ABSTRACT. We study the temporal evolution of energetic particle beams in an intermittently turbulent solar wind environment using the propagating source ...
The Astrophysical Journal, 636:1145–1150, 2006 January 10 Copyright is not claimed for this article. Printed in U.S.A.
PROPAGATION OF ENERGETIC CHARGED PARTICLES IN THE SOLAR WIND: EFFECTS OF INTERMITTENCY IN THE MEDIUM E. Kh. Kaghashvili, G. P. Zank, and G. M. Webb Institute of Geophysics and Planetary Physics, University of California, Riverside, CA 92521 Received 2004 September 1; accepted 2005 September 19
ABSTRACT We study the temporal evolution of energetic particle beams in an intermittently turbulent solar wind environment using the propagating source method developed by Zank and coworkers, which is based on the separation of the total particle distribution function into a main beam component (unscattered part) and a secondary component (produced by the scattered beam particles). We show here that intermittent changes in the turbulence responsible for scattering particles in the radial direction of the solar wind medium can contribute to the generation of fine-scale structure in the intensity profiles of impulsive events. Our aim in this paper is to simplify the problem to the extent that it allows us to demonstrate the suggested process. Implications of a more complicated transport equation are also discussed. We further address the issue of ‘‘dropouts’’ observed by Mazur and coworkers on the basis of transport in an intermittently turbulent medium. Subject headingg s: diffusion — magnetic fields — Sun: particle emission — turbulence
1. INTRODUCTION
of turbulence, distributed randomly in space. The combination of turbulence generated at the Sun at randomly distributed sources and the generation of low-frequency turbulence in the supersonic solar wind in randomly distributed spatial regions will therefore result in a highly random distribution of magnetic field fluctuations (intensity, spectra, etc.) along a given field line or flux tube. (2) The acceleration of particles at the Sun and their subsequent escape and propagation along open field lines can lead to a stochastic distribution of self-excited waves responsible for scattering the particles. This can be understood in terms of stochastic wave growth (see Cairns et al. [2000] for a review, or Zank & Cairns [2000]). In this approach, a beam of particles remains close to a state of marginal stability. The beaming induces wave excitation and growth, which leads to particle scattering that begins to isotropize the beam. The subsequent isotropization of the beam reduces wave growth, allowing the beam to re-form, which will then lead again to wave excitation and growth and further particle scattering. Stochastic wave growth models predict randomly distributed regions of enhanced wave activity (and hence increased particle scattering) interspersed by regions of relative quiescence. In Figure 1, we illustrate the three possible models that may be considered. In Figure 1a, no turbulence is present and particles simply stream ballistically along the Parker interplanetary magnetic field ( IMF). In Figure 1b, magnetic fluctuations are, in a statistical sense, distributed uniformly along and on each magnetic field line; a particle propagating along one field line effectively experiences the same net scattering as one propagating along another. In contrast, Figure 1c shows regions of enhanced turbulence/scattering distributed randomly on each field line, and each line can be different from the others. In this paper we investigate (1) particle propagation along magnetic field lines that possesses distinct localized enhancements in the fluctuation intensity, and (2) the azimuthal distribution of particles for the case in which neighboring field lines possess different particle scattering characteristics. The calculations presented here have some bearing on the ‘‘dropouts’’ observed by the Advanced Composition Explorer (ACE ) spacecraft. Mazur et al. (2000), in examining intensity plots of impulsive events (their Fig. 1), noticed fine-scale structure showing that the particle intensity near Earth increased and
A major problem of interest in space physics is the transport of energetic particles in the presence of a magnetic field. Solar wind observations show that the background magnetic field, the Parker spiral field, is greatly modified by additional fluctuations. These fluctuations exhibit a wide range of amplitudes on all scales and significantly modify the local solar wind properties. For example, it is well known that low-frequency turbulence in high-speed streams possesses different characteristics from that in low-speed streams (Marsch et al. 1996). However, even within the streams themselves, both the intensity and spectral slope of the magnetic fluctuation spectra can vary azimuthally and radially. Because of that, it is expected that the scattering time for particles at various locations in the solar wind plasma will be influenced by differences in the properties of spatially distinct, relatively small-scale magnetic fluctuations. Specifically, we may expect (for a variety of reasons enumerated below) that along a magnetic field line or a flux tube, stretching out from the Sun to 1 AU where particle intensity observations are made, the intensity and spectral characteristics of the turbulence can be different from one radial location to another. Furthermore, the spatial and temporal variations can be different from one field line to another. The