Radiophysics and Quantum Electronics, Vol. 43, Nos. 1–2, 2001

CYCLOTRON WAVE-PARTICLE INTERACTIONSIN THE WHISTLER-MODE WAVEGUIDE

D. L. Pasmanik and V. Yu.Trakhtengerts UDC 533.951

We study the cyclotron interaction of energetic electrons and whistler waves in plasma wave-guides formed by inhomogeneous distribution of cold plasma. Such waveguides can be formedin the Earth’s magnetosphere, e.g., by the plasmapause or by ducts with enhanced background-plasma density. In this paper, we consider a cylindrically symmetric model of a magnetosphericduct with enhanced cold-plasma density in a homogeneous magnetic field. The spatial structureof the eigenmodes of such a waveguide is found. We obtain a set of self-consistent quasilinearequations for cyclotron instability with eigenmode structure taken into account, thus generalizingthe quasilinear theory of a magnetospheric cyclotron maser.

1. INTRODUCTION

The interaction of energetic particles with electromagnetic waves in the Earth’s magnetosphere playsa big role in the formation of the radiation-belt structure and in the generation of low-frequency radiation.One interesting and important problem is cyclotron wave-particle interactions in regions of enhanced cold-plasma density, which also serve as waveguides for the generated waves. Note that in a volume filled withdense plasma, the cyclotron resonance condition is satisfied for a larger number of energetic particles, andwave-energy leakage from the generation region is decreased due to ducting. Therefore, such regions can bevery favorable for cyclotron instability.

The above mentioned guiding structures can be provided by the so-called plasma ducts, i.e., magneticflux tubes filled with a dense cold plasma, or by the plasmasphere boundary called the plasmapause, whichserves as a waveguide for whispering-gallery modes.

It is noteworthy that both the problem of wave-particle interactions in such structures and theproblem of low-frequency electromagnetic wave propagation in plasma waveguides have been studied inseveral papers.

In [1], a model of a flow cyclotron maser was suggested to explain the formation mechanism ofpulsating auroras. This model, based on the self-consistent quasilinear theory of cyclotron instability,allows one to explain some experimental data, namely, the observed dynamics of the spectrum of generatedemissions and of the flux of energetic particles precipitated into the ionosphere. However, the model of wavepropagation in a magnetospheric duct used in that paper was extremely simplified and actually did not takeinto account the real spatial distribution of wave intensity across the waveguide.

On the other hand, some authors studied the problem of low-frequency wave propagation and gener-ation in plasma-waveguide structures in great detail [2]; however, they did not consider the case where thewave energy is supplied by the cyclotron interaction.

This paper is aimed at developing a more rigorous model of the cyclotron interaction of whistler wavesand energetic electrons in magnetospheric plasma waveguides (ducts), which generalizes, in particular, themodel suggested in [1].

Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translatedfrom Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, Nos. 1–2, pp. 127–139, January, 2001.Original article submitted September 13, 2000.

0033-8443/01/431-2-0117$25.00 c© 2001 Plenum Publishing Corporation 117

In Sec. 2, we present the results of eigenmode analysis of a cylindrical plasma waveguide homogeneousalong its axis. In Sec. 3, we derive the equations of the quasilinear theory of cyclotron interactions in sucha waveguide. In Sec. 4, the derived equations are generalized to the case of a smoothly inhomogeneouswaveguide. The main results are discussed in Sec. 5.

2. EIGENMODES OF A CYLINDRICAL PLASMA WAVEGUIDE

To model a magnetospheric duct, we consider a cylindrically symmetric column filled with a coldplasma having an inhomogeneous radial distribution of the number density in an external homogeneousmagnetic field B0 parallel to the duct axis.

In the reference frame whose z-axis is parallel to the geomagnetic field, the plasma dielectric permit-tivity tensor for eiωt-waves with frequencies ωlh � ω < ωB � ωp is written as

ε =

ε ig 0−ig ε 00 0 η

, (1)

where ε = 1 − ω2p/(ω2 − ω2

B), g = −ω2pωB/[ω (ω2 − ω2

B)], η = 1 − ω2p/ω2, ωp and ωB are, respectively, the

electron plasma frequency and gyrofrequency, and ωlh is the lower hybrid frequency.Let us seek the eigenmodes of this system in the form of waves propagating along the duct and having

some transverse structure:E = Φ(r)eiωt−ik0pz, H = iΨ(r)eiωt−ik0pz. (2)

Here k0p = k‖ is the wavevector component along the duct axis, k0 = ω/c, and c is the speed of light. Notethat only axially symmetric modes are considered in this paper, but the results obtained can be generalizedto the case of modes with nonzero azimuthal indices.

