KINETIC THEORY OF PLASMA WAVES

D. Van Eester and E. Lerche

Laboratorium voor Plasmafysica - Laboratoire de Physique des Plasmas

EUROfusion Consortium member

Koninklijke Militaire School - Ecole Royale Militaire

Trilateral Euregio Cluster, Renaissancelaan 30 - B1000 Brussels - Belgium

Tel.: (32 2) 44 14 134, Fax.: (32 2) 735 24 21, e-mail: d.van.eester@fz-juelich.de & ealerche@msn.com

ABSTRACT

In the present paper a very brief introduction is pro-vided to the theory of kinetic waves relevant to the de-scription of wave heating in fusion machines and fo-cussing mostly on radio frequency or ion cyclotron res-onance frequency waves in tokamaks. The text startsby sketching the basic philosophy underlying the stan-dardly adopted methods, describing the interaction of asingle particle with a given wave and the assumptionstypically made to arrive at a trustworthy description ofthe energy exchange, and ends by discussing some of thesubtleties of the modeling of wave-particle interaction ininhomogeneous magnetized plasmas. None of the top-ics will be treated in full detail. Hence, by no means,this text is meant to be all-inclusive. Rather, it aims atproviding a framework that should allow understandingwhat are the difficulties involved, leaving out the de-tailed derivation of the expressions as well as subtletiessuch as relativistic corrections. The interested reader isreferred to the provided references - and the referencesgiven therein - for more in depth information.

I. INTRODUCTION

The interaction between charged particles and elec-tromagnetic waves can be looked at from 2 vantagepoints: From the point of view of the waves ’plasmaheating’ is a process by which they lose energy. Therelevant equation to describe this is the wave equation,derived from Maxwell’s equations. From the point ofview of the particles the same process is viewed as again of energy. The relevant equation to describe thissecond interpretation of the same physical phenomenonis the Fokker-Planck equation, derived starting from thekinetic equation of state. A proper description of thephenomena requires that these 2 aspects are describedon the same footing, which is not at all straightforwardand which only starts to be done now that powerfulcomputers are increasingly available.

Figure 1: Wave-particle interaction: wave point of view(left: fast dynamics) vs. particle point of view (right;slow - net - dynamics) and scheme for modeling bothaspects self-consistently. The wave field (left) is plot-ted in a toroidal cut of a tokamak, the RF heated iondistribution (right) is plotted in terms of the velocitycomponents at the low field side crossing of the equa-torial plane for a prescribed radial position.

The kinetic description of waves in plasmas typi-cally starts from the equation

df

dt= C + S − L (1)

in which f is the distribution function of the chargedparticles being studied, and the right hand side de-scribes how the distribution equation evolves underthe influence of collisions the particles undergo, and ofparticle injection (S=source) and particle loss (L). Inthe high frequency domain and for a sufficiently strongmagnetic field, the left hand side is dominated by pro-cesses on a vastly different time scale than that of thenet effect of collisional interaction and particle loss orgain. Hence on the fastest time scale of the problem theright hand side is negligibly small and can be neglectedto a first approximation. The above equation then sim-ply states that the number of particles is conserved inphase space: Particles can move about and gain or loseenergy - which causes a stretching or squeezing of the

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volume in which a given number of particles resided ata given initial time - but the number of particles in thestretched phase-space volume is always conserved.

II. BASIC PHILOSOPHY [1-11]

II.A. General Formulation

In general, the orbits of particles immersed in elec-tromagnetic fields are not integrable i.e. their motioncannot be described in terms of constants of the motionbut is stochastic and thus ergodically covering parts ofphase space. On top of that, charged particles in mo-tion constitute a current themselves and thus influencethe electromagnetic fields in the fusion machine. Hence,the RF plasma current needs to be carefully accountedfor when solving Maxwell’s equations. Describing theimpact of the charged particles on the fields and theback reaction of the fields on the particles involves thechallenging task of solving a set of coupled nonlinearequations in 6 independent variables in phase space. Asthe temporal and spatial scales cover a range of manyorders of magnitude (ion cyclotron motion involves fre-quencies in the radio frequency - megaHertz - domainwhile net collisional interaction occurs on a time scaleof hundreds of milliseconds in a tokamak such as JET,or seconds in ITER; the macroscopic dimensions of suchfusion machines is several meters, while the ion Larmorradius ρ is of the order of a few millimeters) makingsimplifications is a necessity. The drawback of the widerange of scales is thereby turned into an advantage, al-lowing to set apart phenomena and tackling processeshappening on drastically different scales separately.

It is instructive to have an idea of the relative mag-nitudes of various relevant quantities to understandwhy the ’quasi-linear’ approach and other commonlymade approximations make sense. For typical JET pa-rameters in a D majority plasma (temperature of 5keV ,density of 5 × 1019m−3, magnetic field 3T , major ra-dius of 3m and minor radius of 1m), the ion cyclotronfrequency of the D ions is 23MHz and the electron cy-clotron frequency is 80GHz, the ion thermal velocity is5×105m/s so the typical ion Larmor radius is 3mm, theelectron thermal velocity 3×107m/s so the electron gy-roradius is 0.05mm, the ion collision frequency is 100Hzand the electron collision frequency is 10kHz. For typ-ical RF waves of several MW with electric field valuesof 50kV/m close to the antennas, the RF magnetic fieldis 5× 10−3T and the RF magnetic contribution to theLorentz acceleration |~v × ~BRF | = 2.5kV/m.

Hence the ions travel around the torus in about4× 10−5s, the cyclotron (’gyro’) period τg being muchshorter than the transit (’bounce’ & ’drift’) time τb,d,which itself is much smaller than the collision time τc,

making it senseful to describe the cyclotron motionas much faster than the bounce/transit motion, itselftypically much faster than the collision time and ren-dering a collisionless description senseful. The scalingτg << τb,d << τc is crucial for the customary models.In particular, the ’slower’ phenomena are assumed tobe constant on the faster time scale while the faster- oscillatory - phenomena are treated as being beyondtheir transient state, all quantities merely varying as afunction of time as exp[iωt], where ω is the frequencyat which the external wave launchers are operated. Theparticle motion is essentially imposed by the confiningmagnetic field, the RF field being a small - be it fast -perturbation and the RF electric field effect dominatingthat of the magnetic field. Finally, the Larmor radiusis commonly much smaller than the equilibrium quan-tity gradients, this giving rise to the so-called drift ap-proximation and locally making a quasi-homogeneousdescription senseful. In particular ρ/LBo

<< 1 whereρ is the Larmor radius and LBo

is a typical scalelengthof the variation of the confining magnetic field.

To understand the basic physics of the impact ofthe RF electric field on a test particle, we locally solvethe equation of motion and then use the result to evalu-ate the net energy a charged particle can gain or lose ina rapidly varying electric field along the trajectory it isforced to follow by the fusion machine’s static confiningmagnetic field. We start from a homogeneous plasma,straight magnetic field line analysis and gradually in-clude other effects.

