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non-Gaussian distributions
Accepted Manuscript Charged particle transport and energization by magnetic field fluctuations with Gaussian/nongaussian distributions Pavel Shustov, Anton Artemyev, Egor Yushkov
PII: DOI: Reference:
S0375-9601(14)01236-5 10.1016/j.physleta.2014.12.017 PLA 22997
To appear in:
Physics Letters A
Received date: 21 September 2014 Revised date: 29 October 2014 Accepted date: 3 December 2014
Please cite this article in press as: P. Shustov et al., Charged particle transport and energization by magnetic field fluctuations with Gaussian/nongaussian distributions, Phys. Lett. A (2015), http://dx.doi.org/10.1016/j.physleta.2014.12.017
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Highlights
• Ion transport/acceleration by magnetic fluctuations with different distributions. • The most effective acceleration is for nongaussian magnetic field fluctuations • Both Gaussian/nongaussian distributions give similar energy spectrum shape.
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Charged particle transport and energization by magnetic field fluctuations with Gaussian/nongaussian distributions.
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Pavel Shustov1,2 , Anton Artemyev2 , and Egor Yushkov1,2
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Department of Physics, Moscow State University, Moscow, 119992 Russia. Space Research Institute, Profsoyuznaya 84/32, Moscow 117997, Russia
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Abstract In this paper we investigate charged particle transport and acceleration in a two-dimensional system with an uniform electric field and stationary magnetic field fluctuations. The main idea of this study is to consider dependencies of transport and acceleration rates on properties of distributions of magnetic field fluctuations. We develop a simplified model of magnetic fluctuations with a regulated distribution and apply the test particle approach. System parameters are chosen to simulate conditions typical for ion dynamics in the deep Earth magnetotail. We show that for a fixed power density of magnetic field fluctuations the particle acceleration is more effective in the system where particles interact with small-amplitude (but frequent) fluctuations. In systems with large-amplitude rare fluctuations the particle scattering is less effective and the particle acceleration is weaker.
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Keywords: charged particle acceleration and transport, magnetic field fluctuations, planetary magnetotails
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1. Introduction
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Charged particle acceleration and transport by electromagnetic turbulent fields are considered now as one of the main mechanisms responsible for generation of high-energy particles in many cosmic plasma systems: planetary magnetospaheres [1, 2, 3, 4], solar corona [5, 6], supernova remnants [7]. The important problem of charged particle scattering by electromagnetic turbulence is the relation between rates of particle transport and acceleration. In many systems (planetary magnetotails, solar flares) a turbulent field develops
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Preprint submitted to Physics Letters A
December 10, 2014
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only in compact space domains where plasma instabilities generate magnetic field fluctuations. In this case, the maximum possible energy gained by particles is limited by a time interval which particles can spend in such turbulent regions. However, electromagnetic field fluctuations result also in particle transport. Thus, there is a competition between particle spatial transport and acceleration: particles are accelerated only in the turbulent field domain, but the same field is responsible for a particle escape from this domain. Relationship between the charged particle turbulent transport and acceleration is determined by properties of electromagnetic field fluctuations. For example, the intermittency of fluctuations can provide more effective acceleration for the same transport rate in some particular systems [8]. Thus, the investigation of charged particle interaction with electromagnetic field fluctuations with different spatial and temporal distributions is important and perspective problem of space plasma physics. In-situ spacecraft observations in the near-Earth plasma environment have revealed various properties of electromagnetic field fluctuations [9, 10, 11]. For example, distributions of magnetic field fluctuations in the magnetotail region are often significantly nongaussian [12]. This property should influence on charged particle transport. In this paper we investigate the corresponding effects using the test particle approach and a simple model of magnetic field fluctuations with a controlled level of deviation from the gaussian form. We concentrate on specific plasma system (the distant Earth magnetotail) where a combination of magnetic field fluctuations [13, 14] and a large-scale convection electric field [15, 16] provides the effective energization of solar wind protons penetrated into the magnetosphere.
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2. The magnetic field model and test particle approach
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We consider the simplified system geometry: particles move in the (x, y) plane, while the magnetic field is directed along the z-axis. The magnetic field can be presented as a sum of the constant background field B0 and fluctuationing component δBz (x, y). There is also a constant electric field Ey provided by the solar wind interaction with the planet magnetosphere [16]. In the realistic magnetosphere configuration there is one additional component of the magnetic field Bx (z) [17]. However, this component vanishes at the plane z = 0 and can be omitted for simplicity of calculations of test particle trajectories [18, 8].
