ISSN 00213640, JETP Letters, 2009, Vol. 89, No. 9, pp. 441–447. © Pleiades Publishing, Ltd., 2009. Original Russian Text © A.I. Neishtadt, A.V. Artemyev, L.M. Zelenyi, D.L. Vainshtein, 2009, published in Pis’ma v Zhurnal Éksperimental’noі i Teoreticheskoі Fiziki, 2009, Vol. 89, No. 9, pp. 528–534.
Surfatron Acceleration in Electromagnetic Waves with a Low Phase Velocity A. I. Neishtadta, b, A. V. Artemyeva, L. M. Zelenyia, and D. L. Vainshteina, c a
Space Research Institute, Russian Academy of Sciences, ul. Profsoyuznaya 84/32, Moscow, 117997 Russia email:
[email protected] b Department of Mathematical Sciences, Loughborough University, LE11 3TU Loughborough, United Kingdom c Department of Mechanical Engineering, Temple University, Philadelphia, PA 19122, USA Received March 10, 2009
The possibility of unlimited surfatron acceleration of nonrelativistic charged particles by slow electromag netic waves has been revealed. The capture into such an acceleration regime has been described. The region of the parameters in which the effect of the capture and acceleration exists has been considered. The contri bution of this mechanism to an increase in the energy of charged particles in the Earth’s magnetosphere has been estimated. PACS numbers: 05.45.a, 94.05.Pt DOI: 10.1134/S0021364009090045
The surfatron acceleration of charged particles is their acceleration by a plasma wave along the wave front that is ensured by the presence of a static mag netic field (see, e.g., [1], Ch. 8, Sect. 4). The possibil ity of such an acceleration by a shock wave was pointed out in [2]. Acceleration by an electrostatic wave prop agating perpendicularly to the magnetic field was con sidered in [3–6]. It was assumed that the wave was highfrequency sinusoidal (wave frequency is much higher than the Larmor frequency of particle rotation) and that a particle was initially on the bottom of a potential pit created due to the joint action of the wave and magnetic field. It was demonstrated that the energy of the accelerated nonrelativistic particle increases only to a certain limit [3, 4]. Then, the potential pit disappears and acceleration ceases. For certain relations between the parameters of the prob lem, the ultrarelativistic particle is infinitely acceler ated: its velocity approaches the speed of light and its energy increases infinitely [5] (the term surfatron acceleration was initially proposed when describing this phenomenon). The case where the particle was not initially in the potential pit was also analyzed. In this case, the main action of the wave on the particle occurs near the Cherenkov resonance, when the parti cle velocity projection on the wave propagation direc tion is close to the phase velocity of the wave. The action of the wave on the particle far from resonance can be neglected and the motion of the particle on the Larmor circle can be considered in the leading approximation. However, the particle velocity projec tion on the wave propagation direction varies in this motion and resonance should inevitably appear if the velocity of the particle is higher than the phase velocity
of the wave. Analysis of the phenomena occurring in this case leads to the known general problem of reso nance in systems with fast and slow motions ([7], Ch. 6, Sect. 1.7). In this case, capture into resonance is possible: the particle is trapped in the potential pit and begins to move so as to maintain the resonance state. This is the capture into the surfatron accelera tion regime, which is possible for both the nonrelativ istic [8, 9] and relativistic particles [10]. The results of work [10] was extended in [11, 12] to the case of an electrostatic wave propagating not perpendicularly to the magnetic field. The case of an ultrarelativistic par ticle and an electromagnetic wave was considered in [11, 13]. In this work, we analyze the possibility of the surfa tron acceleration of a nonrelativistic particle by a plasma electromagnetic wave propagating perpendic ularly to the static magnetic field. The main result is that the particle captured into the surfatron accelera tion regime in such a field configuration does not leave the potential pit and is infinitely accelerated. If the parameters of the problem are such that surfatron acceleration is impossible (the potential pit is absent), the energy of the particle changes in each passage through resonance, but the accumulation of these changes leads to a limited increase in energy (the par ticle energy ε cannot exceed a certain value ε*). This paper consists of three parts. The possibility of the capture of the particle by the wave is discussed and the subsequent acceleration of this particle is analyzed in the first part. The particle motion in the absence of capture into resonance is discussed and the diffusion mechanism of an increase in energy is developed in the second part. The third part is devoted to the determi
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introducing the time scale t0 (δ t0δ, bn t0bn), spatial scale l0 (r r/l0, k kl0), and the corre sponding velocity scale u = l0/t0 (v v/u, vφ vφ/u). We supplement the system of equations (1) by · the equation φ = k(vy – vφ). In the resulting system of equations, the variables vx and vy are slow in the case kvφ bn, and the variable φ can be considered as fast far from resonance (vy ≈ vφ). To approximately describe the dynamics far from resonance, we average the righthand side of the resulting system of equations with respect to the fast variable φ. As a result, we arrive at the equations v· x = bnvy and v· y = –bnvx describing 2
2
the Larmor motion. The Larmor circle v x + v y = φs
Fig. 1. Phase portrait of a pendulum with torque for the case vx > 0.
