. The periphery includes a scrape-off layer (SOL) adjacent to the core and jet-like structures extending from the ends, which provide a natural connection to remote solid surfaces. Fully kinetic analyses have indicated that particle end-loss through the SOL is sensitive to the mirror ratio and electrical biasing, which highlights the importance of the end region magnetic structure (Dettrick et al. Reference Dettrick2021; Gota et al. Reference Gota2024; Steinhauer & Nicks Reference Steinhauer and Nicks2025). End mirrors can influence the axial end-loss in the open-field edge/SOL (Dettrick et al. Reference Dettrick2021; Gota et al. Reference Gota2024; Steinhauer & Nicks Reference Steinhauer and Nicks2025). Previous edge experiments have shown that the particle end-loss time increases with the mirror ratio in a high-mirror-ratio regime (Ohtsuka, Okada, & Gota Reference Ohtsuka, Okada and Goto1999; Dettrick et al. Reference Dettrick2021; Gota et al. Reference Gota2024).

Figure 1.

Schematic of a field-reversed configuration.

A diagram of a field-reversed configuration showing the structure of a plasma with labeled regions and flow directions.
Figure 1. Long description

A diagram of a field-reversed configuration. The diagram shows an elongated and axisymmetric compact-toroid plasma structure. It comprises a closed-field core surrounded by an open-field periphery. The diagram includes several labeled regions: the scrape-off layer (SOL), the separatrix, and the X-point. The scrape-off layer (SOL) is the outer region, while the separatrix is a boundary that separates the closed-field core from the open-field periphery. The X-point is a specific point on the separatrix. Arrows indicate the direction of plasma flow within these regions. The diagram also includes axes labeled with r, r_w, r_s, R, theta, and z, which represent different spatial dimensions and positions within the plasma structure.

Because the confining magnetic field is primarily poloidal, FRCs can sustain a very high volume-averaged plasma pressure–magnetic pressure ratio ( beta $\beta$ ). Owing to the simply connected topology of the core, FRCs are also linear systems that allow axial translation because no coils link through the plasma centre (Steinhauer, Roche & Steinhauer Reference Steinhauer, Roche and Steinhauer2020). However, this combination of a high beta $\beta$ and simply connected structure accompanied by an open-field periphery makes it difficult to infer internal quantities such as the core size and trapped flux from external magnetic measurements alone. Therefore, equilibrium reconstruction methods that combine a plasma model with standard diagnostic data are important (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020).

At Nihon University, FRCs are produced in the FRC amplification via translation–collisional merging (FAT-CM) device, which is illustrated in figure 2. In FAT-CM, two FRC-like plasmoids are independently formed by the field-reversed theta-pinch (FRTP) method and are translated towards a central confinement section. The plasmoids collide near the midplane with relative speeds of approximately 200–500 km s–1, and notable plasma heating as well as increases in trapped poloidal flux and excluded-flux have been observed during and after the collisional merging process (Asai et al. Reference Asai2019; Watanabe et al. Reference Watanabe, Asai, Takahashi, Kobayashi and Harashima2021). Experiments have indicated that, following the seemingly destructive violence of a super-Alfvénic collision, a quiescent FRC structure can emerge in a self-organised manner, which is accompanied by a marked increase in the excluded-flux compared with that in the two formation sections (Asai et al. Reference Asai2019).

Figure 2.

Schematic view of FAT-CM.

A schematic view of a plasma confinement device with labeled sections and coils.

The axial magnetic field profile, particularly the mirror magnetic field strength in the confinement section, is expected to influence both the translation dynamics and final merged FRC state. Internal magnetic probe (IMP) measurements in FAT-CM have shown that the confinement section requires a sufficient mirror magnetic field strength for two colliding FRCs to fully merge into a single FRC (Gota et al. Reference Gota2018). In related translation experiments, an FRC injected into a weaker and quasi-static magnetic field in the confinement section underwent super-Alfvénic acceleration, which was driven primarily by an axial magnetic pressure gradient acting on the slender plasmoids in the formation sections. The results of these experiments were qualitatively consistent with two-dimensional resistive magnetohydrodynamic (MHD) simulations (Kobayashi & Asai Reference Kobayashi and Asai2021). These observations indicate the need for a systematic evaluation on how the mirror magnetic field strength in the confinement section governs whether a closed magnetic flux FRC equilibrium can be realised after merging.

The quantitative interpretation of translated and merged FRCs has often relied on simplified assumptions and standard formulae based on limited magnetic measurements. For high- beta $\beta$ plasmas, however, the magnetic field structure can be strongly reshaped by the plasma pressure, and inferring the trapped flux and true core geometry calls for a reconstruction technique that remains robust even when simplified approximations fail (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020). Therefore, for experimental systems with end mirrors and open-field peripheries, an analysis framework that retains essential two-dimensional geometry while maintaining computational efficiency is valuable for routine data interpretation.

