- Email: [email protected]
On the electron dynamics in the field of a whistler wave propagating along a magnetic field in a weakly inhomogeneous plasma
ARTICLE IN PRESS
Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 969–972 www.elsevier.com/locate/jastp
Rapid Communication
On the electron dynamics in the field of a whistler wave propagating along a magnetic field in a weakly inhomogeneous plasma V.L. Krasovsky Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117997, Russia Received 21 December 2006; received in revised form 10 April 2007; accepted 14 April 2007 Available online 19 April 2007
Abstract The relativistic motion of electrons in the field of a finite amplitude circularly polarized whistler propagating along a constant magnetic field in a plasma with longitudinal inhomogeneity is considered. It is shown that the equations of the particle motion exhibit a constant of the motion despite the spatial dependence of the wave parameters. The existence of the constant allows one to reduce the equations of motion to a canonical form describing one-dimensional oscillations of a particle with the Hamiltonian slowly varying in the process of the oscillations. r 2007 Elsevier Ltd. All rights reserved. PACS: 52.65 Cc; 52.35 Hr; 94.20 wj; 94.30 Hn Keywords: Gyroresonance; Whistler; Integral of motion; Triggered emissions; Inhomogeneous plasma
1. Introduction The studies of nonlinear gyroresonant wave–particle interaction in space plasmas call for a careful analysis of charged particle motion in the field of a finite amplitude wave (Dungey, 1963; Helliwell, 1983; Omura et al., 1991; Rycroft, 1991). In the case of a nonuniform plasma dielectric constant, the amplitude of the wave and wavenumber are varying in space. This leads to certain difficulties in the description of the particle dynamics, as the equations of motion become ‘‘nonintegrable’’. In such a situation, approximate methods to describe the Tel.: +7 495 3334167; fax: +7 095 3331248.
E-mail address: [email protected]. 1364-6826/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jastp.2007.04.002
particle trajectories take on great significance. Approximate equations of the motion of an electron moving in the field of a circularly polarized electromagnetic wave propagating in a longitudinally nonuniform magnetic field were used even in the early work devoted to resonant interaction of VLF waves with energetic electrons in the Earth radiation belts (see e.g. Dysthe, 1971). Similar equations have also been exploited repeatedly in subsequent studies on the nonlinear gyroresonant phenomena in the magnetosphere (e.g. Nunn, 1974; Karpman, 1974; Roux and Pellat, 1978; Bell and Inan, 1981; Molvig et al., 1988; Carlson et al., 1990; Shklyar et al., 1992). In essence, the simplified description of the particle dynamics (Dysthe, 1971) reduces to a change of rigorous equations of motion
ARTICLE IN PRESS 970
V.L. Krasovsky / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 969–972
for equations averaged over the Larmor gyration of the particle. In doing so, the forces exerted by the wave on the electron are introduced into the equations after the averaging procedure. Strictly speaking, such an approach is justified only in the limiting case of vanishing amplitude of the wave. In this limit, or in the absence of the wave, the electron magnetic moment is conserved within the framework of the approximate (averaged) equations proposed by Dysthe (1971), whereas in the case of a finite amplitude wave, these equations do not exhibit any integrals of the motion. A more general approach based on the averaging of the rigorous equations, taking into account the terms associated with the field of the wave, leads to a modification of the corresponding approximate equations of the motion (Krasovsky and Matsumoto, 1998). In particular, in contrast with the equations postulated by Dysthe (1971), the approximate equations derived by Krasovsky and Matsumoto (1998) have the integral of the motion even at a finite amplitude of the wave L¼
1 2 mco0 ðv þ v2? Þ 2 z 2eB0 " 2 # eA eA ¼ const, v2? 2v? cos x þ mc mc
ð1Þ
where vz and v? are the electron velocities parallel and perpendicular to the external magnetic field B0 ¼ B0 ðzÞez , respectively, A ¼ AðzÞ is the amplitude of the vector-potential of the wave A, and x is the angle between the vectors v? and A. Numerical solutions of the rigorous equations of motion have confirmed that the approximate invariant of motion L is really conserved with a very high accuracy in the process of the gyroresonant interaction between electrons and a quasi-monochromatic whistler propagating in the geomagnetic trap (Krasovsky et al., 2002). This paper is devoted to relativistic motion of electrons in the field of a whistler wave propagating along a uniform magnetostatic field in a plasma with a longitudinal inhomogeneity. In this case, somewhat lengthy procedure of the averaging of the input equations (carried out by Krasovsky and Matsumoto, 1998) is unnecessary, so that the description of the electron dynamics becomes even simpler in comparison with the case of a nonuniform magnetic field discussed in the mentioned paper. It is remarkable that, despite the spatial dependences of the wavenumber and amplitude of
the wave due to the plasma inhomogeneity, even the rigorous (rather than averaged) equations of the electron motion exhibit, as shown below, an integral of the motion, and it coincides, in essence, with (1) in the non-relativistic limit. 2. Canonical equations of motion Let us consider the relativistic motion of an electron in the field of a right-hand circularly polarized whistler wave propagating along a uniform external magnetic field B0 ¼ B0 ez in a collisionless plasma. In the presence of a weak longitudinal inhomogeneity of the plasma dielectric constant, the field of the wave can be described with the help of the vector-potential Ax ¼ AðzÞ cos c; Ay ¼ AðzÞ sin c, Z z c ¼ o0 t dz0 kðz0 Þ.