Because the magnetic field is imposing a clearasymmetry in the dynamics along as opposed to per-pendicular to the magnetic field lines, the discussion ofthe wave-particle interaction is most easily describedwith reference to the direction along ~e// = ~Bo/Bo and2 independent directions perpendicular to the staticmagnetic field ~Bo. Neglecting the equilibrium electricfield related to the ohmic circuit, the solution of theequation of motion of a charged particle immersed ina homogeneous, static magnetic field can be written asv⊥,1 = v⊥cosφ, v⊥,2 = v⊥sinφ where φ = φo−Ω(t− to)while v// = ct, with Ω = qBo/m (Bo the confiningfield, q the charge and m the mass of the species) thecyclotron frequency, which can further be integratedto get the particle position: x⊥,1 = x⊥,1,GC − ρsinφ,x⊥,2 = x⊥,2,GC + ρcosφ in which the Larmor radius isgiven by ρ = v⊥/Ω and ’GC’ refers to the guiding cen-tre position. Assuming the electric field is a plane wavecharacterized by a wave vector ~k, defining ψ as the anglebetween ~k and ~e⊥,1 (k⊥,1 = k⊥cosψ, k⊥,2 = k⊥sinψ),the work the electric field does on a particle can bewritten

q ~E.~v = qN=+∞∑

N=−∞LNexp[−Nφ] (2)

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in which

LN = [v⊥2

(E−JN+1eiψ + E+JN−1e

−iψ)

+E//v//JN ]eiNψ

is the Kennel-Engelmann operator [17] and where theelectric field is evaluated at the guiding center ~xGC =[x⊥,1 +ρsinφ]~e⊥,1 +[x⊥,2−ρcosφ]~e⊥,2 +[v//(t− to)]~e//rather than at the particle position and in which the ar-gument of the Bessel functions is k⊥ρ. In doing so themost rapidly varying contribution (the cyclotron oscil-lation) is isolated from all slower contributions. Fig-ure 2 illustrates that using the guiding center positionrather than the particle position as the reference posi-tion makes the bookkeeping much simpler when study-ing heating: In the particle frame, integration over areference volume entails integrating over all orbits withvarious speeds and guiding centers that are intersectingthe reference volume. When particles are in coherentmotion with a wave and are periodically exchangingenergy with it, this exchange is not considered to be’heating’ although the energy streaming into the refer-ence volume in ~x will increase at some times and de-crease at others. In the guiding center ~xGC frame thepicture is much clearer, simpler and more symmetrical,as there is no leaking of particles into or out of refer-ence volumes. On top of that, the fastest evolution hasbeen separated out, a non-negligible advantage whensearching for equations that will need to be solved nu-merically as it implies a significant speed-up of the com-putations. Finally, as will be seen later, expressing thefields in terms of guiding center coordinates allows in-terfacing to the Fokker-Planck equation describing thenet impact of the fields on the particles (rather than theimpact of the particle motion on the fields) in a natu-ral way, allowing to make wave and particle equationsmore easily compatible. From the equation of motionone readily finds that the change of the particle energyis ε = dε

dt = q ~E.~v which, using the above found expres-sion, can be written more explicitly as

ε = q

+∞∑

−∞LN ( ~EGC(to))exp[i(NΩ + k//v// − ω)(t− to)].

For most frequencies ω the right hand side is periodicand hence the energy transfer between the electric fieldand the particles is merely oscillating around an averagevalue but no net acceleration is taking place. At theDoppler shifted cyclotron resonances ω = NΩ + k//v//the exponential time dependent factor associated to aspecific cyclotron harmonic N on the right hand sideis constant and hence - in spite of all other terms stilloscillating as a function of time - there is a net energytransfer.

Net heating takes place when NΩ + k//v// = ω, inwhich the Doppler shift term k//v// is usually a cor-rection to NΩ, except when N = 0 in which case it iscrucial. In the radio frequency domain (tens of MHz)and for typical magnetic field strengths of current-daymagnetic fusion machines (a few Tesla), the resonancecondition for the ions can easily be satisfied for N 6= 0i.e. they undergo cyclotron heating, while that of theelectrons requires N = 0 i.e. they feel the Cerenkoveffect. As a consequence, ions and electrons react verydifferently to waves driven at frequencies in the ion cy-clotron frequency range: For not too energetic particles,the argument k⊥ρ of the Bessel function is small so thatJ0 ≈ 1 and JM << 1 when M 6= 0. Hence, the ionsare mainly accelerated in the perpendicular directionby the perpendicular components of the electric fieldwhile the contribution of the parallel electric field has aminor impact on them; on the other hand, the parallelelectric field gives the electrons a net pull in the paral-lel direction (Landau damping). Cerenkov interactionequally involves the perpendicular electric field compo-nents, an effect known as transit time magnetic pump-ing (TTMP). Whereas Landau damping causes parallelacceleration and is present even when the electric fieldis spatially uniform, TTMP affects the perpendicularenergy and requires inhomogeneity of the field. An ele-gant discussion of the wave-particle interaction can befound in [12].

Since collisons are infrequent but non-absent, it iscustomary to interpret the frequency ω in the resonantdenominator as a complex quantity with a very small,positive imaginary part iν, ν loosely being interpretedas the collision frequency that would appear in the par-ticle equation of motion if collisions would be accountedfor in a simple way. This gives a recipe for how to encir-cle the poles at the resonances to ensure causality. Thecontribution of the energy from events in the far past(to → −∞) is then absent and only the end contribu-tion of the time integral at time t survives. The needfor the elimination of the far past history is of partic-ular interest to ensure there is net heating. It will bediscussed separately later.

II.B. The Quasilinear Approach: The RF PerturbedDistribution and the Quasilinear Diffusion Operator

The time evolution equation (1) is rewritten mak-ing use of the fact the confining magnetic field is muchlarger than the fastly varying purely oscillatory electro-magnetic perturbation, driven at the antenna frequencyω i.e. proportional to ∝ exp[iωt]: Both the distributionitself and the Lorentz force are separated into a largeterm only involving slowly varying quantities (referredto with a subscript ’o’), and a small but rapidly varying

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Figure 2: Cyclotron motion as seen in the particle (top)and guiding centre (bottom) reference frame.

contribution (related to the driven RF fields):

df

dt=df

dt|o +

df

dt|RF =

dFodt|o +

dFodt|RF +

dfRFdt|o +

dfRFdt|RF = C + S − L

with ddt |o = ∂

∂t + ~v.∇~x + qm [ ~Eo + ~v × ~Bo].∇~v and

ddt |RF = ~aRF .∇~v = q

m [ ~ERF + ~v × ~BRF ].∇~v, Fo theslowly varying and fRF the rapidly varying distribu-tion function. The first, zero order term in the aboveonly varies on the slowest time scale, the next 2 termsare first order corrections which oscillate at frequencyω, while the most rapidly varying terms in the last,second order term contains factors that oscillate at fre-quency 2ω. Since Fo only depends on the constantsof the motion, dFo

dt |o can be simplified to ∂Fo

∂t . The 2linear terms yield an expression for the RF perturbeddistribution i.e. for the evolution on the fast time scale,known as the Vlasov equation:

~aRF .∇~vFo +dfRFdt|o = 0

i.e.

fRF = −∫ t

orbit

dt′~aRF .∇~vFo (3)

which can be inserted in the fourth term of the evolu-tion equation. Averaging < ... > the 4 terms over a fulloscillation period for all oscillatory aspects of the mo-tion and the driven response, yields an equation for theslow time variation, known as the Fokker-Planck equa-tion. The first term stays untouched, the second andthird term as well as the oscillatory parts of the fourthterm vanish while a constant, second order contributionsurvives. This yields

∂Fo∂t

=< C > + < S > − < L > + < Q > (4)

in which < Q >=< ∇~v.~a∗RF∫ torbit

dt′~aRF .∇~vFo > isthe quasi-linear diffusion operator, acting on the slowlyvarying distribution function.

II.C. The Wave Equation & The Conductivity Tensor

Combining Maxwell’s equations for the evolutionof the electric field and the magnetic field, and assum-ing the waves are driven at a frequency ω, the waveequation can be written in terms of the electric field ~Eonly,

∇×∇× ~E − k2o~E = iωµo[ ~Jantenna + ~Jplasma], (5)

in which ko = ω/c with c the speed of light. The fields

are excited by the current density ~Jantenna flowing onthe antennas typically located close to the edge of theplasma. The plasma current ~Jplasma is composed of thecontributions from the various plasma constituants s,~Jplasma =

∑s qα

∫d~v~vfRF,s, and is fully defined when

the perturbed distributions of all species are known.Strictly, the plasma current contains an ohmic contri-bution ( ~Jplasma = ~Johmic + ~JRF ) aside from the fast-varying RF contribution. It has been neglected in thepresent text.

An elegant way to solve the wave equation is rely-ing on variational techniques, by multiplying the equa-tion with a test function vector and integrating overthe volume of interest. Performing partial integrationto remove the highest order derivatives from ~E not onlyallows to chose lower order base functions for a givendesired numerical accuracy when solving the equation,it also allows to obtain a more symmetrical formulationin which the test function vector ~F and the electric field~E play a similar role. The resulting equation is

∫d~x[k2

o~F ∗. ~E − (∇× ~F )∗.(∇× ~E)] +W =

−[

∫

surface

d~S. ~F ∗ ×∇× ~E + iωµo

∫d~x~F ∗. ~Jantenna]

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with W/[iωµo] =∫d~x~F ∗. ~JRF = q

∫d~xd~v ~F ∗.~vfRF .

