wave-particle interaction at double resonance

19
Wave-particle interaction at double resonance A. Zaslavsky, 1 C. Krafft, 1 L. Gorbunov, 2 and A. Volokitin 3 1 Laboratoire de Physique et Technologie des Plasmas, Ecole Polytechnique, 91128 Palaiseau Cedex, France 2 Lebedev Physical Institute, Russian Academy of Sciences, Leninskiy Prospekt 53, 119991, Moscow, Russia 3 IZMIRAN, Troitsk, Moscow Region, Russia Received 29 November 2007; published 22 May 2008 This paper is devoted to studying wave-particle interaction at “double resonance” condition, i.e., when two waves interact resonantly with the same group of charged particles. A theoretical Hamiltonian model and a symplectic numerical code are built to describe the three-dimensional interactions of wave spectra with reso- nant electrons in a magnetized plasma. Related simulations on the evolution of two waves of close parallel phase velocities interacting resonantly with particles’ fluxes have been performed, which reveal some common features which do not depend on the kind of waves, instabilities, and particles’ distributions: after the stage of linear instability, when the waves’ amplitudes saturate due to particle trapping, a nonlinear process takes place which is characterized by a quasiperiodical exchange of energy between the waves, depending in particular on the value of the mismatch between the waves’ resonant velocities. In order to explain such observations, a simple Hamiltonian model describing the interaction of two different waves of close resonant velocities with a periodical train of bunches of trapped particles moving synchronously has been built. It allows one to describe the nonlinear characteristics of this process as well as to estimate analytically its time scale and shows a good agreement with the numerical simulation results. DOI: 10.1103/PhysRevE.77.056407 PACS numbers: 52.35.g, 52.65.y I. INTRODUCTION This paper is devoted to studying wave-particle interac- tion at “double resonance” condition, when two waves inter- act resonantly with the same group of charged particles in a collisionless plasma. The interest in such a problem is mani- fold, as discussed below. First, the study of wave-particle interaction at double resonance can be considered as an approach to the investiga- tion of the nonlinear stage of the so-called “sideband insta- bility.” It is well-known that an unstable monochromatic wave of frequency which grows in a plasma usually satu- rates by trapping resonant particles 1,2. The system formed by this wave and the particles that it traps can be perturbed by the presence of other waves, with resonance velocities close to that of the main wave, that can interact with the trapped particles. In particular, a sideband instability can de- velop, for which the most unstable satellite waves have fre- quencies equal to n tr , where tr is the bounce fre- quency of an electron oscillating in the main wave potential well and n is an integer. If among all the satellite waves one has a growth rate significantly larger than the others, the nonlinear stage of the instability can be reduced to the inter- action of two waves with one group of resonant particles. Note that the sideband instability was studied analytically by considering groups of electrons with close velocities and phases interacting with two or more waves at Landau 1,36 and cyclotron resonances 7, as well as investigated experi- mentally see, e.g., 8, and more recently 9. Another reason to study the problem of two or more waves interacting resonantly with the same group of particles is aimed at understanding fundamental physical processes of wave-particle interaction in plasmas which arise in the theory of nonlinear scattering of waves on particles 2,10. In this case, one has usually to perform nonlinear perturbative calculations where third-order charge density terms present the following general structure: dv k 1 1 - k 1 · v · v k 2 2 - k 2 · v · v k 1 - k 2 1 - 2 - k 1 - k 2 · v · v f v . 1 Equation 1 refers to the case of two waves of frequencies 1 and 2 in an unmagnetized plasma; = 1 - 2 is the frequency of one of the beating modes; k 1 , k 2 and k = k 1 - k 2 are the corresponding wave numbers; v is the velocity; and f v usually represents the particles’ velocity distribution function. It is commonly supposed that the integrations over the poles in Eq. 1 can be performed separately, taking into account the contribution of one pole only and calculating the other integrals using their principal values. It means that the number of particles which are simultaneously in resonance with each wave, that is, whose velocities v verify both the conditions 1 k 1 · v and 2 k 2 · v, is smaller than the total number of particles resonantly interacting with any wave. For example, in the frame of the theory of induced scattering of waves on particles, only the pole defined by 1 - 2 - k 1 - k 2 · v = 0 is considered. In a plasma without ambient magnetic field and described by a Maxwellian velocity dis- tribution, the group of particles which can be in resonant interaction simultaneously with two waves is strongly re- duced. Indeed, considering the picture in the velocity space, the particles which interact resonantly with the wave 1 , k 1 lie on a plane, whereas those interacting resonantly with the wave 2 , k 2 lie on another one; the intersection of the two planes reduces to a straight line, where only an extremely small number of particles can interact with both waves. However, in a magnetized plasma, it is not difficult to find PHYSICAL REVIEW E 77, 056407 2008 1539-3755/2008/775/05640719 ©2008 The American Physical Society 056407-1

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Page 1: Wave-particle interaction at double resonance

Wave-particle interaction at double resonance

A. Zaslavsky,1 C. Krafft,1 L. Gorbunov,2 and A. Volokitin3

1Laboratoire de Physique et Technologie des Plasmas, Ecole Polytechnique, 91128 Palaiseau Cedex, France2Lebedev Physical Institute, Russian Academy of Sciences, Leninskiy Prospekt 53, 119991, Moscow, Russia

3IZMIRAN, Troitsk, Moscow Region, RussiaReceived 29 November 2007; published 22 May 2008

This paper is devoted to studying wave-particle interaction at “double resonance” condition, i.e., when twowaves interact resonantly with the same group of charged particles. A theoretical Hamiltonian model and asymplectic numerical code are built to describe the three-dimensional interactions of wave spectra with reso-nant electrons in a magnetized plasma. Related simulations on the evolution of two waves of close parallelphase velocities interacting resonantly with particles’ fluxes have been performed, which reveal some commonfeatures which do not depend on the kind of waves, instabilities, and particles’ distributions: after the stage oflinear instability, when the waves’ amplitudes saturate due to particle trapping, a nonlinear process takes placewhich is characterized by a quasiperiodical exchange of energy between the waves, depending in particular onthe value of the mismatch between the waves’ resonant velocities. In order to explain such observations, asimple Hamiltonian model describing the interaction of two different waves of close resonant velocities with aperiodical train of bunches of trapped particles moving synchronously has been built. It allows one to describethe nonlinear characteristics of this process as well as to estimate analytically its time scale and shows a goodagreement with the numerical simulation results.

DOI: 10.1103/PhysRevE.77.056407 PACS numbers: 52.35.g, 52.65.y

I. INTRODUCTION

This paper is devoted to studying wave-particle interac-tion at “double resonance” condition, when two waves inter-act resonantly with the same group of charged particles in acollisionless plasma. The interest in such a problem is mani-fold, as discussed below.

First, the study of wave-particle interaction at doubleresonance can be considered as an approach to the investiga-tion of the nonlinear stage of the so-called “sideband insta-bility.” It is well-known that an unstable monochromaticwave of frequency which grows in a plasma usually satu-rates by trapping resonant particles 1,2. The system formedby this wave and the particles that it traps can be perturbedby the presence of other waves, with resonance velocitiesclose to that of the main wave, that can interact with thetrapped particles. In particular, a sideband instability can de-velop, for which the most unstable satellite waves have fre-quencies equal to ntr, where tr is the bounce fre-quency of an electron oscillating in the main wave potentialwell and n is an integer. If among all the satellite waves onehas a growth rate significantly larger than the others, thenonlinear stage of the instability can be reduced to the inter-action of two waves with one group of resonant particles.Note that the sideband instability was studied analytically byconsidering groups of electrons with close velocities andphases interacting with two or more waves at Landau 1,3–6and cyclotron resonances 7, as well as investigated experi-mentally see, e.g., 8, and more recently 9.

Another reason to study the problem of two or morewaves interacting resonantly with the same group of particlesis aimed at understanding fundamental physical processes ofwave-particle interaction in plasmas which arise in thetheory of nonlinear scattering of waves on particles 2,10. Inthis case, one has usually to perform nonlinear perturbative

calculations where third-order charge density terms presentthe following general structure:

dvk1

1 − k1 · v·

v

k2

2 − k2 · v

·

v

k1 − k21 − 2 − k1 − k2 · v

·

vfv . 1

Equation 1 refers to the case of two waves of frequencies1 and 2 in an unmagnetized plasma; =1−2 is thefrequency of one of the beating modes; k1, k2 and k=k1−k2 are the corresponding wave numbers; v is the velocity;and fv usually represents the particles’ velocity distributionfunction. It is commonly supposed that the integrations overthe poles in Eq. 1 can be performed separately, taking intoaccount the contribution of one pole only and calculating theother integrals using their principal values. It means that thenumber of particles which are simultaneously in resonancewith each wave, that is, whose velocities v verify both theconditions 1k1 ·v and 2k2 ·v, is smaller than the totalnumber of particles resonantly interacting with any wave.For example, in the frame of the theory of induced scatteringof waves on particles, only the pole defined by 1−2− k1−k2 ·v=0 is considered. In a plasma without ambientmagnetic field and described by a Maxwellian velocity dis-tribution, the group of particles which can be in resonantinteraction simultaneously with two waves is strongly re-duced. Indeed, considering the picture in the velocity space,the particles which interact resonantly with the wave 1 ,k1lie on a plane, whereas those interacting resonantly with thewave 2 ,k2 lie on another one; the intersection of the twoplanes reduces to a straight line, where only an extremelysmall number of particles can interact with both waves.However, in a magnetized plasma, it is not difficult to find

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two different waves with the same resonance velocities evenfor the same wave branch, if one takes into account thefrequency dependence on the angle between the wave vec-tors and the ambient magnetic field B0. In this case, almostall resonant particles can interact with the waves simulta-neously, so that original physical features can be pointed outduring the wave-particle interaction processes. Thus for thecase of induced scattering on particles of two waves with thesame Landau resonance velocity vR=1 /kz12 /kz2 butwith different frequencies 12, the resonance velocityof the beating mode, vR = 1−2 / kz1−kz2, coincides withvR kz1 and kz2 are the wave vector components directedalong B0. Then a conventional perturbative approach to theprocess seems to be not adequate because the terms such asEq. 1 contain such singularities which are not considered inthe usual theory.