To address this need, Grushenka was developed as a fast tool for reconstructing fully two-dimensional (2-D) equilibria consistent with routine magnetic data (e.g. wall-mounted flux loops and magnetic probes) that uses a rotating fluid-based model augmented to capture several kinetic effects (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020). Grushenka finds that the solution bifurcates, yielding either an FRC (i.e. closed magnetic flux) or a high- beta $\beta$ mirror (HBM) (i.e. no closed magnetic flux) depending on the experimental inputs. Further, it was shown that FRC equilibria exist only within a limited radius–length shape domain (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020). Such capabilities make it suitable for assessing equilibrium consistency in collisional merging experiments where the external field profile is a key control factor.

In this study, we applied Grushenka to the 2-D equilibrium analysis of collisional merging plasmas in FAT-CM and investigated how the mirror magnetic field strength in the confinement section affects the reconstructed equilibrium solutions. Because FAT-CM allows flexible control of the mirror magnetic field strength, which is a key input parameter for Grushenka, and provides internal diagnostics, it offers an appropriate platform for assessing the validity of the equilibrium reconstruction. We determined the mirror magnetic field conditions under which the reconstruction yields a closed magnetic flux FRC equilibrium and identified conditions where the solution instead transitions to a HBM equilibrium. We compared reconstructed equilibria with independent magnetic diagnostics and examined the implications for interpreting collisional merging FRC experiments. The remainder of this paper is organised as follows. Section 2 summarises the experimental set-up and dataset. Section 3 describes the reconstruction method and inputs. Section 4 presents the reconstructed equilibria and their dependence on the mirror magnetic field strength. Section 5 discusses the implications of the findings and their limitations, and § 6 concludes the paper.

2. Experimental set-up and dataset

FAT-CM is a linear facility with a total length of approximately 8 m and a coil diameter of approximately 1 m. The device comprises formation sections at each end and a central confinement section. Each formation section contains a quartz radius with an inner radius of 128 mm that is surrounded by a theta-pinch coil and produces an FRC-like plasmoid by the FRTP method. The two plasmoids are then translated towards the confinement section and collide near the midplane ( z equals 0 $z=0$ ) before merging (Asai et al. Reference Asai2019; Watanabe et al. Reference Watanabe, Asai, Takahashi, Kobayashi and Harashima2021). The confinement section comprises a cylindrical flux conserver with an inner radius of approximately 0.39 m and external straight coils that generate an axial guide field. The ends of the flux conserver are conical and mirror coils are installed at both ends to provide axial compression.

In the present experiments, the mirror magnetic field strength was scanned by varying the charging voltage V c of the mirror coils between 1.5 and 3.0 kV. Charging voltages of 1.5, 2.0, 2.5 and 3.0 kV corresponded to vacuum mirror ratios of 3.37, 4.18, 4.94 and 5.93, respectively. We analysed the collisional merging discharges obtained under these four conditions. The reference time t = 0 was defined as the onset of the rapidly rising main theta-pinch field in the FRTP sequence. We focused on the quasi-stationary phase at t equals 57.5 mu s $t = 57.5\, \unicode{x03BC} \text{s}$ , which was selected as a representative time slice for equilibrium analysis. Table 1 summarises the shot conditions and representative measured parameters at t $t$ = 57.5 µs, which included the axial magnetic field at the midplane ( upper B 0 $B_{0}$ ), the axial magnetic field at the far end ( upper B Subscript normal m $B_{\mathrm{m}}$ ) and the effective mirror ratio ( upper R Subscript normal upper M ${R}_{\mathrm{M}}$ ).

Table 1.

Shot conditions and representative plasma parameters ( t $t$ = 57.5 µs).

A table summarizing shot conditions and plasma parameters for different charging voltages.

Note that the vacuum mirror ratio was evaluated under vacuum conditions (i.e. no plasma), while the effective mirror ratio upper R Subscript normal upper M ${R}_{\mathrm{M}}$ was evaluated during the plasma discharge. Here, upper R Subscript normal upper M ${R}_{\mathrm{M}}$ was defined as upper R Subscript normal upper M Baseline identical to upper B Subscript normal m Baseline divided by upper B 0 ${R}_{\mathrm{M}}\equiv B_{\mathrm{m}}/B_{0}$ , where upper B 0 identical to upper B Subscript z Baseline left parenthesis z equals 0 comma t right parenthesis $B_{0}\equiv B_{z}(z=0,t)$ is the axial magnetic field at the midplane and upper B Subscript normal m Baseline identical to upper B Subscript z Baseline left parenthesis z equals upper Z Subscript max Baseline comma t right parenthesis $B_{\mathrm{m}}\equiv B_{z}(z=Z_{\max },t)$ is the axial field in the mirror region at the same time. The mirror position ( upper Z Subscript max $Z_{\max }$ ) corresponded to the end plane of the mirror region. Figure 3(a) shows the locations of measurement instruments used to collect data. Wall-mounted magnetic probes for collecting the axial magnetic field ( upper B Subscript z $B_{z}$ ) were installed along the confinement section at z = −1.275, −1.185, −1.075, −0.90, −0.60, −0.30, 0, 0.30, 0.60, 0.90, 1.075, 1.185 and 1.275 normal m $\mathrm{m}$ . These measurements were used to characterise the vacuum magnetic field profile and provide key inputs for the equilibrium analysis. An IMP array was positioned at the midplane (z = 0) to provide three-component magnetic field measurements ( upper B Subscript x Baseline comma upper B Subscript y Baseline comma upper B Subscript z Baseline $B_{x},B_{y},B_{z}$ ) at 12 radial locations from r = −0.08 to 0.36 m in 0.04 m increments. These data were used to examine the internal field structure (e.g. field reversal) and to identify the quasi-stationary phase used for analysis of the representative time slice. A heterodyne laser interferometer with three chords at r = 0, 0.12, and 0.20 m was used to obtain the line-integrated electron density, as shown in figure 3(b). The line-averaged electron density from each chord was used as an input for the reconstruction. The total (ion + electron) temperature ( upper T Subscript normal t normal o normal t $T_{\mathrm{tot}}$ ) was estimated by combining interferometer measurements of the electron density with the estimated thermal pressure from magnetic diagnostics using the formula p equals n k Subscript normal upper B Baseline upper T Subscript normal t normal o normal t $p=n{k}_{\mathrm{B}}T_{\mathrm{tot}}$ (assuming quasi-neutrality).