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For magnetic field fluctuations δBz (x, y) we use the model proposed in
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[1]: Nk Nθ B1 δBz = √ Pk cos (Φ) Nθ Nk n =1 n =1 θ
k
Φ(k, θ, x, y) = k(x cos θ + y sin θ) + φnk ,nθ 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73
where B1 is an amplitude of fluctuations, Nk = 100, Nθ = 100 are number of harmonics, Pk ∼ k −1 is the spectrum taken from [3], φnk ,nθ ∈ [0, 2π] is randomly distributed initial phases. The angle θ is distributed uniformly as θ = 2π(nθ −1)/(Nθ −1), while the wave vector is k = Δknk and Δk = 0.1/L0 , L0 is a typical spatial scale of magnetic field fluctuations (below we introduce L0 through parameters of charged particles). We consider a case of stationary magnetic field fluctuations when particles along their trajectories interact with spatially varying static magnetic fields. Thus, magnetic field fluctuations given by Eq. (1) do not depend on time. These fluctuations have a Gaussian distribution. To check this we calculate the differences ΔBi = δBz (x+(i+1)Δx, y+(i+1)Δy)−δBz (x+iΔx, y+iΔy) with Δx = δ0 cos η, Δy = δ0 sin η, η ∈ [0, 2π] is a random value, δ0 is the step size of magnetic field calculations along a virtual line (here we set δ0 = 1), i = 0, 1, 2.... The distribution of ΔBi for model (1) is shown in Fig. 1(λ = 1 line). To obtain a nongaussian distribution of magnetic field fluctuations (λ) we modify model (1): δBz (x, y) = Cλ sign(δBz (x, y))|δBz (x, y)|λ where δBz (x, y) is calculated with Eq. (1). The constant coefficient Cλ is used to normalize the power density of magnetic field fluctuations for different λ: L L Cλ2 =
−L −L
L L −L −L
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δBz2 (x, y)dxdy
|δBz (x, y)|2λ dxdy
where L = 10/Δk. Distributions of magnetic field fluctuations ΔB for different λ are shown in Fig. 1. One can see that the increase of λ results in modification of the ΔB distribution: for larger λ the distribution contains more high values of ΔB. Roughly speaking, in the distribution of ΔB with large λ the small amplitude fluctuations are suppressed, while the large amplitude fluctuations are amplified. On the other side, we can modify the ΔB 3
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distribution by decreasing the λ parameter. In this case, we suppress the large amplitude fluctuations and amplify the small amplitude fluctuations. The resulting ΔB distribution is also nongaussian (see Fig. 1). Distributions shown in Fig. 1 are typical for the Earth magnetosphere and solar wind [19, 12, 20]. Spacecraft measurements of quasi-stationary magnetic field fluctuations transform spatial-scales to time-scales because a velocity of plasma (and frozen-in magnetic field) convection is significantly larger than the spacecraft velocity. Thus, spatial scales of magnetic field inhomogeneity are measured by spacecraft as time-scales of magnetic field variations. These time-scales are inverse proportional of the plasma convection velocity. In this case, the higher velocity of plasma convection (i.e. more disturbed conditions in the magnetosphere [21, 22]) corresponds to a smaller time-scale δ0 of calculations of magnetic field differences ΔB. Indeed, the Gaussian distribution of ΔB obtained in our model with λ = 1 approximates well the magnetic field fluctuations measured during quiet geomagnetic conditions (i.e. measurements with larger δ0 ), while λ = 1 corresponds to magnetic field fluctuations measured in more disturbed magnetosphere (i.e. measurements with smaller δ0 ). The corresponding evolution of measured distributions of magnetic field fluctuations with δ0 can be found in [12, 20].
Figure 1: Distributions of magnetic field fluctuations ΔB for different system parameters. Total number of points in each distribution is 106 . 99 100 101
To investigate the charged particle interaction with fluctuating magnetic field we use the test particle approach. Equations of motions of a nonrelativistic ion with the charge q and the mass m can be written as x¨ =
qB0 (1 + b(x, y)) y˙ mc 4
qEy qB0 − (1 + b(x, y)) x˙ m mc where b(x, y) = δBz /B0 . We introduce the dimensionless time t → tqB0 /mc, the dimensionless velocities (vx , vy ) = (x, ˙ y)/ ˙ 2H0 /m where H0 is an initial particle energy. √ The spatial coordinates are normalized as: (x, y) → (x, y)/L0 where L0 = 2H0 mc2 /qB0 . We also introduce the dimensionless particle 2 2 energy ε = (vx + vy )/2 and the drift velocity VD = cEy /(B0 2H0 /m) characterizing the electric field intensity. For the typical conditions in the deep Earth magnetotail we can estimate B0 ∼ 1 − 3 nT [23], δBz ∼ 0.1 − 1 nT [13, 14], ion energy H0 ∼ 1 − 2 keV [24] (and corresponding L0 ∼ 5000 km), electrostatic field Ey = 0.1 − 0.3 mV/m [25]. Thus, we have following values of dimensionless parameters: b ∼ 0.1, VD ∼ 0.1 − 0.5. Serval examples of particle trajectories are shown in Fig. 2. In the case of the electric field absence (VD = 0), a particle randomly walks in the plane (x, y) with the constant energy. This motion can be considered as a combination of the particle gyrorotation around the background magnetic field B0 and the particle scattering by magnetic field fluctuations δBz . In the system with a finite VD the particle gains energy and a spatial scale of its gyrorotation (i.e. the particle gyroradius) increases with time. In this case, the particle motion can be considered as a combination of the regular drift along the x-axis with velocity VD and the random walking in the plane (x, y) due to the interaction with magnetic field fluctuations. For larger value of λ the particle more often interacts with large magnetic field fluctuations, but between such interactions the particle moves in almost constant background magnetic field due to absence of small amplitude fluctuations (see distributions in Fig. 1). As a result, the drift ∼ VD becomes more important in the system with λ > 1. To statistically characterize the charged particle interaction with spatially fluctuating magnetic field we numerically integrate the ensemble of 106 trajectories. For this ensemble we calculate averaged particle displacements in geometrical and velocity spaces: N R(t) = N1 (xi (t) − x0i )2 + (yi (t) − y0i )2 i=1 N 2 2 V (t) = N1 vxi (t) + vyi (t) y¨ =
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i=1
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where the subscript 0 corresponds to initial values. We also collect statistics 5
Figure 2: Four examples of particle trajectories and the corresponding profiles of the particle energy are shown for different system parameters. All simulations are carried out with B1 /B0 = 0.1.
