Pitch Angle Space Research Articles

Electron Heating by Magnetic Pumping and Whistler-mode Waves

The investigation of mechanisms responsible for the heating of cold solar wind electrons around the Earth’s bow shock is an important problem in heliospheric plasma physics because such heating is vitally required to run the shock drift acceleration at the bow shock. The prospective mechanism for electron heating is magnetic pumping, which considers electron adiabatic (compressional) heating by ultralow-frequency waves and simultaneous scattering by high-frequency fluctuations. Existing models of magnetic pumping have operated with external sources of such fluctuations. In this study, we generalize these models by introducing the self-consistent electron scattering by whistler-mode waves generated due to the anisotropic electron heating process. We consider an electron population captured within a magnetic trap created by ultralow-frequency waves. Periodical adiabatic heating and cooling of this population drives the generation of whistler-mode waves scattering electrons in the pitch-angle space. The combination of adiabatic heating and whistler-driven scattering provides electron acceleration and the formation of a suprathermal electron population that can further participate in the shock drift acceleration.

Simulations of energetic particle driven instabilities in CFQS

A nonlinear simulation of the energetic particle driven instabilities in the Chinese First Quasi-axisymmetric Stellarator (CFQS) has been conducted for the first time. MEGA, a hybrid simulation code for energetic particles interacting with a magneto-hydrodynamic fluid, was used in the present work. Both the m/n = 3/1 energetic-particle-mode (EPM) like mode and the m/n = 5/2 toroidal Alfvén eigenmode (TAE) were found, where m is the poloidal mode number and n is the toroidal mode number. Four important results were obtained as follows. First, the instability in the CFQS in three-dimensional form was shown for the first time. Second, strong toroidal mode coupling was found for the spatial profiles, and it is consistent with the theoretical prediction. Third, the resonant condition caused by the absence of axial symmetry in CFQS was demonstrated for the first time. The general resonant condition is f mode = Nf ϕ − Lf θ , where f mode, f ϕ , and f θ are mode frequency, particle toroidal transit frequency, and particle poloidal transit frequency, respectively; N and L are arbitrary integers, represent toroidal and poloidal resonance numbers. For EPM-like mode, the dominant and subdominant resonant conditions are f mode = 3f ϕ − 7f θ and f mode = f ϕ − f θ , respectively. For TAE, the dominant and subdominant resonant conditions are f mode = 4f ϕ − 9f θ and f mode = 2f ϕ − 3f θ , respectively. On the one hand, the toroidal resonance numbers are different from the toroidal mode numbers by 2. This indicates that the two-fold rotational symmetry affects the resonance condition. On the other hand, the subdominant resonances satisfy N = n, which is expected for the axisymmetric plasmas and most of the toroidal plasmas including stellarators. Fourth, the nonlinear frequency chirpings in CFQS were demonstrated for the first time. Hole and clump structures were formed in the pitch angle and energy phase space, and the particles comprising the hole and clump were kept resonant with the modes during the mode frequencies chirping.

Toroidal modeling of energetic passing particle drift kinetic effects on tearing mode stability

Drift kinetic effects of the neutral beam injection induced passing energetic particles (EPs) on the linear stability of the n = 1 tearing mode (TM) (with the dominant poloidal harmonic of m = 2) are numerically investigated utilizing the MARS-K code (Liu et al 2008 Phys. Plasmas 15 112503), in a tokamak plasma with finite equilibrium pressure and anisotropic thermal transport. In the low plasma pressure regime, it is found that co- (counter-) passing EPs stabilize (destabilize) the TM, agreeing with previous studies. However, as the plasma pressure increases beyond a critical value, it is found that co-passing EPs also destabilize the mode. An in-depth analysis reveals that the net effect of co-passing EPs is a result of competition between the stabilizing contribution from the non-adiabatic drift kinetic terms and the destabilizing contribution associated with adiabatic terms, with the latter becoming more dominant at higher equilibrium pressure. Non-perturbative magnetohydrodynamic-kinetic hybrid modeling also finds that co- and counter-passing EPs modify the TM eigenfunction differently, with the counter-passing EPs enhancing the sideband harmonics. Furthermore, effects of the plasma resistivity and toroidal rotation, as well as that of the equilibrium distribution of EPs in the particle pitch angle space, are also investigated, showing asymmetric results on the TM stability between the co- and counter-passing EPs. The first order finite orbit width correction is found to be stabilizing with co-passing EPs and destabilizing with counter-passing particles. Finally, drift resonances between passing EPs and the TM induce finite frequency to the mode and generate finite net torques inside the plasma, due to the neoclassical toroidal viscosity and the Reynolds stress associated with 3D perturbations.

