Test particle acceleration in three-dimensional Hall MHD turbulence

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, A12110, doi:10.1029/2006JA011988, 2006

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Test particle acceleration in three-dimensional Hall MHD turbulence P. Dmitruk1 and W. H. Matthaeus1 Received 25 July 2006; revised 2 October 2006; accepted 13 October 2006; published 21 December 2006.

[1] Numerical experiments of test particle acceleration are performed using turbulent

magnetic and electric fields obtained from direct numerical solutions of the compressible three-dimensional Hall magnetohydrodynamic (MHD) equations. Comparisons are made of the results for the test particle momentum distribution function with and without the Hall term in the MHD solution. Electrons and protons are considered for the test particles at short times before particles leave the simulation box (with length size of the order of a few turbulent correlation lengths). The particles momentum distribution functions develop long tails in a short time. For electrons, no substantial difference it is found between results with and without the Hall term in the MHD solution, while for protons differences are seen in the low momentum part of the distributions. When a background uniform magnetic field (guide field) is added, the particle acceleration is anisotropic. Electrons develop large parallel momentum, while protons develop large perpendicular momentum. A discussion of the basic particle acceleration mechanisms in this system is made. Citation: Dmitruk, P., and W. H. Matthaeus (2006), Test particle acceleration in three-dimensional Hall MHD turbulence, J. Geophys. Res., 111, A12110, doi:10.1029/2006JA011988.

1. Introduction [2] One possible way to efficiently accelerate particles in a plasma is by the effect of turbulent electric and magnetic fields. In particular, magnetohydrodynamic (MHD) turbulence can have both stochastic and coherent effects for particle energization. On long timescales (compared with the evolution of the fields), stochastic effects play the most important role, momentum diffusion being the relevant mechanism. Since large length scales are involved (many turbulent correlation lengths), this type of problem is difficult to study through a direct approach and often turbulence is modeled with phenomenologies (spectral form) and as a collection of waves with random phases [Miller et al., 1997]. The importance of coherent effects (correlated phases) has however been pointed out [Dmitruk et al., 2003; Arzner et al., 2006] for the particle energization problem, using direct numerical simulations to obtain the turbulent fields. On short timescales, coherent effects are even more relevant, particles can be accelerated in localized structures and the acceleration by turbulent fields becomes even more related to the problem of particle acceleration by magnetic reconnection [Ambrosiano et al., 1988]. [3] Test particle simulations are a useful tool to gain insight into this problem, as the typical quasi-linear analytical approach may not adapt easily to the highly inhomogeneous and irregular fields in turbulence. The test 1 Bartol Research Institute, University of Delaware, Newark, Delaware, USA.

Copyright 2006 by the American Geophysical Union. 0148-0227/06/2006JA011988$09.00

particle approach is not self-consistent (particles do not feed back into the fields) but remains valid when the number density of particles they statistically represent (and their total energy content) is small as compared to the background plasma number density (thermal population). [4] Reconnecting fields for pushing test particles can be given analytically [Heerikhuisen et al., 2002] or from MHD simulations [Schopper et al., 1999; Birn and Hesse, 1994]. For a two-dimensional reconnecting structure, the effect of turbulence, enhancing the particle energization, has been shown by Ambrosiano et al. [1988]. In these mentioned studies, a single reconnecting structure (current sheet) is considered. A recent study where test particles move in multiple current sheets is performed by Turkmani et al. [2006]. In previous papers [Dmitruk et al., 2003; Dmitruk et al., 2004] we have studied the particle energization problem in three-dimensional MHD turbulence where many reconnection-like structures (current channels) result as a natural evolution of the fields. The relation between MHD turbulence and reconnection [Matthaeus and Lamkin, 1986] or whether how one leads to the other is actually a complex subject. A single reconnecting current sheet with initial fluctuations can evolve to a turbulent state [Onofri et al., 2006] but reconnection-like features can also be found in the evolution of MHD turbulence [Dmitruk and Matthaeus, 2006]. The purpose of the present paper is to extend on the results of Dmitruk et al. [2004], by considering the relativistic equations of motion for the test particles and the inclusion of the Hall effect in the MHD equations. [5] The Hall effect in Ohm’s law has been shown to be important in establishing the rate of reconnection by a

