Kinetic modeling of nonlinear electron cyclotron resonance heating › AG › APP › PLASMA › PUBLIC › OWN_PUBLICATI… · is established by Coulomb collisions, and leads to

PHYSICS OF PLASMAS VOLUME 10, NUMBER 1 JANUARY 2003

Kinetic modeling of nonlinear electron cyclotron resonance heatingRichard KamendjeInstitut fur Theoretische Physik, Technische Universita¨t Graz, Petersgasse 16, A-8010 Graz, Austria

Sergei V. KasilovInstitute of Plasma Physics, National Science Center ‘‘Kharkov Institute of Physics and Technology,’’Ul. Akademicheskaya 1, 61108 Kharkov, Ukraine

Winfried Kernbichler and Martin F. HeynInstitut fur Theoretische Physik, Technische Universita¨t Graz, Petersgasse 16, A-8010 Graz, Austria

~Received 8 July 2002; accepted 8 October 2002!

A numerical method is proposed for modeling electron cyclotron resonance heating~ECRH! andcurrent drive in fusion devices with, generally speaking, nonlinear wave–particle interaction.Considered is the case of second harmonic extraordinary wave resonance. The method is applied tomodel the electron distribution function in a simplified magnetic field geometry. Various combinedeffects of wave–particle interaction and Coulomb collisions are demonstrated. The model iscompared to the bounce-averaged quasilinear Fokker–Planck equation for the case of perpendicularpropagation of the wave beam with respect to the magnetic field. In addition, the absorptioncoefficients computed from these two models are compared to the results of both the linear and theadiabatic model. Significant differences between results are found for present-day ECRH powerlevels and fusion device parameters. ©2003 American Institute of Physics.@DOI: 10.1063/1.1525796#

in

iserr-

elopthra, tnd

iniod

toeityo

u

rn

ex-hethee

in-thethe.llelngthticto.

ngeag-ely,ac-at-antfndex-

ts-ab-ear

stri-fi-di-ch

I. INTRODUCTION

The electron cyclotron resonance heating~ECRH! is anauxiliary plasma heating method which is routinely usedmodern toroidal fusion devices.1,2 Together with electron cy-clotron current drive~ECCD!, this heating method is nowused for a variety of different tasks. In particular, ECCDviewed as an important tool for tokamak steady-state option on the basis of fully noninductive external driven curent. Such a performance has been recently achieved onTCV ~Tokamak a` Configuration Variable! tokamak.3 Due totheir short wavelength which enables a well localized wavparticle interaction in space, waves from the electron cyctron frequency range allow for a rather accurate power desition and current generation at desired positions inplasma volume. This gives the possibility to control thedial profile of the plasma current, and, as a consequenceprofile of the rotational transform angle in tokamaks astellarators~see, e.g., Refs. 4–7!. In addition, ECRH andECCD are also applied to stabilize neoclassical tearmodes8,9 which may cause a severe confinement degradat

The problem of numerical modeling of ECRH anECCD can formally be separated into two different parts:~i!the problem of wave propagation and absorption and~ii ! theevolution of the resonant particle distribution function dueradio frequency~rf! wave–particle interaction. To solve thfirst problem it is necessary to know the rf current densThis quantity has mainly been calculated within the framethe linear theory of plasma oscillations,10–13 where a linear-ized Vlasov equation for a small perturbation of the distribtion function is solved. The unperturbed~background or non-resonant! distribution function is not specified by lineatheory. Usually this function is assumed to be Maxwellia

751070-664X/2003/10(1)/75/23/$20.00

Downloaded 14 Jan 2003 to 129.27.161.29. Redistribution subject to AI

a-

the

–-

o-e-he

gn.

.f

-

.

Due to the short wave scale, the current density can bepressed in terms of the dielectric permittivity tensor for thomogeneous plasma in a uniform magnetic field. Withinapplicability of the linear absorption theory and within thframework of geometrical optics, wave absorption in anhomogeneous plasma in a nonuniform magnetic field oftoroidal fusion devices is well described in most cases bywave absorption coefficients resulting from this tensor1,2

Some exceptions are the case of relatively long parawavelength as compared to the phase decorrelation ledue to parallel inhomogeneity of the main magnefield,14–16 and the fact that ballistic mode conversion duethe linear echo effect17,18 is completely lost by this approachThe last effect, however, takes place only in a narrow raof wave propagation angles with respect to the main mnetic field and the cyclotron resonance plane, respectivand is easily destroyed by nonlinear wave–particle intertion effects which are important in the case of plasma heing. Thus, the linear theory of wave absorption by resonparticles with a Maxwellian distribution function is one othe main tools for the computation of power deposition awave-generated current density profiles in a majority ofisting experimental devices.

In reality, the unperturbed~background or nonresonan!distribution function does not stay Maxwellian in the preence of resonant wave–particle interaction even if linearsorption theory stays valid. Instead, it satisfies the quasilinevolution equation19–22,11,12,2,23–26where the effect of theresonant wave–particle interaction on the unperturbed dibution function is described by diffusion terms with coefcients which are quadratic in the wave amplitude. This adtional diffusion modifies the balance in velocity space whi

© 2003 American Institute of Physics

P license or copyright, see http://ojps.aip.org/pop/popcr.jsp

tioe

iana

mr–b

ca

easHo

ueon

roinicAs

e

iclr-anps

erineheuinalodmeaticr-e

ticaw

bee-ye

arile

e,Did-of

m-

so-then-–

the

no-n-d,ingra-n.ngeoc-

par-ia-ere

par-en-

ofeenb

thpu-heto

ov-n

nu-r-a--thec-

ofnt,iveve

crip-pli-h as

76 Phys. Plasmas, Vol. 10, No. 1, January 2003 Kamendje et al.

is established by Coulomb collisions, and leads to the formtion of quasilinear plateau regions. As a result, the absorpis reduced by quasilinear effects as compared to the lintheory with the assumption of an unperturbed Maxwelldistribution function. Therefore, whenever the effect of qusilinear relaxation of the distribution function becomes coparable with the effect of Coulomb collisions, the FokkePlanck-type quasilinear diffusion equation with a Coulomdiffusion term has to be solved which, in most cases,only be done numerically~see, e.g., Refs. 25, 27–29, 7!.

The linear theory of wave absorption and the quasilintheory of the evolution of the distribution function are preently the main tools for a quantitative description of ECRand ECCD in fusion devices. However, the applicabilitythese theories is violated for some ECRH~ECCD! scenariosin typical experimental conditions. In particular, this is trfor one of the basic scenarios where the second harmelectron cyclotron resonance for the extraordinary mode~Xmode! is used.

Indeed, if particles interact resonantly with a monochmatic plane wave of finite, but not very high amplitude,course of time, the nonlinear change of the wave–partphase causes resonant particle trapping by the wave.result, in most cases all particles perform the finite~quasi!periodic motion with a ‘‘bounce’’ periodtbE in a limiteddomain of phase space, and, on average, neither gain enfrom the wave nor lose energy to the wave.30–32 This takesplace when phase space regions occupied by parttrapped by the wave in vicinities of different cyclotron hamonic resonances are narrow as compared to the distbetween these trapping regions. For very high wave amtudes these regions may overlap, leading to the stochaenergy exchange between particles and waves.33–36 Suchhigh amplitudes, however, are not typical for modern expments using gyrotrons. In magnetic traps, the cyclotronteraction of particles with the wave is localized in a rathnarrow region of space due to both, the finite width of twave beam and the magnetic field inhomogeneity. Thwave–particle interaction becomes limited to short timetervals which, depending on the case, may be much smthan tbE so that nonlinear effects during the single actinteraction become negligible. This does not mean immeately that the linear and quasilinear approximations becoapplicable, since the wave–particle phase memory betwsingle acts of interaction causes a different kind of resonparticle trapping by the wave which is called superadiabaity effect.37,38,11This effect in tokamaks corresponds to paticle trapping in the vicinities of ‘‘bounce’’ resonances of thunperturbed particle motion in the toroidal magnefield.39–41 However, these kinds of resonances overlaprelatively low values of the wave amplitude, much belovalues typical for the experiments, and particle motioncomes diffusive37,38,11,41as if the phase memory effect is dstroyed. In addition, the phase memory effect is destroeven faster by Coulomb collisions.42–44,15,45,16Therefore,single acts of wave–particle interaction in most casesstatistically independent. Therefore, the quasilinear desction of wave–particle interaction remains valid if the particflight time through the radiation beam,t f , is small compared


a-nar

--

n

r-

f

ic

-

lea

rgy

es

celi-tic

i--r

s,-lerfi-e

ennt-

t

-

d

rep-

to the oscillation period of a particle trapped in the wavtbE . ~Usually, in the most often used ECRH and ECCscenarios with waves launched close to the tokamak mplane, the small radiation beam width is the main causethe limitation of the wave–particle interaction time as copared to the parallel magnetic field inhomogeneity.! How-ever, during ECRH using the second harmonic X-mode renance, this condition is satisfied only for particles wirelatively large parallel velocities as compared to the perpdicular velocity~see Sec. II!. Therefore, the nonlinear waveparticle interaction effects are essential for this scenario.

These effects have been extensively studied inframework of the adiabatic theory46–55 in the opposite limit-ing case,t f@tbE . Within this theory, the canonical actiointegral, which is an exact invariant of resonant particle mtion ~omitting the effects of nonresonant cyclotron harmoics! in the field of a plane wave in a uniform magnetic fielstays adiabatic invariant with respect to the slowly changmain magnetic field and the wave amplitude during the tversing of the cyclotron interaction region by the electroThis adiabatic invariant undergoes a discontinuous cha~jump! when the separatrix between phase space regionscupied either by trapped or untrapped orbits crosses aticular particle orbit determined as a line of constant adbatic invariant ~particle trapping and detrapping by thwave!. Two different possibilities of particle detrapping ahandled with the help of probabilities.56 It should be men-tioned here that the adiabatic limit,t f@tbE , is realized incase of typical second harmonic X-mode ECRH~ECCD!scenarios only in a narrow region of phase space whereallel electron velocities are small compared to the perpdicular velocity ~see Secs. II and IV!. Moreover, unlike incomputations within the quasilinear theory, calculationsthe absorbed power within the adiabatic theory have bperformed without taking into account the effect of Coulomcollisions on the particle distribution function together wisuch an effect of wave–particle interaction. In these comtations the distribution function of electrons which enter tinteraction region~wave beam! has usually been assumedbe Maxwellian.49,51,52

Since the domain of phase space which is neither cered by the~quasi!linear nor by the adiabatic approximatiooccupies a significant part of this space~see Sec. II!, theproblem of wave–particle interaction has to be treatedmerically. Numerical computations of nonlinear paticle orbits are also widely used for various applictions.47,34,35,57,52,54,58–60In particular, the numerical computations of the absorbed power by those particles enteringwave beam with a Maxwellian distribution show the redution of power absorption by nonlinear effects,47,52,60as onecould expect from analytical estimates.61,62

In the computations where the effect of the distortionthe electron distribution function are taken into accouwhich are ultimately needed, e.g., in case of current drmodeling, particle stochastic dynamics outside the wabeam should also be taken into account. The proper destion of this dynamics constitutes a separate, rather comcated numerical problem even in symmetric systems suctokamaks, because the problem of localized ECRH~ECCD!


ti

o-thesei

ere

incuthaic

thav

ohfm

olomininee

p-

i

thrlyntes

heedist

atth

quge

deerasuae

trie

he.

asenceribe

there-

onen-

aineffi-

re-th-n-

-f a

e,r-hebe

sid-use-heicu-fed.

ss,ec-

77Phys. Plasmas, Vol. 10, No. 1, January 2003 Kinetic modeling of nonlinear ECRH

becomes high dimensional, e.g., if one considers the heanear rational magnetic surfaces63 which is of interest forECRH ~ECCD! tearing mode suppression, or if toroidal rtation frequency of trapped particles is comparable tocollision time.64,65This ‘‘outer’’ problem becomes even morcomplex in applications where the issue of convective losof ECRH-generated fast electrons in stellaratorsaddressed.66–69 In addition, one cannot use the bounce avaging procedure employed in various Fokker–Planck solvwhen the wave–particle interaction becomes nonlinear, sthe strong changes in the distribution function do not acmulate slowly after many passings of a particle throughwave beam, but happen instantaneously after the first ping. Thus, one has to rely upon Monte Carlo methods whare weakly sensitive to the problem dimension.44,70,71,65,67,68

At the same time, the straightforward method of solvingequations of motion every time a particle enters the wbeam is rather ineffective and does not permit resolutionthe distribution function in reasonable computing time. Tdifficulty here is similar to the difficulty in the problem osolving the drift-kinetic equation in realistic magnetic geoetries in the long mean-free path regime.67,68 There, in orderto handle modifications of stochastic orbits by Coulomb clisions which take place on rather long time scales as cpared to the magnetic field parallel inhomogeneity traverstime, one has to spend the computing time for reproducalmost the same regular drift motion. This problem has bresolved by the stochastic mapping technique65,72,73,69,74,75

~SMT!, where the modeling of stochastic processes is serated from the modeling of drift orbits which are precomputed and used in the form of Poincare´ maps.