Figure 3.

Positions of measurement instruments in FAT-CM: (a) x equals 0 $x=0$ cross-section and (b) midplane ( z equals 0 $z=0$ ) cross-section.

Diagram of measurement instruments in a plasma structure.

3. Analysis methods

3.1. Conventional quasi-1-D equilibrium analysis

Conventional quasi-one-dimensional (quasi-1-D) equilibrium analysis is based on the tendency of FRCs to be elongated in the axial direction. Accordingly, the plasma structure can be modelled as an infinitely long cylinder with a purely axial magnetic field. Under this assumption, the radial pressure is balanced by equating the external vacuum magnetic pressure to the total pressure (thermal plus magnetic) inside the FRC. Similarly, the axial forces are balanced by equating the contributions of magnetic tension, thermal pressure and magnetic pressure in a vacuum periphery to the magnetic pressure of the vacuum magnetic field at infinity. This simplified model ignores plasma pressure outside the separatrix and presumes the SOL to be thin relative to the radius of the plasma core.

A related limitation arises in the conventional excluded-flux method for estimating the FRC radius. With these assumptions, the wall flux in a vacuum ( normal v $\mathrm{v}$ ) is

(3.1) upper Phi Subscript normal v Baseline equals normal upper Pi r Subscript normal w Baseline Superscript 2 Baseline upper B Subscript normal v \begin{equation}\varPhi _{\mathrm{v}}={\Pi} {r_{\mathrm{w}}}^{2}B_{\mathrm{v}}\end{equation}

where upper B Subscript normal v $B_{\mathrm{v}}$ is the vacuum magnetic field (radially uniform) and r Subscript normal w $r_{\mathrm{w}}$ is the wall radius. The wall flux in the presence of plasma ( p $p$ ) is

(3.2) phi Subscript normal p Baseline equals normal upper Pi left parenthesis r Subscript normal w Baseline Superscript 2 Baseline minus r Subscript normal upper Delta phi Baseline Superscript 2 Baseline right parenthesis upper B Subscript normal p Baseline comma \begin{equation}\phi _{\mathrm{p}}={\Pi} \big({r_{\mathrm{w}}}^{2}-{r_{{\Delta} {\phi} }}^{2}\big)B_{\mathrm{p}},\end{equation}

where r Subscript normal upper Delta phi $r_{{\Delta} {\phi} }$ is the excluded-flux radius and upper B Subscript normal p $B_{\mathrm{p}}$ is the magnetic field outside the separatrix (also radially uniform). If the wall flux is ‘frozen’, then phi Subscript normal v $\phi _{\mathrm{v}}$ (vacuum shot) is identical to phi Subscript normal p $\phi _{\mathrm{p}}$ (plasma shot). Then, the excluded-flux radius is given by

(3.3) r Subscript normal upper Delta phi Baseline equals r Subscript normal w Baseline StartRoot 1 minus StartFraction upper B Subscript normal v Baseline Over upper B Subscript normal p Baseline EndFraction EndRoot period \begin{equation}r_{{\Delta} {\phi} }=r_{\mathrm{w}}\sqrt{1-\frac{B_{\mathrm{v}}}{B_{\mathrm{p}}}}.\end{equation}

In this no-SOL model, the excluded-flux radius ( r Subscript normal upper Delta phi $r_{{\Delta} {\phi} }$ ) is identical to the separatrix radius ( r Subscript normal s $r_{\mathrm{s}}$ ). For this reason, r Subscript normal upper Delta phi $r_{{\Delta} {\phi} }$ is commonly used as a proxy for r Subscript normal s $r_{\mathrm{s}}$ . However, if an SOL with a finite thickness is considered, then r Subscript normal s $r_{\mathrm{s}}$ is less than r Subscript normal upper Delta phi $r_{{\Delta} {\phi} }$ , which presumably depends on the relative thickness of the SOL. In addition, accounting for 2-D effects such as the field-line curvature and influence of a mirror magnetic field at the ends may lead to r Subscript normal s $r_{\mathrm{s}}$ being substantially less than r Subscript normal upper Delta phi $r_{{\Delta} {\phi} }$ .

