The Work Sessions of the Plasma Theory The Group of Plasma Theory (F. Spineanu, Madalina Vlad, Andreea Croitoru, Dragos Palade, Virgil Baran, Alexei Zubarev) Abstract A work session gathers participants with a prealable knowledge of the Plasma Theory, interested to clarify, for them and for the others, problems related to this …eld. The Sessions will consist of a presentation followed by discussions on the various approaches, derivation and approximations, etc. The …nal result of this e¤ort will be an enhanced knowledge but, more importantly, original work to improve previous approaches and to o¤er explanation to experiments. For some of the Work Sessions the presentations will be made to coincide with the program of the Doctoral School in plasma physics. The present text will be modi…ed by successive updates and adaptations. [‡orin.spineanu@in‡pr.ro]

1

The Work Sessions of Plasma Theory. Introduction

Subjects from: Fusion, astrophysical, solar, quantum plasma Processes: free energy, waves and instabilities, radiation, transport, structures We will focus on con…ned plasma. The physical picture for a Tokamak reactor plasma is not fully elaborated. Some aspects, like the H-mode, the pedestal, Edge Localized Modes, turbulent transport are not yet clear, but they are essentially involved in the success of the reactor plasma. Other aspects, like the particle pinch, the current hole, the fast penetration of heavy impurities to the center, etc. are not understood, although models have been proposed and are currently used in calculations. A scenario for the 1

evolution of plasma in reactor (heating, fuelling, current drive) must relay on a good, logically integrated, physical picture. Note. The complex picture of plasma (in almost any practical case) prevents a systematic de…nition of a single path along which a presentation should be organized. It is an illustration of a hypertext, where the ensemble of facts is connected not in juxtaposition but multiply cross-connected, as a graph extending in all the "volume". As all other …elds, the theory of plasma is a structure and its content of "truth" (as we know it now) should probably be represented as being continuous. Many textbooks (especially those that are intended to teaching) start by discretizing this continuum since they are constrained to adopt a point of start and a progressive logical construction but nevertheless they meet di¢ culties. Simply stated, in such approaches it is recommended to return again and again and reconstruct the connections. It seems there is no other way and we really do not know another one.

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Basic description (kinetic, ‡uid)

This will be a simple introduction to the basic elements of description of plasma processes in terms of a distribution function, respectively in terms of ‡uid quantities. It is not the right place for a detailed theory, but in order to proceed further we need to know few elementary things. We will touch several subjects. The Liouville Theorem, the ChapmanKolmogorov probability equations, the Fokker-Planck equation, the Boltzmann equation, the Vlasov equation. For practical applications, we discuss moments of the Fokker-Planck equation. The equations of conservation of density, momentum, energy, angular momentum. We contrast the kinetic aspect, able to account for the resonances wave-particle, but approximative, to the ‡uid equations always exact but unable to separate into distinct parts, all the complex behaviors. Space time scales, the geometry, initial and boundary conditions and some knowledge of what we look for are necesary. Gyration and the multiple space/time scales. Parameters of expansion, for toroidal plasma. Diamagnetic ‡ow. P…rsch Schluter current. Illustration of the danger arising from using the noncollisional diamagnetic ‡ow, versus the ‡ows resulting from the neoclassical drifts of particles. Intrinsic ‡ows in a toroidal plasma. Intrinsic non-stationarity. 2

The rate of radial transport associated to this equilibrium. The famous q2. Resistivity vs. Landau damping

3

Neoclassic

This is the most characteristic manifestation of the geometry of toroidal magnetic con…nement. It is speci…c to Tokamak, Stellarator and similar devices (e.g. Large Helical Device - Nagoya, torsatron, many others, being interesting but closed due to the current dominant success of the Tokamak). The name comes from the fact that it can be seen as a development from the classical transport, which is that of a collisional plasma immersed in a straight magnetic …eld (let us say: cylindrical). The basic motion is gyration and collisions will produce on the average a space step of the order of the radius of the gyration. The toroidality induces …rst of all the space variation of the magnitude of the con…ning magnetic …eld. Since the lines are helical, a particle must travel from the outer region (where the magnetic …eld is small) to the inner region (closer to the vertical, - main - , symmetry axis of the torus) where the magnetic …eld is larger. Then the particles that do not have enough energy in their parallel motion will be re‡ected back and cannot follow the full magnetic line. The orbit takes the shape of a banana and this changes fundamentalyy the transport: the collisional random space step is no more the Larmor gyration radius, but is the width of a banana, and this is much larger. The transport, the loss of particle and energy, is much enhanced. The toroidal geometry of the magnetic …eld. Magnetic surfaces. Rational lines and ergodicity. The basic invariants of particle motion: energy, magnetic moment, longitudinal invariant. The integral of the ‡ux through the area of a moving banana. The average over the magnetic surface. Operator of annihilation by integration along the line.