In the present paper the approach is based on the simidea and is complimentary to SMT.76,77 Instead of recalcula-tion of particle orbits each time a particle passes throughwave beam, these orbits are precomputed and used foconstruction of a transition probability density which fuldescribes the orbit change by the wave–particle interactiothe wave–particle phase is random when the particle enthe beam. As discussed above, this assumption is well jufied in case of realistic heating scenarios.

The structure of the paper is as follows. In Sec. II tderivation of Hamiltonian equations of motion is recallwith the main purpose to introduce the notation and to dcuss the necessary approximations. Also in this section,limiting cases described by the quasilinear and adiabtheories are considered. In Sec. III the model geometry ofmain magnetic field is introduced and the mapping techniis briefly outlined. In Sec. IV the mapping relation linkinthe particle distribution function on different sides of thwave beam by nonlinear particle orbits in the beam isrived. The deterministic mapping relation is replaced thby the stochastic relation using the wave–particle phdecorrelation between successive particle passings throthe beam. Also in this section, the transition probability mtrix which describes the stochastic mapping is introducand the numerical method for the computation of this mais outlined. In Sec. V the formal mapping relation linking thdistribution function on different sides of the beam by tstochastic particle orbits outside the beam is introduced


ng

e

ss-rsce-ess-h

eef

e

-

--

ggn

a-

lar

ethe

ifrsti-

-heicee

-eegh-dx

It

should be mentioned that this relation in each practical cshould take into account the specific device geometry. Sithe main purpose of this work is a proper method to descthe wave–particle interaction~Sec. IV!, this geometry~Sec.III ! has been chosen as simple as possible. In Sec. VIMonte Carlo approach is discussed and in Sec. VII thesults of the computation of the electron distribution functiin case of second harmonic X-mode resonance with perpdicular wave beam propagation with respect to the mmagnetic field are presented. The power absorption cocients and absorbed power profiles are compared to thesults of linear, quasilinear, and adiabatic theory with or wiout taking into account the distortion of the distributiofunction by heating. Finally, in Sec. VIII the results are summarized.

II. NONLINEAR WAVE–PARTICLE CYCLOTRONINTERACTION

A. Hamiltonian equations

In the following, the wave electric field within the electron cyclotron resonance zone is assumed in the form owave beam:

E5E0~r !Re@ fF~r !ei (k"r2vt)#, ~1!

where E0(r ) is the amplitude which is constant along thmagnetic field,f[E/uEu is the complex polarization vectork is the wave vector, andv is the wave frequency. The paallel form factor F(r ) describes the beam shape along tmagnetic field. The main magnetic field is assumed touniform. Here, a Cartesian coordinate system (x,y,z) is usedsuch thatB5ezB0 andk5exk'1ezki , where (ex ,ey ,ez) areunit vectors along the coordinates. Since the effects conered in this paper occur in a localized spatial region, theof the simplified form, Eq.~1!, instead of the eikonal representation is justified. In the following, it is assumed that telectron Larmor radius is small compared to the perpendlar scale ofE0 , f, andF. In addition, the small variation othe polarization vector along the magnetic field is neglectThus,E05const,k5const,f5const, andF(r )5F(z).

Introducing the generalized momentumP and the vectorpotentials of the main magnetic fieldA0 and of the wavefield A

P5p1e

cA, A5A01A, ~2!

A05eyB0x, A5E0c

vIm@ fF~z!ei (k'x1kiz2vt)#, ~3!

the Hamiltonian of the system is

H5H~x,z,P,t !5m0c2g,

g5A111

m02c2 S P2

e

cAD 2

, ~4!

wheree, m0 , c, andp are the electron charge, the rest mathe speed of light, and the kinematic momentum, resptively. Here, the radiation gaugew50 is used.


he

,

is

oin

otio

Adic

for

lesn

rel-tic,

der

l

o-o-

ofter-

toredayters

ing

is

altohe

linedo-e-allst

.ce,

icle

.s


The Hamiltonian is expanded up to linear order in tperturbed vector potential

H'm0c2g01H5m0c2g02e

cv"A, ~5!

where

g05A111

m02c2 S P2

e

cA0D 2

, v51

m0g0S P2

e

cA0D .

~6!

The omitted quadratic term in~5! is of the ordervE /v'

;cE0 /(v'B0)!1 with respect to the retained linear termwherevE[eE0 /(m0v).

After a canonical transformation(x,y,z,Px ,Py ,Pz)°(f,Y,Z,J' ,PY ,PZ) with the generat-ing function78

F3~Px ,Py ,Pz ,f,Y,Z!51

m0vc0S Px

2 tanf

22PxPyD

2PyY2PzZ,

instead ofx and Px , a new pair of canonical variablesintroduced

f5arctanS Py2m0vc0x

PxD , J'52

Px2

2m0vc0 cos2 f.

~7!

Away from the resonance,f corresponds to the usual gyrphase andJ' corresponds to the perpendicular adiabaticvariant, J'52p'

2 /(2m0vc0). Here, vc05eB/(m0c),0 isthe nonrelativistic electron cyclotron frequency. The restthe variables are not affected by the canonical transforma(Y5y, etc.!. Using Eq.~6! and Eq.~7!, one obtains

k"r5kiZ1a sinf1c0 , a[2k'v'

vc.0,

c0[k'PY

m0vc05const, ~8!

where

v'5A22m0vc0J'

m0g0, vc5

vc0

g0,

g05A111

m02c2 ~PZ

222m0vc0J'!. ~9!

Since the Hamiltonian is independent ofY, the canonicalmomentumPY and, as a consequence,c0 are integrals ofmotion. Therefore, below they are treated as parameters.sumingv'n0uvcu, the perturbed Hamiltonian is expandeinto Bessel functions where only the resonant terms whcorrespond to then0

th harmonic resonance are kept

H'2eE0

vF~Z!ImF S v i f iJn0

1v'

2~Jn021f 2

1Jn011f 1! Dei (kiZ1c01n0f2vt)G . ~10!


-

fn

s-

h

The account of nonresonant terms which are responsiblethe ponderomotive interaction~see, e.g., Ref. 79! is usuallyperformed using the averaging method.31,80,81This results insome modifications of the definition of independent variaband of the ‘‘unperturbed’’ part of the equations of motiowhich are small in the case of the resonant interactionevant here. Assuming the electrons to be weakly relativisv/c!1, and considering electromagnetic waves,N5ck/v;1, the Bessel functions can be replaced with leading orterms in the expansion overa;N'v' /c!1

H'2eE0

2v

v'

~n021!! S a

2D n021

u f 2uF~Z!cos~kiZ1c1

1n0f2vt !, ~11!

where c15c01argf22p/2. With the help of a canonicatransformation using the generating function,78 F25(n0f2vt1c1)I'1PzZ, the system can be reduced to an autonmous system where the Hamiltonian is an invariant of mtion

H5m0c2g02vI'2vEI'

n0/2F~z!cos~w1kiz!, ~12!

I'5J'

n0, w5n0f2vt1c1 , Pz5PZ , ~13!

vE5vEu f 2uu2m0vc0n0un0/2

2~n021!! S k'

2m0vc0D n021

. ~14!

The main goal of this section is to reduce the numberindependent parameters describing the wave–particle inaction to an amount which allows us to precompute and sthe solution on some grid in parameter space. On presentcomputers, the maximum reasonable number of parameis four. This goal can actually be achieved by approximatthe two-dimensional autonomous system~12! through a trun-cated one-dimensional nonautonomous system. Thisachieved by replacing the variablez in the perturbed Hamil-tonian simply byv it, wherev i5Pz /m0 is the nonrelativisticinitial parallel velocity of electrons. As a result, the canonicmomentumPz becomes a constant of motion. In orderverify this approximation, one has to check, first, that tnonlinear change of the wave–particle phasec5w1kiz dueto the nonconservation ofPz in the original two-dimensionasystem is small compared to its change due to the retaeffects, in particular due to the relativistic shift of the cycltron frequency. In addition, the variation of the parallel vlocity due to the wave–particle interaction must be smcompared to its initial value. As will be seen below, the lacondition is always violated for particles with very smallv i .However, the amount of these particles is very small, too

Starting from this point, the second harmonic resonann052, for the extraordinary mode,u f 2u;1, is assumed. Theequation for the evolution of the phase of a resonant partis

w5]H

]I'

52n0vc2v'v2

n0m0c2 dI' , ~15!

where dI' is the difference ofI' from its resonant valueThe bounce timetbE ~period of the nonlinear oscillation


fa

n

y

te

llshe

tsr-

f-b

umis

cel-

lo-nn-ncyforopa-

onic

ef-n

o-.-

s

ia-

e

eto

on


inside the wave beam! is derived fromdw;wtbE;1. In thesame way, the following estimates are obtained:

dI'; I'tbE;U]H

]wU;vEI'tbE , ~16!

dPz; PztbE;U]H

]zU;vEI'tbE maxS ukiu,1

LbD , ~17!

where ]F(z)/]z'1/Lb has been used, withLb being thebeamwidth. Combining Eq.~15! and Eq.~16!, one obtains

tbE;c

vA m0

vEI'

, dI';c

vAm0vEI',

dPz;c

vAm0vEI' maxS ukiu,

1

LbD . ~18!

The change of the wave–particle phasec due to the variationof Pz appears to be small for waves with propagation notfrom perpendicular

d~kiz!;dPzki

m0tbE;Ni

2 maxS 1,1

kiLbD!1, Ni5

cki

v.

~19!

Using Eq.~17!, the condition that the effect of the variatioof v i is small, gives

v i

v'

@AN'

E0

B0Ni maxS 1,

1

kiLbD . ~20!

Since, in experimentsE0;10 kV cm21 andB0 is of the or-der of a few Tesla,E0 /B0;1023. Therefore, condition~20!is violated only for a small fraction of particles with versmall parallel velocities.

Performing the canonical transformation of the truncasystem with the generating functionF25(w1kiv it)I' , ex-pandingg0 , Eq. ~9!, up to fourth-order terms in the smaparameterv/c @see also~13!#, and retaining only those termwhich containI' ~since the omitted terms do not enter tnontrivial pair of equations of motion!, the Hamiltonian isreduced to

H5Xkiv i2n0vc0S 12v i

2

2c2D 2v CI'2n0

2vc02

2m0c2 I'2

2vEI'

n0/2F~v it !cosc. ~21!

Finally, assuming the beam shape to be Gaussian suchF(z)5exp$2z2/(2Lb

2)%, and introducing the dimensionlestime t5uv iut/(&Lb), the dimensionless energy of the pependicular particle motionw is introduced with the help othe scaling factors, I'5sw. In order to preserve the Hamiltonian form of the equations, the Hamiltonian has toscaled too,H5Huv ius/(&Lb). The following scaling factor,s5221/2uv ium0c2Lb

21n022vc0

22 results in ~compare to Refs.49, 50, 82!

H5Vw2 12 w21«we2t2

cosc, ~22!

where


r

d

hat

e

«5&LbuvEu

uv iu,

V5&Lb

uv iuS kiv i2n0vc0S 12

v i2

2c2D 2v D . ~23!

Thus, the desired equation set is

dw

dt5« w e2t2

sinc, ~24!

dc

dt5V2w1« e2t2

cosc. ~25!

It should be noted that the wave–particle parallel momenttransfer due to the magnetic Lorenz force from the wavefully neglected in this set. This effect is of minor importanfor ECCD with Ni

2!1 as compared to the effect of the colisional relaxation time dependence on the energy.83 How-ever, this effect is of prime importance in the case of cyctron autoresonance,Ni51, when the change of the cyclotrofrequency due to particle interaction with the wave is caceled by the corresponding change in the Doppler frequeshift. This effect has been first demonstrated in Ref. 84the case of the fundamental resonance with the wave prgating along the magnetic field in vacuum,Ni5N51, andthen, in Ref. 85, in the general case of the resonant harmindex and total wave refractive indexN relevant here. Inparticular, this autoresonance effect can be used in newfective ECCD scenarios.29 Within the present approach, iorder to take this effect into account, the nontruncated twdimensional autonomous system~12! would have to be used

In the following, the set of nonlinear ordinary differential equations is solved numerically. The solutions att5t0

which satisfy the initial conditionsw5w0 and c5c0 at t

52t0 are denoted with w5UABw (w0 ,c0) and c

5UABc (w0 ,c0). The valuet055 is used in the computation

such that the exponential in~24! and ~25! is negligible.

