INTRODUCING THE NOVATRON, A NOVEL REACTOR CONCEPT A PREPRINT Jan Jäderberg, Jan Scheffel, Katarina Bendtz, Rickard Holmberg, Kristoffer Lindvall, Per Niva, Robin Dahlbäck, Katherine Dunne Novatron Fusion Group Stockholm, Sweden [email protected] October 27, 2023 ABSTRACT A new magnetic mirror design is described – the Novatron – with the potential to confine fully stable compact mirror-cusp fusion plasmas. The novel design features an axisymmetric magnetic field configuration with concave curvature throughout, as seen from the outside. Moreover, the Novatron is designed to be self-stabilized against the drift cyclotron loss-cone modes (DCLC). This paper describes the fundamental topology of the magnetic field and presents an outline for the design of the magnet system. Keywords: Novatron, mirror, cusp, minimum-B, MHD-stable, axisymmetric magnetic mirror, high-β, fusion reactor concept, compact pilot reactor 1 Introduction 1.1 Background and history Since the beginning of fusion research, several promising reactor designs have been proposed and studied. Two principal branches of magnetic confinement emerged during this time: designs with closed field lines, like tokamaks and stellarators, and designs with open field lines, such as mirror machines and cusps. By the mid-1980s, both branches made steady progress El-Guebaly [2010], but major funding bodies decided to narrow focus to the tokamak –which at the time was outperforming mirror machines. However, the past four decades of mirror machine research have created significant breakthroughs Simonen et al. [2008], paving the way for mirrors and cusps to potentially become viable, cost-efficient devices. A challenge for mirror confinement is the interchange (flute) instabilities occurring at plasma surfaces with “unfavorable” curvature Teller [1981a]. Unfavorable curvature, as opposed to favorable curvature, is defined as decreasing magnetic field in the direction of decreasing plasma pressure. This corresponds to convex magnetic field lines, as seen from the plasma region. The unfavorable curvature is indeed an intrinsic problem for configurations based on the classic magnetic mirror design. Another problem for mirror designs is axial loss, that is loss through the mirror throat, which occurs for particles with a too high axial velocity component. The region in the velocity space corresponding to the plasma which leak out is referred to as the loss cone Ryutov [1988]. Drift cyclotron loss-cone (DCLC) modes is a kinetic electron drift wave instability in the azimuthal direction. Properties of the plasma causing DCLC modes to develop include gradients in the velocity distribution as well as anisotropy Post and Rosenbluth [1966], giving rise to excessive losses if present. A common way of DCLC modes to appear is by fast electrons escaping from the end plasma (through the loss cone) resulting in the formation of a plasma potential which is negative outside the mirrors and positive inside the mirror, commonly referred to as ambipolar potential El-Guebaly [2010]. The positive plasma potential expels low energy ions, causing a void in the ion velocity distribution referred to as an ambipolar hole. These anisotropic, non-Maxwellian ion distributions can thus drive these micro-instabilities. The paper is organized as follows. In the remainder of the introduction, the first two sections give a brief history of the mirror/cusp experiments at Lawrence Livermore National Laboratory (LLNL) and Nagoya University, respectively. The Novatron concept is then introduced followed by other mirror concepts. Section 2 is devoted to a more detailed layout of the Novatron design, where the first sections focus on the fundamental topology, making a comparison to a classical mirrors, cusps and mirror-cusps. The latter sections discuss the magnetic domains, the aspect ratio and number of Larmor radii, the magnetic null (B = 0) region, the mirror ratio, and the expanders. Section 3 provides the conclusions and the paper is finalized by the authors’ acknowledgements and contributions. 