Interaction between inclined current sheets and the heliospheric termination shock



[1] Using two-dimensional hybrid simulations we study the interaction between the Heliospheric Current Sheet (HCS) and the solar wind termination shock (TS). Hot flow anomalies (HFA) are regions of hot, deflected and disturbed plasma flow which may form at the intersection of a shock and current sheet. We study the role played by the inclination, θCn, of the current sheet relative to the shock normal. As previously found, low values of θCn are associated with HFA formation. We find that as θCn increases the HFA is modified, until at θCn = 60° it disappears completely. Thus, HFAs are unlikely to be formed near the TS since the HCS is highly inclined relative to the radial direction (θCn > 60°). We also find that some suprathermal particles, particularly interstellar pickup ions, are trapped near the intersection point of the shock and current sheet where they gain considerable energy, and subsequently drift along the current sheet upstream of the shock. This process is more effective for larger values of θCn. Thus, we expect that upstream of the TS there are likely to be high-energy particles (energized pickup ions) associated with crossings of the HCS, but only of the particular magnetic reversal polarity associated with HFAs. This may be relevant to recently reported Voyager observations.

1. Introduction

[2] A hot flow anomaly (HFA) is a limited disruption of the surface of a shock where a discontinuity, typically a tangential discontinuity (TD), embedded in the flow convects into the shock. At the Earth’s bow shock [Schwartz et al., 1985; Thomsen et al., 1986; Schwartz et al., 2000] they are seen as events with elevated temperature, strongly disturbed flow and magnetic field, and usually embedded in undisturbed solar wind with compression regions on their edges. They are predominantly observed close to the quasi-perpendicular bow shock, and their duration is of order minutes. HFAs are one of a broader family of upstream disturbance events, which also includes density holes [Wilber et al., 2008] and foreshock cavities [Sibeck et al., 2002].

[3] Simulations [Burgess, 1989; Thomas et al., 1991] indicate that HFAs are formed by the disruption of a quasi-perpendicular shock by a TD which produces a channel of hot magnetosheath-like plasma into the upstream region along the TD. The expansion of this plasma produces the flow deflection, and compression regions on its edges. As suggested by the modeling (and confirmed observationally [Thomsen et al., 1993]), in order to trigger an HFA, the orientation of the TD has to be such that the motional electric field on both sides is directed towards the TD. This configuration ensures that reflected-gyrating ions from the quasi-perpendicular shock stream upstream by being focussed back towards the TD, which guides them into the upstream and away from the shock, thus disrupting the shock.

[4] The question arises whether HFA events can occur at the termination shock (TS) due to the heliospheric current sheet (HCS), and, if they do, what would be their effect. It is improbable that a HFA would be directly observed at the TS since that requires simultaneous intersection with the HCS. However, immediately before its crossing of the TS in late 2004, Voyager 1 crossed a HCS sector boundary [Burlaga et al., 2005, Figure 1]. Richardson et al. [2006] reported observations of sporadic intensity spikes of low-energy anomalous cosmic rays coincident with crossings of the HCS of a particular magnetic polarity reversal (the one in which the drift of protons is inward, towards the Sun, along the HCS). This is the same configuration for which a HFA would form close to the shock.

[5] The interaction of the HCS and TS is different from the situation at the Earth’s bow shock in two major ways: the HCS is highly inclined relative to the shock normal because of the wavy nature of the HCS [e.g., Jokipii and Thomas, 1981], and the shock is modified by the presence of a significant number of interstellar pickup ions. This paper addresses both these issues. Although the chosen parameters are representative of the distant solar wind, our results are generally applicable to other scenarios as well.

2. Numerical Model

[6] We use two-dimensional, self-consistent hybrid simulations to study the interaction of inclined current sheets and shocks, particularly in the presence of interstellar pickup ions. The ions are treated kinetically, while the electrons are a charge-neutralizing, adiabatic (adiabatic index of 5/3), massless fluid. The model is similar to earlier simulations of HFAs [Thomas et al., 1991], and the particular code used here has also been used to study the origin of low-energy anomalous cosmic rays at the TS [Giacalone and Decker, 2010].

[7] The model uses a two-dimensional Cartesian geometry (x-z). Plasma flows in the x direction, reflects from a rigid wall at x = xmax, and the shock propagates in the −x direction. The magnetic field points in the y direction (either positive or negative). All quantities are periodic in the z direction.

[8] Two thin current sheets, with opposite magnetic polarity reversals, are included in the initial and upstream boundary conditions. This is necessitated by the periodic boundary conditions in the z direction. Across each current sheet, the magnetic field changes sign, while maintaining a constant magnitude. The thickness of each current sheet is initially taken to be one ion inertial length. The angle between both current sheets and the shock normal, θCn, is varied from 0° to 60°.

