Ion beams with energies of the order of several tens of keV are frequently observed in the Earth's magnetotail. Here we consider two possible acceleration mechanisms, the cross tail electric field Ey and the stochastic acceleration due to the electromagnetic fluctuations present in the magnetotail. Electromagnetic perturbations are generated by random oscillating “clouds” moving in the x − y plane. A test particle simulation has been performed in order to reproduce the interaction between protons and electromagnetic fluctuations and the constant dawn-dusk electric field, Ey, in the magnetotail current sheet. Protons are accelerated via a stochastic Fermi-like process and, by varying the features of the electromagnetic fluctuations, the combined effect of Ey and of the moving clouds can explain the typical energy range of ion observations from 20 to 100 keV, and the characteristic acceleration times of several minutes.
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 The acceleration of particles in space plasmas remains one of the most challenging problems for theorists. A first attempt to explain the energies of cosmic rays was due to Fermi , who proposed a model based on stochastic acceleration of relativistic particles colliding with magnetized clouds present in space. During these encounters, particles can either gain (head-on collision) or loose (tail-on collision) energy, depending on the relative sign between the particle velocity and the cloud velocity. However, the details of how electromagnetic fluctuations influence the acceleration of charged particles are poorly understood. Up to now many mechanisms have been proposed, ranging from magnetic reconnection [Litvinenko and Somov, 1993; Pritchett and Coroniti, 2004; Zharkova and Gordovskyy, 2004; Lui et al., 2005] to second-order stochastic Fermi acceleration [Miller et al., 1996; LaRosa et al., 1996; Petrosian and Liu, 2004].
 Here we propose that the additional (with respect to Ey) energization mechanism is a Fermi-like acceleration due to the presence of moving magnetic structures, which mimic the magnetic fluctuations observed by spacecraft in the distant magnetotail. Protons are assumed to move in the neutral sheet along Speiser orbits, and the two dimensional (2D) numerical model described by Perri et al.  is used in order to capture the essentials of particle acceleration in the neutral sheet, while the motion perpendicular to the sheet is neglected. In order to directly compare the effect of the stochastic acceleration due to the moving magnetic structures, and of the direct acceleration due to the dawn-dusk electric field, we adopt a 2D model, since both these mechanisms are at work inside the current sheet. By varying the features of electromagnetic fluctuations, we show that the combined effect of Ey and of the moving clouds can explain a range of energetic ion observations, including the typical energies and the typical acceleration times.
2. Numerical Model
 We use a test particle numerical code in which protons interact with the electromagnetic fields generated by N random positioned clouds in the x − y plane [Perri et al., 2007]. We use the field equations, B = ∇ × A and E = −∇ϕ − ∂A/∂t, in the gauge where −∇ϕ = Eyey, with Ey = const. The vector potential has components in the x − y plane, i.e., A = (Ax(r, t), Ay(r, t), 0), that are given by Ax = Ay = A0Σnψ(ξn), where the sum is over the N clouds, ψ(ξn) = e−ξn and ξn = ∣r − rn(t)∣/lcl, with ∣r − rn(t)∣ the difference between particle and cloud positions and lcl the typical size of the clouds. The electromagnetic field equations are [Perri et al., 2007]:
where δEx = δEy = δE and
The fluctuating terms in Equations 1 and 2 are generated by the clouds oscillating in the x − y plane according to xn(t) = xn0 + a cos(ωt + αn) and yn(t) = yn0 + a sin(ωt + βn), where xn0 and yn0 are the initial random coordinates of the n-th cloud, a is the oscillation amplitude, ω is the oscillation frequency, and αn and βn are the initial random phases along x and y, respectively. For simplicity, the amplitude a of the oscillation has the same value as lcl. Also, N = 100 clouds are put in the simulation box.