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of energy distributions for different moments of time, and distributions of particle spatial jumps Δt . The latter one is calculated as a difference between particle positions Δt = (x(ti+1 ) − x(ti ))2 + (y(ti+1 ) − y(ti ))2
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for two fixed moments of time ti and ti+1 = ti + Δt.
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3. Particle transport and energization
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To calculate profiles of R(t) and V (t) For several sets of system parameters (λ and VD ) we numerically integrate 106 trajectories with initially random positions. For the simplest case with VD = 0 and the √ Gaussian distribution of magnetic field fluctuations (λ = 1) we obtain R ∼ t (see the first panel in Fig. 3). This simple behavior of the particle displacement R(t) resembles the classical Brownian motion [26]. Charged particles interact with random fluctuations of the magnetic field and drift stochastically due to the magnetic field inhomogeneity [see 27, 28, 29]. The time profile R(t) changes substantially if we add a constant electric field VD to the system (see the second panel in Fig. 3). In this case, there are two processes: the spatial diffusion of particles and the drift with the
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velocity VD directed along the x-axis (the electric field is directed along yaxis). Fig. 3 shows that for reasonable amplitudes of the electric field and magnetic field fluctuations the drift effect is stronger and R(t) is about ∼ VD t. However, there is also some displacement in y direction due to magnetic field fluctuations (see trajectories in Fig. 2). The only exception is the system with λ < 1. In this case, the small-amplitude and very frequent magnetic field fluctuations significantly influence the charged particle transport. As a result, the displacement R(t) grows with time slower than in the systems with λ ≥ 1. In contrast to R(t) profiles,√ the displacement in the velocity space V (t) increases with time as V (t) ∼ t. In the system with the Gaussian distribution of fluctuations (λ = 1) the acceleration process is more effective than in systems with nongaussian large-amplitude fluctuations (λ = 5). The charged particle energy is conserved in the course of the cross-field drift in the constant electric and magnetic fields. Thus, the acceleration occurs only when particles meet magnetic field fluctuations. Profiles V (t) clearly show that it is more effective for particles to frequently interact with small-amplitude fluctuations (i.e. with the Gaussian distribution of fluctuations or with the λ < 1 distribution) than rarely interact with large amplitude fluctuations (i.e. with the nongaussian distribution of fluctuations corresponding to λ > 1). It is rather unexpected result showing that the distribution of magnetic field fluctuations can change the acceleration rate even if the total power density of magnetic field fluctuations is fixed. The last panel of Fig. 3 demonstrates that for the same displacement R the acceleration V is more effective for λ = 1 and λ < 1 than for λ > 1. Thus, in a localized region of magnetic field fluctuations the particles can gain more energy if these fluctuations are more frequent (even if fluctuation amplitudes are small).
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4. Particle statistics
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In this section we consider particle energy distributions and distributions of particle spatial jumps Δt . In Fig. 4 we show two sets of energy distributions for λ = 1 and λ = 5 (these distributions are used to plot V (t) profiles in Fig. 3). For both values of λ the energy distributions have a Gaussian-line exponential tail ∼ exp(−ε). The difference between two sets of distributions mainly corresponds to the average energy (the position of the distribution peak or plateau) and the energy maximum (the position where the exponential tail drops to small values ∼ 1 − 10). For λ = 1 the typical and maximum
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Figure 3: Profiles of particle displacements in geometry and velocity spaces R, V for different system parameters (see the text for details). All simulations are carried out with B1 /B0 = 0.1. 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199
energy values are somewhat larger than for λ = 5 (the same time moments). Distributions shown in Fig. 4 clearly demonstrate a very fast (exponential ∼ exp(−ε)) drop of number of accelerated particles with the energy increase. This property significantly differs obtained energy distributions from results of spacecraft observations in the Earth magnetotail where power law spectra are often collected [30, 31, 32]. We show that the charged particle acceleration by the convection electric field cannot reproduce heavy tails of energy spectra independently on a distribution of magnetic field fluctuations in the magnetometric. Thus, additional mechanisms of charged particle acceleration are necessary to modify the modeled energy spectra. This important conclusion points out to the significant role of inductive electric fields in generation of a high-energy ion population in the magnetotail, while the convection electric field and charged particle scattering by stationary magnetic field fluctuations can result only in heating of plasma (see, also [18, 4]). In Fig. 5 we show distributions of spatial jumps Δt of charged particles interacting with magnetic field fluctuations. There is a clear difference be8
Figure 4: Energy distributions for two values of λ and several time moments (indicated by colors). All simulations are carried out with VD = 0.5 and B1 /B0 = 0.1. 200 201 202 203 204 205 206 207 208 209 210 211
tween Δt -distributions obtained for λ = 1 and λ = 5. For Gaussian magnetic field fluctuations (λ = 1) we observe more dispersive Δt -distributions than for nongaussian fluctuations (λ = 5). If particles interact with frequent smallscale fluctuations then their spatial displacements change in the diffusive-like manner: corresponding Δt -distributions are gaussian-like with a wide core and an exponential tail (this is especially well seen for the small drift velocity VD = 0.1). In contrast, particles interacting with large amplitude rare fluctuations simply drift with an almost constant velocity VD . The corresponding Δt -distribution is very narrow with the main peak at Δt ∼ VD Δt. This is drastically difference of the charged particle transports in turbulent magnetic field with the fixed power of fluctuations but with different distributions of fluctuations.