B. Limiting cases

Introducing the electron time of flight across the radtion beam,t f5Lb /v i , the nonlinearity parameter53 definedas eNL5t f /tbE @see Eq.~18!# is related to essential devicand plasma parameters as

eNL5Lbuvc0u

c

v'

uv iuAE0N

B0;Pb

1/4B01/2Lb

1/2 tanx, ~26!

where x (tanx5v' /vi) is the pitch angle of particles. Th~quasi!linear and the adiabatic limiting cases correspondeNL!1 andeNL@1, respectively.

In the quasilinear case where the productw«;eNL2 is

small

w «;N'Lb

2vc02

c2

E0

B0tan2 x'eNL

2 . ~27!

Equations~24! and~25! can be solved by a series expansiof w andc using also the fact that«!1 ~typically w>1)

w5w01w11w21¯ , ~28!


-it

ca

iorty

-arisa

cnea

diapmgdioee

as

es.

theof

e

forcityronv-

er.of

f theA

theular

e,de

so-

ior

forspond

,and


c5c01~V2w0!t1c11¯ , ~29!

wherew05const andc05const. In the first-order approximation, the solution of the corresponding equation set winitial conditionsw150, c150, at t52` is obtained

w15« w0E2`

t

dt1e2t12sin@c01~V2w0!t1#, ~30!

c15« w0E2`

t

dt1e2t12~t12t!sin@c01~V2w0!t1#

1«E2`

t

dt1e2t12cos@c01~V2w0!t1#. ~31!

Introducing the notation

Dw1,25w1,2~`! ~32!

for the averages over the initial phasec0 , one gets

^Dw1&50, ~33!

^Dw12&5

p

2«2w0

2e2(V2w0)2/2, ~34!

^Dw2&5S 2

w01V2w0D ^Dw1

2&2

. ~35!

The following relation for the averages is easily derived:

]

]w0

^Dw2&2

']

]w0

^Dw12&

25^Dw2&'^Dw&. ~36!

Therefore, if successive passes of the beam are statistiindependent such that the initial phasec0 is random for eachpassing, the distribution function is described by a diffustype equation~see Sec. IV C!, namely, by the quasilineakinetic equation with purely diffusive transport in velocispace due to wave–particle interactions.10,16

In the caseeNL@1, the period of nonlinear particle oscillation ~bounce time! becomes short compared to the chacteristic time of the variation of the wave amplitude. In thcase, the adiabaticity condition is fulfilled and there existsadiabatic invariant86,80,50,53

J5 R w~H,c!dc. ~37!

Here,w(H0 ,c) is obtained from the equationH5H(w,c)5H0 with e5« exp$2t2% assumed constant in Eq.~22! andthe integral is taken over a period of this function. The funtion w(H0 ,c) describes exact particle orbits in the plawave where three classes of them, co-, counterpassing,trapped, exist. Near the separatrix line which subdividesferent classes of particles, the validity of the adiabaticproximation is violated because the bounce time logarithcally diverges. In this region the orbits change their topolowhen passing through the separatrix. In this case, the abatic invariant changes by a finite amount, since its definitis different for the orbits with different topologies. Thanalysis of these transitions is similar to the one performin Refs. 50 and 53 and leads to similar results. In the c«.V, all particles retain their energyw after passing


h

lly

n

-

n

-

ndf--i-yia-n

de

through the wave beam. In the opposite case,«,V, this istrue only for nonresonant particles with initial energyw0

satisfying the relationuw02Vu.Dw0 , where

Dw052

p S V arcsinA«

V1A«~V2«! D . ~38!

Resonant particles satisfyinguw02Vu,Dw0 either retain orchange their energy to a new value with equal probabilitiThe final values of energy are given as

w5V6uw02Vu. ~39!

In order to demonstrate regions in velocity space whererespective limit cases are applicable, the level contourseNL are plotted in Fig. 1 for two values of the power in thbeam,Pb5400 kW andPb5100 kW, B052.55 T, andLb

52 cm. This picture reveals that, for parameters relevantpresent day experiments, there is a wide region in velospace where none of the limit cases of electron cyclotwave–particle interaction is valid. Note that the depicted leels are only slightly modified when changing the powTherefore, the change of the power within the rangepresent day experimental values does not make either olimiting cases fully applicable. For the points labeled fromto E around the resonance linev52uvcu in Fig. 1, within theresonance zone in velocity space, Fig. 2 and Fig. 3 showdependence of the normalized energy of the perpendicparticle motion after one pass through the wave beam,WN

5UABw (w0 ,c)/wres, versus the initial wave–particle phas

c, for the case of perpendicular propagation of the X mowith the beam powerPb5400 kW. Here,wres correspondsto the energy of the perpendicular particle motion at the renance line. With increasingeNL ~from point A to point E inFig. 1!, WN moves away from the nearly harmonic behav

FIG. 1. Domains of validity of different approaches in velocity spacetypical experimental parameters. Dashed and dashed-dotted lines correto the values of the power in the beamPb5400 kW andPb5100 kW,respectively. The values ofeNL are given in the plot. Markers A, B, C, Dand E correspond to starting points of the particle orbits plotted in Fig. 2Fig. 3.


ftly

thti

thaldow

mit

san-is-ese

av

ih-

av-

isheThefor

the


which is characteristic for the~quasi!linear interaction re-gime~picture A of Fig. 2!. In the quasilinear case, the orbit oa particle on its way through the rf wave field is just slighmodified, whereas with increasing value ofeNL this modifi-cation becomes stronger. In contrast, the prediction ofstrong nonlinear behavior covered by the adiabaapproximation50,49,53,51,52,46,55,59,48,47is reproduced in pictureF of Fig. 3. One of the main assumptions of this theory isinformation loss about the initial phase of rapid nonlineoscillations of particles which are trapped in the wave fieIn other words, any phase memory effect is neglected. H

FIG. 2. Normalized energy of the perpendicular particle motion after tr

eling through the wave beam,WN5UABw (w0 ,c)/wres, versus initial wave–

particle phasec. The labels A, B, and C indicate that the respective curveplotted for its pertinent starting point in velocity space marked in Fig. 1. Tcorresponding pitch angle values arex511.47°, 41.8°, and 56.2°, respectively.


ec

er.-

ever, from the present computations in the adiabatic li~picture E of Fig. 3! it is seen that this effect~effect of‘‘phase bunching’’57! is clearly present and that it influencethe process of energy gain or energy loss of a particle trsiting the wave beam. In addition, instead of distinct redtribution between two energy levels, a certain spread of thlevels over energy is also observed.87

-

se

FIG. 3. Normalized energy of the perpendicular particle motion after tr

eling through the wave beam,WN5UABw (w0 ,c)/wres, versus initial wave–

particle phasec. The labels D and E indicate that the respective curveplotted for its pertinent starting point in velocity space marked in Fig. 1. Tcorresponding pitch angle values are 79.6° and 84.6°, respectively.picture labeled with F gives a qualitative result of the adiabatic theorytwo different initial particle energies~solid, dashed-dotted!. Dashed linesindicate the maximum and minimum value of the separatrix betweenresonant and nonresonant particles.


e-sab

eris

eacmtio. Itintrb

te

uls

ngat

tbethcto

tri

re

Ticth

ely.thendis

rloaseent

is

ithin

nsA

thisecle

is

rfc-g to

en setgralthe

ari-ing

n (ble,

ow-r-

ri-

s,


III. PROBLEM GEOMETRY AND MAPPINGTECHNIQUES

During ECRH, the electron distribution function is dtermined by resonant wave–particle interaction proceswhich take place in the small power deposition regionwell as by the effects of particle drift motion and Coulomcollisions in the main plasma volume. Therefore, the genproblem of computing the electron distribution functioncomplicated and requires a separate consideration forspecific magnetic field geometry which determines the peliarities of the particle drift motion outside the wave beaNevertheless, the main features of wave–particle interacare basically the same for many toroidal fusion devicesthe present study, the main interest is in the developmena proper model of wave–particle interaction, which canturn be used for particular cases of magnetic field geomeThe influence of the device geometry in its full extent canhandled by the stochastic mapping technique72,74,75 ~SMT!.Also in the present paper, a formulation of equations adopin the SMT approach is used.

Thus, a simplified geometry is assumed which shoqualitatively represent typical experimental conditionWithin this model, the main magnetic field is directed alotheZ axis and a narrow Gaussian radiation beam propagacross the main magnetic field in theX–Z plane~see Fig. 4!.The system is periodic in theZ direction with the periodL.Such a simple model geometry is representative, e.g., formagnetic axis of a tokamak. With more proximity, it canused also for the case of tokamak midplane heating withresonance located at the high field side and with safety favalues distant from low-order rationals. The effectstrapped particles are discarded in this case.

For the purpose of numerical computations of the disbution function, cuts are introduced at positionsz5const~vertical lines in Fig. 4!. These cuts, labeled A and B, aplaced at each side of the beam so that the relationLb!dL!L is realized. Here,Lb anddL are the width of the beamand the distance between the cuts A and B, respectively.kinetic equation is reduced then to a set of relations whmaps the pseudoscalar particle flux densities throughneighboring cuts,G5uv iuJ f , whereJ and f are the phase

FIG. 4. Problem geometry.


ess

al

chu-.n

nof

y.e

d

d.

es

he

eorf

-

hehe

space Jacobian and the distribution function, respectivCombined together using the periodicity of the problem,relations which map the flux through the wave beam athrough the outer region form an integral equation, whichthen solved using the Monte Carlo method.

It should be noted that the advantage of the Monte Camethod is the possibility to use simultaneously several phspace coordinate systems, either for modeling of differphysical processes or within different spatial regions. Thisused in the multiple coordinate system approach,88 whereseveral local magnetic coordinate systems are coupled wthe same algorithm.

In the presented geometry, wave–particle interactiotake place only in the narrow inner region between cutsand B, where the rf beam is launched. Since the width ofregion,dL, is small, collisions have a negligible effect on thparticle orbits when compared to the effect of wave–partiinteractions. Therefore, the collisionless Vlasov equationsolved in this region~Sec. IV!. On the other hand, in theouter region between cuts A and B the amplitude of thefield is small. Therefore, the effect of wave–particle interation is neglected there and stochastic changes accordinCoulomb collisions~pitch angle and energy scattering! aretreated by solving the drift kinetic equation89 ~Sec. V!.

IV. MAPPING RELATION FOR THE CASEOF WAVE–PARTICLE INTERACTION

A. Deterministic mapping

In the following, the wave–particle interaction will bassumed to be described by the 2D autonomous equatiosuch that the Jacobian of independent variables is an inteof motion. These properties are characteristic of bothoriginal problem described by the Hamiltonian~12! and tothe truncated problem described by the equation set~24!.This equation set can also be made autonomous if the vablet is viewed as an independent spatial variable satisfythe equation of motiont5221/2L21v i , wherev i5Pz /m0 .In both cases the system possesses an integral of motioHor Pz) which is convenient to use as an independent variathus introducing the new noncanonical set of variablesxi ,x5(I' ,w,H,z) in the first case, andx5(w,c,Pz ,z) in thesecond case. The Jacobian of this set

J5]~ I' ,w,Pz ,z!

]~x!, ~40!

does not have to be the constant of motion anymore. Hever, the quantityv iJ is a constant of motion. The characteistic functions Xi(x,t) of the equations of motionxi

5Vi(x) @where, in particular,x35V3(x)50 and z5V4(x)5v i(x)], satisfy

]

]tXi~x,t!5Vi~Xi~x,t!!, Xi~x,0!5xi . ~41!

With the help of these functions, a Lagrangian set of vaables (u,t) associated with the cutA is introduced asxi

5Xi(xA(u),t) for i 51, . . . ,4. Here,ui5xAi for i 51,2,3 are

the values of the first three variables on the cutA, and thelast variablexA

45zA is a parameter. For these new variable


lsis

d

heth

ses

aam-cu

-

f

d

t

n

ere

themap

edtioni-eters

tothe

tionver-of

n-

and


the first three velocity components are zero,ui5Vmi

5(]ui /]xj )Vj50, i 51,2,3, and the last component equaone,t5Vm

4 5(]t/]xj )Vj51. Therefore, the Jacobian of thLagrangian variable set

Jm[]~x!