3.1

Full set of equations of charged motion of particles

The motion of particles in the toroidal magnetic …eld of a Tokamak becomes essential for transport and we have to consider it in detail. It is an old problem and we are going to discuss it from the basic formulations.

3

Simpli…ed treatments of the transport induced by the instabilities are possible, with exact knowledge of the trajectories replaced by general characteristics (for example: width of a banana, the poloidal angle of re‡ection, the time of bounce etc.). A full representation of the orbit is required in the study of e¤ect of heating, in particular the change of the orbit due to a di¤erent 2 =2), which distribution between the longitudinal vk2 and perpendicular (v? is important for the rotation of the plasma. Massive Monte-Carlo numerical calculations of the processes in the pedestal or at the X point or in Scrape-O¤ Layer are currently carried out. There is a list, for the set of equations of motion. 1. Wong Burrell, Galeev Berk 2. Galeev Sagdeev; Hahm Fong. Explicit expressions. 3. Boozer; Hamiltonian form, plus ripple. banana orbits and circulating orbits. Parameters, and in the velocity space. Bounce averaging versus initial condition problem (transients) in the orbit integration. The drift of particles. Polarization drift: classical (Spitzer) and neoclassical (Hinton Robertson) @E=@t. Numerical integration of the equations of motion. Orbit, and our’s.

3.2

The drift-kinetic equation: derivation.

The kinetic point of view. Return to Liouville theorem, Fokker-Planck equation. The BBGKY hierarchy. Boltzmann equation. Why we can talk of a distribution function of only one-particle, and at what cost. Including the theory of particle motion in magnetic …eld. Change of variables to exhibit the guiding center and the gyration. Expansion and gyroaveraging. Rederivation of the equations of motion by expanding for guiding center.

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4

The drift-kinetic equation: solution

Solutions of Galeev Sagdeev Rosenbluth Hazeltine Hinton; solutions of the drift equation in trapped/circulating regimes; Hazeltine Hinton, Hirschman Sigmar, Helander Sigmar. Hinton Santarius, numerical The neoclassical non-uniformity of potential (1) fj (r; ) on magnetic surfaces. Stringer.

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(1)

(r; ) and of density

The collision operator

It is the …rst source of transport. Collisions have a typical time scale, the inverse of the frequency. The spatial scale is given by the geometry, it can be the Larmor radius in the main con…ning magnetic …eld, or the Larmor radius rede…ned for the poloidal …eld, with the typical spatial deviation at a scattering being …xed by the banana width. The di¤usion is the square of the spatial scale divided by the time scale. The collisions can take a trapped particle into a circulating one. [Various regimes]. Exchange of momentum; exchange of energy; the parameters , . Pitch-angle scattering. Transition trapped-circulating Energy. Application to NBI = Neutral Beam Injection. Krook, Lorentz, Fokker Planck, Landau The collisions and the di¤usion is the velocity space during radiation heating and current drive. Numerical realization.

5.1

Viscosity

It is important to make the distinction: the resistive dissipative mechanism (almost always: collisions) is irreversible, it is a real dissipation of the energy from a regular process and transfer of energy to degrees of freedom that before were not excited. They will never return it back to the regular process. Entropy is increasing. No other process that is NOT irreversible (increasing the entropy) should be considered. The turbulence of an ideal ‡uid cannot increase the entropy, is not dissipative and is not able to mediate reconnection of ‡uid or magnetic …eld. It may do that, but it is reversible. 5

most explicitly, the viscosity is simply the transport of the momentum directed along one axis in the direction perpendicular of that axis. It is NOT a dissipation, is a transport of momentum. However, since the viscosity is associated to collisions, the dissipation and the viscosity is frequently not separated, once viscosity exists, we expect that some dissipation also exists. But this is NOT neccessary. The viscosity is behind the Entrainment, as for example in the atmospheric convection, between rasing cloud and static environment or at the surface oceanatmosphere. viscosity underlying the poloidal magnetic damping. Stix, Chang C.S. Radial current Jr associated with the magnetic damping. Shaing, di¤erence (explicit) pk p? . gyroviscosity Simple formulas for viscosity

