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A basic feature of fluid mechanics concerns the frictionless phase-space dynamics of particles in an incompressible fluid. The issue, besides its theoretical interest in turbulence theory, is important in many applications, such as the pollutant dynamics in the atmosphere, a problem relevant for civil protection monitoring of air quality. Actually, both the numerical simulation of the ABL (atmospheric boundary layer) portion of the atmosphere and that of pollutant dynamics may generally require the correct definition of the Lagrangian dynamics which characterizes arbitrary fluid elements of incompressible thermofluids. We claim that particularly important for applications would be to consider these trajectories as phase-space trajectories. This involves, however, the unfolding of a fundamental theoretical problem up to now substantially unsolved: {\it namely the determination of the exact frictionless dynamics of tracer particles in an incompressible fluid, treated either as a deterministic or a turbulent (i.e., stochastic) continuum.} In this paper we intend to formulate the necessary theoretical framework to construct such a type of description. This is based on a phase-space inverse kinetic theory (IKT) approach recently developed for incompressible fluids (Ellero \textit{et al.}, 2004-2008). {\it Our claim is that the conditional frictionless dynamics of a tracer particles - which corresponds to a prescribed velocity probability density and an arbitrary choice of the relevant fluid fields - can be exactly specified}., Contributed paper at RGD26 (Kyoto, Japan, July 2008)

The gyrokinetic description of particle dynamics faces a basic difficulty when a special type of canonical variables is sought, i.e., the so-called \textit{gyrokinetic canonical variables}. These are defined in such a way that two of them are respectively identified with the gyrophase-angle, describing the fast particle gyration motion around magnetic field lines, and its canonically conjugate momentum. In this paper we intend to discuss the conditions of existence for these variables., Comment: Contributed paper at RGD26 (Kyoto, Japan, July 2008)

Fundamental aspects of inverse-kinetic theories for the incompressible Navier–Stokes equations [M. Tessarotto, M. Ellero, RGD24, Italy, July 10–16, 2004, AIP Conf. Proc. 762 (2005) 108; M. Ellero, M. Tessarotto, Physica A 355 (2005) 233] include the possibility of defining uniquely the kinetic equation underlying such models and, furthermore, the construction of a kinetic theory implying also the energy equation. The latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations. These issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory. Purpose of this work is to prove that under suitable prescriptions the inverse-kinetic theory can be determined to satisfy such requirements.

A remarkable feature of standard quantum mechanics is its analogy with classical fluid dynamics. This has motivated in the past efforts to formulate phase-space techniques based on various statistical models of quantum hydrodynamic equations. In this work an inverse kinetic theory for the Schr\"odinger equation has been constructed in order to formally describe the standard quantum dynamics by means of a classical dynamical system (to be denoted as phase-space Schr\"odinger dynamical system). It is shown that the inverse kinetic theory can be (non)uniquely determined under suitable mathematical prescriptions. In particular, when the quantum linear momentum is identified with a suitable linear kinetic momentum, it follows that the fluctuations of the position vector and the kinetic linear momentum satisfy identically the Heisenberg theorem.

Detailed properties of the electrostatic Shukla–Nambu–Salimullah potential in an inhomogeneous magnetoplasma in the presence of ion streaming due to diamagnetic drift as in a laboratory discharge plasma have been examined analytically. The potential becomes a sensitive function of the external static magnetic field, the scalelength of inhomogeneity, and the diamagnetic ion streaming velocity. For a decreasing ion density gradient, there is a limit of existence of this static modified shielding potential.

A new class of magnetic fields, deduced from the extension of force-free fields to the complex domain, is presented. The compatibility of such kind of fields with the magnetic diffusion equation for harmonic fields, in absence of plasma flow velocity, is shown. This allows to produce a static Lorentz force density in a resistive plasma and possibly to balance pressure forces in the momentum equation. An explicit example of equilibrium solution for a cylindrical plasma column with scalar pressure is presented and its connections with previous experimental results are discussed. Some discussion is also devoted to the question of the existence of this kind of closed equilibria.