 Test particles are injected into an L × L simulation box, L being the typical size corresponding to the cross tail width, with initial energies of the order of 100 eV. This is a characteristic value for protons coming from the magnetospheric mantle and reaching the magnetotail. The equations of motion are integrated until a particle leaves the simulation box; at this point another particle is injected in order to keep the total number of particles constant. Assuming a size for the simulation box of L = 105 km, comparable with the actual width of the magnetotail, and a normalization magnetic field B0 = 2 nT, in agreement with the observations in the distant Earth's neutral sheet [Hoshino et al., 1994], we obtain a normalization proton gyrofrequency ω0 = 0.2 s−1, a normalization velocity v0 = ω0L = 2 × 104 km/s and a normalization electric field E0 = B0ω0L = 40 mV/m. The typical proton gyroperiod is nearly 30 s. The equations of motion are integrated numerically for an ensemble of 5000 test particles having initial random positions and velocities extracted from a Gaussian distribution with vth = 120 km/s.
 We consider that these magnetized clouds correspond to electromagnetic perturbations present in the neutral sheet, so that we can assume a typical velocity corresponding to the Alfvén wave velocity, i.e., VA ≃ 500 km/s [Hoshino et al., 1994; Voros et al., 2007]. In the present simulations we have been varying just one parameter, that is the cloud size. Three cases are reported here, i.e., lcl = 0.016L, 0.032L, 0.08L corresponding to oscillation frequencies ω = VA/lcl. These frequencies fall in the range of those observed by Geotail in the distant magnetotail [Hoshino et al., 1994]. We also define the dimensionless cloud size R = lcl/L. Finally, the typical value of the dawn-dusk electric field has been fixed to Ey = 0.2 mV/m, in the range of the observed values [Hoshino et al., 1994; Grigorenko et al., 2009], while the order of magnitude of the fluctuating electric field is 1 mV/m. Figure 1 shows the time behavior of the magnetic field Bz obtained from the numerical model for a time of 20 ω0−1 at a fixed position, in the case when the cloud dimension is R = lcl/L = 0.032. The profile in Figure 1 compares rather well with the irregular oscillations of the normal component of the magnetic field observed, for instance, by Geotail in the distant tail [see Hoshino et al., 1994, Figure 1].
3. Numerical Results and Conclusions
 Particle trajectories are integrated numerically in the electromagnetic fields prescribed above and several runs are performed by varying the parameter R, in order to compare the influence of the two acceleration mechanisms, the dawn-dusk electric field and the Fermi-like acceleration. In the first case shown, we consider the smallest value for the cloud dimensions, which corresponds to R = 0.016. Figure 2a displays the velocity distribution functions in log-lin axes along the y direction for three different times from the simulation starting time (t = t0). It can be seen that particles gain in few minutes a large bulk velocity along positive the y direction due to the presence of the dawn-dusk electric field. After that, different transient stages are observed and the system reaches a stationary state after 10–15 minutes. This time corresponds to the typical lifetime of protons into the simulation box. When the clouds are small, particles interact weakly with the electromagnetic fluctuations, so that the steady electric field plays the main role in the particle dynamics.
 In the second case, a magnetic cloud size of R = 0.032 was considered. The corresponding velocity distribution functions are plotted in Figure 2b. We can see that the distribution is broader, because the stronger interaction with the electromagnetic clouds provides a randomization of the velocities, and the evolution of the PDF is faster with respect to the case displayed in Figure 2a. Also, the PDF is anisotropic and displaced toward positive values of vy.
 Finally, in the third case, the size of the clouds is increased, corresponding to R = 0.08. Looking at Figure 2c, the situation drastically changes. Now the main acceleration for protons is due to their interaction with the clouds, while the dawn-dusk electric field has a minor importance for the particle dynamics. Indeed, the bulk velocity is negligible with respect to the previous cases; on the contrary, the PDFs of vy are wider and nearly isotropic, due the randomizing effect of stochastic acceleration; particles in the tails gain velocities of the order of 2000 km/s, or even more, in about 5 minutes. It is worth noticing that an acceleration time of 5 minutes is not much longer than the time a proton needs to cross the magnetotail under a free acceleration by Ey, therefore the Fermi acceleration mechanism can be considered to be very fast. We notice that the larger R, the larger the chance for particles to interact with the moving clouds. Also, the used values of R are consistent with the spatial scales inferred from the magnetic field observations [Hoshino et al., 1994; Borovsky and Funsten, 2003].
