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  • Momentum transport by waveparticle interaction

    This article has been downloaded from IOPscience. Please scroll down to see the full text article.

    2011 Plasma Phys. Control. Fusion 53 054007

    (http://iopscience.iop.org/0741-3335/53/5/054007)

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  • IOP PUBLISHING PLASMA PHYSICS AND CONTROLLED FUSION

    Plasma Phys. Control. Fusion 53 (2011) 054007 (11pp) doi:10.1088/0741-3335/53/5/054007

    Momentum transport by waveparticle interaction

    T Hellsten

    Alfven Laboratory, School of Electrical Engineering, KTH, Association VR-Euratom, SE-100 44Stockholm, Sweden

    E-mail: thel@kth.se

    Received 29 September 2010, in final form 12 January 2011Published 7 April 2011Online at stacks.iop.org/PPCF/53/054007

    AbstractEnergy and momentum can be transported across the plasma by waves emitted atone place and absorbed at another. Exchange of momentum and energy betweenthe particles and the waves change the drift orbits, which may give rise to a non-ambipolar particle transport. The main effect of the non-ambipolar transport andquasi-neutrality is a toroidal precession of the trapped particles, which togetherwith the changes in the parallel velocities of the passing resonant particlesconserve the toroidal momentum. Non-resonant interactions can give rise to anet change of the local wave number in spatial inhomogeneous plasmas witha resulting force on the medium. Both resonant and non-resonant interactionshave to be taken into account in order to have a consistent description ofthe momentum transported by the waves. The momentum transfer is, inparticular, important for waves with short wave length and low frequency, andmay explain the enhanced rotation seen in some mode conversion experiments,when the fast magnetosonic wave is converted to an ion-cyclotron wave. Theapparent contradiction that the wave momentum may change due to non-resonant waveparticle interactions without changing the energy in geometricoptics is explained.

    1. Introduction

    Waves can be used to heat plasmas to thermonuclear temperatures and drive currents requiredfor confinement or stabilization of macroscopic modes. They play an important role inanomalous transport of thermal and supra-thermal particles. In order to predict the performanceof wave heating and wave induced transport it is important to understand the interactionsbetween waves and charged particles. Waveparticle interactions displace the guiding centresof the particles across the magnetic field and accelerate them along the field lines. Nettransfer of momentum from one species to another species or from one location to anotheror between trapped and passing particles can affect the flow and the currents in toroidalplasmas. Momentum transport is, in particular, important for low frequency waves and

    0741-3335/11/054007+11$33.00 2011 IOP Publishing Ltd Printed in the UK & the USA 1

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    waves with high wave numbers such as slow kinetic waves and drift waves. Waves canalso affect the confinement by direct transport of energy and momentum from one locationwhere they are emitted to another where they are absorbed. Transfer of energy and momentumbetween particles and waves take place by resonant and non-resonant interactions. Non-resonant interactions have received considerable interest [13] in order to explain the improvedconfinement when heating the plasma with ion-Bernstein waves [4, 5] and the increased rotationobserved during ion-cyclotron heating in mode conversion experiments [6]. The conclusionfrom these studies of uniform plasmas was that the ponderomotive force cancelled the forcefrom the Reynolds stress [13]. Thus, it was not possible to explain the increased rotationby non-resonant interactions. Recently, it has been shown that a net force appears from non-resonant interactions in spatial dispersive non-uniform plasma for Alfven waves due to thechange in the parallel wave number [7] with the potential of driving flow. This force agreeswith the reactive force arising from the change in momentum of the wave due to refractionin geometric optics, which in toroidal plasmas produce the up- and downshift of the parallelwave numbers.

    How waves are affected by waveparticle interactions can be seen by solving the waveequation, using the appropriate dielectric tensor. A powerful tool to study wave propagationin weakly inhomogeneous plasmas is geometrical optics, with which the wave field can becalculated. The effects on the particles caused by waveparticle interactions are obtained bysolving the equation of motion with the appropriate wave field. However, the effects of thenon-linear responses by an inhomogeneous wave field become less intuitive. The consistencyof the two approaches arises because of the dielectric tensor, whereby the response functionis derived from the equation of motion including the wave field. Here we reconcile the forcearising due to changes in wave number and momentum density of the wave, due to refractionin geometrical optics with the ponderomotive force and the force from the Reynolds stressfor the fast magnetosonic wave propagating across an inhomogeneous magnetic field. Thesenon-linear forces arise through non-resonance interactions and affect all particles. The changesof the momentum by these forces are of the same order as those by wave absorption throughresonance interactions and have to be included on the same footing in order to have a consistentdescription of the momentum transport by the waves.

