D. F. Escande

Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory

The universal description of orbits in the domain swept by a slowly varying separatrix is provided through a symplectic map derived by means of an extension of classical adiabatic theory. This map connects action-angle-like variables of an orbit when far from the instantaneous separatrix to time-energy variables at a reference point of the orbit very close to the corresponding separatrix. When the separatrix pulsates periodically with a small frequency epsilon , the authors combine this map with WKB theory to obtain a description of the structure underlying chaos: the homoclinic tangle related to the hyperbolic fixed point whose separatrix is pulsating. For each extremum of the area within the pulsating separatrix, an initial branch of length O(1/ epsilon ) of the stable manifold is explicitly constructed, and makes O(1/ epsilon ) transverse homoclinic intersections with a similar branch of the unstable manifold.

Quasi-single helicity states in the reversed field pinch: Beyond the standard paradigm

This paper reports experimental results showing that coherent helical structures are formed in the reversed field pinch (RFP) self-organizing plasma core as a result of transitions to states where the n-spectrum of the m=1 modes is dominated by a single (1,nmax) geometrical helicity. These states are dubbed quasi-single helicity (QSH) states. Their magnetic and thermal properties measured in the reverse field experiment (RFX) [G. Rostagni, Fusion Eng. Des. 25, 301 (1995)] device are described. The present theoretical understanding of QSH states is discussed and some recent theoretical results are presented. The role of aspect ratio is discussed. These results represent a significant step to open a path beyond the standard paradigm that a bath of magnetohydrodynamic (MHD) modes is intrinsic to the RFP.

Magnetic order and confinement improvement in high-current regimes of RFX-mod with MHD feedback control

The RFX-mod machine (Sonato et al 2003 Fusion Eng. Des. 66 161) recently achieved, for the first time in a reversed-field pinch, high plasma current up to 1.6 MA with good confinement. Magnetic feedback control of magnetohydrodynamic instabilities was essential to reach the goal. As the current is raised, the plasma spontaneously accesses a new helical state, starting from turbulent multi-helical conditions. Together with this raise, the ratio between the dominant and the secondary mode amplitudes increases in a continuous way. This brings a significant improvement in the magnetic field topology, with the formation of helical flux surfaces in the core. As a consequence, strong helical transport barriers with maximum electron temperature around 1 keV develop in this region. The energy confinement time increases by a factor of 4 with respect to the lower-current, multi-helical conditions. The properties of the new helical state scale favourably with the current, thus opening promising perspectives for the higher current experiments planned for the near future.

Internal and external electron transport barriers in the RFX-mod reversed field pinch

An interesting result of magnetic chaos reduction in RFX-mod high current discharges is the development of strong electron transport barriers. An internal heat and particle transport barrier is formed when a bifurcation process changes the magnetic configuration into a helical equilibrium and chaos reduction follows, together with the formation of a null in the q shear. Strong temperature gradients develop, corresponding to a decreased thermal and particle transport. Turbulence analysis shows that the large electron temperature gradients are limited by the onset of micro-tearing modes, in addition to residual magnetic chaos. A new type of electron transport barrier with strong temperature gradients develops more externally (r/a = 0.8) accompanied by a 30% improvement of the global confinement time. The mechanism responsible for the formation of such a barrier is still unknown but it is likely associated with a local reduction of magnetic chaos. These external barriers develop primarily in situations of well-conditioned walls so that they might be regarded as attempts towards an L–H transition. Both types of barriers occur in high-current low-collisionality regimes. Analogies with tokamak and stellarators are discussed.

Microscopic dynamics of plasmas and chaos: the wave–particle interaction paradigm

The wave–particle interaction is central to microscopic plasma dynamics. A paradigm of such an interaction is one occurring during the weak warm beam–plasma instability: a Langmuir turbulence sets in and saturates by the formation of a plateau in the particle distribution function. A new approach permits us to deal with the regular and chaotic aspects of this problem using the classical mechanics of the corresponding N-body problem only. The classical Landau-van Kampen theory is recovered by using mathematical tools not more intricate than a finite Fourier sum. A single calculation yields spontaneous emission and the particle dynamics as well; classical explicative models of Landau damping are found to be misleading. Recent tools of Hamiltonian chaos enable us to derive the quasilinear equations in the regime of saturation of the instability. The calculations are readable by graduate students and provide a simple solution to a 20 year old controversy in the Vlasovian frame. As a result, the macroscopic irreversible evolution of a plasma is described by fully accounting for its microscopic reversible mechanics; for the first time, an old dream of the 19th century comes true: the irreversible evolution of an N-body problem is described by taking into account the true character of its chaotic motion.

