Search

The relativistic guiding center dynamics of charged particles is described in terms of noncanonical variables. The gyrokinetic transformation is obtained using the perturbative Lagrangian approach with a fully relativistic, four-dimensional covariant formulation. It is shown that the definition of the ignorable gyrophase (as well as those of the magnetic moment and the gyrocenter energy) is not unique, and allows for several free functions in the gyrokinetic transformation. This freedom can be interpreted as a choice of the reference frame. One of these frames, namely that moving with the relativistic version of the E_×B_-drift velocity, generates the simplest and intuitive description.

A Lagrangian approach to relativistic particle guiding-center dynamics in strongly magnetized plasmas, based on a covariant formulation, is presented, proving that a four-dimensional transformation is required in order to define the correct relativistic gyrophase-averaged Lagrangian. Comparisons with previous approaches show that these become potentially inconsistent, particularly for magnetoplasmas exhibiting relativistic mass-flow velocities and/or deeply relativistic particles, causing the paradox of hyper-relativistic guiding-center velocities. To avoid this inconvenience, the relativistic velocity addition law is adopted to describe the particle motion with respect to the reference frame locally at rest with the fluid. Based on the adoption of the relativistic Hamilton principle, new expressions are determined for the relativistic guiding-center velocity transformation, the relativistic magnetic moment and the relativistic gyrokinetic equations of motion.

An approximate model for calculating the average lifting force and absorbed power, for a magnetically levitated metallic sphere, is presented. It is shown that by imposing suitable boundary conditions, expressed in terms of matching conditions for the magnetic field, it is possible to express the average absorbed power in terms of appropriate physical parameters. For a given conductive sphere, three cases of interest are analyzed, namely: (i) fixed frequency and resistivity; (ii) fixed external currents and resistivity; (iii) fixed external currents and frequency. The analysis displays the relevant parametric dependence and proves the existence of minima for the absorbed power in parameter space. The results are of potential interest for improving the temperature control of the metallic sample, which is of fundamental importance both in levitation melting and undercooling processes.

An analytical method based on two-scale analysis is developed to solve the combined boundary and initial value problem describing the ballooning modes with frequency comparable to the parallel ion acoustic one in toroidally rotating plasmas. A sonic rotation is shown possibly to be harmful for the stability. A subsonic rotation is anticipated to always play a role of stabilizing the strong ballooning modes. But, the stabilizing effect of a subsonic rotation on the weak ballooning modes is negligible. A comparison with the ideal magnetohydrodynamic (MHD) analysis is presented. The scheme for including the kinetic effects into the MHD analysis is given. The different features of the ion temperature gradient modes [L.-J. Zheng and M. Tessarotto, Phys. Plasmas 3, 4610 (1996)] and the ballooning modes in the presence of toroidal rotation are also discussed.

With the two time scale feature revealed, an analytical method is developed to solve the combined eigenvalue and initial value problem describing the ion temperature gradient (ITG) modes in the toroidally rotating plasmas. With full gyrokinetics and toroidal geometry effects being taken into account in an ordering consistent manner, dispersion relations are obtained for ITG modes both of toroidal and slab‐like branches. The results show that the sonic rotation can be harmful for the stability, while subsonic rotation with flow shear can be expected to be stabilizing. The existence of the flow shear leads the time evolution of the fluctuation signals to display two different time scales—a fast oscillating (either fast or slow growing) signal with its amplitude modulated by a slowly varying envelope, resembling fishbone‐like fluctuations.

Conversion of the magnetohydrodynamic (MHD) stability conditions to fully kinetic ones is required, since the primary assumption for MHD description that particles move collectively as fluid cells is violated for nearly collisionless fusion plasmas. A unified kinetic singular layer equation is derived in this paper for implementing this conversion. The scheme outlined in the paper makes the conversion able to be realized in a rather simple way for both low and high n modes (n the mode number). Moreover, the equation covers a wide parameter domain. The frequency ranges from the ion transit frequency to that to be much lower than the ion bounce frequency. In particular, the case with mode frequency comparable to the magnetic drift one can also be treated; and the large aspect ratio assumption is not required. The application to investigate the energetic particle effect on the MHD modes is also outlined in the paper. The results are discussed, with the importance of the q (safety factor) value on the stabili...