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DMITRUK AND MATTHAEUS: TEST PARTICLES IN HALL MHD TURBULENCE

number of studies [Shay et al., 2001; Wang et al., 2001], as part of the Geospace Environment Modeling (GEM) reconnection challenge [Birn et al., 2001] and also identified of importance in laboratory reconnection experiments [Ji et al., 1998; Cothran et al., 2005]. These type of studies involve a single set up reconnecting structure. In a turbulent regime, however, it has been shown [Matthaeus et al., 2003] that global energy decay rates are not sensibly affected by the presence of the Hall term. We have recently also studied the structure of the electromagnetic field in Hall MHD turbulence [Dmitruk and Matthaeus, 2006] and found that the electric field is appreciably modified at small scales (of the order of the ion skin depth) but the effect on the magnetic field (and velocity field) is comparatively small. This is for a case where the ion skin depth is set up to be slightly larger than the turbulent dissipation length scale. Some of these results are briefly reviewed in the next section of this paper. We remark however that we address here the effects of the Hall term in a global sense but, as mentioned, comparatively large effects are found in local current sheet reconnection analysis [Shay et al., 2001; Wang et al., 2001]. Also, if, unlike the strong turbulence MHD case we consider here, the focus is on dynamical scales of the order or smaller than the ion skin depth (as in the approximation known as EMHD) or high-frequency waves (like whistlers) the presence of the Hall term is relevant [see Ng et al., 2003; Galtier, 2006]. [6] It is then of interest to study the effect of the Hall term in MHD turbulence on the test particle acceleration problem at short timescales, and this is one of the main subjects of the present paper. Results of computation of the momentum distribution function will be shown for different types of test particles, electrons and protons, according to their skin depth (inertial scale). The case with a strong guide field is also considered, which introduces anisotropic effects in the particle acceleration. The general mechanisms by which particles are accelerated in this system are also discussed.

2. Magnetic and Electric Fields [7] The fields for pushing the test particles are obtained from a direct numerical solution of the compressible threedimensional resistive Hall MHD equations in a turbulent regime. The dimensionless resistive Hall MHD equations are @v 1 JB 1 1 2 rp þ v þ þ v rv ¼ þ r rr v @t rgMs2 r R 3 ð1Þ

@B JB 1 2 rB ¼ r ðv BÞ r þ @t r Rm

ð2Þ

Here B is the magnetic field, v is the velocity field, J = r B is the current density, r is the density, and p is the pressure. The continuity equation @r/@t + r (rv) = 0 and a polytropic equation of state p rg completes the system. [8] The magnetic and velocity fields are expressed in Alfven speed units, a characteristic plasma speed v0 =

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pffiffiffiffiffiffiffiffiffiffi dB/ 4pr0 is used for velocity unit and the root mean square magnetic field dB = hB2i1/2 is used for magnetic field unit. No background magnetic field is considered in this case, that is hBi = 0, but a uniform magnetic field B0 can be added (see section IV). The density r is expressed in units of r0, a reference (initial) plasma density. A characteristic length scale L of the order of the turbulence correlation length is used as length unit (the simulation box is set up to be 2pL) and the unit timescale is an eddy turnover time t0 = L/v0. equa[9] The dimensionless numbers appearing in the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tions are the Mach number Ms = v0/cs, where cs = gp0 =r0 is the sound speed, the Reynolds number R = v0L/n (with n the kinematic viscosity), the magnetic Reynolds number Rm = v0L/m (with m = c2h/(4p) the magnetic diffusivity), and the Hall parameter ¼

rii L

ð3Þ

where rii = c/wpi is the ion skin depth or ion inertial length and wpi = (4pne2/mi)1/2 is the plasma frequency (n = plasma number density, e ion charge, mi ion mass). [10] The Hall parameter appears in front of the Hall term in the normalized equations which becomes important at scales smaller than the ion skin depth rii. For space plasmas, is in general a small quantity. Examples are < 102 for the Earth magnetotail, 104 for the solar wind and < 106 for the solar corona. This express the disparity of scales between kinetic effects, which start to play a role at the ion skin depth rii, and the macroscopic behavior of the plasma at the large turbulent scales L. There is a direct relation between parameters in the test particle equation of motion and the Hall parameter, and this will be discussed in the next section. [11] The Hall MHD equations are numerically solved with a Fourier pseudospectral code. The scheme ensures exact energy conservation for the continuous time spatially discrete equations. The discrete time integration is done with a second-order Runge-Kutta method. Resolution of 2563 Fourier modes is considered. The kinetic and magnetic Reynolds numbers (limited by the numerical resolution) are set to R = Rm = 1000 at the initial state, Mach number is Ms = 0.25. The initial state of the fields consists of nonzero fluctuation amplitudes for the velocity and magnetic field (in equipartition and with total mean squares normalized to 1) random-phased in the k-space (wave vector) shell 1 jkj 4 (with k in units of 1/L). After a few eddy turnover times, the fields have developed a broad range of scales (see Figure 1 below). The MHD state at t = 2t0 is taken for pushing the test particles, as described in the next section. Only the short time behavior of the particles is studied, which justifies the use of a fixed MHD state. [12] Two types of MHD solutions are computed. In one case, the value of the Hall parameter is set to = rii/L = 1/32, while in another case (no Hall) = 0. The value of chosen for the Hall MHD case corresponds to a Hall wave number of kH = 32. The turbulent dissipation scale at t = 2t0 is kd = 115, while kmax = 128 is limited by the numerical resolution. The wave numbers are then chosen to satisfy kH < kd < kmax, that is the Hall length scale or ion skin depth is slightly larger than the turbulent dissipation length scale.

