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This study examines the impact of low and zero magnetic shear, along with rational values of safety factor, on linear mode behavior in tokamak plasmas. We focus on how magnetic field line topology and parallel domain length influence linear mode structures and growth rates as magnetic shear decreases. Our results show that as magnetic shear is reduced to low values, linear modes can transition between different branches, significantly affecting their characteristics. At zero magnetic shear, accurate modeling requires precisely capturing magnetic topology, as small deviations near rational surfaces can greatly impact growth rates. [ABSTRACT FROM AUTHOR]

Previously, tokamak research has been focused mainly on large aspect ratio devices where the vessel/plasma major radius is about a factor three larger than the plasma radius. This research culminated in the design and construction of the international thermonuclear experimental reactor, ITER. Spherical tokamaks (ST), with aspect ratio below two, represent an attractive alternative to large aspect ratio tokamaks as, in our opinion, provide a faster, more economical and compact solution on the path to a fusion reactor. STs are the focus of research at Tokamak Energy Ltd with its present device ST40 in operation and the first ST reactor being designed, taking advantage of the high temperature superconductor (HTS) technology. HTS allow to design a ST with magnetic field/comparable or exceeding that of present-day large aspect ratio tokamaks. However, plasma studies carried out so far on compact, low aspect ratio tokamaks have been limited to small, low magnetic field, low plasma-current devices and therefore the data available for extrapolating to large scale ST plasmas is limited. This paper addresses the problem of scaling the results of large aspect ratio tokamak and existing ST plasmas to a high field ST reactor using plasma-similarity arguments in order to mitigate its design and operational risks. The role of the plasma aspect ratio in scaling burning plasmas as well as conventional experiments in deuterium is highlighted. We find that the scaling for fusion-reactor plasmas exhibit a stronger dependence on the magnetic field and aspect ratio than the one of conventional non-burning plasmas. The parameters of a ST having the same fusion gain Q fus of ITER under different confinement assumptions and for different aspect ratios are presented and discussed. [ABSTRACT FROM AUTHOR]

It is possible to generate the low-divergence ion bunches through the interaction of equally or slightly different counter-propagating laser pulses. In this paper, in the framework of plasma beat wave, we have numerically simulated the simultaneous acceleration of carbon/hydrogen ion beams. Using a two-dimensional particle-in-cell simulation, we have shown that in an HC plasma mixture with an optimal hydrogen ratio of 1:5, the laser reflectivity coefficient reduces up to about 4.6%. This condition may provide the acceleration of low-divergence H+ and C4+ ions beam. The cut-off energy for protons and C4+ ions are about 27 MeV and 410 MeV, respectively at n H = 0.2 n C. The values increase by about 15.6% for protons and 21% for C4+ at n H = 0.7 n C. In the presence of an axial magnetic field, the energy absorption arrives at its maximum at the values of Ωs/ ω p = 0.1, where Ωs and ω p are the gyro-frequency and plasma frequency, respectively. Here, the average kinetic energy of the accelerated ions raises by 17.9% and 7.3% for carbon and proton ions, respectively. Compared to the magnetic field-free case, divergence angles were suppressed at approximately 24.2% and 20.3% for the carbon ions and hydrogen ions, respectively. [ABSTRACT FROM AUTHOR]

The paper presents the latest results and summarizes the study of ion heat transport in the Globus-M spherical tokamak in regimes with neutral beam injection. Ion heat transport was investigated in discharges with various toroidal magnetic fields and plasma currents. Measurements of the ion temperature were carried out using charge-exchange recombination spectroscopy and a neutral particle analyzer. Plasma simulations using transport code ASTRA have shown that ion heat transport in Globus-M is predominantly neoclassical even for low collisionality range. The local transport analysis indicates that plasma core is stable to ion scale instabilities. [ABSTRACT FROM AUTHOR]

The Weibel instability, a subject of relevance to many fields of physics ranging from inertial confinement fusion to some astrophysical scenarios, is usually probed with side-on proton radiography. In order to diagnose the strength and wavelength of the spontaneous fields, two concerns that must be settled are how to distinguish the coexisting electric and magnetic fields, and how to overcome the counteracting of deflections in radiographing the filamentary structured fields. In this paper, proton radiography of the Weibel instability in two counterstreaming plasma flows is studied by simulation. It suggests that the electric field dominates the deflection of probe protons, whereas the contribution from the magnetic field is negligible. To resolve the deflectioncounteracting problem, the spatial spectrum of the electric field energy is found to be related to the deflection velocity of the probe beam by theoretical analyses. The strength and wavelength of the electric field are then obtainable from the proton flux on the detection plane, whereas the strength and wavelength of the magnetic field can be deduced through the equilibrium between the electric field and the magnetic field pressure gradient after the linear growth stage of the instability. Both numerical and experimental verifications suggest that our method performs well in extracting the strength and wavelength of the spontaneous electric and magnetic fields of the Weibel instability from proton radiography. [ABSTRACT FROM AUTHOR]

By driving hot electrons between two metal plates connected by a wire loop, high power lasers can generate multi-tesla, quasi-static magnetic fields in miniature coil targets. Many experiments involving laser-coil targets rely on proton deflectometry directed perpendicular to the coil axis to extract a measurement of the magnetic field. In this paper, we show that quantitative measurements using perpendicular probing are complicated by the presence of GV m−1 electric fields in the target that develop on sub-ns timescales. Probing parallel to the coil axis with fiducial grids is shown to reliably separate the electric and magnetic field measurements, giving current estimates of I ≈ 5 kA in 1 mm- and 2 mm-diameter wire loops. An analytic model of proton deflection in electric and magnetic fields is used to benchmark results from the particle-in-cell code and help deconvolve the magnetic and electric field deflections. Results are used to motivate a new experimental scheme that combines a single-plate target with axial proton probing and direct current measurements. This scheme has several important advantages over the traditional target and diagnostic set-up, enabling the robust measurement of coil magnetic fields and plasma properties, as well as making it easier to validate different theoretical models at a range of laser intensities. [ABSTRACT FROM AUTHOR]