The charge reduction effect, produced by the nonlinear Debye screening of high-Z charges occuring in strongly-coupled plasmas, is investigated. An analytic asymptotic expression is obtained for the charge reduction factor which determines the Debye-Hueckel potential generated by a charged test particle. Its relevant parametric dependencies are analyzed and shown to predict a strong charge reduction effect in strongly-coupled plasmas., 4 figures

The classical Debye-H\"{u}ckel screening effect of the electrostatic field generated by isolated charged particles immersed in a plasma is reviewed. The validity of the underlying mathematical model, and particularly of the weak-field approximation, are analyzed. It is shown that the presence of the plasma sheath around test particles and the resulting effect of charge screening are essential for the description of plasmas which are strongly coupled., Comment: 3 figures

A fundamental aspect of CFD for incompressible fluids and magnetofluids is the algorithmic complexity which characterizes both conventional direct simulation methods and kinetic approaches based on microscopic or phenomenological models based on asymptotic kinetic theories. A possible solution to this problem, can be provided by so‐called inverse kinetic theories suitably constructed in such a way to avoid the computational complexity of previous numerical approaches, in particular due to the Poisson equation for the fluid pressure, which typically (for example, for so‐called fractional step due to Kim and Moin), involves an algorithmic complexity of order N2, being N the grid number characterizing the numerical solution. This may explain why, in massive parallel DNS simulations of turbulent flows, the maximum grid number remains substantially limited. Goal of the present Note is propose a solution to this problem, pointing out a new inverse kinetic theory which does not require the solution of Poisson eq...

An inverse kinetic theory applying specifically to incompressible Newtonian fluids which permits us to avoid the N 2 algorithmic complexity of the Poisson equation for the fluid pressure is presented. The theory is based on the construction of a suitable kinetic equation in phase space, which permits us to determine exactly the fluid equations by means of the velocity moments of the kinetic distribution function. It is found that the fluid pressure can also be determined as a moment of the distribution function without solving the Poisson equation, as is usually required in direct solution methods for the incompressible fluid equations. Finally, the dynamical system, underlying the incompressible Navier–Stokes equations and advancing in time the fluid fields, has been also identified and proven to produce an unique set of fluid equations.

Recently, an increasing interest in astrophysical as well as laboratory plasmas has been manifested in reference to the existence of relativistic flows, related in turn to the production of intense electric fields in magnetized systems. Such phenomena require their description in the framework of a consistent relativistic kinetic theory, rather than on relativistic MHD equations, subject to specific closure conditions. The purpose of this work is to apply the relativistic single-particle guiding-center theory developed by Beklemishev and Tessarotto, including the nonlinear treatment of small-wavelength EM perturbations which may naturally arise in such systems. As a result, a closed set of relativistic gyrokinetic equations, consisting of the collisionless relativistic kinetic equation, expressed in hybrid gyrokinetic variables, and the averaged Maxwell's equations, is derived for an arbitrary four-dimensional coordinate system., Comment: 6 pages, 1 figure. Contributed to the Proceedings of the 24th International Symposium on Rarefied Gas Dynamics, July 10-16, 2004 Porto Giardino Monopoli (Bari), Italy

The well-known gyrokinetic problem regards the perturbative expansion related to the dynamics of a charged particle subject to fast gyration motion due to the presence of a strong magnetic field. Although a variety of approaches have been formulated in the past to this well known problem, surprisingly a purely canonical approach based on Lie transform methods is still missing. This paper aims to fill in this gap and provide at the same time new insight in Lie-transform approaches., Comment: 6 pages, 1 figure. Contributed to the Proceedings of the 24th International Symposium on Rarefied Gas Dynamics, July 10-16, 2004 Porto Giardino Monopoli (Bari), Italy