]~u,t!J~x!5Jm~u!5v i~xA!J~xA!, ~42!

is a constant of motion due to Liouville’s theorem,Jm

5]Jm /]t5](JmVm4 )/]t5](JVi)/]xi50.

The ‘‘bounce’’ time,tb5tb(u).0, defined for particleswith v i.0 as the travel time from cutA to cutB, is given bythe solution toX4(xA(u),tb)5zB . With this, the map fromcut A to cut B is introduced as

UABi ~u![Xi~xA~u!,tb~u!!, i 51,2,3, ~43!

where the mapping of the invariant of motionu3 is trivial,UAB

3 (u)5u3. The definition of the inverse mapUBAi (u),

which is a map from cutB to cutA, is the same as~43! up toexchanging indicesA andB (tb is then negative!. These twomaps are linked by the relationsUBA(UAB(u))5u andUAB(UBA(u8))5u8, where u are the variables associatewith cut A andu85UBA(u) are associated with cutB.

In the following, the truncated model is adopted for tstudy. In this model, particles which are reflected frombeam and return to the same cut do not exist@for the non-truncated model, the amount of such particles is small,~20!#. Therefore, it is sufficient to consider only particlewith v i.0. The corresponding results forv i,0 can be ob-tained using symmetry.

In order to obtain the relation between the phase spflux densities of particles entering the region of the bethrough cutA, GA

(H)(u)[v i(xA)J(xA) f (xA), and the respective density of particles leaving the same region throughB, GB

(H)(u)[v i(xB)J(xB) f (xB), wherexBi 5u8 i for i 51,2,3

andxB45zB , the following relation is derived using the con

servation of the JacobianJm along the orbit:

]~u8!

]~u!v i~xB~u8!!J~xB~u8!!5v i~xA~u!!J~xA~u!!. ~44!

Taking a product of the relation~44! with the relationf (xB(u8))5 f (xA(u)) which follows from the constancy othe distribution function along the orbit,f (X(x,t))5 f (x),and integrating over d3u both sides of the result weightewith d(u92UAB(u)), the explicit ‘‘unperturbed’’ mappingrelation is obtained~see also Ref. 72!

GB(H)~u!5E d3u8d~u2UAB~u8!!GA

(H)~u8!. ~45!

Here, the following renotation has been made after the ingration,u9→u, u→u8. A more explicit form of~45! is ob-tained by performing the trivial mapping over the invariaof motion u35x3

GB(H)~w,c!5E

0

`

dw8E2`

`

dc8 d~w2UABw ~w8,c8!!d~c

2UABc ~w8,c8!!GA

(H)~w8,c8!, ~46!


e

e

ce

t

e-

t

where the explicit notation of the variables,u15x15w andu25x25c, corresponds to the truncated problem, and whthe dependencies on the invariant~parameter! u35x3 havebeen omitted. Note that these dependencies, as well asdependencies on other problem parameters, enter theUAB(u8) only through the dimensionless parameters« andV, Eq. ~23!. Therefore, together with the argumentsw8 andc8, the dimension of storage of these functions discretizon a given mesh is four. However, the smooth reconstrucof these functions with the help of interpolation is a formdable task because the dependencies on problem paramare very steep~see Fig. 2 and Fig. 3!; the changes in thederivatives may reach ten orders of magnitude.

B. Stochastic mapping in the discretized form

The problem is simplified, however, if one takes inaccount the phase relaxation during the motion outsidewave beam~see, e.g., Ref. 16 and references therein!. As aresult, GA

(H) becomes independent ofc, GA(H)(w,c)

5GA(H)(w). Consequently, the phase dependence ofGB

(H) isof no interest since it will be smeared out by phase relaxabefore the particles re-enter the beam. Thus, taking the aage of~46! over c, the system can be described in termsphase averaged flux densities

GB(H)~w!5E

0

`

dw8 PAB~w,w8!GA(H)~w8!, ~47!

where

PAB~w,w8!51

2p E2`

`

dc8E0

2p

dc d~w

2UABw ~w8,c8!!d~c2UAB

c ~w8,c8!!, ~48!

defines the transition probability density fromw8 to w.Due to collisions, the flux density of particles which e

ter the beam,GA(H), has a smooth behavior on the scaleDw

!At f /tcw, wheret f is the typical flight time through theregion outside the beam andtc is the collision time. There-fore, GA

(H) can be considered constant over this scalerelation ~47! can be discretized. Introducing levels inw, wi

5 iDw, wherei 50,1,2,. . . , anddefining box averaged fluxdensities as

^GA(H)& i[

1

Dw Ewi

wi 11dwGA

(H)~w!, ~49!

the flux density is discretized as follows:

GA(H)~w!'(

j 50

`

^GA(H)& jQ~w2wj !Q~wj 112w!, ~50!

whereQ(x) is the Heaviside step function. Substituting~50!in ~47! and calculating GB

(H)& i as defined in~49!, yields

^GB(H)& i5(

j 50

`

P ABi j ^GA

(H)& j , ~51!

where


lss

elth

t

gh

-

mo-

emal

trix

e

i-

rv-

ide

thehe

the

ithn

tes

B


P ABi j 5

1

2pDw Ewi

wi 11dwE

0

2p

dcEwj

wj 11dw8

3E2`

`

dc8 d~w2UABw ~w8,c8!!d~c

2UABc ~w8,c8!!, ~52!

is the transition probability matrix normalized to 1

(i 50

`

P ABi j 51. ~53!

This matrix is obtained by mapping ‘‘bands’’ between levein velocity space through the ECRH beam. It is defined aratio of the intersection area~see Fig. 5! of the image on cutA of the band between levelswi andwi 11 on cut B with theband between the levelswj and wj 11 on cut A, divided bythe total band area.

Note that, in all realizable cases, the number of levcan be chosen in such a way that the distance betweenlevels,Dw, is much less than the standard deviation duecollisions after one passing outside the beam,Dwc

;At f /tcw. Considering, for instance, the case of a hitemperature and high density plasma withTe510 K, ne

51014 cm23, for particles with thermal perpendicular motion, Dwc can be estimated asDwc'4•1022wT , wherewT

FIG. 5. Overlapping area of bands after mapping of one band from cutcut A.


a

seseo

is the dimensionless thermal energy of the perpendiculartion ~here,t f52pR/vT has been used wherevT is the ther-mal velocity andR5200 cm). In the worst case of thwave–particle resonance zone width equal to the therenergy, choosingDw53.3•1023wT ~i.e., 300 levels! is suf-ficient to satisfyDw!Dwc .

In order to calculate the intersection areas, the maP AB

i j is presented in the form

P ABi j 5

1

2pDw~Ai , j 112Ai , j2Ai 11,j 111Ai 11,j !, ~54!

with

Ai , j5E EVB

dw dcE EVA

dw8 dc8 d~w

2UABw ~w8,c8!!d~c2UAB

c ~w8,c8!!, ~55!

where the regionsVA andVB on cutsA andB, respectively,are defined as follows:

E EVA

dw8dc8[Ewj

`

dw8E2`

`

dc8, ~56!

E EVB

dw dc[E0

widwE

0

2p

dc. ~57!

The imageVB8 on cutA of the regionVB on cutB is definedso that the pointu8[(w8,c8)PVB8 if the image pointu[(w,c)PVB . Here, pointsu andu8 are linked by the directand inverse mappings,u5UAB(u8) and u85UBA(u), re-spectively. Thus, integrating withd-functions, Eq.~55! canbe rewritten as

Ai , j5(k

E E(VAùVB8 )k

dw8 dc85(k

R](VAùVB8 )k

w8 dc8,

~58!

where (VAùVB8 )k is thekth intersection of regionsVA andVB8 ~image on cutA of the regionVB on cutB; see Fig. 6!.The last expression in~58! which contains integrals over thboundaries ](VAùVB8 )k of the intersection regions(VAùVB8 )k follows from the Stokes theorem. Due to perodicity, the linesp18p48 andp38p28 in Fig. 6 are the same up toa shift of 2p in c8. Therefore, one can build an area preseing transformation of the regionVB8 to a region with straightside boundaries shown in Fig. 7. With the straight sboundaries, linesCE and FG, one can move all figuresformed by the original side boundaries (p18p48 andp38p28) andthe straight side boundaries, into the region betweenstraight boundaries~these are regions surrounded in tcounterclockwise direction by the linesCp18C, EDp48E, andHGH). As a result, the intersection areas transform toshaded areas shown in Fig. 7. The areas labeled with ‘‘1 ’’ inFig. 7 are accounted for with the plus sign and those w‘‘ 2 ’ ’ are accounted for with the minus sign. The directioof contour integration is opposite for ‘‘1 ’’ and ‘‘ 2 ’’ areas.Therefore, everything is taken into account if one integra

to


eso in-

s,

ma-e-

-


along the contoursMNp18M and OPp28O without need totake care of self-interactions while computingAi , j 1

. The con-tributions from the line integrals along the side boundariagain, compensate each other. Thus, one has to integratethe image of the linew5wi ~this is the linep18p28) and overthe line w5wj only. Finally, using the value ofc as lineparameter,Ai , j is expressed as

Ai , j5E0

2p

dc]UBA

c ~wi ,c!

]c~ UBA

w ~wi ,c!

2wj !Q~ UBAw ~wi ,c!2wj !. ~59!

FIG. 6. ImageVB8 of the areaVB and intersection regions (VAùVB8 )1 and(VAùVB8 )2 with areaVA .

FIG. 7. Direction of contour integration.


,verThis formula contains the contributions from all possibletersection regions (VAùVB8 )k . Note that here theinverse

mapping functionUBA is present~but notUAB!).For the preloading of transition probability matrice

functions UBAw (wi ,c) and UBA

c (wi ,c) are obtained by nu-merical integration on an adaptive grid ofc values such thatthe strong dependence onc is resolved~see Fig. 8 and Fig.9!.

Using now the discrete probabilitiesP ABi j of ~52!, the

discretized analog forPAB(w,w8) of ~48! is constructed as

P ABD ~w,w8!5 (

i , j 50

`

d~w2w81wj2wi !

3P ABi j Q~w82wj !Q~wj 112w8!. ~60!

Thus, the mapping relation is obtained in its final form as

GB(H)~w,Pz!5E dw8E dPz8 P AB

D ~w,w8!

3d~Pz2Pz8!GA(H)~w8,Pz8!. ~61!

The dependencies of the transition probability matrixP ABD on

the parameters are obtained by a precomputation of thistrix using a grid of parameter values. A simple, piecewisconstant dependence is used in further computations~no in-

FIG. 8. Images on cut B,w8(c)[UBAw (wi ,c), of the canonical perpendicu

lar action~a!, and,c8(c)[UBAc (wi ,c), of the wave–particle phase~b!.


r-sea

ecic

tenoor

spnn

cle

le

a

ssedl-

-d tor-asre-eri-es

.q.

d-

etheveof

giontingm,

ut

hatre-


terpolation!. This matrix is fully defined by the parameters«and V ~23!. This, again, results in a four-dimensional stoage. In the simple problem geometry assumed in the prepaper, it is sufficient to perform the precomputation onequidistant grid of the parameter 1/v i .

It should be mentioned that the main results of this stion apply also to heating and current drive scenarios whare close to autoresonance,84,85,29 Ni'1. In this case, thenontruncated system has to be considered. For this syshowever, the full parametrization with four parameters ispossible. However, if the wave amplitude, the form factthe wave vector, and the polarization vector~see Sec. II A! aswell as the magnetic field strength stay unchanged for acific problem geometry~in particular, the simple geometry ithis paper!, the dimension of the storage is three. In additiocare should be taken about the small amount of partiwhich are reflected from the beam.

The mapping relation for particles with negative paralvelocity ~negativePz) which expressesGA

(H) throughGB(H) is

obtained in full analogy with~61!. Forming a vectorGH

5(GA(H) ,GB

(H)), both integral relations can be written insymbolic form

FIG. 9. Image on cut B of the canonical perpendicular action,w8(c)

[UBAw (wi ,c), versus the image of the wave–particle phase,c8(c)

[UBAc (wi ,c), i.e., w8(c8). ~d! is the same as~c! with zoomed turnover

region located aroundw850.005 andc850.


ntn

-h

m,t,

e-

,s

l

GH8 5PHGH . ~62!

Here, GH denotes particles which enter the beam andGH8denotes particles which leave the beam. It should be strethat relation~62! maps one half of the hypersurface subvoumes~‘‘area’’ of cuts! to a different half, which has no intersections with the first one. These subvolumes correspondifferent signs of particle velocity normal to the cuts. In oder to form the equation for the flux density, this relation hto be combined with the mapping relation for the outergion derived in the next subsection. Due to the system podicity, the combined relations finally map the subvolumonto themselves, thus forming a mapping equation~see Sec.VI !.