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taken, after the solution of the magnetic and velocity field is obtained, from the normalized Ohm’s law as E ¼ v B þ

Figure 1. (top) Power spectra of the magnetic field and velocity field (dashed) for the Hall and non-Hall (light gray) solution at t = 2t0. (bottom) Power spectra of the electric field for the Hall and non-Hall (light gray) solution. The straight lines indicate the value of the Hall wave number kH and the dissipative wave number kd.

Both length scales are smaller than the turbulent characteristic (large) scale L. [13] The choice of ion skin depth of the order of the turbulent dissipation length scale is crucial in establishing the interpretation of the test particle results: The assertion is that collisionless kinetic plasma physics becomes important at or near the ion inertial scale. It is, for example, well established in the laboratory [Ren et al., 2005; Matthaeus et al., 2005] that the Hall effect is the first kinetic effect that become influential in this range of length scales. Furthermore, theoretical arguments suggest that dissipative effects become important at wavelengths smaller than this. For example, in linear Vlasov theory, for many parameters one sees a steep increase of the linear damping rate for Alfven waves at scales smaller than the ion inertial scale [Gary and Nishimura, 2004]. It is likely that nonlinear Vlasov damping also sets in at a similar scale. Finally, this picture is consistent with solar wind observational results [e.g., Leamon et al., 1998], which show spectral steepening at scales near the ion inertial scale. Our approach is to assume that as one moves to smaller scales, just after the Hall effect becomes significant, dissipative effects, here parameterized by a scalar resistivity, also become important. The same approach was taken and discussed in our previous test particle study [Dmitruk et al., 2004]. The electric field is

JB 1 J þ r Rm

ð4Þ

[14] Figure 1 shows the power spectra of the magnetic and velocity field (top) and of the electric field (bottom) for both the Hall and non-Hall (light gray lines) solution. The Hall wave number kH and turbulent dissipation wave number kd are indicated for reference. There is almost no difference between the magnetic field for the Hall and nonHall solution (and similarly for the velocity field), only at scales of the dissipation wave number some differences can be seen. The electric field shows instead large differences at scales of the order of the Hall wave number, that is, there is a difference in the electric field for the Hall solution at length scales of the order of the ion skin depth. Figure 2 shows three-dimensional views of the electric field absolute value jE(x, y, z)j for the Hall (top box) and the non-Hall (bottom box) solution. The electric field is an intermittent quantity [see also Milano et al., 2003], with large values occupying a small fraction of the box volume. Differences can be seen at small scales, which corresponds to the ion skin depth. The main reason for the differences in the electric field for the Hall and non-Hall solution comes from the additional term in the generalized Ohm’s law for the Hall case. A more complete discussion of the properties of the magnetic and electric field for the Hall and non-Hall solution is made by Dmitruk and Matthaeus [2006].

3. Test Particles [15] The relativistic equations of motion for the charged particles in the fields obtained from MHD (in CGS units) are dp 1 ¼ q uBþE ; dt c

dx ¼u dt

ð5Þ

where p is the particle momentum, u is the particle velocity, x is the position, q is the charge, and c is the speed of light. Momentum and velocity are related by p = Gm0u, with G = (1 u2/c2)1/2 and m0 the rest mass. [16] If the fields are expressed in the MHD units (Alfven speed units) described in the previous section, and the particle momentum and time units are taken as m0v0, and t0 = L/v0, respectively (with v0 the characteristic MHD plasma speed), the equations of motion for the particles can be written in a dimensionless form dp ¼ aðu B þ EÞ; dt

dx ¼u dt

ð6Þ

where a is a dimensionless quantity which relates particles parameters with MHD fields parameters [see also Ambrosiano et al., 1988; Dmitruk et al., 2003] a¼

q v 0 t0 mi L pffiffiffiffiffiffiffiffiffiffi ¼ Z m0 rii m0 c 4pr0

ð7Þ

where Z is the charge number,pmffiffiffiffiffiffiffiffiffi i is ffi the ion mass, and the ion inertial length rii = mic/(e 4pr0 ) is introduced.

