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Cyclotron amplification of whistler waves
Adv. Space Res. Vol. 24, No. 1, pp. 95-98, 1999 Q 1999 COSPAR. Published by Elsevier Science Ltd. AI1 rights reserved
Pergamon
Printed in Great Britain 0273-1177/99 $20.00 + 0.00
www.elsevier.nl/locate/asr
CYCLOTRON Y. Hobars’.‘.
PII: SO273-1177(99)00432-9
AMPLIFICATION
V. Y. Trakhtengcrts
OF WHISTLER
2, A. G. Demekhov’,
WAVES
and M. Hayakawa’
’ Department of Electronic Engineering, The University of Electra-Communications. Chofu. 182-8585 Tokyo, Japan 21nstitute of Applied Physics, 46 Ulyanov Street, 603600 Nixhny Novgorod, Rus.yia
1-5-l.
Chofugaoka.
ABSTRACT Cyclotron wave-particle interactions in the case of well-organized distributions of encrgct,ic electrons in an inhomogeneous magnetic field are studied. Step and 6 function distributions of field-aligned velocit,y are considered. The one-hop amplification of whistler waves is calculated analytically and by numerical computation. In rigorous approach, taking into account. the third-order terrn in the spatial dependence of the electron phase with respect to the wave, some new features of the one-hop amplification rate r as function of frequency and electron beam parameters are obtained. I’ exhibits a quasi-periodic structure as fimction of frequency or characteristic electron parallel velocity. For the step-like dist,ribution it, remains positive. For the h-function it changes sign. The dependence of l-’ on the total energy, characteristic parallel velocity, position of the injection point in relation to the equator, and dispersion in parallel velocity of energetic electrons such as is discussed. 01999 COSPAR. Published by Elsevier Science Ltd. INTRODUCTION The cyclotron interaction of whistler waves with beams of energetic electrons in an inhomogeneous magnetic field attracts a lot of interest in connection with some important problems of magnetospheric physics. such as triggered ELF/VLF emissions, chorus generation. wave generation in the aurora1 ZOIIC. The beam is represented by parallel velocity distribution functions with as step-like, and 6 functions, rcspect,ivcly. Step-like distribution is formed by quasi-linear relax&ion of the beam-plasma and Cyclotron inst,abilit,ies (Ivanov, 1977; Trakhtengerts et aZ. 1986). Dirac-type distribution funct,ion is produced in the process of cyclotron interaction of quasi-monochromatic whistler packet with radiation belt electrons (Karpman et al. 1974: Nunn, 1974). Both types of distribution functions arc suit,ablc for wave generation with fine structures. In this paper we analyze the cyclotron inst,ability in the presence of a beam with arbitrary injection point along can inhomogeneous magnetic field. A small dipersion spread of the field-aligned velocity component, which can exist in a beam source or appears urldcr the expansion of a beam in the inhomogeneous geomagnetic field is taken into account. The problem is solved in the regime of stationary injection and stationary amplification, when the second-order cyclotron resonance effects are absent,.
FORMULATION
Cyclotron
Amplification.
We shall consider
the case when the beam
density
is small
and the change
of phase
between
wave and
96
Y. Hobara et al.
particle is determined by the mismatch of the cyclotron (z) in an inhomogeneous magnetic field: w -
WH
resonance
condition
along the particle
trajectory
kV(,
=
(1)
_ where w and k are frequency and wave vector of a whistler wave (Ic 11H, 2 is the geomagnetic field), WH is the electron gyrofrequency, and ~11is the electron velocity component along the magnetic field. In Eq.1, WH, k and ~11depend on the coordinate z along g. into the form (Tmkhtengetis et al. 1996)
In this case the one-hop
amplification
rate I can be brought
(3) and A=w - WH - kvil. The minimum resonant energy (WHI, - w)/kL. The variables W and the first adiabatic
w = Y(vi+vf): N,,, is the whistler wave refractive index, determined frorn the equation A=O. Distribution
WRL = invariant
is achieved II are qwer by
at t,he equator
WWRL/~
2
II
_
y;,
q = -(W
H is the magnetic
-
Igy”
field amplitude,
(VRI,
112
(> ;
=
(41
and the stationary
point ,zPt is
Functions.