Figure 5: Distributions of spatial jumps Δt for two values of λ, two values of VD and several time moments (indicated by colors). All simulations are carried out with B1 /B0 = 0.1.
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5. Conclusions
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In this paper we have considered the influence of distribution form of magnetic field fluctuations on charged particle transport and acceleration in the 2D system with the constant electric field. We use the test particle approach and compare averaged displacements in spatial and velocity spaces R(t), V (t) of particle ensembles obtained in the cases of Gaussian and nongaussian distributions of magnetic field fluctuations. The model of magnetic field fluctuations is constructed so that the total power density of fluctuations is fixed, while the parameter λ > 0 regulates properties of the distribution of fluctuations: larger λ corresponds to larger amplitudes of fluctuations. Thus, due to the conservation of the total power density, the larger λ corresponds to smaller number of fluctuations, i.e. the increase of λ results in the suppression of small amplitude fluctuations and the amplification of large amplitude fluctuations. For λ = 1 the distribution of fluctuations is Gaussian, for λ < 1 this distribution contains small-amplitude fluctuations, for λ > 1 the distribution contains large amplitude fluctuations (see Fig. 1). Integration of a large number of particle trajectories in this magnetic field model allow us to obtain the following results 1. The larger λ value corresponds to weaker acceleration of particles within the fixed space domain: the profile V (R) grows slower for larger λ. The most effective acceleration for finite R is obtained for nongaussian distribution of magnetic field fluctuations with amplified smallamplitude fluctuations (λ < 1). Particles gain more energy if they interact with magnetic fluctuations more frequently, even if amplitudes of these fluctuations are small. 2. The shape of energy distributions of accelerated particles does not significantly depend on the distribution of magnetic field fluctuations. For both Gaussian and nongaussian distributions of fluctuations we obtain similar energy spectra with different locations of peaks. 3. The distributions of spatial jumps Δt are significantly different for Gaussian and nongaussian distributions of fluctuations. In case of small amplitude frequent fluctuations we have the gaussian-like distribution of Δt , while for high-amplitude rare fluctuations we have a transport of particles with the almost constant drift velocity VD .
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Obtained results show that the power spectral density of magnetic field fluctuations cannot alone characterizes the efficiency of charged particle scattering in space plasma systems. For example, in this paper we consider the 10
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conditions typical for the deep magnetotail of the Earth magnetosphere. On the basis of our model results we can mention that large amplitude fluctuations with nongaussian distributions correspond to the almost unperturbed particle cross-field drift with a weak acceleration. In contrast, the frequent small-amplitude fluctuations should result in the effective particle diffusion in space. In the latter case, the larger energy can be gained by particles for fixed size of the spatial domain filled by fluctuating magnetic field. The authors are thankful to Prof. L. M. Zelenyi and Dr. I. Y. Vasko for useful discussions. The work was supported by the Russian Scientific Fund (project # 14-12-00824). [1] P. Veltri, G. Zimbardo, A. L. Taktakishvili, L. M. Zelenyi, Effect of magnetic turbulence on the ion dynamics in the distant magnetotail, J. Geophys. Res. 103 (1998) 14897–14916. doi:10.1029/98JA00211. [2] A. Greco, P. Veltri, G. Zimbardo, A. L. Taktakishvilli, L. M. Zelenyi, Numerical simulation of ion dynamics in the magnetotail magnetic turbulence: On collisionless conductivity, Nonlinear Processes in Geophysics 7 (2000) 159–166. [3] L. M. Zelenyi, A. Artemyev, H. Malova, A. V. Milovanov, G. Zimbardo, Particle transport and acceleration in a time-varying electromagnetic field with a multi-scale structure, Physics Letters A 372 (2008) 6284– 6287. doi:10.1016/j.physleta.2008.08.035. [4] S. Perri, G. Zimbardo, A. Greco, On the energization of protons interacting with 3-D time-dependent electromagnetic fields in the Earth’s magnetotail, J. Geophys. Res. 116 (2011) A05221. doi:10.1029/2010JA016328. [5] P. Dmitruk, W. H. Matthaeus, N. Seenu, Test Particle Energization by Current Sheets and Nonuniform Fields in Magnetohydrodynamic Turbulence, Astrophys. J. 617 (2004) 667–679. doi:10.1086/425301. [6] V. Petrosian, Stochastic Acceleration by Turbulence, Space Sci. Rev. 173 (2012) 535–556. , doi:10.1007/s11214-012-9900-6. [7] A. M. Bykov, Y. A. Uvarov, D. C. Ellison, Dots, Clumps, and Filaments: The Intermittent Images of Synchrotron Emission in Random Magnetic Fields of Young Supernova Remnants, The Astrophysical Journal 689 (2008) L133–L136. , doi:10.1086/595868. 11
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PII: DOI: Reference:
S0375-9601(14)01236-5 10.1016/j.physleta.2014.12.017 PLA 22997
To appear in:
Physics Letters A
Received date: 21 September 2014 Revised date: 29 October 2014 Accepted date: 3 December 2014
Please cite this article in press as: P. Shustov et al., Charged particle transport and energization by magnetic field fluctuations with Gaussian/nongaussian distributions, Phys. Lett. A (2015), http://dx.doi.org/10.1016/j.physleta.2014.12.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
• Ion transport/acceleration by magnetic fluctuations with different distributions. • The most effective acceleration is for nongaussian magnetic field fluctuations • Both Gaussian/nongaussian distributions give similar energy spectrum shape.