Recent progress in reversed field pinch research in the RFX experiment

The article presents an overview of recent experimental results obtained on the RFX device. The authors obtained and studied a reversed field pinch plasma with a plasma current of up to 1 MA, negligible radiation losses and low effective charge. The local power and particle balance shows that in standard operation the plasma core is dominated by magnetic turbulence and that the global confinement is mainly provided by the edge region, where a strongly sheared radial electric field is present. With poloidal current drive the amplitude of magnetic fluctuations and the thermal conductivity of the plasma core are reduced, leading to improved confinement. Reduced heat transport is also observed when the width of the n spectrum of magnetic fluctuations is reduced.

Impact of helical boundary conditions on nonlinear 3D magnetohydrodynamic simulations of reversed-field pinch

Helical self-organized reversed-field pinch (RFP) regimes emerge both numerically—in 3D visco-resistive magnetohydrodynamic (MHD) simulations—and experimentally, as in the RFX-mod device at high current (IP above 1 MA). These states, called quasi-single helicity (QSH) states, are characterized by the action of a MHD mode that impresses a quasi-helical symmetry to the system, thus allowing a high degree of magnetic chaos healing. This is in contrast with the multiple helicity (MH) states, where magnetic fluctuations create a chaotic magnetic field degrading the confinement properties of the RFP. This paper reports an extensive numerical study performed in the frame of 3D visco-resistive MHD which considers the effect of helical magnetic boundary conditions, i.e. of a finite value of the radial magnetic field at the edge (magnetic perturbation, MP). We show that the system can be driven to a selected QSH state starting from both spontaneous QSH and MH regimes. In particular, a high enough MP can force a QSH helical self-organization with a helicity different from the spontaneous one. Moreover, MH states can be turned into QSH states with a selected helicity. A threshold in the amplitude of MP is observed above which is able to influence the system. Analysis of the magnetic topology of these simulations indicates that the dominant helical mode is able to temporarily sustain conserved magnetic structures in the core of the plasma. The region occupied by conserved magnetic surfaces increases reducing secondary modes amplitude to experimental-like values.

Vlasov equation and N-body dynamics

Difficulties in founding microscopically the Vlasov equation for Coulomb-interacting particles are recalled for both the statistical approach (BBGKY hierarchy and Liouville equation on phase space) and the dynamical approach (single empirical measure on one-particle (r,v)-space). The role of particle trajectories (characteristics) in the analysis of the partial differential Vlasov-Poisson system is stressed. Starting from many-body dynamics, a direct derivation of both Debye shielding and collective behaviour is sketched.

Proof of quasilinear equations in the strongly nonlinear regime of the weak warm beam instability

Quasilinear (QL) theory was developed in 1962 to describe the saturation of the weak warm beam-plasma instability, which involves the development of a Langmuir turbulence and the formation of a plateau in the electron velocity distribution function. The original derivations assume that particle orbits are weakly perturbed (quasi linear description), though the plateau formation is the result of a strong chaotic diffusion of the beam particles. Over two decades a controversy has developed about the validity of QL equations in the chaotic saturation regime within the Vlasovian description of the problem, and is not yet settled. Here a proof of these equations is proposed that does not resort to this description. Instead the Langmuir wave-beam system is described as a finite-number-of-degrees-of-freedom Hamiltonian system. The calculation of the chaotic drag on a beam particle is performed using Picard fixed point equation derived from Hamilton equations, and by making approximations justified by the spatial...

Uniform derivation of Coulomb collisional transport thanks to Debye shielding

The effective potential acting on particles in plasmas being essentially the Debye-shielded Coulombpotential, the particles collisional transport in thermal equilibrium is calculated for all impact pa-rameters b, with a convergent expression reducing to Rutherford scattering for small b. No cutoffat the Debye length scale is needed, and the Coulomb logarithm is only slightly modified.PACS numbers :52.20.-j Elementary processes in plasmas45.50.-j Dynamics and kinematics of a particle and a system of particles52.25.Fi Transport properties52.25.Dg Plasma kinetic equationsKeywords : Coulomb collisions, Coulomb logarithm, Debye screening, collisional transportI. WHAT DO WE CALL COLLISIONS AND WHAT IS THEIR EFFECT ?