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Note that we are assuming that the dominant plasma ion with mass mi is the proton, and the test particle a is computed in terms of the (proton plasma) Hall parameter and the test particle mass and charge. This expression shows that the value for the Hall parameter and the particles parameters are related. The fact that rii L for most space plasmas highlights the disparity of length scales between MHD and particle (kinetic) properties. In our direct study, limited by the numerical resolution, we have set rii/L = 1/32, which implies specific values for a for a given test particle. For instance, ap = 32 for protons, and ae = 58752 for electrons. The value of a for a given test particle is not arbitrary in this interpretation. [18] Timescales for particles and MHD can also be compared through a. The nominal pffiffiffiffiffiffiffiffiffiffi particle gyroperiod t (based on a magnetic field v0/ 4pr0 in terms of the plasma alfvenic speed) is t ¼ 2p

Figure 2. Three-dimensional views of the absolute value electric field jE(x, y, z)j for the Hall case (top box) and the non-Hall case (bottom box). Light tones correspond to high values of jEj, dark tones correspond to low values of jEj. The same tone scale is used in both cases. [17] Since the Hall parameter is = rii/L, it can be seen that a¼Z

mi 1 m0

ð8Þ

m0 c q

pffiffiffiffiffiffiffiffiffiffi 4pr0 1 m0 rii 2p ¼ 2p t0 ¼ t0 Z mi L a v0

ð9Þ

[19] Fifty thousand test particles are followed in the simulation box, starting from an initial isotropic maxwellian distribution function for the particles momentum, with a mean speed of 0.1v0 (that is, particles start with low momentum). Particles initial positions are chosen at random in the box. A value of v0/c = 104 is taken, recalling here that v0 is the plasma characteristic speed, which is of the order of the thermal speed (when plasma beta 1). The initial particle population is then subthermal. An additional simulation is made with a larger value of v0/c = 102 to attain larger relativistic energies. Particle equations of motion are solved with an adaptive fourth-order RungeKutta method. The values of the fields are linearly interpolated from the MHD simulation grid points to obtain the field values at the particle position (the electric field is obtained from equation (4)). [20] Two different test particle runs are done for the electrons and protons. The momentum distribution function is computed, and the run is stopped when the mean distance traveled by the particles is of the order of the turbulent correlation length L. Since the periodic box is of length 2pL, there are no periodicity artifacts introduced in these simulations. [21 ] Figure 3 shows the distribution function of a momentum component pj for electrons (results are similar for all three momentum components j = x, y, z, that is, the momentum distribution function is isotropic in this case). This is for a case with v0/c = 104. The top panel corresponds to a very short time, t = 4t e, where t e = 104t0 is the nominal electron gyroperiod. The bottom panel is the distribution function at the end of the run, t = 1200t e = 0.14t0 when the mean distance traveled by the electrons is of order L. The light gray line corresponds to the non-Hall MHD solution. As can be seen, there is not much difference between the distribution functions for the Hall and non-Hall case. At this very short time, particles momentum is of the order of the reference momentum mev0 (that is, particle speeds are of the order of the plasma characteristic speed). Recall that particles started from a low mean speed 0.1v0. Particles acquire the plasma speed in a

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very short time, a result that was obtained by Dmitruk et al. [2004]. At longer times, but still small compared with the turbulent eddy turnover time, long tails have developed in the distribution function, with speeds of many times the plasma characteristic speed (large particle momentum). [22] Figure 4 shows the distribution function for the absolute particle momentum p at two different times. An extended power law is observed for intermediate values, while a steep tail is obtained for the large momentum values. There is not much difference between Hall and non-Hall (light gray) results. [23] Figure 5 shows the distribution function for a momentum component pj for protons (again, results are isotropic). The top panel corresponds to t = 2t p, where t p = 0.2t0 is the nominal proton gyroperiod. The bottom panel corresponds to t = 23t p = 4.5t0 when the mean distance traveled by protons is of order L. Again, at times of the order of the particle nominal gyroperiod, protons have acquired speeds of the order of the plasma characteristic speed. Larger values, of the order of 10 times the plasma speed, are developed at longer times. Results for the Hall MHD case and the non-Hall MHD case (light gray) show

Figure 3. (top) Distribution function of a momentum component pj for electrons at t = 4t e (electron gyroperiod t e) for the Hall MHD fields and non-Hall fields (in light gray). (bottom) Distribution function of a momentum component for electrons at t = 1200t e = 0.14t0 (eddy turnover time t0). The inset plot shows the core part of the distribution functions at this time.

Figure 4. Distribution function of the total momentum p for electrons at t = 4t e and t = 1200t e = 0.14t0 for the Hall MHD case and non-Hall case (light gray).

Figure 5. (top) Distribution function of a momentum component pj for protons at t = 2t p (proton gyroperiod t p) for the Hall MHD fields and non-Hall fields (in light gray). (bottom) Distribution function of a momentum component for protons at t = 23t p = 4.5t0 (eddy turnover time t0).

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Figure 6. Distribution function of the total momentum p for protons at t = 2t p and t = 23t p = 4.5t0 for the Hall MHD case and non-Hall case (light gray).