The management of excessive heat flux to tokamak divertors is one of the most important issues, especially for long-pulse operations of EAST. Different simulation codes based mainly on fluid models have been developed to study the physical processes in edge plasma, including the heat flux to divertor target plates. In these simulations, only the neutral region is included and the heat flux is estimated by using energy transmission coefficients based on a simplified sheath model. In this paper, particle-in-cell simulations are conducted to study the heat flux flowing to the EAST divertor target plates by taking the sheath into account. It is found that the parallel electron velocity distribution function (VDF) in the sheath is a cut-off Maxwellian distribution so the electron sheath energy transmission can be obtained analytically, while ion VDFs in the sheath strongly depend on the incident angle between the magnetic field and divertor target plates. For a small incident angle, in the sheath, the ions move along open orbits and are accelerated mainly in the direction perpendicular to the plane formed by the magnetic field and target plates due to the Lorentz force. An analytical expression for the ion heat flux to the target plates is presented and the obtained results are in good agreement with the simulation results. [ABSTRACT FROM AUTHOR]

Spherical tokamaks (STs) have features that make them a potentially attractive option for fusion power production compared to conventional tokamaks (CTs) including operation at high beta and high self-driven 'bootstrap' current. The thermal energy confinement time (τΕ) also typically has a stronger dependence on toroidal magnetic field and a weaker dependence on plasma current, but so far it has not been established how this difference impacts performance under reactor conditions. This aspect is explored in this paper. Using empirical data from NSTX and MAST, and from multiple CTs, we investigate analytically and by using established fusion codes the potential fusion performance, characterised by the fusion triple product, nTτΕ, and fusion power gain, Qfus, where n and T are the density and temperature respectively. We find that for similar values of field and fusion power, but smaller volume, STs can have nTτΕ up to a factor of three higher and Qfus an order of magnitude higher than CTs. We identify the origin of this enhanced performance and outline a measurement to advance this finding. Potentially our results open an alternative and faster route to fusion power based on relatively small, low power STs. [ABSTRACT FROM AUTHOR]

During magnetic reconnection, such as in the magnetopause of magnetized planets, the upstream plasma conditions between the two inflow regions are usually different. In this paper, we demonstrate that such a kind of asymmetric reconnection can be studied in the laboratory using the recently proposed experimental scheme where reconnection is driven by laser-powered capacitor coils. Two-dimensional particle-in-cell simulations on the plane in a cylindrical coordinate are conducted to study magnetic reconnection with the inflow along the direction. Magnetic reconnection is found to be asymmetric with a stronger magnetic field in the inner (small) inflow region and a weaker magnetic field in the outer (large) inflow region due to the cylindrical symmetric geometry. Electron crescent velocity distributions are observed near the flow stagnation point while ion crescent velocity distributions are observed in the region with Larmor electric field. The out-of-plane Hall magnetic field is asymmetric between the two inflow regions with a larger spatial scale in the outer inflow region. This asymmetric Hall magnetic field configuration is different from that in previous studies. The typical reconnection rate increases with a stronger driver and the highest rate is around 0.2 , where is the typical value of the magnetic field and is the Alfven speed. This study provides a new method to experimentally study asymmetric reconnection in the laboratory and has potential applications regarding magnetic reconnection in the magnetopause of magnetized planets. [ABSTRACT FROM AUTHOR]

The evolution of a deformed subcritical fast magnetosonic shock front is compared between two two-dimensional particle-in-cell simulations with different orientations of the magnetic field relative to the simulation box. All other initial and simulation conditions are kept identical. Shock boundary oscillations are observed in the simulation where the magnetic field direction is resolved. This oscillation is caused by the reformation of the shock front. One part of the front acts as a shock, while the other functions as a magnetic piston, with both halves changing their states in antiphase. The oscillation period corresponds to the time required for one shock wave to grow as the other collapses. In contrast, the corrugated fast magnetosonic shock does not oscillate in the second simulation, where the magnetic field is oriented out of the simulation plane. This dependence on magnetic field orientation suggests that the shock oscillation is induced by magnetic tension, which is only effective in the first simulation. In both simulations, the shock perturbation does not grow over time, indicating that the shocks are stable. The potential relevance of these findings for the Alfvénic oscillations of the supercritical Earth's bow shock, detected by the MMS multi-spacecraft mission, is also discussed. [ABSTRACT FROM AUTHOR]

Recent numerical calculations have shown that while strong toroidal rotation can increase the external kink limit of tokamak plasmas, the associated rotation shear can drive a Kelvin-Helmholtz like global instability in the plasma, if the rotation frequency exceeds a threshold value (Chapman et al 2011 Plasma Phys. Control. Fusion 53 125002; Chapman et al 2012 Nucl. Fusion 52 042005). On the basis of a large aspect ratio toroidal expansion of the magnetohydrodynamic stability equations, the present paper investigates analytically various properties of this instability in tokamak plasmas with sonic toroidal flows and low magnetic shear in the core region. We also compare the analytical results with numerical code calculations. Many characteristic features and parameter dependences of the instability can be understood from the analytical theory, such as an eigenmode structure peaking at the position of largest rotation shear, and insensitivity of the growth rate to the plasma beta and to the precise value of the safety factor in the region of low magnetic shear. From an algebraic expression for the growth rate, valid asymptotically at large rotation frequencies, the drop in the dynamic pressure associated with the flow in the plasma can be identified as a major driving mechanism of the instability. For modes with (dominant) poloidal mode number m > 1, and rotating equilibria with isothermal magnetic surfaces, another driving mechanism of the instability is related to the centrifugally induced density variation along the magnetic field lines. [ABSTRACT FROM AUTHOR]

ITER as a superconducting fusion machine needs efficient wall conditioning techniques for application in the presence of the permanent high toroidal magnetic field for (i) reducing the in-vessel impurity content, (ii) controlling the surface hydrogen isotopic ratio and (iii) mitigating the in-vessel long-term tritium inventory build-up. Encouraging results recently obtained with ion-cyclotron wall conditioning (ICWC) in the present-day tokamaks and stellarators have raised ICWC to the status of one of the most promising techniques available to ITER for routine inter-pulse and overnight conditioning with the ITER main ICRF heating system in the presence of the permanent high toroidal magnetic field. This paper is dedicated to a milestone experiment in ICWC research: the first simulation of ICWC operation in an equivalent ITER full-field scenario and the assessment of the wall conditioning effect on the carbon wall in the largest present-day tokamak JET. In addition, we address in this paper the following topics: (i) an analysis of the radio frequency (RF) physics of ICWC discharges, (ii) the optimization of the operation of ICRF antennas for plasma startup and (iii) an outlook for the performance of ICWC in ITER using the ICRF heating system. Important operational aspects of the conventional ICRF heating system in JET (the so-called A2 antenna system) for use in the ICWC mode are highlighted: (i) the ability of the antenna to ignite the cleaning discharge safely and reliably in different gases, (ii) the capacity of the antennas to couple a large fraction of the RF generator power (>50%) to low-density (1016-1018 m-3) plasmas and (iii) the ICRF absorption schemes aimed at improved RF plasma homogeneity and enhanced conditioning effect. Successful optimization of the JET-ICWC discharge parameters (BT = 3.3T, f = 25 MHz) resulted in a reliable operation of the JET A2 antennas and a high conditioning efficiency in a scenario imitating closely ITER full-field operation (BT = 5.3T, f = 40 MHz) with the fundamental ion-cyclotron resonance for deuterium (ω = ΩD+ ) located on-axis. Numerical modelling with the 3D electromagnetic code Micro Wave Studio, a 1D RF full wave code and a 0D plasma code allows extrapolating the results obtained on JET and other present-day tokamaks to ITER and provides good prospects for the use of the ITER ICRF antennas for ICWC purposes. [ABSTRACT FROM AUTHOR]