ð2Þ
1
The frequency of the wave o0 is assumed to be constant. The wavenumber k and amplitude A are varying along the direction of wave propagation z due to the inhomogeneity of the plasma parameters, for example, as a consequence of a spatial dependence of the plasma density n0 ¼ n0 ðzÞ. For brevity, we will use the following units of measurement to describe the physical variables ½t ¼ o1 0 ;
½z ¼ c=o0 ;
½p ¼ mc;
½E ¼ ½B ¼ mco0 =jej,
½A ¼ ½f ¼ mc2 =jej;
½v ¼ c;
½k ¼ o0 =c,
½E ¼ mc2 .
Then, the set of necessary equations is dr=dt ¼ v;
dp=dt ¼ ðE þ v BÞ,
E ¼ qA=qt rf; p ¼ Ev;
B ¼ hez þ r A,
E ¼ ð1 v2 Þ1=2 ¼ ð1 þ p2 Þ1=2 ,
(3) (4) (5)
where h jejB0 =mco0 ¼ const. Considering the plasma with the longitudinal inhomogeneity, we need, generally speaking, to take into account the longitudinal electrostatic field E z ¼ dfðzÞ=dz, although it is usually weak as a consequence of plasma quasi-neutrality. The equations of motion (3) take a compact form if we go over from the transverse momentum p? to the vector q defined by q ¼ p? A ¼ Ev? A,
(6)
ARTICLE IN PRESS V.L. Krasovsky / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 969–972
and then represent q in the polar coordinates q and g qx ¼ q cos g;
qy ¼ q sin g.
(7)
With the help of (2)–(7) it is not difficult to obtain dq hA ¼ sin a, dt E da h þ kpz hA ¼ sin a, 1þ dt qE E dpz df kAq A þ q cos a dA þ sin a , ¼ dz E E dz dt
(8) (9)
(10)
dz pz ¼ , (11) dt E where a g c is the angle between the vectors q and A. According to (8)–(10) the variation of the energy E ¼ ð1 þ p2z þ q2 þ A2 þ 2qA cos aÞ1=2 is given by the equation dE 1 df ¼ pz þ Aq sin a . (12) dt E dz From (8) and (12), it follows directly that the derivative of the quantity q2 , (13) 2h where W ¼ E f is the total electron energy, vanishes, dL=dt ¼ 0, i.e. L is a constant of the motion. Eliminating q from the set of Eqs. (8)–(11) with the help of (13), and treating the z coordinate as the independent variable instead of time, we come to the following canonical equations: L¼W
dW qH Aq ¼ ¼ sin a, dz qa pz
(14)
da qH h W f hA ¼ ¼kþ þ cos a, dz qW pz qpz
(15)
with the Hamiltonian H ¼ HðW ; a; zÞ ¼ kðW þ fÞ pz ,
(16)
where pz ½ðW þ fÞ2 1 q2 A2 2Aq cos a1=2 and q ½2hðW LÞ1=2 . Thus the availability of the integral of the motion, L ¼ const, allows one to treat the electron dynamics as a one-dimensional motion. In the absence of the plasma inhomogeneity, H ¼ HðW ; aÞ ¼ const; f ¼ 0, this motion has been discussed in Roberts and Buchsbaum (1964) and Lutomirski and Sudan (1966), and we will not reproduce the corresponding calculations here. It will
971
suffice to mention that the motion may be interpreted in terms of one-dimensional oscillations of a particle in an effective (pseudo) potential well. In the non-relativistic limit, v2 51, using the change of variables q sin a ¼ v? sin x;
q cos a ¼ v? cos x A,
(17)
where x is the angle between v? and A, we find L ’ 1 þ L0 , with 1 2 ðv þ v2? Þ 2 z 1 ðv2? 2Av? cos x þ A2 Þ f. ð18Þ 2h Expression (18) differs from (1) only by the dimensionless form and by the last term fðzÞ, which has not been taken into account in the paper by Krasovsky and Matsumoto (1998), where an inhomogeneity of the magnetic field h ¼ hðzÞ was considered instead of the plasma inhomogeneity. Despite the formal coincidence of (1) and (18), it should also be noted that (1) is an exact constant of the electron motion averaged over the Larmor gyration of the particle and is merely an approximate invariant of the real motion of the particle described by the input equations, whereas L determined by (13) or (18) is an exact constant of the rigorous input equations (3). Furthermore, the canonical equations (14) and (15) describe the dynamics of nonresonant particles with rapidly oscillating wave–particle phases a or x as well as the motion of particles near gyroresonance in contrast with the analogous equations (50) and (51) of the above-mentioned paper. L0 ¼
