The HCS (or sector boundary) is usually associated with multiple crossings (marked by a change in the sign of magnetic field components), reflecting the flapping and waving HCS and comprising a number of secondary thinner current sheets. However, it was possible to find a rare case of the sector boundary with a relatively clear single boundary crossing that occurred from 9 to 13 October 2007, measured by the Solar TErrestrial RElations Observatory Ahead of the Earth orbit (STEREO-A) spacecraft shown in Figs. –. The HCS crossing is represented by the change in the IMF sign from positive to negative (see Fig. c–e). The process of the HCS crossing at 1 AU usually lasts for several days .

The main HCS crossing occurred on the 12 October 2007, indicated by the red vertical line in Figs. –, in accordance with crossings of the zero line in the IMF components shown in Fig. c, d and e. This crossing was accompanied by a large decrease in the IMF strength (Figs. d and d) and a sharp change in the IMF azimuthal (clock) angle ϕ (see Fig. e). It is remarkable that the area of sharp changes in the direction of motion for the suprathermal electrons, seen in Fig. a and b as a wide section dominated by green and yellow, at pitch angles >100∘ is shifted from the main HCS crossing indicated by the signatures in the IMF (see Fig. c–e). Such shifts of electron U turns have represented a big puzzle for HCS identification for a long time until they were recently explained by the passage of bounced electrons through the reconnecting HCS with a strong guiding field . We will discuss this phenomenon further in Sect.  after describing in more detail the simulation of particle motion in a current sheet.

As mentioned above, the boundary crossing in this event was not fast, and there was a wide interface region, predominantly in the positive direction, full of the secondary current sheets observed after the first change in the IMF vector. This interface region is indicated by the two horizontal grey arrows in Fig. c–e and around the red vertical line, corresponding to the HCS crossing. Magnetic islands separated by small-scale current sheets effectively scatter electrons inside this region, which looks like a prolonged heat flux dropout in suprathermal electron flux histograms (the blue areas from 9 to 12 October 2007 in Fig. a–b). This crossing was accompanied by the high plasma beta (see Fig. c), which is one of the distinct signs of a current sheet occurrence.

During the examined period the solar-wind velocity was rather low (as shown in Fig. a). The high-speed stream approached the spacecraft several days later, so this was a clean HCS crossing not contaminated by the corotating interacting region (CIR). The slow solar-wind interaction with a current sheet is not shown, while we assume it exists since it supplies the particles entering into the HCS from the solar wind. As seen in Fig. e, the fluxes of energetic ions in the 0.15–0.6 MeV and 1.8–10 MeV ranges significantly change near the main HCS crossing. However, their maxima correspond to neither the HCS position (shown in Fig.  by the red vertical line) nor to the solar-wind density (Fig. b) and velocity (Fig. a) changes. The best correlation is seen between the flux variations and the plasma beta, which reflects a dominance in this HCS crossing of the plasma pressure over the magnetic pressure (Fig. c).

Figures  and demonstrate the acceleration of particles at least up to several MeV for ions and 0.1 MeV for electrons near the HCS. The variations in the ion flux of solar wind in the energy range of up to several MeV and electron flux of up to 0.4 MeV are shown in Fig. a and b, respectively. The IMF strength and the IMF azimuthal angle can be found in Fig. c and d. The main HCS crossing and the intermediate zone containing numerous secondary current sheets and magnetic islands are indicated by the red vertical line, as in Fig. .

The first distinct pattern in variations in the ion flux is a clear shift of its maximum from the main HCS crossing (sector boundary) further away from this boundary. The second pattern indicates a significant difference between the distributions of electrons and ions, showing the electron flux increase to spread to a much wider area than the ion flux. However, the electron flux density increase is observed to be approximately symmetrical regarding the main HCS crossing (the purely red area in Fig. b).

Hence, in the case of the single HCS crossing presented here, the fluxes of both types of solar wind particles – ions and electrons – are found to have significant topological differences, with the energies of both species increasing significantly to suprathermal level.