    2. Waveparticle interaction

    Maxwells equations and the equation of motion define the waves in plasmas and how theyinteract with the particles. Understanding of wave-propagation and interaction with particlescan be achieved by perturbations of low amplitude waves in homogeneous plasmas, for whichthe equation of motion can be linearized, and an algebraic wave equation can be obtained byFourier transforming the equations with respect to time and space. The linearized equationdefines the waves, their polarization, how they propagate, etc. The linear Lorentz forceaveraged over wave periods and gyro-motion vanishes except at resonances, where they giverise to collisionless absorption and emission processes. An insight into how waves propagateand how momentum and energy are transported across the plasma by waves can be obtainedfrom geometrical optics; Hamiltons equation for wave quantum. The momentum of thequantum is proportional to the wave number. Changes in the wave numbers due to refractionas the wave propagates in a spatial inhomogeneous dispersive medium result in forces on theplasma.

    Particles together with their momentum and energy are transported across the magnetic fluxsurfaces by changes in the particle orbits caused by collisions and waveparticle interactions;either by changes of the invariants or displacements to other orbits with the same invariants.

    2

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    Energy and momentum are transferred between the waves and particles by waveparticleinteractions. To determine the transport it is important to know how the waveparticleinteractions affect the orbits. The drift orbits are determined by stationary magnetic andelectric fields or fields varying on a time scale well below the cyclotron frequency. For timeindependent fields the drift orbits in axisymmetric plasmas can be defined by three invariants ofmotion and the initial conditions. When describing the motion of nearly collisionless particlesin a plasma, it is convenient to describe the drift orbits in terms of guiding centre orbit invariants.For an axisymmetric torus we can use (W , , P , ), where W is the energy, is an adiabaticinvariant, = B0/W , P is the canonical angular momentum, is a label distinguishingparticles with the same invariants but different orbits, is the magnetic moment, which is anadiabatic invariant, and B0 the magnetic field at an arbitrary position, here chosen to be at themagnetic axis.

    Non-ambipolar flows across the flux surfaces due to waveparticle interactions generateelectric fields. The change in the electric field leads to a direct change in the drift velocitiesof all particles within the time scale of a gyro-period by the polarization drift and later on thetime scale of the bounce time by the neoclassical polarization drift. Of particular importancefor momentum conservation in toroidal plasma is the precession of the trapped particles by thenormal component of the electric field.

    The polarization currents nearly cancel the change in the electric field, but a small fractionwill remain such that the change in momentum of the plasma due to the EB-drift is consistentwith conservation of toroidal and poloidal momentum. To assess the effects of waveparticleinteractions requires calculation of the change in the electric equilibrium field, as well as thechange in the orbit invariants.

    3. Resonant interactions

    Absorption and emission of waves take place by resonant interactions. A resonance is heredefined as a number of nearly periodic oscillations producing a net acceleration of the particleduring a decorrelation time. The change in the wave momentum, Pw, by resonant waveparticle interactions in a homogeneous plasma is related to the change in energy of the wave,Ww,

    Pw = (k/)Ww, (1)where k is the wave number and the angular wave frequency. A resonant interaction resultsin an acceleration parallel to the magnetic field and a displacement of the guiding centre acrossthe field, x, of the particle given by

    x = (k B/qB2)W, (2)where q is the charge of the particle, B the magnetic field and W the change in energy ofthe particle due to the waveparticle interaction. For localized resonances the interactions takeplace in real space, in the neighbourhood of the points on the drift orbit where the resonancecondition is satisfied. Non-localized resonances have to be treated differently. From therelation between the changes in parallel momentum and energy, (mv) = (k/)W , at theresonance, the displacement across the flux surfaces of the guiding centre in an axisymmetricplasma and the change in the parallel velocity become

    res = Wq

    [n

    B2B2

    kR

    BB

    B2

    ]res

    (3a)

    vres = Wm

    [n

    R

    B

    B+ k

    B

    B

    ]res

    , (3b)

    3

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    where m is the mass of the particle, R the major radius, n toroidal mode number, the subscriptres denotes the location of the resonance and k = Bn/BR + Bk/B. Since is not anignorable variable in a torus, a local approximation of the poloidal wave number, k , is used,which has to satisfy

    20 k d = m , where m is an integer. In an axisymmetric plasma the

    changes in the orbit invariants P and are related to the change in energy [8]:P = (n/)W (4a) = (nc0/ )W/W, (4b)

    where c0 is the cyclotron frequency for a magnetic field B0. The poloidal mode number kdoes not affect the change in P , but will change v and .

