Modeling the propagation of solar energetic particles in corotating

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A11, 1404, doi:10.1029/2003JA009928, 2003

Modeling the propagation of solar energetic particles in corotating compression regions of solar wind L. Kocharov,1 G. A. Kovaltsov,2 J. Torsti,1 A. Anttila,1 and T. Sahla3 Department of Physics, University of Turku, Turku, Finland Received 8 March 2003; revised 31 July 2003; accepted 3 September 2003; published 22 November 2003.

[1] We present the first modeling of solar energetic particle (SEP) events inside corotating

compression regions. We consider gradual compressions in the interplanetary magnetic field brought on by interaction of the solar wind streams of different speed. The compression model is similar to that previously suggested for the acceleration of lowenergy particles associated with corotating interaction regions (CIRs). In the framework of focused transport, we perform Monte Carlo simulations of the SEP propagation, adiabatic deceleration and reacceleration. A trap-like structure of the interplanetary magnetic field modifies the SEP intensity-time profiles, energy spectra, and anisotropy. Particle diffusion and adiabatic deceleration are typically reduced. For this reason, at a corotating vantage point the SEP event development after the intensity maximum is slower than would be expected based on the modeling in the standard, Archimedean spiral field. At the noncorotating spacecraft the magnetic tube convection past the observer becomes more important. The numerical model forms a basis on which to interpret SEP observations made by present and future spacecrafts at the longitude-dependent speed of solar wind. In particular, the modeling results are similar to the patterns observed with the ERNE particle telescope on board SOHO in August 1996. In the proton anisotropy data, we find a signature of the magnetic mirror associated with the CIR. A relation is established between the spectra observed at 1 AU and the SEP injection spectrum near the INDEX TERMS: 2118 Interplanetary Physics: Energetic particles, solar; 2104 Interplanetary Physics: Sun. Cosmic rays; 2102 Interplanetary Physics: Corotating streams; 2134 Interplanetary Physics: Interplanetary magnetic fields; KEYWORDS: solar energetic particles, acceleration of particles, transport of particles Citation: Kocharov, L., G. A. Kovaltsov, J. Torsti, A. Anttila, and T. Sahla, Modeling the propagation of solar energetic particles in corotating compression regions of solar wind, J. Geophys. Res., 108(A11), 1404, doi:10.1029/2003JA009928, 2003.

1. Introduction [2] Propagation of accelerated solar ions in the interplanetary medium has long been studied, providing a theoretical basis for the computations of interplanetary transport functions, which in turn are necessary for fitting the particle measurements by interplanetary spacecrafts [e.g., Toptygin, 1985; Bieber et al., 2002, and references therein]. Energetic protons, traveling in the interplanetary medium, suffer a scattering at the short-scale inhomogeneities in the interplanetary magnetic field and a focusing by the long-scale component of the field. Concurrent scattering and focusing may lead to an adiabatic deceleration, which has been incorporated into different numerical techniques [Ruffolo, 1995; Kocharov et al., 1998; Kallenrode, 2001, and refer1 Also at the Va¨isa¨la¨ Institute for Space Physics and Astronomy, University of Turku, Turku, Finland. 2 Also at the Ioffe Physical-Technical Institute, St. Petersburg, Russia. 3 Now at the Rados Technology, Turku, Finland.

Copyright 2003 by the American Geophysical Union. 0148-0227/03/2003JA009928$09.00

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ences therein]. Those techniques assume the interplanetary magnetic field (IMF) to be of the Archimedean spiral configuration [Parker, 1958]. The classic model of solar wind is a steady state prediction that avoids discussion of interplanetary dynamic processes such as overtaking highspeed streams or falling behind slow-speed solar wind. However, such processes can change the magnetic field and affect the propagation of solar energetic particles. [3] An effect of the nonstandard interplanetary magnetic field on the transport of energetic particles has already been employed for a qualitative explanation of few observations [e.g., Tan et al., 1992; Anderson et al., 1995]. Bieber et al. [2002] developed a numerical model of interplanetary transport that includes the first treatment of focused transport through a magnetic bottleneck associated with a coronal mass ejection (CME) situated beyond the Earth’s orbit. A nonstandard interplanetary magnetic field may be associated also with corotating interaction regions (CIRs). Propagation of solar energetic particles (SEPs) inside a CIR was detected with the ERNE instrument on board SOHO [Torsti et al., 1999a, 1999b]. The ERNE can measure protons in the range from
1 MeV to