cRadio frequency (RF) heating experiments have recently been conducted in JET (3He)-H plasmas. This type of plasmas will be used in ITER's non-activated operation phase. Whereas a companion paper in this same PPCF issue will discuss the RF heating scenario's at half the nominal magnetic field, this paper documents the heating performance in (3He)-H plasmas at full field, with fundamental cyclotron heating of 3He as the only possible ion heating scheme in view of the foreseen ITER antenna frequency bandwidth. Dominant electron heating with global heating efficiencies between 30% and 70% depending on the 3He concentration were observed and mode conversion (MC) heating proved to be as efficient as 3He minority heating. The unwanted presence of both 4He and D in the discharges gave rise to 2 MC layers rather than a single one. This together with the fact that the location of the high-field side fast wave (FW) cutoff is a sensitive function of the parallel wave number and that one of the locations of the wave confluences critically depends on the 3He concentration made the interpretation of the results, although more complex, very interesting: three regimes could be distinguished as a function of X[3He]: (i) a regime at low concentration (X[3He] < 1.8%) at which ion cyclotron resonance frequency (ICRF) heating is efficient, (ii) a regime at intermediate concentrations (1.8 < X[3He] < 5%) in which the RF performance is degrading and ultimately becoming very poor, and finally (iii) a good heating regime at 3He concentrations beyond 6%. In this latter regime, the heating efficiency did not critically depend on the actual concentration while at lower concentrations (X[3He] < 4%) a bigger excursion in heating efficiency is observed and the estimates differ somewhat from shot to shot, also depending on whether local or global signals are chosen for the analysis. The different dynamics at the various concentrations can be traced back to the presence of 2 MC layers and their associated FW cutoffs residing inside the plasma at low 3He concentration. One of these layers is approaching and crossing the low-field side plasma edge when 1.8 < X[3He] < 5%. Adopting a minimization procedure to correlate the MC positions with the plasma composition reveals that the different behaviors observed are due to contamination of the plasma. Wave modeling not only supports this interpretation but also shows that moderate concentrations of D-like species significantly alter the overall wave behavior in 3He-H plasmas. Whereas numerical modeling yields quantitative information on the heating efficiency, analytical work gives a good description of the dominant underlying wave interaction physics. [ABSTRACT FROM AUTHOR]

Measurement of plasma parameters during the ablation phase in the Z-pinch is crucial for investigating the dynamic behaviors. In this study, optical Thomson scattering was employed to measure the temperature and velocity of the ablation plasma in a double-wire Z-pinch. The scattering spectra profiles were fitted using a model that considered the velocity distribution. The experimental results revealed the energy evolution of ablation plasma, the establishment of the global magnetic field and the development of axial non-uniformities. The precursor plasma was found to play a key role in strengthening the global magnetic field. A resistive layer near the wire core with a size of 1.5 mm was observed in the ablation plasma after the precursor plasma column formed. The plasma underwent rapid heating in this layer, the electron temperature rises from 17 eV to 22 eV. Upon leaving this layer, electron the temperature is stable at around 22 eV. The radial distribution of the ablation rate increases and decreases, indicating the axial motion of the ablation plasma, which could be caused by the tilt motion of the stream and the secondary modulation of the natural wavelength. [ABSTRACT FROM AUTHOR]

The virtual-casing principle is used in plasma physics to convert a Biot–Savart integration over a current distribution into a surface integral over a surface that encloses the current. In many circumstances, use of virtual casing can significantly speed up the computation of magnetic fields. In this paper, a virtual-casing principle is derived for a general vector field with arbitrary divergence and curl. This form of the virtual-casing principle is thus applicable to both magnetostatic fields and electrostatic fields. The result is then related to Helmholtz’s theorem. [ABSTRACT FROM AUTHOR]

Modulated lower hybrid waves, are injected into the Experimental Advanced Superconducting Tokamak (EAST), to investigate the modulated evolutions of pedestal structure and unstable edge-localized modes (ELMs). The spectrum patterns of show that the ELM amplitude only slightly decreases while the ELM frequency becomes almost 3–5 times larger as the lower hybrid radio frequency (LHRF) jumps into the lower power level. Pedestal structure analysis indicates that higher LHRF power injection can increase the plasma stored energy and core plasma parameters, accompanying with the pedestal shift and gradient increase. Ion saturation contour current plots suggest that LHRF power can deposit in the plasma edge region to profoundly change the three-dimensional (3D) magnetic topology, similarly to the effect of resonant magnetic perturbations (RMPs), by inducing the helical current filaments (HCFs) following along the magnetic field lines in the scrape-off layer (SOL), which may cause the density pump-out effect by increasing the radial plasma transport. The modulated higher temperature and lower density of pedestal top-values can reduce the pedestal collisionality, which will strongly increase the bootstrap current density and gradually shift the Peeling-Ballooning mode (PBM) boundary. ELITE code stability analysis indicates that LHRF injection can lead to the relatively unstable ELMs with a higher pedestal gradient, simultaneously accompanying with the normalized pressure gradient and edge bootstrap current density increase. All of these results imply that there is a significant correlation between the ELM behavior and pedestal structure modulation because of LHRF dissipation at the plasma edge for the unstable ELM modulated case. [ABSTRACT FROM AUTHOR]