Weak Turbulence and Quasilinear Diffusion for Relativistic Wave-Particle Interactions Via a Markov Approach

We derive weak turbulence and quasilinear models for relativistic charged particle dynamics in pitch-angle and energy space, due to interactions with electromagnetic waves propagating (anti-)parallel to a uniform background magnetic field. We use a Markovian approach that starts from the consideration of single particle motion in a prescribed electromagnetic field. This Markovian approach has a number of benefits, including: 1) the evident self-consistent relationship between a more general weak turbulence theory and the standard resonant diffusion quasilinear theory (as is commonly used in e.g. radiation belt and solar wind modeling); 2) the general nature of the Fokker-Planck equation that can be derived without any prior assumptions regarding its form; 3) the clear dependence of the form of the Fokker-Planck equation and the transport coefficients on given specific timescales. The quasilinear diffusion coefficients that we derive are not new in and of themselves, but this concise derivation and discussion of the weak turbulence and quasilinear theories using the Markovian framework is physically very instructive. The results presented herein form fundamental groundwork for future studies that consider phenomena for which some of the assumptions made in this manuscript may be relaxed.

Wave Excitation by Power-law-Distributed Energetic Electrons with Pitch-angle Anisotropy in the Solar Corona

Abstract Radio waves from the Sun are emitted, as a rule, due to energized electrons. Observations infer that the related energized electrons follow (negative) power-law velocity distributions above a break velocity U b . They might also distribute anisotropically in the pitch-angle space. To understand radio wave generation better, we study the consequences of anisotropic power-law-distributed energetic electrons in current-free collisionless coronal plasmas utilizing 2.5-dimensional particle-in-cell simulations. We assume that the velocity distribution f u of the energized electrons follows a plateau (∂f u /∂u = 0) and a power-law distribution with spectral index α for velocities below and above U b , respectively. In the pitch-angle space, these energized electrons are spread around a center μ c = 0.5. We found that the energetic plateau-power-law electrons can more efficiently generate coherent waves if the anisotropy of their pitch-angle distribution is sufficiently strong, i.e., a small pitch-angle spread μ s . The break velocity U b affects the excitation dominance between the electrostatic and electromagnetic waves: for larger U b electrostatic waves are mainly excited, while intermediate values of U b are required for an excitation dominated by electromagnetic waves. The spectral index α controls the growth rate, efficiency, saturation, and anisotropy of the excited electromagnetic waves as well as the energy partition in different wave modes. These excited electromagnetic waves are predominantly right-handed polarized, in X- and Z-modes, as observed, e.g., in solar radio spikes. Additionally about 90% of the kinetic energy loss of the energetic electrons is dissipated, heating the ambient thermal electrons. This may contribute to the coronal heating.

A Focused Transport-based Kinetic Fractional Diffusion-advection Equation for Energetic Particle Trapping and Reconnection-related Acceleration by Small-scale Magnetic Flux Ropes in the Solar Wind

Abstract Analysis of energetic particle inner heliospheric spacecraft data increasingly suggests the existence of anomalous diffusion phenomena that should be addressed to achieve a better understanding of energetic particle transport and acceleration in the expanding solar wind medium. Related to this is fast-growing observational evidence supporting the long-standing prediction from magnetohydrodynamic (MHD) theory and simulations of the presence of an inner heliospheric, dominant quasi-two-dimensional MHD turbulence component that contains coherent contracting and merging (reconnecting) small-scale magnetic flux rope (SMFR) structures. This suggests that energetic particle trapping in SMFRs should play a role in anomalous diffusion in the solar wind that warrants further investigation. However, progress in studying such anomalous energetic particle transport phenomena in the solar wind is hampered by the lack of a fundamental derivation of a general fractional kinetic transport equation linking macroscopic energetic particle fractional transport to the microscopic physics of energetic particle interaction with SMFR structures. Here, we outline details of how one can derive a closed ensemble-averaged focused transport equation in the form of a general kinetic fractional diffusion-advection equation from first principles following the nonlinear Eulerian correlation function closure approach of Sanchez et al. With this equation one can model the anomalous diffusion of energetic particles in ordinary, momentum, and pitch-angle space in response to particle trapping in numerous SMFRs advected with the solar wind flow.