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above 100 MeV and also ions and relativistic electrons. The high-energy detector of ERNE (HED) is a particle telescope which allows precise measurements of the ion flight trajectory in the energy range above
14 MeV nucl1. The description and details of the ERNE instrument are given by Torsti et al. [1995]. At around 1630 UT on 13 August 1996, when the ERNE was registering the CIR accelerated protons of low energies, 1.6– 8 MeV, the high-energy proton intensity started to rise in the energy range through 50 MeV. A detailed analysis of the high-energy event onset indicates its solar origin. However, no modeling of a SEP event observed inside CIR has been yet performed. [4] In general terms the solar wind speed is a function of heliographic latitude and longitude. As the Sun rotates, flows of different speeds become radially aligned in an inertial frame of reference. Faster wind runs into slower wind ahead while simultaneously outrunning slower trailing wind. Since these radially aligned parcels of plasma originate from different positions on the Sun at different times, they are threaded by different magnetic field lines and are thus prevented from interpenetration. As a result, a CIR bounded by two shocks forms [e.g., Gosling and Pizzo, 1999; Lazarus et al., 1999]. Previously, the CIR-associated energetic particles observed at 1 AU were explained in terms of particle acceleration at the forward and reverse shocks bounding the CIR at distances >2 AU. The shockaccelerated particles must then propagate back to the observer at 1 AU, resulting in the low-energy roll-over in the particle energy spectra, but observations do not seem to fit this picture. For this reason, Giacalone et al. [2002] suggested that energetic particles are accelerated close to 1 AU within the transition region from slow to fast solar wind, which is a more gradual change than the near instantaneous jump across the shock at >2 AU. We will employ a similar model of the corotating transition region in solar wind for the Monte Carlo simulations of the SEP propagation from the near-Sun source to a space craft at 1 AU. [5] We start with a formulation of the nonstandard SEP transport model (section 2) and a general investigation of its properties (section 3). Then, as an example of the model application, the results of simulations will be compared with the ERNE observations of the August 1996 event (section 4). Our new findings are summarized in section 5.

2. Numerical Model [6] Giacalone et al. [2002] constructed a simple model of a corotating interaction region, applicable for heliocentric distances ]2 AU, where the forward and reverse shocks have not yet been formed. In this model, the solar wind flow at each point is radial in a nonrotating heliocentric frame, with a speed U, and in steady state in a frame that corotates with the Sun at a rate . In the corotating frame, the flow velocity, Uc, is not radial but comprises also the azimuthal speed  r sin q, along with the radial component U, which is identical to that in the nonrotating frame. The spherical coordinates, r, f, and q, are defined in the frame that corotates with the Sun. The azimuthal angle is counted anticlockwise, if observed from the North pole of the Sun.

The radial flow speed is assumed to be a function of the radial distance, r, and the azimuth angle, f:    1
fc   r=W  f U ðr; fÞ ¼ Us þ Uf  Us tanh 2
fc ! frf   r=W  f 1  ðUf  Us Þ tanh ;
frf 2

ð1Þ

where Us and Uf are the slow and fast solar wind speeds; the angular parameters fc(rf ) and
fc(rf ) respectively control a position and an azimuthal width of the compression (rarefaction) region of solar wind; the parameter W is a speed at which the disturbance moves radially outward in the nonrotating frame of reference. Thus U varies as a function of f from Us to Uf and back to Us (but with different widths over which the variation takes place). [7] The solar wind density, r(r, f), was obtained by solving the continuity equation for density in the corotating frame: r ðrU c Þ ¼ 0

ð2Þ

[see Giacalone et al., 2002, equation (5)]. Then a divergence-free magnetic field that is frozen-in to plasma can be introduced as B ¼ rU c ;

ð3Þ

where is a function that obeys the equation b r ¼ 0;

ð4Þ

the unit vector b is parallel to the magnetic field line: b

B Uc : ¼ B Uc

ð5Þ

Magnetic field introduced with equation (3) is divergencefree, because r B = rUcb r + r (rUc) = 0, and frozen-in to plasma, because r (Uc B) = 0 [e.g., Priest, 1982, equation (2.53a)]. [8] Equation (4) suggests that the function does not change along a magnetic field line, but may take different values on different lines. This property can be used to fit the boundary condition on the surface of the Sun. For simplicity sake consider only magnetic field lines situated in the equatorial plane, q = 90. Substituting a solution of equation (2), r(r, f) by Giacalone et al. [2002], into equation (3) yields r 2 i  U ðr; fÞ; U ðr; fÞ  W r  i r 2 Bf ðr; fÞ ¼ 

 r; U ðr; fÞ  W r

Br ðr; fÞ ¼

ð6Þ

where the function U(r, f) is given by equation (1), and few constants are grouped together as i. The normalization factor i is defined for a magnetic line ‘i’ as a constant that maintains at the line foot point (r, fi) the magnetic field