Substituting these expressions into Maxwell’s equations, we obtain the following equations for theelectric and magnetic field components of a wave in the cylindrical coordinate system (r, ϕ, z):

Ψr = ipΦϕ, Ψϕ =1

k0 (ε− p2)

(k0pgΦϕ + ε

dΦz

dr

), Ψz =

1k0r

d(rΦϕ)dr

,

Φr =i

p2 − ε

(gΦϕ +

p

k0

dΦz

dr

),

∆⊥Φϕ − Φϕ

r2+ k2

0

(g2

p2 − ε− p2 + ε

)Φϕ =

k0gp

p2 − ε

dΦz

dr,

∆⊥Φz +p2

p2 − ε

ε′

ε

dΦz

dr− k2

0

η

ε(p2 − ε)Φz =

k0p

ε(p2 − ε)

1r

ddr

rgΦϕ

p2 − ε. (3)

Here ε′ ≡ dε/dr.It follows from analysis of this system of equations [2] that its exact explicit analytic solution can

be found only if the distribution of the number density of the cold plasma is homogeneous or piecewise-homogeneous. Therefore, one can obtain analytic formulas for the eigenmodes of a duct with piecewise-constant plasma profile (Fig. 1). However, due to the presence of a sharp duct boundary, any eigenmode ofsuch a system comprises two components: the whistler-mode component and the small-scale electrostaticone. The whistler wave propagates inside a waveguide formed by the duct and vanishes exponentially outsideit, while the electrostatic wave propagates in the entire space and is responsible for the energy leakage fromthe waveguide.

If the plasma distribution is homogeneous, the whistler-mode and electrostatic waves are independent.However, we consider the case with a sharp boundary at which the coupling between the whistler wave andthe electrostatic wave takes place; hence, these waves are no longer independent. Therefore, even if there is a

118

Fig. 1. A duct with piecewise-constant distributionof the background plasma.

Fig. 2. A duct with a smooth boundary.

source exciting a pure whistler mode in the system, the energy is transformed to the electrostatic componentand thus leaks from the duct. The analysis of the Q-factor of modes of such a system showed that the lossescan be quite significant.

The efficiency of this wave transformation is determined by the relation between the characteristicscale of the inhomogeneity and the wavelength of the electrostatic wave, and the transformation is practicallyabsent if the inhomogeneities are sufficiently smooth [3].

The sharp boundaries are absent under real magnetospheric conditions. Therefore, the duct modelconsidered above does not apply, and it is necessary to consider a smooth profile of the cold plasma density.Since there is no exact explicit analytic solution for such plasma distributions, we use an approximatesolution of the system (3).

Let the cold plasma density have a sufficiently smooth profile (ε′/ε � k0q, where k0q = k⊥ is thetransverse wavevector component with respect to the magnetic field); moreover, let us consider the low-frequency waves (ω � ωB � ωp), whose propagation angle with respect to the magnetic field is not toolarge (q . p).

We take into account that in this frequency range, the relations p2 ∼ g, qw . p, and qel � p hold forthe wavevector components, where the subscripts “w” and “el” correspond to the whistler and electrostaticmodes, respectively. Therefore, the electrostatic wave does not satisfy the adopted approximation. We alsouse the fact that ε � g � η in the low frequency range and, hence, p2 � ε.