Figure 4.

Computational domain of Grushenka.

Diagram of a computational domain illustrating magnetic field profiles and regions.

3.2. Fully 2-D equilibrium analysis

We employed Grushenka to realise the fully 2-D equilibrium reconstruction of the flux variable psi left parenthesis r comma z right parenthesis $\psi (r,z)$ . In FAT-CM, the mirror magnetic field strength can be varied by adjusting the charging voltage of the mirror coils. Grushenka is well suited for analysing such configurations because it allows the magnetic field strength at the end of the computational domain, where the mirror coils are installed, to be specified explicitly. This model introduces the mirror magnetic field upper B Subscript normal m $B_{\mathrm{m}}$ at the end region and reconstructs the curvature of the external magnetic field, which allows the influence of the mirror magnetic field on the overall equilibrium to be considered. Another advantage of this approach is that intrusive IMPs are not required as direct inputs for the equilibrium reconstruction. Conventionally, equilibrium reconstruction depends on routine non-intrusive measurements at the device wall from which the excluded-flux size and external field profile are inferred. In the present study, however, we employed an IMP array to identify the quasi-stationary phase used for analysis of the representative time slice and to provide an independent validation of the reconstructed midplane magnetic field profile and equilibrium branch classification.

A key feature of Grushenka is that it admits two equilibrium branches in mirror magnetic field geometry: an FRC branch with a closed magnetic flux and an HBM branch with no closed magnetic flux. Under the given boundary conditions and input parameters, Grushenka can automatically determine to which branch the equilibrium converges. As shown in figure 4, the computational domain was defined in two dimensions: a central section with a fixed wall radius ( r Subscript normal w $r_{\mathrm{w}}$ ) and conical end sections. We assumed midplane symmetry so that only one side of the midplane needed be addressed. Boundary conditions were imposed on the magnetic flux psi left parenthesis r comma z right parenthesis $\psi (r,z)$ as follows (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020).

  1. 1. A Dirichlet condition was imposed at the radial boundary by prescribing the wall flux upper Psi left parenthesis r Subscript normal w Baseline left parenthesis z right parenthesis comma z right parenthesis equals upper Psi Subscript normal w Baseline left parenthesis z right parenthesis ${\varPsi}(r_{\mathrm{w}}(z),z)=\varPsi _{\mathrm{w}}(z)$ .

  2. 2. We assumed that midplane ( z equals 0 $z=0$ ) symmetry implies the Neumann condition partial differential upper Psi divided by partial differential z equals 0 $\partial \varPsi /\partial z=0$ .

  3. 3. At the end plane ( z equals upper Z Subscript max $z=Z_{\max }$ ), a periodic condition partial differential upper Psi divided by partial differential z equals 0 $\partial \varPsi /\partial z=0$ (Neumann) was imposed, which is a common artifice in equilibrium computations.

The two-dimensionality of the computational domain allowed us to consider both the finite length of the FRC as well as the effects of the mirror magnetic field.

The following expression was adopted for the wall flux: upper Psi Subscript normal w Baseline left parenthesis z right parenthesis equals upper B Subscript normal v Baseline left parenthesis z right parenthesis r Subscript normal w Superscript 2 Baseline left parenthesis z right parenthesis divided by 2 $\varPsi _{\mathrm{w}}(z)=B_{\mathrm{v}}(z){r}_{\mathrm{w}}^{2}(z)/2$ . The nominal vacuum field is expressed as follows (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020):

(3.4) upper B Subscript normal v Baseline left parenthesis z right parenthesis equals upper B 0 plus normal upper Delta upper B Subscript normal m Baseline z squared divided by upper Z Subscript max Superscript 2 Baseline comma \begin{equation}B_{\mathrm{v}}(z)=B_{0}+{\Delta} B_{\mathrm{m}}z^{2}/{Z}_{\max }^{2},\end{equation}

where the adjustable parameters are the base magnetic field strength at the midplane ( upper B 0 $B_{0}$ ) and the increment of the mirror magnetic field strength ( normal upper Delta upper B Subscript normal m ${\Delta} B_{\mathrm{m}}$ ) at the end.

Figure 5.

Calculation flowchart of Grushenka.

Flowchart illustrating the calculation process of Grushenka.