KOCHAROV ET AL.: SOLAR ENERGETIC PARTICLES IN CIR

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Figure 1. Illustration of the solar wind model. Magnetic field lines for the (a) forward and (b) reverse compression regions. The disturbance speed W is (left) 1200 km s1 and (right) 300 km s1. The fast and slow solar wind speeds, Uf and Us, respectively, are 800 km s1 and 400 km s1 for all panels. Roots of two neighboring magnetic field lines are separated by 15 in the longitude i. The reference longitude, i = 0, corresponds to the long-dash-dotted line. Pairs of heavy solid lines illustrate the interplanetary magnetic traps; the highlighted lines in the case of forward compression are i = 30 and 45 (Figure 1a), and in the case of the reverse compression i =  45 and 60 (Figure 1b). Figures 1c – 1h show scans of solar wind parameters, as the vantage point moves in the corotating frame around the Sun in the 1 AU orbit, which is illustrated with a circle in the corresponding top panel. Shown in Figures 1c – 1h are the magnetic field strength, B, and the reciprocal of the focusing length, LB (Figures 1c and 1d); the radial speed of solar wind at the vantage point, U, and at the magnetic line root, U0 (Figures 1e and 1f ); the magnetic line length between the root and the vantage point, ZE, and the longitude of the magnetic line root in respect to the solar disc center (in radians), l0 (Figures 1g and 1h). A zero time corresponds to crossing the magnetic line i = 0.

strength equal to a boundary value B(r, fi). Near the Sun the magnetic field is almost radial, Br(r, f) B(r, fi), and hence the first equation (6) yields

i ðU ðr ; fi Þ  W Þ

Bðr ; fi Þ : U ðr ; fi Þ

ð7Þ

In what follows, the model field will be computed basing on equations (6) and (7) for the uniform boundary field B(r, fi) = 1 G. [9] The model may reproduce either a ‘‘reverse’’ or ‘‘forward’’ compression in the solar wind, depending on the value of the parameter W. This is illustrated with Figure 1. If the disturbance speed, W, is faster than the fast

solar wind speed, Uf, then the disturbance will be a forward compression (Figure 1a); whereas for W slower than the slow solar wind speed, Us, the disturbance will be a reverse compression (Figure 1b). Throughout the paper, we adopt common angular parameters of the corotating region:
fc = 1,
frf = 25, and fc  frf = 70. A relative longitude of the magnetic line root will be measured in respect to the compression longitude, fc: i = fifc. The reference magnetic line, which is rooted at fi = fc (i = 0), is shown with a long-dash-dotted curve in Figures 1a and 1b. Figures 1c – 1h show scans of solar wind parameters that would be observed at a fixed point in the nonrotating frame, e.g., on board SOHO. Such a point orbits the Sun in the corotating frame as illustrated with the 1 AU circle in Figures 1a and 1b. The solar wind parameters comprise

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the radial speed of solar wind, U, the magnetic field strength, B, and the reciprocal of the focusing length, 1 1 @B ¼ ; LB B @x

ð8Þ

where x stands for the magnetic line length measured from the Sun. We additionally show the solar wind value at the root of the connected-to-observer magnetic line, U0, the line length between the Sun and the vantage point, ZE, and also a longitude of the magnetic line root in respect to the longitude of the 1 AU observer, l0 = fi  f, or what is the same, in respect to the solar disc center. [10] Solar energetic particles traveling in the interplanetary medium experience focusing or mirroring in the long-scale magnetic field and scattering at small-scale irregularities (turbulence) in the solar wind. Our Monte Carlo simulations of the SEP propagation and energy change are based on the model of focused transport in the guiding center approximation. We trace particles along the interplanetary magnetic field lines taking into account magnetic focusing in a fixed frame corotating with the Sun. However, the particle scattering is modeled in the local solar wind frame. A particle experiences numerous successive small-angle scatterings with a scattering frequency which is independent of the particle pitch angle (the isotropic small-angle scattering). The scattering frequency is related to the parallel mean free path, , which is used for parametrization of the scattering. We consider the mean free path to be independent of spatial coordinate and proportional to a power of the proton momentum, p:   ¼ o

p po

a ;

ð9Þ

where o and po respectively are the mean free path and the momentum of the 1 MeV proton. We consider three different values of the power law index: a = 0, 13, 1. A numerical implementation of particle transport is similar to that previously described by Kocharov et al. [1998], but the numerical codes have been newly written and tested to take into account the nonstandard model of solar wind. [11] As magnetic field is frozen-in to the plasma and in steady state in the corotating frame, the magnetic field is linked to the fluid velocity by the equation r  ðU c  BÞ ¼ 0:

ð10Þ

This equation can be recast into the form r U c ¼ ðb rÞUc 

Uc ðb rÞB: B

ð11Þ

The divergence of solar wind velocity, which is the left member of this equation, appears also in the energychanging term of the well-known diffusion-convection equation of the cosmic ray transport [Parker, 1965; Toptygin, 1985, equation (8.17)]. The right-hand side of equation (11) comprises a derivative of the solar wind velocity along the magnetic field line, which is a part of the velocity divergence, and a rate of the magnetic field change

in the local solar wind frame, which accounts for the rest of the velocity divergence. Hence both the velocity and magnetic field gradients along the magnetic line contribute to the energy change of solar energetic particles. The solar wind velocity gradient affects the particle energy through the scatterings at different Uc, whereas the magnetic field change affects particles through the betatron effect in the local solar wind frame. A rising profile of the magnetic field strength and a falling profile of the fluid speed can cause particle acceleration. However, similar to Giacalone et al. [2002] we do not use the diffusion-convection approach, because the particle mean free path may be comparable to the compression scale. Note also that the betatron effect is implicit in our numerical scheme, because we trace particles in the corotating frame where magnetic field is static. [12] In contrast to Giacalone et al. [2002], we are interested in the solar energetic particles. For this reason, all particles are injected near the Sun, at rinj = 0.01 AU, either with a fixed energy (monoenergetic injection) or with the injection energy randomly distributed according to an assumed injection spectrum. We adopt an instantaneous injection of all particles at the injection time tinj = 0, and a normalization to the total, time-energy-integrated number of injected protons Ninj = 1032/(4p) particles per sr of heliocentric solid angle near the IMF line foot point. However, the injection parameters for the 13 August 1996 event fitting will be separately specified in section 4.

3. Results of Simulations [13] In a standard, Archimedean spiral field, the magnetic field strength, B, monotonically decreases along a magnetic field line, whereas the fluid speed in corotating frame, Uc, monotonically increases. Consequently, an energetic particle experiences an average adiabatic deceleration during its trip from near the Sun to beyond the Earth, as exemplified in Figure 2b (bottom). In contrast, a longitude-dependent speed of solar wind suggests nonmonotonic profiles of Uc and/or B (Figure 2a). Magnetic enhancements appear in both forward (W = 1200 km s1) and reverse (W = 300 km s1) compressions, setting up a kind of magnetic mirror away from the Sun (Figures 1a and 1b). As a result of mirroring, a solar particle may be temporally trapped and somewhat reaccelerated, as exemplified in Figure 2b (top). This reacceleration modifies the average energy change, depending on the longitude of the SEP injection into the CIR. [14] Figure 2c shows the average energy of protons observed on different magnetic lines. The modified energy profiles can be compared to the profile of the standard adiabatic deceleration shown with dash-dotted lines. It is seen, for example, that propagation of protons along the i = 60 line in the reverse compression (Figure 1b) results in a very slow deceleration, excluding only first few hours of the event (Figure 2c, bottom). In the case of i = 90, the deceleration-dominating phase is followed by a phase dominated by reacceleration. The initial deceleration is caused by the adiabatic deceleration near the Sun, where solar wind is nearly standard. Then particles spend more time at
1 –2 AU, where the solar wind is compressed. In a vicinity of the compression a particle may bounce between the scattering centers (waves) in the high-speed wind in the

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Figure 2. Illustration of the particle propagation in the solar wind compression regions. (a) Relevant solar wind speed (Uc) and magnetic field (B) along the length (x) of a selected magnetic line (i) for the cases of forward (W = 1200 km s1) and reverse (W = 300 km s1) compressions. (b) Two patterns of particle trajectories in the reverse compression, at the strongly distorted line, i = 60, and at the nearly Archimedean spiral line, i = 180. (c) Average energy of protons arriving at 1 AU for different magnetic lines versus time (the magnetic line parameter i labels the curves; all protons were injected from the Sun with the same energy: Einj = 2 MeV). (d) Energy-angle-integrated intensity of protons registered at 1 AU for the same, monoenergetic injection from the Sun. The proton mean free path  = 0.3 AU with a = 0 in all cases. The fast and slow solar wind speeds, Uf and Us, respectively, are 800 km s1 and 400 km s1. upstream region and the scattering centers in the downstream region, where the wind speed is lower, and also between the flowing scattering centers and the magnetic mirror in rest. Those converging reflections produce a kind of the first-order Fermi acceleration which may partly compensate adiabatic deceleration in the rest of solar wind or sometimes even dominate. [15] In significant sections of the corotating regions the standard adiabatic deceleration is modified to somewhat lower values. However, the modification is more significant for the reverse compression, because in the forward com-

pression the velocity Uc never decreases (Figure 2a). On the other hand, the magnetic trapping effect makes intensitytime profiles very flat (Figure 2d). Note a distinctive, short peak in the beginning of the i = 60 event (Figure 2d, bottom). It is a signature of the not very long magnetic field line ZE as compared to the mean free path , an enhanced focusing inside the Earth’s orbit, and magnetic mirroring behind the Earth (Figure 1b). [16] Figure 3 shows energy spectra of solar protons (intensities) registered at 1 AU in the case of a monoenergetic injection, dNinj/dE / d(EEinj), a kind of Green’s