Figure 8. Distribution function of a momentum component pj for protons for the case with v0/c = 102 at t = 23t p = 4.5t0 for the Hall MHD fields and non-Hall fields (in light gray).

some noticeable differences at low momentum (lower distribution function values for the Hall case) and at intermediate momentum (higher distribution function values for the Hall case). Figure 6 shows the absolute value momentum p distribution function at two different times. A tail is also developed here but at comparatively smaller speeds than for electrons. Tails are slightly larger for the Hall MHD case and instead a decrease is observed at low momentum. [24] Results with a larger value of v0/c = 102 are shown in the next set of figures. Here the particles, especially electrons in the upper range of momenta, are becoming moderately relativistic. Figure 7 shows the distribution function of a momentum component pj for the electrons in this case, for the Hall and non-Hall cases. Figure 8 shows the distribution function of a momentum component pj for the protons. Results are qualitatively similar to the low v0/c = 104 case. Figure 9 is the

distribution function for the energy of the electrons (in excess of the rest energy) at the final time. Figure 10 is the distribution function for the energy of the protons (in excess of the rest energy) at the final time. Relativistic energies are obtained here. [25] The general mechanisms by which particles (electrons and protons) are accelerated in this system are discussed in section 5, after results for the case with a strong guide field are shown in the next section.

Figure 7. Distribution function of a momentum component pj for electrons for the case with v0/c = 102 at t = 1200t e = 0.14t0 for the Hall MHD fields and non-Hall fields (in light gray).

Figure 9. Distribution function of the energy for electrons (in excess of the rest energy) for the case v0/c = 102 at the final time t = 1200t e = 0.14t0 for the Hall MHD case and non-Hall case (light gray).

4. Case With a Guide Magnetic Field [26] When a background magnetic field B0 is added, then MHD turbulence develops anisotropy, in the sense that fluctuations vary faster in the perpendicular direction than in the parallel direction to the background magnetic field, a well-known result from MHD turbulence theory and simulations [Shebalin et al., 1983; Oughton et al., 1994].

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Figure 10. Distribution function of the energy for protons (in excess of the rest energy) for the case v0/c = 102 at the final time t = 23t p = 4.5t0 for the Hall MHD case and nonHall case (light gray).

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MHD equations, with = 1/32 as considered in the previous section. The direction of the background field is indicated for reference. At this time the ratio between the rms fluctuating field dB = hb2i1/2 and the background field is dB/B0 = 1/10. Light tones (yellow in color) correspond to upward currents, while dark tones (blue in color) correspond to downward currents. Current channels can be seen along the direction of the background magnetic field. The current density changes rapidly in the perpendicular direction while it changes smoothly in the parallel direction. Anisotropy is evident in this figure. [29] The same parameters are considered for the test particles (electrons and protons) as in the previous section. The value v0/c = 104 is considered here. Since a strong guide field is present, the nominal gyroperiods are smaller. For electrons it is t e = (2p/a) (dB/B0)t0 105 t0. The top of Figure 12 shows the distribution function of two components of the test particles momentum, px and pz,

[27] Test particle acceleration is anisotropic as well, as investigated by Dmitruk et al. [2004]. Results for test particles are reported here for the case of relativistic equations of motion for the particles and the Hall effect term in MHD. [28] The total magnetic field is now B = b + B0^z where b is the fluctuating field, and a background field is considered in the z direction. At t = 2t0 a view of the current density Jz (x, y, z) is shown in Figure 11. This is a solution of the Hall

Figure 11. Three-dimensional view of the current density in the parallel direction Jz(x, y, z) for the Hall MHD solution with a background magnetic field B0. Light tones (yellow) correspond to upward currents and dark tones (blue) correspond to downward currents.

Figure 12. (top) Distribution function for the momentum parallel component pz (continuous line) and a momentum perpendicular component px (dashed line) of electrons at t = 4t e for the MHD fields with a background magnetic field B0 for the Hall and non-Hall solution (light gray). (bottom) Distribution function for the parallel pz and transverse momentum component px (dashed) of electrons for Hall and non-Hall solutions (light gray) at t = 8200 t e = 0.1 t0. The inset plot shows the core part of the distribution functions at this time.

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Figure 13. Distribution function of the total momentum p for electrons at t = 0.1t0 for the Hall MHD case and nonHall case (light gray), with a background magnetic field B0.

Figure 14. (top) Distribution function for the momentum parallel component pz (continuous line) and a momentum perpendicular component px (dashed line) of protons at t = 2t p for the MHD fields with a background magnetic field B0 for the Hall and non-Hall solution (light gray). (bottom) Distribution function for the parallel pz and transverse momentum component px (dashed) of protons for Hall and non-Hall solutions (light gray) at t = 90 t p = 1.7 t0. The inset plot shows the core part of the distribution functions at this time.