3. Discussion and conclusions The previous work (Krasovsky and Matsumoto, 1998; Krasovsky et al., 2002) in combination with the above-stated results leads to the conclusion that the constant of motion determined by (1), (13) or (18) is fairly general by its nature. The existence of the constant L allows one to treat the electron dynamics in terms of a one-dimensional motion of the particle in an effective potential well not only in the absence of the plasma inhomogeneity, at the constant wavenumber k and amplitude A, but also in the case of a spatial dependence of the wave parameters. Therefore, the constancy of L may be exploited in a variety of analytical as well as in numerical studies of the nonlinear gyroresonant interaction between particles and quasi-monochromatic whistler mode, including the relativistic consideration.
ARTICLE IN PRESS 972
V.L. Krasovsky / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 969–972
In the absence of the inhomogeneity, ð1=kÞðdk=dzÞð1=AÞðdA=dzÞ ¼ 0, the technique to solve the equations of electron motion may be found in Roberts and Buchsbaum (1964). Lutomirski and Sudan (1966) have also investigated the kinetic structure of exact nonlinear whistler mode presenting a counterpart of the well-known electrostatic Bernstein–Greene–Kruskal (BGK) wave (Bernstein et al., 1957). In the presence of a weak longitudinal plasma inhomogeneity, a0, the Hamiltonian (16) (or analogous expression in Krasovsky and Matsumoto, 1998 as applied to the case of nonuniform magnetic field) depends weakly on the ‘‘time’’ z, so that the electron motion can be treated as one-dimensional oscillations of the particle in a slowly varying effective potential. Under these conditions, it is natural to take advantage of an adiabatic approximation to solve the equations of motion (14) and (15) or to solve the corresponding Vlasov equation for the electron distribution function. Then, a self-consistent analytical description of the ‘‘adiabatic’’ gyroresonant interaction of the finite amplitude whistler wave with high energy electrons becomes possible on the basis of sufficiently accurate analysis demonstrated, for example, for slowly varying BGK waves in (Krasovsky, 1995). Hence, the equations of motion (14) and (15) may be used for a closed self-consistent description of the spatial evolution of the nonlinear whistler mode (Lutomirski and Sudan, 1966) propagating in a weakly inhomogeneous plasma. Finally, we will touch briefly on a more general type of the space–time dependence of the wave parameters. If the amplitude and frequency of the wave depend on time, as is observed, for example, in the course of the generation of triggered VLF radio emissions (experimental data have been presented, for example, by Carlson et al., 1990), one need to put A ¼ Aðz; tÞ and to add a correction jðtÞ to the phase of the wave c in (2). Then, reproduction of the mathematical treatments carried out in Section 2 leads to the result dL qA qj ¼ v? cos x þ A sin x a0, (19) dt qt qt i.e. the quantity L is not an exact constant. However, the right-hand side of (19) contains a double smallness since, in addition to the smallness of the amplitude A, the partial derivatives with respect to time must be proportional to a small number of resonant electrons. In view of the above, the approximate constancy of L might be used also
in studies (q=qta0).