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Figure 3. Energy spectra of the initially 2 MeV protons after the interplanetary transport from near the Sun to 1 AU (Green’s functions). Figures 3a – 3c show a time development of the spectra, for the five successive 4 hour intervals indicated in Figure 3a, in the case of (a) nearly standard, Archimedean spiral field (i = 180, W = 1200 km s1), (b) the forward compression (i = 45, W = 1200 km s1), and (c) the reverse compression (i = 60, W = 300 km s1). In Figures 3a – 3c, the mean free path  = 0.3 AU with a = 0. Figures 3d– 3f illustrate dependencies of the time-average spectra on the mean free path value and index of the rigidity dependence, a in equation (9). Green’s functions are plotted in a linear log scale as functions of E(a+1)/2. (d) A dependence on the mean free path value for the rigidity-independent : a = 0. (e) and (f ) A dependence on the index a at o = 0.1 AU. Straight lines in Figures 3e and 3f show an exponential spectrum in the form of equation (12) with a = 1/3 and a = 1, respectively. In Figures 3d– 3f, i = 60 and W = 300 km s1. The fast and slow solar wind speeds, Uf and Us, respectively, are 800 km s1 and 400 km s1 in all cases.

functions for the interplanetary transport. Figures 3a – 3c show broadening of the proton distribution as time elapses. Even in the standard field the distribution does not remain monoenergetic (Figure 3a), because different particles travel along different trajectories and consequently experience different energy losses. In the compressed field, some particles may be even accelerated, and a high-energy tail develops (Figures 3b and 3c). The steepness of the highenergy distribution depends on the profiles of the fluid speed Uc and the magnetic field B, which may be different on different lines and in different compressions. [17] In Figures 3d– 3f we show the spectra averaged over a longer time period, in order for a time-integrated spectrum to be closer to the steady state one. Figure 3d illustrates how the spectrum slope depends on the magnitude of the particle

mean free path , at a = 0. Monte Carlo simulations also indicate that shapes of the high-energy distributions are different for different rigidity dependencies of , controlled by the parameter a (Figures 3e and 3f ). Inspection of the results shown in Figures 3d, 3e, and 3f reveals that the highenergy part of distributions is shaped approximately as h i IðEÞ
exp sEðaþ1Þ=2 ;

ð12Þ

where the coefficient s depends on the mean free path o and the fluid speed and magnetic field profiles. Equation (12) was verified against Monte Carlo simulations for a = 0 – 1 and o = 0.1 –0.3 AU, in the energy range shown in Figure 3.

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case of a standard tube, shown in Figure 4b. It is seen from Figure 4a that starting with the second time interval, the >3 MeV spectrum at 1 AU does not differ very much from the injection spectrum. The same is true also for the maximum intensity spectrum. A slow softening of intervalintegrated spectra still proceeds due to a faster leakage of the higher energy particles. However, this softening is much weaker than would be expected based on the standard model of solar wind. Thus the SEP event development after the intensity maximum is dominated by the conservation of SEPs inside the corotating compression region (CIR).

4. Discussion

Figure 4. Relative spectrum of protons defined by equation (14), at corotating vantage points (a) i = 30 and (b) i = 180, for the five successive 4 hour intervals (curves 1 –5), starting from the particle injection time, and for the maximum intensity time (curve M). Power law index of the injection spectrum is S = 3.1. The fast and slow solar wind speeds, Uf and Us, respectively, are 500 km s1 and 360 km s1; the speed of the forward compression W = 600 km s1; the mean free path o = 0.1 AU with a = 1/3. The magnetic field line i = 180 represents a standard Archimedean spiral at solar wind speed 360 km s1, shown for a comparison. Minor irregularities in the spectra are statistical artifacts. [18] In the next set of simulations, we inject SEPs with a power law spectrum: N ðEÞ ¼ Ninj

  S  1 Eo S ; E Eo

ð13Þ

where N(E ) is a number of injected protons per sr of solar surface and unit of energy; the minimum energy of injection is Eo = 1 MeV. An intensity ‘‘observed’’ at 1 AU, I(E), is compared to the injection spectrum as " cðEÞ ¼ IðEÞ

Z

1

dEo GðE; t  to ; Eo ÞQðEo ; to Þ:

dto 0

;