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perpendicular and parallel to the guide field, respectively, at a very short time t = 4t e. Results for the Hall and non-Hall (in light gray) are shown. The perpendicular component momentum distribution is shown with the dashed line. At this short time, perpendicular momentum is larger than the parallel momentum, and of the order of the plasma reference value mev0. The bottom of Figure 12 shows the distributions at a longer time t 8200t e 0.1t0 when the mean square distance traveled by electrons is of the order of the turbulent correlation length L. At this time, both the parallel and perpendicular momentum distributions show long tails, but values are considerably larger in the parallel direction. The anisotropy of the momentum distribution for electrons is evident. There are no big differences between Hall and non-Hall (light gray) results, the non-Hall case shows slightly larger values at the distribution tails. The inset plot shows the core part of the distribution functions. Total momentum p distribution at this same time is shown in Figure 13. [30] Figure 14 shows the distribution function of two components of the momentum px, pz for protons. The top of Figure 14 corresponds to a short time t = 2t p, where t p = (2p/ap)(dB/B0)t0 0.02t0 is the nominal proton gyroperiod in presence of the background magnetic field. Results are shown for the Hall and non-Hall (in light gray) cases. The dashed line corresponds to the perpendicular momentum px, the continuous line corresponds to the parallel momentum pz. The particles momentum is of the order of the plasma reference value mpv0 at this time. The bottom of Figure 14 shows the distributions at t 90t p 1.7t0, when the mean distance traveled by protons is of the order of a turbulent correlation length L. Extended tails have developed in the distributions, the perpendicular values are larger than the parallel values, by a factor of two or more. The distribution functions are anisotropic, but in opposite way as for the electrons. Some differences can be seen between Hall and non-Hall results at low momentum (lower distribution function values for the Hall case) and high momentum (higher distribution function values for the Hall case). The inset plot shows the core part of the distribution functions. Figure 15 shows the total momentum p distribution at this

Figure 15. Distribution function of the total momentum p for protons at t = 1.7t0 for the Hall MHD case and non-Hall case (light gray), with a background magnetic field B0.

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same time. A power law is seen at intermediate values, while a slightly steeper tail is observed for the large values.

5. Discussion [31] Results for test particles in the MHD fields shown in the previous sections can be qualitatively explained with simple arguments. The expression for the normalized electric field in equation (4) is considered: E ¼ v B þ

JB 1 J þ r Rm

ð10Þ

so that the (dimensionless) equations of motion for the test particles are dp ¼a dt

J 1 Bþ J uvþ r Rm

ð11Þ

where, recall, here u is the particle velocity and v is the plasma velocity. [32] First, we consider the case of electrons with a background magnetic field plus fluctuations. The equations of motion can be separated into perpendicular and parallel components Jk J? u? v? þ Bk þ u k v k þ : r r 1 B? þ J? Rm

dr? ¼ ae dt

drk ¼ ae dt

J? u? v? þ r

1 B? þ Jk Rm

ð12Þ

ð13Þ

Neglecting magnetic field fluctuations in the parallel direction (or absorbing them into the background field), then it can be assumed that Bk B0^z. The first term in the right side of the equation for the perpendicular direction equation (12) describes a perpendicular drift motion, plus a motion around the guide field B0. This happens in a very short timescale of the order of t e/t0 = (2p/ae)(dB/B0) 1 (t e/t0 105 with the values used). In this short timescale the electrons acquire a perpendicular momentum of the order of me(v? + J?/r), where v? is the local (perpendicular) plasma velocity and J? is the local plasma current, plus a gyromotion (also with momentum amplitude of order mev? if particles started from a small initial velocity) around the strong magnetic field B0. A small difference can be expected at this very initial stage between Hall and non-Hall MHD ( = 0) due to the additional term J?/r for the Hall case. However, 1, and J? is smaller and incoherent (not organized in structures) as compared with Jk (see the anisotropy effect shown in Figure 11 for the MHD current density). The very short time behavior of the electrons results in the distribution function observed in Figure 12. [33] In turn, the first term in the right side of equation (13) for the particle parallel momentum involves the difference between the perpendicular velocities of the particle and the plasma (plus the Hall term). This is given by the gyromotion around the background magnetic field B0. For electrons, this is an oscillatory motion in a very fast timescale and in a very short length scale (gyroradius). Electrons will essentially see the same field B? over many