By investigating the conservation laws of guiding‐center mechanics, an extension of quasi‐helical static magnetohydrodynamic equilibria to incorporate strong flow is obtained, subject to suitable ordering assumptions. Such equilibria correspond to self‐consistent Fokker‐Planck kinetic equilibria in the presence of a quasi‐helical magnetic field. The effect of small symmetry‐breaking contributions in the magnetic field is investigated.

The explicit form of the weakly‐relativistic Vlasov and the Fokker‐Planck collision operators for a multispecies plasma are evaluated by using a systematic expansion in β_=v_/c and retaining corrections up to the second order. Properties of these operators are investigated.

The kinetic singular layer equation is derived from the ballooning representation formalism. The mode frequency and the ion‐ion collision frequency are assumed to be lower than the ion transit frequency. It is proven that the balance between the instability drives and the small‐parallel‐ion‐velocity (SPIV) effect, instead of inertia effect, has to be imposed on determining the eigenfrequency. Consequently, it is found that the resistive effect is ignorable, while the dominant collisional effect appears to be the collisional contribution to the SPIV effect, leading to the conclusion that the customary resistive magnetohydrodynamic modes, such as the resistive interchange, resistive ballooning, and tearing modes, are nonexistent in this description.

The purpose of this paper is to investigate the classical scalar‐pressure magnetostatic equilibrium problem for nonsymmetric configurations in the framework of a Hamiltonian approach. Requiring that the equilibrium admits locally a family of nested toroidal magnetic surfaces, the Hamiltonian equations describing the magnetic flux lines in such a subdomain are obtained for a general canonical curvilinear coordinate system. The properties of such a coordinate system are investigated and a representation of the magnetic field is obtained. Its basic feature is that the magnetic field must fulfill suitable periodicity constraints to be imposed on arbitrary rational magnetic surfaces for general nonsymmetric toroidal equilibria, i.e., it is quasisymmetric. Implications for the existence of magnetostatic equilibria are pointed out. In particular, it is proven that a generalized equilibrium equation exists for such quasisymmetric equilibria, which extends the Grad–Shafranov equation to fully three‐dimensional configurations. As an application, a representation is obtained for generalized helically symmetric equilibrium, extending the definition given by Nuhrenberg and Zille [Phys. Lett. A 129, 113 (1988)]. Since the new representation overcomes the inconsistency exhibited by the previous representation near the magnetic axis, pointed out by Garren and Boozer [Phys. Fluids B 3, 2805, 2822 (1991)], it appears potentially useful to interpret the numerical findings of quasihelical equilibria obtained so far.

A new non‐perturbative and non‐variational solution method is proposed for the gyrokinetic equation, based on a suitable approximation for the linearized Fokker‐Planck collision operator. The approach, which allows, in principle, the accurate evaluation of arbitrary moments of the distribution function for a weakly collisional magnetoplasma with toroidal equilibria of arbitrary aspect‐ratio, is susceptible of numerous significant applications, both for transport calculations, i.e., in particular, leading to the evaluation of neoclassical fluxes, as well for the investigation of linear stability analyses of dissipative perturbations.

Collisional ballooning mode theory in the banana regime is developed for tokamak configurations from the gyrokinetic formalism. A general dispersion relation is obtained, which in principle can deal with a collision operator of any type. However, investigation of an approximate Fokker–Planck collision operator developed in recent neoclassical transport theory is detailed. The most significant feature of the present theory as compared to the customary treatment lies in that the distinction between particle and fluid velocities is made in the ordering analyses. This reveals that the eigenfrequency of modes is determined by balancing the small‐parallel‐ion‐velocity (SPIV) effect [L.‐J. Zheng and M. Tessarotto, Phys. Plasmas 1, 3928 (1994)], instead of the fluid inertia one, with the instability drives. Since the parallel‐electric‐field effect is found to be negligible as compared to the SPIV effect, in contrast to the customary resistive ballooning mode picture, the leading collisional effect is demonstrated...