nation of the regions of the parameters in which cap ture and acceleration are observed. In this work, we consider the parameters typical for geophysics, whereas most works on surfatron acceleration are devoted to the relativistic limits and, thereby, the effects characteristic of interstellar space [14–16]. The problem under consideration has a planar geometry: particles move in the (x, y) plane and the magnetic field has a single component Bz(x, y, t). This field is the sum of the static component B0 and the ˜ 1 sin(kr – ωt). The electromagnetic wave field B1 = B electric field is determined from Maxwell’s equations (curlE = ⎯c ⎯1ez∂Bz/∂t), is parallel to the (x, y) plane, and has the components Ex(x, y, t) and Ey(x, y, t). The equations of motion of a charged particle with mass m and charge e in these fields have the form mdv/dt = eE + eBz(v × ez)/c and dr/dt = ∇. CAPTURE OF THE PARTICLE BY THE WAVE Let the wave propagate along the y axis: k = key with a phase velocity vφ = ω/k (the case of the wave with kvφ bn, bn = eB0/mc is considered analytically, but numerical results will be present for a wider set of parameters). Then, the equations of motion of the particle have the form v· x = δv φ sin ( φ ) + ( b n – δ sin ( φ ) )v y , (1) v· y = – ( b n – δ sin ( φ ) )v x . ˜ 1 /mc are the Here, φ = k(y – vφt) + φ0 and δ = –e B phase and amplitude of the wave, respectively. Sys tem (1) is reduced to the dimensionless variables by
2
v 0 = const in the (vx, vy) plane intersects the reso nance straight line vy = vφ at two points (we assume that v0 > vφ > 0). To describe the dynamics near reso · nance, we take into account that φ ≈ 0 and use φ and · φ instead of y and vy: v· x = b n v φ ,
(2)
·· φ = – k ( b n – δ sin ( φ ) )v x .
In this system, the variable vx changes more slowly · than a pair of φ and φ . For this reason, we first con sider the second equation in system (2) at a fixed vx value. This is the equation of a nonlinear pendulum with a constant torque. The phase portrait of such a pendulum for the case δ > bn is shown in Fig. 1. The thick solid line is the separatrix. In order to derive the formula for the area S under the separatrix, which is important for subsequent analysis, we write the Hamiltonian for the second equation of system (2): ·2 H = φ /2 + kvx(bnφ + δcos(φ)). The energy on the separatrix can be expressed as Hs = kvx(bnφs + δcos(φs)), where φs is the phase at the saddle point. · The separatrix intersects the axis φ = 0 at the points φs and φ2; the latter can be determined from the equation Hs = kvx(bnφ2 + δcos(φ2)). Then, S=2
3/2
k vx δ
φ2
×
∫
( φ s – φ )b n /δ + ( cos ( φ s ) – cos ( φ ) ) dφ.
(3)
φs
Further, we take into account a slow linear increase in vx with time according to the first equation of system (2): vx ~ bnvφt. The motion of particles located inside the region bounded by the separatrix (oscillation region marked as A in Fig. 1) can be represented as the superposition of motion along the phase contours of the Hamiltonian H and the evolution of these con tours. In this case, owing to the presence of fast and JETP LETTERS
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vy
Fig. 2. Capture of the particle by one wave for three wavenumber values. The parameters of system (1) are bn = π/4, δ = π, and vφ = 1.