The surface term needs to vanish at the metallic wallto ensure no electromagnetic flux leaks away. A supple-mentary advantage of this formulation is that it readilyyields the associated energy conservation theorem whensubstituting the test function vector by the electric field(see further for the expression for the absorbed powerdensity shared by the wave and particle descriptions).

The perturbed current density ~Jplasma and the elec-

tric field ~E are related by the conductivity tensor ~~σ:In Fourier space ~Jplasma,~k′ = ~~σ~k′,~k.

~E~k which is closely

related to the dielectric tensor~~K = ~~1 + iωµo~~σ.

For a plasma in thermal equilibrium, theterm q ~F ∗.~vfRF can be written more explicitly as−q ~F ∗.~v

∫ tdt′q ~E.~v Fo

kT in which the last factor can beshifted in front of the particle history integral since theslowly varying distribution only depends on the con-stants of the motion. One gets

W = ωµoq22π

∫d~xdv⊥dv//v⊥

FokT

∑

N

LN (~F )∗LN ( ~E)

NΩ + k//v// − ω(6)

Isolating the various contributions from the testfunction vector and the electric field in this expressionyields an expression for the conductivity tensor.

The velocity space integrals in Eq. (6) can beperformed to yield a compact expression for the di-electric response in a Maxwellian plasma. The inte-gral over the parallel velocity yields the Fried-Conteplasma dispersion function Z(ζ), which - aside fromthe hot plasma corrections to the wave propagation -describes the process of collisionless damping. The ar-gument of the Fried-Conte function is ζ = ω−NΩ

k//vth. Fig-

ure 3 depicts this function for Im(ζ) → 0+. The realpart asymptotically approaches the cold plasma limitRe[Z] ≈ −1/ζ, but bends the resonant crossing from+∞ to −∞ at ζ = 0 into a smooth transition behavinglike Re[Z] ≈ −2ζ. The imaginary part is a Gaussian.Physically its width is determined by the scalelengthover which the cyclotron frequency Ω varies, and thefactors contributing to the Doppler shift, namely theparallel wave number k// and the thermal velocity vth.Away from the cold plasma resonance damping fadesaway quickly while the reactive part stays significantmuch further from ζ = 0.

As long as k⊥ρ << 1 is satisfied, the Bessel func-tions can easily be approached by their truncated Tay-lor series expansion and the perpendicular integrals caneasily be integrated. Retaining all finite Larmor radiuseffects yields modified Bessel functions (see e.g. [3]).Although the perpendicular (cyclotron gyration) dy-namics seems more daunting than the parallel dynam-ics, it is the latter that is most challenging: In strong

magnetic fields, the cyclotron motion moves the parti-cle only a small distance - the Larmor radius ρ - awayfrom the guiding center, hence equilibrium quantitiestypically vary little between the particle and the guid-ing center positions. But the guiding centers themselvesoften sample large regions of the machine since theirmotion is only restricted by the magnetic field topol-ogy. Taylor series expansions are routinely used for theperpendicular dynamics but have to be used with carefor the parallel dynamics.

Figure 3: The Fried-Conte plasma dispersion functionand its leading order Taylor and asymptotic series re-presentation.

Expressions have also been derived to account forarbitrary Fo (see e.g.[7]). The Fried-Conte function isnow replaced by other (in general numerically evalu-ated) functions. For a sufficiently refined velocity grid,the distribution function can locally be approximatedwith bi-linear functions and the partial integral can beevaluated analytically, yielding a logarithmic contribu-tion. Upon crossing the resonance, the logarithm picksup a ’switch-on kick’ imaginary contribution: It is thedelta function contribution at the pole of the originalintegrand that represents the discontinuous Heavisidestep energy ’kick’ when picking up the energy due tocrossing the resonance. The kick shows up in the uni-form plasma description as a resonance crossing in ve-locity space. In non-uniform plasmas the kick can justas well be described by integrating along the orbit.

II.D. The Cold Plasma Limit

To get a feeling of how drastically a plasma changesthe wave characteristics of the electromagnetic wavesthat exist in vacuum, it is already sufficient to simplyconsider the cold plasma limit. This may seem a dras-tic oversimplification but since k⊥ρ is small when thetemperature is low, the cold plasma limit yields a rea-sonable description of the fate of the waves launchedfrom RF antennas, to the obvious exception of the col-lisionless damping processes themselves which are an

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inherently kinetic - as opposed to fluid - effect.Although it is sufficient to take the asymptotic limit

Z(ζ) → −1/ζ and J0(k⊥ρ) → 1, while JM → 0 forBessel function with order M > 1 to retrieve the coldplasma limit, it is much easier to directly rederive theconductivity tensor starting from the solutions of theequation of motion. Using the Stix notation [3], thecold plasma dielectric tensor can be written

~~K. ~E =

S −iD 0iD S 00 0 P

.

E⊥,1E⊥,2E//

in which S = (R+ L)/2, D = (R− L)/2, with

R = 1−∑

s

ω2p,s/ω(ω + Ωs),

L = 1−∑

s

ω2p,s/ω(ω − Ωs)

P = 1−∑

s

ω2p,s/ω

2

where the sum is on the various types of species s theplasma is constituted of and ωp is the plasma frequency.

II.E. Dispersion Equation Roots

Waves in a cold plasma are electromagnetic in char-acter i.e. their energy is carried purely by the Poyntingflux. When the plasma density goes to zero, their dis-persion roots join the vacuum roots k2

⊥ = k2o − k2

//.

With respect to ~Bo, one of the 2 cold plasma rootsis essentially transverse electric, and the other essen-tially transverse magnetic in character. Referring tothe group (energy propagation) velocity, the former isknown as a ’fast’ wave while the other is a ’slow’ wave.The former allows to carry wave power across mag-netic surfaces and is the preferred candidate to heatthe plasma core in the ion cyclotron domain, while thelatter tends to propagate along magnetic surfaces. Fi-nite temperature effects add kinetic corrections to thesemodes, and introduce supplementary wave branches.For not too energetic particles, the dielectric tensor isusually truncated at second order effects in the Larmorradius. This results in a supplementary mode appearingin the dispersion equation: the (first) Bernstein wave.This wave is essentially electrostatic in nature i.e. itsenergy is carried by particles in coherent motion withthe wave, while its Poynting flux is negligible. Figure4 shows a dispersion plot of the fast wave exciting theBerstein wave at the place where the decoupled coldplasma fast wave has a resonance (S = k2

//). This be-

ing very close to the ion-ion hybrid layer (S = 0) sincek2// << |S| in sufficiently dense plasmas, the mode con-

version layer is often labeled as the ion-ion hybrid layer.

Figure 4: Fast and (first) ion Bernstein wave dispersionequation roots for 3 different central temperatures us-ing a dielectric description retaining all finite Larmorradius corrections. Note that the fast wave root hardlychanges while the Bernstein wave root - a root absentin a cold plasma description - depends sensitively onthe temperature.

Strictly speaking, the Berstein wave cannot be de-scribed by a dispersion resulting from a truncated Tay-lor series expansion in k⊥ρ since k⊥,Bernρ is of order 1,although such a model does correctly locate the placeswhere the fast wave excites it for up to second cyclotronharmonic terms. At higher frequencies and/or for moreenergetic particles, the customary truncation of the di-electric tensor is not even rigorous for the fast wave any-more. Hence, higher order finite Larmor radius termshave to be retained. A hot plasma supports an infinityof hot plasma modes, adding supplementary Bernsteinmodes. Whether they actually play a role depends onwhether or not they are excited. Increasing the fre-quency while keeping the magnetic field fixed bringshigher harmonics into the plasma. Higher Bernsteinwave modes can be excited but the fast and Bernsteinwaves are gradually more decoupled at higher frequen-cies.