of
the
nonstationary
phenomena
References Bell, T.F., Inan, U.S., 1981. Transient nonlinear pitch angle scattering of energetic electrons by coherent VLF wave packets in the magnetosphere. Journal of Geophysical Research 86, 9047–9063. Bernstein, I.B., Greene, J.M., Kruskal, M.D., 1957. Exact nonlinear plasma oscillations. Physical Review 108, 546–550. Carlson, C.R., Helliwell, R.A., Inan, U.S., 1990. Space–time evolution of whistler mode wave growth in the magnetosphere. Journal of Geophysical Research A 95, 15073–15089. Dungey, J.W., 1963. Loss of Van Allen electrons due to whistlers. Planetary and Space Science 11, 591–595. Dysthe, K.B., 1971. Some studies of triggered whistler emissions. Journal of Geophysical Research 76, 6915–6931. Helliwell, R.A., 1983. Controlled stimulation of VLF emissions from Siple Station, Antarctica. Radio Science 18, 801–814. Karpman, V.I., 1974. Nonlinear effects in the ELF waves propagating along the magnetic field in the magnetosphere. Space Science Reviews 16, 361–388. Krasovsky, V.L., 1995. Adiabatic wave–particle interaction in a weakly inhomogeneous plasma. Soviet Physics JETP (English Translation) 80, 420–432. Krasovsky, V.L., Matsumoto, H., 1998. On the resonant particle dynamics in the field of a finite-amplitude circularly polarized wave propagating along the axis of a magnetic trap. Physics of Plasmas 5, 2210–2216. Krasovsky, V.L., Matsumoto, H., Omura, Y., 2002. Approximate invariant of electron motion in the field of a whistler propagating along the geomagnetic field. Geophysical Research Letters 29(12), 10.1029/2001GL014638. Lutomirski, R.F., Sudan, R.N., 1966. Exact nonlinear electromagnetic whistler mode. Physical Review 147, 156–165. Molvig, K., Hilfer, G., Miller, R.H., Myczkowski, J., 1988. Selfconsistent theory of triggered emissions. Journal of Geophysical Research A 93, 5665–5683. Nunn, D., 1974. A self-consistent theory of triggered VLF emissions. Planetary and Space Science 22, 349–378. Omura, Y., Nunn, D., Matsumoto, H., Rycroft, M.J., 1991. A review of observational, theoretical and numerical studies of VLF triggered emissions. Journal of Atmospheric and Terrestrial Physics 53, 351–363. Roberts, C.S., Buchsbaum, S.J., 1964. Motion of a charged particle in a constant magnetic field and a transverse electromagnetic wave propagating in the field. Physical Review A 135, 381–389. Roux, A., Pellat, R., 1978. A theory of triggered emissions. Journal of Geophysical Research 83, 1433–1441. Rycroft, M.J., 1991. Interaction between whistler-mode wave and energetic electrons in the coupled system formed by magnetosphere, ionosphere and atmosphere. Journal of Atmospheric and Terrestrial Physics 53, 849–858. Shklyar, D.R., Nunn, D., Smith, A.J., Sazhin, S.S., 1992. An investigation into the nonlinear frequency shift in magnetically propagated VLF pulses. Journal of Geophysical Research A 97, 19389–19402.
Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 969–972 www.elsevier.com/locate/jastp
Rapid Communication
On the electron dynamics in the field of a whistler wave propagating along a magnetic field in a weakly inhomogeneous plasma V.L. Krasovsky Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117997, Russia Received 21 December 2006; received in revised form 10 April 2007; accepted 14 April 2007 Available online 19 April 2007
Abstract The relativistic motion of electrons in the field of a finite amplitude circularly polarized whistler propagating along a constant magnetic field in a plasma with longitudinal inhomogeneity is considered. It is shown that the equations of the particle motion exhibit a constant of the motion despite the spatial dependence of the wave parameters. The existence of the constant allows one to reduce the equations of motion to a canonical form describing one-dimensional oscillations of a particle with the Hamiltonian slowly varying in the process of the oscillations. r 2007 Elsevier Ltd. All rights reserved. PACS: 52.65 Cc; 52.35 Hr; 94.20 wj; 94.30 Hn Keywords: Gyroresonance; Whistler; Integral of motion; Triggered emissions; Inhomogeneous plasma
1. Introduction The studies of nonlinear gyroresonant wave–particle interaction in space plasmas call for a careful analysis of charged particle motion in the field of a finite amplitude wave (Dungey, 1963; Helliwell, 1983; Omura et al., 1991; Rycroft, 1991). In the case of a nonuniform plasma dielectric constant, the amplitude of the wave and wavenumber are varying in space. This leads to certain difficulties in the description of the particle dynamics, as the equations of motion become ‘‘nonintegrable’’. In such a situation, approximate methods to describe the Tel.: +7 495 3334167; fax: +7 095 3331248.
E-mail address: [email protected]. 1364-6826/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jastp.2007.04.002
particle trajectories take on great significance. Approximate equations of the motion of an electron moving in the field of a circularly polarized electromagnetic wave propagating in a longitudinally nonuniform magnetic field were used even in the early work devoted to resonant interaction of VLF waves with energetic electrons in the Earth radiation belts (see e.g. Dysthe, 1971). Similar equations have also been exploited repeatedly in subsequent studies on the nonlinear gyroresonant phenomena in the magnetosphere (e.g. Nunn, 1974; Karpman, 1974; Roux and Pellat, 1978; Bell and Inan, 1981; Molvig et al., 1988; Carlson et al., 1990; Shklyar et al., 1992). In essence, the simplified description of the particle dynamics (Dysthe, 1971) reduces to a change of rigorous equations of motion
ARTICLE IN PRESS 970
V.L. Krasovsky / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 969–972
for equations averaged over the Larmor gyration of the particle. In doing so, the forces exerted by the wave on the electron are introduced into the equations after the averaging procedure. Strictly speaking, such an approach is justified only in the limiting case of vanishing amplitude of the wave. In this limit, or in the absence of the wave, the electron magnetic moment is conserved within the framework of the approximate (averaged) equations proposed by Dysthe (1971), whereas in the case of a finite amplitude wave, these equations do not exhibit any integrals of the motion. A more general approach based on the averaging of the rigorous equations, taking into account the terms associated with the field of the wave, leads to a modification of the corresponding approximate equations of the motion (Krasovsky and Matsumoto, 1998). In particular, in contrast with the equations postulated by Dysthe (1971), the approximate equations derived by Krasovsky and Matsumoto (1998) have the integral of the motion even at a finite amplitude of the wave L¼
1 2 mco0 ðv þ v2? Þ 2 z 2eB0 " 2 # eA eA ¼ const, v2? 2v? cos x þ mc mc
ð1Þ
where vz and v? are the electron velocities parallel and perpendicular to the external magnetic field B0 ¼ B0 ðzÞez , respectively, A ¼ AðzÞ is the amplitude of the vector-potential of the wave A, and x is the angle between the vectors v? and A. Numerical solutions of the rigorous equations of motion have confirmed that the approximate invariant of motion L is really conserved with a very high accuracy in the process of the gyroresonant interaction between electrons and a quasi-monochromatic whistler propagating in the geomagnetic trap (Krasovsky et al., 2002). This paper is devoted to relativistic motion of electrons in the field of a whistler wave propagating along a uniform magnetostatic field in a plasma with a longitudinal inhomogeneity. In this case, somewhat lengthy procedure of the averaging of the input equations (carried out by Krasovsky and Matsumoto, 1998) is unnecessary, so that the description of the electron dynamics becomes even simpler in comparison with the case of a nonuniform magnetic field discussed in the mentioned paper. It is remarkable that, despite the spatial dependences of the wavenumber and amplitude of
the wave due to the plasma inhomogeneity, even the rigorous (rather than averaged) equations of the electron motion exhibit, as shown below, an integral of the motion, and it coincides, in essence, with (1) in the non-relativistic limit. 2. Canonical equations of motion Let us consider the relativistic motion of an electron in the field of a right-hand circularly polarized whistler wave propagating along a uniform external magnetic field B0 ¼ B0 ez in a collisionless plasma. In the presence of a weak longitudinal inhomogeneity of the plasma dielectric constant, the field of the wave can be described with the help of the vector-potential Ax ¼ AðzÞ cos c; Ay ¼ AðzÞ sin c, Z z c ¼ o0 t dz0 kðz0 Þ.