A famous aspect of discrete dynamical systems defined by area-preserving maps is the physical interpretation of stochastic transitions occurring locally which manifest themselves through the destruction of invariant KAM curves and the local or global onset of chaos. Despite numerous previous investigations (see in particular Chirikov, Greene, Percival, Escande and Doveil and MacKay) based on different approaches, several aspects of the phenomenon still escape a complete understanding and a rigorous description. In particular Greene's approach is based on several conjectures, one of which is that the stochastic transition leading to the destruction of the last KAM curve in the standard map is due the linear destabilization of the elliptic points belonging to a peculiar family of invariants sets {I(m,n)} (rational iterates) having rational winding numbers and associated to the last KAM curve. Purpose of this work is to analyze the nonlinear phenomena leading to the stochastic transition in the standard map and their effect on the destabilization of the invariant sets associated to the KAM curves, leading, ultimately, to the destruction of the KAM curves themselves., Comment: 6 pages, 1 figure. Contributed to the Proceedings of the 24th International Symposium on Rarefied Gas Dynamics, July 10-16, 2004 Porto Giardino Monopoli (Bari), Italy

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

J.APPL.PHYS.

J.PHYS.SOC.JAPAN

Based on recent theoretical developments (Cremaschini and Tessarotto, 2011–2013), in this paper the issue is addressed of the first-principle construction of the nonlocal relativistic radiation-reaction (RR) equation for classical spherical-shell, finite-size particles and antiparticles. This is achieved invoking the axioms of Classical Electrodynamics by means of the Hamilton variational principle. In connection with this, the Lagrangian conservation laws, together with the possible existence of adiabatic invariants, and the transformation laws of the RR equation with respect to CPT and time-reversal transformations are investigated. The latter properties make possible the parametrization of the RR equations, holding respectively for particles and antiparticles of this type, in terms of the same coordinate time t and the investigation of the qualitative properties of their solutions. In particular, in both cases the RR self-force is found to have the same signature, which implies that the dynamics of classical finite-size antiparticles is equivalent to that of classical extended particles of opposite charge. Therefore, in the framework of Classical Mechanics, a distinction between particles and antiparticles cannot be made based solely on the electromagnetic interactions associated with electromagnetic RR phenomena. As a basic application of the theory, the Lagrangian conservation laws and symmetry properties for the Hamiltonian asymptotic approximations of the exact RR equation are also addressed.

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.

PHYS.PLASMAS

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.

A variational theory is developed for the study of collisional transport in a multispecies plasma immersed in a toroidal and axisymmetric equilibrium magnetic field. The approach is based on a perturbative expansion in terms of an appropriate admimensional parameter which characterizes the inhomogeneities of the equilibrium magnetic field. Main purposes of the paper are to set up a general mathematical formalism for a plasma in the weakly collisional regime and to determine an interpolation formula for the leading-order neoclassical transport coefficients.

A model of collision operator is presented for weakly collisional symmetric magnetoplasmas and its predictions are compared with those stemming from a variational approach. It is proven that the two methods yield for the particle and kinetic energy fluxes results which are in agreement up to terms of first order with respect to an appropriate expansion parameter which characterizes the inhomogeneities of the equilibrium magnetic field.

A collisional transport theory for finite β magnetoplasma immerse in an equilibrium magnetic field, defined in a largely arbitrary bounded domain, is presented. Perturbative solutions of the Fokker-Planck equation are determined in the limit of the weakly collisional regime. The results are used to evaluate explicit expressions for the particle and kinetic energy fluxes.

A detailed stability analysis is carried out for high-frequency whistler waves, as to typical conditions of the s.w. plasma. At low frequencies (\(|\omega | \ll \Omega _i ,{\text{ }}\Omega _i \) being the ion cyclotron frequency) an analytical solution of the general linear dispersion relation is obtained, and a stability analysis again performed, determining the threshold condition for the occurrence of instability.

