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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Relativistic Electron Precipitation Events Driven by Solar Wind Impact on the Earth's Magnetosphere

Certain forms of solar wind transients contain significant enhancements of dynamic pressure and may effectively drive magnetosphere dynamics, including substorms and storms. An integral element of such driving is the generation of a wide range of electromagnetic waves within the inner magnetosphere, either by compressionally heated plasma or by substorm plasma sheet injections. Consequently, solar wind transient impacts are traditionally associated with energetic electron scattering and losses into the atmosphere by electromagnetic waves. In this study, we show the first direct measurements of two such transient-driven precipitation events as measured by the low-altitude Electron Losses and Fields Investigation CubeSats. The first event demonstrates storm-time generated electromagnetic ion cyclotron waves efficiently precipitating sub-relativistic and relativistic electrons from >300 keV to 2 MeV at the duskside. The second event demonstrates whistler-mode waves leading to scattering of electrons from 50 to 700 keV on the dawnside. These observations confirm the importance of solar wind transients in driving energetic electron losses and subsequent dynamics in the ionosphere.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Collisionless cooling of perpendicular electron temperature in the thermal quench of a magnetized plasma

Thermal quench of a nearly collisionless plasma against an isolated cooling boundary or region is an undesirable off-normal event in magnetic fusion experiments, but an ubiquitous process of cosmological importance in astrophysical plasmas. Parallel transport theory of ambipolar-constrained tail electron loss is known to predict rapid cooling of the parallel electron temperature $T_{e\Vert}$ although $T_{e\Vert}$ is difficult to diagnose in actual experiments. Instead direct experimental measurements can readily track the perpendicular electron temperature $T_{e\bot}$ via electron cyclotron emission. The physics underlying the observed fast drop in $T_{e\bot}$ requires a resolution. Here two collisionless mechanisms, dilutional cooling by infalling cold electrons and wave-particle interaction by two families of whistler instabilities, are shown to enable fast $T_{e\bot}$ cooling that closely tracks the mostly collisionless crash of $T_{e\Vert}$. These findings motivate both experimental validation and reexamination of a broad class of plasma cooling problems in laboratory, space, and astrophysical settings.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Propagation of ion cyclotron emission in the DIII-D tokamak

Experiments in L- and H-mode plasmas performed on the DIII-D tokamak explored ion cyclotron emission (ICE) propagation via the recently upgraded ICE diagnostic. The distance between the plasma and the outer wall was scanned to alter the evanescent region, which was seen to impact the detection of runaway electron-driven whistler modes in a comparable frequency range to typical ICE harmonics (f≳10fci) [Heidbrink et al., Plasma Phys. Controlled Fusion 61, 014007 (2018)]. In L-mode plasmas, core ICE did not exhibit clear trends as the plasma-wall distance was altered. Instead, inadvertent changes to the fast ion density correlated with different ICE fine structures; the most spectral bands (spaced ∼150–200 kHz apart) were observed at the highest fast ion density, and a just single band when nfast decreased by ∼ 40%. The full-wave Petra-M code simulated core ICE propagation through the plasma and to ICE probes, agreeing with experimental findings that these are likely compressional fast waves, which propagate to probe locations with no dependence on plasma-wall gap. Edge ICE in H-mode plasmas was similarly unaffected by plasma-wall distance. Unlike in core ICE cases, there were negligible changes to the edge fast ion distribution and edge bulk plasma profiles. However, changes in ELM frequency seemed most closely tied to ICE behavior. Ultimately, core and edge ICE did not exhibit clear dependencies on the plasma-wall distance, and ICE harmonics were observed with even the largest plasma-wall gaps. This independence bodes well for similar diagnostics in future tokamaks, which might be placed further from high radiation areas without adversely impacting measurement capabilities.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