The impact of hyper-resistivity on non-ideal ballooning modes (BMs) is studied in the presence of resonant magnetic perturbation (RMP) through considering the hyper-resistivity, resistivity and diamagnetic effect in the BM model with an equilibrium distorted by RMP, which is stable for ideal BMs. Similar to the resistivity, the hyper-resistivity is also destabilizing for the BMs, but RMPs make the mode spectrum of the BMs destabilized by the hyper-resistivity move towards the low toroidal mode number side on the flux surface with a safety factor slightly larger than the RMP resonance safety factor, where the growth rates of the BMs destabilized by the resistivity decrease due to RMP. When both the hyper-resistivity and the resistivity are considered, there is a sort of competitive relationship between them in determining the properties of BMs. If either of the hyper-resistivity term and the resistivity term is much larger than the other one, the instability of BMs is mainly determined by the larger one, and the effect of the smaller one is masked. The destabilizing mechanisms of the hyper-resistivity and the resistivity on BMs are similar, namely, the diffusion and dissipation of current and magnetic field weaken the stabilizing effect of magnetic field line bending. The research results may be important for understanding the enhancement of plasma transport and the mechanism of small edge localized mode (ELM) during ELM control with RMP. [ABSTRACT FROM AUTHOR]

Many models for anomalous transport consider the turbulent E × B transport arising from electrostatic micro-instabilities. In this paper we investigate whether the perturbed magnetic field that is associated with such instabilities at small but finite values of β can lead to significant stochastic magnetic field transport. Using the tearing parity, long wavelength ion temperature gradient (ITG) modes in a plasma slab with magnetic shear as an example, we calculate the amplitude of the perturbed magnetic field at the resonant surface that results. The plasma model consists of a Braginskii description for the electrons with the dissipation at the resonant surface evaluated for the semi-collisional regime, while a finite ion Larmor radius kinetic model is invoked for the ions. The resulting stochastic field transport is estimated and also compared with an estimate for the E × B transport due to the ITG mode. [ABSTRACT FROM AUTHOR]

In this paper, the results of the study of the influence of the applied external axial magnetic field on the dynamics of pinch and neutron production are presented, following from measurements using x-ray, interferometry and neutron diagnostics performed on the plasma-focus PF-1000 device with deuterium as the filling gas at the current of 2 MA and neutron yield above 1010. The permanent magnets with a magnetic field of a few hundredths of tesla were used both inside the anode body and in front of the end of the dense column. This magnetic field decreases the neutron yield, depresses the implosion velocity and the velocity of the transformations of internal structures, stabilizes the pinch column, increases its axial symmetry and indirectly confirms the existence of internal closed currents inside the pinch structures. [ABSTRACT FROM AUTHOR]

Relativistic electron beam propagation through solid density plasma is a rich area for magnetic field dynamics. It is well known that Ohmic heating of the background plasma caused by the beam significantly affects magnetic field generation, primarily through changes in the resistivity. In particular, temperature changes in the background plasma leads to the generation of a magnetic field that acts to deflect relativistic electrons from the beam axis. This 'beam hollowing' field could have disastrous implications for the fast ignitor scheme. In this paper, the effects of background heat flow on magnetic field generation are considered, first with a simple analytic investigation, and then with 1D Vlasov Fokker-Planck and classical transport simulations using a rigid beam for the fast electrons. It is shown that the thermal conduction of the background plasma acts to diffuse the temperature, reducing both the temperature gradients and the beam hollowing field. This gives rise to the re-emergence of a collimating magnetic field. The influence of the background heat flux is also investigated in the context of solids with imposed resistivity gradients, and is shown to significantly enhance the magnetic field present. More exotic transport effects, such as an enhanced Nernst velocity (due to non-local heat flux) and double peaked temperature profiles (due to distortion of the heating and heat-flow profiles by the magnetic field), are also reported. [ABSTRACT FROM AUTHOR]

The last decade has seen the development of new experimental devices to explore the physics of magnetized dusty plasmas. Because of the small charge-to-mass ratio of the charged microparticles, it is necessary to operate these experiments at high magnetic fields of several Tesla in order to observe the direct effect of the magnetic forces on the transport properties of the charged microparticles. While the study of magnetized dusty plasmas is still the ultimate goal, these experiments have also provided new opportunities to studies regimes of strongly magnetized, low temperature, laboratory plasmas that have not been extensively explored. Experiments show the formation of new types of self- and imposed-ordered structures that form in both the plasma and among the microparticles. This paper summarizes recent experimental observations of plasma filamentation (in the plasma) and will discuss possible connections to 'dust gridding' phenomena that are observed in the magnetized dusty plasma experiment device. [ABSTRACT FROM AUTHOR]

We study the diffusion of stochastic isotropic and anisotropic magnetic field lines in turbulence with average poloidal magnetic field component. In the current work we employ the so-called DeCorrelation Trajectory method in order to compute the running and asymptotic diffusion coefficients of stochastic magnetic field lines with average poloidal magnetic field component. A modification of the diffusion coefficients is caused by the average magnetic field component parameter in the presence of trajectory trapping. We calculate the field line diffusion coefficients Dxx and Dyy for different values of the magnetic Kubo number K, the average magnetic field component parameter Kav and the anisotropy parameter Λ. Furthermore we compare our new findings with those already obtained in papers dealing with the calculus of the diffusion coefficients for a stochastic sheared magnetic field. [ABSTRACT FROM AUTHOR]

The problem of inferring the internal inductance ℓ i from external magnetic measurements in tokamaks is considered. It was practically resolved in JET (Barana O et al 2002 Plasma Phys. Control. Fusion 44 2271), but several questions remain to be addressed before extrapolation of that method on other tokamaks. These naturally arise because of the integral nature of ℓ i determined by unknown distribution of the magnetic field inside the plasma. Without universal solutions, the quality of approximations should be examined along with potential experimental enforcements of the existing procedures. This is the main goal of the study that is fully analytical. Several relevant quantities are calculated for the large-aspect-ratio plasma with elliptical shifted magnetic surfaces. Usually the plasma elongation K is mentioned as a key factor in the task. Here it is explicitly shown that not only K , but its radial derivative too plays a role. The latter parameter did not appear in the JET scaling and in the accompanying discussions. It is also shown that the K -dependent term representing there the first iteration actually brings a systematic error. Therefore, several improvements to that scaling can be proposed. The study is based on the use of virial relations for the toroidal axisymmetric configurations. [ABSTRACT FROM AUTHOR]

