Neoclassical offset toroidal velocity and auxiliary ion heating in tokamaks

In conditions of ideal axisymmetry, for a magnetized plasma in a generic bounded domain, necessarily toroidal, the uniform absorption of external energy (e.g., RF or any isotropic auxiliary heating) cannot give rise to net forces or torques. Experimental evidence on contemporary tokamaks shows that the near central absorption of RF heating power (ICH and ECH) and current drive in presence of MHD activity drives a bulk plasma rotation in the co-I _p direction, opposite to the initial one. Also the appearance of classical or neoclassical tearing modes provides a nonlinear magnetic braking that tends to clamp the rotation profile at the q-rational surfaces. The physical origin of the torque associated with P _RF absorption could be due the effects of asymmetry in the equilibrium configuration or in power deposition, but here we point out also an effect of the response of the so-called neoclassical offset velocity to the power dependent heat flow increment. The neoclassical toroidal viscosity due to internal magnetic kink or tearing modes tends to relax the plasma rotation to this asymptotic speed, which in absence of auxiliary heating is of the order of the ion diamagnetic velocity. It can be shown by kinetic and fluid calculations, that the absorption of auxiliary power by ions modifies this offset proportionally to the injected power thereby forcing the plasma rotation in a direction opposite to the initial, to large values. The problem is discussed in the frame of the theoretical models of neoclassical toroidal viscosity.     

Effect of perturbations on magnetic fields as a possible problem for implementing toroidal magnetic configurations with a zero rotational transform

Approved reliable methods and a unified quasi-resonant approach are used to reproduce the results characterizing the effect of dangerous perturbations on three particular types of toroidal magnetic configurations with a zero rotational transform (ι/2π = 0). An analysis is made of the perturbations whose geometry and amplitudes correspond to imperfections and defects in the assembly of the actual magnetic systems of conventional stellarators. The effect of perturbations on magnetic configurations with ι/2π = 0 is of the same quasi-resonant nature as that for systems with ι/2π ≠ 0 and a zero magnetic shear. However, in contrast to shearless configurations with a nonzero rotational transform, the most dangerous perturbations are those with n = 0 and m = 1, 2, 3, … (where m is the poloidal mode number and n is the toroidal mode number) and the effect of harmonics with the smallest poloidal and toroidal numbers is more destructive and nonlocal even for the amplitudes corresponding to imperfections and defects in the assembly of conventional stellarators. Such an extremely high sensitivity of magnetic systems with ι/2π = 0 to the perturbations under consideration may be a serious obstacle for their practical implementation.     

Anomalous Fast-Ion Transport During Neutral Beam Injection on EAST

The fast-ion transport during neutral beam injection on the Experimental Advanced Superconducting Tokamak (EAST) is studied. Based on the NUBEAM and TRANSP codes, it is found that fast-ion transport is anomalous when the minimum safety factor (q_min) is about 2, while it is neoclassical when q_min is around 1. Neutral beam injection heating efficiency, plasma stored energy, and the total heating power are reduced when the fast ion transport is anomalous. The Alfvén continuum spectrum and the mode structures of toroidal Alfvén eigenmodes (AEs) are also calculated for comparison between neoclassical fast-ion transport and anomalous fast-ion transport. High-q_min discharge with anomalous fast-ion transport has more AE activity than that of lower q_min discharge with neoclassical fast-ion transport.     

Physical Parameters of a Reactor-Stellarator with Small Ripples of the Helical Magnetic Field

The paper describes the calculation data on the physical parameters of a reactor-stellarator, where the nonuniformities of the helical field are smaller than the toroidal magnetic field nonuniformities: ε_h < ε_t. Unlike the previous studies, where the ion-component transport coefficients had the collision frequency dependence proportional to ν^1/2, this being equivalent to the ε_h > ε_t case, in the present calculations, these coefficients were assumed to be in proportion to the first power of the collision frequency, D_i ∝ ν for ν_eff < 2ω_E, and to D_i ∝ ν^?1 for the inverse inequality. Here, ω_E is the rotation frequency of plasma in the radial electric field. As before, the plasma electrons corresponded to the mode of D_e ∝ ν^?1. As initial parameters for numerical calculations, a reactor with R = 8 m, r_p = 2 m, and B₀ = 5 Т was taken. A numerical code was used to solve the set of equations that describes the plasma space?time behavior in the reactor-stellarator under the conditions of equal diffusion fluxes. The start of reactor operation in the mode of thermonuclear burning was provided by heating sources with a power of several tens of megawatts. Steady-state operating conditions of a self-sustained thermonuclear reaction were attained by maintaining the plasma density through DT fuel pellet injection into the plasma.     