II.F. The Fokker-Planck Equation [17-23]

Electromagnetic waves cannot directly be observedexperimentally so their behavior is indirectly studiedthrough e.g. the response of temperature and density tosudden changes in the externally launched power level(see e.g. [22]). On the other hand, multiple diagnos-tics exist to monitor aspects of fast particle populationspresent in the plasma and to cross-check against theo-retical predictions.

As briefly discussed before, when all fast scale dy-namics are removed from the description by averagingover all oscillatory aspects of the motion and driven res-ponse, the Fokker-Planck equation (4) results. Whereasthe wave equation is commonly tackled by integrating

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Figure 5: Fast wave dispersion root at f = 300MHzand k//,o = 5/m in a D − T − (α) − (DNBI) DEMOplasma; Bo = 5.74T . The top curve shows the realpart of the fast wave root; the bottom plot depicts theimaginary part in which ion cyclotron heating at the6th, 7th and 8th harmonic is observed.

over velocity space so that the independent variablesare spatial coordinates, the Fokker-Planck equation isnecessarily solved in terms of constants of the motion.The distribution function of a given plasma species rep-resented in terms of the constants of the motion (e.g.energy, magnetic moment) is, by definition, the samealong the trajectory. However, because of the magneticfield inhomogeneity, the same distribution expressed interms of its local velocity components (v⊥, v//) looksdifferent depending on the location one looks at it (seeFig.6). Hence, interpretation of experimental data re-quires careful analysis: As diagnostics focus on differ-ent aspects of a same distribution, they may seeminglycontradict but in truth corroborate one another.

The Coulomb collision operator for a uniformplasma is known. A convenient, symmetrical form isdue to Landau (see e.g. [1, 2, 18]):

∑

s

C(Foa, Fos) = ∇~v.∑

s

~Sa/sC

~Sa/sC =

q2aq

2s lnΓa/s

8πε2oma

∫d~v′

u2~~1− ~u~uu3

[Foams

∂Fos∂~v′−Fosma

∂Foa∂~v

]

in which ’a’ refers to the species under examination andthe sum is over all species ’s’ in the plasma; ~u is the rel-ative velocity ~v−~v′. Since the species of type ’a’ is one

of the species in the sum, the collision operator is anon-linear integro-differential operator. If the species’a’ is a small minority, its selfcollisions can be neglectedand the Fokker-Planck equation becomes a linear equa-tion in Foa, but if it is one of the main constituants thenonlinear collision operator has to be retained.

Figure 6: 3 representations of the same RF heated

beam distribution energy density mv2

2 Fo: (a) as a func-tion of the constants of the motion velocity v and (nor-malized) magnetic moment xn, and as a function of(v⊥, v//) at (b) the low field side midplane and (c) thehigh field side midplane (see [23]).

Again, analytical expressions are available for thecase the distribution function of the species ’s’ isMaxwellian, in which case the collision operator canbe written in terms of the error function. In caseFos is isotropical, the integrals that need to be eval-uated reduce to 1-dimensional integrals and in the fullyanisotropical case the operator acting on Foa can bewritten in terms of the Rosenbluth potentials. Thestep from the uniform plasma collision operator C tothe operator < C > averaged over all fast aspects ofthe motion is a nontrivial step, the fully rigorous treat-

345

ment of which is still awaited.In view of the fact that the various species inter-

act with each other collisionally, and that several typesof species can simultaneously be heated by electromag-netic waves, a series of coupled Fokker-Planck equationsrather than a single one should be solved. This can bedone iteratively, taking the distributions obtained in theprevious iteration to compute the collision operator inthe current step. Provided convergence is reached, thisallows accounting for the non-linear collision operatorwithout making use of a non-linear system solver. Fig-ure 7 shows a simplified 1-dimensional case in which itwas assumed that all distributions are isotropic. It de-picts an ITER example for the conditions foreseen forwave heating of the D−T plasma during the activatedphase of operation of the machine: the majority of Tions is heated at its second harmonic cyclotron layer,while a minority of 3He is simultaneously heated atits fundamental cyclotron resonance to help crankingup the fusion reactivity; unavoidably, the electrons areheated by Landau and TTMP damping.

0 200 400 60010

15

20

25

30

35

40

45

iteration

T eff (k

eV)

Electrons Deuterium Tritium 3He

(a)

Figure 7: ITER D − T − (3He) heating: (a) Effectivetemperatures and (b) electron power balance.

II.G. A Note on Selfconsistency

A rigorous treatment requires that the Fokker-Planck (FP ) and wave (W ) equations are solved si-multaneously and on the same footing. Their intimateconnection is exemplified by the 2 expressions of theabsorbed power density:

Pabs,FP =∂

∂t[

∫d~vd~xεFo]|RF =

∫d~vd~xε

∂Fo∂t|RF =

1

2Re

∫d~vd~xε∇~v.~a∗RF

∫ t

−∞dt′~aRF .∇~vFo

=q

2Re

∫d~vd~x~E∗.~vfRF =

q

2Re

∫d~x~E∗. ~JRF = Pabs,W

in which Pabs,FP is the RF power density written inthe way it is used in the Fokker-Planck equation (withε the energy, and ∂Fo

∂t |RF the RF diffusion operator,and Pabs,W the RF power density as written in thewave equation, involving the RF perturbed current den-sity ~JRF and fRF the perturbed distribution function;~aRF = q

m [ ~ERF +~v× ~BRF ] is the Lorentz force accelera-tion/decelaration caused by the small but rapidly vary-ing electromagnetic field driven at frequency ω. For-mally writing down the above expression is immediate.To come up with practical expressions for the coeffi-cients to be used in the wave and Fokker-Planck equa-tions is less trivial, at least when the equations are trulytreated on the same footing i.e. when the 2 problemsposed in 6-dimensional phase space are solved removing3 of the 6 independent variables to arrive at an equationin the remaining 3 variables using the same approxima-tions for both equations. Getting the proper coefficientsrequires integrating (a) on the velocity space variablesto obtain the dielectric response coefficients needed inthe wave equation and (b) on the gyro, bounce anddrift motions to find an expression for the quasi-lineardiffusion operator. Ideally, the same elementary ’build-ing blocks’ are used and the relevant integrations areperformed on them.

III. SOME ASPECTS OF NONUNIFORM PLASMAMODELING

III.A. Mode Coupling [38-51]

Before commenting on the particular issues broughtabout by the impact of the plasma inhomogeneities onthe orbits of the particles and the challenges this leadsto when trying to write down a rigorous expression forthe dielectric response, a simplified problem is lookedat first, namely that of the wave propagation in a toka-mak in absence of a poloidal field i.e. where the guid-ing center orbits are assumed to simply being given byϕ(t) = ϕ(to) + v//(t − to). Starting from Eq. (6), butretaining the full wave spectrum and toroidal curvaturewhile assuming that the various species are Maxwellianand that the toroidal angle as well as the distance frommidplane are ignorable variables (allowing to isolate in-dividual n toroidal modes and kZ) yields

W = ωµo(2π)3

∫dRRdv⊥dv//v⊥

FokT

∑

N

[∫kR′LN (~F )]∗[

∫kRLN ( ~E)]

NΩ + k//v// − ω

346

which is fully symmetrical w.r.t. the test function vec-tor ~F and the electric field ~E, guaranteeing a positivedefinite power density for a plasma in thermal equilib-rium. To arrive at a practical expression one of thefollowing 2 approaches is used:

• Assuming that k⊥ρ << 1 so that the Bessel func-tions in the Kennel-Engelmann expressions can beapproximated by a truncated Taylor series expan-sion around the origin, which upon realizing that

dm

dRm~E(R) =

∫dkR(ikR)mexp[ikRR] ~EkR

allows to write down an expression for the dielec-tric response W and the purely electromagnetic(curl) term to be used in the Galerkin form of thewave equation; it is customary to truncate the Tay-lor series at terms of second order in k⊥ρ. Remov-ing the differential operators from the test func-tion vector components ~F by partial integrationsallows to find the corresponding expression for thedielectric tensor, and the so obtained surface termsimmediately provide the expression for the kineticflux [38].