Electron Diffusion and Advection During Nonlinear Interactions With Whistler‐Mode Waves

AbstractRadiation belt codes evolve electron dynamics due to resonant wave‐particle interactions. It is not known how to best incorporate electron dynamics in the case of a wave power spectrum that varies considerably on a “sub‐grid” timescale shorter than the computational time‐step of the radiation belt model ΔtRBM, particularly if the wave amplitude reaches high values. Timescales associated with the growth rate of thermal instabilities are very short, and are typically much shorter than ΔtRBM. We use a kinetic code to study electron interactions with whistler‐mode waves in the presence of a thermally anisotropic background. For “low” values of anisotropy, instabilities are not triggered and we observe similar results to those obtained in Allanson et al. (2020, https://doi.org/10.1029/2020JA027949), for which the diffusion roughly matched the quasilinear theory over short timescales. For “high” levels of anisotropy, wave growth via instability is triggered. Dynamics are not well described by the quasilinear theory when calculated using the average wave power. Strong electron diffusion and advection occur during the growth phase (≈100 ms). These dynamics “saturate” as the wave power saturates at ≈ 1 nT, and the advective motions dominate over the diffusive processes. The growth phase facilitates significant advection in pitch angle space via successive resonant interactions with waves of different frequencies. We suggest that this rapid advective transport during the wave growth phase may have a role to play in the electron microburst mechanism. This motivates future work on macroscopic effects of short‐timescale nonlinear processes in radiation belt modeling.

Influence of Off-Axis Neutral Beam Injection on Resistive Wall Mode Stability

Investigated are the effects of energetic particles (EPs), arising from off-axis neutral beam injection, on the resistive wall mode (RWM) instability in tokamaks. The analytic results for a cylindrical plasma show that EPs contribute the strongest kinetic damping to the RWM, as the deposition radius approaches the peak location of the mode eigenfunction. Toroidal computations are also carried out utilizing the MARS-K code, which includes EPs with anisotropic equilibrium distribution in the particle pitch angle space. These self-consistent numerical results confirm the analytic finding, that the RWM growth rate decreases as the peak of the EPs’ radial density profile approaches the maximum of the mode displacement. This study thus demonstrates the possibility of the RWM stabilization by the off-axis neutral beam injection in tokamak experiments.

Using Dimensionality Reduction and Clustering Techniques to Classify Space Plasma Regimes

Collisionless space plasma environments are typically characterised by distinct particle populations. Although moments of their velocity distribution functions help in distinguishing different plasma regimes, the distribution functions themselves provide more comprehensive information about the plasma state, especially at times when the distribution function includes non-thermal effects. Unlike moments, however, distribution functions are not easily characterised by a small number of parameters, making their classification more difficult to achieve. In order to perform this classification, we propose to distinguish between the different plasma regions by applying dimensionality reduction and clustering methods to electron distributions in pitch angle and energy space. We utilise four separate algorithms to achieve our plasma classifications: autoencoders, principal component analysis, mean shift, and agglomerative clustering. We test our classification algorithms by applying our scheme to data from the Cluster-PEACE instrument measured in the Earth's magnetotail. Traditionally, it is thought that the Earth's magnetotail is split into three different regions (the plasma sheet, the plasma sheet boundary layer, and the lobes), that are primarily defined by their plasma characteristics. Starting with the ECLAT database with associated classifications based on the plasma parameters, we identify 8 distinct groups of distributions, that are dependent upon significantly more complex plasma and field dynamics. By comparing the average distributions as well as the plasma and magnetic field parameters for each region, we relate several of the groups to different plasma sheet populations, and the rest we attribute to the plasma sheet boundary layer and the lobes. We find clear distinctions between each of our classified regions and the ECLAT results.