C. Limiting cases

The simplest form for the transition probability in Eq~47! is obtained in the adiabatic approximation. Using E~39!, it can be written as

PAB~w,w8!5 12 Q~Dw02uDw8u!@d~Dw2uDw8u!

1d~Dw1uDw8u!#

1Q~ uDw8u2Dw0!d~w2w8!, ~63!

where Dw0 is given by ~38!, Dw5w2V and Dw85w82V. Consequently, the transition probability matrixPAB

i j be-comes simple,PAB

i j 5 12(d i , j1d i ,2k02 j ) for kmin,i,j,kmax and

PABi j 5d i , j otherwise. Here,wk0

5V is the position of theresonance zone in the linear case;wkmin

5V2Dw0 andwkmax

5V1Dw0 define the boundaries of the nonlinearly broaened resonance zone.

In the quasilinear limit, a significant distortion of thdistribution function is achieved in the long-mean-free-paregime only, when also collisions only weakly modify thparticle velocity during the time needed to re-enter the wabeam. In this case, the equation governing the evolutionthe distribution function is a diffusion equation containintwo diffusion operators, a quasilinear operator and collisoperator. The quasilinear operator can be obtained neglecfirst the collisions during the particle flight outside the beathus, assuming the phase space particle density on cutA atthe time momentt1tb to be the same as the density on cB at the time moment t, GA

(H)(t1tb ,w)5GB(H)(t,w)

[G(t,w), wheretb5L/uv iu is the particle return time to thebeam~‘‘bounce’’ time!. Using ~47!, it can be rewritten as

G~ t,w!51

2p E0

`

dt0E0

`

dw0E2p

p

dc8 d~ t2tb2t0!

3d„w2UABw ~w0 ,c8!G~ t0 ,w0!…, ~64!

where

UABw ~w0 ,c8!5w01Dw~w0 ,c0!,

Dw~w0 ,c0!5Dw1~w0 ,c0!1Dw2~w0 ,c0!. ~65!

Here, the last two quantities given by~32! are small com-pared to their characteristic scale of variation. Assuming tG has the same or even a bigger scale of variation with


l.on

,

a-

nohe

,y

of

ctionntity


spect tow, the change ofG during the bounce time is smalTherefore, using the usual Fokker–Planck procedure,can expand the right-hand side of~64! up to the linear orderwith respect to the new integration variablesDt5t2t0 andup to the quadratic order with respect toDw5w2w0

G~ t,w!'G~ t,w!2tb

]G~ t,w!

]t2

]

]w^Dw&G~ t,w!

1]2

]w2

^Dw2&2

G~ t,w!. ~66!

FIG. 10. ~Color! Color density plots of the transition probability matrixP AB

i j , from the quasilinear case~top! to the nonlinear case~bottom! withdecreasing values ofv i . The corresponding values of the nonlinearity prameter,eNL at w5V are 0.17 (w50.39), 1.3 (w58.3), 2.1 (w518.5),respectively. Abscissas correspond to the value of the dimensionless cacal perpendicular actionw when entering the rf beam and ordinates to tvalue of the same quantity after the transition.


eTaking into account the fact thattb , v i , andJ are indepen-dent of w and using ~65!, ~33!, and ~36!, the bounce-averaged quasilinear equation is obtained

] f

]t5

]

]w S ^Dw12&

2tb

] f

]wD5

1

v'

]

]v'S v'DQL

v'v'] f

]v'D[LQLf , ~67!

ni-

FIG. 11. ~Color! Color density plots of the transition probability matrixP AB

i j , from the nonlinear case~top! to predictions of the adiabatic theor~bottom! with increasing values of the nonlinearity parameter,eNL , i.e.,decreasing values ofv i . For the two first plots, the corresponding valuesthe nonlinearity parameter,eNL at w5V, are 3.7 (w534.3), 5.8 (w553.7), respectively. The last plot corresponds to valueseNL@1. Abscissascorrespond to the value of the dimensionless canonical perpendicular aw when entering the rf beam and ordinates to the value of the same quaafter the transition.


croceteno

m

eeaeonrdt

-llya-on

ym

on

r1

e

ak

larofig.ac-in

thebot-ce

the

e-In

-sor

alalar


DQLv'v'[

pe2E02u f 2u2k'

2 v'2 Lb

2

8v2m02v i

2tb

3expS 2Lb

2~v2kiv i22uvcu!2

v i2 D . ~68!

One can check that up to a renotation, Eq.~68! can also beobtained from Eqs.~60! and~68! of Ref. 25. In the approxi-mation used above, as well as in Ref. 25, relativistic effeare taken into account in the expression for the cyclotfrequency only. The effect of the magnetic Lorentz forwhich causes parallel acceleration of electrons is neglecThe general case which, together with the magnetic Loreforce, includes also the main magnetic field parallel inhomgeneity can be found, e.g., in Refs. 26 and 90.

It should be stressed that the quasilinear operatorLQL inEq. ~67! contains no convection term due to the relation~36!between the variance,^Dw2&, and the deviation,Dw&, ofparticle phase space coordinatew after the interaction withthe wave. This deviation is not taken into account in soMonte Carlo procedures, where only the change ofv' whichis linear in the electric field is included in the model. At thsame time, often, during the derivation of the quasilinequations~see, e.g., Ref. 25!, the absence of the convectivterm in the diffusion equation is assumed, and, again,proceeds only with linear changes of phase space coonates. However, the convection term is generally presenthe drift kinetic quasilinear equation.90 This term, which de-scribes ponderomotive effects,79 vanishes only after averaging over the complete collisionless motion in the axiasymmetric systems~including, e.g., the banana toroidal rottion in tokamaks!. In cases when such an averaging is npossible~nonaxisymmetric systems, regimes with collisiofrequency higher that the banana rotation frequency in smetric systems! the ‘‘convection-free’’ form of the quasilin-ear diffusion operator cannot be assumed.

In Fig. 10 and Fig. 11 color density plots of the transitiprobability matrixPAB

i j computed for differentv i values arepresented. In these figures, the three different interactiongimes can be easily distinguished. The top picture in Fig.characterizes the~quasi!linear regime where the particl


tsn

d.tz-

e

r

ei-

in

t

-

e-0

transverse energy is just slightly modified due to the weinteraction with the wave field. With decreasingv i , theprobability for a change of the energy of the perpendicuparticle motion increases together with the nonlinearitythe wave–particle interaction. The two last pictures in F11 show results for strongly nonlinear wave–particle intertion where the adiabatic theory is formally valid. The onethe middle is the result of the computation according tomethod presented in Sec. IV B, whereas the one on thetom is the result predicted by adiabatic theory. The differenin their structure is due to the particle phase memory inbeam ~‘‘phase bunching’’57!, an effect which is discardedwithin adiabatic theory.

V. MAPPING RELATION FOR THE OUTER REGION

In the region outside the beam,zB,z,zA , where zB

50 andzA5L, besides the drift, the particle velocity expriences only stochastic changes by Coulomb collisions.this region, the drift kinetic equation

] f

]t1v i

] f

]z5LCf ~69!

is to be solved, where

LC51

v2

]

]vv2Dvv~v !S ]

]v1

v

vT2D 1

Dxx~v !

sinx

]

]xsinx

]

]x~70!

is the Coulomb collision operator. Here,v is modulus of thevelocity andx5arctan(v' /vi) is the pitch angle. The expressions for the components of the Coulomb diffusion tenDvv andDxx are given, e.g., in Ref. 91. The operatorLC canbe presented also in cylindrical coordinates (v' ,v i) using anappropriate covariant tensor transformation. In cylindriccoordinates, the mapping relations for the pseudoscphase space flux densities through the cutsA and B, GA,B

5GA,B(v' ,v i ,t)5v'uv iu f (zA,B ,v' ,v i ,t), can be formallywritten through propagators~transition probability densities!as follows:

GB~v' ,v i ,t !5E0

`

dv'0E0

`

dv i0E2`

`

dt0 PBBC ~v'0 ,v i0 ;v' ,v i ,t2t0!

3GB~v'0 ,v i0 ,t0!1E0

`

dv'0E2`

0

dv i0E2`

`

dt0 PABC ~v'0 ,v i0 ;v' ,v i ,t2t0!GA~v'0 ,v i0 ,t0!,

GA~v' ,v i ,t !5E0

`

dv'0E0

`

dv i0E2`

`

dt0 PBAC ~v'0 ,v i0 ;v' ,v i ,t2t0!GB~v'0 ,v i0 ,t0!

1E0

`

dv'0E2`

0

dv i0E2`

`

dt0 PAAC ~v'0 ,v i0 ;v' ,v i ,t2t0!GA~v'0 ,v i0 ,t0!. ~71!


lu

-end

s

te

tu

areor

-

c-thevnte-

e-

r-ro-e of

ewdia-si-

x-drlo

ga-m

Forta-be

on

e

u-


These propagators are connected with the sotions of Eq. ~69! in terms of Green functionsg6

5g6(v'0 ,v i0 ;z,v' ,v i ,t) which satisfy the boundary conditions g650 for t520 and at infinite boundaries of thvelocity space. In the coordinate space, the boundary cotions are

g1~v'0 ,v i0 ;0,v' ,v i ,t !5d~v'2v'0!d~v i2v i0!d~ t !,

for v i.0, ~72!

g2~v'0 ,v i0 ;L,v' ,v i ,t !5d~v'2v'0!d~v i2v i0!d~ t !,

for v i,0, ~73!

g1~v'0 ,v i0 ;L,v' ,v i ,t !50, for v i,0, ~74!

g2~v'0 ,v i0 ;0,v' ,v i ,t !50, for v i.0. ~75!

The connection between the transition probabilityP and theGreen functiong is the following:

H PBAO

PBBO

PAAO

PABOJ 5

uv'v iuuv i0v'0u

¦

g1~v'0 ,v i0 ;L,v' ,v i ,t2t0!,

v i0.0, v i.0;

g1~v'0 ,v i0 ;0,v' ,v i ,t2t0!,

v i0.0, v i,0;

g2~v'0 ,v i0 ;L,v' ,v i ,t2t0!,

v i0,0, v i.0;

g2~v'0 ,v i0 ;0,v' ,v i ,t2t0!,

v i0,0, v i,0.

~76!

Forming again, in analogy with Sec. IV B, a vectorGO

5(GA ,GB), the sum-integral relation~71! can be written in asymbolic form

GO5POGO8 . ~77!

Here, the prime denotes the same half-hypersurface a~62!.

VI. MONTE CARLO ALGORITHM

The mapping relations~62! and ~77! correspond to dif-ferent coordinate systems in the phase space, coordinau5(w,c,Pz ;Py ,y) and v5(v' ,f,v i ;x,y), respectively~here, the coordinates after the semicolon correspond tocoordinates of the intersection of the field line with the cand are treated as parameters!. Therefore, the pseudoscalphase space flux densities,G, are not identical for the samphysical point, since they differ by the Jacobian of the codinate change. Introducing the operatorM of the coordinatechange fromv to u5U(v), one obtains

GH5MGO5E d3vd~u2U~v!!GO5]~u!

]~v!GO . ~78!

Writing the inverse coordinate mapping asM21 and combin-ing relations~62! and ~77!, the mapping equation in a symbolic form is

GO5POM21PHMGO[PGO . ~79!


-

i-

in

s

het

-

This integral form of the kinetic equation is a Fredholm seond kind integral equation which can be solved usingMonte Carlo ~MC! method by averaging over a Markochain. The corresponding solution of the homogeneous igral equation is

GO5 limK→`

1

K (k51

K

PkGO(0) , ~80!

whereGO(0)5(GA

(0) ,GB(0)) is some initial value ofGO . For K

being large enough, the result ofk steps,PkGO(0) , does not

depend on the initial value. Following only one chain corrsponding toGA

(0)5v'0uv i0ud(v'2v'0)d(v i2v i0) is suffi-cient for the computation.