Collisionless kinetic ballooning mode theory is presented in the case of the frequency of modes, ω, being lower than the ion magnetic drift frequency, ωDi, in the outer region of the ballooning representation space. A fluid‐like kinetic ballooning mode equation is obtained. Previous results are pointed out to be inconsistent due to the particle velocities not being distinguished from the fluid ones in the ordering analyses. In our new theory, it is found that the small parallel ion velocity (SPIV) effect, which was first discovered in the regime ωDi≪ω≪ωsi, with ωsi the ion acoustic frequency along the field lines [L.‐J. Zheng and M. Tessarotto, Phys. Plasmas 1, 3928 (1994)], is a dominant effect in ordering with respect to any other kinetic effects previously considered, such as the finite Larmor radius and the bounce‐average trapped particle effects. It is concluded that the eigenfrequency of the modes should be determined by the SPIV effect rather than by the fluid inertia one, which suggests that those...

The derivation of the fluid‐like kinetic ballooning mode equation from the gyrokinetic and Maxwell equations in the low‐frequency regime is revisited. The traditional results are pointed out to be ordering‐inconsistent. With the theory developed in the paper, two new effects are discovered: the small parallel‐ion‐velocity (SPIV) and the apparent mass (AM) effects. The SPIV effect is induced by the ions with parallel velocities much smaller than the thermal velocity, while the AM effect mainly by the ions with velocities comparable with the thermal velocity. The trapped‐particle contribution to the SPIV effect can be interpreted as a harmonic trapped‐particle effect, which can be even more important than the traditional bounce‐average trapped‐particle effect. The SPIV effect is shown to be more significant in ordering than the inertia effect with the AM and finite Larmor radius modifications. The application of the equation is discussed. The comparison with the ideal magnetohydrodynamic theory is made.

The collisionless kinetic ballooning mode equations are obtained from the gyrokinetic and Maxwell equations for the comparable frequency regime (i.e., for frequency of the modes comparable with the ion acoustic frequency along the field lines). A two length‐scale analysis is made to reduce the kinetic ballooning mode equations to a fluid‐like one. The finite Larmor radius (FLR) effect and particle–wave resonance (Landau damping) are taken into account. The trapped‐particle effect is proven to be negligible in the ordering scheme adopted in this paper. It is emphasized that the non‐FLR kinetic effects, especially the parallel electric field effect, are significant in the comparable frequency regime rather than the intermediate one. The non‐FLR kinetic effects are found to be coupled to the vorticity equation through the apparent mass term, which suggests that the toroidal effect is essential to the kinetic mechanism in the singular layer (i.e., the outer region of ballooning mode representation space). The...

A general method for constructing Monte Carlo collision operators is formulated based on the introduction of a stochastic Liouville equation. The approach is applied to the investigation of a suitable discretized gyrokinetic equation describing the dynamics of a strongly rotating multispecies plasma, in a toroidally axisymmetric configuration. Improved expressions are obtained for the Monte Carlo collision operators for general non‐normal (in v space) coordinate systems. As an application, the important case of the bounce‐averaged gyrokinetic equation is discussed, in various cases of interest. Finally, using an approximate collision operator, the friction coefficients are evaluated and expressed in terms of energy and momentum restoring coefficients, thus yielding for the Monte Carlo operators a representation convenient for their numerical implementation.

Braginskii’s two‐fluid equations, with collisional effects neglected and electron fluid assumed to be isothermal along the magnetic field lines, are employed to obtain the singular mode equations. While the ordering analysis in the paper indicates that, in the intermediate frequency regime (i.e., that with the frequency of the modes higher than the ion acoustic frequency along the magnetic field lines, but lower than the electron one), the stabilizing finite Larmor radius (FLR) effect on the modes dominates over other collisionless, two‐fluid effects, the paper is devoted to the investigation of the comparable frequency regime (i.e., that with the frequency of the modes comparable to the ion acoustic frequency along the field lines). It is found that, in such a regime, the parallel electric field is excited and the plasma behaves nonadiabatically. The singular mode equations consist of two coupled second‐order partial differential equations, in which the FLR, the plasma compression, and the parallel elect...