Further examples of energetic particles in the solar wind are shown in Fig. , obtained from the ACE and Cluster 1 measurements. This ICME event occurred on 21 January 2005 and was considered in detail for Cluster 1 by . We are mainly interested in examining the unusually wide leading ICME edge of the compressed plasma, indicated by the grey arrows in Fig.  that shows a comparison of the ACE and Cluster measurements of the total IMF. The event was previously discussed by on the basis of the Cluster data only. This area of the ICME was full of multiscale current sheets, which look like sharp decreases followed by increases in the magnetic field B and form small-scale magnetic islands that demonstrate the full set of possible signatures of magnetic reconnection in the vicinity of the ICME.

The first sharp increase in the IMF strength in this event corresponds to a shock wave, where the solar-wind velocity changes sharply by up to 1000 km s-2 and density experiences an increase of 1 order of magnitude. As seen from Fig. , despite a spatial difference between the spacecrafts ACE and Cluster 1 of ≈200Re, the IMF structure inside the leading edge was very stable and changed with distance only inside the magnetic cloud, after the passage of the region under investigation. This means that the current sheets associated with the ICME leading front discussed by are not local but represent stable structures propagating together with the body of the ICME.

Taking into account this fact and using the ACE instrumental possibilities, in addition to facts already known about this event, one can find a few additional features of particle acceleration. Let us show that some of them may be attributed to the current sheets. The pitch angle distribution functions of suprathermal electrons measured by the Solar Wind Electron, Proton, and Alpha Monitor (SWEPAM) onboard of the ACE spacecraft at different energies are compared with the three components of the IMF in Fig. . The blue colour in Fig. a–c corresponds to low values and the red to high values of the distribution function. Detailed explanations of the data are given at http://www.srl.caltech.edu/ACE/ASC/DATA/level3/swepam/index.html.

Again, similarly to the ion flux in the HCS, an increase in high-energy particle flux occurs earlier than the ICME-associated shock comes (seen in the far right corner of each plot in Fig. a); this cannot, then, be responsible for this particle acceleration. Also, there is a sharp change in the direction of motion by suprathermal electrons observed at the moment of the shock crossing and after the passage of the leading edge, indicated by grey arrows in Fig. . The fluxes of all energies are increased inside the investigated area (wide red section in Fig. a–c), which means that the acceleration of particles occurs in this region. It is easy to see multiple crossings of the neutral lines (current sheets) during this period. The zero line is represented in red in Fig. d–f in order to indicate such crossings.

The most remarkable feature in this particular ICME is the presence of the bidirectional “strahl” electrons located at the both sides of the ICME front. The strahls look like the two horizontal ribbons seen in the pitch angle spectrograms along 0 and 180∘ (Fig. a–c). They are particularly pronounced in the highest-energy channel (Fig. a). Strahls are known to be suprathermal electron beams, travelling along the magnetic field direction. Their origin is still unclear; however, their properties have been studied for many years . It is known that strahls predominantly appear anti-sunward in regular solar wind, although near ICMEs counterstreaming strahls may be also observed. We present their proposed interpretation below in Sect. .

It can be observed that the accelerated ions and electrons of high (MeV-range) energies occurred several hours earlier than the ICME's leading front reached the Earth orbit, as seen in Fig. . Acceleration at the shock is seen as the first sharp increase in both ion and electron flux of different energies. The leading edge of the ICME, indicated by the arrows, is clearly reflected in the ion and electron flux changes.

The local increase in both ion (Fig. a) and electron (Fig. b) fluxes, corresponding to the shock arrival (the left arrow in Fig. ), indicates the point at which particles experience reflection and acceleration. This mechanism is responsible for a gradual increase in ions and electrons fluxes, forestalling the ICME arrival. Electrons, being lighter and faster, go further from the source of acceleration (ICME front) than ions. A local increase in the ion flux is observed approximately from 15:00 UT (the foreshock area) to 18:30 UT, corresponding to the border of the leading edge. As a result, the temporal profiles of ion and electron fluxes observed in the vicinity of the ICME become essentially different, resembling the ones occurring in the solar-wind particles during their crossing of the HCS, as shown Fig. .