1.1.1 Mirrors at the Lawrence Livermore National Laboratory (LLNL) In the early 1950s, an ambitious magnetic mirror fusion research program Bishop [1958], led by Richard F. Post, was launched at the LLNL. The research program included the so-called Toy Top Magnetic Mirrors Teller [1981b] which achieved MHD stability to the interchange mode using Ioffe bars. The Toy Top experiments lead to the development of the three consecutive 2X machines Coensgen et al. [1972], where instead of Ioffe bars, Baseball coils were employed to address the interchange instabilities. These coils additionally improved upon the mirror ratio Teller [1981b]. The succeeding 2XII machine Coensgen et al. [1972], equipped with more compact (i.e. even higher mirror ratio) Yin-Yang coils El-Guebaly [2010], demonstrated an improved suppression of interchange instabilities and increased confinement time. The upgraded version 2XIIB Coensgen et al. [1976] (B for beams) was equipped with 12 neutral beam modules of 20 kV, a total of 12 MW. The neutral beams raised the temperature and density of the plasma (exceeding 10^20 m^-3), but the confinement time was limited by DCLC instabilities and the inability to reach fusion relevant temperatures. A solution involving injecting a stream of plasma near the mirrors to control the DCLC instabilities was implemented Post [1987]. For all the minimum-B configurations, i.e. with field-line averaged favorable curvature, such as Ioffe bars, Yin-Yang and baseball coils, the stabilizing minimum-B property was achieved by abandoning axisymmetry, thus introducing neoclassical, radial transport Teller [1981a]. In 1975, 2XIIB became the first fusion experiment to reach 100 million degrees Fowler [1997], three years before the Princeton Large Torus (PTL) Tokamak. Furthermore, 2XIIB still holds the record for mirror machines in terms of ion temperature, triple product and high β. The conclusion made by the end of the 2XIIB experiment was that the qualitative feasibility of mirror fusion containment had been established, instabilities could be adequately controlled and a dense plasma at thermonuclear temperatures could be created and sustained Kesner [1983]. However, further improvement of energy confinement time τE and of the triple product was limited by the cooling effect from the flow of lower energy plasma needed to suppress DCLC modes Coensgen et al. [1972], Fowler et al. [2017], Baker et al. [1984], the low mirror ratio, neoclassical transport as well as structure mechanical complications in designing the Yin-Yang coils Teller [1981a]. 1.1.2 Mirror-cusps at Nagoya University Radio Frequency (RF) plugging is an approach to prevent axial losses by making use of the ponderomotive force acting on the particles when an oscillating electrical field is applied. In the 1960's a research program was initiated at the Institute of Plasma Physics at Nagoya University, eventually demonstrating the success of RF plugging in cusps with densities of 10^18 m^-3 and 0.4 keV Sato [1985]. A series of experiments and configurations were studied, starting with the spindle cusp TPD-II Hiroe et al. [1975] and finalized by the RFC-XX-M Okamura et al. [1984]. While the TPD experiments consisted of a single cusp, the RFC-XX experiments had a central cell (with a uniform or a mirror field) and the cusps were placed at the ends, introducing anchor functionality (see Sec. 1.3.1). RFC-XX-M proved to be stable under a range of plasma conditions despite the fact that the inside of the plasma has unfavorable curvature Baker et al. [1984]. 1.2 Introduction to the Novatron The Novatron, a mirror-cusp magnetic topology, provides the interchange stabilizing features of the successful Yin-Yang coils, but in an axisymmetric system with a geometry that has a much higher mirror ratio and larger plasma volume than the classical cusp. The Novatron design has a low aspect ratio, that is ratio between length and diameter (short and wide) enabling many Larmor radii across the plasma radius. Table 1: Critical issues for mirror machines and Novatron's approach to address them. rp here denotes the plasma radius and rL the Larmor radius. Challenge | Novatron approach Interchange instability | Favorable curvature DCLC instability | High rp / rL ratio Neoclassical transport | Axial symmetry Axial losses | High mirror ratio This provides an alternative stabilization against the DCLC modes which limited performance in earlier mirror devices Teller [1981b]. A high mirror ratio, rM ≈ 16 is planned to mitigate axial losses, which the RFC-XX was missing. The design features and the problems they address are summarized in Table 1. and will be elaborated in more detail in Section 2. Furthermore, the compact and simple design of the magnets allows for a cost-efficient and reliable reactor concept. Figure 1: The shape of the vacuum magnetic field lines for a basic Novatron design with its characteristic features. All boundary flux surfaces passing through