Localization of magnetized electrons in current filaments as a fundamental cause of Coulomb explosion

Mechanisms for generating current filaments in a dense plasma under the action of focused laser pulses and in a Z-pinch configuration are discussed. The main properties of current filaments with a zero and nonzero electron vorticity Ω _e =B?(c/e)?×p _e that originate at magnetic fields in the range 4πn _e m _e c²?B²?4πn _i m _i c² are investigated under the conditions of Coulomb explosion at currents below the ion Alfvén current. A study is made of the equilibrium configurations of nonquasineutral current filaments in a purely longitudinal (B_z) and a purely azimuthal (B_θ) magnetic field and also in a more general case of a helical magnetic field, having two components, under conditions such that the charge separation occurs on a spatial scale on the order of the magnetic Debye radius r_B ? |B|/(4πen_e. It is shown that strong electric fields generated in the current filaments are comparable in magnitude to the atomic field and are capable of accelerating ions to energies of several tens of megaelectronvolts. The ion dynamics in strong electric fields of the filaments is calculated numerically and is shown to lead to the formation of collisionless shock waves on time scales on the order of several inverse ion plasma frequencies ω _pi ^?1 . The possible formation of current filaments on different spatiotemporal scales is considered.     

Nonlinear dynamics of beam–plasma instability in a finite magnetic field

The nonlinear dynamics of beam–plasma instability in a finite magnetic field is investigated numerically. In particular, it is shown that decay instability can develop. Special attention is paid to the influence of the beam?plasma coupling factor on the spectral characteristics of a plasma relativistic microwave accelerator (PRMA) at different values of the magnetic field. It is shown that two qualitatively different physical regimes take place at two values of the external magnetic field: B ₀ = 4.5 kG (Ω ~ ωB _p ) and 20 kG (Ω _B ? ω_p). For B ₀ = 4.5 kG, close to the actual experimental value, there exists an optimal value of the gap length between the relativistic electron beam and the plasma (and, accordingly, an optimal value of the coupling factor) at which the PRMA output power increases appreciably, while the noise level decreases.     

Alfvén eigenmode evolution computed with the VENUS and KINX codes for the ITER baseline scenario

A new application of the VENUS code is described, which computes alpha particle orbits in the perturbed electromagnetic fields and its resonant interaction with the toroidal Alfvén eigenmodes (TAEs) for the ITER device. The ITER baseline scenario with Q = 10 and the plasma toroidal current of 15 МА is considered as the most important and relevant for the International Tokamak Physics Activity group on energetic particles (ITPA-EP). For this scenario, typical unstable ТАЕ-modes with the toroidal index n = 20 have been predicted that are localized in the plasma core near the surface with safety factor q = 1. The spatial structure of ballooning and antiballooning modes has been computed with the ideal MHD code KINX. The linear growth rates and the saturation levels taking into account the damping effects and the different mode frequencies have been calculated with the VENUS code for both ballooning and antiballooning TAE-modes.     

On the minimum electron transport coefficients in tokamaks in the range of low collision frequencies

There are two close empirical scalings, namely, the T-11 and neo-Alcator ones, that provide correct estimates for the energy confinement time in tokamaks in ohmic heating regimes in the linear part of the dependence τ _E (\(\bar n_e \)) in the range of low values of \(\bar n_e \) and 〈ν _e ^* 〉 ≤ 1. The similar character of electron energy confinement in this range, which expands with increasing magnetic field B ₀, has stimulated the search for dimensionless parameters and simple physical models that would explain the experimentally observed dependences χ _e ～ 1/n _e and τ _Ee ～ \(\bar n_e \). In 1987, T. Okhawa showed that the experimental data were satisfactorily described by the formula χ_e⊥ = (c ²/ω _pe ² )ν _e /qR, in deriving of which the random spatial leap along the radius r on the electron trajectory was assumed to be the same as that in the coefficient of the poloidal field diffusion, while the repetition rate of these leaps was assumed to be ν _e /qR. In 2004, J. Callen took into account the decrease in the fraction of transient electrons with increasing toroidal ratio ? = r/R and corrected the coefficient c ²/ω _pe ² in Okhawa equation by the factor σ _‖ ^Sp /σ _‖ ^neo . If one takes into account this correction and assumes that the frequency of the stochastic process is equal to the reciprocal of the half-period of rotation of a trapped electron along its banana trajectory, then the resulting expression for χ_e⊥ will coincide with the T-11 scaling: χ _e ^an ∞ ?^1.75(T _e /A _i )^0.5/(n _e qR) at A i = 1. If the same stochastic process also involves ions, it may result in the opening of the orbit of a trapped ion at the distance ～(c/ω _pe )(m _i /m _e )^1/4. In this case, the calculated coefficient of electron and ion diffusion D is close to D ^an ≈ χ _e ^an /2.     