• In reality, k⊥ρ << 1 is not satisfied for all modesthat the plasma supports and thus that assump-tion should not be made if such modes are excited.Bernstein modes are finite temperature modes forwhich k⊥ρ ≈ 1 and even the cold plasma slowmode violates the smallness condition. Hence ifshort wavelength branches are excited - either di-rectly at the plasma edge or at ion-ion hybrid lay-ers [3] - a more rigorous treatment is needed toensure the predicted fate of the shorter wavelengthmodes is correctly described. The easiest way to dothis is to rewrite the Fourier integrals as discretesums and to use locally constant base functions[H(kR−kR,i)][H(kR,i+1−kR)]. The Galerkin formof the wave equation is hereby transformed intoa system of linear equations allowing to find theelectric field Fourier components in the discretizedFourier space.

Figure 8 shows an example of the integration of the1D integrodifferential wave equation. The top figure de-picts the perpendicular wave components. An incomingfast wave carries energy into the region of interest fromthe right. At the ion-ion hybrid layer at R ≈ 3m modeconversion to the Bernstein wave takes place, althoughpart of the fast wave energy simply tunnels throughthe confluence layers and makes it to the high field side(left on the plot) as a fast wave. Note that the Bern-stein wave is efficiently absorbed, its amplitude hav-ing shrunk again to zero about 0.3m towards the highfield side. The bottom figure shows the corresponding

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5−20

−15

−10

−5

0

5

10

15

20

TOMCAT−U JET:5% H in D, Bo=3.45T, f=51MHz, n=26, kZ=10/m

R [m]

Elec

tric

Fiel

d [a

.u.]

Re[ER]Im[ER]Re[EZ]Im[EZ]

−1000 −500 0 500 1000

0.1

0.2

0.3

0.4

0.5

kR [/m]

Elec

tric

Fiel

d Sp

ectru

m [a

.u.]

Abs[ER,k]Abs[EZ,k]

Figure 8: Bernstein wave excitation by the fast magne-tosonic wave at the ion-ion hybrid layer: electric fieldcomponents (top) and Fourier spectrum (bottom).

kR Fourier spectrum of the perpendicular electric field.The Bernstein wave is a backward, electrostatic wave:Its main field component is the component in the di-rection of the background gradient, and for a leftwardpropagating wave that carries energy from the conflu-ence layer towards the high field side it is the kR > 0spectrum that is significantly non-zero. The 2 peaksin the low kR part of the spectrum correspond to theincoming fast wave (highest amplitude for kR < 0 asthe fast wave is a forward wave carrying energy in thesame direction as the phase velocity) and the reflectedwave (somewhat smaller peak, and in the kR > 0 regionsince the reflected wave necessarily carries less energythat the incoming wave).

In two dimensions poloidal as well as radial modecoupling occurs. Figure 9 gives an example of 2Dwave equation modeling in which the geometry and thepoloidal magnetic field has been accounted for. In thisITER example the short wavelength modes are not ex-cited.

III.B. Orbit topology [24-32]

The motion of a charged particle in an axisymmet-rical tokamak is characterized by 3 constants of the mo-tion and by 3 periodic aspects of the motion. The 3 con-

stants of the motion often used are the energy ε = mv2

2 ,

the magnetic moment µ =mv2⊥2Bo

and the toroidal angu-lar momentum Pϕ = mRvϕ − qΨ/2π (ϕ is the toroidalangle, q the charge and Ψ the poloidal magnetic flux)but suitable other sets of 3 independent functions of thecustomary 3 can equally well be used. In order of de-

347

Figure 9: Poloidal electric field component for the RFheating scenario foreseen for the activated ITER phase;3% 3He in a balanced D − T plasma, f=53MHz andBo = 5.3T .

creasing oscillation frequency, the 3 oscillatory aspectsare the cyclotron motion, the bounce motion and thetoroidal drift motion. Figure 10 gives a schematic viewof the various oscillatory aspects of the motion for atrapped particle in a tokamak.

Figure 10: Schematic representation of the particle or-bits in a tokamak (JET-EFDA figure JG05.537-4).

Even on a single particle level, adding the poloidalfield to the description vastly changes the complexityof the wave-particle interaction problem since the guid-ing center orbits are now no longer on R = ct surfacesbut have become poloidally closed loops. Rather thansampling a unique value of the confining magnetic field,the guiding centers sample regions of varying toroidalfield strength. Whereas in a uniform plasma a parti-cle either is ’in resonance’ or ’out of resonance’ at alltimes, the resonances in inhomogeneous plasmas arelocalized i.e. the resonance condition is satisfied onlylocally at some positions along the orbit. The phasefactor exp[i(NΩ+k//v//−ω)(t−to)] in the earlier men-

tioned evolution equation for the particle energy is nowgeneralized to an integral over ~k space of terms of theform exp[iΘ(t)] in which Θ = −Nφ+~k(t).~xGC(t)−ωt.In the neighbourhood of the resonance the phase in theexponential can be approximated by a truncated Tay-lor series expansion, Θ(t) ≈ Θ(to) + Θ(to)(t − to) +12 Θ(to)(t − to)2 + 1

3!

...Θ(to)(t − to)3. The corresponding

exponential factor generally oscillates very quickly sothat its integral does not accumulate a net contribu-tion. Close to stationary phase points (Θ = 0) thephase variation slows down and the integral picks up afinite contribution. Figure 11 depicts the relevant inte-gral for a regular stationary phase point (Θ 6= 0) andfor a higher order stationary phase point (Θ = 0). Theformer is representative for a standard resonance cross-ing while the latter is representative for a resonance ata turning point of the orbit, where 2 resonances merge(strictly, the higher order stationary phase point is abit separated from the turning point: v// = 0 does notcoincide with vθ = 0). The linear line corresponding tothe uniform plasma case for which the particle alwaysstays in resonance is indicated as well.

Figure 11: The energy kick felt by the particle alongthe orbit for resonance at a regular point (Θ 6= 0) andat a tangent resonance point (Θ = 0).

In spite of the fact that energetic ions have guid-ing center orbits that deviate significantly from mag-netic surfaces, the difference between the toroidal an-gular momentum Pϕ and the poloidal flux function Ψis often neglected (’zero drift’ or ’zero banana width’approximation). Aside from the fact that this is anacceptable approximation in large enough machines orfor low enough temperatures, the main motivation forthis approximation is that it hugely simplifies the equa-tions while keeping poloidal mode coupling and particletrapping/detrapping, two of the most important inho-mogeneity effects, intact. Since the dielectric responsewritten earlier was using the electric field at the guidingcenter rather than the particle position and since guid-ing centers stay on magnetic surfaces in the zero drift

348

approximation, the parallel gradient can be written asan algebraic rather than as a differential operator whenexpressing the various quantities in terms of their (dis-crete) toroidal and poloidal Fourier series expansions:

∇// =cosα

|∂~x/∂ϕ|∂

∂ϕ+

sinα

|∂~x/∂θ|∂

∂θ

= cosαintorR

+ sinαimpol

|∂~x/∂θ| = ik//

for each individual poloidal mode mpol and toroidalmode ntor; α is the angle between the total magneticfield and the toroidal direction. The denominator resul-ting from the particle history integral is now no longera constant and net resonant interaction only takes placeat the poloidal angle that satisfies NΩ + k//v// = ω inwhich the cyclotron frequency, the parallel wave num-ber and the parallel velocity now all vary along the or-bit. Although the density and temperature are constantalong the zero-drift guiding center trajectory, poloidalmode coupling takes place because of the magnetic fieldand geometrical inhomogeneity the guiding center ex-periences along its orbit. This has one mild and onemore important consequence:

• The mild consequency is that the perpendiculardifferential operator in the expression LN due toKennel-Engelmann requires retaining the differen-tial character in both independent perpendiculardirections. The resulting expressions yield a doublesum over poloidal modes, and differential operatorsin the direction perpendicular to the magnetic sur-faces. For heating scenarios in which short wave-length branches are excited, a proper description ofthe poloidal coupling requires accounting for a verylarge number of poloidal modes and couplings, anda large number of radial grid points. In an axisym-metrical tokamak there is no toroidal coupling andthus a single sum on the toroidal mode spectrumremains; in a real tokamak - in which magneticripple occurs since a discrete number of toroidalmagnetic field coils are installed - and in a stel-larator, also the toroidal modes are coupled. Evenin the zero drift limit, solving the wave equation in2 or 3 dimensions requires powerful computers.