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Charged particle transport and energization by magnetic field fluctuations with Gaussian/nongaussian distributions.
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Pavel Shustov1,2 , Anton Artemyev2 , and Egor Yushkov1,2
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1
Department of Physics, Moscow State University, Moscow, 119992 Russia. Space Research Institute, Profsoyuznaya 84/32, Moscow 117997, Russia
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Abstract In this paper we investigate charged particle transport and acceleration in a two-dimensional system with an uniform electric field and stationary magnetic field fluctuations. The main idea of this study is to consider dependencies of transport and acceleration rates on properties of distributions of magnetic field fluctuations. We develop a simplified model of magnetic fluctuations with a regulated distribution and apply the test particle approach. System parameters are chosen to simulate conditions typical for ion dynamics in the deep Earth magnetotail. We show that for a fixed power density of magnetic field fluctuations the particle acceleration is more effective in the system where particles interact with small-amplitude (but frequent) fluctuations. In systems with large-amplitude rare fluctuations the particle scattering is less effective and the particle acceleration is weaker.
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Keywords: charged particle acceleration and transport, magnetic field fluctuations, planetary magnetotails
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1. Introduction
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Charged particle acceleration and transport by electromagnetic turbulent fields are considered now as one of the main mechanisms responsible for generation of high-energy particles in many cosmic plasma systems: planetary magnetospaheres [1, 2, 3, 4], solar corona [5, 6], supernova remnants [7]. The important problem of charged particle scattering by electromagnetic turbulence is the relation between rates of particle transport and acceleration. In many systems (planetary magnetotails, solar flares) a turbulent field develops
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Preprint submitted to Physics Letters A
December 10, 2014
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only in compact space domains where plasma instabilities generate magnetic field fluctuations. In this case, the maximum possible energy gained by particles is limited by a time interval which particles can spend in such turbulent regions. However, electromagnetic field fluctuations result also in particle transport. Thus, there is a competition between particle spatial transport and acceleration: particles are accelerated only in the turbulent field domain, but the same field is responsible for a particle escape from this domain. Relationship between the charged particle turbulent transport and acceleration is determined by properties of electromagnetic field fluctuations. For example, the intermittency of fluctuations can provide more effective acceleration for the same transport rate in some particular systems [8]. Thus, the investigation of charged particle interaction with electromagnetic field fluctuations with different spatial and temporal distributions is important and perspective problem of space plasma physics. In-situ spacecraft observations in the near-Earth plasma environment have revealed various properties of electromagnetic field fluctuations [9, 10, 11]. For example, distributions of magnetic field fluctuations in the magnetotail region are often significantly nongaussian [12]. This property should influence on charged particle transport. In this paper we investigate the corresponding effects using the test particle approach and a simple model of magnetic field fluctuations with a controlled level of deviation from the gaussian form. We concentrate on specific plasma system (the distant Earth magnetotail) where a combination of magnetic field fluctuations [13, 14] and a large-scale convection electric field [15, 16] provides the effective energization of solar wind protons penetrated into the magnetosphere.
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2. The magnetic field model and test particle approach
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We consider the simplified system geometry: particles move in the (x, y) plane, while the magnetic field is directed along the z-axis. The magnetic field can be presented as a sum of the constant background field B0 and fluctuationing component δBz (x, y). There is also a constant electric field Ey provided by the solar wind interaction with the planet magnetosphere [16]. In the realistic magnetosphere configuration there is one additional component of the magnetic field Bx (z) [17]. However, this component vanishes at the plane z = 0 and can be omitted for simplicity of calculations of test particle trajectories [18, 8].