    The changes of the particle orbits result in changes of the electric field, which becomes non-trivial to calculate. However, when the changes in the orbits due to the change in the field aresmall one may locally add up the displacements caused by waveparticle interactions, includingthe averaged change of the charge density due to the changes in the orbit invariants. Forinteractions with trapped particles the changes in the radial charge density can be approximatedby the changes of the radial location of the turning points, which are determined by the changein P , with contributions from parallel acceleration and radial displacement

    res = 1q

    n

    W. (5)

    The displacements of the turning points of the trapped particles give rise to particle flux,

    (t) () = lim

    0n

    q

    t :

    Wt

    tFJ dW d dP n

    qP

    (t)RF () (6)

    where (t) () denotes the particle flux of trapped particles of species , F denotes thedistribution function expressed in the orbit invariants (W , , P , ), J the Jacobian andP

    (t)RF () is the flux surface integrated power absorbed by the trapped particles. The integral is

    taken over a toroidal flux shell with a width . For passing particles we neglect for simplicitythe change in the orbits and only include the local radial displacement, given by equation (3a),which gives rise to a particle flux [9]

    (p)

    () = lim0

    1q

    p:

    Wp

    t

    [n

    B2B2

    kR

    BB

    B2

    ]res

    FJ dW d dP

    1q

    [n

    B2B2

    kR

    BB

    B2

    ]res

    P(p)

    RF (). (7)

    For passing particles the poloidal mode number, which depends on the position of theinteraction, is important for the particle flux. Note for waves with k 0 the flux of passingparticles becomes (p) nP (p)RF /q .

    The polarization current, arising when the plasma accelerates to the E B drift,nearly cancels the field, but a small residual field remains. The resulting E B-driftproduces a toroidal precession of the trapped particles, a poloidal and a toroidal drift ofthe passing particles. The ratio between the toroidal precession velocity of the trappedparticles and the toroidal component of the E B drift of the passing particles is givenby v(t) /v(p) = B2/B2 , where the brackets v(t) and v(p) denote averaging over thetoroidal velocity over the trapped and passing particles of species , respectively. The fractionof trapped particles for an isotropic plasma with circular flux surfaces is given by

    2r/R0.

    Thus, the change in the toroidal momentum of the plasma is approximately given by the sum

    4

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    of the changes in the precession of trapped particles and the toroidal component of the changesin the parallel velocity of the resonant passing particles. The latter are given by

    mRv

    (p)

    = B2

    B2

    P

    (p)

    , (8a)

    or expressed in terms of changes in energy of the resonant particlesmRv

    (p)

    = B2n

    B2Wp. (8b)

    For waves with n = 0 and k = 0 the changes in the toroidal velocity of the plasma conservingP when assuming R R0 becomes

    R0(Er)n

    B

    =i,e

    ntm

    = resonantparticles

    B2B2

    P (p) + P (t) , (9a)

    where (Er)n is the contribution to the radial electric field by waves with finite n and nt isthe density of trapped particles of species . Expressing equation (9a) in terms of changes inenergy of the resonant particles gives

    (Er)n

    B

    =i,e

    ntm

    = resonantparticles

    n

    R0

    (B2B2

    Wp + Wt ). (9b)

    For waves with k = 0 and n = 0 the flux of resonant particles is dominated by passingparticles such that P is unchanged. The changes in the parallel velocities of the resonantpassing particles are given by

    mv(p)

    = B

    B

    k

    Wp. (10a)

    The changes in the toroidal velocities of the resonant passing particles becomemv

    (p)

    = BBkB2

    Wp. (10b)The waveparticle interactions result in displacements across the flux surfaces and accelerationparallel with the magnetic field conserving the canonical angular momentum P . The resultingchange in the electric field gives rise to a change in precession of the trapped particles, suchthat the total angular momentum is conserved