The interior structure is governed by a combination of an extended fluid model for the current density j Subscript theta Baseline left parenthesis r comma z right parenthesis $j_{\theta }(r,z)$ , which is linked to the magnetic flux ( psi $\psi$ ) through Ampere’s law:

(3.5) StartLayout 1st Row 1st Column normal upper Delta Superscript asterisk Baseline psi 2nd Column equals minus mu 0 r j Subscript theta Baseline 2nd Row 1st Column normal upper Delta Superscript asterisk 2nd Column equals r squared bold nabla bold dot left bracket left parenthesis StartFraction 1 Over r squared EndFraction right parenthesis bold nabla right bracket period EndLayout \begin{align}{\Delta}^{*}\psi & =-\mu_{0}rj_{\theta}\nonumber\\[5pt]{\Delta}^{{*}} & =r^{2} \boldsymbol{\nabla }\boldsymbol{\cdot }\left[\left(\frac{1}{r^{2}}\right)\boldsymbol{\nabla }\right]\!.\end{align}

A flexible current density profile was adopted:

(3.6) j equals r upper P prime left parenthesis psi right parenthesis exp left parenthesis StartFraction m Subscript i Baseline upper Omega squared r squared Over 2 k Subscript normal upper B Baseline upper T Subscript normal t normal o normal t Baseline EndFraction right parenthesis comma \begin{equation}j=rP'\left(\psi \right)\exp\! \left(\frac{{m}_{i}\varOmega ^{2}r^{2}}{2{k}_{\mathrm{B}}T_{\mathrm{tot}}}\right)\!,\end{equation}

where upper P left parenthesis psi right parenthesis $P(\psi )$ is a surface function of psi $\psi$ and upper P prime left parenthesis psi right parenthesis $P'(\psi )$ denotes the derivative with respect to psi $\psi$ . The exponential factor corresponds to the centrifugal effect: upper Omega $\varOmega$ is the rotation frequency, m Subscript i ${m}_{i}$ is the ion mass, k Subscript normal upper B ${k}_{\mathrm{B}}$ is Boltzmann’s constant and upper T Subscript normal t normal o normal t $T_{\mathrm{tot}}$ is the total temperature (ion + electron). For simplicity, both upper Omega $\varOmega$ and upper T Subscript normal t normal o normal t $T_{\mathrm{tot}}$ were treated as constants. In the absence of rotation ( upper Omega equals 0 $\varOmega =0$ ), combining (3.5) and (3.6) results in the familiar Grad–Shafranov equation for static plasma equilibria. The pressure function has a two-branch form: one in the core ( psi less than 0 $\psi \lt 0$ ) and one in the SOL ( psi greater than 0 $\psi \gt 0$ ). Each is the composite of two rigid rotor-like elements ( p equals normal upper C normal e Superscript alpha psi $p=\mathrm{Ce}^{\alpha \psi }$ ). Another kinetic effect related to a finite Larmor radius is to enforce continuous p left parenthesis psi right parenthesis $p(\psi )$ , p prime left parenthesis psi right parenthesis $p'(\psi )$ and p double prime left parenthesis psi right parenthesis $p''(\psi )$ at the separatrix. This gives rise to a form for the current density with two free parameters. The solution algorithm is an iterative procedure that automatically adjusts the free parameters while converging to conformity with (3.5) and (3.6). Figure 5 shows a flowchart illustrating the iterative solution procedure. Among the key input parameters were the external magnetic field strength (i.e. the midplane magnetic field strength upper B 0 $B_{0}$ and the mirror-coil increment normal upper Delta upper B Subscript normal m ${\Delta} B_{\mathrm{m}}$ ) and the plasma size. The plasma size was characterised by the excluded-flux radius at the midplane ( upper R Subscript phi $R_{{\phi} }$ ) and by the axial distance over which the excluded-flux radius decreased to 2 divided by 3 upper R Subscript phi $2/3R_{{\phi} }$ , which was defined as the half-length ( upper Z Subscript phi $Z_{{\phi} }$ ). From these data, a fitting function for the external magnetic field strength was constructed and target parameters were corrected for the plasma size. Satisfying these target parameters together with the prescribed external magnetic field strength function obtained the magnetic flux function and associated correction terms that satisfied the theta $\theta$ component of Ampère’s law. From the resulting magnetic flux function, various physical quantities such as the density and energy were subsequently evaluated.

4. Results

4.1. Representative reconstructions

Table 2 presents the geometry and boundary conditions for FAT-CM that were implemented in Grushenka. By varying the target excluded-flux dimensions StartSet upper R Subscript upper Phi Baseline comma upper Z Subscript upper Phi Baseline EndSet $\{R_{\varPhi }, Z_{\varPhi }\}$ , two branches of equilibrium solutions were obtained. Figure 6(a) shows an HBM equilibrium solution comprising solely open magnetic field lines, while figure 6(b) shows an FRC equilibrium solution characterised by a closed magnetic flux. Figure 7 presents the radial distributions of the magnetic field at the midplane ( z equals 0 $z=0$ ) for each case. In the case of the FRC equilibrium solution, the sign of upper B Subscript z Baseline left parenthesis r right parenthesis $B_{z}(r)$ clearly reversed. Figure 8 shows the excluded-flux radius (red dashed line), separatrix radius (red solid line) and half-length (black dashed line) within the computational domain. For the HBM equilibrium solution, no separatrix radius is shown because there was no closed magnetic flux.