A Monte Carlo approach for investigating the dynamics of quiescent collisional magnetoplasmas is presented, based on the discretization of the gyrokinetic equation. The theory applies to a strongly rotating multispecies plasma, in a toroidally axisymmetric configuration. Expressions of the Monte Carlo collision operators are obtained for general v‐space nonorthogonal coordinates systems, in terms of approximate solutions of the discretized gyrokinetic equation. Basic features of the Monte Carlo operators are that they fullfill all the required conservation laws, i.e., linear momentum and kinetic energy conservation, and in addition that they take into account correctly also off‐diagonal diffusion coefficients. The present operators are thus potentially useful for describing the dynamics of a multispecies toroidal magnetoplasma. In particular, strict ambipolarity of particle fluxes is ensured automatically in the limit of small departures of the unperturbed particle trajectories from some initial axisymmet...

The purpose of this paper is to formulate the transport problem for a multispecies rotating toroidal magnetoplasma in the so‐called runaway regime, which is defined by an appropriate ordering of relevant characteristic frequencies, in particular, the Larmor frequency, the characteristic acceleration frequency due to the applied electric field and the effective collision frequency, all evaluated at some characteristic speed v0. A suitable form of the gyrokinetic equation is obtained to describe the time‐dependent, multispecies plasma response to an applied electric field, in toroidal geometry and for a strongly rotating, quiescent, and collisional plasma. Its moment equations are proven to imply the reduction of the energy equation to Joule’s law, as well as consequences on the form of Ohm’s law and of the Grad–Shafranov equation. To construct an approximate solution of the gyrokinetic equation and to evaluate all relevant fluxes, appearing in the moment equations, a general variational solution method is ...

A reduced gyrokinetic equation is derived for a multispecies toroidal axisymmetric plasma with arbitrary toroidal differential rotation speeds and in the presence of a finite induced electric field. The kinetic equation obtained, extending previous results obtained by Hinton and Wong [Phys. Fluids 28, 3082 (1985)] and by Catto et al. [Phys. Fluids 30, 2784 (1987)], has a form suited for transport applications, via variational techniques; in particular it exhibits the feature that all source terms, including the Spitzer source term, carrying the contribution due to the inductive electric field, appear to be acted upon by the collision operator. Moreover, the equation displays a new contribution due to ‘‘explicit’’ velocity perturbations, here proven to be consistent with transport ordering, whose evaluation appears relevant for transport calculations. In addition, general expressions are obtained for the neoclassical fluxes in terms of a variational principle, as well as for the classical ones, retaining, ...

Astrophysical plasmas in the surrounding of compact objects and subject to intense gravitational and electromagnetic fields are believed to give rise to relativistic regimes. Theoretical and observational evidence suggest that magnetized plasmas of this type are collisionless and can persist for long times (e.g., with respect to a distant observer, coordinate, time), while exhibiting geometrical structures characterized by the absence of well-defined spatial symmetries. In this paper the problem is posed whether such configurations can correspond to some kind of kinetic equilibrium. The issue is addressed from a theoretical perspective in the framework of a covariant Vlasov statistical description, which relies on the method of invariants. For this purpose, a systematic covariant variational formulation of gyrokinetic theory is developed, which holds without requiring any symmetry condition on the background fields. As a result, an asymptotic representation of the relativistic particle magnetic moment is obtained from its formal exact solution, in terms of a suitably-defined invariant series expansion parameter (perturbative representation). On such a basis it is shown that spatially non-symmetric kinetic equilibria can actually be determined, an example being provided by Gaussian-like distributions. As an application, the physical mechanisms related to the occurrence of a non-vanishing equilibrium fluid 4-flow are investigated., Comment: 16 pages