RF antenna helicity dependent particle heating in a helicon source

Experiments have demonstrated that ion phenomena, such as the lower hybrid resonance, play an important role in helicon source operation. Damping of the slow branch of the bounded whistler wave at the edge of a helicon source (i.e. the Trivelpiece-Gould mode) has been correlated with the creation of energetic electrons, heating of ions at the plasma edge, and anisotropic ion heating. Here we present ion velocity distribution function measurements, electron density and temperature measurements, and magnetic fluctuation measurements on both sides of an m = |1| helical antenna in a helicon source as a function of the driving frequency, magnetic field strength, and magnetic field orientation relative to the antenna helicity. Significant electron and ion heating (up to two times larger) occurs on the side of the antenna consistent with the launch of the m = +1 mode. The electron and ion heating occurs within one electron skin depth of the plasma edge, where slow wave damping is expected. The source parameters for enhanced particle heating are also consistent with lower hybrid resonance effects, which can only occur for Trivelpiece-Gould wave excitation.

wave modes↗
DOE OSTI2024

First high-power helicon results from DIII-D

Abstract More than 0.6 MW of rf power at 476 MHz has been coupled to DIII-D plasmas by launching helicon (whistler) waves with a traveling-wave antenna (comb-line) in the fast-wave polarization (Van Compernolle et al 2021 Nucl. Fusion 61 116034) which resulted in the observation of electron heating of the core plasma with single-pass absorption based on ray-tracing in L-mode discharges. The coupling performance of the 1.5 m wide 30-element comb-line traveling-wave antenna has been consistent with expectations based on the 2015–2016 experiments on DIII-D with a low-power 12-element prototype (Pinsker et al 2018 Nucl. Fusion 58 106007). The conditioning process that was necessary to carry out high-power experiments is discussed; rf-specific impurities have not been observed. Parametric decay instabilities have been observed and are being investigated as a potential edge absorption mechanism (Porkolab et al 2023 AIP Conf. Proc. 2984 070004).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Stability of superthermal strahl electrons in the solar wind

We present a kinetic stability analysis of the solar wind electron distribution function consisting of the Maxwellian core and the magnetic-field aligned strahl, a superthermal electron beam propagating away from the sun. We use an electron strahl distribution function obtained as a solution of a weakly collisional drift-kinetic equation, representative of a strahl affected by Coulomb collisions but unadulterated by possible broadening from turbulence. This distribution function is essentially non-Maxwellian and varies with the heliospheric distance. The stability analysis is performed with the Vlasov–Maxwell linear solver leopard. We find that depending on the heliospheric distance, the core-strahl electron distribution becomes unstable with respect to sunward-propagating kinetic-Alfvén, magnetosonic, and whistler modes, in a broad range of propagation angles. The wavenumbers of the unstable modes are close to the ion inertial scales, and the radial distances at which the instabilities first appear are on the order of 1 au. However, we have not detected any instabilities driven by resonant wave interactions with the superthermal strahl electrons. Instead, the observed instabilities are triggered by a relative drift between the electron and ion cores necessary to maintain zero electric current in the solar wind frame (ion frame). Contrary to strahl distributions modelled by shifted Maxwellians, the electron strahl obtained as a solution of the kinetic equation is stable. Our results are consistent with the previous studies based on a more restricted solution for the electron strahl.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Collisionless ion-electron energy exchange in magnetized shocks

Energy partition between ions and electrons in collisionless shocks has been a long-standing unsolved fundamental physical question. Here, we show that kinetic simulations of moderate Alfv´enic Mach number, magnetized, collisionless shocks reveal rapid, faster-than-Coulomb, energy exchange between ions and electrons when the plasma is sufficiently magnetized. Using kinetic and multi-fluid models with counter-streaming ions, we identify resonances between electron whistler and ion magnetohydrodynamic waves that account for this rapid energy exchange.

High-energy-density plasmas↗
DOE OSTI2025

Theoretical investigation of plasma wave generation by pulsed electron beams in space

Here, we report theoretical calculations of plasma wave generation in the whistler modes and in the extraordinary modes, by pulsed electron beams in a magnetized plasma. The numerical simulations of the wave generation take into account the longitudinal expansion of the electron beam due to the space charge force and the energy spread. The work presented in this article provides predictions for the wave generation performance of the beam plasma interactions experiment (Beam PIE), where pulsed electron beams were produced by a spaceborne radio frequency (RF) linear accelerator. We also theoretically explore the desirable properties of the pulsed electron beam for future space experiments, which will be the next step toward eventually demonstrating the radiation-belt remediation (RBR).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Development of High-efficiency and Cost-effective Forged Ingot Niobium Technology for Science Frontiers and Accelerator Applications