Using the above relations, one can estimate the terms entering the system (3) (see Appendix 1).Neglecting the small terms, we obtain the following system of equations:

Ψr = ipΦϕ, Ψϕ = gΦϕ

p , Ψz = 1k0r

d(rΦϕ)dr , Φr = igΦϕ

p2 , Φz = − 1k0pηr

d(rgΦϕ)dr ,

and ∆⊥Φϕ − Φϕ

r2 + k20

(g2

p2 − p2)

Φϕ = 0. (4)

An analytic solution of the system can be found for the following cold plasma density profile (Fig. 2):

N2c =

N2d , r ≤ a;

∆(N2)a2

r2+ N2

∞, r > a,(5)

where Nd and N∞ are the plasma densities inside and outside the duct, respectively, ∆(N 2) = N2d − N2∞,

and a is the characteristic radius of a duct. For this cold-plasma distribution, the solution of Eq. (4) for Φϕ

119

is represented in terms of the eigenmode series

Φkϕ(r) ={

BkJ1(k0qkr), r ≤ a;CkKνk

(k0skr), r > a,(6)

where J and K are the Bessel and Macdonald functions, respectively, q2k = g2

d/p2k − p2

k, s2k = p2

k − g2∞/p2k,

ν2k = 1 − k2

0a2∆(g2)/p2

k, gα = g(Nα), ∆(g2) = g2(Nd) − g2(N∞), and k is the mode index. Using theboundary conditions at r = a, we obtain the following relations for pk, qk, sk, νk, and Ck/Bk:

J1(k0qka)(rJ1(k0qkr))′r=a

=Kνk

(k0ska)(rKνk

(k0skr))′r=a

andCk

Bk=

J1(k0qka)Kνk

(k0ska). (7)

Note that the waveguide eigenmodes satisfy the following orthogonality conditions:

∞∫0

cr dr

4π

([Φk,Ψ∗

k′ ]z + [Φk′ ,Ψ∗k]z)

= δk,k′ (8)

and ∞∫0

r dr

4π

((∂(ωε)∂ω

Φk,Φ∗k′

)+ (Ψk,Ψ∗

k′))

=Wkδk,k′ . (9)

Here δk,k′ is the Kronecker symbol, and the asterisk denotes the complex conjugation. In Eqs. (8) and (9),we used the normalization condition

∞∫0

r dr([Φk,Ψ∗

k]z + [Φk,Ψ∗k]z)

=4π

c. (10)

3. WAVE-PARTICLE INTERACTIONS

3.1. Equation for wave-energy transfer

Now we consider the excitation problem for the waveguide eigenmodes. Since the hot plasma compo-nent has a much smaller density than the cold one under magnetospheric conditions, one may assume thatthe wave propagation is determined only by the cold-plasma parameters, while the cyclotron interactionwith energetic particles serves as a wave-energy source.

In this case, the solution can be sought using the method of slowly varying amplitudes. Let usrepresent the wave field as a set of wave packets:

E =∑n,k

Ank(z, t)Enk =∑n,k

Ank(z, t)Φnk(r)eiωnt−ik0pnkz, (11)

where ωn are given (carrier) frequencies, the values of parallel wavevectors pnk and transverse wave-packetprofiles Φnk(r) correspond to the waveguide eigenmodes at frequencies ωn, obtained in the previous section,and Ank are slow functions of their arguments:∣∣∣∣ 1A ∂Ank

∂t

∣∣∣∣� ω−1,

∣∣∣∣ 1A ∂Ank

∂z

∣∣∣∣� (k0pnk)−1.

Substituting the above expressions into Maxwell’s equations with the source in the form of theresonant-particle current JR, we obtain Eq. (A2.15) (see Appendix 2) in which Enk(z, t, r) ≡ Ank(z, t)Φnk(r).

Following the standard derivation of the quasilinear-theory equations [4], we assume that the phases

120

ϑn of the waves with different frequencies (Ank = |Ank|eiϑn) are random and independent. This assumptionyields

〈Ank〉 = 0, 〈AnkA∗n′k′〉 = Inkk′δn,n′ ,

where the angular brackets denote averaging over phases ϑn, and Inkk′ , according to the chosen normalization(10), is the energy-flux spectral density.