Z

t

IðE; tÞ ¼

#1

N ðEÞ v ð1 AUÞ3 4p

[19] Interaction of solar wind flows of different speed may create a kind of magnetic trap (shown with heavy solid curves in Figures 1a and 1b). Successive scatterings at the turbulence in solar wind flow and mirrorings at the downstream magnetic mirror may increase the particle energy. For instance, Figure 2b (top) shows a case when the particle mean free path exceeds the compression scale, so that a particle can experience a kind of the first-order Fermi acceleration between the scattering centers flowing with the speed Uc 960 km s1, on the one side, and the magnetic mirror in rest and the scattering centers flowing with the speed Uc 690 km s1, on the other (see Figure 2a, bottom). However, in the most of space the adiabatic deceleration dominates. Acceleration, deceleration, and concurrent random walk of particles widen the particle distribution, which may gain a high-energy tail, even if the mean energy decreases. The upper row of Figure 3 exemplifies the proton distributions at 1 AU in the case of impulsive, monoenergetic injection, i.e., a Green’s function, G(E, t  to, Eo), for the injection energy Eo = 2 MeV and the injection time to = 0. The tail development is clearly seen in Figures 3b and 3c, despite the most likely energy does not increase. The high-energy spectrum depends on the index of rigidity dependence of the particle mean free path, as illustrated with equation (12). The physical understanding of the spectral tail is similar to that given for a CME acceleration by Kocharov and Torsti [2003]: the exponential spectra come from a combination of the compressive acceleration and the diffusive escape of particles from the acceleration region. [20] A Green’s function should be convolved with the source spectrum, Q(Eo, to), to get the SEP event spectrum at 1 AU: ð15Þ

0

ð14Þ

where v is the proton velocity. As an example we adopt S = 3.1, which is close to the slope of the 15– 50 MeV proton spectrum in the 13 August 1996 event. Figure 4 shows the relative spectrum c(E) for the five successive time intervals, 4 hours each, and also a similar ratio for the maximum intensity time. In Figure 4a we show a case of the magnetic tube compressed beyond the Earth, which is a trap-like magnetic configuration (i = 30 in a forward compression, like in Figure 1a). It may be compared to the

One can see that a high-energy tail of the Green’s function cannot significantly affect the convolved spectrum as long as it is not harder than the source spectrum injected from the Sun, whereas the mean energy is always essential for the convolution. Note that for the goals of data fitting, we will assume that Q(Eo, to) is a separable function of its arguments: Q(Eo, to) = N(Eo)q(to). A convolution over the energy Eo will be implicitly incorporated into the basic Monte Carlo code, so that the Green’s function R will appear as already integrated over Eo, g(E, t  to) = N(Eo)G(E, t  to, Eo) dEo, to be shown in Figure 6 below.

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profile to the gradient of B in the parallel to field direction (focusing length) and by this expedient to the negative anisotropy observed on 14 August 1996. Figure 5c shows the mean free path to focusing length ratio, /LB, for two values of . The -to-LB ratio is known to control the SEP anisotropy in a steady state, which is expected to occur shortly after the maximum intensity time. The corresponding pitch angle distribution is of the exponential form [Bieber et al., 1986]: IðmÞ ¼ Ao expðbo mÞ;

Figure 5. Modeled and observed solar wind parameters for 14 August 1996: (a) interplanetary magnetic field intensity; (b) solar wind speed; (c) the proton mean free path to focusing length ratio; and (d) the root longitude of the magnetic field line met at the moment. Curves show the model parameters for a forward compression. The fast and slow solar wind speeds, Uf and Us, respectively, are 500 km s1 and 360 km s1; the compression speed W = 600 km s1. Histograms in Figures 5a and 5b show experimental data of Wind. Points in Figure 5c refer to the SOHO/ERNE anisotropy data fitted with equation (16).

[21] It is not often that observations of CIRs and solar particle events coincide, but when they do SEPs may be used as a diagnostics tool for the CIR. On the other hand, a CIR may trap solar particles and thus enhance the count statistics in a particle instrument. Concurrent CIR and SEP events were observed during 7 – 18 August 1996 [Torsti et al., 1999a]. Figure 5 shows profiles of the interplanetary magnetic field, B, and the solar wind speed, U, observed on board Wind [Lepping et al., 1995; Ogilvie et al., 1995] at around the time of the SEP event recorded on board SOHO. In Figure 5 we plot also the model profiles corresponding to a forward compression with parameters Uf = 500 km s1, Us = 360 km s1, and W = 600 km s1. [22] From an empirical standpoint, the observed change in B might be completely due to the gradient of B in the perpendicular to magnetic line direction, but the 2-D model of solar wind allows us to link the spacecraft observed