purpose of this paper is to investigate particle transport in a highly inhomogeneous scattering environment, especially at relatively early times where the diffusion approximation is inappropriate. Spatial and temporal variations (‘‘intermittency’’) of the fluctuations responsible for particle scattering on a particular magnetic field line can result from several factors. (1) Turbulence in the solar wind within 1 AU results from turbulence generated both at the Sun and in the supersonic solar wind itself, the latter by stream shear, shock wave propagation, and unstable plasma populations (fluid instabilities and kinetic instabilities). We can expect turbulence originating from the Sun to be highly temporally variable because of (a) highly irregular photospheric motions associated with random fluid motions, and (b) spatially and temporally complex plasma motions and plasma distributions in the corona. Thus, a particular magnetic field line will have, superimposed on its mean field, regions of both intense and relatively quiet levels 1145
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Fig. 1.— Geometry of solar magnetic field lines. Three possible cases are shown in the figure: (a) no turbulence, (b) ‘‘conventional’’ turbulence, and (c) spatially and temporally varying turbulence (intermittent turbulence).
decreased repeatedly and rapidly. Two explanations have been advanced for the fine-scale structure. (1) Both Mazur et al. (2000) and Giacalone et al. (2000) suggest that fluid motions in the photosphere induce magnetic field line random walking. For a small, highly localized particle source (a flare), field lines originating in the source region are filled with particles unlike neighboring flux tubes. The random walk of these field lines can lead to the convection past the spacecraft of flux tubes that are sometimes filled and sometimes empty, giving rise, according to Mazur et al. (2000) and Giacalone et al. (2000), to ‘‘dropouts’’ in the intensity spectrum. However, Ruffolo et al. (2003) argue that the random walk needed to explain the dropouts is too slow to explain the very rapid perpendicular diffusion of particles within the inner solar system observed by, e.g., Interplanetary Monitoring Platform 8 (IMP-8) and Ulysses (McKibben et al. 2001). Rapid perpendicular diffusion would obviously eliminate the small-scale gradients associated with dropouts. (2) An alternative explanation was formulated by Ruffolo et al. (2003). By assuming a two-component model of solar wind turbulence, a superposition of slab and twodimensional fluctuations (Zank & Matthaeus 1992), Ruffolo et al. (2003) argued that the random walk of field lines in the vicinity of two-dimensional O-points was suppressed (i.e., the diffusion rate is approximately that of slab turbulence), whereas the field lines away from islands experienced rapid diffusion by two-dimensional turbulence. Thus, observed particle intensity profiles correspond to filled flux tubes associated with O-points/islands close to the source regions. While an attractive explanation, the Ruffolo et al. (2003) idea also presents some complications. Fine-scale structure, although not always as clearly exhibited as in the examples shown in Mazur et al. (2000), was nevertheless present in almost all examples examined by those researchers (J. Mazur 2005, private communication). The fineness of the structure (which was not noticed prior to Mazur et al. 2000) suggests that the two-
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dimensional O-points are closely packed, having a high filling factor. A high filling factor for O-points may reduce the efficiency of perpendicular diffusion. Also, the correlation between source size and the presence of dropouts is very unclear and may depend on the ‘‘typical size’’ of islands. These are details that remain to be elucidated in the theory advanced by Ruffolo et al. (2003). As we show below, low-frequency turbulence that has an intermittent character can introduce intermittency into the observed particle distributions, manifested as fine-scale structure in the arrival profile of low-energy impulsive solar energetic particles (SEPs). Particle transport in intermittent turbulence may therefore complement the two mechanisms (see also Ragot & Kahler 2003) already suggested to explain the dropouts, and may, at a subtle level, be related to the Ruffolo et al. (2003) idea (i.e., intermittency may have some relation to the number and distribution of O-points). We consider a simple picture of particle transport in temporally and spatially varying turbulence, and neglect variation normal to the solar magnetic field. This makes our problem one-dimensional, as the spatial changes occur along a particular magnetic field line. When particles propagate through regions where the scattering time is small, beam particles experience trapping. Particles that escape from a region of small scattering time to a region of large scattering time then propagate further ‘‘quasi-ballistically,’’ until they encounter another region of enhanced scattering. The propagation of particles along a magnetic field is governed by a Boltzmann equation, the general form of which can be expressed as @f f þ Lf ¼ þ S L; ð1Þ @t t c where L embodies all the operators modifying the distribution function (focusing, energy changes, etc.), except for collisions, and S and L are source and loss terms, respectively. In what follows, we neglect the source and loss terms, and follow the temporal evolution of an impulsively