Nonquasineutral relativistic current filaments and their X-ray emission

Nonquasineutral electron current filaments with the azimuthal magnetic field are considered that arise due to the generation of electron vorticity in the initial (dissipative) stage of evolution of a current-carrying plasma, when the Hall number is small (σB/en _e c ? 1) because of the low values of the plasma conductivity and magnetic field strength. Equilibrium filamentary structures with both zero and nonzero net currents are considered. Structures with a zero net current type form on time scales of t < t _sk = (r ₀ω _pe /c)² t _st, where t _sk is the skin time, t _st is the typical time of electron-ion collisions, and r ₀ is the radius of the filament. It is shown that, in nonquasineutral filaments in which the current is carried by electrons drifting in the crossed electric (E _r ) and magnetic (B _θ) fields, ultrarelativistic electron beams on the typical charge-separation scale r _B = B/(4πen _e ) (the so-called magnetic Debye radius) can be generated. It is found that, for comparable electron currents, the characteristic electron energy in filaments with a nonzero net current is significantly lower than that in zero-net-current filaments that form on typical time scales of t < t _sk. This is because, in the latter type of filaments, the oppositely directed electron currents repel one another; as a result, both the density and velocity of electrons increase near the filament axis, where the velocities of relativistic electrons are maximum. Filaments with a zero net current can emit X rays with photon energies ? ω up to 10 MeV. The electron velocity distributions in filaments, the X-ray emission spectra, and the total X-ray yield per unit filament length are calculated as functions of the current and the electron number density in the filament. Analytical estimates of the characteristic lifetime of a radiating filament and the typical size of the radiating region as functions of the plasma density are obtained. The results of calculations are compared with the available experimental data.     

Measurements of the plasma density in the FTU tokamak by a pulsed time-of-flight X-wave refractometer

On-line control over the plasma density in tokamaks (especially, in long-term discharges) requires reliable measurements of the averaged plasma density. For this purpose, a new method of density measurements—a pulsed time-of-flight plasma refractometry—was developed and tested in the T-11M tokamak. This method allows one to determine the averaged density from the measured time delay of nanosecond microwave pulses propagating through the plasma. For an O-wave, the measured time delay is proportional to the line-averaged density and is independent of the density profile (f?f _p ) τ_o ≈ k _o \(\tfrac{1}{{f^2 }}\mathop \smallint \limits_l \) N(x dx. A similar formula is valid for an X-wave: τ_X = ≈ k _x \(\tfrac{{f^2 + f_c^2 }}{{(f^2 - f_c^2 )^2 }}\mathop \smallint \limits_l \) N(x)dx. Here, f is the frequency of the probing wave, f _p is the plasma frequency, l= 4 a is the path length for two-pass probing in the equatorial plane, a is the plasma minor radius, k _O and k _X are numerical factors, f _c is the electron-cyclotron frequency at the axis of the plasma column, and f _p ?f _c , f. Measurements of the time delay provide the same information as plasma interferometry, though they do no employ the effect of interference. When the conditions f _p ?f _c , f are not satisfied, the measured time delay depends on the shape of the density profile. In this case, in order to determine the average density regardless of the density profile, it is necessary to perform simultaneous measurements at several probing frequencies in order to determine the average density. In ITER (Bt ～ 5T), a spectral window between the lower and upper cutoff frequencies in the range of 50–100 GHz can be used for pulsed time-of-flight X-wave refractometry. This appreciably simplifies the diagnostics and eliminates the problem of the first mirror. In this paper, the first results obtained in the FTU tokamak with a prototype of the ITER pulsed time-of-flight refractometer are presented. The geometry and layout of experiments similar to the planned ITER experiments are described. The density measured by pulsed time-of-flight refractometry is shown to agree well with the results obtained in FTU with a two-frequency scanning IR interferometer. The results obtained are analyzed, and the future experiments are discussed.     