Development of Forged Ingot Niobium Technology: Worlds science frontier programs and SRF accelerator applications demand high performance and cost-effective SRF accelerator technology [1-8]. Fine-grain (FG) and Large-grain (LG) ingot niobium technologies have been very well developed and implemented in all the present-day accelerator projects. However, forged ingot niobium technology which is the focus of this development proposal will be much more cost-effective and expected to have several technical advantages. FG niobium sheet production is very complex involving more than ten processing steps making them prone to contamination. As a result, they are very expensive to produce and require stringent QA procedure to be ready for SRF cavity production. The accelerating cavity process steps are also numerous and require strict procedures in order to achieve high accelerating gradients and quality factors needed for science frontier programs. LG niobium disc production, directly sliced from the ingot, is relatively simple and straight forward to keep surface cleanness. The disc production cost is significantly low compared to FG niobium sheet production. However, there are (some) draw backs due to non-homogeneity of the grain boundary distribution, resulting in non-uniform mechanical properties and complex cavity fabrication, although the LG cavities achieve the expected high-gradient performance goals with lower cost. Medium-grain (MG) niobium disk production may be realized with a new approach/process, the disc directly sliced from the forged ingot, involves a simpler process steps contributing major production cost reduction [9]. These discs are expected to be superior as they tend to be homogenous with uniform sub millimeter grains and mechanical properties. We are eagerly looking forward to developing the forged ingot niobium SRF accelerator technology for the benefit of the world-wide science frontier programs, green energy subcritical nuclear energy systems and a wide variety of industrial applications including the production of radio isotopes and nuclear transmutation applications. Measurement of thermal characteristics of the forged ingot niobium: Measurement of the thermal diffusivity, D, of superconducting MG niobium is important to be understood in comparison with FG sheet and LG disc, as well as that of advanced composite material of Nb3Sn film sputtered on forged ingot Nb, which will be determined using transient pump-probe thermo-modulation [10,11]. Transient thermo-modulation is based on using an ultrafast laser pulse to heat the superconducting materials by a few K, then a synchronized laser pulse probes the reflectance of the heated material. For thin films on a substrate (e.g, 100-nm Nb3Sn on Nb), it takes <100 ps for the heat to reach the substrate by diffusion. Therefore, an ultrafast method is needed to probe D of the studied material. D will be measured from room temperature to liquid helium temperatures. Fundamental Research: The increase of rf loss in SRF cavities is related to trapping of residual magnetic field during the cavity cool-down. The study suggested that the micro structure plays the role in flux trapping sensitivity. The research includes the flux trapping and expulsion study as forged ingot niobium goes through several mechanical deformations, crystallization leading to the optimal SRF cavity performance. Furthermore, the cavity made from FG, MG, and LG sheet/disc will be compared to understand the optimal re-crystallization temperature that cavity needed to be heat treated which minimize the flux trapping and increase SRF cavity performance. References: 1. S. Belomestnykh, Overview of recent SRF developments for ERLs, presented at the SRF 2015, Whistler, BC, Canada (2015). 2. A. Yamamoto, M. Yamanaka and G. Myneni, Ingot Nb based SRF Technology for the International Linear Collider, in Science and Technology of Ingot Niobium for Superconducting Radio Frequency Applications, AIP Conf. Proc. 1687, 030005-1 ? 03005-6, 2015 3. Report of the Workshop on Energy and Environmental Applications of Accelerators, DOE Workshop Report. https://science.osti.gov/-/media/hep/pdf/Reports/2020/CASM_WorkshopReport.pdf? la=en&hash=AEB0B318ED0436B1C5FF4EE0FDD6DEB84C2F15B2 4. G. Ciovati, et al., Design of a cw, low-energy, high-power superconducting linac for environmental applications, Phys. Rev. Accel. Beams 21, 091601 (2018). 5. P. Dhakal, et al., Effect of high temperature heat treatments on the quality factor of a large-grain superconducting radio-frequency niobium cavity, Phys. Rev. ST Accel. Beams 16, 042001 (2013). 6. P. Kneisel et al., Review of ingot niobium as a material for superconducting radio frequency accelerating cavities, Nuclear Instruments and Methods in Physics Research A 774, 133 (2015). 7. G. Ciovati, P. Dhakal, and G. R, Myneni, Superconducting radio-frequency cavities made from medium and low-purity niobium ingots, Supercond. Sci. Technol. 29, 064002 (2016). 8. M. Drury et al., commissioning of the prototype C75 cavities in a CEBAF cryomodule, in Proc. of IPAC 2018, Vancouver, BC, Canada (2018) 9. Feasibility of forged-ingot niobium disc and SRF cavity fabrication technology recently demonstrated in cooperation of ATI, BSCE, and KEK (2020). 10. Hani E. Elsayed-Ali, ?Measurements of heat transport in thin films by ultrafast laser-based techniques,? 3rd International Conference on Thermal Issues in Emerging Technologies Theory and Applications, Cairo, Egypt, pp. 347 ? 350 (2010). DOI: 10.1109/ThETA17616.2010 11. W. M. G. Ibrahim, H. E. Elsayed-Ali, M. Schinn, and C. A. Bonner, Jr., ?Ultrafast investigation of electron dynamics in multilayer metals,? Int. J. Heat and Mass Transfer, 47(10?11), 2261?2268 (2004).