introduced energetic particle beam. For simplicity, we assume that the particle energies of interest are sufficiently large that we need retain only the propagation term in L, i.e., v@f /@r. We also assume a relaxation or BGK form for the scattering operator, noting that for this case, the quasilinear small-angle scattering operator and the BGK operator do not give significantly different results (Kaghashvili et al. 2004). Since the problem we consider is one-dimensional, and all changes happen along a particular magnetic field, any effects introduced by perpendicular gradients in the system are eliminated. The outline of the paper is as follows. Section 2 presents a basic description of the propagating source method used here to describe the propagation of energetic particles from the Sun. In x 3, simulation results for a particle beam propagating in the solar wind are presented and analyzed, and x 4 concludes the report. 2. PROPAGATING SOURCE METHOD: FORMALISM A new approach called the ‘‘propagating source method’’ has been developed recently (Zank et al. 1999, 2000) to solve the time-dependent Boltzmann equation. The underlying physics of the method is as follows. The total particle distribution function ( f ) is split into two parts, one that represents the unscattered population of beam particles (F ), and another ( f s ) produced by the scattering of original beam particles. The propagating source method has been used to study the temporal behavior of pick-up
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ions and impulsively introduced energetic particle distributions governed by both the BGK Boltzmann equation and the case when collisions are described by a small-angle scattering operator (Lu et al. 2001; Lu & Zank 2001; Kaghashvili et al. 2004). In the latter case, both isotropic and anisotropic scattering operators were considered. Here, we consider the simplest case possible, a one dimensional isotropic BGK scattering form of the Boltzmann equation @f @f h f i f þ v ¼ ; @t @r
ð2Þ
where the velocity-space distribution function of the solar energetic particles, f ¼ f (r; t; ; v), is the function of position, r, time, t, pitch angle cosine, ¼ cos , and beam velocity. The general form of the mean scattering time is ¼ (r; t; v; ). Here, since we are analyzing the energetic particle distribution at a given energy, the simplified model can only describe the scattering of particles residing in a constant energy shell in velocity space. In earlier treatments of the propagating source method, the general form of the mean free scattering time was taken (Zank et al. 2000) to be constant, representing a spatially averaged scattering time (¼0 ). In this case, the initial energetic particle beam distribution decayed exponentially in time, with a characteristic decay time scale 0 . In this paper, we generalize the propagating source method, which allows allow us to map the temporal evolution of the primary beam component more precisely at a given location. The result of this change is that it alters the source term in the scattered particle distribution equation and modifies the spectrum. As discussed in the introduction, the intermittent character of turbulence in the solar wind allows us to assume that the particle scattering time is spatially and temporally dependent, i.e., ¼ (r; t). The resulting temporal behavior of the beam can then be quite different from that of beam propagation in a plasma where the scattering time is constant spatially. In the next section, we use the propagating source method procedure in the f2 approximation (e.g., Zank et al. 1999, 2000) to study the evolution of the energetic charged particles propagating in the solar wind. 3. NUMERICAL SIMULATIONS Two qualitatively different cases are considered. First, we consider inhomogeneities (intermittency) in the scattering medium along a given magnetic field. In this case, we explore changes in the energetic charged particle distribution as it propagates and compare to the case of a uniform scattering medium. The second case addresses the effects of cross-field gradients on the propagation of energetic charged particles. We assume that adjacent magnetic flux tubes exhibit slightly different levels of wave activity, thereby introducing slightly different particle scattering characteristics along each. 3.1. Intermittency Along the Magnetic Field Consider the propagation of an energetic charged particle beam when the characteristic changes in the scattering medium occur along the field line. In our simulation, the actual simulation parameters correspond to a beam of 0.1 MeV nucleon1 particles, the transition timescale of which is about 12 hr. To demonstrate the differences in the temporal evolution of the distribution function, we have chosen a few representative cases for the scattering time. Figure 2 demonstrates the propagation of the solar energetic particles in a medium where is constant everywhere
Fig. 2.—Propagation of energetic particles in a medium for which the scattering time is uniform in space. In all plots, the radial direction is along the x-axis and the time evolution is shown along the y-axis. The omnidirectional particle distribution intensities are represented by different colors. The top plot shows the evolution of the initial beam distribution, the middle plot represents the evolution of the scattered particle distribution, and the bottom plot is that for the total particle distribution. The spatially uniform scattering time, in normalized units, is ¼ 5.