    (Er)k

    B

    =i,e

    ntm

    = resonantparticles

    BBk

    B2Wp. (11)

    For waves with finite n and k we have

    Er (Er)k + (Er)n

    1=i,e

    ntm

    {( BB

    2

    B2k

    +

    B3B2

    n

    R0

    ) = resonant

    particles

    Wp

    +nB

    R0

    = resonant

    particles

    Wt }, (12)

    5

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    where k is calculated at the resonance position. Thus, both the toroidal and the poloidal wavenumbers contribute to the radial electric field. The main effect of a non-ambipolar transportby waveparticle interactions is a toroidal precession of all trapped particles. The sum of thechange in the toroidal momentum of the trapped particles and the toroidal component of theparallel velocity of the resonant passing particles equals the total absorbed toroidal momentum.Collisions, not included here, will try to reduce the differences in toroidal velocities betweentrapped and passing particles. In the above calculations we have for simplicity used anequilibrium with concentric circular magnetic surfaces.

    4. Non-resonant interactions

    The momentum density of a wave is proportional to the wave number. In geometrical opticsthe change in the wave number is caused by refraction and is given by

    dk/dt = M/x, (13)where = M is the dispersion relation. The necessary conditions for geometric opticsto be fulfilled is L and the sufficient Fresnel condition is l W 2/, where l is thelength of the ray trajectory, the wave length, L the inhomogenity scale length and W thetransverse wave packet size [10]. According to geometrical optics in a spatial dispersiveinhomogeneous medium the wave momentum can change as the wave propagates withoutchanging the wave energy. The change in the wave number, and hence the momentum, givesrise to a reactive force on the medium. The reaction force is described on the microscopiclevel by non-resonant interactions of single particles with a wave field. However, the changein momentum by non-resonant interactions without a corresponding change in energy leadsto an apparent contradiction because of momentum and energy conservation of waveparticleinteractions. The changes in energy and momentum of the particles can be calculated fromthe equation of motion. In the first order the averaged changes in energy and momentumover a wave period vanishes. In the next order, which includes quadratic terms of the wavefield, a finite contribution may appear for non-uniform, spatial dispersive media. A changein the parallel wave number and the associated change in the wave amplitude give rise to aparallel acceleration of the particle, which changes the energy of the particle. From the energyconservation of waveparticle interactions there should be a corresponding change in energy ofthe wave. A change in the perpendicular wave number produces a perpendicular acceleration ofthe particle that changes the guiding centre position of the particle. Averaging the accelerationperpendicular to the magnetic field over the gyro-motion produces a net change of the guidingcentre position, but no change in the perpendicular energy.

    The apparent contradiction caused by acceleration parallel to the magnetic field can beexplained as follows. From conservation of momentum of waveparticle interactions thechange in momentum by the wave quantum corresponds to an acceleration of the particle, v,such that mv = hk. The acceleration changes the energy of the particle with mvv.This change in energy corresponds to a change in frequency of the wave quantum by h.The change in energy of the wave, lost to the particle, corresponds to the change in frequencydue to the change in Doppler shift = vk. Thus, in the coordinate system, where theparticle, or a medium, is at rest the wave frequency and wave energy are unchanged explainingthe apparent contradiction that non-resonant interactions do not change the energy of thewave. For a symmetric distribution function, for which the positive and negative contributionscancel, there will be no net change in energy of the wave. The change in momentum of the wavewithout a corresponding change in energy for non-resonant interactions differs from resonantinteractions. Another difference is that non-resonant interactions take place at a given position

    6

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    with all particles while resonant interactions take place only with the resonant ones. If thewhole plasma would achieve a net acceleration, there would be a net change in the frequencyof the wave corresponding to the change in Doppler shift caused by the change in rotation.When calculating the distribution function with a FokkerPlanck code the acceleration of theparticles by the non-resonant force should in principle be included; however, it may in manycases be negligible. The changes in energy and momentum of the particle because of the lowvelocity of the non-resonant particles, which are mostly thermal, are of second order and ofthe same order as the changes caused by resonant interactions.