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gyroperiods, so this term does not produce a net increase in the momentum of the particles. [34] The second term in equation (13) for the particle parallel motion involves the current density in the parallel direction. The current is organized in sheet-like structures, being very coherent along the direction of the strong magnetic field (see Figure 11). The ohmic current term can be very small for the electric field seen by the particles (due to the factor 1/Rm) but it can still be the relevant term in the sheet structures, since remains coherent for long distances and is not damped by the strong magnetic field which is aligned in the same direction of the formed currents. Electrons which start with small momentum have short gyroradii and they will move along the current channels and increase their momentum in response to an accelerating field that is almost constant over a correlation length in the parallel direction (of order L, the turbulence correlation length). The parallel current density is not directly affected by the Hall effect, and as suggested by Figure 1 (for the magnetic field power spectrum) only changes at scales of the order of the small turbulent dissipation scale. This explains why there is not much difference in the results with Hall and without the Hall term. The large parallel momentum acquired by electrons is evidenced in Figure 12. The most energetic electrons will be those that find the more intense currents. This behavior was investigated by Dmitruk et al. [2004] where electron trajectory plots are also shown. [35] Electrons in the current channels will see a weak transverse magnetic field B? but the reconnection-like geometry observed in MHD turbulence is not as symmetric as idealized 2-D models, and the particles may see a nonnegligible transverse magnetic field, at which point, if the parallel momentum is large, the electron can pitch angle scatter and transfer part of the parallel momentum to a perpendicular direction. So, after many gyroperiods some of the electrons can also have large perpendicular momentum due to this effect, as shown in Figure 12. [36] The main difference in the behavior of protons in the turbulent MHD fields is that their gyroradius is not very small, even when they start with small momentum. The same form of the equations of motion can be applied for the protons, but with ae replaced by ap. In a timescale of the order of t p/t0 = (2pap) (dB/B0) (with the values considered t p/t0 0.02t0) protons acquire a momentum of the order mp(v? + J?/r) which is the plasma reference value, plus a contribution from the Hall effect. Results in the top panel of Figure 14 show this short time behavior. Once protons acquire this momentum the gyroradius is not small in the sense it is for electrons, and the protons start to sample variations in the MHD fields and their motion is more complex. Eventually some of the protons can see variations in the fields at turbulent length scales (larger than the dissipation length) resulting in perpendicular kicks due to a transverse electric field, coming from the term v? B0 in the electric field. This effect is described by Dmitruk et al. [2004] and the same argument can be applied here, with the addition of the terms coming from the Hall effect in the electric field. The protons can acquire large perpendicular momentum through this type of motion. Pitch angle scatter contributes in term to increase the parallel momentum too. This is shown in the bottom panel of Figure 14 for the longer time (many gyroperiods) behavior of protons.

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[37] The fact that electrons have very small gyroradius (smaller than the ion skin depth c/wpi), as compared to the MHD length scales included in the description, explains why the electron distribution functions for the Hall and non-Hall case are much more similar than in the case of the protons, where some differences are observed at both the low and high momentum values. This is the analog, for test particles, of the familiar statement that, in the collisionless regime, electron and ion motions become decoupled at scales on the order of (or smaller than) the ion inertial scale [e.g., Shay et al., 2001; Wang et al., 2001]. If other terms in the generalized Ohm’s law allowing electron decoupling from the flow were included, the behavior of the electrons could be modified. [38] The behavior of test particles in the MHD fields without a background magnetic field (section 3) can be qualitatively understood in the same way as in the previous paragraphs. The current structures clearly observed for the case with a background magnetic field (see Figure 11) also occur locally in the case with no background field. This is discussed by Dmitruk and Matthaeus [2006]. There are reconnection-like structures with a local nonzero guide magnetic field. Particles locally respond to this configuration in a similar way as in the case with a background magnetic field. In short timescales, they acquire speeds of the order of the plasma reference speed v0, while at longer times electrons develop large parallel momentum (to the local guide field) while protons develop large perpendicular momentum. Orientation of this local current structures is however not aligned in a preferential direction as with the case with a background magnetic field, so the global behavior of particles is isotropic. Note that a Boltzman statistical model of energization by a collection of this kind of randomly oriented, randomly positioned reconnection sites was presented [leRoux et al., 2002] earlier as a possible explanation of distributed heliospheric particle energization [Gloeckler, 2003; Gosling et al., 2004]. [39] We have previously suggested [Dmitruk et al., 2004] that the particle acceleration mechanisms shown here could provide a way of direct energy dissipation in coronal heating models, where dissipation of perpendicular turbulent fluctuations via kinetic effects remains a challenge. The anisotropic proton distribution functions in the case of a background magnetic field could be related to large perpendicular ion temperatures observed in coronal holes [Kohl et al., 1997]. The test particle approach however lacks self-consistency and the disparity of scales for the coronal heating problem is perhaps a more important obstacle for the direct approach taken here. The results however could be useful as a first qualitative insight into the problem of dissipation of MHD turbulence by kinetic effects. A recent model addressing this issue is discussed by Markovskii et al. [2006].