At last, the study of two waves interacting simultaneouslywith an electron beam should reveal a mechanism of nonlin-ear transformation of energy between the waves, through theparticles. If one considers the resonant interaction of a modu-lated flux of energetic particles or a periodic train of par-ticles’ bunches with two waves of different frequenciesand/or natures, there exists an interesting possibility to trans-fer the energy carried by one wave to the other one and, forexample, to transform electrostatic energy into electromag-netic energy, or inversely. At the same time, it is possible tomodify the waves’ frequencies and to increase their energydensities. In our opinion, the following cases are worthstudying: i both waves are in Landau resonance with a trainof electron bunches moving with the velocity vzvR=vR vRand vR are the resonance velocities of the two waves; thiscase is mentioned above as nonlinear induced scattering onelectrons, with vz1−2 / kz1−kz2; and ii one of thewaves interacts at Landau resonance 1=kz1vz with themodulated beam whereas the other one interacts with it atcyclotron resonance, 2nc=kz2vz c is the electron cy-clotron frequency; assuming that kz1=kz2 leads to the pro-cess of nonlinear Raman scattering at 2=1nc on Lar-mor rotators.

In a magnetized plasma, the double resonance conditionbetween two waves of resonance velocities vR1 and vR2 and agroup of particles, i.e., vzvR1vR2, can be realized in thefollowing particular cases. A first simple example corre-sponds, in a strongly magnetized plasma pc , p is theelectron plasma frequency, to a quasielectrostatic lower hy-brid wave and a right-polarized electromagnetic wave whis-tler with dispersion 1pkz1 / k1 at kz1 k1 and 2ckz2

2 c2 /p2 at kz2 k2, respectively k2 is the perpen-

dicular component of the wave vector k2 ; the double reso-nance condition vz1 /kz12 /kz2 can be fulfilled for kz1kz2 if the lower hybrid wave satisfies kz1k1c2p

3 /c andkz1 / k1p /c, which follows from c2k12p

2. Moreover,the parameters required to achieve the double resonance con-dition can be obtained more accurately by solving numeri-cally both the resonance interaction conditions and the wavedispersion equations. Figure 1 shows the linear dispersioncurves calculated for several types of waves propagating in acold magnetized plasma, for fixed c /p and k /kz k isthe perpendicular component of the wave vector. The twooblique lines labeled “L” and “AC” represent the Landau and

the anomalous cyclotron resonance conditions, kz= /vz andkz= +c /vz, respectively. The existence, for the samevalue of kz, of crossing points between these lines and thedispersion curves of the lower hybrid and the slow extraor-dinary wave branches labeled “LH” and “SX” illustratesthe possibility of finding two waves interacting simulta-neously with the same group of resonant particles.

In this paper, the double resonance problem is solved ana-lytically using a simple one-dimensional 1D Hamiltonianmodel describing the resonant interaction of two waves withthe particles they trap, supposing that the particles move allquasisynchronously in the potential wells of the waves. Thismodel allows one to describe the nonlinear characteristics ofthe process at work as well as to estimate analytically itstime scale. Moreover, these results are shown to fit qualita-tively those provided by three-dimensional 3D numericalsimulations performed by means of a symplectic code basedon a Hamiltonian model describing the 3D interactions ofwave spectra with resonant electrons in a magnetizedplasma. The paper presents several relevant examples ofphysical situations where the double resonance phenomenonoccurs; for example, when electrostatic waves interact i atLandau resonances with an electron beam bump-on-tail in-stability and ii at normal anomalous cyclotron reso-nances with energetic electron fluxes through ring fan in-stability. In all cases the simulations reveal that the timeevolution of the amplitudes of the two waves with closeresonant velocities exhibits some common features consist-ing of a slow and quasiperiodical exchange of energy be-tween the waves.

The paper is organized as follows. After a short presenta-tion of the theoretical 3D Hamiltonian model used to de-scribe resonant wave-particle interaction in a magnetizedplasma Sec. II, we focus, in Sec. III, on the situation in

FIG. 1. Dispersion relations in a cold magnetized plasma:ckz /p as a function of /p. The labels LH, SX, O, and Eindicate the lower hybrid, the slow extraordinary, the ordinary, andthe electromagnetic light =ckz modes dispersion curves,whereas L and AC represent the Landau and the anomalouscyclotron resonance conditions, respectively; the vertical lines indi-cate the lower and upper hybrid frequencies. The existence of cross-ing points black dots between the horizontal line ckz /p

=const and the curves illustrates the possibility to find a so-called”double resonance condition” for two different waves. The fixedparameters are k /kz=0.224 and c /p=0.75.

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which two waves are interacting simultaneously with thesame population of resonant particles. If the waves have thesame or very close resonant velocities, we show analyti-cally the occurrence of a nonlinear process characterized byquasiperiodical energy exchanges between the waves; theseresults are obtained using a Hamiltonian description wherethe electron population consists of bunches of trapped par-ticles moving quasisynchronously in the waves’ potentialwells. Section IV presents 3D numerical simulations ofwave-particle interactions under various double resonanceconditions, showing that the energy exchanges predicted inSec. III are observed. Finally, Sec. V is devoted to discussour results and to state our conclusions; in particular, thedependence of the time scale of the nonlinear wave oscilla-tions as a function of the electron flux intensity is considered.

II. THEORETICAL MODEL

The numerical simulation results and the analytical devel-opments discussed in the next sections are based on a theo-retical model 11,12 which describes in 3D geometry theevolution of waves resonantly interacting with particles in amagnetized plasma nonlinear wave-wave interactions areneglected compared to wave-particle ones. Such models, themain assumptions and properties of which will be exposed inthis section, have been first used in the field of plasma phys-ics for studying the nonlinear evolution of the 1D beam-plasma instability 13,15. They turned out to be of Hamil-tonian nature 16, which enables one to study the self-consistent wave-particle interaction as a classicalmultidimensional mechanical system, e.g., with the help ofperturbation and averaging techniques developed in theframe of Hamiltonian mechanics. Such methods were shownto be fruitful, allowing one to recover in a simple and naturalway several classical plasma physics results, as the Landaugrowth/damping rate, for example, but also to provide inter-esting physical results that cannot be recovered using theusual Vlasov-Poisson approach 17,18. A complete textbookdealing with such a Hamiltonian approach has been recentlypublished 19, in which the interested reader can find moredetails. The theoretical model exposed hereafter is an exten-sion of the 1D Hamiltonian model to 3D geometry with amagnetic field, the derivation and applications of which canbe found in 11,12,17,20. Let us now present the main fea-tures of this model.

We suppose that the plasma electrons can be divided intwo groups: a bulk component i.e., particles moving adia-batically in the field of the waves with the density n0, and aflux of energetic resonant particles with a much smaller av-erage density nresn0. The bulk component, which deter-mines the waves’ dispersion, is described in the linear ap-proximation using hydrodynamic equations. However, theresonant particles have to be considered owing to a kineticapproach which takes into account their full nonlinear dy-namics in the waves’ fields.

For electrostatic oscillations in a homogeneous magne-tized plasma, the electric field E is derived from the scalarpotential =Re kktexpik ·r−kt, which consists inthe sum of plane waves with slowly varying amplitudes

kt, so that the average electric field energy density for thewave k ,k is given by Ek

2 /8= kk2 /16; k is thefrequency of the wave with Fourier component Ek and wavevector k. Using a time averaging procedure to remove thefast oscillating terms from the Poisson equation 11, we ob-tain

k2kk k2 k

k − kk

k2 k

ki

d

dtk −

4e

V

V

eikt−k·rnresr,tdr ,

2

where nresr , t is the density of the resonant electrons ionsare considered as motionless; −e0 is the electron charge;k is the dielectric permittivity for electrostatic waves in acold magnetized plasma; V=LzL

2 is the volume of the sys-tem which consists of a periodic box of sizes Lz=2 /kz minand L=2 /k min along and across the ambient magneticfield B0=B0z, respectively z is the unit vector along B0; kzand k are the parallel and perpendicular wave vectors, andkz min and k min are their minimum values. Then the slowevolution of the wave amplitude kt due to resonant inter-action with particles is described by

id

dt

ek

mev2 −

p2

k2v2 k

k−12nres

n0

1

Np=1

N

eikt−k·rp, 3

where me is the electron mass; v is a fixed velocity used forthe normalization see below; and rp is the position of theparticle p. We substituted, in the Fourier transform of nres,the integral over the phase space volume by the sum on the Nresonant particles located in the volume V N=nresLzL

2 ,according to

V

eikt−k·rnresr,tdr

V=

V

d2rdz

LzL2 dvfresv,r,teikt−k·r

→1

Np=1

N

eikt−k·rp. 4

The distribution function fres of the resonant particles is nor-malized so that the average density is nres= fresv ,r , tdvdr. The motion of an electron p is describedby the Newton-Lorentz equation

dvp

dt+ vp zc =

e

me

=e

meRe

kikkteik·rp−kt,

drp

dt= vp, 5

where vpvp ,vzp is the velocity of the particle p. Equations3 and 5 lead to the conservation of the total energy H andparallel momentum Pz of the system in the volume V:

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d

dtH

=d

dtV nres

N

pmevp

2

2− e Re

kkeik·rp−kt +

kWk

= 0, 6

dPz

dt=

d

dtVk

kz

kWk +

nres

N

p

mevzp = 0, 7

where Wk=kk /kk2k2 /16 is the energy densityof the wave k ,k and mevp

2 /2−e Re kk expik ·rp−kt is the total energy of the particle p in the wave field.The dynamical system formed by Eqs. 3 and 5 can alsobe described using the Hamiltonian formalism developed inAppendix B see also Refs. 12,20, which allows one tobuild a symplectic numerical code for the efficient and accu-rate calculation of the wave-particle evolution. It allowed theauthors to perform convergent calculations that ensure theconservation of the dynamical invariants of the system with aprecision better than 10−3 whereas using time steps of theorder of c =0.1–0.2. The symplectic algorithm andmover are presented in Ref. 17.