6

Rotation

The rotation is probably the most important subject in relation with the con…nement in Tokamak. The reason is a combination of two facts: the most e¢ cient energy transport is usually produced by long radially extended eddies of some instability (Ion Temperature Gradient driven instability and turbulence): the radially elongated eddies transport plasma from the hot regions (close to the center) toward the cold edge. the sheared poloidal rotation is able to tear appart the radially elongated eddies; the di¤usion decreases with the square of the factor of radial reduction of the typical step. Less transport means better con…nement. If you want better con…nement, for the same amount of input power, look for poloidal plasma rotation. Hazeltine: equilibrium neoclassical poloidal rotation determined by a gradient of the temperature. The ‡uid description: surface-averaged toroidal viscosity is zero (Rozhansky Tendler) The kinetic description : 6

equations for dV =dt drift-kinetic equation solution with PV and function. See also Stringer - nonuniformity of (1) fj and of (1) in the surfaces. The Landau singularity. Rotation with speed comparable with cs Hinton Wong. Centrifugal e¤ect on ions. Coriolis e¤ect, on heavy impurities. "Banana pile-up" (Hsu, Wiesen). Decay of poloidal rotation. The Transit Time Magnetic Pumping. The parallel viscosity. (Rosenbluth, Stix, Shaing, Taguchi, Hassam Drake) Rotation produced by 1. ICRH; ECH; ECCD; LHCD 2. ionization of neutrals, transitory (gas pu¤, pellets) 3. trapped / untrapped changes 4. Stringer mechanism (spontaneous spin-up, shock formation) E¤ect of sheared rotation on the instabilities. Suppression of turbulence linear, shift of the mode relative to the resonant surface and coupling to damping higher modes DIA (Biglari Diamond Terry) Universal connection between sheared rotation (vorticity) and density (Ertel’s theorem) The MHD invariant (Sagdeev, Moiseev, Tur, Yanovskii), see further.

6.1

Bootstrap current

The theory; (Taylor, Connor, Hastie; Cordey). The need for a non-zero current on the axis. The advanced tokamak. Which is "leaning" on what ? The lesson of the diamagnetic "‡ow". About bootstrap current in the "pedestal’: how much space need the banana orbits? 7

7

Transport

7.1

Neoclasic

Forces and ‡uxes; the Onsager symmetric matrix. The general expressions for -‡uxes in terms of K and L. Derivation of the relationship between parallel friction forces and perpendicular ‡uxes. The anisotropy of the pressure. Rosenbluth Hazeltine Hinton 1973 The minimum rate of entropy production. The functional. Numerical: The code NCLASS Coupled with the transport code SIMTR.

7.2

Anomalous (turbulent)

The main source of energy loss in Tokamak is the anomalous transport. It is produced by the instabilities which are excited due to the strong radial gradients (sources of free energy) and evolve into turbulent regime. The Review of Connor Wilson, the collection of expressions for the thermal di¤usion coe¢ cient. All instabilities. The H-mode. The pedestal. The Internal Transport Barrier The zonal ‡ows assumed to be excited by Reynolds stress. Examination of alternatives: Stringer mechanism; The Edge Localized Modes. Remedy: impurity seeding to control the amplitude of ELMs; or, magnetic stochasticity Resonant Magnetic Perturbation

8

MHD, tearing modes

For Tokamak the MHD formulation is an approximation. We can use it for global invariants and for a localized description of magnetic reconnection and formation of magnetic islands. basic tearing: Furth, Killeen, Rosenbluth. Boundary layer formulation, constant- approximation. magnetic islands, e¤ect on transport The Reduced MHD (Strauss, Biskamp) The Grad-Shafranov equation 8

saw-tooth instability The magnetic stochasticity

9

Invariants of the ‡uid and of the two-‡uid plasma

9.1

Invariant connecting the vorticity, the density and the current

The Ertel’s theorem The extension to the Sagdeev Moiseev Tur Yanovskii invariant. The consequences of the existence of this invariant the H- mode layer is vorticity/current the zonal ‡ows are associated with strong deformation of the current pro…le, including the current-hole. The role of ECH, ECCD, ICRH.

9.2

Turbulent equipartition

Yankov, basic Nycander Yankov, Isichenko, Diamond

10 10.1

Instabilities Drift waves

Basic Propagator, the inverse of the operator of time derivative along the particle’s orbit. Finite Larmor Radius The magnetic shear The story: Krall Rosenbluth

9

more than the residuum at the resonant surface (Ross and Mahajan, Tang): at the resonant surface kk ! 0 the phase velocity of the electric perturbation is higher than any electron speed, and there is no extraction of energy from the electron component to let the drift wave to increase. The Schrodinger-like eq. has a convex potential, no bound eigenmode. the radiative condition at the ion’s turning point on the radius (Berk Pearlstein). No need for bound state of the Schrodinger-like potential. The radiation condition ensures the existence of the eigenmodes. the scattering of trajectories around the resonant surface is su¢ cient to ensure extraction of energy from the electron population (Hirschman Molvig). The basis is: Dupree, Tetreault, DuBois, Balescu, Misguich. Further, the Renormalization Group modi…es the propagator. The review of Horton. The Hasegawa Mima equation, cuasi-three dimensional. The ion polarization drift. Comparison with the Euler equation. Hasegawa-Wakatani. Work of Diamond, Itoh, Itoh, centered on zonal ‡ow by "predator-pray" analysis.