6. Conclusions [40] Test particles moving in electromagnetic fields obtained from direct numerical simulations of the MHD equations experience rapid acceleration as evidenced by tails in the distribution function of their momenta. In a first stage, over a few gyroperiods, particles acquire momentum values of the order of their mass times the plasma charac-

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teristic speed v0. The subsequent behavior of particles depends on their properties. Electrons, with small gyroradii, get accelerated in the direction parallel to the magnetic field, which can be a background magnetic field or a local guide magnetic field. This acceleration comes from the ohmic component of the electric field, which is coherent along the magnetic field direction. Protons, instead with larger gyroradii, get accelerated from variations of the MHD fields at turbulent length scales. This acceleration acts mainly in the direction perpendicular to the magnetic field. When speeds are large, the possibility of pitch angle scatter effects can transfer momentum between parallel and perpendicular directions. Tails of the distribution function show that particles can acquire large momentum values corresponding to speeds of many times the plasma speed v0. [41] In this short timescale behavior, particles are more affected by coherent effects in MHD. Formation of structures like current channels which have a reconnection-like geometry seems to be relevant for the acceleration in this stage. The turbulent regime shows many reconnecting-like current channels with a complex structure. Since the Hall effect has been identified of importance in reconnection, it is here incorporated in a Hall MHD description. The ion skin depth is set to be slightly larger than the turbulent dissipation length scale for this case. For electrons, no substantial difference it is found between results with and without the Hall term in the MHD solution while for protons differences are seen in the low momentum part of the distributions. The acceleration of electrons is mostly due to parallel currents, which are not sensibly affected by the inclusion of the Hall effect, while the acceleration of protons is mostly due to variations of the plasma magnetic and velocity fields at large turbulent length scales. The additional Hall term can modify slightly the initial drift acquired by test particles in the very short time behavior. [42] Although the test particle approach is not selfconsistent, it can provide good insight into the problem of particle acceleration by turbulent fields. Additional effects could include the incorporation of an energy loss model mechanism that would allow the study of the acceleration problem on longer timescales. In addition, self-consistent plasma models of sufficient resolution may be able to identify the processes described here, as well as the self consistent kinetic responses that limit the dynamical tendencies of test particles. [43] Finally, we remark on the possibility of comparison of the present results and energetic particle and plasma properties in space plasmas. In this regard the two most relevant parameters are the ratio of ion inertial scale to turbulence correlation scale, designated above (see equation (3)). The value of = 1/32 was established in our numerical simulations to correspond approximately to the actual length scales in our Hall MHD simulation. This value is not far from the value of 1/100 that we estimate for the magnetotail and may be comparable to values near the magnetosheath. For solar wind ( = 104) or coronal ( = 106) application, maintaining correspondence of the dissipation scale and ion inertial scale would require use of a numerical code with sufficient resolution to allow 4 or 6 decades, respectively, of inertial range spectral resolution. This is well beyond our range of available computing resources. The second important ratio, of Alfveń speed to

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light speed, was set in the simulations to either 102 or 104, in the two cases we computed. This range is not too different from what would be needed to correspond to solar wind, coronal, or magnetospheric values. An additional issue is that in comparing distribution functions and energies (see, e.g., Figures 9 and 10) from simulations with observed particle distributions, energy loss mechanisms and escape mechanisms can become important, as we mentioned above. This is related to the issue of timescales. Here we terminated the test particle calculations when the particle had traveled, on average, a characteristic length scale L, roughly the turbulence correlation scale [Ambrosiano et al., 1988]. In our simulation, for electrons, this corresponded to a very short timescale, 104 of a macroscopic timescale (sound crossing time of distance L). For protons, the corresponding time was 1/5 of the macroscopic timescale. However in a real application this temporal cutoff might change, for example, because the macroscopic length scale L is much larger (solar wind or corona) than what we simulated here. However, the applicability of our time cutoff might also change because the actual escape mechanisms or realistic geometry of field lines might come in to play, making the correct time cutoff either larger or smaller than the assumption we made here in our idealized numerical experiments. For order-of-magnitude comparisons of test particle energy in simulations with observed energetic particles in space plasmas, we point to the ‘‘VBL’’ scaling that has been confirmed in turbulent reconnection simulation [Ambrosiano et al., 1988] and in turbulence simulations [Dmitruk et al., 2003]. This approach has been used to arrive at reasonable characteristic energies for energetic particles for magnetospheric, coronal, and interplanetary parameters [Goldstein et al., 1986], in astrophysical plasmas [Makishima, 1999], and in laboratory experiments [Brown et al., 2002], even though the specific details of acceleration and escape may differ in various cases. [44] Acknowledgments. Research was supported by NSF ATM0539995 and DOE DE-FG02-98ER54490. [45] Amitava Bhattacharjee thanks Loukas Vlahos and Jacob Heerikhuisen for their assistance in evaluating this paper.

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P. Dmitruk and W. H. Matthaeus, Bartol Research Institute, University of Delaware, Newark, DE 19716, USA. ([email protected]; whm@ udel.edu)

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