Observe that figure 7 also shows the pressure profiles for representative examples. Keep in mind that the pressure is at once removed ((3.5) and (3.6)) from the strictly magnetic results because it depends on the selection of a suitable pressure function family. In any case, care should be taken not to overinterpret this computed profile, e.g. by extracting quantitative details from it.

Table 2.

Geometry and boundary conditions for FAT-CM.

A table showing geometry and boundary conditions for FAT-CM.
Figure 6.

Contours of the magnetic fields calculated by Grushenka for the (a) HBM equilibrium solution and (b) FRC equilibrium solution.

Heat maps comparing magnetic fields for two equilibrium solutions.

4.2. Dependence on the mirror ratio

Previous experiments on FAT-CM indicated that the plasma did not relax to an FRC when the mirror magnetic field strength was reduced (Gota et al. Reference Gota2018). Therefore, FRC merging experiments were performed at charging voltages of 1.5, 2.0, 2.5 and 3.0 kV, which corresponded to vacuum mirror ratios of 3.37, 4.18, 4.94 and 5.93, respectively. Figure 9 shows the IMP measurements in individual shots, which served as an independent validation of the midplane field reversal behaviour reconstructed by Grushenka. For shots with a smaller effective mirror ratio, no clear sign reversal of upper B Subscript z Baseline left parenthesis r right parenthesis $B_{z}(r)$ was observed, which indicated that the plasma state did not have a tendency to relax to an FRC equilibrium. In contrast, a clear sign reversal was observed for shots with a larger vacuum mirror ratio, which suggests relaxation towards an FRC equilibrium. The time at which the magnetic fluctuations relaxed ( t $t$ = 57.5 µs) was regarded as corresponding to a quasi-equilibrium state. Figure 10 presents the excluded-flux radius and half-length calculated from wall-mounted magnetic probe measurements ( t $t$ = 57.5 µs). As the effective mirror ratio increased, the plasma length decreased. A slight increase in the plasma radius was also observed, although the change was small. These trends are evident both in the shot-by-shot data and in the standard deviations evaluated for each condition.

Figure 7.

Radial distributions of the magnetic fields calculated by Grushenka for the (a) HBM equilibrium solution and (b) FRC equilibrium solution.

Two line graphs compare magnetic field distributions in different plasma structures.

Figure 8.

Plasma size parameters calculated by Grushenka for the (a) HBM equilibrium solution and (b) FRC equilibrium solution.

Two line graphs compare plasma size parameters for different equilibrium solutions.

Figure 9.

Time evolution of the magnetic field at the midplane measured by IMPs for a representative shot.

Four heat maps show the time evolution of the magnetic field at different charging voltages.

Figure 10.

Excluded-flux radius and half-length determined from wall-mounted magnetic probe measurements. The solid line shows a representative shot and shaded regions denote ±1σ, where σ is the sample standard deviation.

Four line graphs depict the relationship between radius and length for different shots and voltages.

To provide experimental evidence for the mirror-field dependence of the post-merging configuration, we quantify the plasma size using the excluded-flux radius r Subscript normal upper Delta upper Phi $r_{{\Delta}\varPhi }$ at the midplane and the corresponding axial half-length upper Z Subscript normal upper Delta upper Phi $Z_{{\Delta} \varPhi }$ obtained from the wall-mounted magnetic probes at t equals 57.5 mu normal s $t = 57.5\, \unicode{x03BC} \mathrm{s}$ . In addition, the internal probe array at z equals 0 $z=0$ is used to extract the radial extent and strength of field reversal (e.g. the zero-crossing radius of upper B Subscript z Baseline left parenthesis r right parenthesis $B_{z}(r)$ and the minimum value of upper B Subscript z $B_{z}$ ), which provides an independent indicator of the core ‘thickness’. These experimentally defined metrics collectively show that increasing mirror-field setting shifts the configuration towards larger radial extent and shorter axial extent.

In the experiment, it was difficult to vary the mirror magnetic field strength continuously. Moreover, changing the mirror magnetic field strength affected other parameters such as the translation velocity. Therefore, we scanned the target plasma dimensions { upper R Subscript phi Baseline comma upper Z Subscript phi Baseline right brace $R_{{\phi} },Z_{{\phi} }\}$ and the mirror magnetic field increment ( normal upper Delta upper B Subscript normal m ${\Delta} B_{\mathrm{m}}$ ). The other input parameters were fixed to typical conditions for the FAT-CM device, which are listed in table 3. Figure 11 shows the results, which were categorised into three outcomes: relaxation to an FRC equilibrium, relaxation to an HBM equilibrium and no convergence (NC). Figure 11(a) shows the effects of the mirror magnetic field strength on the plasma half-length ( upper Z Subscript phi $Z_{\phi}$ ) when the plasma radius upper R Subscript phi $R_{{\phi} }$ was fixed at 0.2 m. As the mirror magnetic field strength increased, the plasma state tended to converge to a shorter FRC. In addition, the range of the mirror magnetic field strength over which an FRC was obtained increased as upper Z Subscript phi $Z_{{\phi} }$ decreased. These results suggest that FRC equilibrium is favoured for plasma states with a shorter axial length and that the formation of a shorter FRC requires a higher mirror magnetic field strength. Figure 11(b) shows the effects of the mirror magnetic field strength on upper R Subscript phi $R_{{\phi} }$ when upper Z Subscript phi $Z_{{\phi} }$ was fixed at 0.7 m. As the mirror magnetic field strength increased, the plasma state tended to converge to an FRC with a larger upper R Subscript phi $R_{{\phi} }$ . The range of mirror magnetic field strengths over which an FRC was obtained increased with upper R Subscript phi $R_{{\phi} }$ . These results indicate that FRC equilibrium is favoured for plasma states with a larger radius.