Within the algorithm, it is convenient to sample the paticle position stepwise, performing samplings of each pcess in its convenient coordinate system. The sequenccorresponding samplings is given in Eq.~79! by the inversesequence of operators. The algorithm for sampling the nvalue of the particle energy after it passed through the ration beam is determined by the transition probability denties ~TPD! P AB

D (w,w8) and P BAD (w,w8), which have been

precomputed before the actual Monte Carlo run~see, e.g.,Ref. 92 for sampling algorithms for processes with an eplicit TPD!. For particle motion in the outer region, insteaof a precomputed propagator, a conventional Monte Caprocedure is used to solve the drift kinetic equation.93 In thesimple problem geometry of the present paper, the propators in ~76! can be sampled by a few steps of this algorith~depending on the collisionality regime! because the colli-sionless particle motion can be described analytically.more realistic geometries, the limitations of the computional speed caused by integrating the drift motion canavoided in the long mean-free-path~LMFP!-regime using thestochastic mapping technique.72,74,75

Denoting the cut index and the velocity componentsthe cut after thekth complete Monte Carlo step withCk

where Ck5A or B and v'k and v ik , respectively~here k51,2,3,. . . ,K), the statistical estimate of the solution~80! iswritten as

GC~v' ,v i!51

K (k51

K

d~v'2v'k!d~v i2v ik!dC,Ck, ~81!

whereC is eitherA or B. Practically, various phase spacintegrals

^A&[ (C5A,B

2pE0

`

dv'E2`

`

dv i AC~v' ,v i!GC~v' ,v i!

52p

K (k51

K

(C5A,B

ACk~v'k ,v ik!, ~82!

are computed within the Monte Carlo procedure. In particlar, box averages of the phase space flux densityG are com-puted using a (V' i , Vi j ) mesh in velocity space with

AC~v' ,v i!51

DV'DViQ~v'2V' i !Q~V' i 11

2v'!Q~v i2Vi j !Q~Vi j 112v i!. ~83!


an

ndloe-sn

g

an

s

eis

or

seo

de

r

ap

rohetu

-

dize

ofnge

nndthe

nlnce


Here,DV'5V' i 112V' i andDVi5Vi j 112Vi j .The separation of the computation of the nonlinear p

ticle orbits from the computation of the distribution functio~particle phase space flux density! permits a fast computationof this quantity with good resolution in phase space.

The field line integrated absorbed power density,Pabs

[*dz Pabs, is computed by directly evaluating the secomoment of the distribution function within the Monte Carprocedure. In fact,Pabs corresponds to the difference btween the incoming and the outgoing energy flux densitie~asimilar expression for the absorbed power density is giveRef. 52! in between the cuts A and B of Fig. 4

Pabs52pE0

`

dv'E2`

`

dv i~GB~v' ,v i!

2GA~v' ,v i!!mv2

2signv i . ~84!

For the purpose of comparison, also the bounce-averaquasilinear equation

] f

]t5LCf 1LQLf , ~85!

has been solved, where the Coulomb collision operatorthe bounce-averaged quasilinear operator,LC and LQL aregiven by Eqs.~70! and ~67!, respectively. A Monte Carloprocedure has been used, which is similar to procedureRefs. 70, 71, and 94.

VII. RESULTS OF THE MODELING

Integrating the energy conservation law,¹•S1Pabs50,with S being the Poynting flux, along the magnetic field lina nonlinear differential equation for the wave amplitudeobtained

Pabs52]

]xSx,

Sx[E dz Sx5cLb

8ApN~x!~ u f yu21u f zu2!E0

2~x!. ~86!

Here, f y and f z are components of the polarization vect@see Eq.~1!#. Equation~86! is solved numerically using aRunge–Kutta algorithm. Here, the model assumption is uthat the beam shape is not changed by the absorption. Cbining the results of Eq.~84! and Eq.~86! allows for a self-consistent computation of profiles of the wave amplituE0 , of the absorbed power density,Pabs, of the nonlinearabsorption coefficient,60 aNL5Pabs/Sx, and of the opticalthickness,t5 1

2*0xaNL(x8)dx8, as functions of the majo

radius.For computing the profiles of power deposition, optic

thickness, and other quantities associated with wave progation and absorption, the magnetic field geometry intduced in Sec. III has been extended to slab geometry wthe magnetic field, the plasma density, and the temperaare assumed to vary as


r-

in

ed

d

in

,

dm-

,

la--rere

B5B~x!5B0

R0

R, R5R01x, ~87!

ne5ne~x!5n0S 12x2

a2D , ~88!

Te5Te~x!5Te0S 12x2

a2D , ~89!

respectively, whereB052.5 T corresponds to the nonrelativistic second harmonic resonance zone, 2vc01v50. Thebeamwidth is taken to beLb52 cm. Other parameters usefor numerical computations are typical for a medium sfusion device with a plasma major radius ofR05200 cm, aminor radius of a520 cm, and a central temperatureTe053 keV. The electron densities are chosen in the ra1013– 1014 cm23.

A. The electron distribution function

In Fig. 12, results for the electron distribution functiof (v' ,v i) are shown for computations with the nonlinear athe quasilinear model using identical parameter values. Innonlinear case, the distribution function is given on cutB. As

FIG. 12. ~Color! Contour lines of the electron distribution functiof (v' ,v i) for the nonlinear model~on cutB) and for the quasilinear modefor the same input power. The black line marks the position of the resonaline v52uvcu.


s,ic

liteamhe

li-n

eioinreisicl

es

of

bytri-

s

am.

oftionthe

aniffu-

lowies

uen

nce


expected for present day power levels, in both caseplateau-like structure is formed in the resonance zone whbecomes more pronounced with decreasing collisionaThe width of the plateau in the nonlinear case is larger duthe nonlinear broadening of the resonance zone. At the stime, due to collisions the distribution is asymmetric in tnonlinear case because, on cutB, particles with negativeparallel velocity arrive from the outer region where colsions tend to thermalize the distribution function. In additiothe distribution function of particles withv i.0 locally haspositive derivatives overv' which is the consequence of thcombined effects of the nonlinear wave–particle interactand collisions. Without collisions this derivative is zerothe resonance zone. With increasing collisionality, thegions with positive] f /]v' become more pronounced. Theffect is a common feature of the nonlinear wave–partinteraction~see, e.g., Ref. 55!. Here, it can be explained interms of the adiabatic theory using the mapping relation~63!.This relation transforms the distribution function of particlentering the beam,f A(w), to a distribution function of out-going particles,f B(w)5( f A(w)1 f A(V22w))/2, which issymmetric around the resonance point,w5V. The functionf B(w) has a region with positive derivative wheneverf A(w)

FIG. 13. ~Color! Contour lines of the quantityP1 , Eq. ~90!, for the non-linear and quasilinear models. Red corresponds to power gain and blpower loss by particles. The black line marks the position of the resonaline v52uvcu.


ah

y.to

e

,

n

-

e

is not a linear function. Generally speaking, the presencesuch regions may cause the electron Bernstein wave~EBW!instability, although this question still has to be clarifiedstudying the EBW increments corresponding to such disbutions, but this is not the subject of the present paper.

In Fig. 13, the distribution of power gained by particlefrom the radiation beam,P1 , is plotted as a function ofparticle positions in phase space before they enter the beIn the general case, this quantity is given as

P1~v' ,v i!52pGC~v' ,v i!^DW'~v' ,v i!&, ~90!

where^DW'(v' ,v i)& is the average over the initial phasethe change of the energy of the perpendicular particle moafter it has passed through the beam as a function ofparticle velocity before this passing. Here,C5A for v i.0andC5B for v i,0. In the quasilinear case, this quantity cbe expressed through the bounce-averaged quasilinear dsion coefficient~68! as

P1~v' ,v i!52pLm0f ~v' ,v i!]

]v'

v'2 DQL

v'v' . ~91!

For both models, on average, particles with energies bethe resonant value gain energy, while particles with energ

toce

FIG. 14. ~Color! Contour lines of the quantityPS for the nonlinear andquasilinear models. The black line indicates the position of the resonaline v52uvcu.


tline

ityl.sr-.

cis

lense

f

sab

am.theisre-y

be-te-enantig.

theoner.rted

ro-s a

ar-ersthe

ticac-iedich

inofon-d, in

ernt

inod

ide.ated


above the resonant value lose energy. The difference inwidth of the energy exchange region arises from the nonear broadening of the resonance zone. In the quasilinmodel, it shrinks to a point with decreasing parallel velocwhereas it stays of constant width in the nonlinear mode

In Fig. 14 the half-sum of the power gained by particlePS5(P11P2)/2, is plotted, taken as a function of the paticle position before,P1 , and after,P2 , passing the beamIn the quasilinear case this quantity is given by

PS~v' ,v i!522pLm0v'2 DQL

v'v']

]v'

f ~v' ,v i!. ~92!

It corresponds to the energy flux density in velocity spadue to rf diffusion. In the long-mean-free-path regime thquantity corresponds to the power loss from resonant etrons to the background plasma through Coulomb collisioOne can observe that this quantity is positive in both caconsidered.

In Fig. 15 integrals ofP1 , P2 , andPS over the velocitymodulusv taken with the factorv2/v' in the subintegrandversus the pitch anglex are shown. In Fig. 16 integrals oP1 and P2 over the perpendicular velocityv' versus theparallel velocityv i are shown. The first kind of integralcorresponds to the pitch angle distribution of the power

FIG. 15. ~Color! Integrals overv from P1 , P2 ~top! and PS ~bottom!weighted withv2/v' as functions of the pitch anglex. Curves correspond-ing to P with subscripts1, 2 andS are labeled with the same symbolsthe legends. Curves corresponding to the nonlinear and quasilinear mare also indicated in the legend.


he-ar,

,

e

c-s.s

-

sorbed by electrons after a single passing of the wave beA striking difference is seen between the nonlinear andquasilinear model. In particular, a large amount of energycirculating between the wave beam and electrons in thegion 60°,x,120°. At the same time, the effect of energcirculation in velocity space cannot be seen in Fig. 16cause of averaging over the perpendicular velocity. The ingrals ofP1 andP2 over the perpendicular velocity are thsame and correspond to the actual power lost by resoparticles to the background due to collisions. Note that F16 is similar to the last picture~bottom! in Fig. 15. Themaximum of absorption in these figures corresponds toregion where the effects of rf interaction and of collisionsthe distribution function approximately match each othFor higher parallel velocities~pitch angles closer to 0° o180°), the rf interaction is weak and the amount of deposipower is reduced, whereas for small parallel velocities~pitchangles closer to 90°) the rf interaction dominates and pduces the plateau on the distribution function which, aresult, leads to a strong reduction of the absorbed power~thelinear theory predicts the maximum of the absorption by pticles with pitch angles equal to 90°). Note that the powabsorption in the region of small parallel velocitieis increased in the nonlinear case when compared toquasilinear.

In the case of plasma heating in a nonuniform magnefield of fusion devices, the process of wave–particle intertion in the region of pitch angles close to 90° is accompanby the generation of magnetically trapped particles whmay cause convective particle and energy transportstellarators.68 The importance of the detailed structuresuch a suprathermal particle source for the profiles of cvective particle and energy fluxes has been demonstrateparticular, in Ref. 75.

B. Power absorption

In order to study the reduction of the absorbed powdensity Pabs and, respectively, of the absorption coefficie

els

FIG. 16. Integrals overv' of P1 andP2 weighted withv2/v' as functionsof v i . Curves corresponding toP with subscripts1 and2 are labeled withthe same symbols in the legends. All three kinds of integrals coincCurves corresponding to the nonlinear and quasilinear models are indicin the legend.


thdiftt taxarisut

ddi-the

sb-tiv-he

ed

s -


aNL as compared to results of the linear theory, besidesnonlinear model of this paper and the quasilinear modelcussed in Sec. IV C, the following models have been usedthe computation ofa shown in Fig. 17 and Fig. 18. The firstwo models correspond to the often used assumption thadistribution of electrons which enter the wave beam is Mwellian. The respective curves in Fig. 17 and Fig. 18denoted as ‘‘~Maxwell!’’ in the legends. To some extent, thassumption models the exact particle dynamics in the o

FIG. 17. Profiles of the absorption coefficienta along the major plasmaradiusR for the input power valuePb54 kW and different plasma densitieand constant amplitude of the wave electric field.


es-or

he-e

er

region in the case of a short mean-free-path regime. In ation, the adiabatic model has been examined withinpresent approach where the TPDPAB has been taken in thesimplified ‘‘adiabatic’’ form ~63!. The corresponding curveare labeled with ‘‘adiabatic’’ in the legends. The linear asorption coefficient has been obtained using the fully relaistic computation of Refs. 95 and 96 modified here for tfinite plasma density case.