Formation of a Z-pinch during electromagnetic compression of a nonquasineutral current filament

A study is made of the structure of a relativistic current filament with the azimuthal magnetic field B_θ in the range 4πn _e m _e c²?B _θ ² 4πn _i m _i c², when the plasma quasineutrality near the filament axis is violated and a narrow peak in electron density is formed there. The ion dynamics in a strong radial electric field of the filament on time scales of about several inverse ion plasma frequencies ω _pi ^?1 is investigated. The initial ion pressure prevents the ion plasma component from compression to infinitely high densities under the action of the electric field and leads to the formation of a dense hot plasma core near the axis of the Z-pinch on time scales of about a dozen ω _pi ^?1 . The compression of the ion component in the axial region gives rise to a collisionless “unloading” shock wave that propagates away from the axis and is accompanied by the vanishing of the radial ion velocity behind the shock front, the accumulation of positive charge near the axis, and the formation of a steady-state ion density profile. It is shown theoretically that ion-ion dissipation manifests itself as the destruction of the hot core of the formed Z-pinch on picosecond time scales. This may serve to explain the explosions of “hot points” in a current-carrying plasma.     

Transition into the improved core confinement mode as a possible mechanism for additional electron heating observed in the lower hybrid current drive experiments at the FT-2 tokamak

In experiments on lower hybrid current drive (LHCD) carried out at the FT-2 tokamak, a substantial increase in the central electron temperature T _e (r = 0 cm) from 550 to 700 eV was observed. A complex simulation procedure is used to explain a fairly high LHCD efficiency and the observed additional heating, which can be attributed to a transition into the improved core confinement (ICC) mode. For numerical simulations, data obtained in experiments with deuterium plasma at 〈n _e 〉 = 1.6 × 10¹⁹ m^–3 were used. Simulations by the GRILL3D, FRTC, and ASTRA codes have shown that the increase in the density and central temperature is apparently caused by a significant suppression of heat transport in the electron component. The mechanism for transition into the improved confinement mode at r < 3 cm can be associated with the broadening of the plasma current channel due to the lower hybrid drive of the current carried by superthermal and runaway electrons. In this case, the magnetic shear s = (r/q)(dq/dr) in the axial region of the plasma column almost vanishes during the RF pulse. In this study, the effect of lower hybrid waves on the plasma parameters, resulting in a transition into the ICC mode, is considered. New experimental and calculated data are presented that evidence in favor of such a transition. Special attention is paid to the existence of a threshold for the transition into the ICC mode in deuterium plasma.     

Turbulent dynamics of helical perturbations in the edge plasma of the T-10 tokamak

Turbulent dynamics of the edge plasma in the T-10 tokamak is simulated numerically by solving nonlinear MHD equations in the framework of the four-field {?, n, p _e , p _i } reduced two-fluid Braginskii hydro-dynamics. It is shown that the transition from ohmic to electron-cyclotron heating is accompanied by a decrease in the amplitudes of turbulent fluctuations in plasma. This is caused by the enhancement of longitudinal dissipation due to the increase in the electron temperature. However, phase relations between potential fluctuations of different modes change in such a way that the Reynolds turbulent force increases, which leads to an increase in the poloidal velocity in the direction of ion diamagnetic drift. Since the poloidal and ion diamagnetic drift velocities enter into the equation of the radial force balance for ions with different signs, the radial electric field decreases. The simulation results agree qualitatively with the results of experiments in the T-10 tokamak. The dependence of the radial electric field on the plasma density, ion pressure, and neutral density is also calculated.     

Determination of the drag coefficient of a toroidal plasma vortex in air

Forces acting on toroidal vortices in an unbounded medium (plasma vortices in air and vortex rings in air and water) are investigated. A solution to the equations describing such votrices is obtained. It is shown that this solution satisfactorily agrees with experiment. Based on the experimental results and the solution obtained, the drag coefficient C _x of such vortices is found. For the same Reynolds numbers, the value of C _x may be much less than the drag coefficient of a drop-shaped axisymmetric body (0.045), which is known to be the best streamlined object.     