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For magnetic field fluctuations δBz (x, y) we use the model proposed in
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[1]: Nk Nθ B1 δBz = √ Pk cos (Φ) Nθ Nk n =1 n =1 θ
k
Φ(k, θ, x, y) = k(x cos θ + y sin θ) + φnk ,nθ 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73
where B1 is an amplitude of fluctuations, Nk = 100, Nθ = 100 are number of harmonics, Pk ∼ k −1 is the spectrum taken from [3], φnk ,nθ ∈ [0, 2π] is randomly distributed initial phases. The angle θ is distributed uniformly as θ = 2π(nθ −1)/(Nθ −1), while the wave vector is k = Δknk and Δk = 0.1/L0 , L0 is a typical spatial scale of magnetic field fluctuations (below we introduce L0 through parameters of charged particles). We consider a case of stationary magnetic field fluctuations when particles along their trajectories interact with spatially varying static magnetic fields. Thus, magnetic field fluctuations given by Eq. (1) do not depend on time. These fluctuations have a Gaussian distribution. To check this we calculate the differences ΔBi = δBz (x+(i+1)Δx, y+(i+1)Δy)−δBz (x+iΔx, y+iΔy) with Δx = δ0 cos η, Δy = δ0 sin η, η ∈ [0, 2π] is a random value, δ0 is the step size of magnetic field calculations along a virtual line (here we set δ0 = 1), i = 0, 1, 2.... The distribution of ΔBi for model (1) is shown in Fig. 1(λ = 1 line). To obtain a nongaussian distribution of magnetic field fluctuations (λ) we modify model (1): δBz (x, y) = Cλ sign(δBz (x, y))|δBz (x, y)|λ where δBz (x, y) is calculated with Eq. (1). The constant coefficient Cλ is used to normalize the power density of magnetic field fluctuations for different λ: L L Cλ2 =
−L −L
L L −L −L
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δBz2 (x, y)dxdy
|δBz (x, y)|2λ dxdy
where L = 10/Δk. Distributions of magnetic field fluctuations ΔB for different λ are shown in Fig. 1. One can see that the increase of λ results in modification of the ΔB distribution: for larger λ the distribution contains more high values of ΔB. Roughly speaking, in the distribution of ΔB with large λ the small amplitude fluctuations are suppressed, while the large amplitude fluctuations are amplified. On the other side, we can modify the ΔB 3
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distribution by decreasing the λ parameter. In this case, we suppress the large amplitude fluctuations and amplify the small amplitude fluctuations. The resulting ΔB distribution is also nongaussian (see Fig. 1). Distributions shown in Fig. 1 are typical for the Earth magnetosphere and solar wind [19, 12, 20]. Spacecraft measurements of quasi-stationary magnetic field fluctuations transform spatial-scales to time-scales because a velocity of plasma (and frozen-in magnetic field) convection is significantly larger than the spacecraft velocity. Thus, spatial scales of magnetic field inhomogeneity are measured by spacecraft as time-scales of magnetic field variations. These time-scales are inverse proportional of the plasma convection velocity. In this case, the higher velocity of plasma convection (i.e. more disturbed conditions in the magnetosphere [21, 22]) corresponds to a smaller time-scale δ0 of calculations of magnetic field differences ΔB. Indeed, the Gaussian distribution of ΔB obtained in our model with λ = 1 approximates well the magnetic field fluctuations measured during quiet geomagnetic conditions (i.e. measurements with larger δ0 ), while λ = 1 corresponds to magnetic field fluctuations measured in more disturbed magnetosphere (i.e. measurements with smaller δ0 ). The corresponding evolution of measured distributions of magnetic field fluctuations with δ0 can be found in [12, 20].
Figure 1: Distributions of magnetic field fluctuations ΔB for different system parameters. Total number of points in each distribution is 106 . 99 100 101
To investigate the charged particle interaction with fluctuating magnetic field we use the test particle approach. Equations of motions of a nonrelativistic ion with the charge q and the mass m can be written as x¨ =
qB0 (1 + b(x, y)) y˙ mc 4
qEy qB0 − (1 + b(x, y)) x˙ m mc where b(x, y) = δBz /B0 . We introduce the dimensionless time t → tqB0 /mc, the dimensionless velocities (vx , vy ) = (x, ˙ y)/ ˙ 2H0 /m where H0 is an initial particle energy. √ The spatial coordinates are normalized as: (x, y) → (x, y)/L0 where L0 = 2H0 mc2 /qB0 . We also introduce the dimensionless particle 2 2 energy ε = (vx + vy )/2 and the drift velocity VD = cEy /(B0 2H0 /m) characterizing the electric field intensity. For the typical conditions in the deep Earth magnetotail we can estimate B0 ∼ 1 − 3 nT [23], δBz ∼ 0.1 − 1 nT [13, 14], ion energy H0 ∼ 1 − 2 keV [24] (and corresponding L0 ∼ 5000 km), electrostatic field Ey = 0.1 − 0.3 mV/m [25]. Thus, we have following values of dimensionless parameters: b ∼ 0.1, VD ∼ 0.1 − 0.5. Serval examples of particle trajectories are shown in Fig. 2. In the case of the electric field absence (VD = 0), a particle randomly walks in the plane (x, y) with the constant energy. This motion can be considered as a combination of the particle gyrorotation around the background magnetic field B0 and the particle scattering by magnetic field fluctuations δBz . In the system with a finite VD the particle gains energy and a spatial scale of its gyrorotation (i.e. the particle gyroradius) increases with time. In this case, the particle motion can be considered as a combination of the regular drift along the x-axis with velocity VD and the random walking in the plane (x, y) due to the interaction with magnetic field fluctuations. For larger value of λ the particle more often interacts with large magnetic field fluctuations, but between such interactions the particle moves in almost constant background magnetic field due to absence of small amplitude fluctuations (see distributions in Fig. 1). As a result, the drift ∼ VD becomes more important in the system with λ > 1. To statistically characterize the charged particle interaction with spatially fluctuating magnetic field we numerically integrate the ensemble of 106 trajectories. For this ensemble we calculate averaged particle displacements in geometrical and velocity spaces: N R(t) = N1 (xi (t) − x0i )2 + (yi (t) − y0i )2 i=1 N 2 2 V (t) = N1 vxi (t) + vyi (t) y¨ =
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i=1
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where the subscript 0 corresponds to initial values. We also collect statistics 5
Figure 2: Four examples of particle trajectories and the corresponding profiles of the particle energy are shown for different system parameters. All simulations are carried out with B1 /B0 = 0.1.