As one could expect, absorption coefficients obtain

FIG. 18. Profiles of the absorption coefficienta along the major plasmaradiusR for the input power valuePb5400 kW and different plasma densities and constant amplitude of the wave electric field.


it

t

’’I

tuom

thne

ant

qu

f-fi-

sll

teiaths

ornas18

t.ithtsgtelygivenlin-rvedtestdelthatin

paththeuch

of

onwithrelin-

t


using a Maxwellian only weakly depend on plasma densthrough such a dependence of the polarization vectoru f 2uand of the refractive indexN' . The reduction of this coeffi-cient is purely due to nonlinear effects57,49,50which limit themaximum change in perpendicular velocity of electronsthe width of the resonance zoneDv';(cvE)1/2. With chang-ing power from 400 to 4 kW, the ‘‘adiabatic Maxwellianabsorption coefficient increases approximately 3 times.order to analyze these dependencies, it is sufficient to sthe case when both the linear width of the resonance zover perpendicular velocity which is due to the finite beawidth, Lb , and the nonlinear width are much smaller thanvalue of the perpendicular velocity on the resonance liv'r . Integrating Eq.~90! overv' , and dividing the result bySx, Eq. ~86!, one obtains

a i8'22puv iuv3s3

Sxm0v'r

] f M

]v'rE

0

`

dw~V2w!^Dw&, ~93!

where the Maxwellian distribution function,f M , has beenexpanded up to the linear order term over (v'2v'r), andthe absorption for distribution functions which are constwith respect tov' is set to zero. Here, in the factor in fronof the integral,n0vc0'v was used, and the scaling factorsis defined in Sec. II A. In the adiabatic limit,Dw&5(V2w) for uw2Vu,Dw0'4AV«/p @see Eq. ~38!# and^Dw&50 otherwise. In the quasilinear limit,^Dw& is givenby ~36!. Calculating the integral for both cases (V@1 shouldbe used in the quasilinear case!, one obtains

a iA8 52213/2

3p2Sx

m0v'r2 uv iu~cvEu f 2uN'!3/2

] f M

]v'r

, ~94!

a iL8 52p5/2

4Sx

m0v'r3 vLbvE

2 u f 2u2N'2

] f M

]v'r

~95!

in the adiabatic and the quasilinear case, respectively. Etion ~95! follows also from the integration of Eq.~92!. Ab-sorption coefficientsaA and aL are defined by integrals othe pertinenta i8 over v i . Computing the ratio of the adiabatic Maxwellian absorption coefficient to the linear coefcient, one obtains the scaling

aA

aL5

211/2c3/2

p9/2vLb~vEu f 2uN'!1/2;eNL21 , ~96!

whereeNL is given by~26!. This ratio scales with power aP21/4, which explains the increase in the adiabatic Maxweian absorption coefficientA10'3 times with decreasingpower. At the same time, the absorption coefficient compuwithin the approach of the present paper for a Maxwelldistribution does not follow this simple scaling becauseadiabatic theory is never applicable in velocity space awhole.

The effects of the nonlinear~quasilinear! plateau forma-tion appear to have much stronger influence on the abstion coefficient in the cases considered here. The reductiothe nonlinear absorption coefficient in the high power c(P5400 kW) as compared to the linear coefficient is 44,and 5.6 times for the density values 1013 cm23, 3


y

o

ndyne

e,

t

a-

-

dnea

p-ofe,

•1013 cm23, and 1014 cm23, respectively.~The absorptioncoefficients are compared atR519 cm, which correspondsto the maximum of the linear absorption coefficien!Roughly, the nonlinear absorption coefficient scales wdensity, i.e., it is proportional to the collisionality. Resulwith a non-Maxwellian distribution function obtained usinthe quasilinear or the adiabatic model have approximathe same scaling. In the cases considered here, theyvalues of the same order, as compared to the general noear case. At the same time, the discrepancy can be obseat the highest density which corresponds to the shormean-free-path regime. In this case, the quasilinear mounderestimates the power absorption. It should be noted‘‘long mean-free-path’’ regimes here are not the same asthe neoclassical transport theory because the mean-freeshould be compared to the length which is needed byfield line to re-enter the wave beam. This length can be mlonger than the tokamak connection length in the caseoff-axis heating.

Finally, in Figs. 19 and 20 results of the computatiwhere the wave absorption has been taken into accounthelp of Eqs.~84! and ~86! are presented. The results acompared with the predictions of the linear and the quasi

FIG. 19. Profiles of the self-consistently computed absorption coefficiena

and field line integrated absorbed power densityPabs. The results are com-pared with those of the linear and the quasilinear approximations.R corre-sponds to the plasma major radius.


abfilie

th

sgipo

.

hinaticth.thl-RH,

anex-nckdis-f-ry.tailsrivetel-earltsar,

ffi-for

i-tronx-

theha-

mag-use.sing, an-arynh

eldof a

tic

ifi-thee

less re-hileso.hus,ofwer

redtion,ki-

pu-the

ainly-ali-

flu-with

-h


ear approximations. As a result of the reduction of thesorption coefficient, a broadening of the absorption prooccurs which is more pronounced for lower plasma densitAs a consequence, for the density valuene53•1013 cm23,the radiation ‘‘shine-through’’ occurs in the case wherelinear theory predicts the complete wave absorption~opticaldeptht522.5; see Fig. 20!. It should be noted that, in thiparticular case, the nonlinear and the quasilinear modelsimilar results despite the different pictures of power desition in the velocity space~see Sec. VII A!. This coinci-dence, of course, is not always the case~see Fig. 17 and Fig18!.

VIII. DISCUSSION AND CONCLUSION

A numerical approach for modeling of ECRH whicconsistently takes into account nonlinear wave–particleteraction has been developed. The main focus of thisproach has been on the proper description of wave–parinteraction. Therefore, the particle dynamics outsidewave beam has been assumed as simple as possibleresults of computations with such a model show thatdistortion of the particle distribution function from Maxwelian is strong for parameters typical for present day ECexperiments. This leads to a reduction of the absorption

FIG. 20. Profiles of the optical thicknesst and the wave electric field amplitude E along the major plasma radiusR. The results are compared witthose of the linear and the quasilinear approximations.


-es.

e

ve-

-p-leeThee

a

consequent broadening of the absorption profile or evenincomplete absorption. Such effects can, in principle, bepected from the quasilinear theory where a Fokker–Plaequation is assumed to be valid. At the same time, thetortion of the particle distribution function is essentially diferent from what is expected from the quasilinear theoTherefore, effects which are essentially dependent on deof the wave absorption in phase space such as current dor the generation of suprathermal trapped particles in slarators have to be properly described using the nonlinmodel of the rf interaction. For example, preliminary resuof current drive computations using the linear, the nonlineand the quasilinear models97 show a significant difference inthe values of the current density and the current drive eciency from all three models for parameter values typicalmodern experiments.

The positive derivative of the distribution function indcates that nonlinear effects of ECRH may cause the elecBernstein wave instability. In fact, in high power ECRH eperiments on the stellarator W7-AS,98 bursts of the nonther-mal electron cyclotron emission have been observed. Atsame time, they cannot be uniquely attributed to the mecnism described here, because in those experiments thenetic drift of the suprathermal electrons also could be a caof positive derivatives of the electron distribution function

It should be noted that, besides the case of heating uthe second harmonic X-mode resonance considered hereother practically important case, the fundamental ordinmode resonance,49,50,82 is also described by a Hamiltoniawith two parameters as in Eq.~22!. Therefore, the approacof this paper is fully applicable also in that case.

The assumptions on the geometry of the magnetic fiused in this paper are not essential and the considerationrealistic situation is straightforward. Especially for realis

magnetic field configurations, the propagatorPc can besampled using the stochastic mapping technique.72,75 Notethat the account of a realistic geometry may change signcantly the physical picture of absorption. For example,distribution function of magnetically trapped particles will bessentially different from the distribution of passing particbecause, in the region of the wave beam, these particleenter the wave beam during each bounce oscillation, wpassing particles need many bounce oscillations to doEffectively, this means that the ‘‘connection length’’ is mucshorter for trapped particles than for passing particles. Thone can expect a plateau formation on the distributiontrapped particles and a consequent strong reduction of poabsorption by these particles. Such an effect is fully ignoin bounce-averaged Fokker–Planck codes where, in addiaveraging over the toroidal angle is also applied to thenetic equation.

In the present paper, a model assumption for the comtation of absorbed power profiles has been used wherebeam does not change its shape due to absorption. Cert~see Ref. 49!, this assumption will not always hold in realistic situations; therefore, these results may have only a qutative nature. There are two mechanisms which would inence the beam shape. The first one is connected


e-

ba-av

if-ntr

centas

hergaio

intfirfi-

co

lin

ago

foo

nk

onethtrsm

ion

oll

-es.

.

,

,

V.

.

,

ch.

.

ys.

B

s.

d.


different values of the field line integrated absorbed powdensity for field lines crossing the beam with different impact parameters~the samex but different y coordinates!.Since the value ofE0 is maximal fory50, the degradationof the absorption coefficient with respect to the linear asorption coefficient is maximal for the ray with impact prametery50. Thus, the beam will be absorbed strongerthe edges than in the center, and, as a result, will shrink oy. Finally, this may result in such a narrow profile that dfraction effects will become important. Note that differedegradation of the absorption coefficient for rays with diffeent impact parameters need not be a pure consequennonlinear effects only. If the distribution function is differefor different field lines, the difference in quasilinear degrdation will also be essential. This is the case in tokamaktoroidally trapped particles play a role in the absorption.

Another type of wave–particle interaction effects on tbeam profile is due to the nonlocal redistribution of eneby resonant particles along the magnetic field line. In pticular, this kind of effect may cause the nonlinear refractof the O-mode beam.49

These effects of beam deformation can be takenaccount within the present approach. For example, thekind of ‘‘beam shrinking’’ effects does not require a modication of equations of motion~24! and~25! ~and of the stor-age!, because the beam shape in they direction can be arbi-trary @see Eq.~1!#.

It should be stressed that nonlinear wave–particle effewill be increasingly important in reactor-scale applicationsECRH and ECCD. This can be easily seen from the scaof the nonlinearity parameter, Eq.~26!, which increases withthe size of the device and with the value of the main mnetic field. Therefore, a proper theoretical descriptionthese effects is necessary.

ACKNOWLEDGMENTS

The authors are grateful to Professor K. N. Stepanovuseful discussions, to Dr. I. V. Pavlenko for benchmarkingour quasilinear computations, to Dr. M. Rome´ for providingsome important references, and to Dr. N. B. Marushchefor useful comments.

This work has been carried out within the AssociatiEURATOM-OAW, under Contract No. P13495-TPH with thAustrian Science Foundation and also with funding fromFriedrich Schiedel Stiftung fu¨r Energietechnik. The contenof the publication is the sole responsibility of its publisheand does not necessarily represent the views of the Comsion or its services.

1V. V. Alikaev, A. G. Litvak, E. V. Suvorov, and A. A. Fraiman, inHigh-Frequency Plasma Heating, edited by A. G. Litvak~American Institute ofPhysics, New York, 1992!.

2M. Bornatici, R. Cano, O. De Barbieri, and F. Engelmann, Nucl. Fus23, 1153~1983!.

3O. Sauter, C. Angioni, S. Codaet al., Phys. Plasmas8, 2199~2001!.4V. Erckmann, U. Gasparino, and H. Maassberg, Plasma Phys. ContrFusion34, 1917~1992!.

5F. Alladio, B. Angelini, M. L. Apicella et al., Plasma Phys. ControlledFusion41, A323 ~1999!.


r

-

ter

-of

-if

yr-n

ost

tsfg

-f

rf

o

e

is-

ed

6K. A. Razumova, V. V. Alikaev, A. A. Borschegovskiiet al., Plasma Phys.Controlled Fusion42, 973 ~2000!.

7R. Dumont, G. Giruzzi, and E. Barbato, Phys. Plasmas7, 4972~2000!.8F. Leuterer, S. Gu¨nter, M. Maraschek, F. Ryter, W. Suttrop, R. Wolf, TheASDEX-Upgrade-Team, G. Gantenbein, and H. Zohm, Fusion Eng. D53, 277 ~2001!.

9R. Wolf, J. Hobirk, G. D. Conwayet al., Nucl. Fusion41, 1259~2001!.10A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, A. G. Sitenko, and K. N

Stepanov,Plasma Electrodynamics~Pergamon, Oxford, 1975!, Vol. 1.11T. H. Stix, Waves in Plasmas~American Institute of Physics, New York

1992!.12M. Brambilla, Kinetic Theory of Plasma Waves~Oxford University Press,

New York, 1998!.13D. G. Swanson,Plasma Waves~Academic, New York, 1989!.14A. V. Timofeev and G. N. Chulkov, Sov. J. Plasma Phys.5, 712~1979!, in

Russian.15S. V. Kasilov, A. I. Pyatak, and K. N. Stepanov, Nucl. Fusion30, 2467

~1990!.16S. V. Kasilov, A. I. Pyatak, and K. N. Stepanov, inReviews of Plasma

Physics, edited by B. B. Kadomtsev~Consultants Bureau, New York1997!, Vol. 20.