slow variables, the system has an adiabatic invariant: evolution occurs so that the area I bounded by a phase contour remains unchanged in the leading approxi mation [17]. The area bounded by the separatrix increases in this motion at vx > 0, because this area is proportional to v x , and |vx | increases. Therefore, the particles located in the oscillation region at vx > 0 cannot leave it and are infinitely accelerated according to the law vx ~ bnvφt. For the same reason, new parti cles that execute Larmor motion and approach reso nance at vx > 0 can be captured into the oscillation region. Thus, of two points on the Larmor circle at which the particle is in resonance with the wave, only one point with vx > 0 remains at which the particle can be captured into resonance. Of an ensemble of parti cles approaching resonance at a certain value vx > 0, only a fraction P is captured into the oscillation region. In the leading order in 1/k 1, P = 2 (∂S/∂vx) v· x /(2πkbnvx) = Svφ/4πk v x . This expression follows from the general formula for the probability of capture into resonance (the capture probability appears because the initial data for the trajectories with and without capture are mixed) [7]. The capture probability is calculated as the ratio of the phase vol ume coming to the oscillation region to the sum of the phase volumes coming to the oscillation region and passing through resonance without the capture and is of the order of 1/ k . The numerical simulation for system (1) are pre sented in Fig. 2, where the particle trajectories are shown in the velocity space (the capture into reso nance occurs at vx > 0, in agreement with the theory). While the case of the wave with ω bn (correspond ingly, with k 1) was considered above, it is worth noting that, as seen in Fig. 2, similar effects also occur for the case of intermediate wavelengths and low fre JETP LETTERS
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quencies (ω ≈ bn and k ≈ 1). One can see in Fig. 3 that the velocity vx after capture increases linearly with time with the same coefficient bnvφ for all k and ω val · ues. The phase trajectory in the ( φ , φ) plane for the particle after capture is also shown in Fig. 3. Thus, the vx velocity component for the trapped particles increases linearly as vx ~ bnvφt, whereas the x coordinate increases quadratically as x ~ bnvφt2/2. Since I = const in the process of motion inside the oscillation region, the amplitude of oscillations of φ decreases with increasing time owing to an increase in vx. With each turn, the particle approaches the bottom of the potential well φ = φ0 and the amplitude of oscil · lations of φ increases (as clearly seen in the right panel in Fig. 3). Correspondingly, the amplitude of oscilla tions of y decreases and the amplitude of oscillations of vy increases. Let us determine the rate of the devel opment of these processes. When the amplitude of oscillations of φ becomes sufficiently small, the expan sion of the potential energy near the bottom of the potential well for the Hamiltonian can be taken in the quadratic approximation; i.e., the potential energy is ~kvxδq2/2, where q = φ – φ0 is the deviation of φ from the well bottom coordinate. Therefore, the oscillations of φ have the frequency Ω ~ kv x δ . In this approxi mation, the adiabatic invariant I is determined from the relation 2πI = H/Ω. At the time of capture into resonance, I ~ k . Hence, the amplitude of oscilla tions of φ satisfies the relation
k ~ kvxq2/ kv x δ . –1/4
Thus, oscillations of φ have the amplitude q ~ v x , which corresponds to oscillations in the y coordinate – 1/4 with the amplitude ~v x /k. Using again the conser vation of the adiabatic invariant, we conclude that the
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Fig. 3. (Left panel) Time dependence of the velocity vx for the solutions presented in Fig. 2 and (right panel) the particle trajectory · in the (φ, φ ) plane near resonance for the case k = 100. 1/4 · amplitude of oscillations of φ is ~v x and the ampli 1/4
tude of oscillations of vy is ~v x /k.
In order to analyze an increase in the energy with time for the ensemble of particles, we consider the geometry of the motion of the particle at the given field values. In the leading approximation, a phase point in the (vx, vy) plane moves along the Larmor circle with
DIFFUSION MECHANISM OF ENERGY GAIN Capture is impossible at δ < bn, because the poten tial energy of the Hamiltonian H has no minimum. However, in this case jumps of energy occur when passing resonance. The first moment of the statistical distribution of energy jumps is zero (i.e., the directed increase in energy is absent). The second moment is nonzero and, thereby, energy diffusion is possible (an increase in the standard deviation of energy from its initial value). The energy jump at one passage through resonance can be determined as a function of the parameters of the system. To this end, we use the equa tion for the particle energy: ε· = δvφvxsin(φ). Integrat ing over time and replacing the time integral with the integral over the phase φ (energy changes only near · resonance where φ = ± v x kδ f(φ, φ*), f(φ, φ*) = · 2 ( φ* – φ )b n /δ + cos ( φ* ) – cos ( φ ) and where φ = ± v x kδ f(φ, φ*) and φ* is the phase value at the time of passage through resonance), we obtain the follow ing expression for the energy jump: φ*
∆ε ∼ v φ v x δ/k
∫ sin ( φ ) dφ/f ( φ, φ* ).