Particle‐in‐Cell Experiments Examine Electron Diffusion by Whistler‐Mode Waves: 2. Quasi‐Linear and Nonlinear Dynamics

AbstractTest particle codes indicate that electron dynamics due to interactions with low amplitude incoherent whistler mode‐waves can be adequately described by quasi‐linear theory. However there is significant evidence indicating that higher amplitude waves cause electron dynamics not adequately described using quasi‐linear theory. Using the method that was introduced in Allanson et al. (2019, https://doi.org/10.1029/2019JA027088), we track the dynamical response of electrons due to interactions with incoherent whistler‐mode waves, across all energy and pitch angle space. We conduct five experiments each with different values of the electromagnetic wave amplitude. We find that the electron dynamics agree well with the quasi‐linear theory diffusion coefficients for low amplitude incoherent waves with (Bw,rms/B0)2≈3.7·10−10, over a time scale T of the order of 1,000 gyroperiods. However, the resonant interactions with higher amplitude waves cause significant nondiffusive dynamics as well as diffusive dynamics. When electron dynamics are extracted and analyzed over time scales shorter than T, we are able to isolate both the diffusive and nondiffusive (advective) dynamics. Interestingly, when considered over these appropriately shorter time scales (of the order of hundreds or tens of gyroperiods), the diffusive component of the dynamics agrees well with the predictions of quasi‐linear theory, even for wave amplitudes up to (Bw,rms/B0)2≈5.8·10−6. Quasi‐linear theory is based on fundamentally diffusive dynamics, but the evidence presented herein also indicates the existence of a distinct advective component. Therefore, the proper description of electron dynamics in response to wave‐particle interactions with higher amplitude whistler‐mode waves may require Fokker‐Planck equations that incorporate diffusive and advective terms.

Galactic Cosmic-Ray Anisotropies: Voyager 1 in the Local Interstellar Medium

Abstract Since crossing the heliopause on 2012 August 25, Voyager 1 observed reductions in galactic cosmic ray count rates caused by a time-varying depletion of particles with pitch angles near 90°, while intensities of particles with other pitch angles remain unchanged. Between late 2012 and mid-2017, three large-scale events occurred, lasting from ∼100 to ∼630 days. Omnidirectional and directional high-energy data from Voyager 1’s Cosmic Ray Subsystem are used to report cosmic ray intensity variations. Omnidirectional (≳20 MeV) proton-dominated measurements show up to a 3.8% intensity reduction. Bidirectional (≳70 MeV) proton-dominated measurements taken from various spacecraft orientations provide insight about the depletion region’s spatial properties. We characterize the anisotropy as a “notch” in an otherwise uniform pitch angle distribution of varying depth and width centered about 90° in pitch angle space. The notch averages 22° wide and 15% deep, signifying a depletion region that is broad and shallow. There are indications that the anisotropy is formed by a combination of magnetic trapping and cooling downstream of solar-induced transient disturbances in a region that is also likely influenced by the highly compressed fields near the heliopause.

Toroidal Alfvén eigenmode triggered by trapped anisotropic energetic particles in a toroidal resistive plasma with free boundary

The toroidal Alfvén eigenmode (TAE), excited by trapped energetic particles (EPs), is numerically investigated in a tokamak plasma, using the non-perturbative magnetohydrodynamic-kinetic hybrid formulation based MARS-K code (Liu et al 2008 Phys. Plasmas 15 112503). Compared with the fixed boundary condition at the plasma edge, a free boundary enhances the critical value of the EPs kinetic contribution for driving the TAE. Free boundary also induces finite perturbations at the plasma edge as expected. An anisotropic distribution of EPs, in the particle pitch angle space, strongly enhances the instability and results in a more global mode structure, compared with the isotropic case. The plasma resistivity is also found to play a role in the EPs-destabilized TAE. In particular, the mode stability domain is mapped out, in the 2D parameter space of the plasma resistivity and a quantity defining the width of the particle distribution in pitch angle (for anisotropic distribution). A resonance layer in the poloidal mode structure, with the layer width increasing with the plasma resistivity, appears at the large width of the particle distribution in pitch angle space. A mode conversion, from the modified ideal kink by the EPs kinetic effect to the TAE, is also observed while increasing the birth energy of EPs. Computational results suggest that the TAE mode structure can be modified by certain key plasma parameters, such as the EPs kinetic contribution, the equilibrium pressure, the plasma resistivity, the distribution of EPs, as well as the birth energy of EPs. Such modification of the eigenmode structure can only be obtained following the non-perturbative hybrid approach (Wang et al 2013 Phys. Rev. Lett. 111 145003, Wang et al 2015 Phys. Plasmas 22 022509), as adopted in this study. More importantly, numerical results show that near the marginal stability point, the dominant poloidal harmonics of the TAE overlap with each other, and are localized at the tip positions of the Alfvén continua. This kind of TAE structure in high beta plasma with unstable ideal kink is substantially different from that of the conventional TAE.