Quasi-single helicity (QSH) state has been widely observed in reversed field pinch devices to sustain thermal structures, enclosed by steep temperature gradients interpreted as electron internal transport barriers (eITB). Within the evolution of a QSH cycle, three sub-states can be identified in the RFX-mod experiment, named DAx, SHAxn and SHAxw, where the subscript n and w indicate narrow and wide thermal structures, respectively. Such sub-states feature different magnetic topologies as well as different width of the thermal structures. This paper presents a characterization of the three QSH sub-states mainly devoted to describe the properties of the eITB temperature gradients. The analyses show a decreasing trend of the absolute value of the thermal gradient as the dominant mode increases. Further study on the thermal diffusivity confirms that these behaviors are essentially due to the local change of the heat transport. Besides, the thermal gradient in DAx and SHAxn groups shows an increasing trend with the decrease of secondary modes. Such behavior has been studied and it can be explained by stochastic transport in agreement with what found for SHAXw (Lorenzini et al 2012 Nucl. Fusion52 062004). As for SHAxw group, the thermal gradient suddenly decreases with respect to DAx and SHAxn values. This behavior can still be explained in the framework of stochastic transport considering the structure of magnetic perturbations at the location of thermal gradients (Lorenzini et al 2016 Phys. Rev. Lett.116 185002). The total energy confinement time has also been evaluated, adopting a more appropriate method. The result shows that the energy confinement time increases up to 40% from DAx to SHAxw. [ABSTRACT FROM AUTHOR]

The destabilization mechanism of the collisional microtearing mode driven by an electron temperature gradient is studied using theoretical analyses and gyrokinetic simulations including a comprehensive collision model, in magnetized slab plasmas. The essential destabilization mechanism of the microtearing mode is the lag of the parallel inductive electric field behind the magnetic field owing to the time-dependent thermal force and inertia force induced by the velocity-dependent electron–ion collisions. Quantitative measurements of the collision effects enable us to identify the unstable regime against collisionality and reveal the relevance of the collisional microtearing mode with existing toroidal experiments. A nonlinear simulation demonstrates that the microtearing mode does not drive magnetic reconnection with the explosive release and conversion of the magnetic energy. [ABSTRACT FROM AUTHOR]

Magnetic reconnection is the topological reconfiguration of the magnetic field in a plasma, accompanied by the violent release of energy and particle acceleration. Reconnection is as ubiquitous as plasmas themselves, with solar flares perhaps the most popular example. Other fascinating processes where reconnection plays a key role include the magnetic dynamo, geomagnetic storms and the sawtooth crash in tokamaks. Over the last few years, the theoretical understanding of magnetic reconnection in large-scale fluid systems has undergone a major paradigm shift. The steady-state model of reconnection described by the famous Sweet–Parker (SP) theory, which dominated the field for  ∼50 years, has been replaced with an essentially time-dependent, bursty picture of the reconnection layer, dominated by the continuous formation and ejection of multiple secondary islands (plasmoids). Whereas in the SP model reconnection was predicted to be slow, a major implication of this new paradigm is that reconnection in fluid systems is fast (i.e. independent of the Lundquist number), provided that the system is large enough. This conceptual shift hinges on the realization that SP-like current layers are violently unstable to the plasmoid (tearing) instability—implying, therefore, that such current sheets are super-critically unstable and thus can never form in the first place. This suggests that the formation of a current sheet and the subsequent reconnection process cannot be decoupled, as is commonly assumed. This paper provides an introductory-level overview of the recent developments in reconnection theory and simulations that led to this essentially new framework. We briefly discuss the role played by the plasmoid instability in selected applications, and describe some of the outstanding challenges that remain at the frontier of this subject. Amongst these are the analytical and numerical extension of the plasmoid instability to (i) 3D and (ii) non-magnetohydrodynamics (MHD) regimes. New results are reported in both cases. [ABSTRACT FROM AUTHOR]

In this paper, a numerical tool, called the inverse equilibrium tool (IET), for the solution of the inverse equilibrium engineering problem is presented. IET is developed in a MATLAB environment and allows for the computation of the coil currents needed to obtain a predetermined plasma shape with well defined plasma global parameters (i.e. total plasma current and total poloidal magnetic flux at the boundary) by solving a constrained minimization problem. IET can be used for the characterization of an existing plasma boundary or for the full design of a new one. Thus, it can be used to generate families of equilibrium configurations and to determine if the desired plasma shape is within the engineering capabilities of the device. The typical features of IET can be summarized as follows: (a) the plasma shape can be arbitrarily defined or characterized by means of a compact analytical functional form; (b) the fixed boundary plasma equilibrium solver allows the user to arbitrarily define the plasma current density profile for both tokamak and reversed field pinch magnetic configurations; (c) the minimization problem for the computation of the equilibrium coil currents is solved with both single and multi-objective optimization approaches. IET was validated considering both limiter and lower single null ITER-like plasma configurations and an upper single null experimental plasma in the RFX-mod tokamak. For all the cases under analysis, IET leads to accurate results for determining the currents that are able to reproduce the direct equilibrium reconstruction. The accuracy of the solution and the simplicity of both user interface and computational algorithm are the key features of the IET computational tool. [ABSTRACT FROM AUTHOR]

The paper considers the dynamics of the plasma generated by the explosive-emission cells of the cathode spot of a vacuum arc in a magnetic field. It is shown that the expansion of the (high-density) plasma in a transverse magnetic field may cause asymmetry in the plasma density distribution at the cathode spot boundary. The asymmetry, in turn, increases the probability of the ignition of new explosive-emission cells in the region of a stronger magnetic field in the direction of B × I. The disturbed plasma density distribution estimated in the magnetohydrodynamic approximation is presented. In addition, the velocity of the ‘retrograde’ spot motion (ignition of new explosive-emission cells) in the stronger field region is estimated as a function of the external magnetic field strength. The velocity estimates (a few to tens of m s−1 T−1) are shown to agree with experimental data. [ABSTRACT FROM AUTHOR]

The time evolution of the target temperature and ionization degree during the laser-target interaction is of primary importance to understanding the transition between solid and plasma. When the interaction lasts a few tens of femtoseconds, the target resistivity is not well known as in the Spitzer regime, and therefore approximated information must be used from experiments and/or from models. The calculation of the target temperature and the magnetic fields produced inside the target after the propagation of a fast electron current is performed in this paper accounting for the pulse temporal envelope and making use of a complete resistivity model. Analytic calculations of temporal and spatial varying magnetic fields are also presented. Finally, a novel interpretation of the beam hollowing phenomenon is given based on the outcomes of the model developed. [ABSTRACT FROM AUTHOR]