Myneni, G.↗
DOE OSTI2023

Can strong substorm-associated MeV electron injections be an important cause of large radiation belt enhancements?

It has become well-established that strong outer radiation belt enhancements are due to wave-driven electron energization by whistler-mode chorus waves. However, in this study, we examine strong MeV electron injections on 10 July 2019 and find substantial evidence that such injections may be a crucial contributor to outer radiation belt enhancement events. For such an examination, it is essential to precisely separate temporal flux changes from spatial variations observed as Van Allen Probes move along their orbits. Employing a new “hourly snapshot” analysis approach, we discover unprecedented details of electron flux evolutions that suggest that for this event, the outer belt enhancement was not continuous but instead intermittent, mostly composed of 4 large discrete injection-driven flux increases. The injections appear as sharp flux increases when observed near apogee. Otherwise, by comparing hourly snapshots for different times, we infer injections and infer temporally stable fluxes between injections, despite strong and continuous chorus emission. The fast and intermittent electron flux growth successively extending earthwards implies cumulative outer belt enhancement via a series of repetitive inward transport associated with injection-induced electric fields.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Simulation of Plasma Emission in Magnetized Plasmas

The recent Parker Solar Probe observations of type III radio bursts show that the effects of the finite background magnetic field can be an important factor in the interpretation of data. In the present paper, the effects of the background magnetic field on the plasma-emission process, which is believed to be the main emission mechanism for solar coronal and interplanetary type III radio bursts, are investigated by means of the particle-in-cell simulation method. The effects of the ambient magnetic field are systematically surveyed by varying the ratio of plasma frequency to electron gyrofrequency. The present study shows that for a sufficiently strong ambient magnetic field, the wave–particle interaction processes lead to a highly field-aligned longitudinal mode excitation and anisotropic electron velocity distribution function, accompanied by a significantly enhanced plasma emission at the second-harmonic plasma frequency. For such a case, the polarization of the harmonic emission is almost entirely in the sense of extraordinary mode. On the other hand, for moderate strengths of the ambient magnetic field, the interpretation of the simulation result is less clear. The underlying nonlinear-mode coupling processes indicate that to properly understand and interpret the simulation results requires sophisticated analyses involving interactions among magnetized plasma normal modes, including the two transverse modes of the magneto-active plasma, namely, the extraordinary and ordinary modes, as well as electron-cyclotron-whistler, plasma oscillation, and upper-hybrid modes. At present, a nonlinear theory suitable for quantitatively analyzing such complex-mode coupling processes in magnetized plasmas is incomplete, which calls for further theoretical research, but the present simulation results could provide a guide for future theoretical efforts.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Planar Collisionless Shock Simulations with the Semi-implicit Particle-in-cell Model FLEKS