    The force due to the non-resonant interactions can be obtained by calculating the forceon the guiding centres of ions and electrons including quadratic terms of the wave field. Thereactive force due to refraction can be calculated by calculating the change in the momentumdensity of the wave due to the change in the wave number. Here we shall show that thesetwo forces are the same for the fast magnetosonic wave in a plane slab. To prove the generalcase is out of the scope of this study. Here we consider a single ion species plasma slab withthe magnetic field in the z-direction, which increases in the x-direction, and calculate the netforces in the x-direction.

    While the geometric optics describe momentum and energy transport by waves in atransparent way, the non-linear forces on individual particles from the equation of motiondue to changes in the wave numbers are more difficult to assess. For this purpose it is betterto use the momentum equation, although it is less transparent. From geometrical optics weconclude that there will only be a net change in the momentum of the wave for spatial dispersivewaves in inhomogeneous plasmas. Focusing of the wave in a homogeneous medium will onlyconcentrate the wave field and momentum density. Changes of the momentum parallel to themagnetic flux surfaces can cause particle transport across the flux surfaces. In an axisymmetricplasma only the poloidal variation of the equilibrium can affect the spatial dispersion e.g. byvariation of the magnetic field along the flux surfaces.

    The force on a charged particle due to non-resonant interactions is obtained by expandingthe equation of motion to second order in amplitude and first order in inhomogeneity. In a coldplasma including an inhomogeneous background, the equation of motion for a single particleis in the first order given by

    mv(1)

    t= qE + qv(1) B0. (14)

    The first order perturbation of the velocity and displacement can be calculated by assumingv(1)/t = iv(1) giving v()i = i0()ij Ej/qn and x()i = 0()ij Ej/qn , where

    ()ij is the cold plasma susceptibility tensor. The inhomogeneity of the plasma is taken into

    account by assuming the wave field to have the form E = E0(x) exp i(k x t). Averagingthe second order terms over a wave period gives

    m

    v(2)

    t

    + mv(1) v(1)

    = qx(1)

    E

    x

    + q

    v(1)

    (x(1)

    B0

    x

    )+ qv(1) B(1) + q

    x(1)

    v(1)

    x B0

    ,

    (15)where iB(1) = E. The brackets denote averaging over wave period and gyro-motion.The left-hand side is a generalized ponderomotive force, here denoted by FP, and the secondterm of the right hand is the convective derivative of the velocity. Transferring the latter termto the right-hand side we write the equation in the form

    mv(2)/t = F (2). (16)7

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    The force, F (2), defines a force on the particles caused by the inhomogenity of the medium andcorresponds to the change in momentum by the wave. Note that momentum of the differentspecies can be affected by taking momentum from one species by waveparticle interactionand transferring it to another without changing the net momentum of the wave.

    In axisymmetric plasmas only the poloidal and perpendicular wave numbers can change.The changes in the guiding centre position and in the parallel velocity cause the total changein P to vanish. If the change in position of the guiding centre is slow compared with thecyclotron motion, the magnetic moment will be conserved. The change in caused by thechange in parallel velocity of the particle will then be given by = W/W . This isthe same relation as for the resonant interactions given by equation (4b) with n = 0.

    The generalized ponderomotive force defined by the right-hand side of equation (15),

    FPx = qx

    Ex

    x

    + q

    vy

    (x

    B0

    xez

    )+ qvyBz, (17)

    gives

    FPx = mB2

    (32 22 2

    )ikxEyEy

    +m

    B2

    (2

    2 22

    x(

    32 22 2

    )1

    E0y

    E0y

    x

    )EyEy, (18)

    where the cyclotron frequency is given by = qB/m . The derivative of the amplitudeof the electric field, appearing in equation (18), can be expressed in terms of derivativesof the equilibrium quantities. The variation of the amplitude of the wave field can becalculated from geometric optics by expanding to first order. Here we use conservationof Poynting flux from which we have kx |Ey |2 = C, where C is a constant, which gives(1/E0y)E0y/x = (1/2kx)kx/x. For the magnetosonic wave we assume k2x 2A/V 2A,which gives (1/E0y)E0y/x = (1/2Bz)Bz/x, where VA =

    B2/0 is the Alfven

    velocity and 0 the permeability of free space. Taking the limit / 0 of equation (18)we obtain

    FPx = mB2

    ikxEyEy 12B

    B

    xEyEy

    . (19)