Study of Plasma Flow Modes in Imploding Nested Arrays

Results from experimental studies of implosion of nested wire and fiber arrays at currents of up to 4 МА at the Angara-5-1 facility are presented. Depending on the ratio between the radii of the inner and outer arrays, different modes of the plasma flow in the space between the inner and outer arrays were implemented: the sub-Alfvénic (V _r < V _А ) and super-Alfvénic (V _r > V _А ) modes and a mode with the formation of the transition shock wave (SW) region between the cascades. By varying the material of the outer array (tungsten wires or kapron fibers), it is shown that the plasma flow mode between the inner and outer arrays depends on the ratio between the plasma production rates ?_in /?_out in the inner and outer arrays. The obtained experimental results are compared with the results of one-dimensional MHD simulation of the plasma flow between the arrays. Stable implosion of the inner array plasma was observed in experiments with combined nested arrays consisting of a fiber outer array and a tungsten inner array. The growth rates of magnetic Rayleigh?Taylor (MRT) instability in the inner array plasma at different numbers of fibers in the outer array and different ratios between the radii of the inner and outer arrays are compared. Suppression of MRT instability during the implosion of the inner array plasma results in the formation of a stable compact Z-pinch and generation of a soft X-ray pulse. A possible scenario of interaction between the plasmas of the inner and outer arrays is offered. The stability of the inner array plasma in the stage of final compression depends on the character of interaction of plasma jets from the outer array with the magnetic field of the inner array.     

Compression dynamics of quasi-spherical wire arrays with different linear mass profiles

Results of experimental studies of the implosion of quasi-spherical wire (or metalized fiber) arrays are presented. The goal of the experiments was to achieve synchronous three-dimensional compression of the plasma produced in different regions of a quasi-spherical array into its geometrical center. To search for optimal synchronization conditions, quasi-spherical arrays with different initial profiles of the linear mass were used. The following dependences of the linear mass on the poloidal angle were used: m _l (θ) ∝ sin^–1θ and m _l (θ) ∝ sin^–2θ. The compression dynamics of such arrays was compared with that of quasi-spherical arrays without linear mass profiling, m _l (θ) = const. To verify the experimental data, the spatiotemporal dynamics of plasma compression in quasi-spherical arrays was studied using various diagnostics. The experiments on three-dimensional implosion of quasi-spherical arrays made it possible to study how the frozen-in magnetic field of the discharge current penetrates into the array. By measuring the magnetic field in the plasma of a quasi-spherical array, information is obtained on the processes of plasma production and formation of plasma flows from the wire/fiber regions with and without an additionally deposited mass. It is found that penetration of the magnetic flux depends on the initial linear mass profile m _l (θ) of the quasi-spherical array. From space-resolved spectral measurements and frame imaging of plasma X-ray emission, information is obtained on the dimensions and shape of the X-ray source formed during the implosion of a quasi-spherical array. The intensity of this source is estimated and compared with that of the Z-pinch formed during the implosion of a cylindrical array.     

Changes in photosynthesis of alpine plant <Emphasis Type="Italic">Saussurea superba</Emphasis> during leaf expansion