the mirror throats will have a favorable curvature, see the text. 1.3 Other current mirror concepts 1.3.1 Techniques to suppress interchange modes, DCLC modes, and axial losses It is generally accepted that the confinement characteristics of a simple mirror device are insufficient for achieving Q factors (produced fusion power / heating power) of relevance for a reactor. Current mirror concepts are addressing the confinement and stability problems related to mirror designs in various ways, with a general focus on axisymmetric designs with unfavorable curvature in the central cell and where the interchange instability is (planned to be) stabilized by other means. One approach is so called vortex stabilization Beklemishev et al. [2010] by which the plasma is stabilized by rotating it around the central axis. Another approach is anchor cells Ryutov et al. [2011], regions outside the central cell with favorable curvature making up for the unfavorable curvature in the central cell. According to the criteria derived by Longmire and Rosenbluth Rosenbluth and Longmire [1957] favorable curvature at one section of the field line can under certain conditions, make up for unfavorable curvature at another section. A third technique, line-tying, makes use of conducting plates at the end of the plasma region allowing for electrons to flow between field lines and thus short-circuit unstable flux tubes Katanuma et al. [2013]. Potential DCLC instabilities, in turn, are being addressed either by filling the ambipolar hole Ryutov [1988] or by a new approach using the centrifugal potential associated with plasma rotation to trap low-energy ions Endrizzi et al. [2023]. The ambipolar hole can be filled by inserting 'warm' (and isotropic) plasma (ions), or by a technique commonly referred to as sloshing ions Fowler [2022], whereby ions are injected at an angle towards the field lines are causing ions to bounce between mirrors, making it possible to create a potential at the adjustable turning point, thereby trapping warm thermal ions Simonen et al. [1983]. (Footnote 1: Sloshing ions can also be created using neutral beams directed at an angle to the field lines.) Expanders are cells placed at the ends of the central mirror in the axial direction where the magnetic field is allowed to "fan out" after the mirror throat (see Fig. 4). By diluting the influx of plasma on the end plates, it suppresses secondary emission electrons from the end plates potentially cooling the plasma Ryutov [2004]. The resulting favorable curvature of the field in the expanders can also be used to stabilize against interchange by the same principle as for anchor cells Ryutov et al. [2011]. Expanders also offer the opportunity to provide a radially electric field using biased collectors. This radial field is used for vortex stabilization in the central cell. Both interchange stabilization and vortex stabilization using expanders rely on a certain degree of plasma flow between the central cell and the expanders. For the vortex confinement, the plasma needs to be in electrical contact with the absorber electrodes in the expanders. To confine the plasma axially, a high mirror ratio can be supplemented with the introduction of electrostatic fields in the so called tandem cells, extending the central cell in the axial direction. This is commonly referred to as electrostatic end-plugging and is intended to confine the ions. Electrostatic end-plugging can be augmented by a modification of the electrostatic potential, commonly referred to as a thermal barrier, extending the confinement to also include electrons. 1.3.2 Gas Dynamic Trap and related experiments The Gas Dynamic Trap (GDT) Ivanov and Prikhodko [2013] developed at the Budker Institute (Novosibirsk), is a classic axisymmetric mirror with a long mirror-to-mirror distance, exceeding the effective mean free path of ion-ion scattering into the loss cone, and with high mirror ratio, yielding the plasma isotropic and Maxwellian. The central cell is supplemented by expanders. To further stabilize against interchange modes, several methods have been researched including line tying, anchor cells as well as vortex confinement using the expanders Ivanov et al. [2016]. The vortex confinement method currently used for stabilizing against interchange modes requires a certain axial loss for the plasma to be in direct electric contact with the absorber electrodes Simonen et al. [2010]. The axial loss constitutes a challenge for reactor size scaling Bagryansky [2023]. Ambipolar electrostatic barriers in the expanders are currently being investigated to address the axial loss problem Soldatkina [2023]. Alternative remedies including several multi-mirrors (e.g. GDMT Skovorodin et al. [2023]) and/or helical confinement extensions (e.g. SMOLA Sudnikov et al. [2017]) of the GDT concept are being/have been investigated. Regarding DCLC, injection of warm plasma with an isotropic distribution has been employed where a few percent of isotropic ions are sufficient for suppression. However, this method reduced the theoretical Q factor from 1 to 0.04 owing to axial losses of the warm ions Bagryansky [2023]. There is to date a lack of agreement between theoretical and experimental accounts of DCLC in the GDT experiment Kotelnikov et al. [2017]. 1.3.3 Gamma 10 The Gamma 10 experiment Inutake et al. [1985] at the University of Tsukuba is the world’s largest existing mirror machine measuring 27 m in the axial direction. It is an axisymmetric mirror equipped with tandem cells used for electrostatic and thermal end plugging, which has been shown to suppress axial losses successfully Inutake et al. [1985]. To stabilize against interchange modes, the design includes baseball coil anchor cells. Interchange instabilities, however, have been observed Minami et al. [2002]. Sloshing ions have been used to stabilize against DCLC Yatsu et al. [1984]. A density limit of 2 × 10^18 m^-3 has been observed Kawamori et al. [2002] due to challenges in Ion Cyclotron Radio Frequency (ICRF) heating for high density plasmas. Currently, this constitutes an active area of research for the experiment. Another challenge for the Gamma 10 is the low electron temperature leading to cooling of the ions. In recent years, the Gamma 10 research program has primarily revolved around investigations relevant to the large international tokamak experiments ITER and DEMO. 1.3.4 The Wisconsin HTS Axisymmetric Mirror (WHAM) Inspired by GDT, the WHAM experiment Endrizzi et al. [2023] at the University of Wisconsin-Madison is conceptualized as a classic axisymmetric mirror with high mirror ratio (up to 16). Several experimental approaches are planned for stabilization of interchange modes including vortex stabilization. To achieve the high mirror ratio and high beta, a strong (17 T) magnetic field will be induced from high temperature superconducting (HTS) magnets. Sloshing ions and centrifugal potential from plasma rotation are planned approaches for suppressing DCLC modes. An active area of research complementing the WHAM experiment is in prediction modeling. The modeling encompasses, among other things, stability analysis of anisotropic MHD equilibrium profiles, Fokker-Planck simulations, kinetic instability analysis, and particle confinement time estimates. 2 The Novatron magnet design 2.1 Axial symmetry and favorable curvature – comparison to the classical mirror and the cusp Two basic fusion concepts, or configurations, within the open field line group are the classic magnetic mirror (partially unfavorable curvature) and the biconic cusp (favorable curvature). The classic magnetic mirror has a magnetic field pointing in the normal direction of the symmetry plane while the biconic cusps have a tangential magnetic field toward the symmetry plane, see Fig. 2. In this context, the Novatron constitutes its own category of magnetic confinement, with a normal magnetic field at the symmetry plane like the classic mirror but featuring “favorable curvature” like the biconic cusp. Figure 2: Coils and field lines for a schematic classic mirror (a), a biconic cusp (b), and a Novatron (c). Current directions in the coils are marked with a cross (×) for current directed into the plane and a dot (·) for current directed out of the plane of view. The Novatron magnetic configuration is shown in Fig 1. The magnetic field design is characterized by fully “favorable curvature” of the outer plasma surface, due to an increasing magnetic field in the outward direction. It is theoretically established Rosenbluth and Longmire [1957] that such a field will provide MHD stability for an inwardly directed plasma pressure gradient. The axisymmetric design prevents neo-classical transport, that is non-collisional particle drifts in the radial direction Fowler et al. [2017]. These losses constituted a problem for non-axisymmetric minimum-B stable configurations like Baseball or Yin-Yang coil devices which had abandoned the axisymmetry. A classic magnetic mirror (Fig. 3a) features a