Then, averaging Eq. (A2.15) over phases and integrating over the duct cross-section with account ofthe orthogonality conditions (8) and (9), we obtain the following equation:

Wnk∂Ink

∂t+

∂Ink

∂z= −

∫r dr dϕ 〈(JR, A∗

nkE∗nk)〉, (12)

where Ink ≡ Inkk and InkWnk is the energy density per mode:

Wnk =∫

r dr dϕ

4π

((∂(ωε)∂ω

Φnk,Φ∗nk

)+ (Ψnk,Ψ∗

nk))

. (13)

3.2. Electron motion in the duct-eigenmode field

Let us consider the motion of an electron in the field of a duct eigenmode (2). The equation ofelectron motion has the form

V = − e

mEnk + [ωB,V] +

e

mc[Hnk,V], (14)

where Enk and Hnk are determined by Eqs. (2), (4), and (6), and e and m are electron charge and mass,respectively. Expanding this equation, we obtain

u = − e

mRe[(

E⊥nk +iVz

cH⊥nk

)e−iφ +

iu

cHznk

],

φ = ωB − e

muIm[(

E⊥nk +iVz

cH⊥nk

)e−iφ

],

Vz = − e

mRe[Eznk − iu

cH⊥nke

−iφ

], (15)

where ueiφ = Vr + iVϕ, E⊥nk = Ernk + iEϕnk, and H⊥nk = Hrnk + iHϕnk.In the linear approximation, this system can be written as

u = − e

mAnkRe

{[F⊥nk(r0) + F ′

⊥nk(r0)ρB cos(ωBt + φ0) +

+u

c

[Φznk(r0) + Φ′

znk(r0)ρB cos(ωBt + φ0)]eiωBt+iφ0

]eiθnk

},

Vz = − e

mAnkRe

{[[Φznk(r0) + Φ′

znk(r0)ρB cos(ωBt + φ0)]eiωBt+iφ0 +

+u

c

[Ψ⊥nk(r0) + Ψ′

⊥nk(r0)ρB cos(ωBt + φ0)] ]

eiθnk

},

φ = ωB − eAnk

muIm{[F⊥nk(r0) + F ′

⊥nk(r0)ρB cos(ωBt + φ0)]eiθnk

}, (16)

where ρB is the electron gyroradius, F⊥nk = Φ⊥nk − Ψ⊥nkVz/c θnk = ωnt − k0pnk (z0 + Vzt) −−ωBt− φ0, and the following expressions for the unperturbed electron trajectory are used:

z = z0 + Vzt and r =[r20 + ρ2

B + 2r0ρB cos(ωBt + φ0)]1/2

.

121

Here z0 and φ0 are, respectively, the particle coordinate and gyrophase at t = 0, and r0 is the coordinateof the center of the electron Larmor circle. The assumption ρB � r0 yields r ≈ r0 + ρB cos(ωBt + φ0);hence f(r) ≈ f(r0) + f ′(r0)ρB cos(ωBt + φ0). Following the standard procedure for solving this systemunder linear approximation (see, e.g., [5]) and using the inequality Φz � Ψ⊥ which follows from Eqs.(4)(Φz ∼ qkgΨ⊥/(p2

kη)), we integrate Eqs.(16) and neglect small terms of the order of ρBf ′/f ∼ ρBk0q toobtain

∆u ≡ u− u0 = − e

mA Re

(F⊥(r)∆−1

eiωt−ik0pz−iφ

),

∆Vz ≡ Vz − Vz0 = − eu

mcA Re

(Ψ⊥(r)∆−1

eiωt−ik0pz−iφ

),

φ = ωBt + φ0, (17)

where ∆−1 = ω − ωB − k0pVz and u0 and Vz0 are the initial values of the corresponding variables.

3.3. Resonant-particle current

Using the formulas obtained above, we find the resonant current JR defined as

JR = −e

∫V ∆F d3V. (18)

Here ∆F is the perturbation of the energetic-electron distribution function F (r, V, t) due to the effect ofwaves:

∆F =∑n,k

∆Fnk =∑n,k

(∂F

∂u∆unk +

∂F

∂Vz∆Vznk

). (19)

Let us consider the quantity Re(JR, A∗nkE

∗nk), which actually is the work of the resonant current on the

given mode. In our approximation, we have

(JR, A∗nkE

∗nk) ≈ −e

∫∆F A∗

nkE∗nk⊥eiφu2 dudVz dφ. (20)

We substitute the formulas for ∆F and Enk⊥ into this equation and use the Plemelj formula