 On the other hand, the shape of the velocity distribution remains close to a Gaussian, that is, no power law tails are developed. This is shown in the inset of Figure 2c, that displays the velocity distributions both at injection and at later times as a function of the standardized variables (vy − 〈vy〉)/σ, being σ the standard deviation of vy. The fact that the PDFs in the inset overlap shows that the Gaussian shape is preserved. It is interesting to notice that, using a different model of magnetic fluctuations, i.e., random-phased plane waves with a power-law amplitude spectrum, Zelenyi et al.  have obtained PDFs with power-law tails [see also Petrosian and Liu, 2004].
 In Figure 3, we compare the steady state energy distribution function for the three values of R, along with the initial Gaussian energy distribution. We can notice that the energy gain increases with the size of the clouds, which means that the interaction with them is stronger. In particular, for R = 0.032 and R = 0.08, the energy acquired by particles is substantially larger than the potential drop of 20 keV in the simulation box, due to Ey and represented by the vertical dashed line in Figure 3. We find that about 2% of injected particles reaches energies much larger than the potential drop.
 It appears that the contributions from stochastic and direct acceleration mechanisms are different: in the first analyzed case, with R = 0.016 (see Figure 2a), the interaction between protons and the cloud electromagnetic fields is weak, and the cross-tail electric field is the main mechanism for particle acceleration. This leads to a beam-like, anisotropic velocity PDF. When the cloud dimension increases, the interaction becomes stronger, so that the Fermi-like mechanism becomes competitive, and could even mask the effect of the constant electric field (see Figure 2c), leading to a randomization of the velocities, to a nearly isotropic velocity distribution, and to an increase in the energy gained by particles. Such energetic and broad distributions compare well with those observed in the PSBL [Keiling et al., 2004; Grigorenko et al., 2009].
 In summary, the dawn-dusk electric field Ey alone cannot explain the energy of 100 keV ion beams observed by spacecraft in the Earth's magnetotail. Here we show that another mechanism can be at work in accelerating particles, i.e., a stochastic Fermi-like process based on moving structures which simulate the electromagnetic perturbations present in the neutral sheet. Assuming realistic values of the parameters, this mechanism allows to reach the required energies in the short times that ions spend in the magnetotail performing Speiser orbits, so that it appears to be very efficient. Increasing the cloud size, the energy grows up to almost 100 keV, in agreement with the observations [Keiling et al., 2004; Grigorenko et al., 2009]. On the other hand, we point out that one or the other acceleration mechanism may be dominating at different times and places in the magnetotail [Zelenyi et al., 2007].
 Stochastic acceleration has been considered widely in different astrophysical contexts. However, we point out that in most cases the features of the accelerating fields are poorly known, and educated guesses have been done. Conversely, in the case of the magnetotail, we have direct experimental access to quantities like the magnetic fluctuation level, the turbulence spectra, that is to the parameters that characterized the accelerating fields, and to the energy of accelerated particles. In this sense, the magnetotail can serve as a benchmark for the stochastic acceleration models.
 In this work we explain some observations in the magnetospheric environment by using a 2D model of stochastic Fermi-like acceleration. We suggest that this model could be also applied to other physical contexts.
 We thank F. Lepreti, V. Carbone, A. Vulpiani, and L. M. Zelenyi for useful suggestions, and M. Dolgonosov and E. Grigorenko for fruitful discussions. This research was partially supported by Italian Space Agency, contract ASI I/015/07/0 “Esplorazione del Sistema Solare”, and by the INTAS 06-1000017-8943 project.