large volume, however with a geometry allowing for detrimental interchange instabilities to develop. By adding a second magnetic coil pair located concentrically outside the first magnet pair (Fig. 3b) and with current running in the opposite direction, the magnetic flux will be squeezed in the ring-shaped volume (mirror throat) between the magnets, thus locally increasing the B field. The shape of the central field lines passing through the symmetry plane has now been transformed from convex to concave, that is from unfavorable curvature to favorable curvature (Fig. 3b). This constitutes the most fundamental step of the Novatron concept. Figure 3: A schematic classic mirror (a). Adding a concentric coil with opposite current (b) will give a favorable curvature of the inner field lines. A solenoidal coil radially outside the plasma volume with the current in the same direction as the second coil system (c) will transform the outer field lines from unfavorable to favorable curvature. Adding a third magnetic coil system (Fig. 3c) located between the two mirror throat volumes with a current running in the same direction as the second magnet coil pair also will transform the outer flux lines from convex to concave so that all field lines that cross the symmetry plane, and enter the mirror throats will have favorable curvature. This results in an increased total flux in the plasma confinement domain. 2.2 Comparison to RFC-XX experiments The RFC-XX magnetic field (see Sec. 1.1.2), though bearing resemblance to the Novatron in that it consists of a central cell with favorable curvature complemented by cusps at the ends, importantly has mirror throats perpendicular to the central axis. By increasing the inclination angle, a higher mirror ratio can be realized. Figure 4: Terminology for the Novatron. The symmetry plane is the Z = 0 plane, the separatrix is the field line separating flux running in the clockwise from those running in the counter clockwise direction. The chimney and the mirror throat are the two different types of mirrors in the Novatron, where the mirror throats are extended by expanders. BM (B0) is the magnetic field strength at the mirror throat (symmetry plane). Furthermore, B = 0 in two regions centered on the z axis. 2.3 Magnetic domains When adding the second magnet system (Fig. 3b), magnetic separatrices - separating flux going in the clockwise direction from flux going in the counterclockwise direction - will appear (Fig. 4). The magnetic domains outside the separatrices with flux lines passing between the center hole (chimney) and the mirror throat will have favorable curvature for plasma with its pressure gradient directed away from the coils. Flux lines from the mirror throat passing through the symmetry plane will extend through the corresponding mirror throat on the opposite side of the symmetry plane. This flux makes up the "inner magnetic domain" (teal in Fig. 5) which is the main plasma domain. The mirror ratio, defining the mirror loss cone, is very high compared to Yin-Yang or other minimum-B coil designs. Figure 5: The Novatron concept offers two distinct magnetic domains, the inner magnetic domain (in teal) and additional outer magnetic domain (in gray). As mentioned, the Novatron concept provides alternative MHD stable domains – the outer magnetic domains (marked in gray in Fig. 5). Flux that passes through the mirror throat and bends back into the chimney, forms the "outer magnetic domain". The chimney constitutes an area with the highest B-field in the system (Fig. 6). The associated favorable curvature and a high mirror ratio is expected to support inner domain plasma confinement. Figure 6: Magnetic field strength vs curve length for three field lines characteristic of the Novatron magnetic field. The top curve is the field line encircling the M1 magnet, the middle curve is a field line close to the separatrix and running within the inner magnetic domain and the bottom curve is the outermost field line, as marked in the figure. Figure 7: Magnetic field strength along symmetry plane (see Fig. 4). 