1∆−1nk

= P

(1

∆−1nk

)+ iπδ(∆−1nk

), (21)

where ∆−1nk= ωn−ωB − k0pnkVz, the symbol P denotes the principal value of the integral, and δ(x) is the

delta-function. Averaging yields the following relation:

〈(JR, A∗nkE

∗nk)〉 = −π2e2

mInk

∫u2 dudVz δ(∆−1nk

)(

∂F

∂uF⊥nkΦ⊥nk +

∂F

∂Vz

u

cΨ⊥nkΦ⊥nk

). (22)

3.4. Equation for the electron distribution function

The kinetic equation for the distribution function of energetic electrons has the form

∂F

∂t+(V,

∂F

∂r

)+(V,

∂F

∂V

)= 0. (23)

We represent the distribution function as the sum of slowly varying component and rapidly oscillatingperturbation related to the interaction of the initial distribution function with waves:

122

F = F0 + f1 and f1 =∑n,k

∆Fnk. (24)

Substituting this expression into the kinetic equation and averaging over random wave phases and over fastoscillations, we obtain

∂F0

∂t+ Vz

∂F0

∂z+(VD⊥ ,

∂F0

∂r⊥

)= −

⟨(V,

∂f1

∂V

)⟩, (25)

where VD = VD⊥+Vzz0 is the slowly varying component of electron velocity, i.e., the guiding-center velocityin the drift approximation of charged-particle motion, and r⊥ is the coordinate across the magnetic fieldlines.

Using Eq. (19) and relations (16) and (17), we obtain

⟨(V,

∂f1

∂V

)⟩=

⟨∑n,k

∑n′,k′

(unk

∂

∂u+ Vznk

∂

∂Vz+ φnk

∂

∂φ

)(∆un′k′

∂

∂u+ ∆Vzn′k′

∂

∂Vz

)F0

⟩=

=πe2

2m2

∑n,k

Ink

(F⊥nk

(∂

∂u+

1u

)+

u

cΨ⊥nk

∂

∂Vz

)δ(∆−1nk

)(F⊥nk

∂

∂u+

u

cΨ⊥nk

∂

∂Vz

)F0. (26)

Combining the results obtained in this section, we can write down the following system of self-consistent equations for the slowly varying component of the energetic electron distribution function F andthe spectral density of wave-energy flux Ink (hereafter the subscript “0” is omitted):

∂F

∂t+ Vz

∂F

∂z+(VD⊥ ,

∂F

∂r⊥

)= DF, (27)

∂Ink

∂t+ VGnk

∂Ink

∂z= γnkInk. (28)

Here

D =πe2

2m2

∑n,k

InkΛnku2δ(ωn − ωB − k0pnkVz)Λnk, (29)

γnk =π2e2

mWnk

∫r dr

∫u3 dudVz δ(ωn − ωB − k0pnkVz)ΛnkΦ⊥nkF, (30)

Λnk =F⊥nk

u

∂

∂u+

Ψ⊥nk

c

∂

∂Vz, (31)

and VGnk = 1/Wnk is the group velocity of the corresponding wave packet.Actually, this system of equations generalizes the quasilinear theory equations by taking into account

the spatial structure of excited electromagnetic waves.

4. A DUCT IN AN INHOMOGENEOUS MAGNETIC FIELD

Let us generalize the system of equations (27)–(31) for the description of wave-particle interactionsin magnetospheric ducts.

For this, we should take into account that the parameters of a duct, such as the external magneticfield B0, the cold plasma density Nc, and the duct radius a, depend on the z-coordinate. However, thesedependences are very smooth as compared to the scales of waves propagating in the duct. Therefore, theproblem of wave propagation in such a duct can be solved using the method of cross-sections [6]. Then theduct eigenmodes have the form

Enk = Φnk(r, z) exp(iωnt−ik0

∫pnk dz) and Hnk = iΨnk(r, z) exp(iωnt−ik0

∫pnk dz), (32)

123

where Φnk(r, z), Ψnk(r, z), and pnk(z) correspond to the solution (6)–(7) for the duct parameters in a givencross-section z. Due to the chosen normalization (10) for the duct modes, the quantity Ink is the adiabaticinvariant in the absence of sources. Note also that, according to the method of cross-sections, the eigenmodesin the vicinity of any cross-section z = z0 can be represented in the form (2) with p replaced by pnk(z0).Therefore, the results obtained in the previous section are correct also for the eigenmodes in the form (32),provided that the duct parameters vary smoothly.