• Whereas the previous section involves supplemen-tary bookkeeping but is not truly posing a prob-lem, the fact that the parallel mode number ap-pears in the resonant denominator gives rise to afundamental problem: Whereas expression (6) is

fully symmetrical in the test function vector ~F andthe electric field ~E and guarantees positive definiteand purely resonant absorption for Maxwellian dis-tributions, which is what is physically expected,the now obtained expressions are symmetrical for

what concerns the perpendicular operator but areasymmetrical for what concerns the parallel dy-namics. As long as k// is modest (as is typically thecase for the fast wave), this is of little consequence.But for short wavelength branches, positive defi-nite absorption for Maxwellian distributions is nolonger guaranteed. A rigorous cure for that flawrequires a much more sophisticated model, as willbe discussed in the next section.

Figure 12: Schematic representation of the impact ofcyclotron heating on a charged particle in a tokamak:The perpendicular energy of the particle gradually in-creases. Initially passing particles become trapped,their banana tip shifting towards the low field side whenv⊥ gradually grows. The interaction of the particle withthe wave stops when the orbit no longer cuts the res-onance. Just prior to that happening, 2 resonancesmerge, giving rise to efficient heating at the tangentresonance.

It was mentioned earlier that for not too ener-getic ions the Doppler shift term k//v// in the res-onance condition NΩ + k//v// = ω is a small cor-rection to the cyclotron term. As the correspondingdistribution is only significant in a restricted regionof velocity space, it implies that the region where cy-clotron interaction takes place is restricted in space aswell: δR/R ≈ δ(k//v//)/ω. Although the electrons areequally resonantly interacting with the field, the reso-nance condition is much less stringent on them sincek//v// = ω is commonly satisfied in a wide region be-cause of the modest steepness of the temperature pro-file. Consequently, it is fairly straightforward in theion cyclotron frequency domain to ensure ion heatingcan only take places at a predetermined location butit is less evident to avoid the often unwanted electronheating. In big, hot and dense machines such as ITERRF waves have already lost a non-negligible fraction oftheir energy by electron Landau and TTMP dampingbefore arriving at the cyclotron layer.

349

III.C. Bounce Motion, Tangent Resonance, ...

Two approaches are commonly used to derive thewave equation (and in particular to find a suitableexpression for the RF perturbed distribution functionfRF ) and the Fokker-Planck equation (and in particularthe quasilinear diffusion operator < Q >). One is thevery intuitive approach in which the governing Lorentzforce can readily be recognized in the expressions andfor which the link with straight magnetic field line uni-form plasm theory is direct (see e.g. [26, 44, 43]). Theother is more formal but more general and allows tobenefit from the action-angle (Hamiltonian) formalism(see e.g. [33, 35, 36, 30]).

Practical expressions proposed by various authorstend to differ somewhat since different variables are cho-sen and different approximations are made. For waveequation studies (focussing on the fast dynamics), thetrajectory integral is most intuitive and therefore mostfrequently adopted but for Fokker-Planck equation, thedetails of the fast dynamics are only indirectly relevantand all has to be expressed in terms of constants of themotion, hence tending to be closer to the action-angletechnique which elegantly allows to retain the slow timescale physics while integrating away the fast phenomenaby suitable averages over the various relevant oscillationperiods. Kaufman showed, however, that the Hamilto-nian description can equally be used to describe the fastscale physics. More importantly still, he stressed that arigorous description of both aspects of the wave-particleinteraction requires making the same approximationsin both equations if one wants to describe the physicsself-consistently. If applied rigorously, the path inte-gral and action-angle methods are fully equivalent; for asomewhat more detailed discussion, see [34]. However,and in spite of Kaufman’s visionary paper and presentlyavailable powerful computers, a fully rigorous descrip-tion of the plasma heating process by electromagneticwaves is still awaited and a fully selfconsistent descrip-tion based on a sufficiently rigorous footing is a projectstill to be tackled ...

The drift approximation and quasilinear approachmake sense because of the vastly different time and spa-tial scales to describe the wave-particle interaction byfirst computing the zero order motion in absence of therapid but small perturbation, and to account for thecorrections relying on perturbation theory. In an ax-isymmetric tokamak in absence of perturbations, theparticle motion can be described in terms of 3 con-stants of the motion ~Λ and 3 angles ~Φ that describethe periodic aspects of the particle motion. Kaufmanproposed to rely on action-angle variables but in theliterature a wide variety of constants of the motion wassuccessfully used. In contrast, the choice of the anglesas used in the Hamiltonian theory is much more appeal-ing than any other choice since these angles vary lin-

early with time and thus time history integrals becometrivial: Formally, the integrals are like those appear-ing in the uniform plasma case since - once functionsonly involving constants of the motion have been pulledout of time history integrals (since d

dt |o = ∂∂t + ~ω. ∂∂Φ

e.g. fRF = −∫dt′~aRF .∇~vFo can simply be written as

fRF = −∂Fo

∂~Λ.∫dt′Λ), the rapidly varying phase fac-

tor denoting all 4 oscillatory aspects of the driven res-ponse and particle motion is of the form ~m.~Φ(t) − ωtin which ~Φ(t) = ~Φ(to) + ~ω(t− to) and the gyro, bounceand toroidal drift frequencies ~ω = (ωg, ωb, ωd) are only

depending on the constants of the motion ~Λ; the corre-sponding mode numbers are ~m = (mg = −N,mb,md =ntor) in which the bounce mode number mb shouldnot be confused with the poloidal mode number mpol

but the other 2 mode numbers correspond to the cy-clotron mode and the toroidal mode numbers. And sotime history integrals simply yield factors of the form.../[ ~m.~ω−ω] i.e. prescribe that waves and particles res-

onantly interact when the resonance condition ~m.~ω = ω(ω being the generator frequency) is satisfied.

A major simplification of the algebra comes fromthe identity ~aRF = i

ωm [ ddt∇~v − ∇~x]q ~E.~v (see e.g. [5])since it allows to write the various contributions ofwhich the time history integrals needs to be found tocome up with an appropriate expression for the dielec-tric response of the plasma to a rapidly oscillating elec-tromagnetic wave in terms of ε = q ~E.~v. For example

fRF =3∑

j=1

i

ωm

∂Fo∂Λj

[−q ~E.∇~vΛj +

∫ t

dt′DΛjε]

in which DG... = ddt [∇~vG.∇~v...]+∇~vG.∇~x..., hereby es-

sentially reducing the algebraic work to be done to de-scribe the impact of an arbitrary distribution functionFo(~Λ) to the work needed for the case of an isotropicdistribution. For a Maxwellian distribution, it can eas-ily be shown that the net absorption of wave energy bya particle population is positive definite and that theinteraction is resonant in nature:

∑

~m, ~ ′m

< q ~E.~v|∗~ ′m

∫ t

∞dt′q ~E.~v| ~m >=

∑

~m

|q ~E.~v| ~m|2i[ ~m.~ω − ω]

.

Making use of generating functions for the transforma-tion between canonical variables and applying them tothe action-angle ( ~J, ~Φ) variables proposed by Kaufman,one finds DJi = m ∂

∂Φiwhere m is the mass of the

examined type of particles [34]. Whereas the toroidalangular momentum Pϕ and the magnetic moment µare natural variables to use in the computations, thethird Kaufman action - related to the surface enclosedby the poloidal closed drift orbit - is not very practi-cal. Replacing it by the energy ε allows to find a com-pact operator to generalize the expression found for a

350

Maxwellian distribution to that for an arbitrary distri-bution Fo(~Λ): ∂Fo

∂Λ1→ ∂

∂Λ1+ N ∂

∂Λ2+ ntor

∂∂Λ3

when

~Λ = ( εω ,−mv2⊥2Ω , Pϕ = mRvϕ − qΨ

2π ) where Ψ is thepoloidal magnetic flux.