On the propagation of uncertainties in radiation belt simulations

We present the first study of the uncertainties associated with radiation belt simulations, performed in the standard quasi-linear diffusion framework. In particular, we estimate how uncertainties of some input parameters propagate through the nonlinear simulation, producing a distribution of outputs that can be quite broad. Here, we restrict our focus on two-dimensional simulations (in energy and pitch angle space) of parallel-propagating chorus waves only, and we study as stochastic input parameters the geomagnetic index Kp(that characterizes the time dependency of an idealized storm), the latitudinal extent of waves, and the average electron density. We employ a collocation method, thus performing an ensemble of simulations. The results of this work point to the necessity of shifting to a probabilistic interpretation of radiation belt simulation results, and suggest that an accurate specification of a time-dependent density model is crucial for modeling the radiation environment.

Interaction of ULF waves with different ion species: Pitch angle and phase space density implications

AbstractULF waves can accelerate/decelerate the charged particles including the ring current ions via drift‐bounce resonance, which play an important role in the dynamics of ring current during storm times. This study compares the different behaviors of oxygen ions (10.5–35.1 keV) and protons (0.3–12.3 keV) which simultaneously interact with Pc5 ULF waves observed by Cluster on 3 June 2003. The ULF waves are identified as the fundamental mode oscillations. Both oxygen ions and protons show periodic energy dispersion and pitch angle dispersion signatures, which satisfy the drift‐bounce resonance condition of N=2. The different behaviors of oxygen ions and protons include (1) the resonant energy of oxygen ions is higher than that of protons due to mass difference; (2) the phase space density (PSD) of oxygen ions show relative variations (3.6–6.3) that are much larger than that of protons (<0.4), which indicates a more efficient energy exchange between oxygen ions and ULF waves; (3) the PSD spectra show that oxygen ions are accelerated, while protons are decelerated, which depend on the radial gradient of their PSD; (4) the pitch angle distributions (PADs) of the oxygen ions and protons show negative slope and bidirectional field‐aligned features, respectively, which is related to the preexisting state of ion PADs before the interaction with the ULF waves. In addition, the resonant ions with peak fluxes tracing back to the magnetic equator are always collocated with the accelerating (westward) electric field, which indicate that the ions are mainly accelerated near the magnetic equator and the electric field intensity of ULF waves peaks there.

Transport theory in the collisional boundary layer regime for finite aspect ratio tokamaks with broken symmetry

Transport theory in the collisional boundary layer regime for tokamaks with broken symmetry is extended to include the effects of the finite aspect ratio and finite plasma β. Here, β is the ratio of the plasma thermal pressure to the magnetic field pressure. Transport fluxes are calculated on the perturbed magnetic surface where plasma pressure is constant. The extension of the theory to finite aspect ratio tokamaks is made possible because the perturbed particle distribution that contributes to the transport fluxes in the collisional boundary layer regime is localized in the pitch angle space. Invoking the flux-force relation, the transport fluxes can be used for modeling toroidal plasma flow in tokamaks.

Non-perturbative modelling of energetic particle effects on resistive wall mode: Anisotropy and finite orbit width

A non-perturbative magnetohydrodynamic-kinetic hybrid formulation is developed and implemented into the MARS-K code [Liu et al., Phys. Plasmas 15, 112503 (2008)] that takes into account the anisotropy and asymmetry [Graves et al., Nature Commun. 3, 624 (2012)] of the equilibrium distribution of energetic particles (EPs) in particle pitch angle space, as well as first order finite orbit width (FOW) corrections for both passing and trapped EPs. Anisotropic models, which affect both the adiabatic and non-adiabatic drift kinetic energy contributions, are implemented for both neutral beam injection and ion cyclotron resonant heating induced EPs. The first order FOW correction does not contribute to the precessional drift resonance of trapped particles, but generally remains finite for the bounce and transit resonance contributions, as well as for the adiabatic contributions from asymmetrically distributed passing particles. Numerical results for a 9MA steady state ITER plasma suggest that (i) both the anisotropy and FOW effects can be important for the resistive wall mode stability in ITER plasmas; and (ii) the non-perturbative approach predicts less kinetic stabilization of the mode, than the perturbative approach, in the presence of anisotropy and FOW effects for the EPs. The latter may partially be related to the modification of the eigenfunction of the mode by the drift kinetic effects.