in space. In this case, as is expected, the particle distribution diffuses ‘‘uniformly’’ along the field as the particles gradually scatter out of the initial beam and are isotropized (Zank et al. 2000). ( We note that during the large events, the medium through which particles propagate is influenced by the shock-accelerated particles that are released at earlier times [Ng et al. 2003]. This ensures that particles will scatter in ‘‘conventional’’ turbulence.) The beam decays exponentially with a characteristic time scale of ¼ 5 (in transition timescale, 0 , units), propagating at a constant speed, as illustrated in the space-time diagram of Figure 2 (top). The normalized distance, r0 , is 1 AU. The scattered particles form a clear front together with a trailing diffusive tail that fills the spatial region behind the propagating front ( Fig. 2, middle). The combined or total particle distribution is illustrated in the bottom of Figure 2. At early time, there is a fast increase of the scattered distribution of particles. Consider now an intermittently turbulent magnetic field line. For simplicity, we suppose that the medium consists of a series
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Fig. 3.—Spatial variation of the normalized scattering time, , along the background magnetic field.
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of qualitatively different regions where the scattering time changes abruptly from one region to another. The initial distribution is a ring-beam distribution with ¼ 0 ¼ 0:6 (for details see Kaghashvili et al. 2004). The spatial dependence of the scattering time is shown in Figure 3. Figure 4 illustrates the temporal and spatial evolution of particles in this case. Two important processes contribute to the evolution of the particle distribution, these being the enhanced rate of particle scattering out of the beam at regions of strong scattering, and the subsequent trapping of the scattered particles in these regions. Our decomposition of the particle distribution function f into unscattered and scattered components (top left and right panels of Figure 4) respectively, reveals this very clearly. In the top left panel of Figure 4, the beam is released into a strong scattering region (from 1 to 1 in normalized spatial units) and is itself uniformly distributed between 0 and 1. The part of the initial beam distribution well inside the strong scattering region rapidly decays (Fig. 4, top left), generating scattered particles which then themselves have considerable difficulty in diffusing out of this region (the intensity peak of Fig. 4, top right). Conversely, beam particles that were close to 1 (in normalized spatial units), a boundary between
Fig. 4.—Propagation of energetic particles in a medium for which the scattering time is nonuniform in space. Top left: Evolution of the initial beam distribution. Top right: Evolution of the scattered particle distribution. Bottom left: Total particle distribution. Bottom right: Spatial dependence of the total intensity profile at different time steps. The spatial dependence of the normalized scattering time, , is shown in Fig. 3.