This study investigates the applicability of the semi-implicit particle-in-cell code FLexible Exascale Kinetic Simulator (FLEKS) to heliospheric shock simulations. We examine one- and two-dimensional local planar shock simulations, initialized using MHD states with upstream conditions representative of plasmas in the hypersonic, β ∼ 1 regime, for both quasi-perpendicular and quasi-parallel configurations. The refined algorithm in FLEKS proves robust, enabling accurate shock simulations with a grid resolution on the order of the electron inertial length d e . Our simulations successfully capture key shock features, including shock structures (foot, ramp, overshoot, and undershoot), upstream and downstream waves (fast magnetosonic, whistler, Alfvén ion-cyclotron, and mirror modes), and non-Maxwellian particle distributions. Crucially, we find that at least two spatial dimensions are critical for accurately reproducing downstream-wave physics in quasi-perpendicular shocks and capturing the complex dynamics of quasi-parallel shocks, including surface rippling, shocklets, short, large-amplitude magnetic structures, magnetic reconnection, and jets. Furthermore, our parameter studies demonstrate the impact of mass ratio and grid resolution on shock physics. This work provides valuable guidance for selecting appropriate physical and numerical parameters for shock simulations using a semi-implicit PIC method, paving the way for incorporating kinetic shock processes into large-scale collisionless plasma simulations with the MHD-AEPIC model.

plasma astrophysics↗
DOE OSTI2026

Asymptotic-preserving semi-implicit finite volume scheme for extended magnetohydrodynamics

A Finite Volume (FV) scheme is developed for solving the extended magnetohydrodynamic (XMHD) equations, yielding accurate results in the ideal, resistive, and Hall MHD limits. This is accomplished by first re-writing the XMHD equations such that it allows the algorithm to retain the use of ideal MHD Riemann solvers and the constrained transport method to preserve divergence-free magnetic fields. Incorporation of electron inertia and displacement current introduces additional numerical stiffness which motivates a semi-implicit FV scheme that re-formulates the XMHD model as a relaxation system. The equations are then advanced in time using an explicit 2nd-order Runge–Kutta scheme with operator splitting applied to the implicit source term updates at each sub-stage. For additional numerical stability, a density-dependent slope limiter is implemented to increase flux diffusivity at low density regions where non-ideal effects become significant. The algorithm is subsequently implemented in a scalable adaptive mesh refinement (AMR) framework. As the new algorithm retains many aspects of the ideal MHD formulations, it asymptotes naturally to the ideal MHD limit. Moreover, it shows promising results at the resistive and Hall MHD limits. This is verified against reference test problems for ideal, resistive and Hall MHD.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Verifying Ray Tracing Amplitude Methods for Global Magnetospheric Modeling

Abstract Ray tracing is a commonly used method for modeling the propagation of electromagnetic waves in Earth's magnetosphere. To apply ray tracing results to global models of wave‐particle interaction such as energetic electron scattering, it is useful to map the discrete rays to a volume filling mesh. However, some methods have inherent losses of energy from the wave source, or do not account for the full range of wave properties within a sample volume. We have developed and tested a 3D magnetospheric ray tracing code “MESHRAY” which resolves these issues. MESHRAY uses the conservation of Poynting flux through ray triplets with finite volume to determine the local field amplitudes. Electromagnetic wave energy density from all ray data points is mapped to a mesh and verified against the wave source power for energy conservation varying time step length, number of rays, and total time steps. We find that the method is self‐consistent and numerically robust. We further investigate whether the neglect of phase information and superposition has a significant impact on the accuracy of mapping wave intensity to a mesh. We find excellent agreement between the analytic solution for waves emitted by a line source in a plane‐stratified medium and an equivalent ray tracing solution. When phase information is excluded, ray tracing reproduces an average amplitude spread over regions of coherent constructive and destructive interference. This may be an important consideration for interpolating ray tracing results of longer wavelength waves such as magnetosonic, electromagnetic ion cyclotron, or ULF waves.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Upgrades to the ion cyclotron emission diagnostic on the DIII-D tokamak