Table 3.

Typical plasma parameters for FAT-CM used as inputs for Grushenka.

A table listing typical plasma parameters for FAT-CM used as inputs for Grushenka.

Figure 11.

Equilibrium solutions as functions of the mirror magnetic field strength: (a) plasma half-length and (b) plasma radius.

Scatter plot showing equilibrium solutions as functions of mirror magnetic field strength.

5. Discussion

In the present collisional merging experiments at FAT-CM, the mirror magnetic field strength was controlled by the charging voltage of the mirror coils (i.e. nominal vacuum mirror setting). To compare equilibrium solutions, we characterised each discharge at the analysis time by using the effective mirror ratio ( upper R Subscript normal upper M ${R}_{\mathrm{M}}$ ), which was evaluated from the measured magnetic field profile. Experimentally, stronger mirror settings were associated with sign reversal of the midplane axial magnetic field upper B Subscript z Baseline left parenthesis r right parenthesis $B_{z}(r)$ , and with merged plasma states that were radially larger and axially shorter. For weaker mirror settings, no clear reversal of upper B Subscript normal z $B_{\mathrm{z}}$ was observed. Nevertheless, the excluded-flux radius remained substantial at approximately half of the chamber radius, which indicates a strongly diamagnetic plasma even in the absence of a closed magnetic flux. In this regime, Grushenka tended to converge to HBM equilibria with no closed magnetic flux rather than to FRC equilibria with a closed magnetic flux (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020). To interpret these trends, we performed a parametric study on Grushenka by using experiment-like inputs while scanning the target plasma dimensions StartSet upper R Subscript phi Baseline comma upper Z Subscript phi Baseline EndSet $\{R_{{\phi} },Z_{{\phi} }\}$ and the mirror magnetic field strength. The results showed that the region in StartSet upper R Subscript phi Baseline comma upper Z Subscript phi Baseline EndSet $\{R_{{\phi} },Z_{{\phi} }\}$ space that converged to the FRC equilibrium branch depended strongly on the mirror magnetic field strength. A higher mirror magnetic field strength expanded the FRC equilibrium domain and favoured plasma states with shorter axial lengths. This behaviour is qualitatively consistent with the role of end mirrors in providing axial confinement and thereby reducing the internal axial force that must be supplied by core field-line tension (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020). Conversely, when Grushenka converged to HBM equilibria, the magnetic flux remained open, and the plasma state typically occupied a more axially elongated region, which reflects the need to balance the imposed mirror magnetic field without the formation of a closed magnetic flux (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020).

Several caveats should be emphasised. First, FAT-CM produces plasma through a dynamic translation and collisional merging process. Therefore, shot-to-shot reproducibility can be limited, and changing the charging voltage of the mirror coils may also modify dynamic quantities such as translation velocity and collision energy. Such dynamic effects (e.g. super-Alfvénic acceleration driven by magnetic pressure gradients) cannot be captured by static equilibrium reconstruction methods (Kobayashi & Asai Reference Kobayashi and Asai2021). Second, the excluded-flux method provided a convenient estimate of the plasma size, but it can be affected by field curvature and physics in the open-field region. Therefore, the excluded-flux radii should be interpreted with caution, especially when the configuration is not clearly a closed magnetic flux (Tuszewski Reference Tuszewski1980). Finally, the bifurcation between FRC and HBM equilibrium branches can depend on parameters that were held fixed in the present scans (e.g. upper T Subscript normal t normal o normal t $T_{\mathrm{tot}}$ , upper B 0 $B_{0}$ , rotation) as well as on the trapped flux and heating history set by the merging dynamics (Gota et al. Reference Gota2018). Within these limitations, the present analysis using Grushenka reproduced the experimentally observed tendency that a higher mirror magnetic field strength is associated with FRC equilibrium solutions that have a larger plasma radius and shorter plasma length. This supports the use of Grushenka as a tool for rapidly estimating plasma states and exploring the operating windows for forming FRCs in FAT-CM while emphasising that equilibrium-based predictions should be complemented by dynamical analysis.