It is remarkable to note that the flux of suprathermal ions does not fall immediately after the ICME's shock crossing (as one would expect if the particles are to be accelerated by this shock); it fell only after the passage of the current sheet area in the front of the ICME. This effect can not be simply explained by the enhanced solar-wind density because the parameter, characterising the low-energy solar wind, does decrease simultaneously with the ion flux, as shown in Fig. a. Hence, this effect is not a result of the flow's topology. This explicitly indicates the topological difference in the motion and acceleration of particles with the opposite charges, which is the most important property of particle acceleration in a reconnecting current sheet . One can see a noticeable difference in the variations in ions and electrons: the electron flux increases at the same time as the ion flux drops, i.e. electrons anticorrelate, and the centre of this inversion is definitely the leading edge of the ICME, where a current sheet is likely to occur.

The acceleration time during particle motion inside a current sheet is estimated to be of the order of 10-6 s for the electrons and 10-3 s for the protons . This time is much shorter than the time of the magnetic field reconstruction during a reconnection ; thus, the background magnetic field can be assumed to be stationary during the whole simulation.

Also, from the previous test particle simulations in the solar corona, we conclude that the travel distances of accelerating particles along the reconnecting current sheet (RCS) are of the order of 10 km =10Rp at most (for the protons in the corona) , a value obtained by applying the periodic boundary conditions along z direction. On the other hand, it is evident that this travel distance is much shorter than the length scale of the magnetic field variation along the current sheet. In addition, as is generally accepted, we suppose that the magnetic field variation across the current sheet has a much shorter length scale than its variation along the current sheet of length a in the direction z and of b in the direction of y, e.g. d≪a,b.

Let us consider a coordinate system related to the current sheet midplane, which hypothetically can be oriented in any direction in the solar wind. Our simulation domain is a small part of the reconnecting current sheet (see Fig.  for the model cartoon), which is large enough to contain the full trajectories of accelerated particles for the periodic boundary conditions (see Sect. ). All the three components of the background magnetic field are considered; the components are assumed to be stationary and to vary inside this domain only in the x direction across the RCS. The main magnetic field component, Bz, corresponding in most cases to the component Bx in system Geocentric Solar Ecliptic coordinate system (GSE) used for the interplanetary magnetic field (IMF), depends on the simulation coordinate x as follows: Bz(x)=-Bz0tanh⁡xLx.

The transverse magnetic field component Bx corresponding to By (in GSE) is assumed constant inside the simulation domain, e.g. Bx=-Bx0.

The guiding (out-of-plane) magnetic field By corresponding to the Bz component of the IMF (GSE system) is assumed to be maximal in the midplane and to vanish outside the RCS as follows: By(x)=By0sechxLx.

Note that for By0=0 the configuration corresponds to the Harris sheet equilibrium and for By0=Bz0 the equilibrium becomes force-free.

The inflow of plasma into an RCS combined with the condition of the frozen-in magnetic field leads to the induction of the drifted (out-of-plane) electric field Ey. In order to provide the inflow of plasma in our simulation domain, we set up a background electric field that drifted in with velocity Vin by a magnetic diffusion process . Ey=Ey0=Bz0Vin, where Vin is the inflow velocity, which is about 0.1 times the Alfvén speed. It is usually accepted as being equal to the thermal velocity of the ambient plasma . The magnetic field gradient of the main magnetic field component Bz across the current sheet thickness (along the x axis) is ignored .

In Fig.  we plotted the results of the two PIC simulations carried out for a weak guiding field (Fig. a: protons; b: electrons) and for a strong guiding field (Fig. c: protons; d: electrons).