Of course, although the Hamiltonian method offersan elegant framework to do the required evaluations,its simplicity is somewhat misleading:

• The Fourier transformation of the work q ~E.~v doneby the electric field on the particles, written downonly formally in Kaufman’s paper is where the fullcomplexity of the acceleration and deceleration ofparticles on their orbits through an inhomogeneousstatic magnetic field will show up. Happily, thevast difference in time response time of the vari-ous aspects of the motion allows to rely on asymp-totic techniques to perform this step. First, theFourier analysis is performed at a fixed time, andthen the integrals along the orbits are evaluated.The Fourier transform of q ~E.~v is

q ~E.~v| ~m(~Λ) =1

(2π)3

∫d~Φq ~E.~vexp[−i ~m.~Φ].

Formally writing the electric field in terms of its~k spectrum so that, analogously to the uniformplasma Eq. (2), the phase of q ~E.~v is ~k.~xGC−Nφ−ωt (GC=guiding center), it can readily be seen

that the stationary phase points of the ~m Fouriercomponent are given by the condition ~k.~vGC +NΩ = ~m.~ω so that the global resonance condition~m.~ω = ω can be rewritten as ~k.~vGC + NΩ = ω,which reduces to the familiar resonance conditionk//v// + NΩ = ω of the uniform plasma (andmore in particular ρ/LBo = 0 i.e. driftless) limit.It is not a trivial task to rigorously account forthe drift orbit effects since perpendicular correc-tions not only have to be added to the general-ized Kennel-Engelmann operator but they now en-ter the resonance condition as well.

• A supplementary difficulty comes from the factthat there are many thousands of bounce modesthat need to be evaluated to describe wave-particleinteraction accounting for the plasma inhomogene-ity rigorously. This amounts to a significant in-crease of the computation time required to solvethe equations.

III.D. Decorrelation, Superadiabaticity [33-37,52-55]

It was discussed earlier that the work q ~E.~v the elec-tric field does on a charged particle is an oscillatoryfunction of time i.e. can be written as a sum of termsproportional to a phase factor exp[iΘ(t)]. As the guid-ing center orbits in the drift approximation are closed

poloidally, the particles cross every poloidal positionmany times every second (bounce frequency). Mostof the contributions to the work are oscillatory in na-ture and cancel out when integrated over all fast timescales (gyro-, poloidal bounce and toroidal drift mo-tion), yielding a zero net effect. Only the resonant con-tributions possibly give rise to a finite effect. That eventhese do not cancel on average, is not as evident as itmay seem at first sight: In general, the number of cyclesthe work goes through in between 2 successive transitsis not an integer number and thus the phase change isnot a multiple of 2π so the average work done over alonger period of time is the sum of ’energy kicks’ withthe same amplitude but at different phase. Assumingthat the phase difference between 2 successive transitsmodulo 2π is ∆Θ, it can readily be seen that for everyparticular phase at a given crossing, there is anothercrossing in a not too distant past that more or less can-cels out the present contribution since ∆Θ attains anyvalue between 0 and 2π with equal probability. Andso, even if the particle gets an energy kick every timeit crosses the resonance, the net effect of many cross-ings (typically a few thousand per second for standardion temperatures in typical working conditions) is stillzero ... Unless something breaks the pure periodicityand makes the particles somehow ’forget’ about theirencounters in the far past so that rather than a verylarge number of crossings being relevant, only the mostrecent ones are. Collionality does exactly that.

Let us consider the simplest possible ’Krook’ colli-sion operator C(f) = νf , where ν is the dominant col-lision frequency for the species considered, to discussthe principle: Whereas the fast dynamics of the wave-particle interaction is typically described by the Vlasovequation, a somewhat more careful examination of theevolution equation we started from shows that collision-ality can strictly not be omitted when describing thefast time scale: the collision operator C in that origi-nal equation acts on the full distribution f = Fo + fRFand not only on the slowly varying part Fo. Hence theVlasov equation should be extended to contain a smallbut nonzero contribution, reflecting the rare but non-absent collisions the particles undergo along their un-perturbed orbits: d

dt |ofRF +~aRF .∇Fo = C(fRF ). Writ-ing the time derivative along the trajectory in terms ofthe constants of the motion ~Λ and the angles ~Φ andassuming that the perturbed distribution can be writ-ten as the product of a term only involving slow dy-namics and a term involving fast dynamics i.e. fRF =H(~Λ, t)f(~Λ, ~Φ, t), the fast and slow dynamics can be iso-lated: [H d

dt |of + ~aRF .∇~vFo]/f = [− ∂∂tH + νH]. Since

the right hand side of this expression only containsslow dynamics (no fast period response), one can for-mally write that both sides of this equation indepen-dently have to be equal to a slowly varying function,

351

G = G(~Λ, t) which is negligibly small on the fast dy-namics time scale, G ≈ 0. It follows that the factorH can to good approximation be evaluated explicitly:H ∝ exp(νt). And so the Vlasov equation is supple-mented with a ’switch-off’ or ’phase memory loss’ fac-tor: f = −

∫ t−∞ dt′H−1~aRF .∇Fo which - in view of the

result found for H consistent with the Krook collisionoperator - is simply equivalent to the ’causality rule’which prescribes the frequency ω in the driven timeresponse factor exp(iωt) and the resonant denominator(NΩ + k//v//−ω in the uniform plasma expression, or

~ω. ~m−ω in its drift approximation generalisation) to bereplaced by ω+iν. Whereas the collisional contributionis very small, it plays a crucial role in the evaluation ofthe time history integral when integrating over manycrossings through a given point on the closed bounceorbit: It constitutes the ’memory loss’ factor ensur-ing that a finite net contribution is obtained for theresonant contributions to the work done by the drivenelectric field on the particles.

Figure 13: Schematic representation of the importanceof decorrelation.

Why this is crucial and how it works can readilybe seen in Fig. 13: due to the periodic nature of thebounce motion, the sum of the contributions over allbounce modes is only equivalent to the correspondingintegral over (the stationary phase position) bounce an-gles if the collisional broadening of the resonance iswide enough. This brings out a subtle point in theanalysis of the wave-particle interaction: Because ofthe large difference between the bounce and the gyro-frequency (ωb << ωg), it takes thousands of bouncemodes to rigorously account for the magnetic field in-homogeneity i.e. the corresponding stationary phasepoints where the resonant interaction predominantlytakes place (Θ=0) are very closely spaced. Yet thediscrete sum on the bounce modes cannot justifiablybe replaced by a bounce integral unless the decorrela-tion time is short enough i.e. the collision frequency

large enough. And so the very different time scale onwhich the gyro and bounce motions occur is crucial torestore the ’quasi-homogeneous’ nature of localized res-onances, while the decorrelation needs to be sufficientlyfast to ensure that a net interaction takes place at theseresonance locations when averaging over all the fasterprocesses. In view of the typical collision frequencies inhot plasmas, collisions at first sight cannot cause a fastenough decorrelation to guarantee RF heating to havea net effect.

Fully accounting for the actual collision operator inthe right hand side of the ’generalised’ Vlasov equationis not at all a trivial task. Kasilov [53] did the exer-cise of examining more realistic collision operators andfound that the ’switch-off’ factor H can to first approx-imation be taken to be H = exp[(t/τ)n] where n = 3for Cerenkov interaction (Landau damping and tran-sit time magnetic pumping) and n = 5 for cyclotrondamping. He found that the decorrelation times τ aresignificantly shorter than the collison times 1/ν, imply-ing that particles ’lose’ memory of their phase quicklyenough for RF heating to be efficient in magnetic fu-sion devices. Although the details of the impact of thevarious decorrelation functions differs, the net effect isthe same: the ’kick’ particles receive when crossing theresonance is similar (see Fig. 14).

Figure 14: Integrated decorrelation functions for n =1, 2, 3; ζ = τ( ~m.~ω − ω) with τ the decorrelation time.

While for thermal particles it is thought that thedetails of the actual decorrelation are not too relevant(to the important exception of what happens near tan-gent resonance points where two closely spaced reso-nance points merge into a single one), for too energeticparticles, however, the collisions may still be too unfre-quent so that their net effective absorption is reduced.This regime is known as ’superadiabaticity’.

Not only collisions cause a randomization of thephase. Because of the non-integrability of the orbitsand the non-linearity of the problem, stochastization

352

takes place even if collisions would be absent whenlaunching RF waves of a few MW in fusion relevantplasmas. The rich spectrum of modes, each contribut-ing to the full wave-particle interaction and giving riseto fast phase variations of the total work done on theparticles, is likely to trigger sufficient decorrelation.