[9] To simulate the solar-wind plasma in the outer heliosphere where the inclined HCS intersects the TS, we also add freshly ionized interstellar pickup ions to the plasma with a contribution of 20% by number. In the frame moving with the plasma, the pickup ions are uniformly distributed on the surface of a sphere in velocity space with a magnitude equal to the bulk plasma speed.

[10] Three different numerical simulations are performed, differing only in the choice of θCn, the inclination of the current sheets to the shock normal. In all cases the spatial domain is taken to be 800c/ωi × 1000c/ωi (i.e., xmax × zmax), where c/ωi is the ion inertial length. Other important model parameters include: βe = 0.05 (electron plasma beta), βi = 0.05 (ion beta), MA0box = 6 (Mach number of the injected flow measured relative to the simulation frame), Δx = Δz = 0.5c/ωi (grid-cell sizes), Δt = 0.01Ωi−1 (time step), η = 1 × 10 −54π ωi−1 (anomalous resistivity). Note that ωiis the ion plasma frequency, and Ωi is the ion cyclotron frequency. The Alfven Mach number, measured relative to the shock rest frame, is approximately 10. The initial spatially uniform ion distribution was generated using 40 particles per cell (20 each of solar wind and pickup protons, with the statistical weight of each adjusted such that the pickup proton contribution is 20%).

[11] Even though our model ignores a spatial coordinate, since the magnetic field everywhere points in the direction of the ignorable coordinate, the kinetic particles are capable of moving off of individual lines of force, which would otherwise be unphysically restricted due to the reduced dimensionality [e.g., Jokipii et al., 1993]. The reduced dimensionality also influences the ion heating at the shock [Burgess and Scholer, 2007], and within the simulated HFA.

3. Simulation Results

[12] Figure 1 shows the total plasma density, n(x,z), at the end of the simulation (t = 100Ωi−1) for all three simulations, as indicated. In the color-coded representation, blue corresponds to n(x,z) < n1, and red for n(x,z) > 4n1, and the scale is linear between these two extremes. The shock is clearly seen in all three cases. The average density jump is ∼2.5. The current sheets are indicated with dashed lines. In the far left plot, the current sheets remain quasi-stationary, whereas in the two right plots, they move in the positive x direction because the field is frozen into the advecting plasma. In these cases, the intersection points between the current sheet and shock moves upwards, in the positive z direction. Between the two current sheets shown in the far left plot, the magnetic field points out of the paper (Byis negative, but with the chosen geometry, this is out of the paper). The direction of the field is indicated with either a “−” (negative polarity, directed into the paper, By > 0), or “ + ” (positive polarity, directed out of the paper, By < 0). The “active” current sheet with the yellow dashed line has positively charged particles drifting to the left, away from the shock.

Figure 1.

Two-dimensional color-coded representation of the total plasma density (dashed lines). The direction of the magnetic field is indicated with the “−” (into the paper) and “+” (out of the paper). Blue represents the smallest value of n1, and red represents the largest value of 4n1.

[13] The intersection of the active current sheet and the shock produces a HFA which is clearly seen in the far left plot (θCn = 0°). This HFA is similar to that seen in past studies, but differs in that it is dominated by pickup ions. If there were no pickup ions, the HFA would be dominated by specularly reflected solar wind ions. It is important to note that specularly reflected solar wind ions have the same speed (measured in the plasma frame) as pickup ions; thus, the basic physics of HFAs with or without pickup ions is qualitatively similar. The HFA is also visible for the case of θCn = 30°, but is not seen for θCn = 60°. The reason is that the intersection point of the shock and current sheet moves in the z direction with a speed, Vt, that is greater than the speed associated with the gyromotion of the pickup ions, Vg. Consequently, these ions cannot keep up with the motion of the intersection point and do not accumulate enough to create a HFA. This is consistent with the observations reported by Schwartz et al. [2000] at Earth’s bow shock, who also pointed out that when the ratio Vg /Vt = Vg /(U1 tanθCs) is larger than unity, a HFA can form. If we assume that the HFA is made up mostly of pickup ions moving at the speed U1, then for the simulation parameters used in this paper, we find that when θCs < 45°, a HFA can form, but does not form when θCs > 45°. This is consistent with our results. Thomsen et al. [1993] also performed hybrid simulations of the interaction of inclined current sheets with shocks and found that for the relatively low inclinations, HFA formation was not significantly different from that with no inclination. They did not consider a large enough value of θCn to see the HFA essentially disappear, as we find here.