ð2Þ
1
The frequency of the wave o0 is assumed to be constant. The wavenumber k and amplitude A are varying along the direction of wave propagation z due to the inhomogeneity of the plasma parameters, for example, as a consequence of a spatial dependence of the plasma density n0 ¼ n0 ðzÞ. For brevity, we will use the following units of measurement to describe the physical variables ½t ¼ o1 0 ;
½z ¼ c=o0 ;
½p ¼ mc;
½E ¼ ½B ¼ mco0 =jej,
½A ¼ ½f ¼ mc2 =jej;
½v ¼ c;
½k ¼ o0 =c,
½E ¼ mc2 .
Then, the set of necessary equations is dr=dt ¼ v;
dp=dt ¼ ðE þ v BÞ,
E ¼ qA=qt rf; p ¼ Ev;
B ¼ hez þ r A,
E ¼ ð1 v2 Þ1=2 ¼ ð1 þ p2 Þ1=2 ,
(3) (4) (5)
where h jejB0 =mco0 ¼ const. Considering the plasma with the longitudinal inhomogeneity, we need, generally speaking, to take into account the longitudinal electrostatic field E z ¼ dfðzÞ=dz, although it is usually weak as a consequence of plasma quasi-neutrality. The equations of motion (3) take a compact form if we go over from the transverse momentum p? to the vector q defined by q ¼ p? A ¼ Ev? A,
(6)
ARTICLE IN PRESS V.L. Krasovsky / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 969–972
and then represent q in the polar coordinates q and g qx ¼ q cos g;
qy ¼ q sin g.
(7)
With the help of (2)–(7) it is not difficult to obtain dq hA ¼ sin a, dt E da h þ kpz hA ¼ sin a, 1þ dt qE E dpz df kAq A þ q cos a dA þ sin a , ¼ dz E E dz dt
(8) (9)
(10)
dz pz ¼ , (11) dt E where a g c is the angle between the vectors q and A. According to (8)–(10) the variation of the energy E ¼ ð1 þ p2z þ q2 þ A2 þ 2qA cos aÞ1=2 is given by the equation dE 1 df ¼ pz þ Aq sin a . (12) dt E dz From (8) and (12), it follows directly that the derivative of the quantity q2 , (13) 2h where W ¼ E f is the total electron energy, vanishes, dL=dt ¼ 0, i.e. L is a constant of the motion. Eliminating q from the set of Eqs. (8)–(11) with the help of (13), and treating the z coordinate as the independent variable instead of time, we come to the following canonical equations: L¼W
dW qH Aq ¼ ¼ sin a, dz qa pz
(14)
da qH h W f hA ¼ ¼kþ þ cos a, dz qW pz qpz
(15)
with the Hamiltonian H ¼ HðW ; a; zÞ ¼ kðW þ fÞ pz ,
(16)
where pz ½ðW þ fÞ2 1 q2 A2 2Aq cos a1=2 and q ½2hðW LÞ1=2 . Thus the availability of the integral of the motion, L ¼ const, allows one to treat the electron dynamics as a one-dimensional motion. In the absence of the plasma inhomogeneity, H ¼ HðW ; aÞ ¼ const; f ¼ 0, this motion has been discussed in Roberts and Buchsbaum (1964) and Lutomirski and Sudan (1966), and we will not reproduce the corresponding calculations here. It will
971
suffice to mention that the motion may be interpreted in terms of one-dimensional oscillations of a particle in an effective (pseudo) potential well. In the non-relativistic limit, v2 51, using the change of variables q sin a ¼ v? sin x;
q cos a ¼ v? cos x A,
(17)
where x is the angle between v? and A, we find L ’ 1 þ L0 , with 1 2 ðv þ v2? Þ 2 z 1 ðv2? 2Av? cos x þ A2 Þ f. ð18Þ 2h Expression (18) differs from (1) only by the dimensionless form and by the last term fðzÞ, which has not been taken into account in the paper by Krasovsky and Matsumoto (1998), where an inhomogeneity of the magnetic field h ¼ hðzÞ was considered instead of the plasma inhomogeneity. Despite the formal coincidence of (1) and (18), it should also be noted that (1) is an exact constant of the electron motion averaged over the Larmor gyration of the particle and is merely an approximate invariant of the real motion of the particle described by the input equations, whereas L determined by (13) or (18) is an exact constant of the rigorous input equations (3). Furthermore, the canonical equations (14) and (15) describe the dynamics of nonresonant particles with rapidly oscillating wave–particle phases a or x as well as the motion of particles near gyroresonance in contrast with the analogous equations (50) and (51) of the above-mentioned paper. L0 ¼