17A. A. Vodyanitskij, N. S. Erokhin, V. V. Lisitchenko, S. S. Moiseev, andN. Oraevskij, Nucl. Fusion14, 267 ~1974!.

18S. V. Kasilov, Ukr. Fiz. Zh.42, 974 ~1997!.19V. L. Yakimenko, Sov. Phys. JETP17, 1032~1963!.20V. L. Sizonenko and K. N. Stepanov, Sov. Phys. JETP22, 832 ~1966!.21C. F. Kennel and F. Engelmann, Phys. Fluids9, 2377~1966!.22A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, A. G. Sitenko, and K. N

Stepanov,Plasma Electrodynamics~Pergamon, Oxford, 1975!, Vol. 2.23I. B. Bernstein and D. C. Baxter, Phys. Fluids24, 108 ~1981!.24E. V. Suvorov and M. D. Tokman, Plasma Phys.25, 723 ~1983!.25M. R. O’Brien, M. Cox, and D. F. H. Start, Nucl. Fusion26, 1625~1986!.26A. V. Timofeev and M. D. Tokman, Plasma Phys. Rep.20, 336 ~1994!.27G. Giruzzi, Phys. Fluids31, 3305~1988!.28R. W. Harvey, M. McCoy, and G. Kerbel, Phys. Rev. Lett.62, 426~1989!.29A. V. Zvonkov, A. Y. Kuyanov, A. A. Skovoroda, and A. V. Timofeev

Plasma Phys. Rep.24, 389 ~1998!.30T. M. O’Neil, Phys. Fluids8, 2255~1965!.31A. B. Kitsenko, I. M. Pankratov, and K. N. Stepanov, Sov. Phys. Te

Phys.20, 575 ~1976!.32A. B. Kitsenko and I. M. Pankratov, Sov. J. Plasma Phys.10, 250 ~1984!.33V. A. Balakirev, V. A. Buts, A. P. Tolstoluzhskii, and Y. A. Turkin, Sov

Phys. Tech. Phys.28, 1184~1983!.34C. R. Menyuk, A. T. Drobot, K. Papadopoulos, and H. Karimabadi, Ph

Rev. Lett.58, 2071~1987!.35H. Karimabadi, K. Akimoto, N. Omidi, and C. R. Menyuk, Phys. Fluids

2, 606 ~1989!.36D. Farina, R. Pozzoli, and D. Ronzio, Phys. Fluids B5, 104 ~1993!.37A. V. Timofeev, Sov. J. Plasma Phys.1, 47 ~1975!, in Russian.38K. W. Whang and G. J. Morales, Nucl. Fusion23, 481 ~1983!.39A. N. Kaufman, Phys. Fluids15, 1063~1972!.40T. D. Kaladze, A. I. Pyatak, and K. N. Stepanov, Sov. J. Plasma Phy7,

539 ~1981!, in Russian.41A. Becoulet, D. J. Gambier, and A. Samain, Phys. Fluids B3, 137~1991!.42T. H. Stix, Nucl. Fusion15, 737 ~1975!.43B. I. Cohen, R. H. Cohen, and T. D. Rognlien, Phys. Fluids26, 808

~1983!.44T. D. Rognlien, Nucl. Fusion23, 163 ~1983!.45D. van Eester, Y. Louis, P. U. Lamalle, and R. Koch, Phys. Lett. A218, 70

~1996!.46A. I. Neıshtadt and A. V. Timofeev, Sov. Phys. JETP66, 973 ~1987!.47W. M. Nevins, T. D. Rognlien, and B. I. Cohen, Phys. Rev. Lett.59, 60

~1987!.48E. V. Suvorov and M. D. Tokman, Sov. J. Plasma Phys.14, 557 ~1988!.49I. A. Kotel’nikov and G. V. Stupakov, Phys. Fluids B2, 881 ~1990!.50I. A. Kotel’nikov and G. V. Stupakov, J. Plasma Phys.45, 19 ~1991!.51D. Farina and R. Pozzoli, Phys. Lett. A156, 84 ~1991!.52D. Farina and R. Pozzoli, Phys. Fluids B3, 1570~1991!.53D. Farina, R. Pozzoli, and M. Rome´, Phys. Fluids B3, 3065~1991!.54B. I. Cohen, R. H. Cohen, W. M. Newins, and T. D. Rognlien, Rev. Mo

Phys.63, 949 ~1991!.55M. D. Tokman, Plasma Phys. Rep.22, 452 ~1996!.56D. Dobrott and J. M. Greene, Phys. Fluids14, 1525~1971!.57R. Pozzoli, D. Farina, A. Mennella, and D. Ronzio, inProceedings of the


s.

dt-

SS

ain

ma

E.

,

oua

t-

gal

s-

of

gal

,

n,

ory

s

P

F.ics,


Joint Varenna–Lausanne International Workshop~Editrice Compositori,Bologna, Italy, 1990!, pp. 283–300.

58D. Farina, R. Pozzoli, and M. Rome´, in Proceedings of the Joint Varenna–Lausanne International Workshop~Editrice Compositori, Bologna, Italy,1992!, pp. 317–322.

59A. G. Litvak, E. V. Sergeev, M. D. Tokman, and I. V. Khazanov, PhyFluids B 5, 4347~1993!.

60M. F. Heyn, S. V. Kasilov, W. Kernbichler, H. Maassberg, M. Rome´, U.Gasparino, and N. Marushchenko, in26th EPS Conference on ControlleFusion and Plasma Physics, Maastricht~European Phys. Soc., PetiLancy, 1999!, Vol. 23J, pp. 1625–1628.

61A. B. Kitsenko, I. M. Pankratov, and K. N. Stepanov, in10th EuropeanConference on Controlled Fusion and Plasma Physics, Moscow, U~European Phys. Soc., Petit-Lancy, 1981!, Vol. 3, pp. H-12.

62A. V. Zvonkov, Sov. J. Plasma Phys.8, 563 ~1982!.63R. O. Dendy, Plasma Phys. Controlled Fusion27, 1243~1985!.64A. G. Peeters, E. Westerhof, and T. J. Schep, in20th European Conference

on Controlled Fusion and Plasma Physics, Lisboa, Portugal~EuropeanPhys. Soc., Petit-Lancy, 1993!, Vol. 3, pp. 1037–1040.

65S. V. Kasilov, V. E. Moiseenko, M. F. Heyn, and W. Kernbichler, in23rdEPS Conference on Controlled Fusion and Plasma Physics, Kiev, Ukr~European Phys. Soc., Petit-Lancy, 1996!, Vol. 20C, pp. 559–562.

66A. S. Sakharov and M. A. Tereshchenko, Plasma Phys. Rep.21, 93 ~1995!.67S. Murakami, U. Gasparino, H. Idei, S. Kubo, H. Maassberg, N. Nakaji

M. Rome, and M. Okamoto, Nucl. Fusion40, 693 ~2000!.68H. Maassberg, C. D. Beidler, U. Gasparino, M. Rome´, W.-A. Team, K. S.

Dyabilin, N. B. Marushchenko, and S. Murakami, Phys. Plasmas7, 295~2000!.

69S. V. Kasilov, W. Kernbichler, V. V. Nemov, and M. F. Heyn, in26th EPSConference on Controlled Fusion and Plasma Physics, Maastricht, 14–18June 1999, ECA Vol. 23J, edited by B. Schweer, G. Van Oast, andVietzke ~European Phys. Soc., Mulhouse, 1999!, pp. 1629–1632.

70S. V. Kasilov and V. E. Moiseenko, inInternational Conference on OpenPlasma Confinement Systems for Fusion, 14–18 June 1993, NovosibirskRussia, edited by A. Kabantsev~World Scientific, Singapore, 1994!, pp.341–347.~See also Proceedings of the Int. Winterschool and WorkshopPlasma Phys., Pichlmayrgut, Pichl/Schladming, Austria, 22–27 Febr1993!.

71L. G. Eriksson and P. Helander, Phys. Plasmas1, 308 ~1994!.72S. V. Kasilov, V. E. Moiseenko, and M. F. Heyn, Phys. Plasmas4, 2422

~1997!.73S. V. Kasilov, W. Kernbichler, V. V. Nemov, and M. F. Heyn, in25th EPS

Conference on Controlled Fusion and Plasma Physics, 29 June–3 July,Praha, edited by R. M. Pick and G. Thomas~European Phys. Soc., PetiLancy, 1998!, Vol. 22C, p. 1726.

74S. V. Kasilov, W. Kernbichler, V. V. Nemov, and M. F. Heyn, in28th EPS


R

e

,

nry

Conference on Controlled Fusion and Plasma Physics, Madeira, Portu~European Phys. Soc., Petit-Lancy, 2001!, Vol. 25A, pp. 1981–1984.

75S. V. Kasilov, W. Kernbichler, V. V. Nemov, and M. F. Heyn, Phys. Plamas9, 3508~2002!.

76R. Kamendje, S. V. Kasilov, W. Kernbichler, and M. F. Heyn, ProblemsAtomic Science and Technology, Series: Plasma Phys.6, 13 ~2000!.

77R. Kamendje, S. V. Kasilov, W. Kernbichler, and M. F. Heyn, in28th EPSConference on Controlled Fusion and Plasma Physics, Madeira, Portu~European Phys. Soc., Petit-Lancy, 2001!, Vol. 25A, pp. 817–820.

78H. Goldstein,Klassische Mechanik~AULA GmbH, Wiesbaden, 1989!.79C. Grebogi and R. G. Littlejohn, Phys. Fluids27, 1996~1984!.80A. J. Lichtenberg and M. A. Lieberman,Regular and Chaotic Dynamics,

2nd ed.~Springer, New York, 1992!.81A. I. Morozov and L. S. Solov’ev, inReviews of Plasma Physics, edited by

M. A. Leontovich ~Consultants Bureau, New York, 1966!, Vol. 2.82A. V. Zvonkov and A. V. Timofeev, Sov. J. Plasma Phys.12, 238 ~1986!.83N. J. Fisch and A. H. Boozer, Phys. Rev. Lett.45, 720 ~1980!.84V. D. Davydovskii, Sov. Phys. JETP16, 629 ~1963!.85A. B. Kitsenko, Sov. J. Plasma Phys.9, 464 ~1983!.86L. D. Landau and E. M. Lifschitz,Lehrbuch der Theoretischen Physik

Band I, Mechanik, 11 ed.~Akademie, Berlin, 1984!, Vol. 1.87S. V. Bulanov and N. M. Naumova, Phys. Scr.T63, 190 ~1996!.88A. M. Runov, D. Reiter, S. V. Kasilov, W. Kernbichler, and M. F. Hey

Phys. Plasmas8, 916 ~2001!.89D. V. Sivukhin, inReviews of Plasma Physics, edited by M. A. Leontovich

~Consultants Bureau, New York, 1965!, Vol. 1.90S. V. Kasilov, in Proceedings of the Seventh European Fusion The

Conference, edited by A. Rogister~Forshungszentrum Ju¨lich GmbH, Ju-lich, Germany, 1998!, pp. 111–114.

91J. N. Dnestrovskij and D. P. Kostomarov,Numerical Simulation of Plas-mas~Springer, Berlin, 1986!.

92M. H. Kalos and P. A. Whitlock,Monte Carlo Methods. Volume I: Basic~Wiley, New York, Chichester, Brisbane, Toronto, Singapore, 1986!.

93A. H. Boozer and G. Kuo-Petravic, Phys. Fluids24, 851 ~1981!.94S. V. Kasilov, V. E. Moiseenko, M. F. Heyn, and W. Kernbichler, AI

Conf. Proc.403, 321 ~1997!.95S. V. Kasilov and W. Kernbichler, Phys. Plasmas3, 4115~1996!.96W. Kernbichler and S. V. Kasilov, Phys. Plasmas3, 4128~1996!.97R. Kamendje, S. V. Kasilov, W. Kernbichler, I. V. Pavlenko, and M.

Heyn, in29th EPS Conference on Controlled Fusion and Plasma PhysMontreux, Switzerland~European Phys. Soc., Petit-Lancy, 2002!, Vol.26B.

98M. Rome, V. Erckmann, U. Gasparino, H. J. Hartfuss, G. Ku¨hner, H.Maassberg, and N. Marushchenko, Plasma Phys. Controlled Fusion39,117 ~1997!.


Documents

Kinetic modeling of nonlinear electron cyclotron resonance heating › AG › APP › PLASMA › PUBLIC › OWN_PUBLICATI… · is established by Coulomb collisions, and leads to