III. ANALYTICAL INVESTIGATION OF THE DOUBLERESONANCE PROBLEM

The purpose of this paper, as emphasized in the Introduc-tion, is to study the situation in which two waves are inter-acting simultaneously with the same population of resonantparticles in a magnetized plasma. First note that the wave-particle interactions at Landau resonances can be consideredas 1D processes occurring along the direction of the ambientmagnetic field 17, contrary to interactions at cyclotronresonances for which the perpendicular particles’ motion isessential. Thus for reason of simplicity, we shall focus in thissection on 1D interactions, even if the full 3D Hamiltonianpresented in Appendix B allows one to treat the problem ofcyclotron resonances. Moreover, the study of the physicalprocesses at work in the 1D configuration should shed somelight on the nature of the wave energy exchange mechanismsin the more complicated case of 3D geometry. Indeed, in thenext section dealing with numerical simulations based on a3D Hamiltonian model, one can observe the qualitative simi-larity between the process in 1D and 3D geometries.

A classical situation in which two waves are interactingwith the same group of resonant particles in a plasma is thewell-known sideband instability. In this case, the steady statesystem formed by a main wave and the particles that it trapsknown as a Bernstein-Greene-Kruskal BGK wave 21 isperturbed by a second wave of significantly smaller ampli-tude. Analytical calculations can be performed in the linearapproximation 3–5, showing that the second wave alsocalled sideband wave can be driven unstable and determin-ing analytically its frequency and its maximum growth rate.In Appendix A we recover this classical result using the the-oretical model presented above, under the assumption that allthe particles trapped in a potential well are moving synchro-nously like a single macroparticle.

This assumption and the Hamiltonian nature of our modelallow us to go one step further and to consider the full non-linear interaction of two waves having nonvanishing ampli-tudes with a train of particle bunches. The results presentedbelow show that there exists a stable regime for this system,in which both waves periodically exchange energy. On thebasis of numerical simulations, we show in the next sectionthat these energy exchanges can be observed in more realis-tic situations.

A. Two waves interacting with a train of particle bunches

Let us consider the nonlinear evolution of two waves in-teracting resonantly with a periodic train of electron bunches,without supposing that the waves’ amplitudes are small. Inorder to derive analytical results, we simplify the problem byassuming that the motion of the particles in all the bunches issynchronized, which means that all the particles keep thesame phases relative to the waves: the resonant particles aremoving as one single macroparticle, so-called “bunch,” andare described by the same dynamical variables. This is thestrongest assumption done in the below developed model,and it is supported by the simulations results presented in thenext section; it can be true if the parallel wavelengths of thewaves are the same or if the ratio between them is an integer.

It is suitable to describe the 3D electron motion in theambient magnetic and waves’ fields with the help of the fol-lowing pairs of conjugated canonical variables see Appen-dixes A and B for details: the bunch’s longitudinal momen-tum Pb and position Zb in the laboratory frame, its angle ofLarmor rotation b, and its magnetic momentum Jb, as wellas its guiding center in the perpendicular plane,RbXb ,Yb /mec; the dynamics of each wave is de-scribed using the canonical action-angle variables I ,.

If one assumes that the two perpendicular wave vectorsare colinear, k1k2=0, the description of the perpendicu-lar motion of the bunch can be strongly simplified. Indeed,one observes in this case that k ·Rb=const =1,2, sothat the degree of freedom associated with the perpendiculardrift can be avoided. Moreover, the angles kx

=k sin , ky=k cos can also be included as initialconditions in the waves’ phases. Thus after some canonicaltransformations see Appendix B, the Hamiltonian Hb,n seeEq. B14 in Appendix B describing a system of two wavesinteracting at a selected resonance n with a bunch of elec-trons is

Hb,n =P −

kzI2

2Mb+ cJ +

=1,2 − ncI

− qb =1,2

2I1/2Jnkbcos , 8

with

= t − arg − kzZb − k · Rb − nb + , 9

where Mb=Nbme and −qb=−Nbe are the mass and the chargeof the bunch which contains Nb particles; P is a constant ofthe motion which represents the parallel momentum of the

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system see also Eq. B7 in Appendix B; Jn is the Besselfunction of order n; is a parameter see Eq. B4 in Ap-pendix B defined as

=8

Vk2 k,

−1

. 10

The bunch’s Larmor radius b in Eq. 8 can be expressed asfollows:

bI = 2Jb/Mbc1/2 = 2

MbcJ − n

I1/2,

11

where the generalized magnetic momentum J is a constantof the motion

J = Jb + nI = const. 12

One can see from Eq. 8 that the parallel kinetic energy ofthe bunch plays the role of a wave-wave interaction energyand that the coherent motion of the bunched particles canlead to some coupling between the waves. In the case of theLandau resonance, n=0, one has Jb=J=const and thus b=const. Moreover, supposing that the perpendicular wave-lengths are greater than the bunch gyroradius b kb1,i.e., J0kb1, one can describe the system by theHamiltonian Hb,0 see Eq. B18 in Appendix B,

Hb,0 =P −

kzI2

2Mb+ cJ +

=1,2I

− qb2I1/2cos , 13

with =t−arg−kzZb−k ·Rb. Then the Hamiltonequations provide that

dI

dt= − qb2I1/2sin , 14

d

dt= −

kz

MbP −

=1,2kzI − qb/2I1/2cos .

15

B. Equilibrium and small oscillations of the system

The conditions dI /dt=0 and d /dt=0, which deter-mine the stationary points of Eqs. 14 and 15, impose thatsin 1,2=0, i.e., 1,2=0 or 1,2=; only the solution 1,2=0 should be retained as the bunch is considered to betrapped near the bottom of the resulting potential well stableequilibrium is considered. Then the equilibrium pointI1eq , I2eq is obtained from

−kz

MbP − kz1I1eq − kz2I2eq − qb/2Ieq1/2 = 0,

= 1,2. 16

Assuming that the potentials at equilibrium, eq= 2Ieq1/2, satisfy 1eq2eqeq and that the fre-quency shift is small compared to ,

= − qb/2Ieq1/2 = − qbeq−1, ,

17

one can write v=2 /kz2−1 /kz1 in the form

v = qb1/2kz12 I1eq1/2 − qb2/2kz2

2 I2eq1/2, v v ,18

where v is the parallel phase velocity of the wave . ThenEq. 16 imposes that both waves have very close phase ve-locities i.e., whose difference v is of the order of v.To the zeroth order in v /v, the equilibrium condition isgiven by v1=v2, which corresponds to

I1eq

I2eq=

1

2

kz22

kz12 . 19

Neglecting compared to and defining the bunch equi-librium velocity as

Veq = P − kz1I1eq − kz2I2eq/Mb, 20

one obtains from Eq. 16 that

− kzVeq 0. 21

Combining Eqs. 19–21, we get the waves’ actions Ieq0 atequilibrium v=0

I1eq0 =1kz2

kz1

P − Mbv1

2kz1 + 1kz2,

I2eq0 =2kz1

kz2

P − Mbv2

2kz1 + 1kz2. 22

Let us now determine the characteristic frequencies of theoscillations in the system and therefore examine the correc-tion to the zero order stationary state. So let us take intoaccount the small difference in phase velocities v v and assume that Ieq= Ieq0+Ieq, with Ieq Ieq0. Defining 0=−qb /2Ieq01/2 and v0=0 /kz, one obtains from Eq. 18 that

v v201 − I2eq/2I2eq0 − v101 − I1eq/2I1eq0 ,

23

+ 01 − Ieq/2Ieq0 − kzVeq 0, 24

which leads to

kz1I1eq + kz2I2eq − Mbv01 − Ieq/2Ieq0 . 25

Using Eqs. 23 and 25, and neglecting the third-order termproportional to v20v10I1eq, we find Ieq,

Ieq

Ieq0

2kzIeq0v − v20 − v10 − Mbv20v10

kz1v20I1eq0 + kz2v10I2eq0,

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, = 1,2, . 26

So, in the vicinity of the stationary point, the waves’ ampli-tudes perform small oscillations described by the linearizedequations I= Ieq+I and =eq+

d

dtI − qb2Ieq1/2, 27

d

dt

kz

Mbkz1I1 + kz2I2 + qb/81/2Ieq

−3/2I,

28

which can also be written as

d2

dt2 +qb

21

2I1eq+

qbkz12

Mb21I1eq1/21

−qbkz1kz2

Mb22I2eq1/22, 29

d2

dt2 +qb

22

2I2eq+

qbkz22

Mb22I2eq1/22

−qbkz1kz2

Mb21I1eq1/21. 30

Introducing the bounce frequencies of the bunched particlesin the waves’ potential as

tr2 =

ekz2

me2Ieq1/2 =

ekz2

meeq , 31

and defining the frequencies

x2 =

qb2

2Ieq= 2

p2

tr2

kz2

k2 k,

−1nres

n0

Nb

N2

,

32

let us search oscillatory solutions of Eqs. 29 and 30 in theform =0eit. Then

x12 + tr1

2 − 210 −kz2

kz1tr1

2 20, 33

x22 + tr2

2 − 220 −kz1

kz2tr2

2 10, 34

providing that

x12 + tr1

2 − 2x22 + tr2

2 − 2 tr12 tr2

2 . 35

The solutions are linear combinations of oscillationswith frequencies which are roots of Eq. 35, that is, whichsatisfy

22 x12 + tr1

2 + x22 + tr2

2

x12 + tr1

2 − x22 − tr2

2 2 + 4tr12 tr2

2 . 36

Taking into account that x2 tr

2 , a Taylor developmentprovides the two roots of Eq. 35,

12 tr1

2 + tr22 , 37

22

x12 tr2

2 + x22 tr1

2

tr12 + tr2

2 12. 38

The first regime 37 corresponds to particles oscillating atthe frequency 1 in the potential well formed by the twowaves. The slower oscillations, which occur at a much lowerfrequency 21, reveal a more interesting phenom-enon: it describes the low frequency exchanges of energybetween the two waves.