10.2

ITG

The following two kinds of instabilities in the state of turbulence) : Ion Temperature Gradient driven instability (ITG) and Trapped Electron drift (TED), are considered dominant in the outer con…ning region (ITG) and core (TED). slab and toroidal coupling via toroidicity-induced drifts Basic approaches: Horton, Lee Diamond Chen Carreras Cowley Kruskal Liu Rosenbluth sheared toroidal rotation Trapped ion instabilities

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10.3

Trapped Electron Instability

Two-dimensional wave-function balloonning representation

11 11.1

Turbulence E¤ect of scattering of the particles’orbits on the growth rates

Hirschman-Molving, scattering of orbits close to the resonant surface. New chance for drift-waves A complete treatment of a drift wave instability with inclusion of the random scattering of the particle’s orbits.

11.2

Renormalization

Dupree, Tetrault, DuBois, Balescu and Misguich: clumps, etc. Direct Interaction Approximation DIA (Terry Diamond) The Renormalization Group, RNG, for Hasegawa-Mima Di¤usion In contrast: Ballistic and rare events (Levy); percolation.

11.3

The Decorrelation Trajectory Method (Transport turbulent)

This is the main original contribution made by the Group of Plasma Theory in the subject of turbulent transport in plasma. It has been developed along many years and has various applications. Statistical approach The trapping in eddies and induced transport The Decorrelation Trajectory Method The Nested Subensemble Method Various regimes of di¤usion drift waves trapping

11

11.4

Magnetic stochasticity

The most e¢ cient mechanism for radial particle and heat transport. The stochastic instability (Lyapunov exponents of separation of two initially close trajectories). Superposition of islands - Chirikov criterion. Radial current and ambipolarity path integrals test particles Rotation and magnetic stochasticity (Rojanskii Tendler) Other instruments of description of a statistical ensemble of magnetic lines: Gauss linking number.

12

Coherent structures in magnetized plasmas

small scale: drift wave vortices Horton and Meiss; Alfven-drift vortices (Petviashvili, Pokhotelov, de Blank, Schep, Pegoraro) large scale: convective cells, like Rayleigh Benard Observations Shapiro, Rosenbluth, Diamond [convective cells]. Discussion on the vectorial and the scalar nonlinearity. (Sutyrin, Mikhailovskii, etc.) zonal ‡ows : Horton, Diamond. Reynolds stress and large scale balance of energies involved in turbulence ["predator-prey" model by Diamond], zonal ‡ow, transport Similarity with winds of Rayleigh Benard experiment (Howard Krishnamurthi). The suggestion made by Tsinober, a challange for Reynolds stress. Quantitative comparison with the spontaneous Stringer e¤ect. Motion of vortices on a background of gradient of vorticity (Dubin Schecter).

13

Field Theoretical methods

The use of Field Theoretical (FT) methods dramatically enlarge the technical possibilities of the plasma theory. The FT formulations of plasma are similar with problems speci…c to many other …elds. This should be taken seriously. Nobody is denying today the role of the Hamiltonian formulation (and the methods that follow) in problems 12

of plasma or ‡uids. One should accept that this will also happen with the methods of Field Theory. Topological restrictions, transitions, cuasi-coherent structures are familiar problems for FT and speci…c methods have been developed. Formalism for two-dimensional ‡uids. The Euler equation. The Lagrangian: mixed spinor sl (2; C), gauge …eld with Chern-Simons action, interagtion of Coulombian type (generalization of the model of Jackiw Pi for two-dim charges). Equations of motion, selfduality. Exact solution from integrability of sinh-Poisson equation. Comparison with the statistical approach (Montgomery, Taylor, Edwards) Numerical solutions. Possible extension to plasma/atmosphere. Applications to coherent structures Connection with the Charney-Hasegawa-Mima equation. Path integral approach to di¤usion, trapping, Levy ‡ights

14

Special topics

The density snakes in the region of q = 1. The …lamentation of Edge Localized Modes. The current-hole state in tokamak. The pro…le resilliance The self-organization at criticality (avalanches) Vorticity concentration and axial anomaly.

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