    The force due to the convective derivative of the velocity becomes

    mv(1) v(1)x = mB2

    ikxEyEy +

    12B

    B

    xEyEy

    . (20)

    The x-component of the total force F (2) becomes

    F (2)x = FPx mv(1) v(1)x = m

    B3B

    xEyEy. (21)

    The contributions from the generalized ponderomotive force and convective derivative ofvelocity are equal in magnitude. The dominating terms are proportional to ikx , which give riseto oscillating perturbations at twice the frequency, which cancel even before the averaging.Only the terms proportional to the variation of the equilibrium magnetic field remain, givingrise to a non-oscillating net force and a term oscillating with twice the frequency. The latterterm vanishes when averaging over wave period. However, if the dispersion relation is satisfiedfor the wave with twice the frequency, this perturbation will result in coupling to a wave withtwice the frequency, which is not considered here. For the non-oscillating term we have

    8

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    ReEyEy = 0.5|EyEy |. Multiplying with the density we obtain in a cold plasma, the netforce

    nF(2)x =

    nm

    B3B

    xEyEy = c

    20

    2V 2ABB

    x|Ey |2, (22)

    where 0 is the permittivity of free space. In a fluid the convective derivative of the velocity isassociated with the Reynolds stress with the fluctuations given by the plasma motion caused bythe wave. We compare the force by non-resonant interaction given by equation (22) with theforce on the medium caused by refraction in geometrical optics. The change in wave numbergiven by equation (13) gives in the absence of absorption a change in the momentum densityof the wave by

    dPw/dt = (A/x)Ww/. (23)Using the same approximation of the dispersion relation of the magnetosonic wave as earlierwe obtain

    dPxdt

    = kx VAx

    Kx

    , (24)

    where Kx is the x-component of the Poynting vector.dPxdt

    = c2

    2V 2A

    0

    B

    B

    x|Ey |2. (25)

    Thus, the net change in the momentum density of the wave due to refraction in thegeometric optics corresponds to the net non-linear force due to non-resonant interactions of theplasma caused by a generalized ponderomotive force and the force given by the Reynolds stressfrom the plasma motion caused by the wave. Note when neglecting the plasma inhomogeneityand only keeping the dominating term proportional to the wave number, the ponderomotiveforce caused by focusing of the wave field cancelled the force caused by the Reynolds stress[13]. When taking into account the change in wave amplitude due to refraction in theponderomotive force and the force associated with the Reynolds stress a net force remainsthat equals the force caused by the change in momentum density in the geometrical optics.Applying this model locally in an axisymmetric plasma one finds that when the x-componentof the force has a component parallel to the magnetic flux surface, as in the case when thewave propagates above the magnetic axis, the force will produce a flux of particles across theflux surface.

    5. Discussion and conclusions

    Waves affect the confinement by transporting momentum and energy across the plasma andby modifying the drift orbits by waveparticle interactions. Resonant interactions exchangemomentum and energy between the waves and particles. If the wave is damped and emittedlocally by the same species, there is no effect on the particle transport. However, cold particlescan be driven inwards and hot outwards, if they are resonating with the same wave, and thedistribution function gives rise to unstable interactions in one region in the phase space andunstable in another region.

    Waveparticle interactions accelerate the particles along the magnetic field and displacetheir guiding centres across the field, resulting in non-ambipolar particle transport. Also thechange in the orbit width by changes of the perpendicular energy of the particles can giverise to electric fields due to non-ambipolar transport [11]. The residual electric field takinginto account the changes of the field due to polarization drifts gives rise to poloidal as well as

    9

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    toroidal rotation. Both the toroidal and poloidal wave numbers contribute to the electric field,which may stabilize turbulence. The dominating effect on the particles by the non-ambipolartransport by the waveparticle interactions is a change in the toroidal precession of the trappedparticles. The sum of the changes in the toroidal momentum of the trapped particles andthe changes in the toroidal component of the parallel velocity of the resonant passing ionshave to balance the total absorbed toroidal momentum in order to conserve canonical angularmomentum and quasi-neutrality. Collisions will try to relax differences in toroidal velocitiesbetween trapped and passing particles.