The problem of formulating a kinetic treatment for quasi-stationary collisionless plasmas in axisymmetric systems subject to the possibly independent presence of local strong velocity-shear and supersonic rotation velocities is posed. The theory is developed in the framework of the Vlasov-Maxwell description for multi-species non-relativistic plasmas. Applications to astrophysical accretion discs arising around compact objects and to plasmas in laboratory devices are considered. Explicit solutions for the equilibrium kinetic distribution function (KDF) are constructed based on the identification of the relevant particle adiabatic invariants. These are shown to be expressed in terms of generalized non-isotropic Gaussian distributions. A suitable perturbative theory is then developed which allows for the treatment of non-uniform strong velocity-shear/supersonic plasmas. This yields a series representation for the equilibrium KDF in which the leading-order term depends on both a finite set of fluid fields as well as on the gradients of an appropriate rotational frequency. Constitutive equations for the fluid number density, flow velocity, and pressure tensor are explicitly calculated. As a notable outcome, the discovery of a new mechanism for generating temperature and pressure anisotropies is pointed out, which represents a characteristic feature of plasmas considered here. This is shown to arise as a consequence of the canonical momentum conservation and to contribute to the occurrence of temperature anisotropy in combination with the adiabatic conservation of the particle magnetic moment. The physical relevance of the result and the implications of the kinetic solution for the self-generation of quasi-stationary electrostatic and magnetic fields through a kinetic dynamo are discussed.

A kinetic treatment is developed for collisionless magnetized plasmas occurring in high-temperature, low-density astrophysical accretion disks, such as are thought to be present in some radiatively-inefficient accretion flows onto black holes. Quasi-stationary configurations are investigated, within the framework of a Vlasov-Maxwell description. The plasma is taken to be axisymmetric and subject to the action of slowly time-varying gravitational and electromagnetic fields. The magnetic field is assumed to be characterized by a family of locally nested but open magnetic surfaces. The slow collisionless dynamics of these plasmas is investigated, yielding a reduced gyrokinetic Vlasov equation for the kinetic distribution function. For doing this, an asymptotic quasi-stationary solution is first determined, represented by a generalized bi-Maxwellian distribution expressed in terms of the relevant adiabatic invariants. The existence of the solution is shown to depend on having suitable kinetic constraints and conditions leading to particle trapping phenomena. With this solution one can treat temperature anisotropy, toroidal and poloidal flow velocities and finite Larmor-radius effects. An asymptotic expansion for the distribution function permits analytic evaluation of all of the relevant fluid fields. Basic theoretical features of the solution and their astrophysical implications are discussed. As an application, the possibility of describing the dynamics of slowly time-varying accretion flows and the self-generation of magnetic field by means of a \textquotedblleft kinetic dynamo effect\textquotedblright\ is discussed. Both effects are shown to be related to intrinsically-kinetic physical mechanisms., Accepted version for Physics of Plasmas

We propose a classical solution for the kinetic description of matter falling into a black hole, which permits to evaluate both the kinetic entropy and the entropy production rate of classical infalling matter at the event horizon. The formulation is based on a relativistic kinetic description for classical particles in the presence of an event horizon. An H-theorem is established which holds for arbitrary models of black holes and is valid also in the presence of contracting event horizons.

An efficient and systematic treatment of classical and neoclassical transport in all regimes of collisionality is formulated that permits toroidal rotation speeds on the order of the ion thermal speed for arbitrary aspect ratio, cross section, and poloidal magnetic field strength. A more convenient, but somewhat unconventional, form of the reduced kinetic equation is derived that is shown to extend the previous form by properly retaining electric field modifications. The generalized kinetic description is exploited to evaluate explicitly the radial fluxes of toroidal angular momentum and energy in a pure plasma via a variational formulation. The specific results obtained in the Pfirsch–Schluter regime are substantially more general than previous evaluations; also, significant improvements are made in the banana regime.

The classical and neoclassical particle and energy fluxes associated with a slowing down tail, alpha particle distribution function are evaluated for arbitrary aspect ratio e−1, cross section, and poloidal magnetic field. The retention of both electron and ion drag and pitch angle scattering by the background ions results in a large diffusive neoclassical heat flux in the plasma core. This flux remains substantial at larger radii only if the characteristic speed associated with pitch angle scattering, vb, is close enough to the alpha birth speed v0 so that e(v0/vb)3 remains less than some order unity critical value which is not determined by the methods herein. The enhanced neoclassical losses would only have a serious impact on ignition if the critical value of e(v0/vb)3 is found to be somewhat larger than unity.