We shall consider the case of the stationary energetic electrons. Step-like distribution: butions with noise-like
A step-like distribution whistler emissions.
particle
injection
function,
for two types of distribution
appears
in cyclotron
interaction
functions
of smooth
Fo of
distri-
Fstep = bstep.O(W,--W+~lHI)
where the step-like feature corresponds to the boundary of the resonant and nonresonant particles in the velocity space satisfying the equalities vi1 = v* = (2W,/m)‘l” at the injection point. The step height batep which is the funct,ion of adiabatic invariant can be determined from the normalization condition. The value HI is equal to the magnetic field amplitude at the stationary injection point z=zl. Dirac Distribution: A distribution function electrons with a packet of quasi-monochromatic
as 6 function in VII appears whistler waves.
under
cyclotron
interaction
of
where b6 is determined by the normalized condition. Both types of above functions Eq.5 and Eq.7 have no dispersion over VII at the injection point, but dispersion arises at other points due to transformation of vl-dispersion (Eq.6 and Eq.7) into vll-dispersion in the motion of electrons in an inhomogeneous magnetic field. When investigating the influence of the small dispersion in parallel velocity at the injection point, on the whistler wave amplification, we used the following smooth approximation for the Dirac function: b(Y-W*)= Here, AWI, = rn~~~~Avl1is the dispersion
l
GAW,, in parallel
exP [energy.
(%)‘I
(7)
Cyclotron
97
Amplification
RESULTS 1. The one-hop whistler wave amplification I’ is a (or w) for the beam-like distribution function Eq.6, function Eq.5 (cf. Figures l(a) and l(b)). 2. The amplification I’ for step-like and beam-like 21, (or w); the characteristic frequency scale of the
sign-alternative function of the characteristic whereas it remains positive for the step-like distributions oscillations
Aw ---.-N--WAURL W
in 1111 exhibits
an oscillatory
velocity w. distribution
dependence
on
is
(y(ka)-“/3
(8)
I’RC
where the numerical coefficient is 6 CY3 for the first two maxima (Figures l-3). For typical values L N 6 and w z WH[,/2; N, N 7 CI~, one has Aw/w N 10W2, and Af = Aw/2n N 20 Hz. 3. A packet-like oscillating tail appears in addition to the peaks in r near (v, - ?‘Rr,)/?‘K[, = 0 in the case of injection of the beam at a finite distance from the equator (AH # 0, Figures l(a) and l(b)). Near exact resonance, (v, - ZIRL)/ZIRL = 0. I? decreases with AH in both (step and 6) cases. However, r of the oscillating tail decreases in step case and grows in the d case for small AH. 4. An incrcasc of the dispersion Avll in t,he beam source leads to smoothing of all oscillations and decrcasc of the peak growth rate
8000 GoOa 4000 2000 n -2000 -4000 -fxl00 -8000
(a)
(b)
Fig. l(a) One-hop cyclotron amplification with the step-like distribution localized at different points relative to the magnetic equator. Amplification has been normalized to the maximum possible value for a smooth distribution with anisotropy Tl& - 1 = 1, To = o.2(n/2)1’2(nh/n,)(a/Vo)WHL. Plasma parameters are chosen as follows: L = 6, n, = 7 cmm3, We = 1.84 keV(the latter corresponds to the condition wo = ZIRL), and w = 0.5wH~. For these parameters, (Iw)~/~ 2: 431.5 (b) Same as l(a) but for the beam-like distribution.
98
Y Hobara et al. 8000 GooO 4000 2000 0 -2000 -4000 -GO00 -8000 10000 “0
5
ka2f3(u.
t:“v, ,,,, UR,,
15
-5
20
0
(k(ry/yv* 5 Il,yL),Vf(L
15
lo
(a)
Fig. 2(a) Influence of the parallel velocity dispersion on the one-hop cyclotron amplification with the steplike distribution localized at the magnetic equator. Plasma parameters chosen correspond to those in Fig. l(a), (b) Same as 2(a) but for the beam-like distribution.
-5
0
Fig. 3(a) Influence of the parallel velocity dispersion on the one-hop cyclotron amplification with the steplike distribution localized away from the magnetic equator (AH/HL = 0.01). Plasma paramctcrs chosen correspond to those in Fig l(a), (b) Same as (a) but for the beam-like distribution
REFERENCES Ivanov,
A. A., Physics
of Strongly
Karpman, V. I., Y. N. Istomin, packet in inhomogeneous Nunn,
D., A self-consistent
Nonequdibrium Plasmas
(in Russian),
Energoatomizdat,
and D. R. Shklyar, Nonlinear theory of a quasimonochromatic plasma, Phys. Plasmas, 16(8), 685 -703 (1974).
theory of triggered
VLF emissions,
Trakhtengerts, V. Y., V. R. Tagirov, and S. A. Chernous, Geomagn. Aeron., 26(l), 99-106 (1986).
Planet.
Flow cyclotron
Moscow (1977). whistler
Space Sci., 22, 349-378 maser and impulsive
mode
(1974).
VLF emissions,
‘Irakhtengerts, V. Y., M. J. Rycroft, and A. G. Demekhov, Interrelation of noise-like and discrete ELF/VLF emissions generated by cyclotron interactions, J. Geophys. Res., 101 (A6), 13,293.-13,303 (1996).