    Interactions with a wave with a finite poloidal wave number, k , lead to displacementsand parallel accelerations of the guiding centres. The poloidal wave number, in contrast to thetoroidal wave number, is not an invariant and varies as the wave propagates across the plasmadue to refraction. Particles with the same invariants interacting with the same wave at differentpositions along a drift orbit will be accelerated differently in the parallel direction and theirguiding centres will be displaced differently as the poloidal mode numbers at the interactionpoints differ. The interactions lead to the same changes of the guiding centre orbit invariants, and P for the same change in W . However, the modification of the normal component ofthe electric field by the displacement of the guiding centre differs, thereby explaining how thepoloidal momentum can be conserved without explicitly appearing in the expression describingthe changes of the orbit invariants.

    We have shown that the reaction force on a plasma caused by the change in the wave numberby refraction in the geometrical optics for the fast magnetosonic wave for inhomogenities acrossthe magnetic field equals the difference between the ponderomotive force and the force due toReynolds stress. Earlier it was shown that the same holds for Alfven waves with a magneticfield strength varying along the field [7]. That the reaction force from refraction is related tothe ponderomotive force and force due to Reynolds stress may at a first glance appear as asurprise. When neglecting the plasma inhomogeneity and only keeping the dominating termproportional to the wave number, the ponderomotive force caused by focusing of the wavefield cancels the force caused by the Reynolds stress [13]. When taking into account thechange in wave amplitude due to refraction in the ponderomotive force and the force associatedwith the Reynolds stress, a net force remains that equals the force caused by the change inmomentum density in the geometrical optics due to refraction. The change in momentum bythe ponderomotive force and the force from the Reynolds stress tensor are non-linear effectsof the same order as the resonant interactions and has to be included, in order to be consistentwith the changes in momentum density of the wave due to refraction. It is most likely thatunder the conditions of validity of the geometric optics the forces on the media arising fromthe refraction equals the non-linear ponderomotive and force and the force associated with theReynolds stress. We have also resolved the apparent contradiction that a change in momentumof the wave for non-resonant interactions can take place without a change in energy evenfor changes resulting in acceleration parallel to the particle motion. In general, it is easierto evaluate the net force on the medium from the geometric optics than calculating the netnon-resonant force on the particles.

    The up and down shift of the parallel wave number due to refraction can be importantfor current drive when the momentum is taken from one species by non-resonant interactionsand given to another by resonant interactions or when the upshift and absorption does not takeplace on the same magnetic flux surface as is typical for current drive by lower hybrid waves.Stimulated emission of waves followed by damping and changes of the poloidal mode numberby refraction give rise to dipolar like torques for which the effect on the plasma depends onthe radial distance between the processes. The momentum transfer is, in particular, importantfor short wave length modes, such as drift waves and kinetic waves. The latter are often

    10

  • Plasma Phys. Control. Fusion 53 (2011) 054007 T Hellsten

    obtained by mode conversion of the magnetosonic wave, which may explain the flow seenin some mode conversion experiments [6] and enhanced confinement in discharges heated byion-Berstein waves [4, 5]. In earlier studies [13] neglecting the plasma inhomogeneity andonly keeping the dominating term proportional to the wave number, the ponderomotive forcecaused by focusing of the wave field cancelled the force caused by the Reynolds stress, andas a result these studies were unable to explain the flow drive and enhanced confinement dueto poloidal rotation. To explain the increased rotation in mode conversion scenarios an upand down asymmetric change in the poloidal wave number has to take place. In addition thewave absorption should not take place on the same magnetic flux surface at which the poloidalmode number changes due to non-resonant interactions. The computation of the wave fieldand change of the poloidal wave number is non-trivial, but can be done with advanced wavecodes. Because of the differential nature of the resonant and non-resonant forces in connectionwith mode conversion the net effect is difficult to predict. This result is also consistent withthe knowledge that enhanced rotation is only seen in a narrow parameter window [6].

    References

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    Conf. Proc. 1187 625[8] Eriksson L-G and Helander P 1994 Phys. Plasmas 1 308[9] Hellsten T et al 2008 Proc. Theory of Fusion Plasmas, Joint VarennaLausanne Int. Workshop (Varenna, Italy,

    2008) AIP Conf. Proc. 1069 88[10] Pereverzev G V 1998 Phys. Plasmas 5 3529[11] Chang C S et al 1999 Phys. Plasmas 6 1969

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    1. Introduction2. Wave--particle interaction3. Resonant interactions4. Non-resonant interactions5. Discussion and conclusions References