C. Numerical solutions

In order to illustrate the behavior of the system over awider range of parameters, we solved numerically Eqs. 14and 15 for initial conditions when the bunch is trapped inthe potential well of two waves with close kz, that is, whenP=Nbmev+kzI10+ I20, with I0= It=0. All the particlesare considered to be included in the bunch, i.e., Nb /N=1.Close to the equilibrium, i.e., when I10 I20 and 10200 0=t=0, one observes the behavior predicted bythe linearized equations 27 and 28: the waves’ energiesIt oscillate at the frequency 2 while they are perturbedby small amplitude trapping oscillations at the frequency 1.

In order to point out the influence of the bunch’s phaserelative to the waves, Eqs. 14 and 15 have been solvednear the equilibrium point, when varying the difference ofthe phases of the two waves at the initial time, 0= 20−10 see Fig. 2. The results show that the linear approxi-mations 27 and 28 are valid over a large interval aroundthe equilibrium 0=0. For 0, the particle streamsas nearly free the parallel electric field seen by the bunch isvanishing: no periodic oscillations take place and 2 cannotbe defined anymore. As predicted from the solution of thelinearized equations, the variation I of the waves’ actionduring the slow energy exchanges increases with 0:when solving Eqs. 27 and 28 with the initial conditionsI10= I20 and 0=20−10 one obtains that IMb1

20 /2kz2 kz=kz1=kz2, which fits well with the nu-

merical results obtained within the range − /20 /2, as shown by Fig. 2. I is maximum when all theelectrostatic energy carried by the waves is exchanged be-tween them through the slow oscillatory process, i.e., whenIImax=2Ieq; it follows from Eq. 22 that

I1eq = I2eq = Ieq I10 + I20/2. 39

The oscillations with IImax, which are far from theequilibrium state where I=0, occur anyway with a fre-quency which is very close to that predicted by the lineariza-tion and take place for the initial condition 0 /2 seeFig. 2.

Figure 3 shows the trajectories in the phase plane , Ifor =1. Close to the equilibrium point Fig. 3, left pannel,the behavior of the phase and the action is quasiregular andone can observe small amplitude fast oscillations due to themotion of the trapped particles in the bunch as well as largeramplitude oscillations corresponding to the slow energy ex-changes between the waves. Near the point 0 /2 Fig.3, right pannel, the trajectory , I is more complicated: increases almost linearly with trapping oscillations of

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very small amplitude and consequently the action I be-haves as a square cosinus function as expected from Eq.14. When → /2, i.e., I→0 and dI /dt0, as I ispositive, the phase jumps to the value =− /2 due to theterm proportional to /2I1/2cos 2 in Eq. 15.

IV. NUMERICAL SIMULATIONS

On the basis of the analytical investigation presented inthe previous section, we now perform numerical simulationsinvolving two electrostatic waves having the same, or veryclose, resonant velocities and interacting with various elec-tron velocity distributions, for different instabilities and reso-nance conditions. The main attention will be paid to thephysical features which are of special interest for comparisonwith the estimates provided by the analytical developmentspresented in the previous section.

In the numerical simulations, the variables are normalizedas follows:

k → kv/,

k → k/,

k → ek/mev2, 40

where depends on the nature of the waves considered.Then one can present the equation of wave amplitude evolu-tion 3 in normalized variables as

d

dtk = i

pk

k21

Np=1

N

eikt−k·rp, 41

where p is a dimensionless parameter, proportional tonres /n0k /k−1, which characterizes the intensity of thewave-particle interaction.

Electrostatic upper and lower hybrid waves are consid-ered in the simulations. Their dispersion relations can beobtained from the dielectric permittivity k of a magnetizedplasma here thermal effects are neglected by solving theequation kk ,k=0 22, where

−1 −0.5 0 0.5 10.01

0.02

0.03

0.04

0.05

∆φ0/π

Ω2

−1 −0.5 0 0.5 10

5

10

15

20

25

30

∆φ0/π

∆I

Ω2lin

∆I=2Ieq

FIG. 2. Two waves of close kz interacting with a bunch of par-ticles at Landau resonance. Upper panel: normalized frequency 2

characterizing the slow energy exchanges between the two wavescalculated by solving numerically Eqs. 14 and 15 as a functionof the difference of phases of the waves at the initial time,0 /= arg20−arg10 /; the horizontal line represents thenormalized frequency 2lin calculated owing to Eqs. 29 and 30.Lower panel: normalized variation I of the waves’ action duringthe slow energy exchanges, as a function of 0 /; the horizontalline represents the condition I=Imax=2Ieq, when all the energy isexchanged between the waves; the straight oblique lines correspondto the condition I=Mb1

20 /kz22 also see the text. The main

normalized parameters are Mb=1, qb=1, kz1=kz2=kz=1, 1=2

=1, 1=2=0.01, and 10= 20=0.6 i.e., Ieq12.5; arg10=0 and arg20 is varied.

−0.05 0 0.058

10

12

14

16

φ1/π

I 1

−0.4 −0.2 0 0.2 0.4 0.60

5

10

15

20

25

30

φ1/π

I 1

FIG. 3. Color online Two waves of close kz interacting with a bunch of particles at Landau resonance. Trajectories in the phase plane1 , I1 showing the amplitudes of the first wave’s phase and action oscillations. Left panel: the waves’ initial phases are arg10=−arg20=0.1 and the system is close to its equilibrium point 0=0. Right panel: the waves’ initial phases are arg10=−arg20= /4 and the system is near the point 0 /2. The main parameters are the same as in Fig. 2.

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k = 1 −p

2

k2

kz2

k2 +p

2

c2 − k

2

k2

k2 −pi

2

k2 . 42

For a weakly magnetized plasma pc, the upper hybridwaves propagate with the dispersion

k2 p

21 +c

2

p2

k2

k2 , 43

and k /k2 /k. In this case, =p and the corre-sponding parameter p in Eq. 41 is

p =nres

n0 1. 44

Lower hybrid waves propagate in the frequency range lh=pic / c

2+p21/2kc at arbitrary c /p with the

approximate dispersion relation

k2

c2

p2

c2 + p

2

kz2

k2 , 45

and k /k2c2+p

2 /kc2, so that, choosing the nor-

malization frequency =c in Eq. 41, we have

p =p

2

c2 + p

2

nres

n0 1. 46

As shown by the form of the dispersion relations 43 and45, and due to the azimuthal symmetry of the magnetizedplasma considered, all the waves which have the same kz andk have also the same resonance velocity vR= k−nc /kz. It is clear that small differences between the fre-quencies and the perpendicular wave vectors of such wavescan exist without violating the equality of their resonancevelocities. Various electron distributions have been chosen inorder to study wave excitation by different instabilities andresonance conditions; we considered mainly the case whenthe initial parallel distribution fzvz of the resonant particlesconsists in a beam. Figures 4 and 5 show the distributionfzvz resulting from the particles’ interaction with two waves1 ,k1 and 2 ,k2; its evolution with time is typical of the

so-called bump-on-tail instability of a single wave in thehydrodynamic regime see, e.g., 1. Thus the velocity dis-tribution at the saturation stage appears not to depend essen-tially on the fact that two waves and not only one are in-teracting with the beam at the same resonant velocity.

At this step, let us remind some relevant features charac-terizing the spatial self-organization of the resonant particlesduring their interaction with a single wave. If the instabilityis of hydrodynamic type, most of the trapped particles form atrain of bunches periodically distributed in space. All par-ticles have close phases relative to the wave and oscillatesynchronously near the bottom of its potential well. Whenthe instability is of kinetic type, only a small part of the totalamount of trapped particles forms the bunch and most ofthem oscillate in the well with different frequencies; aftersome time, phase mixing occurs and the particles stop toexchange energy with the wave on average. In this state theamplitude of the wave is roughly constant and it can beconsidered as a Bernstein-Green-Kruskal mode see, e.g.,21. For intermediate cases between the hydrodynamic andthe kinetic regimes, a relatively small amount of particlesmove in the well with correlated phases, forming a long timeliving bunch, which in turn supports the wave’s amplitudeoscillations around a saturation level in the asymptotic inter-action stage. In our paper, we consider mainly the situationwhen a significant part of the trapped particles form a well-structured bunch quasihydrodynamic regime.