(4)
–∞
The factor
∫
φ* –∞
ticle through resonance, φ*, and can be treated as a random variable.
sin ( φ ) dφ/f(φ, φ*) depends on the
phase of the wave at the time of the passage of the par
a radius of 2ε (see Fig. 4). The time of motion between the two intersections of the resonance line vy = vφ (for definiteness, we consider motion with vy ≥ vφ) is α/bn, where α = 2arccos(vφ/ 2ε ). The wave phase gain over this time is given by the expres 2 sion ∆φ = (2k/bn)(2ε – v φ )1/2 – ωα/bn, which is obtained by integrating the phase variation rate. When the energy is not too high, this phase gain can be con sidered as a random variable owing to particle energy jumps (since k and ω are large, a small change in the energy leads to a large change in the phase gain; for this reason, the phase values at successive passages through resonance can be treated as independent ran dom variables). Taking into account that vx = (2ε – 2
2
v φ )1/2, we arrive at the equation ∆ε/(2ε – v φ )1/4 ~ vφ δ/k ∆φ. Then, summing energy changes at pas sages through resonance as random variables ( ∆φ ~
∑
t1/2), we obtain the expression ε ~ (const + vφ tδ/k )4/3 for the time variation of the energy. Thus, the energy at the initial stage (until the dependence of the magnitude of the energy jump with 2 the value of the energy itself, ∆ε ~ (2ε – v φ )1/4, is not too pronounced) increases with time as ε ~ t1/2. Upon further increase of energy the exponent in the time JETP LETTERS
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vy
vφ
vx
Fig. 4. (Left panel) Scheme of particle motion in the (vx, vy) plane and (right panel) the time dependence of the energy of the ensemble of particles for system (1) with the parameters bn = 2π, δ = π/2, and vφ = 1.
Fig. 5. Poincaré section (105 points). The parameters of the system are δ = π/2, bn = 2π, k = 100, and vφ = 1.
dependence increases and the asymptotic dependence should be ε ~ t2/3. However, the gain of the wave phase with increasing particle energy between successive passages through resonance becomes more determin istic (α π, ∆ε∂∆φ/∂ε ~ ε–1/4 0). For this rea 2/3 son, the asymptotic behavior ε ~ t were inaccessible. However, the stage ε ~ t1/2, as well as some subsequent stages for which ε ~ tγ (1/2 < γ < 2/3), were obtained numerically. To this end, it is necessary to take various time scales (according to the formula for ∆ε, the energy increase rate decreases with increasing k) (see Fig. 4).
the Poincaré section for the ensemble of particles in the (y, vy) plane by marking the points spaced in time by 2π/ω (see Fig. 5). The energy increases at v0 ~ vφ due to diffusion, but this mechanism is absent at v0 vφ and points form a closed curve. Thus, a particle
Using the KAM theory [7], one can show that there is an energy ε*, such that energy diffusion is absent at ε ε*. To numerically confirm this statement, we plot
Now, we can analyze the contribution of this mech anism to the acceleration of particles in the Earth’s magnetosphere. Let us consider the region at a dis
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with an initial energy ε > v φ /2 in the system with δ < bn gains energy with time until reaching a limit, ε ~ ε* 2 v φ /2. APPLICATIONS AND CONCLUSIONS
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RE
∆r (RE) Fig. 6. (Left panel) Scheme of the interaction of an ion with a magnetosonic wave and (right panel) the energy of ions versus their displacement.