It is convenient to transform the system of equations (27)–(31) to the new variables µ = sin2 θL ≡(BL/B) sin2 θ and v =

√u2 + V 2

z , where θ is the pitch angle and the subscript “L” corresponds to theequatorial cross-section. These variables are the adiabatic invariants of the particle motion in the absenceof interactions with waves. In the new variables, the diffusion coefficient D and the growth rate γnk arewritten as

D =πe2

2m2

∑n,k

InkΦ2⊥nk

ωB

ωBLΛnkv

2µδ(ωn − ωB − k0pnkVz)Λnk, (33)

γnk =π2e2

mWnk

∞∫0

r dr

∞∫0

v4 dv

BL/B∫0

µdµ√1− µωB/ωBL

ω2B

ω2BL

δ(ωn − ωB − k0pnkVz)Φ2⊥kΛnkF, (34)

Λnk = Λ =1v

∂

∂v+

2v2

(ωBL

ωB− µ

)∂

∂µ. (35)

Following [7], we average Eqs. (27) and (28), respectively, over the periods of electron bounce-oscillations between the magnetic mirrors and wave-packet oscillations between the conjugate ionospheres.This procedure is actually reduced to the action of operators T−1

b

∮dz/Vz and T−1

Gnk

∮dz/VGnk, respectively,

on Eq. (27) for the distribution function and on Eq. (28) for the amplitude of the kth wave harmonic. HereTb =

∮dz/Vz is the bounce-period and TGnk =

∮dz/VGnk and VGnk are, respectively, the oscillation period

and group velocity of the wave packet.Finally, we obtain equations similar to Eqs. (27) and (28) in which, however, F and Ik denote their

averaged values:∂F

∂t+(VD⊥ ,

∂F

∂r⊥

)= DF, (36)

∂Ink

∂t= γnkInk. (37)

Here

D =πe2

2m2Tb

∑n,k

InkΦ2⊥nkΛv2µ

leffnkk0presnk

ωresB ωBL

Λ, (38)

γnk =π2e2

mWnkTGnk

∞∫0

r dr

∞∫ωBL−ωnk0pnkL

v5 dv

1−(

ωBL−ωnk0pnkLv

)2∫0

µdµleffnkk0p

resnk

VGnkω2BL

Φ2⊥nkΛF, (39)

and

leffnk =∣∣∣∣(ωB − ωn)ω−2

B

∂

∂z(k0pnkVz + ωB)

∣∣∣∣−1

z=zres

(40)

is the effective interaction length; the subscript “res” shows that the corresponding values are calculated atthe point where the cyclotron resonance condition

ωn − ωB(zres) = k0pnk(zres)v

√1− µ

ωB(zres)ωBL

is satisfied.

124

In the case of cold and dense plasmas where β? = ω2pLv2

0

/(ω2

BLc2) � 1, mv20/2 being the characteristic

energy of energetic particles, the waves are generated mainly at low frequencies, ω ∼ ωBL/β? � ωBL. Usingthis relation, the above equations can be simplified and become

∂F

∂t+(VD⊥ ,

∂F

∂r⊥

)=

1Tb

∂

∂µ

(D

∂F

∂µ

)− δF + J (41)

and∂Ink

∂t= (γnk − νnk) Ink, (42)

where

D =πe2

2m2v2

∑n,k

InkΦ2⊥nk

leffnkk0presnkωBL

ωresB

(43)

and

γnk =π2e2

mcTGnk

presnk

Wnk

∞∫0

r dr

∞∫ωBL

k0pnkL

v3 dv

1−(

ωBLk0pnkLv

)2∫0

µdµleffnk

VGnkωBLΦ2⊥nk

∂F

∂µ. (44)

In the equation for the distribution function, we introduced the energetic-particle source J and theterm δF describing particle losses into the loss cone, where

δ ={

0, µ ≥ µc;δ0 = v/l, 0 ≤ µ ≤ µc.