There are a few important outcomes of the particle acceleration simulation in a reconnecting current sheet. The first one indicates that, for the magnetic topologies with a moderate-to-strong guiding field, there is a strong separation of electron trajectories into one semiplane (say with X>0) and proton trajectories into the opposite one (with X<0), as shown in Fig. c and d for PIC simulations or in the bottom-right close-up of Fig.  obtained with the test particle approach for the given magnetic field topology.

In the case of a weaker guiding field (Fig. a and b), there is a little or no separation of protons (a) and electrons (b); the protons are ejected slightly more to the negative semiplane but some are still ejected to the positive one. Electrons are found cycling around the midplane because they cannot gain sufficient energy to break from the electrostatic force of protons being accelerated in the midplane. As a result, electrons create a cloud, which exists for as long as protons are present in the midplane, and then they follow the protons and become ejected to the same semiplane (X<0 in Fig. a and b).

The second outcome, related to the first, reveals the origin of two beams of the same charge: (1) the transit particles, which enter from the side opposite to the one from which they are to be ejected (for example, protons from the side X<0 and electrons from X>0 in the right-hand close-up in Figs.  and ) and (2) the bounced particles entering the RCS from the same side from which they will be ejected (protons from the side X>0 and electrons from the side X<0).

The transit particles start gaining their energy from a reconnecting electric field Ey, immediately after they are dragged into the diffusion region, while they still travel to the current sheet midplane, where the bulk of their energy is to be gained. Hence, their gyration around the midplane has large radii and they gain much more energy than the bounced electrons, which need to work against the magnetic field topology in order to arrive at the midplane, the only place in an RCS where the particles can be accelerated. Hence, the gyro radii of bounced particles become much smaller than those of the transit particles, and they gyrate closer to the midplane in the case of a weaker guiding field, thus gaining less energy (see Fig. a for protons and b for electrons).

In the case of a stronger guiding field, the transit particles gain more energy than in a weaker one (Fig. c and d), while the bounced electrons cannot even reach the midplane (Fig. d). These bounced electrons are turned around by the magnetic field back in the direction they came from; they then form a shoe-like density distribution, shown in Fig. d, or even take on a medallion-type shape . The distance after which the bounced electrons turn around is inversely proportional to the guiding field magnitude, which was also confirmed earlier by the observation of electron pitch angles for the three cases of the measured magnetic field topologies in the HCS crossings summarised in Table 1 of the paper by .

The third outcome is related to the ambient plasma feedback to a presence of accelerated electrons and protons, whose separation towards the midplane induces a strong polarisation electric field across the current sheet (in the directions X and Z) caused by the particle separation towards the midplane. Some examples of this polarisation electric field Ẽx are shown in Fig.  for the magnetic field topology of the heliosphere for weaker and stronger guiding fields. The magnitude of the polarisation electric field (Ẽx) exceeds the original reconnection electric field Ey by 1–2 orders of magnitude. This electric field defines the direction of motion of the protons and ions passing through any RCS, as discussed in the next section.

The main HCS crossing and the intermediate zone containing numerous secondary current sheets and magnetic islands are indicated by the red vertical line, as in Fig. c–e. Figures  and demonstrate acceleration in the vicinity of the HCS of solar-wind electrons to 100 keV and of ions to several MeV. The ion velocity profile (Fig. ) has a minimum before entering into the HCS, followed by its maximum at some distance after passing through the HCS. Using the other STEREO data, it is possible to find a number of similar cases with a single boundary crossing, a few of which were reported by . The electron pitch angle measurements shown in Fig. a and b also demonstrate the electrons turning their direction of motion by 180∘ at some distance before the sector boundary crossing.

This summary of the particle versus magnetic field measurements allows us to conclude that it is difficult to find any structures in the solar wind potentially responsible for these specific particle acceleration profiles, except for the fact that they pass through the HCS. Thus it is logical to assume that some local re-acceleration can occur at the HCS, and this can be the case in the HCS undergoing a magnetic reconnection.