tance of about 20–30 RE (where RE is the Earth’s radius) from the Earth near the neutral line (in this region, the magnetic field is B0 < 5 nT). The role of electromagnetic wave is played by a strong transverse magnetosonic wave traveling along the magnetotail axis towards or outwards of the Earth (the y axis in our coordinates). The frequency of this wave ω i ω e (ωj = eB0/mj c, where mj with j = i and e are the masses of the ion and electron, respectively) is much higher than the ion cyclotron frequency ωi (i.e., the condition ω bn is satisfied). Since the plasma density in this region is about 0.5 cm–3, the velocity of the magneto sonic waves is vφ < 150 km/s [18]. Since the ion tem perature is about 1–5 keV, i.e., v0 > 300 km/s, the con dition v0 > vφ is also satisfied. Then, we can obtain the dependence of the maximum energy that can be reached by the ions on the distance to which they are displaced by the waves (see Fig. 6). Taking into account the spatial sizes of the region under consider ation (∆x < 40RE), we conclude that this mechanism applied to the magnetosonic waves in the magnetotail of the Earth’s magnetosphere can explain the acceler ation of individual groups of ions up to energies of 0.1–0.3 MeV. Such particle groups are regularly observed in this region and are responsible for the for mation of observed nonMaxwellian energy distribu tions [19]. In the present paper we have analyzed the possibil ity of capture and surfatron acceleration for the system consisting of a particle and a slow electromagnetic wave. The results indicate that the particle can be accelerated in this system and its energy increases with time as (bnvφt)2 and the distance from the initial point is ∆r ≈ 0.5bnvφt2 if the wave amplitude exceeds the external magnetic field. It also has been shown that this mechanism can be responsible for the appearance
of groups of accelerated particles in the Earth’s mag netosphere. We are grateful to A.A. Vasiliev for stimulating dis cussion and to H.V. Malova for assistance in the prep aration of this paper. This work was supported by the Russian Foundation for Basic Research (project nos. 090100333, 080200407, and 070200319) and by the Council of the President of the Russian Federation for Support of Young Scientists and Lead ing Scientific Schools (project no. NSh691.2008.1). REFERENCES 1. G. M. Zaslavskii and R. Z. Sagdeev, Introduction to Nonlinear Physics: From Pendulum to Turbulence and Chaos (Nauka, Moscow, 1988) [in Russian]. 2. R. Z. Sagdeev, Vopr. Teor. Plazmy 4, 23 (1966). 3. R. Z. Sagdeev and V. D. Shapiro, Pis’ma Zh. Eksp. Teor. Fiz. 17, 389 (1973) [JETP Lett. 17, 279 (1973)]. 4. J. M. Dawson, V. K. Decyk, R. W. Huff, et al., Phys. Rev. Lett. 50, 1455 (1983). 5. T. Katsouleas and J. M. Dawson, Phys. Rev. Lett. 51, 392 (1983). 6. R. Sugihara, S. Takeuchi, K. Sakai, et al., Phys. Rev. Lett. 52, 1500 (1984). 7. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathe matical Aspects of Classical and Celestial Mechanics (Encyclopaedia of Mathematical Sciences) (Editorial URSS, Moscow, 2002; Springer, Berlin, 2006). 8. D. L. Vainchtein, E. V. Rovinsky, L. M. Zelenyi, and A. I. Neishtadt, J. Nonlinear Sci. 14, 173 (2004). 9. G. M. Zaslavskii, A. I. Neishtadt, B. A. Petrovichev, and R. Z. Sagdeev, Fiz. Plazmy 15, 368 (1989) [Sov. J. Plasma Phys. 15, 368 (1989)]. 10. A. I. Neishtadt, B. A. Petrovichev, and A. A. Chernikov, Fiz. Plazmy 15, 593 (1989) [Sov. J. Plasma Phys. 15, 593 (1989)]. JETP LETTERS
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SURFATRON ACCELERATION IN ELECTROMAGNETIC WAVES 11. A. A. Chernikov, G. Shmidt, and A. I. Neishtadt, Phys. Rev. Lett. 68, 1507 (1992). 12. A. P. Itin, A. I. Neishtadt, and A. A. Vasiliev, Physica D 141, 281 (2000). 13. A. P. Itin, Fiz. Plazmy 28, 639 (2002) [Plasma Phys. Rep. 28, 592 (2002)]. 14. H. Karimabadi, K. Akimoto, N. Omidi, et al., Phys. Fluids B 2, 606 (1990). 15. N. S. Erokhin, S. S. Moiseev, and R. Z. Sagdeev, Pis’ma Astron. Zh. 15, 3 (1989) [Sov. Astron. Lett. 15, 1 (1989)].
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16. G. N. Kichigin, Zh. Eksp. Teor. Fiz. 119, 1038 (2001) [JETP 92, 895 (2001)]. 17. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 1: Mechanics (Nauka, Moscow, 1982; Per gamon, Oxford, 1988). 18. D. A. FrankKamenetskii, Lectures on Plasma Physics (Atomizdat, Moscow, 1968) [in Russian]. 19. S. P. Christon, D. J. Williams, D. G. Mitchell, et al., J. Geophys. Res. 94, 13409 (1990).
Translated by R. Tyapaev