(45)

Here µc = L−3(4− 3L−1

)−1/2 is the loss-cone boundary in the approximation of dipole magnetic field andl is the length of the magnetic flux tube between the conjugate ionospheres. In the equation for waveamplitudes, wave energy losses due to nonideal reflection from the ionosphere are taken into account by theterm νnkInk.

5. CONCLUSIONS

The analysis performed in this paper yields the system of equations (36)–(39) which generalizesthe known quasilinear theory of magnetospheric cyclotron masers to the case of cyclotron interactions ofelectrons and whistler waves in cylindrical plasma waveguides.

These results can be used, in particular, for generalization of the flow cyclotron maser model andmore exact description of pulsating-aurora formation. Such generalization should help in explaining thedynamics of the spatial structure of energetic-particle precipitation and optical auroral patterns.

Based on Eqs. (41)–(44), one can expect the following qualitative picture of cyclotron instabilitydevelopment in a duct. Since the excitation efficiency (30) of different modes depends on the spatial distri-bution of energetic particles in the duct, a certain mode is excited most efficiently at any time. However,the cyclotron interactions lead to the redistribution of energetic particles and their precipitation from themagnetic trap, which is more intense in the duct regions where the wave field is maximum. Therefore, ifthe particle source is not very strong, a hole in the transverse distribution of trapped energetic particlesin a duct is formed, and the spatial structure of the precipitated electron flux is qualitatively similar tothe transverse amplitude distribution of the excited waveguide mode. This, in turn, causes a decrease inexcitation efficiency of this mode (see Eq. (30)), and another mode is excited more efficiently. Therefore, onecan expect a mode competition accompanied by variations in the wave spectrum and in spatial distributionsof trapped and precipitated energetic particles.

To further develop this model of cyclotron wave-particle interactions in a plasma waveguide, it is

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expedient to generalize it to the case of excitation and propagation of modes with nonzero azimuthal index.It is also necessary to study the interaction of eigenmodes and particles at different cyclotron harmonics(ωn− kzVz − ωB = 0, where m = 0,±1, . . . ). Another important problem is related to the spatial structureand excitation of waveguide modes near the upper cutoff frequency ω = ωB/2 [8].

This work is supported by the Russian Foundation for Basic Research (grant No. 99–02–16175) andby NATO (grant ESR.CLG 975144).

APPENDIX 1

Let us estimate the terms in the last two equations of system (3) assuming that the conditions

p2 ∼ g, q . p, ε � g � η, p2 � ε, ε′/ε � k0q

are satisfied and using the estimate d/dr ∼ k0q.In the last equation of the system (3), we have

∆⊥Φz ∼ k20q

2Φz,p2

p2 − ε

ε′

ε

dΦz

dr≈ ε′

ε

dΦz

dr∼ k0q

ε′

εΦz � k2

0q2Φz

andk2

0

η

ε(p2 − ε)Φz ≈ k2

0

η

εp2Φz � k2

0q2Φz,

k0p

ε(p2 − ε)

1r

ddr

rgΦϕ

p2 − ε≈ k0p

εr

d(rgΦϕ)dr

∼ k20qp

g

ε.

From these relations we obtain the following estimate for the parallel component of the wave electric field:

Φz ∼ qg

pηΦϕ � Φϕ.

The left-hand side terms in the third equation of the system (3) are of the order of k20q

2Φϕ, and onthe right-hand side

k0gp

p2 − ε

dΦz

dr≈ k0g

p

dΦz

dr∼ k2

0gq

pΦz ∼ k2

0

g

η

q2

p2Φϕ � k2

0q2Φϕ.

APPENDIX 2

Let us consider the Maxwell equations for two electromagnetic fields:

rotH1 =1c

∂E1

∂t+

4π

cJ1 +

4π

cJR, (A2.1)

rotE1 = −1c

∂H1

∂t, (A2.2)

rotH∗2 =

1c

∂E∗2

∂t+

4π

cJ∗2, (A2.3)

rotE∗2 = −1

c

∂H∗2

∂t, (A2.4)

where the current densities J1,2 are determined as

J1,2 =

t∫−∞

dt′ σ(t− t′, r)E1,2(t′, r) (A2.5)

(the spatial dispersion is neglected), σ is the complex conductivity tensor [5], and JR is the external current(resonant particle current).