In the case of a reconnecting HCS, one can involve the results of simulations shown in Sect. , where the magnetic field zero is reached at a current sheet midplane (X=0). Then it becomes more evident how the electrons and protons can reach the energies reported in observations after they pass a current sheet. This happens because they gain energy there up to a few MeV for protons and 100 keV for transit electrons (see model results for higher-energy protons and electrons in Fig. a, c and d). This model simulation explains the results reported in Figs.  and .

By contrast, for the low magnetic field of the HCS and the higher guiding field, the bounced electrons cannot approach the midplane of the HCS and are turned around by the magnetic configuration, making them move back to the point at which they were dragged into the HCS, as shown in Fig. d. This explains very naturally the variations in the pitch angle distribution of electrons shown in Fig. a and b and the dependence of the distance at which this turn occurs on the magnitude of the guiding magnetic field, as reported in Table 1 . Hence, by measuring this distance one can derive the simulated ratio between the guiding field By and the main magnetic field component Bz, compare it with the measured one and explain the U turn of solar-wind electrons.

The peculiar profiles of the ion flux velocities with a minimum before and a maximum after the HCS can be naturally explained by the polarisation electric field induced by the separation of accelerated particles of the opposite charge shown in Fig. c and d – the ions simply follow this polarisation electric field during their passage across the HCS. From a profile of the ion velocity across the HCS, one can estimate the polarisation field magnitude using the ratio of the guiding By and transverse Bx magnetic field components.

Of course, everything said above is valid for a single current sheet as modelled. However, the additional scenarios of particle acceleration can occur in magnetic islands formed in a reconnecting current sheet, and this can be valid for the HCS . Additional mechanisms of the acceleration of particles in the vicinity of the merging magnetic islands observed near the HCS and separated by secondary current sheets may explain the existence of a wide area of the HCS filled with energetic particles. If so, the particles can be initially accelerated by the reconnecting electric field of the current sheet discussed here and by and, then, with the evolution of the current sheet caused by tearing instability and the formation of a number of magnetic islands as a result , they can become trapped in closely located magnetic islands, leading to their secondary acceleration .

There is another application of the obtained results for the particles passing through the HCS. It is commonly accepted that, in the absence of SEP events, CIRs are the main source of suprathermal particles in the solar wind . This was not the case in the observations presented in Figs. –; the CIR was very far from the HCS and it came too late to explain the observed effects. It is important to note that, in most cases, the HCS is crossed before the CIR occurrence at 1 AU, i.e. additional acceleration of the solar-wind particles actually takes place in the reconnecting heliospheric current sheet but not at the CIR itself.

In the case of the ICME, additionally to the well-known mechanism of acceleration by shocks, here we suggest particle acceleration by reconnecting current sheets occurring at the leading edge of this propagating structure. The ICME-associated current sheets seem to be relatively stable structures, as seen from a comparison of the Cluster and ACE measurements, and some features of the energetic particles in their vicinity may be attributed to a magnetic reconnection process. Energetic-particle fluxes demonstrate local enhancements around the area filled with current sheets, typical for the particles' movement through reconnecting current sheets. One of the most interesting features of particle acceleration in a current sheet is the anticorrelation of ion and electron flux and their different pitch angle distributions (separation of ions and electrons into the opposite semiplanes), which is consistent with the PIC results presented here (see the first outcome of simulations in Sect. ).

Similarly to the ion flux in the HCS, an increase in high-energy particle flux up to a few MeV occurs in the ICME before the associated shock that is accompanied by a sharp change in the direction of motion by suprathermal electrons observed at the moment of the shock crossing and after the passage of the leading edge, indicated by the grey arrows in Figs. , and seen as the red section in Fig. . It can be seen that these energy gains by ions and electrons as well as the change in direction of motion by electrons can be explained by the mechanism of particle acceleration in a reconnecting current sheet in the front of the ICME, in a similar way as was done above for the HCS.