3. Discussion and conclusions The previous work (Krasovsky and Matsumoto, 1998; Krasovsky et al., 2002) in combination with the above-stated results leads to the conclusion that the constant of motion determined by (1), (13) or (18) is fairly general by its nature. The existence of the constant L allows one to treat the electron dynamics in terms of a one-dimensional motion of the particle in an effective potential well not only in the absence of the plasma inhomogeneity, at the constant wavenumber k and amplitude A, but also in the case of a spatial dependence of the wave parameters. Therefore, the constancy of L may be exploited in a variety of analytical as well as in numerical studies of the nonlinear gyroresonant interaction between particles and quasi-monochromatic whistler mode, including the relativistic consideration.
ARTICLE IN PRESS 972
V.L. Krasovsky / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 969–972
In the absence of the inhomogeneity, ð1=kÞðdk=dzÞð1=AÞðdA=dzÞ ¼ 0, the technique to solve the equations of electron motion may be found in Roberts and Buchsbaum (1964). Lutomirski and Sudan (1966) have also investigated the kinetic structure of exact nonlinear whistler mode presenting a counterpart of the well-known electrostatic Bernstein–Greene–Kruskal (BGK) wave (Bernstein et al., 1957). In the presence of a weak longitudinal plasma inhomogeneity, a0, the Hamiltonian (16) (or analogous expression in Krasovsky and Matsumoto, 1998 as applied to the case of nonuniform magnetic field) depends weakly on the ‘‘time’’ z, so that the electron motion can be treated as one-dimensional oscillations of the particle in a slowly varying effective potential. Under these conditions, it is natural to take advantage of an adiabatic approximation to solve the equations of motion (14) and (15) or to solve the corresponding Vlasov equation for the electron distribution function. Then, a self-consistent analytical description of the ‘‘adiabatic’’ gyroresonant interaction of the finite amplitude whistler wave with high energy electrons becomes possible on the basis of sufficiently accurate analysis demonstrated, for example, for slowly varying BGK waves in (Krasovsky, 1995). Hence, the equations of motion (14) and (15) may be used for a closed self-consistent description of the spatial evolution of the nonlinear whistler mode (Lutomirski and Sudan, 1966) propagating in a weakly inhomogeneous plasma. Finally, we will touch briefly on a more general type of the space–time dependence of the wave parameters. If the amplitude and frequency of the wave depend on time, as is observed, for example, in the course of the generation of triggered VLF radio emissions (experimental data have been presented, for example, by Carlson et al., 1990), one need to put A ¼ Aðz; tÞ and to add a correction jðtÞ to the phase of the wave c in (2). Then, reproduction of the mathematical treatments carried out in Section 2 leads to the result dL qA qj ¼ v? cos x þ A sin x a0, (19) dt qt qt i.e. the quantity L is not an exact constant. However, the right-hand side of (19) contains a double smallness since, in addition to the smallness of the amplitude A, the partial derivatives with respect to time must be proportional to a small number of resonant electrons. In view of the above, the approximate constancy of L might be used also
in studies (q=qta0).
of
the
nonstationary
phenomena
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