6. Summary

Collisional merging experiments at FAT-CM showed that FRC formation depends on the applied mirror magnetic field strength. Conventional quasi-1-D equilibrium analysis does not account for the axial structure of the mirror magnetic field. Therefore, we applied Grushenka to reconstruct fully 2-D equilibria consistent with routine magnetic measurements (Steinhauer et al. Reference Steinhauer, Roche and Steinhauer2020). Experiments were performed at four charging voltages for the mirror coils. At higher charging voltages, a clear sign reversal of the midplane axial magnetic field strength upper B Subscript z Baseline left parenthesis r right parenthesis $B_{z}(r)$ was observed, and the inferred plasma state shifted towards a larger radius extent and shorter axial length. The equilibria reconstructed by Grushenka reproduced these trends: equilibria with a shorter axial extent and larger radial extent were more likely to converge to the FRC branch. In addition, increasing the mirror magnetic field strength tended to produce equilibria that were axially shorter and radially larger. These results indicate that a higher mirror magnetic field strength more readily leads to convergence towards the FRC branch. These results support the use of Grushenka as a rapid tool to estimate the plasma state produced by FAT-CM and to explore the operating windows for FRC formation. Because the merging process is highly dynamic, these equilibrium-based estimates should be interpreted together with shot-to-shot variations and dynamical effects.

Acknowledgements

The authors would like to express their sincere gratitude to the faculty members who supported the experiments and provided valuable discussions, as well as to their laboratory colleagues for their contributions and support.

Editor Cary Forest thanks the referees for their advice in evaluating this article.

Competing interests

The authors report no conflict of interest.

References

Asai, T. et al. 2019 Collisional merging formation of a field reversed configuration in the FAT-CM device. Nucl. Fusion 59, 056024.CrossRefGoogle Scholar
Dettrick, S.A. et al. 2021 Simulation of equilibrium and transport in advanc ed FRCS. Nucl. Fusion 61, 106038.CrossRefGoogle Scholar
Gota, H. et al. 2024 Enhanced plasma performance in C-2W advanced beam-driven field-reversed configuration experiments. Nucl. Fusion 64, 112014.CrossRefGoogle Scholar
Gota, H. et al. 2018 Internal magnetic field measurements of translated and merged field-reversed configuration plasmas in the FAT-CM device. Rev. Sci. Instrum. 89, 10J114.CrossRefGoogle ScholarPubMed
Kobayashi, D. & Asai, T. 2021 Experimental evidence for super-Alfvénic acceleration of the field-reversed configuration due to a magnetic pressure gradient. Phys. Plasmas 28, 022101.CrossRefGoogle Scholar
Ohtsuka, T., Okada, S. & Goto, S. 1999 Particle transport properties in the edge plasma of a field-reversed configuration. J. Plasma Fusion Res. Ser. 2, 222–225.Google Scholar
Steinhauer, L.C. & Nicks, S. 2025 Ion particle confinement in field-reversed configurations. Phys. Plasmas 32, 062506.CrossRefGoogle Scholar
Steinhauer, L.C., Roche, T. & Steinhauer, J.D. 2020 Anatomy of a field-reversed configuration. Phys. Plasmas 27, 112508.CrossRefGoogle Scholar
Tuszewski, M. 1980 Excluded Flux Analysis of a Field Reversed Plasma. Los Alamos Scientific Laboratory.CrossRefGoogle Scholar
Watanabe, T., Asai, T., Takahashi, T., Kobayashi, D. & Harashima, D. 2021 Internal magnetic measurement in collisional merging process of a field-reversed configuration. Rev. Sci. Instrum. 92, 053541.CrossRefGoogle ScholarPubMed
Figure 0

Figure 1. Figure 1 long description.Schematic of a field-reversed configuration.

Figure 1

Figure 2. Schematic view of FAT-CM.

Figure 2

Table 1. Shot conditions and representative plasma parameters (t$t$ = 57.5 µs).

Figure 3

Figure 3. Positions of measurement instruments in FAT-CM: (a) x=0$x=0$ cross-section and (b) midplane (z=0$z=0$) cross-section.

Figure 4

Figure 4. Computational domain of Grushenka.

Figure 5

Figure 5. Calculation flowchart of Grushenka.

Figure 6

Table 2. Geometry and boundary conditions for FAT-CM.

Figure 7

Figure 6. Contours of the magnetic fields calculated by Grushenka for the (a) HBM equilibrium solution and (b) FRC equilibrium solution.

Figure 8

Figure 7. Radial distributions of the magnetic fields calculated by Grushenka for the (a) HBM equilibrium solution and (b) FRC equilibrium solution.

Figure 9

Figure 8. Plasma size parameters calculated by Grushenka for the (a) HBM equilibrium solution and (b) FRC equilibrium solution.

Figure 10

Figure 9. Time evolution of the magnetic field at the midplane measured by IMPs for a representative shot.

Figure 11

Figure 10. Excluded-flux radius and half-length determined from wall-mounted magnetic probe measurements. The solid line shows a representative shot and shaded regions denote ±1σ, where σ is the sample standard deviation.

Figure 12

Table 3. Typical plasma parameters for FAT-CM used as inputs for Grushenka.

Figure 13

Figure 11. Equilibrium solutions as functions of the mirror magnetic field strength: (a) plasma half-length and (b) plasma radius.