Figure 6 shows the variation with time of the square fieldamplitudes Ek2 and of the phase mismatch k=argk1

−argk2

of two lower hybrid waves 1 ,k1 and 2 ,k2;note that k is equal, with accuracy of a constant, to thedifference of the generalized phases 1,2 defined by Eq. 9.The initial values of k have been chosen arbitrarily anddiffer for each simulation. Due to the beam instability, bothwaves’ amplitudes grow exponentially and saturate due totrapping of electrons 14,15. If the two waves behave inde-pendently, the time variation of Ek2 should present, for eachwave, small trapping oscillations around a roughly constantsaturation level. Instead of this, as the two resonant wavevelocities are equal vR=vR1−vR2=2 /kz2−1 /kz1=0, oneobserves the nontrivial interaction predicted in Sec. III be-

1 2 3 4 50

0.02

0.04

0.06

vz

f z(vz)

FIG. 4. Color online Beam instability at the Landau resonance:superposition of the initial solid curve and the final gray patternelectron parallel velocity distributions fzvz; vz is the normalizedparallel velocity. The vertical line indicates the normalized resonantvelocities vR1=vR2=2.2 of the two lower hybrid waves. The nor-malized wave vectors and frequencies are k1= 0.3,0 ,0.3, k2

= −0.3,0 ,0.3, and 1=2=0.67; p=0.03 Eq. 46; =c;p /c3; and N=200 000.

8 10 12 140

0.5

1

vz

f(v z)

FIG. 5. Color online Beam instability at the Landau resonance:superposition of the initial solid curve and the final gray patternelectron parallel velocity distributions fzvz; vz is the normalizedparallel velocity. The vertical lines indicate the resonant velocitiesof the two upper hybrid waves interacting with the electrons. Thenormalized wave vectors and frequencies are k1= kx1 ,ky1 ,kz1= 0.2,0.333,0.125, k2= kx2 ,ky2 ,kz2= −0.2,0.333,0.125; 1

=2=1.4; p=0.05 Eq. 44; =p; and N=100 000.

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tween the waves and the resonant particles, that is, the de-velopment of quasiperiodic exchanges of energy between thetwo waves. The phase mismatch k also performs oscilla-tions, with the same periodicity, around a constant value. Thesame process of energy exchanges between two waves hasbeen evidenced for upper hybrid waves interacting with elec-trons at Landau resonance, as shown by Fig. 7.

This energy exchange process is sensitive to the level ofnonlinearity i.e., to the flux intensity p defined in Eqs. 44and 46 as well as to other parameters, as the direction ofthe wave vectors in the plane perpendicular to B0 or thevelocity mismatch vR=vR1−vR2. Figure 8 shows the timevariation of Ek2 for several values of vR, revealing differ-ent oscillatory behaviors, whose periods decrease when vR

0 500 10000

0.01

0.02

ωct/2π

|Ek|2

0 500 1000

−1.4

−1.2

−1

−0.8

ωct/2π

∆φk/2

π

0 500 10000

0.02

0.04

0.06

ωct/2π

|Ek|2

0 500 1000

−1.4

−1.2

−1

−0.8

ωct/2π

∆φk/2

π

0 500 10000

0.05

0.1

0.15

ωct/2π

|Ek|2

0 500 1000−1.5

−1

−0.5

0

ωct/2π

∆φk/2

π

FIG. 6. Beam instability at the Landau reso-nance: variation as a function of the time ct /2of the normalized square field amplitudes Ek2left panels and of the relative phases k /2right panels of two lower hybrid waves, for p

=0.02, 0.03, and 0.05 from the upper to thelower panels, respectively. Both waves have thesame phase velocity, vR=vR1−vR2=0. The firstwave 1 ,k1 the second wave 2 ,k2 is rep-resented as a black solid thin line. The electronvelocity distribution and the waves are the sameas in Fig. 4.

0 100 2000

0.2

0.4

ωpt/2π

|Ek|2

0 100 200

−1.2

−1

−0.8

−0.6

−0.4

ωpt/2π

∆φk/2

π

0 100 2000

0.02

0.04

0.06

ωpt/2π

|Ek|2

0 100 2000.2

0.4

0.6

0.8

ωpt/2π

∆φk/2

π

0 50 100 1500

0.5

1

ωpt/2π

|Ek|2

0 50 100 150

0

2

4

ωpt/2π

∆φk/2

π

FIG. 7. Beam instability at Landau resonance:variation as a function of the time pt /2 of thenormalized square field amplitudes Ek2 leftpanels and of the relative phases k /2 rightpanels of two upper hybrid waves, for three dif-ferent cases: i upper panels: normalized fre-quencies and wave vectors are 1=2=1.4,k1= 0.2,0.333,0.125, k2= −0.2,0.333,0.125,and vR=0, p=0.05, N=100 000; ii middlepanels: same as i but p=0.02; and iii lowerpanels: the two waves present a small phasevelocity mismatch vR0.05, with the normal-ized phase velocity vR1,2=1,2 /kz1,211.2;1=1.39, 2=1.4, k1= 0.2,0.286,0.125, k2

= 0.4,0.143,0.125; p=0.1; and N=100 000.The electron velocity distribution is the same asin Fig. 6.

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increases. Note that for an extremely small value of vRvR /vR0.005, the period of the energy exchanges is di-vided by a factor of 2 compared to the case when vR=0,and the oscillations are no more symmetric. For larger valuesthat is, when vR /vR0.01–0.02, the waves’ amplitudesperform oscillations of smaller amplitude and period—different from the trapping oscillations—around some aver-age value. This feature, as well as the dependence on p, arecoherent with the analytical results presented above, basedon the hypothesis that a coherent particle bunch formed dur-ing the instability process 15 is simultaneously trapped inthe potential wells of the two waves.

One should note that the oscillatory behavior which takesplace during the saturation stage of the wave-particle insta-bility does not depend on the kind of waves considered in thesimulations: the same qualitative behavior is observed for theupper hybrid waves Figs. 5 and 7 and for the lower hybridwaves Figs. 4, 6, and 8. Moreover, even if this case has notbeen explicitly investigated in the previous section, simula-tions of lower hybrid waves destabilized at cyclotron reso-nances by other types of instabilities have also been per-formed. Relevant results are sketched in Figs. 9–12 for thefan ring instability at the anomalous normal cyclotronresonance see 11,12,23 and 24, showing similar behav-iors of the wave-particle system: after a stage of exponentialgrowth, the instability saturates due to trapping of resonantparticles, after which very slow compared to the bounceperiod tr

−1 exchanges of energy occur between the twowaves. Note that the simulation results agree with those pro-vided by the numerical solution of the equations presented inthe previous section, for the case when the waves exchangebetween each other almost all their energy.

Finally, note that in Figs. 6–8 and 10 the saturation levelof one of the waves’ amplitudes just after the linear stage ofthe instability is much larger than that of the other wave.This feature can be explained by the mechanism of particletrapping: when a wave gets enough energy to trap a largeamount of electrons, the perturbations of their motion due totrapping cause the linear growth of the other wave to saturateas both waves are sharing the same population of resonantparticles. The random fluctuations Schottky noise that de-termine the amplitude of the waves before the instabilityoccurred are responsible for the fact that one wave trapselectrons before the other. Anyway, it occurs also that bothwaves saturate around the same level of energy, in particularfor the fan instability case, as shown in Fig. 11.

V. DISCUSSION AND CONCLUSION

The same oscillatory behavior characterizing the variationwith time of the waves’ amplitudes has been shown, at leastqualitatively, to result from the 3D simulations and from thenumerical solution of Eqs. 14 and 15 for 0 /2i.e., when the waves exchange between each other almost alltheir energy. Let us now compare in more details the ana-lytical results derived in Sec. III and the numerical simula-tions of Sec. IV. One of the most interesting parameters is thetime period T2=2 /2 characterizing the slow energy ex-changes between the waves. In order to obtain clear power

laws for T2 as a function of the basic physical parameters, letus choose two waves which differ one from the other only bythe direction of their perpendicular wave vectors k sothey have the same kz, , and k.

For lower hybrid waves, Eqs. 10, 45, and 46 lead to

=4

V

c2

c2 + p

2

k2 =

4n0

N

c2

p2

p

k2 , 47

so that the frequency 2 defined in Eq. 38 can be expressedas

2 =qb

eq=

c2me

eeqp

k2

Nb

N, 48

where we took into account that qb=eNb and eq= 2Ieq1/2= 102+ 202 /21/2.

0 200 400 600 800 10000

0.05

|Ek|2

0 200 400 600 800 10000

0.05

|Ek|2

0 200 400 600 800 10000

0.05

|Ek|2

0 200 400 600 800 10000

0.05

|Ek|2

0 200 400 600 800 10000

0.05

ωct/2π

|Ek|2

FIG. 8. Beam instability at Landau resonance: variation as afunction of ct /2 of the normalized square field amplitudes Ek2of two lower hybrid waves, for different resonant velocity mis-matches vR between the waves. The parameters are the same as inFig. 4, except of the second wave for which the perpendicular wavevector k2= kx2 varies as follows from the upper to the lowerpanel: kx2=−0.300, −0.303, −0.305, −0.307, and −0.310, corre-sponding to vR /vR=0, 0.005, 0.008, 0.011, and 0.016, respec-tively. The first wave 1 ,k1 the second wave 2 ,k2 is repre-sented by a solid thin line. The electron velocity distribution andthe waves are the same as in Fig. 4.