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of energy distributions for different moments of time, and distributions of particle spatial jumps Δt . The latter one is calculated as a difference between particle positions Δt = (x(ti+1 ) − x(ti ))2 + (y(ti+1 ) − y(ti ))2
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for two fixed moments of time ti and ti+1 = ti + Δt.
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3. Particle transport and energization
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To calculate profiles of R(t) and V (t) For several sets of system parameters (λ and VD ) we numerically integrate 106 trajectories with initially random positions. For the simplest case with VD = 0 and the √ Gaussian distribution of magnetic field fluctuations (λ = 1) we obtain R ∼ t (see the first panel in Fig. 3). This simple behavior of the particle displacement R(t) resembles the classical Brownian motion [26]. Charged particles interact with random fluctuations of the magnetic field and drift stochastically due to the magnetic field inhomogeneity [see 27, 28, 29]. The time profile R(t) changes substantially if we add a constant electric field VD to the system (see the second panel in Fig. 3). In this case, there are two processes: the spatial diffusion of particles and the drift with the
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velocity VD directed along the x-axis (the electric field is directed along yaxis). Fig. 3 shows that for reasonable amplitudes of the electric field and magnetic field fluctuations the drift effect is stronger and R(t) is about ∼ VD t. However, there is also some displacement in y direction due to magnetic field fluctuations (see trajectories in Fig. 2). The only exception is the system with λ < 1. In this case, the small-amplitude and very frequent magnetic field fluctuations significantly influence the charged particle transport. As a result, the displacement R(t) grows with time slower than in the systems with λ ≥ 1. In contrast to R(t) profiles,√ the displacement in the velocity space V (t) increases with time as V (t) ∼ t. In the system with the Gaussian distribution of fluctuations (λ = 1) the acceleration process is more effective than in systems with nongaussian large-amplitude fluctuations (λ = 5). The charged particle energy is conserved in the course of the cross-field drift in the constant electric and magnetic fields. Thus, the acceleration occurs only when particles meet magnetic field fluctuations. Profiles V (t) clearly show that it is more effective for particles to frequently interact with small-amplitude fluctuations (i.e. with the Gaussian distribution of fluctuations or with the λ < 1 distribution) than rarely interact with large amplitude fluctuations (i.e. with the nongaussian distribution of fluctuations corresponding to λ > 1). It is rather unexpected result showing that the distribution of magnetic field fluctuations can change the acceleration rate even if the total power density of magnetic field fluctuations is fixed. The last panel of Fig. 3 demonstrates that for the same displacement R the acceleration V is more effective for λ = 1 and λ < 1 than for λ > 1. Thus, in a localized region of magnetic field fluctuations the particles can gain more energy if these fluctuations are more frequent (even if fluctuation amplitudes are small).
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4. Particle statistics
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In this section we consider particle energy distributions and distributions of particle spatial jumps Δt . In Fig. 4 we show two sets of energy distributions for λ = 1 and λ = 5 (these distributions are used to plot V (t) profiles in Fig. 3). For both values of λ the energy distributions have a Gaussian-line exponential tail ∼ exp(−ε). The difference between two sets of distributions mainly corresponds to the average energy (the position of the distribution peak or plateau) and the energy maximum (the position where the exponential tail drops to small values ∼ 1 − 10). For λ = 1 the typical and maximum
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Figure 3: Profiles of particle displacements in geometry and velocity spaces R, V for different system parameters (see the text for details). All simulations are carried out with B1 /B0 = 0.1. 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199
energy values are somewhat larger than for λ = 5 (the same time moments). Distributions shown in Fig. 4 clearly demonstrate a very fast (exponential ∼ exp(−ε)) drop of number of accelerated particles with the energy increase. This property significantly differs obtained energy distributions from results of spacecraft observations in the Earth magnetotail where power law spectra are often collected [30, 31, 32]. We show that the charged particle acceleration by the convection electric field cannot reproduce heavy tails of energy spectra independently on a distribution of magnetic field fluctuations in the magnetometric. Thus, additional mechanisms of charged particle acceleration are necessary to modify the modeled energy spectra. This important conclusion points out to the significant role of inductive electric fields in generation of a high-energy ion population in the magnetotail, while the convection electric field and charged particle scattering by stationary magnetic field fluctuations can result only in heating of plasma (see, also [18, 4]). In Fig. 5 we show distributions of spatial jumps Δt of charged particles interacting with magnetic field fluctuations. There is a clear difference be8
Figure 4: Energy distributions for two values of λ and several time moments (indicated by colors). All simulations are carried out with VD = 0.5 and B1 /B0 = 0.1. 200 201 202 203 204 205 206 207 208 209 210 211
tween Δt -distributions obtained for λ = 1 and λ = 5. For Gaussian magnetic field fluctuations (λ = 1) we observe more dispersive Δt -distributions than for nongaussian fluctuations (λ = 5). If particles interact with frequent smallscale fluctuations then their spatial displacements change in the diffusive-like manner: corresponding Δt -distributions are gaussian-like with a wide core and an exponential tail (this is especially well seen for the small drift velocity VD = 0.1). In contrast, particles interacting with large amplitude rare fluctuations simply drift with an almost constant velocity VD . The corresponding Δt -distribution is very narrow with the main peak at Δt ∼ VD Δt. This is drastically difference of the charged particle transports in turbulent magnetic field with the fixed power of fluctuations but with different distributions of fluctuations.