Pergamon
Printed in Great Britain 0273-1177/99 $20.00 + 0.00
www.elsevier.nl/locate/asr
CYCLOTRON Y. Hobars’.‘.
PII: SO273-1177(99)00432-9
AMPLIFICATION
V. Y. Trakhtengcrts
OF WHISTLER
2, A. G. Demekhov’,
WAVES
and M. Hayakawa’
’ Department of Electronic Engineering, The University of Electra-Communications. Chofu. 182-8585 Tokyo, Japan 21nstitute of Applied Physics, 46 Ulyanov Street, 603600 Nixhny Novgorod, Rus.yia
1-5-l.
Chofugaoka.
ABSTRACT Cyclotron wave-particle interactions in the case of well-organized distributions of encrgct,ic electrons in an inhomogeneous magnetic field are studied. Step and 6 function distributions of field-aligned velocit,y are considered. The one-hop amplification of whistler waves is calculated analytically and by numerical computation. In rigorous approach, taking into account. the third-order terrn in the spatial dependence of the electron phase with respect to the wave, some new features of the one-hop amplification rate r as function of frequency and electron beam parameters are obtained. I’ exhibits a quasi-periodic structure as fimction of frequency or characteristic electron parallel velocity. For the step-like dist,ribution it, remains positive. For the h-function it changes sign. The dependence of l-’ on the total energy, characteristic parallel velocity, position of the injection point in relation to the equator, and dispersion in parallel velocity of energetic electrons such as is discussed. 01999 COSPAR. Published by Elsevier Science Ltd. INTRODUCTION The cyclotron interaction of whistler waves with beams of energetic electrons in an inhomogeneous magnetic field attracts a lot of interest in connection with some important problems of magnetospheric physics. such as triggered ELF/VLF emissions, chorus generation. wave generation in the aurora1 ZOIIC. The beam is represented by parallel velocity distribution functions with as step-like, and 6 functions, rcspect,ivcly. Step-like distribution is formed by quasi-linear relax&ion of the beam-plasma and Cyclotron inst,abilit,ies (Ivanov, 1977; Trakhtengerts et aZ. 1986). Dirac-type distribution funct,ion is produced in the process of cyclotron interaction of quasi-monochromatic whistler packet with radiation belt electrons (Karpman et al. 1974: Nunn, 1974). Both types of distribution functions arc suit,ablc for wave generation with fine structures. In this paper we analyze the cyclotron inst,ability in the presence of a beam with arbitrary injection point along can inhomogeneous magnetic field. A small dipersion spread of the field-aligned velocity component, which can exist in a beam source or appears urldcr the expansion of a beam in the inhomogeneous geomagnetic field is taken into account. The problem is solved in the regime of stationary injection and stationary amplification, when the second-order cyclotron resonance effects are absent,.
FORMULATION
Cyclotron
Amplification.
We shall consider
the case when the beam
density
is small
and the change
of phase
between
wave and
96
Y. Hobara et al.
particle is determined by the mismatch of the cyclotron (z) in an inhomogeneous magnetic field: w -
WH
resonance
condition
along the particle
trajectory
kV(,
=
(1)
_ where w and k are frequency and wave vector of a whistler wave (Ic 11H, 2 is the geomagnetic field), WH is the electron gyrofrequency, and ~11is the electron velocity component along the magnetic field. In Eq.1, WH, k and ~11depend on the coordinate z along g. into the form (Tmkhtengetis et al. 1996)
In this case the one-hop
amplification
rate I can be brought
(3) and A=w - WH - kvil. The minimum resonant energy (WHI, - w)/kL. The variables W and the first adiabatic
w = Y(vi+vf): N,,, is the whistler wave refractive index, determined frorn the equation A=O. Distribution
WRL = invariant
is achieved II are qwer by
at t,he equator
WWRL/~
2
II
_
y;,
q = -(W
H is the magnetic
-
Igy”
field amplitude,
(VRI,
112
(> ;
=
(41
and the stationary
point ,zPt is
Functions.