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First, in order to determine the power of the scaling lawT2 p

using the 3D numerical simulations, we consideredthe simple case when the two waves have finite initial am-plitudes i.e., much larger than the wave thermal energy orthe Schottky noise, whereas an electron beam—with a smallenough density to avoid wave amplification—is launchedwith a velocity equal to the waves’ phase velocities. In thiscase, the waves’ amplitudes are oscillating regularly and Fig.13 shows the variation of cT2 /2 as a function of p, theinitial waves’ amplitudes being kept constant when varyingp. The dots represent the results provided by the 3D numeri-cal simulations, whereas the straight line indicates the ana-lytical estimate derived from Eq. 48, where Nb /N0.08has been chosen so that the theoretical curve fits quantita-tively the simulation data. When the formed bunch resultsfrom the instability of a beam initially homogeneous inspace, not all the bunched particles are moving with a perfect

synchronism, some of them are being totally phase mixed.So Nb /N should be understood as the approximate proportionof bunched particles with respect to the total number of par-ticles. Note that in this case, the scaling T2 p

−1 is verified,under the assumption that Nb /N does not depend on p.

The more complicated situation, when the energy ex-changes between the waves take place during the saturationstage of the instability, has also been investigated numeri-cally. In this case, the bunch is created by the instabilitymechanism and the motion of the bunched particles remainscoherent during a long time i.e., compared to the bounceperiod of the trapped particles in the waves’ potential well.As discussed previously, the waves’ action at equilibrium,Ieq, is approximately equal to the average value of the satu-rated waves’ action i.e., Eq. 39 is still valid when replacingI0 by Is, where Is is the action of the wave at satura-tion, as the bunch formed by the trapped particles obviously

3 4 5 60

0.02

0.04

0.06

0.08

vz

f z(vz)

3 4 5 60

2

4

6

8

vz

v ⊥

200

400

600

800

FIG. 10. Color online Ring instability at the normal cyclotron resonance. Left panel: superposition of the initial solid curve and thefinal gray pattern velocity distributions fzvz. The vertical line indicates the normalized resonance velocity of the two lower hybrid waves,vR1=vR2= k−c /kz=4.1. The initial distribution function consists of a beam drifting along the parallel and perpendicular directions withthe normalized drift velocities vzb=4 and vb=5, respectively; corresponding thermal velocities are vthz=vth=0.7. Right panel: phase spacevz ,v at the final simulation time. The parameters are the same as in Fig. 9, for p=0.05.

0 500 10000

0.02

0.04

0.06

ωct/2π

|Ek|2

0 500 1000

0

0.2

0.4

0.6

0.8

ωct/2π

∆φk/2

π

0 500 10000

0.05

0.1

ωct/2π

|Ek|2

0 500 1000

0

0.2

0.4

0.6

0.8

ωct/2π

∆φk/2

π

0 500 10000

0.1

0.2

ωct/2π

|Ek|2

0 500 1000

0

0.2

0.4

0.6

0.8

ωct/2π

∆φk/2

π

FIG. 9. Ring instability at the normalcyclotron resonance: variation as a function ofct /2 of the normalized square field amplitudesEk2 of two lower hybrid waves left panels andof their relative phases k /2 right panels,for p=0.04, 0.05, and 0.08 from the upperto the lower panels, respectively. The firstwave 1 ,k1 the second wave 2 ,k2 is rep-resented by a solid thin line. The normalizedparameters are 1=2=0.41, k1= 0.3,0 ,−0.14,k2= −0.3,0 ,−0.14, p /c=3, and N=200 000.

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moves with a parallel velocity approximately equal tov= /kz, at the bottom of the potential well of the satu-rated waves.

Let us now determine the scaling law of T2 using thenumerical simulations of two unstable driven lower hybridwaves interacting with a train of bunches. One can expectthat the equilibrium values Ieq of the waves’ actions at satu-ration should be related to the flux intensity p 11,12. Theupper lower panels of Fig. 14 present the variations of T2keq

2 as a function of p, for beam, fan, and ring instabili-ties, showing that the results are coherent with the estimatesof Ref. 11, obtained using the conservation of the parallelmomentum of the wave-particle system B7 and the as-sumption that the linear growth rate of the instability mustvanish at wave saturation. For the fan instability, the simula-tion results are in good agreement with the analytical predic-

tions, that is, keq2 = k1

s2+ k2

s2 /2 p

3 ks is the poten-tial at saturation and T2 p

−1. For the beam and the ringinstabilities, two regimes have to be distinguished. Indeed,for p0.01, only a small part of the beam particles stronglyinteract with the waves kinetic limit of the wave-particleinteraction; for larger values of p p0.03, nearly all theparticles are trapped coherently by the waves hydrodynamiclimit when all the particles are moving like a single macro-particle in the waves’ potential well. For the beam instabil-ity, no simple power law can be determined: for large p,the numerical simulations provide that keq

2 p1.7, which

is not far from what is obtained analytically by consideringa purely hydrodynamic instability i.e., keq

2 p4/3 11;

note that our values of p are not small enough to allow usto investigate the range of purely kinetic regime wherekeq

2 p4 11. For the ring instability, the power laws

4 5 6 70

0.01

0.02

vz

f z(vz)

4 5 6 70

0.5

1

1.5

2

vz

v ⊥

100

200

300

FIG. 12. Color online Fan instability at the anomalous cyclotron resonance. Left panel: superposition of the initial solid curve and thefinal gray pattern velocity distributions fzvz. The vertical line indicates the normalized resonance velocity of the two lower hybrid waves,vR1=vR2= k+c /kz=5.6. The initial distribution function consists in a nearly flat suprathermal tail extending from vz=4 to 7 in theparallel direction and in a Maxwellian of thermal velocity vth=0.7 in the perpendicular direction. Right panel: phase space vz ,v at thefinal simulation time. The parameters are the same as in Fig. 11, for p=0.03.

0 1000 20000

0.5

1

1.5

x 10−4

ωct/2π

|Ek|2

0 1000 2000

−0.2

−0.1

0

0.1

ωct/2π

∆φk/2

π

0 1000 20000

2

4

6x 10

−4

ωct/2π

|Ek|2

0 1000 2000

0.8

1

ωct/2π

∆φk/2

π

0 1000 20000

1

2

3

x 10−3

ωct/2π

|Ek|2

0 1000 2000

−0.2

−0.1

0

0.1

ωct/2π

∆φk/2

π

FIG. 11. Fan instability at the anomalous cy-clotron resonance: variation as a function ofct /2 of the normalized squared field ampli-tudes Ek2 of two lower hybrid waves left pan-els and of their relative phases k /2 rightpanels, for p=0.02, 0.03, and 0.05 from theupper to the lower panel, respectively. The firstwave 1 ,k1 the second wave 2 ,k2 isrepresented by a solid thin line. The norma-lized parameters are k1= 0.3,0 ,0.3, k2

= −0.3,0 ,0.3, 1=2=0.67, p /c=3, and N=200 000.

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keq2 p

2.5 and keq2 p

1.2 obtained numerically for the ki-netic and hydrodynamic regimes are not very far from thecorresponding theoretical estimates, that is, keq

2 p3 and

keq2 p

2, respectively 11.

Moreover, one can write using Eq. 48 that

2p p

eqpNbp

N, 49

and, owing to the power laws obtained above, one can deter-mine the scaling of T2. Then, for the beam instability, theperiod T2bT2 scales as

T2b,h p−0.15 N

Nbp,

T2b,k p0.25 N

Nbp, 50

where the subscripts h and k indicate the hydrodynamic andthe kinetic regimes, respectively. Supposing that Eq. 48 isalso valid for instabilities taking place at the cyclotronresonances—we are aware that this assumption is not obvi-ous at all—one gets for the fan instability T2f T2

T2f ,k p0.5 N

Nbp, 51

and for the ring instability T2rT2

T2r,h p0.4 N

Nbp,

10-2 10-110

2

103

ωcT

2/2π

Beam

10-2 10-1

100

10-1

10-2

|k|2 eq

p

10-2 10-110

2

103

Fan

10-2 10-1

10-2

10-1

10-3

p

10 -110 -210

2

103

Ring

10 -2 10 -1

100

10-1

10-2

p

ppp-p

pp

p-1.5 p-1.5

p2.5

p1.7

p3

p2.5

p1.2

*

*

* * *

*

* *

*

*

*

*

*

ϕ

-1-1

0.3

FIG. 14. Color online Interaction between two lower hybrid waves and a nonequilibrium electron distribution. Upper panels: variationof cT2 /2 as a function of p, for different kinds of instabilities: beam, fan, and ring, as indicated in the figure. Lower panels: variation ofthe normalized mean value keq

2 around which the waves’ amplitudes are oscillating at saturation, as a function of p, for the three types ofinstabilities. Corresponding p

dependences are indicated as straight lines all the axes have logarithmic scales. The main parameters are thesame as in Fig. 4 for the beam instability, Fig. 9 for the ring instability, and Fig. 11 for the fan instability.

10-3 10-2

102

103

p

ωcT

2/2

π

*

FIG. 13. Color online Interaction between two lower hybridwaves and an electron beam: variation of the normalized periodcT2 /2 of the energy exchanges between the waves as a functionof the flux intensity p Eq. 46. The dots correspond to the resultsprovided by 3D numerical simulations and the straight line to thetheoretical estimate of T2 Eq. 48 Nb /N being fixed to 0.08,as explained in the text. The initial waves’ amplitudes are 10=1.0 and 20=0.1. The main normalized parameters are k1

= 0.2,0 ,0.1, k2= −0.2,0 ,0.1, and 1=2=0.42.