ð16Þ

where m is the pitch angle cosine, bo = /LB, and scattering is assumed to be isotropic. The observed pitch angle distributions were fitted to the exponential form similar to equation (16). Points in Figure 5c show the ERNE/HED anisotropy data as the best fit estimates of the parameter bo. The values of  in theoretical curves were selected to fit the anisotropy observed during the first few hours of 14 August. Figure 5d additionally identifies a model magnetic line met by the spacecraft at the moment. It is seen that between the SEP event onset and 1000 UT 14 August the spacecraft sweeps over the magnetic lines rooted at the relative longitudes i
30. [23] A rising phase of the August 1996 SEP event was qualitatively similar to many events observed without CIRs [Torsti et al., 1999b]. Injection of first particles was estimated to occur at about 1610 UT on 13 August. After the first 6 hours, the intensity development became similar in all energy channels, from the lowest proton energy 1.6 MeV to the highest observed energy
50 MeV. The particle anisotropy observed during the first 5– 6 hours of 14 August indicates the mean free path value  = 0.1– 0.15 AU, at nearly standard focusing length (Figure 5c). However, at around 0700 UT a dramatic change in the anisotropy was observed: it became negative, sunward directed. This pattern is similar to the expectation based on the compression model by Giacalone et al. [2002], which suggests a negative focusing length in the beginning of forward compression (Figure 1c). We consider this observation as a proof of a local magnetic mirror, which is the essential ingredient of the CIR acceleration model proposed by Giacalone et al.. [24] The solar wind parameters of Figure 4a were reserved for fitting the August 1996 data shown in Figures 5 and 6. From Figure 4a one can learn that excluding the first 4 hour period, the period-integrated spectra of >3 MeV protons at 1 AU are close to the injection spectrum at the Sun. Torsti et al. [1999a] found the observational spectra to be nearly exponential: I(E) = Av3/4 exp(aE1/4), during the periods B2 and B3 of Figure 6a. Recall that such an exponent might appear as high-energy asymptote of the modified Bessel function [e.g., Forman et al., 1986]. Note that this refers to the main SEP production, which is different from a weak, early stage production previously discussed by Torsti et al. [1999b]. Hence, in an effort to simulate the main phase of the 13 August 1996 event, we can adopt the SEP injection spectrum in the form   N ðEÞ ¼ A exp aE1=4 ;

ð17Þ

where N(E) is a total (time-integrated) number of injected particles per MeV and per sr of heliocentric solid angle near

KOCHAROV ET AL.: SOLAR ENERGETIC PARTICLES IN CIR

Figure 6. (a) Proton intensity-time profile in the energy channel 12.5– 25 MeV and (b) the 1.6– 50 MeV proton energy spectra for the periods B1, B2, and B3 during the 13 – 14 August 1996 event observed by the ERNE instrument on board SOHO. Boundaries of the three selected periods are indicated with vertical dash-dotted lines in Figure 6a. Points refer to the ERNE measurements. Results of numerical modeling are shown with curves. The model parameters are the same as in Figure 4a, but the injection spectrum is exponential in E1/4 (see equation (17) at a = 5.25 MeV1/4), and the SEP source is positioned at the heliocentric distance of 0.1 AU. An intensity-time profile which would be observed at 1 AU after the impulsive injection, the Green’s function g(E, tto), is exemplified by the dotted curve in Figure 6a for to = 1800 UT, 13 August. An injection profile of the source q(t), that being convolved with Green’s function fits the data, is shown in arbitrary units with dashed line in Figure 6a. The total number of protons injected above 1 MeV near the Sun is Ninj = 2  1032 particles per sr of heliocentric solid angle. Convolved intensities are shown with solid lines in Figures 6a and 6b, and also with dashed and dash-dotted lines in Figure 6b. The CIR spectrum is added as exp(3.5E2/3), where the proton kinetic energy E is in MeV (dotted curve in Figure 6b). the IMF line foot point (more certainly we inject particles at 0.1 AU from the center of the Sun); E is proton kinetic energy. The injection time profile (the source q(to)) is adjusted so that its convolution with Green’s function fits the