[14] Although a HFA does not form when the current sheet is highly inclined relative to the shock normal, a small fraction of pickup ions remain near the intersection point on the shock, as their gyromotion repeatedly brings them across the current sheet and shock. These particles gain considerable energy because they move along the shock with at least the speed of the intersection point which increases with increasing θCs (more specifically, Vt = U1 tanθCs). Thus, efficient particle acceleration may occur when the current sheet is highly inclined relative to the shock.

[15] Figure 2 shows the density of particles with energies greater than 10 times the plasma-ram energy, normalized to the upstream density, n1, for the case θCn = 60°. Clearly the interaction of the inclined current sheet creates a significant number of energetic particles. Again, this is for the current sheet where the proton drift is directed to the left, along the current sheet and away from the shock. There are no energetic particles at the other current sheet with the opposite polarity reversal.

Figure 2.

Two-dimensional color-coded representation of the number density of energetic particles (E > 10Ep, where Ep is the plasma ram energy) for the simulation using θCs = 60°.

[16] Figure 3 shows the density of energetic particles (E > 10Ep) and magnetic field at two different z values, as shown. The region of x does not include the shock, but does include the two separate current sheets upstream of it. There is a rise in energetic particles associated with one current sheet (two plots on the left), but not the other (two on the right). The polarity reversal of the field determines whether or not there are energetic particles associated with it. A spacecraft initially located between the shock and “active” current sheet, would first observe By > 0 (magnetic field directed into the paper of Figure 3, or negative polarity), followed by the current sheet passage and associated energetic particles, followed by By < 0 (field directed out of the paper, or positive polarity). For the opposite polarity reversal there are no energetic particles produced. This is consistent with the Voyager 1 observations of energetic particles seen to be associated with crossings of the HCS reported by Richardson et al. [2006]. In both the observations, and our simulations, the current-sheet drift of protons for this particular current-sheet magnetic polarity reversal is directed upstream.

Figure 3.

Cross sections of energetic-particle density and magnetic field for two values of z, as indicated. Only the upstream portion is shown which does not include the shock.

[17] Figure 4 shows the differential energy spectrum of all ions upstream of the shock (all z, 0 < x < 250c/ωi) at the end of the simulation for the two cases θCs = 30°, 60°, as indicated. The left plot is for solar wind protons and the right plot is for pickup ions. Focusing on the pickup ions, it is clear that the more-inclined current sheet produces a larger maximum energy compared to the lower inclination. Note also that thermal solar wind ions are also accelerated, but with much less efficiency than the pickup ions.

Figure 4.

Energy spectra for solar wind and pickup protons averaged over the entire region upstream of the shock for the two cases θCn = 30° (dashed) and 60° (solid), as indicated.

4. Summary

[18] We have performed hybrid simulations of the interaction of the HCS with the TS. A two-dimensional Cartesian geometry was considered in which the magnetic field everywhere points in the direction of the ignored coordinate. Planes separating oppositely directed magnetic fields defining our current sheets (two because of the assumed period boundaries) were inclined relative to the shock-normal direction. In addition to solar wind ions, we included freshly ionized interstellar pickup ions, contributing 20% to the total number density. These pickup ions dominate the plasma pressure.

[19] From our analysis, we conclude the following: first, hot-flow anomalies (HFA) change their character considerably as θCn, the angle of inclination of the current sheet and shock normal, is increased. The size of the HFA is the largest when θCn = 0, becomes much smaller as θCn is increased, and disappears when θCn = 60°. This is consistent with the analysis of Schwartz et al. [2000] and the reason is that the intersection point moves along the shock too rapidly for particles to become trapped in sufficient number to form the HFA. Thus, we do not expect to find a HFA at the intersection point of the HCS and TS because the HCS is highly inclined relative to the radial direction in the outer heliosphere [Jokipii and Thomas, 1981]. However, as θCn is increased, we have also found that a significant number of particles can be accelerated to high energies. Some particles are able to remain near the shock / current-sheet intersection point as it moves across the surface of the shock, and these particles are the ones that are significantly energized. Moreover, since the interstellar pickup ions dominate the plasma pressure, they are the dominant species accelerated. Thus, upstream of the TS, we expect that energetic interstellar pickup ions may be associated with crossings of the HCS. This has been reported by Richardson et al. [2006]. As a final conclusion, we note that as θCn is increased, the maximum energy attainable also increases.


[20] We acknowledge useful discussions with Marcia Neugebauer on this topic. This work was supported, in part, by NASA under grant NNX07AH19G, and NNX10AF24G, by the NSF under grant ATM0447354, and by STFC (UK) under grant PP/E001424/1.