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T2r,k p0.2 N

Nbp. 52

In order to recover the scaling laws obtained for T2 as aresult of the numerical simulations see Fig. 14, one hasconsidered that the ratio Nbp /N should lie within therange

p1.5

NbpN

p0.7, p 1, 53

an assumption which is reasonable, as for p=0.1 p=0.01,0.03Nb /N0.2 0.001Nb /N0.04.

As a conclusion, the simple model of two waves interact-ing resonantly with an electron bunch, which was presentedin Sec. III, is shown to be in good qualitative agreement withthe results provided by the 3D numerical simulations and todescribe adequately the nontrivial dynamics of bunched reso-nant particles interacting simultaneously with two waves ofclose phase velocities. This result allows us to expect that thecontrol of the processes of periodic exchanges of energy be-tween waves via a common group of particles strongly inter-acting with them should lead to interesting applications, asthe transformation of the energy carried by an electrostaticwave into electromagnetic radiation. Indeed, as explained inthe Introduction, so-called “double resonance conditions”can be found for waves of different natures and are capableof different interaction mechanisms with resonant particles.

ACKNOWLEDGMENTS

The authors acknowledge the Centre National de la Re-cherche Scientifique CNRS, France and the Russian Acad-emy of Sciences Grant No. 06-02-72560-NCNIL_a fortheir financial support.

APPENDIX A: INSTABILITY OF A WAVEWITH TRAPPED PARTICLES

Here we suppose that the particles’ motion can be de-scribed in the frame of 1D geometry and limit ourselves tothe case of the Landau resonance, =kzvz. The motion of anelectron in the potential =Re1e−i1t−k1z+2e−i2t−k2z oftwo waves 1 ,k1 and 2 ,k2 is described by

dpz

dt= me

dvz

dt= e Reik11e−i1−k1vz0t+i1

+ ik22e−i2−k2vz0t+i2 ,

dz

dt= vz, A1

where =kz−vz0t is the phase of the particle relatively tothe wave in the coordinate system moving with the phasevelocity vz0=1 /k1. Defining the electron velocity perturba-tion, k1vz=k1vz−vz0=d1 /dt, and expressing 2 with thehelp of 1, we get

d2

dt2 =ek1

2

meRei1ei + i

k2

k12eit+ik2/k1 , A2

where =k2vz0−2. It is suitable to add the index p to inorder to distinguish each particle one from the other, and tointroduce the notations

= ei,

tr2 =

ek2 me

. A3

So, defining =kvz0−, and with the help of Eq. 3see text, we get

d2p

dt2 = tr12 Reiei1+p + tr2

2 Reik1

k2ei2+t+pk2/k1 ,

A4

d

dt i

8e

k2 /

nres

N p

e−it+pk/k1, A5

where is the dielectric constant of the wave . Let usconsider an instability of the wave 1 ,k1, supposing that itremains in a steady state with the trapped particles and that itcan be considered as a nonlinear BGK wave 21. Below,only some properties of such a solution will be examined andused in the instability analysis. So, neglecting the influenceof the second wave 2 ,tr2→0, the following equationshave to be satisfied:

d2p

dt2 tr12 Reiei1+p , A6

d

dt+ i

d1

dt1 − i

8e

k12 1/1

nres

N

p

e−ip+1 0.

A7

Further we suppose that the bunches are well-formed, i.e.,that the oscillations of the trapped particles are small, p+11, so that in the first approximation the solution of Eq.A6 can be presented as oscillations at the bounce frequencytr1

p + 1 ap costr1t + p, ap 1, A8

where the amplitudes ap and the phases p p=1,N aredistributed to satisfy Eq. A7, that is

1d1

dt

8enres

k12 1/1

,

d

dt1

8enres

k12 1/1

1

N

p

p + 1 . A9

Thus, for a steady state, the main wave 1 ,k1 acquires anonlinear frequency shift

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d1

dt= nl

8enres

1k12 1/1

, A10

which is due to the trapped particles. Note that there is nounique correspondence between the frequency shift nl andthe wave amplitude1. This fact is related to the features ofthe BGK mode solutions 21. Then, according to well-known results on the asymptotic stage of the Landau damp-ing of a finite amplitude wave 14, one can assume that theinitial phases p of the particles are distributed randomly sothat

1

N

p

p + 1 ap costr1t + p = 0, A11

showing that ddt 10 Eq. A9 is satisfied. The exact

value of nl in the existing steady state is not essential forthe instability analysis, and below we exclude it from ourconsiderations by choosing a reference frame moving withthe wave phase velocity corrected by the nonlinear frequencyshift. In this coordinate frame, 1=1−nlt is constant andso we can put 1=0.

In the presence of a second wave of small amplitudetr2tr1, 20, the oscillations of the trapped particlesare slightly perturbed

p ap costr1t + p + p, A12

where the small perturbation p satisfies

d2p

dt2 + tr12 p tr1

2 k2

k1Rei

2

1eit+pk2/k1 .

A13

Then, keeping only terms which are linear on p, we getfrom Eq. A5 =2

d2

dt i

8e

k22 2/2

nres

Ne−it

p1 − i

k2

k1ap costr1t + p

+8e

k22 2/2

k2

k1

nres

Ne−it

p

pe−ik2/k1ap costr1t+p.

A14

The first term in the right-hand side of Eq. A14 does notdepend on the perturbations of the particles’ trajectories

i8e

k22 2/2

nres

Ne−it

p1 − i

k2

k1ap costr1t + p

i8enres

k22 2/2

e−it A15

and represents a forced response of the second wave to themodulated flux of trapped particles. It corresponds to theknown fact that a nonlinear BGK wave is not a pure har-monic wave; the amplitudes of the harmonics k=mk1, m=2, . . . are found from Eq. A5,

k = −8e

mk2 /k

nres

N

p

e−imt+k/k1p,

A16

and are supposed to be small, k 1, if m= 1−k−vz0k1−k nl. Here it is necessary to note that ifthe ratio k2 /k1 of the second wave vector to the main wavevector is not an integer, such term vanishes, as it can beeasily shown by calculating the corresponding Fourier har-monic of the trapped particle density nez , t=n,pz−n−vt−zp,

nk = nez,te−ikzdz

= n,p z − n − vt − zpe−ikzdz

= e−ikvtn

e−iknp

e−ikzp, A17

where =2 /k1. As in the unperturbed state the coordinatezp does not depend on the bunch number n, one can see thatnkne−ikn=ne−2ink/k1=0 if k /k1 is not an integer. Ifthe trapped particles’ oscillations are perturbed by the secondwave, the perturbation zp depends on the bunch number andnke−ikvtne−iknpzpn does not vanish. This correspondsto the last term in Eq. A14.

One should also note that in the 1D case, if two waveshave identical wave vectors, they have also the same orslightly different frequencies and it is not possible to con-sider them as two different waves: there is only one singlewave; but, for a 3D geometry and in the presence of a back-ground magnetic field, two waves can have the same kz butdifferent frequencies. This is exactly what we consider herein this simplified 1D description; so we can assume that thetwo waves are different even if k1=k2. In this case it is notpossible to omit the first term in the right-hand side of Eq.A14, but if the frequency mismatch is large enough, it canbe excluded from the instability analysis because in the lin-ear approach it produces a partial nongrowing solution of Eq.A14. Only the case 0 requires a special consideration,but it corresponds to the condition of double resonance thatis examined in this paper on the basis of a nonlinear descrip-tion of the wave-particle interaction.

Let us consider the situation when all particles oscillatenear the bottom of the potential of the main wave, supposingthat, in a first approach, it is possible to neglect the amplitudeof the unperturbed displacement of the trapped particles.Thus we get from Eqs. A13 and A14 that

d2p

dt2 + tr12 p

k2

k1tr1

2 Rei2

1eit , A18

d2

dt

8e

k22 2/2

k2

k1

nres

Ne−it

p

p. A19

Moreover, in the above assumption all particles are movingsynchronously, and so 1

Nppp. Then, introducing thesolution of Eq. A19,

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2eit t 8enres

k22 2/2

k2

k1eit−tptdt, A20

in Eq. A18, one gets

d2p

dt2 + tr12 p tr1

2 k22

k12

8enres

1k22 2/2

Reit

eit−tptdt .

A21

Searching growing solutions of the form p=Ae−it+c.c.,one finds that

tr12 − 2Ae−it + c.c.

− tr12 k2

2

k12

8enres

1k22 2/2

Re Ae−it

+ +

Aeit

− A22

−1

2tr1

2 k22

k12

8enres

1k22 2/2

1

+

+1

− Ae−it + c.c. , A23

where A is the complex conjugate of A . Finally weget the dispersion relation which determines the growth rateof the second sideband wave and the perturbation of theparticles’ oscillations

2 − tr12 tr1

2 k2

k1

4enres

1k22 2/2

1

+ +

1

− .

A24

Taking into account the difference of notations and defini-tions, one can see that Eq. A24 has some similarity with theequation obtained for the sideband instability in Ref. 3.Indeed, noting that we use hereafter 2

2 + ,k2 =

2 + k2vz0 − 2,k2

+ k2vz0,k2 , A25

and owing to the property of the dielectric permittivity− ,−k= ,k 25, one gets

+ k2vz0 − 21,k2 − 2k1

21 − k2vz0 − ,2k1 − k2

221 − k2vz0 − − 2

2vz02k1 − k2 − − 2

2 − , A26

where we took into account that k1k2 and neglected thesmall imaginary part of the dielectric permittivity . Then,defining +k2vz0→ and p,res

2 =4e2nres /me, Eq. A24can be rewritten at k1 /k2=1 in the form presented in Ref. 3.

1 p,res

2

2 − tr12 1

,k2+

1

− 21,k2 − 2k1 .