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proton intensity-time profile observed with SOHO/ERNE (Figure 6a). A choice of the single spectral parameter, a, allows reasonably good fits of proton spectra for all three time intervals, B1, B2, and B3 (Figure 6b). The direct modeling confirms that the injection spectrum is close to the exponent of equation (17) and yields the spectral index value a = 5.25 MeV1/4. Injection profile is estimated to peak at 1800 UT on 13 August 1996, when according to SOHO/ LASCO observations, the CME leading edge was at about 13 r. Recall that this event was associated with a back-side solar eruption, most likely at
150W, and with metric type II burst at around 1500 UT on 13 August 1996 [Torsti et al., 1999b]. Hence the spectrum of equation (17) was produced at the east flank of coronal mass ejection, far from the flare but still not very far from the Sun. The spectrum shape does not rule out a stochastic acceleration, but not necessarily suggests it. We note a formal coincidence of equations (12) and (17), which might indicate a physical similarity of the acceleration mechanisms. Those mechanisms may produce exponential spectra at different compressive motions operating near the Sun and throughout the interplanetary medium. SEPs of the 13 August 1996 event were accelerated by the CME near the Sun. Then the CIR shaped the SEP event mainly through the modification of adiabatic losses and time-intensity profiles. [25] During 3 hours next to the period B3, the proton intensities in different energy channels simultaneously dropped by 1– 2 orders of magnitude. An extrapolation of the model intensity-time profile shown in Figure 6a indicates that the fall observed after B3 is much faster than expected in magnetic tubes of the period B1 – B3. Hence the proton injection function and/or transport parameters in the tubes met by SOHO after B3 should be very different. The CIR stream interface is a well-known obstacle to energetic particle propagation [Conlon and Simpson, 1977]. Hence one may expect the mean free path to drop as the stream interface approaches. Previously, an abrupt decrease of the mean free path in magnetic tubes passing SOHO was observed and successfully fitted in the SEP event of 9 July 1996 [Kocharov et al., 1997]. However, in the case of the 13– 14 August 1996 event the anisotropy data (Figure 5c) are not sufficient to select a mean free path value for the period after 1000 UT on 14 August. For this reason, our modeling covers only a period of forward compression that does not extend beyond B3. [26] The modeling indicates that a postintensity maximum development at a corotating vantage point would not be fast, but SOHO does not corotate. For this reason, we conclude that the late phase intensity development was dominated by scanning of magnetic tubes differently populated by solar accelerated particles. Judging from Figure 5d, the 3-hourlong fall of the intensities suggests that the SEP populations may be very different on the magnetic lines separated by only 5 in the longitude i, and a cross-field transport during the first 20 hours of the event has spread particles over less or about few degrees in the heliolongitude.

5. Conclusion [27] Analysis of SEP data requires a modeling of the particle transport under different conditions in solar wind. We have performed the first ever modeling of the SEP

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KOCHAROV ET AL.: SOLAR ENERGETIC PARTICLES IN CIR

propagation and energy change in the corotating compression regions of solar wind. The numerical modeling of SEP transport in the CIR model proposed by Giacalone et al. [2002] exemplifies how nonstandard interplanetary conditions can affect SEP events observed on board a space craft at corotating or noncorotating vantage point. We conclude the following: [28] 1. Gradual compressions of solar wind can significantly modify the energy, anisotropy, and time profiles of solar particles. The effects of compressions should be taken into account for an analysis (fitting) of SEP events observed at the longitude dependent speed of solar wind rising near the Earth by more or about 100 km s1 during few hours. [29] 2. Interaction of solar wind streams of different speed can produce a trap-like structure of the interplanetary magnetic field which affects SEPs. Particle diffusion and adiabatic deceleration are typically reduced. For this reason, at a corotating vantage point the SEP event development after the intensity maximum is slower than would be expected based on the modeling in the standard, Archimedean spiral field. At the noncorotating spacecraft the magnetic tube convection past the observer becomes more important. [30] 3. In application to a SEP event observed on board SOHO during 13– 14 August 1996, the model successfully reproduces magnitudes of enhancements in the solar wind speed and in the magnetic intensity profiles both along the spacecraft orbit and along the magnetic field line (the latter comes via focusing length). Our Monte Carlo simulations reveal a significant modification of the SEP transport functions as compared to the case of standard, Archimedean spiral field. [31] 4. The model reproduces the main properties of the August 1996 event, including a period of the negative (sunward directed) anisotropy observed on 14 August. Fitting of the SEP event shows that the proton injection spectrum is nearly exponential function of E1/4 in the energy range 3 – 50 MeV, produced at a CME flank near the Sun. [32] 5. We find in the SEP anisotropy data an evidence of the CIR associated magnetic mirror existing near the Earth’s orbit. Such a mirror may be a source of the CIR particle event concurrently detected at the low-energy range of ERNE. [33] Our modeling forms a basis on which to interpret observations of solar energetic particles by present and future missions, such as SOHO, STEREO, and Solar Orbiter. [34] Acknowledgments. We thank SOHO/LASCO, Wind/MFI, and Wind/SWE teams for the solar data available in the SOHO and Wind archives. This work was supported by the Academy of Finland under grant 53543. G.A.K. was supported also by the RFFI grant 00-02-1703. SOHO is an international cooperation project between ESA and NASA. [35] Shadia Rifal Habbal thanks both referees for their assistance in evaluating this paper.

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A. Anttila, L. Kocharov, G. A. Kovaltsov, and J. Torsti, Department of Physics, University of Turku, Turku FIN-20014, Finland. (leon.kocharov@ srl.utu.fi) T. Sahla, Rados Technology, Turku, Finland.