This paper presents the effect of self-consistent plasma backgrounds, including plasma–neutral interactions, on the dynamics of filament propagation. The principle focus is on the influence of the neutrals on the filament through both direct interactions and through their influence on the plasma background. Both direct and indirect interactions influence the motion of filaments. A monotonic increase of filament peak velocity with upstream electron temperature is observed, while a decrease with increasing electron density is observed. If ordered by the target temperature, the density dependence disappears and the filament velocity is only a function of the target temperature. Smaller filaments maintain a density dependence as a result of the density dependence of the plasma viscosity. The critical size δ*, where filaments are at their fastest, is shifted to larger sizes for higher densities due to the plasma viscosity. If the density dependence of the plasma viscosity is removed, δ* has no density dependence, but rather a temperature dependence. [ABSTRACT FROM AUTHOR]

The exponentiation of a small seed of energetic electrons into a strong current of relativistic electrons during disruptions is a major threat to the ITER mission; the adequacy of the plan for the protection of ITER remains far from clear. The hope that issues involving relativistic electrons can be resolved in the non-nuclear phase of ITER operations has a fundamental flaw. In the nuclear phase two mechanisms exist for producing seed electrons: remnant and steady production. In the non-nuclear phase only the more easily avoidable remnant source exists. Whether the plasma current in ITER must be severely constrained to avoid unacceptable machine damage rests on issues discussed in this paper. An adequate understanding of fast magnetic reconnection is one of these issues. The loss of poloidal field energy during a disruption occurs through two distinct processes: (1) a quasi-ideal fast magnetic reconnection, which conserves magnetic helicity and accounts for large and sudden drops in the internal inductance, and (2) the resistive dissipation, which dissipates the helicity and is required to quench the current. To correctly separate these two processes requires spatial and temporal resolutions far beyond those of existing simulations. The assumption that stochastic magnetic field lines have a diffusive radial motion allows simple simulations of existing experiments and predictions for ITER that go far beyond existing analyses. [ABSTRACT FROM AUTHOR]

We use beam tracingâ€"implemented with a newly-written code, Scottyâ€"and the reciprocity theorem to derive a model for the linear backscattered power of the Doppler backscattering (DBS) diagnostic. Our model works for both the O-mode and X-mode in tokamak geometry (and certain regimes of stellarators). We present the analytical derivation of our model and its implications for the DBS signal localisation and the wavenumber resolution. In determining these two quantities, we find that it is the curvature of the field lines and the magnetic shear that are important, rather than the curvature of the cut-off surface. We also provide an explicit formula for the hitherto poorly-understood quantitative effect of the mismatch angle. Consequently, one can use this model to correct for attenuation due to mismatch, avoiding the need for empirical optimisation. This is especially important in spherical tokamaks, since the magnetic pitch angle is large and varies both spatially and temporally. [ABSTRACT FROM AUTHOR]

Solar coronal mass ejections (CMEs) are large-scale eruptive events in which large amounts of plasma (up to 1013—1016 g) and magnetic fields are expelled into interplanetary space at very high velocities (typically 450 km/s, but up to 3000 km/s). When sampled in situ by a spacecraft in the interplanetary medium, they are termed interplanetary CMEs. They are nowadays considered to be the major drivers of space weather and the associated geomagnetic activity. The detectable space weather effects on Earth appear in a broad spectrum of time and length scales and have various harmful effects for human health and for our technologies on which we are ever more dependent. Severe conditions in space can hinder or damage satellite operations as well as communication and navigation systems and can even cause power grid outages leading to a variety of socio-economic losses. Therefore, the International Space Environment Service has set up a collaborative network of space weather service-providing warning centres around the globe, delivering coordinated operational space weather services for the benefit of the extensive user community. In order to improve the forecasts and predictions, NASA, ESA and other agencies have set-up space weather modelling frameworks. Here, we discuss how such frameworks enable to run and couple different space weather models, and to validate their results by comparing them with those of other similar models and, where possible, to in situ data. Examples of such frameworks are the Community Coordinated Modeling Center (CCMC, NASA GSFC), the Space Weather Modeling Framework at the Center for Space Environment Modeling at the University of Michigan, and ESAs novel Virtual Space Weather Modeling Centre that is being developed. The latter one includes space weather models that are geographically distributed and is the focus of this paper. [ABSTRACT FROM AUTHOR]

The poloidal correlation reflectometry diagnostic operated in ordinary mode with additional radial correlation channel is applied in this paper to investigate the correlation of the turbulent density fluctuations. The perpendicular and radial correlation lengths, l⊥ and lr, the perpendicular velocity v⊥ and the dissipation (mutation) time τd are measured simultaneously from the outer core to edge in the L-mode plasmas of ASDEX Upgrade. It is shown that in the outer core region (0.6 < ρpol < 0.9) the measured correlation lengths scale with the drift wave length, l⊥ ≈ 5ρs and lr ≈ 10ρs, while the dissipation time is inversely correlated with the velocity τd ≈ 40/v⊥(τd is in μs and v⊥ in km s–1). In the pedestal region (0.925 < ρpol < 0.98), where the E × B shear flows are present, a loss of measured correlation is observed which can be explained by a combination of small propagation velocity and an additional reduction of τd. In the Er well region (ρpol ≈ 0.99), the measured perpendicular correlation length increases and the radial correlation length decreases lr ≈ 4ρs compared to the outer core values. The correlation measurements are interpreted in the frame of the linear regime of reflectometry (applied only to ρpol < 0.9). Using the Born approximation we show that the finite wavenumber sensitivity of the reflectometer increases the measured l⊥and lr, but does not affect the measured τd. By the including diagnostic correction the real correlation lengths l⊥ ≈ lr ≈ 3ρs are estimated. [ABSTRACT FROM AUTHOR]

This paper studies the effect of ion-neutral collisions on the propagation of kinetic Alfvén waves (KAWs) in inhomogeneous magnetized plasma. The inhomogeneity in the plasma imposed by background density in a direction transverse as well as parallel to the ambient magnetic field plays a vital role in the localization process. The mass loading of ions takes place due to their collisions with neutral fluid leading to the damping of the KAWs. Numerical analysis of linear KAWs in inhomogeneous magnetized plasma is done for a fixed finite frequency taking into consideration the ion-neutral collisions. There is a prominent effect of collisional damping on the wave localization, wave magnetic field, and frequency spectrum. A semi-analytical technique has been employed to study the magnetic field amplitude decay process and the effect of wave frequency in the range of ion cyclotron frequency on the propagation of waves leading to damping. [ABSTRACT FROM AUTHOR]