Figure 5: Distributions of spatial jumps Δt for two values of λ, two values of VD and several time moments (indicated by colors). All simulations are carried out with B1 /B0 = 0.1.
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5. Conclusions
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In this paper we have considered the influence of distribution form of magnetic field fluctuations on charged particle transport and acceleration in the 2D system with the constant electric field. We use the test particle approach and compare averaged displacements in spatial and velocity spaces R(t), V (t) of particle ensembles obtained in the cases of Gaussian and nongaussian distributions of magnetic field fluctuations. The model of magnetic field fluctuations is constructed so that the total power density of fluctuations is fixed, while the parameter λ > 0 regulates properties of the distribution of fluctuations: larger λ corresponds to larger amplitudes of fluctuations. Thus, due to the conservation of the total power density, the larger λ corresponds to smaller number of fluctuations, i.e. the increase of λ results in the suppression of small amplitude fluctuations and the amplification of large amplitude fluctuations. For λ = 1 the distribution of fluctuations is Gaussian, for λ < 1 this distribution contains small-amplitude fluctuations, for λ > 1 the distribution contains large amplitude fluctuations (see Fig. 1). Integration of a large number of particle trajectories in this magnetic field model allow us to obtain the following results 1. The larger λ value corresponds to weaker acceleration of particles within the fixed space domain: the profile V (R) grows slower for larger λ. The most effective acceleration for finite R is obtained for nongaussian distribution of magnetic field fluctuations with amplified smallamplitude fluctuations (λ < 1). Particles gain more energy if they interact with magnetic fluctuations more frequently, even if amplitudes of these fluctuations are small. 2. The shape of energy distributions of accelerated particles does not significantly depend on the distribution of magnetic field fluctuations. For both Gaussian and nongaussian distributions of fluctuations we obtain similar energy spectra with different locations of peaks. 3. The distributions of spatial jumps Δt are significantly different for Gaussian and nongaussian distributions of fluctuations. In case of small amplitude frequent fluctuations we have the gaussian-like distribution of Δt , while for high-amplitude rare fluctuations we have a transport of particles with the almost constant drift velocity VD .
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Obtained results show that the power spectral density of magnetic field fluctuations cannot alone characterizes the efficiency of charged particle scattering in space plasma systems. For example, in this paper we consider the 10
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conditions typical for the deep magnetotail of the Earth magnetosphere. On the basis of our model results we can mention that large amplitude fluctuations with nongaussian distributions correspond to the almost unperturbed particle cross-field drift with a weak acceleration. In contrast, the frequent small-amplitude fluctuations should result in the effective particle diffusion in space. In the latter case, the larger energy can be gained by particles for fixed size of the spatial domain filled by fluctuating magnetic field. The authors are thankful to Prof. L. M. Zelenyi and Dr. I. Y. Vasko for useful discussions. The work was supported by the Russian Scientific Fund (project # 14-12-00824). [1] P. Veltri, G. Zimbardo, A. L. Taktakishvili, L. M. Zelenyi, Effect of magnetic turbulence on the ion dynamics in the distant magnetotail, J. Geophys. Res. 103 (1998) 14897–14916. doi:10.1029/98JA00211. [2] A. Greco, P. Veltri, G. Zimbardo, A. L. Taktakishvilli, L. M. Zelenyi, Numerical simulation of ion dynamics in the magnetotail magnetic turbulence: On collisionless conductivity, Nonlinear Processes in Geophysics 7 (2000) 159–166. [3] L. M. Zelenyi, A. Artemyev, H. Malova, A. V. Milovanov, G. Zimbardo, Particle transport and acceleration in a time-varying electromagnetic field with a multi-scale structure, Physics Letters A 372 (2008) 6284– 6287. doi:10.1016/j.physleta.2008.08.035. [4] S. Perri, G. Zimbardo, A. Greco, On the energization of protons interacting with 3-D time-dependent electromagnetic fields in the Earth’s magnetotail, J. Geophys. Res. 116 (2011) A05221. doi:10.1029/2010JA016328. [5] P. Dmitruk, W. H. Matthaeus, N. Seenu, Test Particle Energization by Current Sheets and Nonuniform Fields in Magnetohydrodynamic Turbulence, Astrophys. J. 617 (2004) 667–679. doi:10.1086/425301. [6] V. Petrosian, Stochastic Acceleration by Turbulence, Space Sci. Rev. 173 (2012) 535–556. , doi:10.1007/s11214-012-9900-6. [7] A. M. Bykov, Y. A. Uvarov, D. C. Ellison, Dots, Clumps, and Filaments: The Intermittent Images of Synchrotron Emission in Random Magnetic Fields of Young Supernova Remnants, The Astrophysical Journal 689 (2008) L133–L136. , doi:10.1086/595868. 11
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