We shall consider the case of the stationary energetic electrons. Step-like distribution: butions with noise-like
A step-like distribution whistler emissions.
particle
injection
function,
for two types of distribution
appears
in cyclotron
interaction
functions
of smooth
Fo of
distri-
Fstep = bstep.O(W,--W+~lHI)
where the step-like feature corresponds to the boundary of the resonant and nonresonant particles in the velocity space satisfying the equalities vi1 = v* = (2W,/m)‘l” at the injection point. The step height batep which is the funct,ion of adiabatic invariant can be determined from the normalization condition. The value HI is equal to the magnetic field amplitude at the stationary injection point z=zl. Dirac Distribution: A distribution function electrons with a packet of quasi-monochromatic
as 6 function in VII appears whistler waves.
under
cyclotron
interaction
of
where b6 is determined by the normalized condition. Both types of above functions Eq.5 and Eq.7 have no dispersion over VII at the injection point, but dispersion arises at other points due to transformation of vl-dispersion (Eq.6 and Eq.7) into vll-dispersion in the motion of electrons in an inhomogeneous magnetic field. When investigating the influence of the small dispersion in parallel velocity at the injection point, on the whistler wave amplification, we used the following smooth approximation for the Dirac function: b(Y-W*)= Here, AWI, = rn~~~~Avl1is the dispersion
l
GAW,, in parallel
exP [energy.
(%)‘I
(7)
Cyclotron
97
Amplification
RESULTS 1. The one-hop whistler wave amplification I’ is a (or w) for the beam-like distribution function Eq.6, function Eq.5 (cf. Figures l(a) and l(b)). 2. The amplification I’ for step-like and beam-like 21, (or w); the characteristic frequency scale of the
sign-alternative function of the characteristic whereas it remains positive for the step-like distributions oscillations
Aw ---.-N--WAURL W
in 1111 exhibits
an oscillatory
velocity w. distribution
dependence
on
is
(y(ka)-“/3
(8)
I’RC
where the numerical coefficient is 6 CY3 for the first two maxima (Figures l-3). For typical values L N 6 and w z WH[,/2; N, N 7 CI~, one has Aw/w N 10W2, and Af = Aw/2n N 20 Hz. 3. A packet-like oscillating tail appears in addition to the peaks in r near (v, - ?‘Rr,)/?‘K[, = 0 in the case of injection of the beam at a finite distance from the equator (AH # 0, Figures l(a) and l(b)). Near exact resonance, (v, - ZIRL)/ZIRL = 0. I? decreases with AH in both (step and 6) cases. However, r of the oscillating tail decreases in step case and grows in the d case for small AH. 4. An incrcasc of the dispersion Avll in t,he beam source leads to smoothing of all oscillations and decrcasc of the peak growth rate
8000 GoOa 4000 2000 n -2000 -4000 -fxl00 -8000
(a)
(b)
Fig. l(a) One-hop cyclotron amplification with the step-like distribution localized at different points relative to the magnetic equator. Amplification has been normalized to the maximum possible value for a smooth distribution with anisotropy Tl& - 1 = 1, To = o.2(n/2)1’2(nh/n,)(a/Vo)WHL. Plasma parameters are chosen as follows: L = 6, n, = 7 cmm3, We = 1.84 keV(the latter corresponds to the condition wo = ZIRL), and w = 0.5wH~. For these parameters, (Iw)~/~ 2: 431.5 (b) Same as l(a) but for the beam-like distribution.
98
Y Hobara et al. 8000 GooO 4000 2000 0 -2000 -4000 -GO00 -8000 10000 “0
5
ka2f3(u.
t:“v, ,,,, UR,,
15
-5
20
0
(k(ry/yv* 5 Il,yL),Vf(L
15
lo
(a)
Fig. 2(a) Influence of the parallel velocity dispersion on the one-hop cyclotron amplification with the steplike distribution localized at the magnetic equator. Plasma parameters chosen correspond to those in Fig. l(a), (b) Same as 2(a) but for the beam-like distribution.
-5
0
Fig. 3(a) Influence of the parallel velocity dispersion on the one-hop cyclotron amplification with the steplike distribution localized away from the magnetic equator (AH/HL = 0.01). Plasma paramctcrs chosen correspond to those in Fig l(a), (b) Same as (a) but for the beam-like distribution
REFERENCES Ivanov,
A. A., Physics
of Strongly
Karpman, V. I., Y. N. Istomin, packet in inhomogeneous Nunn,
D., A self-consistent
Nonequdibrium Plasmas
(in Russian),
Energoatomizdat,
and D. R. Shklyar, Nonlinear theory of a quasimonochromatic plasma, Phys. Plasmas, 16(8), 685 -703 (1974).
theory of triggered
VLF emissions,
Trakhtengerts, V. Y., V. R. Tagirov, and S. A. Chernous, Geomagn. Aeron., 26(l), 99-106 (1986).
Planet.
Flow cyclotron
Moscow (1977). whistler
Space Sci., 22, 349-378 maser and impulsive
mode
(1974).
VLF emissions,
‘Irakhtengerts, V. Y., M. J. Rycroft, and A. G. Demekhov, Interrelation of noise-like and discrete ELF/VLF emissions generated by cyclotron interactions, J. Geophys. Res., 101 (A6), 13,293.-13,303 (1996).