A dedicated experiment during simultaneous lower hybrid (LH) and ion cyclotron range-of-frequencies (ICRF) operations is carried out to evaluate and understand the effects of ICRF power on the scrape-off-layer (SOL) density profiles and on the resultant LH coupling for a wide range of plasma parameters on Alcator C-Mod. Operation of the LH launcher with the adjacent ICRF antenna significantly degrades LH coupling while operation with the ICRF antenna that is not magnetically connected to the LH launcher minimally affects LH coupling. An X-mode reflectometer system at three poloidal locations adjacent to the LH launcher and a visible video camera imaging the LH launcher are used to measure local SOL density profile and emissivity modifications with the application of LH and LH + ICRF power. These measurements confirm that the density in front of the LH launcher depends strongly on the magnetic field line mapping of the active ICRF antenna. Reflectometer measurements also observe both ICRF-driven and LH-driven poloidal density profile asymmetries, especially a strong density depletion at certain poloidal locations in front of the LH launcher during operation with a magnetically connected ICRF antenna. The results indicate that understanding both LH-driven flows and ICRF sheath driven flows may be necessary to understand the observed density profile modifications and LH coupling results during simultaneous LH + ICRF operation. [ABSTRACT FROM AUTHOR]

We explore streaming instabilities of an electron-ion plasma with relativistic and ultra-relativistic cosmic rays in the background magnetic field using the multi-fluid approach. Cosmic rays can be both electrons and ions. The drift speed of cosmic rays is directed along the magnetic field. In equilibrium, the return current of the background plasma is taken into account. One-dimensional perturbations parallel to the magnetic field are considered. The dispersion relations are derived for transverse and longitudinal perturbations. It is shown that the back-reaction of magnetized cosmic rays generates new instabilities one of which has a growth rate that can approach the growth rate of the Bell instability. These new instabilities can be stronger than the cyclotron resonance instability. For unmagnetized cosmic rays, the growth rate is analogous to the Bell one. We compare two models of the plasma return current in equilibrium with three and four charged components. Some differences between these models are demonstrated. For longitudinal perturbations, an instability is found in the case of ultra-relativistic cosmic rays. The results obtained can be applied to investigation of astrophysical objects such as the shocks by supernova remnants, galaxy clusters, intracluster medium and so on, where interaction of cosmic rays with turbulence of the electron-ion plasma produced by them is of great importance for cosmic-ray evolution. [ABSTRACT FROM AUTHOR]

We investigate how the neoclassical thermal diffusivity of an axisymmetric toroidal plasma is modified by the effect of resonant magnetic perturbations (RMPs), using a drift-kinetic simulation code for calculating the radial thermal diffusivity of ion in the perturbed region under an assumption of zero electric field. Here, the perturbed region is assumed to be generated on and near the resonance surfaces, and is wedged in between the regular closed magnetic surfaces. We find that the dependence of the radial thermal diffusivity on parameters of the toroidal plasma is represented as Χr = Χr(0) {1 + c0 (ωb/νeff∆b²) <∥δBr∥²>/|Bt0|²}, where Χr(0) is the neoclassical thermal diffusivity and c0 is a positive coefficient. Here, ωb is the bounce frequency, νeff is the effective collision frequency, ∆b is the banana width, <∥δBr∥²1/2 is the strength of the RMPs in the radial directions, and |Bt0| is the strength of the magnetic field on the magnetic axis. The value of c0 := c0/(qRax/ √∈t)² is significantly reduced to c0 ~ 10-4 in the simulations because of the drift motion affected by the Coulomb collision, as contrasted with c0 = π predicted by the so-called field-line diffusion theory, where q is the safety factor, Rax is the major radius of the magnetic axis, and ∈t is the inverse aspect ratio. [ABSTRACT FROM AUTHOR]

By adopting the usual cold plasma dielectric tensor, it is demonstrated that a rapidly oscillating electric field gives rise to slow time scale drifts, which cause density modifications near antennas. In the presence of a strong magnetic field, the poloidal gradients of the field are at the origin of radial displacements of the plasma while radial field gradients have the potential to trigger density inhomogeneity along the antenna. The radio frequency-induced plasma drifts are more prominent at higher power and for more evanescent modes. It is discussed that the usual cold plasma dielectric tensor is derived neglecting nonlinear effects and zero-order drifts, and therefore does not uniformly allow the capture of the wave-particle interaction near the antenna self-consistently, necessitating a more detailed description to capture both wave and particle effects on the one hand, and global wave propagation and local sheath effects, on the other. A strategy is proposed to complement the model with other needed ingredients enabling one to capture the dynamics on the fast and slow time scales. [ABSTRACT FROM AUTHOR]

By employing a compound transport model the problem of charged test-particle scattering perpendicular to a mean magnetic field is investigated. In comparison with previous approaches a full time-dependent description is provided. According to these new results there are at least four different transport regimes, namely, the ballistic regime, the strong diffusive regime, the FLRW limit and the subdiffusive regime. It is also shown that the FLRW limit, and therefore quasilinear theory, can only be correct for intermediate time-scales and for high energetic particles. For low energetic particles there is a strong diffusive regime with k[?]/k[?] [?] 1. These results are very important for understanding solar wind observations and recent test-particle simulations. [ABSTRACT FROM AUTHOR]

This paper reviews measurements of edge plasma turbulence in toroidal magnetic fusion devices with an emphasis on recent results in tokamaks. The dominant feature of edge turbulence is a high level of broadband density fluctuations with a relative amplitude dn/n [?] 5-100%, accompanied by large potential and electron temperature fluctuations. The frequency range of this turbulence is [?]10 kHz-1 MHz, and the size scale is typically [?]0.1-10 cm perpendicular to the magnetic field but many metres along the magnetic field, i.e. the structure is nearly that of 2D 'filaments'. Large intermittent bursts or 'blobs' are usually observed in the scrape-off layer. Diagnostic and data analysis techniques are reviewed and the main experimental results are summarized. Recent comparisons of experimental results with edge turbulence theory are discussed, and some directions for future experiments are suggested. [ABSTRACT FROM AUTHOR]