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Fig. 5.—Two-dimensional spatial dependence of the scattering time. The Sun is located at the origin of the system. Everywhere, except in a little region at r ¼ 3:0, the scattering time is equal to its average value, 0 .
strong and weak scattering, propagate rapidly outward and only scatter slowly out of the beam, generating scattered particles which have a large mean free path and are therefore comparatively mobile. At the next region of enhanced scattering (2–3), the beam particles again experience rapid isotropization, producing trapped scattered particles. Evidently, as illustrated in the bottom left panel of Figure 4, the process of beam decay, scattered particle generation, and trapping yields distributions that possess somewhat complicated spatial and temporal characteristics, illustrated clearly in the line plot of the bottom right panel of Figure 4. 3.2. Effects of Cross-Field Gradients The second part of our study concerns spatial inhomogeneities normal to the background magnetic field. This is the case considered by Mazur et al. (2000), Giacalone et al. (2000), and Ruffolo et al. (2003). Unlike those papers, we do not assume that convection of alternately filled and empty flux tubes past a spacecraft provides the cross-field gradients. In our model, adjacent flux tubes also carry particles, but each flux tube can exhibit different levels of wave activity at a particular location. Figure 5 shows the example solar wind environment we took for this simulation. The Sun is located at the origin of the twodimensional plane, and everywhere but in a localized area in space the scattering time is assumed to be constant, corresponding the averaged value for the interplanetary space: ¼ 5. The place where changes in the scattering time occur is separated from the origin by 3 normalized units. One can assume complete symmetry, where the spiral magnetic field originates at the Sun and crosses the structure as it fans out. We have chosen the spacecraft orbit such that it crosses the area of reduced characteristic scattering time. Each data point that we record as an observational distribution function seen by spacecraft represents the result of a simulation along a given magnetic line that starts at the Sun and crosses the spacecraft location at each particular time. Figure 6 shows the total distribution function the spacecraft would observe on the magnetic field as it travels through the center of the structure of reduced scattering time. Since the scattering time decrease and recovery is a smooth function, we can see
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Fig. 6.—Intensity plots of the omnidirectional total distribution function f observed at a given position r ¼ 3:0 (in normalized units) as the spacecraft travels through the lowered scattering time region. Time is measured in 0 units.
that the distribution function measured by spacecraft also changes smoothly, reaching a minimum value at the position where the local scattering time is the smallest. In real situations, of course, it is not expected that there will be a smooth change in the scattering time. This would lead to a more complex behavior of the distribution function seen by spacecraft, which would manifest itself as fine-scale structure in intensity plots. 4. CONCLUSIONS In the previous sections, we demonstrated the profound effects that the short-scale wave activity could have on the propagation of the energetic charged particles. Two major cases for the inhomogeneities in the interplanetary space were considered separately: (1) a medium consisting of a series of qualitatively different regions along the background magnetic field where scattering time changes abruptly from one region to another; and (2) spatial inhomogeneities normal to the background magnetic field. For the first case, there was an enhanced rate of particle scattering in regions of strong scattering with the subsequent trapping of particles. The resulting space and time evolution of the total distribution of particles shown in the bootom left panel of Figure 4 clearly demonstrates the complicated spatial and temporal character of the particle distribution as it evolves. In the latter case, a series of simulations were conducted to imitate the observation distribution function seen by a spacecraft as it travels through the center of a structure with reduced scattering time. Figure 6 illustrates how spatial inhomogeneity in the scattering time is reflected in the energetic particle intensity plot. Evidently, intermittent changes in turbulence along a magnetic field line or flux tube along which particles propagate can significantly alter the initial particle distribution and generate fine-scale structure in the intensity profiles of impulsive events. The model we are using considers only particles of a given energy. The simulations show that due to the variations of the background scattering time, the beam of any given energy would exhibit the same kind of behavior, but using only convective term, our onedimensional model is not able to explain the intensity fall at all energies. We hope that the two-dimensional model with a more
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general form of the Boltzmann equation will allow us to look at the general intensity profiles at various energies. Our goal here was to demonstrate the process in simple terms. In a realistic model, it is also expected that inhomogeneities in both radial and normal directions would contribute to the fine-scale complexity of observed distribution functions.
G. P. Z. and E. K. are supported in part by a NASA grant NAGS-10932. G. M. W. is supported in part by NASA grant NAGS-13451 and NSF grant ATM-0317509. We thank J. Mazur for a useful discussion. We wish to thank the referee for raising some important physical points, which led to a substantially improved paper.
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