We point out a variational method which is applicable to a class of quiescent collisional magnetoplasmas diffusing from a state near the local thermodynamic equilibrium and enbedded in an equilibrium magnetic field with one ignorable coordinate. A perturbative expansion is introduced for the solution of a variational principle stemming from the drift Fokker-Planck equation and the resulting hierarchy of equations is investigated.

We discuss the macroscopic irreversible behaviour of an appropriate class of collisional magnetoplasmas, fulfilling the so-called «small-Larmor-radius ordering», and determine an asymptotic relationship for the average, on an isobaric surface, of the local entropy production rate, in terms of the radial fluxes (particle and kinetic energy) as well as of the source contribution due to Joule dissipation. We point out that the physical interpretation of the various contributions to the entropy production rate is greatly simplified by making use of an exact decomposition which is related to the structure of the (radial) fluxes. Then we discuss a variational principle for the drift Fokker-Planck equation and show how its solution minimizes an appropriate contribution to the entropy production rate, which is related to the geometry of the equilibrium configuration and yields, consequently, a principle of minimum entropy production. Finally we examine the generalization to a class of quiescent nonequilibrium initial states.

An analytical derivation is given of possible electron instabilities in the solar wind, related to the observed forms of the electron distribution functions. The instabilities refer to the frequency regime intermediate between ion and electron cyclotron frequencies. Instability criteria are discussed in terms of the various parameters which have been measured by different solar wind experiments.

The neoclassical transport theory is developed for an axisymmetric toroidal plasma containing highly energetic α-particles as produced by fusion events. It is shown that the α-particle fluxes in this case have a weaker dependence on the aspect ratio than at lower energies since the prevailing collisional effect is a drag. The bootstrap current results to be reduced by the presence of fusion α-particles.

An alternate derivation of Boltzmann's equation from the BBGKY hierarchy is obtained for an infinite classical system of particles interacting with repulsive central force with finite radius of interaction and in the presence of an external force field. It is proved that in an appropriate asymptotic limit this equation is exact for a class of systems called “Boltzmann gases”, except possibly on a set of zero Lebesgue measure in the one particle phase-space.

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.

In this paper an investigation on the linear stability of a locally Maxwellian and collisionless plasma with respect to general electrostatic perturbations of the drift type is proposed. The representation used for the perturbation contains as a special subclass the ballooning mode representation. General stability theorems are found in two asymptotic limits, corresponding to the regimes typical of the collisionless trapped particle and universal drift modes, respectively. In both cases at most two unstable modes can exist, which are related both to the presence of temperature gradients and to unfavourable curvature of the magnetic field. Here a comparison is made with the stability predictions in a sheared slab plasma, for drift waves, and it is found that only in particular limits similar results are recovered in toroidal configurations. Finally analytic stability criteria are derived for the regime of the trapped-particle mode and an application is given for a large aspect ratio with circular cross-section and for radially localized perturbations.

A spatially homogeneous plasma without external fields is considered. Considerations are limited to a one-dimensional system. A perturbation expansion for the average distribution function, neglecting modal coupling effects, is used. The analysis is restricted to a second-order perturbation expansion in the field intensity. It is shown how the weak-velocity-dependence assumption can be relaxed. This leads in turn to a significant modification in the diffusion coefficient and in the dielectric function. (MOW)

Gyrokinetic theory for ballooning modes with frequency lower than the thermal ion bounce frequency is presented. Due to a vast difference of the centrifugal and the Coriolis forces experienced by ions and electrons, the parallel electric field effect is found to be excited even in this low frequency case. Nevertheless, it is proved that, under suitable ordering assumptions, the magnetohydrodynamic (MHD)-structure-like ballooning mode equation can still be obtained. The kinetic effects are found to appear in three aspects: enlarging the magnitude of the apparent inertia, resulting in a real shift of the marginal frequency, and contributing a stabilizing term.

PHYS.PLASMAS

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.