The ion cyclotron emission diagnostic on the DIII-D tokamak comprises seven single-turn loops that measure high-frequency (1–100 MHz) magnetic field fluctuations that are often excited by energetic particles in the plasma. The raw voltage signals induced in the loops in response to these fluctuations travel through a series of cables, isolation transformer DC blocks, low-pass filters, and finally a digitizer before being analyzed in frequency space. The diagnostic has been recently upgraded, most notably to include four additional graphite tile loops and a new eight-channel digitizer. The previous three loops are all on the low-field side of the tokamak. The measurement capabilities of the system have been expanded by the addition of a new horizontally oriented loop on the low-field side, an additional toroidal loop on the low-field side, and two toroidal loops on the high-field side. These loops will be used to provide approximate mode polarization, improved toroidal mode number calculations, and information on modes in inward-shifted plasmas, respectively.

47 OTHER INSTRUMENTATION↗
DOE OSTI2021

Deterministic scattering of relativistic electron beams by off-resonant circularly polarized electromagnetic waves

A mechanism is presented whereby relativistic electron beams localized in phase space are deterministically scattered by coherent circularly polarized electromagnetic waves without stochastic processes. It is shown via an exact single-particle analysis that the condition for maximal scattering is an off-resonant condition, contrary to previous kinetic analyses that predict maximal diffusion or interaction at exact resonance or its harmonics. The mechanism, verified by single-particle simulations, enables a fast, nonlinear redistribution of the beam particles. A possible application of this mechanism to runaway electron suppression is presented.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

A new and alternative look at nonlinear Alfvénic states

The formulation for studying nonlinear Alfvenic states, sustainable in Hall Magnetohydrodynamics (HMHD), becomes considerably simpler and more tractable when circularly polarized Beltrami vectors (the eigenstates of linear HMHD) are used as the basis functions. Nonlinear HMHD is, then, reduced to a rather simple looking set of scalar equations from which a model problem of three interacting Beltrami modes is formulated and analytically solved. The triplet interactions span a variety of familiar nonlinear processes leading to a redistribution as well as periodic exchange of energy. The energy exchange processes (whose strength is measured by an energy exchange/depletion time) will, perhaps, play a dominant role in determining the spectral content of an eventual Alfvenic state. All nonlinearities (sensitive functions of the interacting wave vectors) operate at par, and none is dominant over any substantial region of k-space; their intricate interplay prevents a “universal” picture from emerging; few generalizations on the processes that may, for instance, lead to a turbulent state, are possible. However, the theory can definitely claim: (1) the energy tends to flow from lower to higher k and (2) the higher k z (in the direction of the ambient magnetic field) components of a mode with a given k are depleted/oscillate faster—in some cases much faster. It is noteworthy that the mode coupling is the strongest (with the shortest depletion time) when the participating wave vectors are nearly perpendicular; perhaps, an expected consequence of the curl (cross product) nonlinearities. Numerical simulations will be necessary to help create a fully reliable picture.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Guiding center equations for the magnetic dipole

Since the discovery of Van Allen radiation belts in the 1960s, observations of energetic ions trapped in the Earth's dipole magnetic field have illustrated the remarkable confinement properties of this configuration. As such, it has been used for confining a hot plasma for nuclear fusion studies, starting from the pioneering work of Bo Lehnert and Akira Hasegawa, in the Levitated Dipole Experiment (LDX) at MIT until 2011 and in the RT-1 experiment at the University of Tokyo. More recently, the dipole has been subject to a renewed interest for fusion studies by a couple of startups and for smaller applications as a cold plasma source. While the equilibrium and magneto-hydrodynamic stability of the dipole have been investigated quite in detail, neoclassical properties of the dipole are comparatively much less known: the dipole is more known in geophysics than in fusion science. For this reason, in this paper, we propose a set of Hamiltonian, guiding-center equations to describe the motion of electrons and ions in a magnetic dipole configuration. We also developed a code, and we show the main features of particle motion, benchmarking our results with the analytical solutions for the bounce and precession motion, which are well documented in the literature. We also draw some general conclusions for the neoclassical transport in usual toroidal confinement schemes, such as the tokamak and the stellarator, pointing out the unique advantages of the dipole in confining energetic particles.

Hamiltonian mechanics↗