The native alpine plant Saussurea superba is widely distributed in Qinghai–Tibetan Plateau regions. The leaves of S. superba grow in whorled rosettes, and are horizontally oriented to maximize sunlight exposure. Experiments were conducted in an alpine Kobresia humilis meadow near Haibei Alpine Meadow Ecosystem Research Station (37°29′–37°45′N, 101°12′–101°33′E; alt. 3200 m). Leaf growth, photosynthetic pigments and chlorophyll fluorescence parameters were measured in expanding leaves of S. superba. The results indicate that leaf area increased progressively from inner younger leaves to outside fully expanded ones, and then slightly decreased in nearly senescent leaves, due to early unfavorable environmental conditions, deviating from the ordinary growth pattern. The specific leaf area decreased before leaves were fully expanded, and the leaf thickness was largest in mature leaves. There were no significant changes in the content of chlorophylls (Chl) and carotenoids (Car), but the ratios of Chl a/b and Car/Chl declined after full expansion of the leaves. The variation of Chl a/b coincided well with changes in photochemical quenching (q _P) and the fraction of open PSII reaction centers (q _L). The maximum quantum efficiency of PSII photochemistry after 5 min dark relaxation (F _(v)/F _(m)) continuously increased from younger leaves to fully mature leaves, suggesting that mature leaves could recover more quickly from photoinhibition than younger leaves. The light-harvesting capacity was relatively steady during leaf expansion, as indicated by the maximum quantum efficiency of open PSII centers (\(F_{\text{v}}^{{\prime }}\)/\(F_{\text{m}}^{{\prime }}\)). UV-absorbing compounds could effectively screen harmful solar radiation, and are a main protection way on the photosynthetic apparatus. The decline of q _P and q _L during maturation, together with limitation of quantum efficiency of PSII reaction centers (L _(PFD)), shows a decrease of oxidation state of Q_A in PSII reaction centers under natural sunlight. Furthermore, light-induced (Φ _NPQ) and non-light-induced quenching (Φ _NO) were consistent with variation of L _(PFD). It is concluded that the leaves of S. superba could be classified into four functional groups: young, fully expanded, mature, and senescent. Quick recovery from photoinhibition was correlated with protection by screening pigments, and high level of light energy trapping was correlated with preservation of photosynthetic pigments. Increasing of Φ _NPQ and Φ _NO during leaves maturation indicates that both thermal dissipation of excessive excitation energy in safety and potential threat to photosynthetic apparatus were strengthened due to the declination of q _P and q _L, and enhancement of L _(PFD).     

Suppression of the rayleigh-taylor instability of a low-density imploding liner by a longitudinal magnetic field

The possibility of suppressing the Rayleigh-Taylor instability in a low-density plasma, Π=ω _pi ² Δ²/c²?1 (where Δ is the thickness of the current-carrying slab), is investigated for the case in which the electron currents are much higher than the ion currents. The suppression of this instability in an imploding cylindrical liner by an axial external magnetic field \(B_{0z} \) is considered. It is shown that, for the instability to be suppressed, the external magnetic field \(B_{0z} \) should be stronger than the magnetic field B_0θ of the current flowing through the liner.     

Plans for Liquid Metal Divertor in Tokamak Compass

The COMPASS tokamak (R = 0.56 m, a = 0.2 m, B_T = 1.3 T, I_p ~ 300 kA, pulse duration 0.4 s) operates in ITER-like plasma shape in H-mode with Type-I ELMs. In 2019, we plan to install into the divertor a test target based on capillary porous system filled with liquid lithium/tin. This single target will be inclined toroidally in order to be exposed to ITER-relevant surface heat flux (20 MW/m²). Based on precisely measured actual heat fluxes, our simulations predict (for 45° inclination, without accounting for the lithium vapor shielding) the surface temperature rises up to 700°C within 120 ms of the standard ELMy H-mode heat flux with ELM filaments reaching hundreds MW/m². Significant lithium vaporization is expected. The target surface will be observed by spectroscopy, fast visible and infrared cameras. The scientific program will be focused on operational issues (redeposition of the evaporated metal, ejection of droplets, if any) as well as on the effect on the plasma physics (improvement of plasma confinement, L–H power threshold, Z_eff, etc.). After 2024, a closed liquid divertor may be installed into the planned COMPASS Upgrade tokamak (R = 0.84 m, a = 0.3 m, B_T = 5 T, I_p = 2 MA, P_in = 8 MW, pulse duration ~2 s) with ITER-relevant heat fluxes loading the entire toroidal divertor.     

Adiabatic Heating of Electrons in the Magnetospheric Current Sheet

Electron dynamics and acceleration in an electromagnetic field configuration modeling the current sheet configuration of the Earth’s magnetotail region is investigated. A focus is made on the role of the dawn?dusk magnetic field component B_y in the convection electron heating by an electric field E_y. For numerical integration of a large number of test particle trajectories over long time intervals, the equations of motion written in the guiding center approximation are used. It is shown that the presence of a B_y ≠ 0 magnetic field significantly changes the electron heating and allows electrons with small pitch angles to gain energy much more efficiently than the equatorial electrons. As a result, the convection heating in the current sheet with B_y ≠ 0 leads to the formation of an accelerated anisotropic population of particles with energies higher than a few hundred electronvolts. The obtained results and spacecraft observations in the Earth’s magnetotail are compared, and possible limitations in the proposed model approaches are discussed.