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Multiplying these equations by H∗2, −E∗

2, H1, and −E1, respectively, and summing them up, we get

div([E1,H∗2] + [E∗

2,H1]) +1c

∂

∂t[(H1,H∗

2) + (E1,E∗2)] +

4π

c[(J1,E∗

2) + (J∗2,E1)] = −4π

c(JR,E∗

2). (A2.6)

Now we specify the fields E1 and E2 as series of quasimonochromatic waves:

E1 =∑

n

eiωnt∑

k

e−iknkzEnk(t, z, r) and E2 = eiωn′ te−ikn′k′zEn′k′(t, z, r), (A2.7)

where Enk(t, z, r) is a slowly varying function of t and z. Then

J1 =∑n,k

eiωnt−iknkz

t∫−∞

dt′ σ(t− t′, r)Enk(t′, z, r)eiωn(t′−t)−iknk(z′−z). (A2.8)

Taking into account that Enk(t′) is a slowly varying function, we expand it into a series over t− t′:

Enk(t′) = Enk(t) +∂Enk

∂t(t′ − t) + . . . . (A2.9)

Substituting this expansion into Eq. (A2.8) for the current and using the definition of the complex conduc-tivity tensor σ [5], we obtain

J1 =∑n,k

eiωnt−iknkz

(σ(ωn, r)Enk − i

∂σ(ωn, r)∂t

∂Enk

∂t

)(A2.10)

and, similarly,

J2 = eiωn′ t−ikn′k′z(

σ(ωn′ , r)En′k′ − i∂σ(ωn′ , r)

∂t

∂En′k′

∂t

). (A2.11)

Then we have

(J1,E∗2) + (J∗2,E1) =

∑n,k

ei(ωn−ωn′)t−i(knk−kn′k′)z×

×(

(σnEnk,E∗n′k′) + (σ∗n′E

∗n′k′ ,Enk)− i

(∂σn

∂ω

∂Enk

∂t,E∗

n′k′

)+ i

(∂σn′

∂ω

∂E∗n′k′

∂t,Enk

)), (A2.12)

where σnn′ = σ(ωnn′ , r).

Using the relation−iσ =

ω

4π(ε− δ) (A2.13)

between the tensors σ and ε, where δ is the unit tensor, and taking into account that the tensor ε isHermitian, which implies that σ is anti-Hermitian, we obtain

(J1,E∗2) + (J∗2,E1) =

∑n,k

ei(ωn−ωn′)t−i(knk−kn′k′)z ×

×(

((σn − σn′)Enk,E∗n′k′) +

14π

((∂(ωε)∂ω

)ω=ωn

∂Enk

∂t,E∗

n′k′

)+

+14π

((∂(ωε)∂ω

)ω=ωn′

Enk,∂E∗

n′k′

∂t

)). (A2.14)

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Substituting Eqs. (A2.7) and (A2.14) into Eq. (A2.6) and using the folowing relations valid for anyvector A:

div(Ae−ikz

)=(−ikAz +

∂Az

∂z+ div⊥A

)e−ikz and

∂

∂t

(Ae−iωt

)=(−iωA +

∂A∂t

)e−iωt,

we get

∑n,k

ei(ωn−ωn′ )t−i(knk−kn′k′)z

{(−i (knk − kn′k′) z0 +

∂

∂zz0 + div⊥

)([Enk,H∗

n′k′ ] + [E∗n′k′ ,Hnk])+

+4π

c((σn − σn′)EnkE∗

n′k′) +i

2c(ωn − ωn′) [(Enk E∗

n′k′) + (Hnk,H∗n′k′)]+

+1c

∂

∂t(Hnk,H∗

n′k′) +1c

((∂(ωε)∂ω

)ω=ωn

∂Enk

∂t,E∗

n′k′

)+

((∂(ωε)∂ω

)ω=ωn′

Enk,∂E∗

n′k′

∂t

)}=

= −4π

c(JR,E∗

n′k′)e−iωn′ t+ikn′k′z. (A2.15)

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