2.4 Aspect ratio and number of Larmor radii The low aspect ratio (“short and wide” design) of the Novatron, see Fig. 8, supports a high rp/rL, where rp is the plasma radius in the symmetry plane and rL is the ion Larmor radius, important for DCLC stabilization Coensgen et al. [1976]. Figure 8: The vacuum vessel of the N1 experiment being built at the Alfvén laboratory, KTH Royal Institute of Technology, Stockholm. The DCLC modes are predicted to disappear at a value of 20-100 El-Guebaly [2010], Ferron and Wong [1984]. As an historic comparison, the rp/rL ratio in 2XIIB was typically 2 (except for a large plasma radius test) resulting in DCLC mode activity limiting the performance Hua and Fowler [2004]. DCLC modes were therefore suppressed by a flow of plasma filling the voids in the velocity space for the ions. The injected plasma suppressed DCLC modes but at the cost of cooling the hot central fusion plasma Teller [1981b], Coensgen et al. [1972]. As a more current example, the WHAM experiment is planning for an rp/rL ratio of 3-4 Endrizzi et al. [2023], and is planning to stabilize against DCLC using sloshing ions and plasma rotation, see Sec. 1.3.4). The Novatron will instead suppress DCLC modes mainly by a large value of rp/rL designed to reduce the need for other methods. This has the potential to be a substantial advantage Novatron as compared to i.e. 2XIIB, expected to further enhance the nτE-value. 2.5 B = 0 At two points, where the separatrices cross the axial symmetry axis, the magnetic field is zero (B = 0 in Fig. 4.). Charged particles passing near these points are likely to be scattered into the loss cone and disappear from the plasma region via the separatrix. This may cause the density profile to become hollow, with its minimum on the z-axis. A study of a similar curvature configuration which was investigated at the RFC-XX-M experiment (see Sec. 1.1.2) and featuring a hollow density profile, was performed in the first half of the 1980s Baker et al. [1984]. It was experimentally observed that over a wide range of plasma parameter values the unfavorable pressure gradient plasma in the region close to B=0, was stable to large MHD perturbations in a line-averaged sense. The same study also suggests a possible theoretical account. This phenomenon will be studied in detail in the Novatron geometry by Particle-In-Cell and MHD simulations as well as experimentally by the first Novatron prototypes. 2.5.1 Mirror ratio, RM A core feature of the Novatron design is the high mirror ratio, which again, is crucial in suppressing axial losses. For a reactor sized Novatron, the mirror ratio is planned to exceed 16. Even assuming the conservative nτE relation of log10(RM) Post [1987] as originally suggested by Rosenbluth, a high mirror ratio contributes significantly to the confinement time. 2.6 Expanders The expansion ratio is defined as the ratio of the B-field at the mirrors (BM) compared to the B-field at the end walls (see Figure 4). The expansion ratio is important to prevent cooling of the plasma from secondary electrons emitted off the end plates Soldatkina et al. [2017]. An expansion ratio exceeding the square root of the ion to electron mass ratio (≈ 60 for deuterium plasmas) has been shown to be necessary, notably another limiting factor of previous mirrors such as the 2X experiments. The Novatron design enables a large expansion ratio to the end walls, exceeding the desired expansion ratio of 60. 3 Conclusions The mirror machine concept holds the potential for a compact reactor capable of confining fusion plasmas at high temperatures and high beta values. However, previous mirror machines have encountered challenges due to asymmetry, leading to neoclassical losses, and kinetic instabilities such as DCLC instabilities. The Novatron concept, as discussed in this paper, is designed to address these issues. It aims to suppress DCLC modes without (or with significantly reduced) plasma flow, which would otherwise cool the central plasma. Additionally, its axisymmetric design prevents neoclassical losses. However, the presence of magnetic null regions could potentially lead to nonadiabaticity and the creation of hollow density profiles at the device’s center. Baker et al. [1984] have demonstrated that solutions can be implemented to mitigate these issues. Current efforts are focused on simulations and theoretical studies to optimize plasma shape and position. In conclusion, the Novatron’s innovative approaches to overcoming the challenges faced by mirror machines suggest that it has the potential to achieve record high triple products. This makes it a promising avenue for future research and development in the field of fusion energy. 4 Acknowledgments The authors wish to acknowledge valuable conversations with Professor Kenneth Fowler, and for bringing the author’s attention to relevant literature and experiments. 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