Relativistic electron acceleration by oblique whistler waves

Test-particle simulations of electrons interacting with finite-amplitude, obliquely propagating whistler waves are carried out in order to investigate the acceleration of relativistic electrons by these waves. According to the present findings, an efficient acceleration of relativistic electrons requires a narrow range of oblique propagation angles, close to the whistler resonance cone angle, when the wave amplitude is held constant at relatively low value. For a constant wave propagation angle, it is found that a range of oblique whistler wave amplitudes permits the acceleration of relativistic electrons to O(MeV) energies. An initial distribution of test electrons is shown to form a power-law distribution when plotted in energy space. It is also found that the acceleration is largely uniform in electron pitch-angle space.

Numerical validation of the refined formula of neoclassical toroidal plasma viscosity in tokamaks

Neoclassical toroidal plasma viscosity (NTV) theory in collisionless regimes in tokamaks has been well developed in the past. The NTV evaluated from the connected formula developed by Shaing et al (2010 Nucl. Fusion 50 025020) was in good agreement with the numerical results in most cases. The boundary condition in the superbanana plateau regime has been found to be important in the numerical modelling when the resonant pitch is close to the boundaries of the pitch angle space, but it was not included in the original connected formula. This generates a big discrepancy between the numerical results and those evaluated from the smoothly connected formula as the resonant pitch is close to 0 or 1. Recently, the connected formula was refined. In this paper, we present the method of how to apply this refinement for practical NTV modelling and demonstrate the improvement of this refined formula by comparing the results evaluated from it with the numerical ones. Some techniques are developed to accurately model the NTV with the refined formula. The accuracy of the results modelled from the refined formula is strongly improved over the previous formula as the resonant pitch is close to 0 or 1 and they agree very well with the numerical ones.

Finite-beta effects on quasi-linear diffusion coefficients

[1] One of the most important theoretical tools for modeling the sources and losses of Earth's radiation belt electrons is that based on quasi-linear theory. However, one of the key assumptions made in such efforts is that the waves are described by cold plasma theory. In this paper, quasi-linear diffusion coefficients for hot/warm (or finite-beta) plasmas are formulated and computed on the basis of a recently developed empirical theory in which exact numerical hot/warm plasma dispersion relations are modeled by analytical functions. It is found that finite-beta effects associated with whistler mode waves lead to a reduction of the diffusion coefficients for relativistic electrons near 90° pitch angles, which tends to reduce the efficiency of electron acceleration. Finite-beta effects associated with electron scattering by electromagnetic ion cyclotron (EMIC) waves broaden the range of wave-particle interaction in pitch angle space, and hence increase the efficiency of pitch angle diffusion. The rate of pitch angle diffusion of protons by EMIC waves is also generally enhanced by finite-beta effects.

Modelling of the neoclassical toroidal plasma viscosity torque in tokamaks

Neoclassical toroidal plasma viscosity (NTV) torque induced by non-axisymmetric magnetic perturbation in the collisionless regimes in tokamaks is modelled by solving the bounce-averaged drift kinetic equation numerically. The detailed comparison between the numerical and the analytic solutions of NTV is discussed in this paper. In different asymptotic limits of the collisionless regimes, the numerical solutions are in good agreement with the analytic results. The numerical results are different from the analytic results calculated from the smoothly connected formula in the transit regimes. The pitch angle scattering is especially important in the regime. The final difference between the numerical and the analytic results can be up to a factor of 2 near the transition between the non-resonant and resonant regimes. This reveals the importance of the boundary condition of the pitch angle space. The sign of the electric field is found to be important in the calculation of the NTV torque. It shows that the effect of the resonant particles makes the NTV torque more important for the lower collisionality and lower rotation cases, which are the International Thermonuclear Experimental Reactor relevant conditions. It also shows that the electron NTV torque is important in the low collisionality case. This numerical method can be applied for modelling the NTV torque in different collisionality regimes and their transitions in tokamaks without additional approximations.