This paper explores the effects of particle drifts across the magnetic field in TCV single-null plasmas using the two-dimensional edge plasma transport code UEDGE. In particular, it aims to reproduce a double-peaked density target profile, a feature which has been observed both at JET and in a TCV forward-field ( drift of core ions towards the X-point) discharge. Initial simulations are performed with drift effects turned off. This allows identification of the input parameters that strongly influence the computed steady-state but are not well known from the experiment. Including cross-field drifts self-consistently in the simulations brings computed profiles at the inner target closer to the experiment, with the double peak being reproduced for the density. In agreement with the experiment, simulations of a similar reversed-field shot yield lower ne and higher Te at the inner target, as well as unchanged temperatures at the outer plate. However, several discrepancies remain. In the forward field case, the inner target density peak near the separatrix is much sharper than seen experimentally. Furthermore, simulations show a strong dependence of the outer plate density and of the outer/inner power sharing on toroidal field direction. Experimentally, no such dependencies are observed. In the simulations, the changes of target profiles with field direction are mainly due to E × B drifts in the divertor region. While these simulations highlight the importance of E × B drifts in these TCV plasmas, the remaining differences with the experiment indicate that their role is overestimated in the simulations. [ABSTRACT FROM AUTHOR]

A theoretical study of the propagation of left-hand polarized shear Alfvén waves in spatially decreasing magnetic field geometries near the EMIC resonance, including the spectrum and amplitude of the mode converted EMIC waves and the pitch angle scattering of relativistic electrons transiting the resonant region, is presented. The objective of the paper is to motivate an experimental study of the subject using the UCLA LAPD chamber. The results are relevant in exploring the possibility that shear Alfvén waves strategically injected into the radiation belts using either ionospheric heating from ground based RF transmitters or injected by transmitters based on space platforms can enhance the precipitation rate of trapped relativistic electrons. Effects of multi-ionic composition are also investigated. [ABSTRACT FROM AUTHOR]

Improving parameters of laser-driven proton and ion beams becomes one of the most important goals in the field of laser acceleration in order to fulfill requirements of foreseen applications. This work presents parametric 2D and 3D particle-in-cell simulations of various target designs in order to reduce proton beam divergence without significant drop in maximum energies or in proton number. The optimal target design proved to be a channel-like target which produces not only a long-lasting focusing transverse electric field in contrast to a flat foil, but also a magnetic quadrupole with strong octupole component inside the guiding channel. A combination of both electric and magnetic features results in a strong proton beam divergence reduction, accompanied by a higher uniformity of the beam, which is studied as a function of proton energy. [ABSTRACT FROM AUTHOR]

We show that generation of strong magnetic field filaments due to Weibel instability accompanies the hole-boring ion acceleration by a circularly polarised laser pulse. The magnetic field confines energetic protons accelerated from hole boring in high density collimated jets. A part of energetic protons are reaccelerated by the electric field and focused by the magnetic field generated from the jets. As a result, highly collimated proton jets with high cutoff energy can be produced. [ABSTRACT FROM AUTHOR]

Cold and hot vertical displacement events (VDEs) are frequently related to the disruption of vertically-elongated tokamaks. The weak poloidal magnetic field around the null-points of a snowflake divertor configuration may influence the vertical displacement process. In this paper, the major disruption with a cold VDE and the vertical disruption in the HL-2M tokamak are investigated by the DINA code. In order to better illustrate the effect from the weak poloidal field, a double-null snowflake configuration is compared with the standard divertor (SD) configuration under the same plasma parameters. Computational results show that the weak poloidal magnetic field can be partly beneficial for mitigating the vertical instability of the plasma under small perturbations. For major disruption, the peak poloidal halo current fraction is almost the same between the snowflake and the SD configurations. However, this fraction becomes much larger for the snowflake in the event of a hot VDE. Furthermore, during the disruption for a snowflake configuration, the distribution of electromagnetic force on a vacuum vessel gets more non-uniform during the current quench. [ABSTRACT FROM AUTHOR]

The coupling characteristics of the 4.6 GHz lower hybrid (LH) grill and the factors affecting LH wave coupling, mainly drift, with divertor configuration in anti-clockwise/clockwise magnetic field direction are studied. Based on the Langmuir probe measurement, the relationship between the reflection coefficient (RC) of LH power and the edge density at the grill mouth () measured by Langmuir probes is investigated. The optimum LH wave coupling with RC less than 2% is achieved when is in the range of – , which is quite close to the theoretically predicted value. In addition, the experimental results show that drift in the divertor scrape off layer plays a dominant role in affecting and LH wave coupling. Studies show that the discrepancy in and LH wave coupling in different magnetic field directions is mainly attributed to the different directions of drift, whereas the corresponding discrepancy with different plasma currents is mainly ascribed to the change in drift intensity. Therefore, for the upper single null divertor configuration, a lower/higher plasma currents is more favorable for increasing and improving LH wave coupling in anti-clockwise/clockwise. The above studies are of great significance for optimizing and improving LH wave coupling in the extension of the plasma operation regime. [ABSTRACT FROM AUTHOR]

We report a novel method to control plasma rotation speed, namely, using the rotating magnetic field (RMF), which is a mature technique to form field reversed configuration, to drive the electron rotation and then the ion rotation via electron–ion collisions in a magnetic mirror plasma. It can be observed that the plasma starts rotating if the RMF strength exceeds a threshold value, corresponding to which the value of the magnetization parameter becomes larger than the value of the penetration parameter. The flow speed achieved in this experiment is approximately 0.14 Mach. The rotation is found to start from outside to inside with the propagating time almost equal to the penetration time of the RMFs. Also, a clear sheared flow is observed to have developed in the region of core plasma. It has been further identified that the electron–ion friction is the dominant force to drive the ion spinning up. The rotation speed is primarily determined by the RMF strength, instead of background magnetic field strength, which may offer a method to study the effect of rotation on the confinement in different magnetic field strengths while keeping the rotation profile intact. [ABSTRACT FROM AUTHOR]

A problem arising in several engineering areas is to design magnets outside a volume that produce a desired magnetic field inside it. One instance of this problem is stellarator design, where it has recently been shown that permanent magnets can provide the required shaping of the magnetic field. Here we demonstrate a robust and efficient algorithm REGCOIL_PM to calculate the spatial distribution of these permanent magnets. The procedure involves a small number of fixed-point iterations, with a linear least-squares problem solved at each step. The method exploits the Biot–Savart Law's exact linearity in magnetization density and approximate linearity in magnet size, for magnets far from the target region. No constraint is placed on the direction of magnetization, so Halbach solutions are found naturally, and the magnitude of the magnetization can be made uniformly equal to a target value. [ABSTRACT FROM AUTHOR]