A27

However, one has to stress that the result A27 of Kruer etal. 3 and Eq. A24 are not identical. This can be explainedby the fact that two different problems are investigated.Kruer et al. suppose that at initial time, except of the mainwave trapping particles, there are many other waves withdifferent wave vectors and quite small amplitudes. Amongthis set of waves, there are only two waves which are un-stable and grow. On the contrary, the present paper supposesthat except the main wave, there is only one other wave andits aim is to find the conditions when this wave is unstable.

The growth rate =Im can be calculated supposingthat −Retr1. Assuming that only one satellitewave so-called red wave is growing and neglecting corre-spondingly the second term in the right-hand side of Eq.A24, one can see from

tr12 − 2 + 2itr1 + = −

k1

k2

4e2nres

me /2

A28

that the maximum growth rate max is reached at tr1and is expressed as

max k1

2k2

4e2nres

tr1me /2 p,res

2

2tr1 /2.

A29

Note that the same growth rate can be obtained for the bluesatellite wave if one keeps the second term on the right-handside in Eq. A24. Using the obtained solution, one can seethat the second term omitted in Eq. A24 is small if p,res

2

tr13

2; in the opposite case, both the red and blue satellite

waves overlap, which means that their growth rates becomeof the same order as the difference between their frequencies.

APPENDIX B: HAMILTONIAN DESCRIPTIONOF THE BUNCH-WAVES SYSTEM

1. Hamiltonian formalism for wave-particle interaction

When the nonlinear coupling between the waves can beneglected compared to their interactions with the particles,the Hamiltonian H describing a group of N resonant elec-trons interacting with a packet of Nw electrostatic wavesk ,k in a magnetized plasma is given by 19

H = p=1

N

Hp + k

Nw

Hk − p=1

N

k

Nw

ekp. B1

Let us detail each term of this Hamiltonian, keeping in mindthe notations of Sec. II: rp= xp ,yp ,zp is the position of the

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particle p, vp= vxp ,vyp ,vzp is its velocity, and k is the slowvarying complex amplitude of the potential of the wavek ,k. Hp is the kinetic energy of the nonrelativistic elec-tron p moving in the ambient magnetic field B0=B0z, thatcan be written using the angle-action formalism as

Hp =pzp

2

2me+ cJp, B2

where pzp=mevzp is the parallel momentum of the particleand Jp=mevxp

2 +vyp2 /2c is its magnetic momentum. Hk is

the energy of the plasma eigenmode k ,k, that is

Hk = kIk, B3

where Ik= k2 /2k is the canonical action of the wave andk depends on the wave dispersion as

k =8

Vk2 k,kk

−1

. B4

The last term in Eq. B1 describes the interaction betweenthe particles and the waves and can be written as

kp = 2kIk1/2cosk · rp − k , B5

with k ·rp=k ·Rp+ 2Jp /mec1/2kx cos p+ky sin p andRp= Xp ,Yp /mec ,zp; Yp ,Xp, Jp ,p, and pzp ,zp are theconjugated pairs of momenta and positions; Rp is the guidingcenter position of the particle p, and p=−arctanvxp /vyp isthe azimuthal angle associated to the action Jp. The wavephase k=kt−argk is canonically conjugated to Ik.

In order to make the cyclotron resonances appear explic-itely, we expand the interaction term in Fourier series withrespect to the angle p, using eix sin =n=−

+ Jnxein, whereJnx is the Bessel function of integer order n. Introducingthe angle k so that kx=k sin k and ky =k cos k, as wellas the Larmor radius p= 2Jp /mec1/2 of the particle p, oneobtains

kp = 2kIk1/2 n=−

+

Jnkpcosk · Rp + np + k − k ,

B6

showing that the cyclotron resonance condition for the par-ticle p and the wave k ,k is kzzp+np−kconst, i.e.,kzvzk−nc.

The evolution of the system is then obtained by using theHamilton equations dp /dt=−H /q and dq /dt=H /p,where p ,q are the pairs of canonically conjugated vari-ables. One can check that the parallel momentum P of thesystem is a constant of the motion

P = k

kzIk + p

pzp = const. B7

2. Bunch-waves system

Let us consider the case of one bunch that is, a singlemacroparticle formed by Nb resonant particles moving syn-chronously in the potential of Nw electrostatic waves defined

by the angle-action variables I,, the frequencies andwave vectors ,k, as well as the interaction parameters Eq. B4, where =1, . . . ,Nw. The Hamiltonian B1 ofthis bunch-waves system can be decomposed as follows:

H = p=1

Nb

Hp + p=Nb+1

N

Hp + k

Hk

− ek

p=1

Nb

kp + p=Nb+1

N

kp , B8

where one can assume that ekp=Nb+1N kp0 and

p=Nb+1N Hpconst as the N−Nb resonant electrons which do

not belong to the bunch are submitted to phase mixing. Notethat we ignore here the influence of the trapped phase-mixedparticles on the waves’ evolution and therefore on the motionof the trapped bunched particles. Thus the system can bedescribed by the Hamiltonian Hb,

Hb = p=1

Nb pzp2

2me+ cJp

+

I − ep=1

Nb

Re2I1/2eik·rp− ,

B9

where the momenta and the positions of all the bunchedparticles are equal, i.e., pzp= pz, Jp=J, and rp=r for p=1, . . . ,Nb. Then one can define new canonical variablesPb ,Zb, Jb ,b, and Yb ,Xb representing the bunch’s con-jugate momenta and positions,

Pb = Nbpz,

Jb = NbJ ,

Yb = NbY , B10

Zb =1

Nbp=1

Nb

zp = z ,

b = ,

Xb = X . B11

Avoiding the 3Nb−1 degrees of freedom owing to the3Nb−1 independent constraints pp= Pb and rp=rb, one ob-tains

Hb =Pb

2

2Nbme+ cJb +

I

− eNb Re2I1/2eik·rb− . B12

Note that Hb Eq. B12 is similar to the Hamiltonian de-scribing the interaction between the waves and a single par-ticle of mass Mb=Nbme and charge −qb=−Nbe. Below wewill use such notations and only express the final results interms of me, e, and Nb.

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In order to simplify the problem, let us consider that allthe bunched particles are moving with a velocity close to theresonance velocity defined by kzPbt=0 /me+nc−0,for all the waves what is justified by the trapping mecha-nism, and let us neglect the influence of the other reso-nances by keeping only the contribution of the resonance n,expik ·rb−Jnkbexpik ·Rb+nb+−.Note that in the most realistic case where only two wavesare considered =1, =2, it follows from dk ·R /dt= e /mec2I

1/2sink ·rp−kxky−kykxthat k ·R=const =1,2 if k1k2=0; then the degreeof freedom associated with the perpendicular particles’ driftcan be avoided. The angles verifying kx=k sin andky=k cos can also be included as an initial conditionin the waves’ phases by defining I = I and =

−k ·R−n. Then the Hamiltonian is

Hb =Pb

2

2Mb+ cJb +

I

− qb Re 2I1/2JnkbeikzZb+nb−,

B13

and after performing several canonical transformations, weget omitting primes

Hb,n =P −

kzI2

2Mb+ cJ +

− ncI

− qb2I1/2Jnkbcos , B14

where I= 2 /2 and =t−arg−kzZb−k ·Rb−nb+. The bunch’s Larmor radius b is

bI = 2Jb/Mbc1/2 = 2

MbcJ − n

I1/2.

B15

The generalized momenta J=Jb+nI and P= Pb+kzI are constants of the motion their conjugated vari-ables Zb and b do not appear explicitly in Hb,n. Note that inEq. B14 the parallel kinetic energy of the bunch plays therole of a wave-wave interaction energy and that the coherentmotion of the bunched particles can generate some couplingbetween the waves. The bunch Larmor radius b may alsoplay this role in the case of the cyclotron resonances n0Eqs. B14 and B15.

Let us now focus our attention to the case when only twowaves are present, in order to point out a specific couplingmechanism responsible for the periodic exchanges of energybetween the waves. The Hamiltonian equations provide, us-ing Eq. B14 and taking into account Eq. B15, that

d

dtI = −

Hb,n = − qb2I1/2Jnkbsin ,

B16

d

dt =

I

Hb,n

= − nc −kz

MbP − kz1I1 − kz2I2

− qb cos /2I1/2Jnkb

− 2I1/2dJnkbdI

, B17

where =1,2. For the Landau resonance n=0, the last termin Eq. B17 vanishes, as b= 2J /Mbc1/2=const. Then theonly coupling between the waves results from the parallelmotion of the bunch. Supposing that kb1, i.e.,J0kb1, the system can be described by the Hamil-tonian

Hb,0 =P −

kzI2

2Mb+ cJ

+ I − qb2I1/2cos . B18

The case of cyclotron resonances n0 is more complicated,as the last term in Eq. B17 contributes to the couplingmodification of the frequency detuning due to the particles

d

dI

Jnkb = kJnkbdb

dI

= −nk

MbcbJnkb .

B19

Let us show that, for the resonances n= 1, this coupling isin most cases negligible. For the ring instability at the normalcyclotron resonance n=1, the particles which interactstrongly with the waves verify J1kb0, that is, kb1.8 23; then it is not necessary to take into account thelast term in Eq. B17 and one can suppose that J1kbconst. The validity of this assumption has been checkedwith the help of numerical simulations. For the fan instabilityat the anomalous cyclotron resonance n=−1, assuming thatk is large enough so that kb1 for all particles in thebunch, one can write J−1kb−kb /2 which providesfor Eq. B17 that

d

dt = + c −

kz

MbP − kz1I1 − kz2I2

+ kqb/8I1/2b +1

Mbc

I/21/2

bcos .

B20

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