Furthermore, from the profile of ion velocities and from the distance at which the suprathermal electron changes its direction, one can deduce some information about magnetic and electric field components leading to such particle profiles. There is a noticeable difference in the variations in ions and electrons: the ion flux decreases after the leading edge crossing (Fig. a), while the electron flux, by contrast, increases before this crossing (Fig. b); this can be explained naturally by the separation of particles with opposite charges with respect to the current sheet midplane (similar to the particles shown in Fig. c and d and their consequent ejection to the opposite semiplanes (e.g. the opposite sides of the exhausts of a current sheet)).

The most remarkable feature of this particular ICME is the presence of the bidirectional high-energy electrons (strahls) on both sides of the ICME front. The term strahl is used here according to the definition of , considering strahls to be the narrowest field-aligned beams. The strahls are seen in the highest-energy channel as the two horizontal ribbons in the pitch angle spectrograms along 0 and 180∘ (Fig. ). Indeed, simultaneous changes in the strahl intensity at all energies are commonly observed near the HCS and at the leading edges of CIRs and ICMEs . In the HCS case, they simply travel along the magnetic field lines, predominantly in the anti-sunward direction, but at the ICME edges their symmetric depletions are observed.

The strahls are usually assumed to propagate outward from the corona, i.e. to have a solar origin. An alternative view is to consider strahls as suprathermal electrons accelerated due to the resonant interaction with whistler waves, suggested by , although, this theory cannot be used to explain their bidirectionality near ICMEs. This bidirectionality was interpreted as a consequence of the double magnetic connection of ICMEs to the Sun. However, the bidirectional strahls are observed at corotating shocks very far from the Earth, at 5 AU , where such a supposition is very doubtful because of the long distance from the Sun; it thus requires some additional suppositions, such as mirroring or similar, which are also difficult to maintain.

Meanwhile, our results on the accelerated electron behaviour near a reconnecting current sheet allow an alternative explanation: electron acceleration to the strahl energies occurs locally due to a magnetic reconnection that appears at the HCS, the leading edges of CIRs and ICMEs as well as at corotating shocks, where current sheets are formed. We suggest that the strahls associated with ICME are generated by the most energetic electrons (transit electrons), which are accelerated in a current sheet at the ICME's front to fairly high energies, as shown in Fig. d (upper stream).

Then the dependence of electron trajectories in a current sheet on the magnetic field topology can easily explain the bidirectionality of the ICME-associated strahls because the accelerated electrons have a tendency to move to the opposite sides of the midplane to the diagonal quarters with respect to the planes Z=0 and X=0. It is obvious that, if not disturbed, such a movement of transit electrons must be bidirectional with respect to the plane Z=0, as its direction of motion is dependent on the signs of Bx and By. For different signs of Bz, the magnetic topology will follow either the left- of right-hand rule, leading to the bidirectional move of the energetic transit electrons seen as strahls.

The particle transport problem concerning the time after the electrons are ejected from the current sheets where they are accelerated has not been investigated yet. From the general point of view, one can expect that energetic electrons ejected from the HCS into opposite directions (from and to the Sun) have to undergo some interaction with the ambient plasma. The fraction of electrons which moves towards the Sun, where density increases, will be scattered by the ambient particles and lose their energy in Coulomb collisions and ohmic losses, similar to solar-flare electrons . The difference is that the initial energy of the solar-wind electrons is well below keV; hence, they will lose this energy rather fast while approaching the Sun. By contrast, the ejected electrons moving from the Sun into the rarifying plasma do not meet many particles, and, thus, they can keep propagating as beams of energetic electrons, or strahls.

The geometry of motion of electrons around the ICME's current sheets is slightly different, since the stronger magnetic field of this current sheet in front of the ICME is perpendicular to the weaker interplanetary magnetic field (IMF). The energetic electrons accelerated in this current sheet keep moving along this stronger magnetic field of the ICME's current sheet as long as this current sheet exists. This is why observers can see energetic electrons, or strahls, for a very long time as two electron streams moving in opposite directions as they are ejected from the current sheet. However, a precise particle transport scenario after their ejection from the HCS or the ICME's front current sheet can be only derived from simulations of the process of particle scattering and motion in a given magnetic field topology that will be the subject of a forthcoming paper.