IV. END NOTE

With the dawn of powerful parallel computers, thedegree of realism that can be reached when modelingthe interaction of particles and waves in hot, magne-tized, inhomogeneous plasmas contained in magneticfusion experimental reactors is gradually increasing.Various techniques are available to highlight the studyof specific aspects of the interaction. Even so, the prob-lem to be tackled is challenging and a number of aspectsare only starting to be touched upon.

One aspect of importance in the context of wave-particle interaction is the description of the oppositeof wave heating, namely the onset of instabilities trig-gered by particle distributions: In present-day waveand Fokker-Planck descriptions, it is implicitly assumedthat the RF fields are never powerful enough to makethe factor ∂Fo

∂Λ1+N ∂Fo

∂Λ2+n ∂Fo

∂Λ3that appears in the per-

turbed distribution fRF and hence in the expressionfor < Q > change sign, while experimentally it is wellknown that MHD modes can be triggered when RFheating is efficient.

Another - even more essential - aspect is the rigor-ous accounting of the bounce spectrum ensuring thatthe dielectric response is properly described, withoutartificially giving rise to damping that has to be rejectedon physical grounds: While only the rigorous applica-tion of the procedure proposed by Kaufman guaranteesthe causality to be respected for all modes the plasmasupports, no models based on this procedure are yetavailable.

V. A COMMENT ON THE LIST OF REFERENCES

Although most authors in the reference list com-ment on various subtopics treated in this text - makinga clean separation impossible - the papers most relevantto read up on the general treatment have references [1-11], the wave equation is somewhat more the focus in[12-16] while the Fokker-Planck aspects are the mainsubject in [17-23]. Details on the orbit topology andits role in describing the wave-particle interaction canbe found in [24-32]. The action-angle formalism is dis-cussed in [33-37]. Comments on how to solve the rele-vant set of equations - with a focus on accounting forthe realistic geometry - are given in [38-51] while therole of decorrelation is the key subject in [52-55].

VI. ACKNOWLEDGMENT

The authors gladly thank Dr. P.U. Lamalle for thematerial he provided for this lecture.

REFERENCES

1. B. WEYSSOW, Kinetic Theory, these summerschool proceedings

2. R. BALESCU, Transport Processes in Plasmas,North-Holland, Amsterdam (1988)

3. T.H. STIX, Waves in Plasmas, AIP, New York(1992)

4. B.D. FRIED and S.D CONTE, The Plasma Dis-persion Function, Academic, New York (1961)

5. L.D. LANDAU & E.M LIFSCHITZ, Course ofTheoretical Physics, Pergamon, Oxford (1960)

6. R. KOCH, Phys. Lett. A 157, 399 (1991)

7. D. VAN EESTER, Plasma Phys. Contr. Fusion35, 1309-1319 (1993)

8. D. VAN EESTER, Journal Plasma Phys. 54 1, pp.31-58 (1995)

9. D. VAN EESTER, Journal Plasma Phys. 60 3, pp.627-671 (1998)

10. D. VAN EESTER, Journal Plasma Phys. 65 5, pp.407-452 (2001)

11. D. VAN EESTER, Plasma Phys. Contr. Fusion47 5, pp. 459-481 (2005)

12. R. KOCH, Ion cyclotron, lower hybrid and Alfvenwave heating, these summer school proceedings

13. E. WESTERHOF, Electron cyclotron waves, thesesummer school proceedings

14. P.U. LAMALLE, Phys. Lett. A 175, pp. 45-52(1993)

15. P.U. LAMALLE, Non-local theoretical generalisa-tion and tridimensional numerical study of the cou-pling of an ICRH antenna to a tokamak plasmas,LPP-ERM/KMS Laboratory Report 101, Brussels(1994)

16. P.U. LAMALLE, Plasma Phys. Contr. Fusion 39,1409 (1997)

17. C.F. KENNEL and F. ENGELMANN, Phys. Flu-ids 9, 2377 (1966)

353

18. C.F.F. KARNEY, Computer Physics Reports 4,North-Holland, Amsterdam, 1986 pp 183-244

19. J.D. GAFFEY, J. Plasma Phys. 16 149 (1976)

20. R. KOCH, Fast particle heating, these summerschool proceedings

21. E. WESTERHOF, Current Drive, these summerschool proceedings

22. E. LERCHE et al., Plasma Phys. Contr. Fusion50 035003 (2008)

23. E. LERCHE et al., Plasma Phys. Contr. Fusion51 044006 (2009)

24. H. DE BLANK, Guiding center motion, these sum-mer school proceedings

25. H. DE BLANK, Plasma equilibrium in tokamaks,these summer school proceedings

26. I.B. BERNSTEIN et al., Phys. Fluids 24, 108(1981)

27. V.S. BELIKOV and Y. I. KOLESNICHENKO,Plasma Phys. 24 61 (1982)

28. V.S. BELIKOV and Y. I. KOLESNICHENKO,Plasma Phys. Contr. Fusion 36 1703 (1994)

29. J. HEDIN, Ion Cyclotron Heating in Toroidal Plas-mas, Ph.D. thesis, Royal Institute of Technology,Stockholm (2001)

30. J. HEDIN et al., Nucl. Fusion 42, pp. 527-540(2002)

31. P.U. LAMALLE, Plasma Phys. Contr. Fusion 40,465-479 (1998)

32. P.U. LAMALLE, Plasma Phys. Contr. Fusion 48,433-477 (2006)

33. A.N. KAUFMAN, Phys. Fluids 15 pp. 1063-1069(1972)

34. D. VAN EESTER, Plasma Phys. Contr. Fusion41, L23 (1999)

35. A. BECOULET et al., Phys. Fluids B 3, pp. 137-150 (1991)

36. A. BECOULET et al., Phys. Plasmas 1, pp. 2908-2925 (1994)

37. H.E. MYNICK, Nucl. Fusion 30 357 (1990)

38. D. VAN EESTER et al., Plasma Phys. Contr. Fu-sion 40, 19491975 (1998)

39. L. VILLARD et al., Comput. Phys. Rep. 4, 95(1986)

40. D. SMITHE et al., Phys. Rev. Lett. 60 801 (1988)

41. O. SAUTER, Nonlocal Analyses of Electrostaticand Electromagnetic Waves in Hot, Magnetized,Nonuniform, Bounded Plasmas, LRP-457/92,Ecole Polytechnique Federale de Lausanne (1992)

42. M. BRAMBILLA, Nucl. Fusion 34, 1121 (1994)

43. M. BRAMBILLA, Plasma Phys. Contr. Fusion41, 1 (1999)

44. E.F. JAEGER et al., Phys. Plasmas 8 1573 (2001)

45. E.F. JAEGER et al., Phys. Plasmas 13, 056101(2006)

46. J.C. WRIGHT et al., Phys. Plasmas 11, 2473(2004)

47. D.B. BATCHELOR, Integrated Simulation of Fu-sion Plasmas, Physics Today, 02/05, 35 (2005)

48. J. CARLSSON et al., Nucl. Fusion 37, 719 (1997)

49. J. CARLSON, ICRH Heating and Current Drivein Tokamaks, Ph.D. thesis, Royal Institute of Tech-nology ISBN 97-7107-237-7 (1998)

50. T. HELLSTEN et al., Nucl. Fusion 44, pp. 892-908 (2004)

51. M. DIRICKX and B. WEYSSOW, J. Plasma Phys.59 211 (1998)

52. D. FAULCONER and R. LIBOFF, Phys. Fluids15 1831 (1972)

53. S.V. KASILOV et al., Nucl. Fusion 30 2467 (1990)

54. D. VAN EESTER et al., Phys. Letters A, 218, pp.70-79 (1996)

55. Y. LOUIS, Etude analytique et numerique de lareponse dielectrique non-locale d’un plasma detokamak a une perturbation radiofrequence, ten-ant compte de la toroidicite et de la decorrelation,PhD thesis, Universite Libre de Bruxelles, LPP-ERM/KMS Report 103 (1995)

354