Quasi-perpendicular shocks: Length scale of the cross-shock potential, shock reformation, and implication for shock surfing



[1] One-dimensional (1-D) full particle simulations of almost perpendicular supercritical collisionless shocks are presented. The ratio of electron plasma frequency ωpe to gyrofrequency Ωce, the ion to electron mass ratio, and the ion and electron β (β = plasma to magnetic field pressure) have been varied. Due to the accumulation of specularly reflected ions upstream of the shock, ramp shocks can reform on timescales of the gyroperiod in the ramp magnetic field. Self-reformation is not a low ωpece process but occurs also in (ωpece)2 ≫ 1, low β simulations. Self-reformation also occurs in low ion β runs with an ion to electron mass ratio mi/me = 1840. However, in the realistic mass ratio runs, an electromagnetic instability is excited in the foot of the shock, and the shock profile is considerably changed compared to lower mass ratio runs. Linear analysis based on three-fluid theory (incident ions, reflected ions, and electrons) indicates that the instability is a modified two-stream instability between the decelerated solar wind electrons and the solar wind ions on the whistler mode branch. In the reforming low ion β shocks, part of the potential drop occurs at times across the foot, and part of the potential (∼40%) occurs over a few (∼4) electron inertial lengths in the steepened up ramp. Self-reformation is a low ion β process and disappears for a Mach 4.5 shock at/or above βi ≈ 0.4. It is argued that the ion thermal velocity has to be an order of magnitude smaller than the shock velocity in order for reformation to occur. Since according to these simulations only part of the potential drop occurs for relatively short times over a few electron inertial lengths, it is concluded that coherent shock surfing is not an efficient acceleration mechanism for pickup ions at the low β heliospheric termination shock.

1. Introduction and Motivation

[2] Interstellar pickup ions, i.e., interstellar matter that penetrates into the solar system and is ionized in the inner heliosphere and picked up by the outward moving solar wind, are currently under intensive observational and theoretical investigation. This is partly due to the fact that these ions are most likely the source of the so-called anomalous cosmic ray component [Fisk et al., 1974]. Acceleration of these ions to high energies is most probably at the termination shock of the solar wind [Pesses et al., 1981]. Furthermore, pickup ions have been observed to be efficiently injected into an acceleration process in corotating interaction regions [Gloeckler et al., 1994], which are bounded by a forward and reverse shock pair. Both, the termination shock as well as corotating shocks are basically quasi-perpendicular shocks, i.e., the angle ΘBn between the shock normal and the upstream magnetic field is larger than 45°. There exist a number of competing acceleration mechanisms for pickup ions at quasi-perpendicular shocks. In the classical first-order Fermi acceleration mechanism, the particle must have a sufficiently high energy that it can scatter diffusively across the microstructure and macrostructure of the shock and experience the compression between the converging upstream and downstream flows. One-dimensional (1-D) hybrid simulations of quasi-perpendicular shocks by Liewer et al. [1993] and Kucharek and Scholer [1995] where pickup ions had been included have shown that these ions are only reflected with reasonable rates when the shock normal angle is smaller than ∼60°. Kucharek and Scholer [1995] suggested that these backscattered pickup ions contribute to the injection pool for a diffusive shock acceleration process, although the termination shock finds itself with a ΘBn < 60° for only ∼10% of the time. A somewhat different approach was taken by Giacalone et al. [1994]. They also use a 1-D hybrid simulation of a quasi-perpendicular shock but include an ad hoc scattering perpendicular to the magnetic field. This enables pickup ions to diffuse across field lines. Giacalone et al. [1994] found that when reasonable scattering times are considered pickup ions are injected and accelerated to fairly high energies, whereas solar wind ions are not. Giacalone et al. [1997] have pointed out that stationary shocks, as the termination shock and planetary bow shocks, might have different pickup reflection properties than interplanetary travelling shocks. Pickup ions have in the inertial frame a speed between zero and twice the solar wind speed. Because propagating shocks move only about 100 km/s faster than the 400–700 km/s solar wind, nonaccelerated interstellar pickup ions have already many times the shock ramming velocity when they encounter interplanetary travelling shocks and constitute thus a suprathermal particle population. Giacalone et al. [1997] proposed that in a first step pickup ions are accelerated in the inner heliosphere. These ions are then further accelerated at the termination shock to become the anomalous cosmic rays.

[3] A possible injection and acceleration mechanism at quasi-perpendicular shocks, which strongly favors the injection of pickup ions over solar wind ions, is shock surfing. Based on an idea by Sagdeev [1966], Zank et al. [1996] and Lee et al. [1996] have investigated the possibility that pickup ions are trapped at the shock between the electrostatic potential of the shock and the upstream Lorentz force: part of the pickup population incident on the shock has so little energy in the shock frame that it cannot overcome the electrostatic barrier and is reflected at the shock ramp. Upstream it is again turned around by the Lorentz force. The time spent upstream of the shock determines the maximum energy gain for the trapped pickup ions. This time is limited by two conditions: (1) the particles' incident normal velocity vx must satisfy the condition (m/2)vx2eΦ, where m is the particles' mass, Φ the cross-shock potential, and e the electron charge and (2) the Lorentz force is smaller than the force exerted by the electrostatic potential eΦ, i.e., evyBeΦ/Ls, where B is the magnetic field magnitude and Ls the length scale of the cross-shock potential [Zank et al., 1996]. Here, the electrostatic potential gradient has been approximated by eΦ/Ls. Thus, the maximum energy a particle can reach by shock surfing is inverse proportional to the square of the cross-shock potential length scale. Assuming that the cross-shock potential is of the order of the upstream bulk energy and that Ls is of the order of the electron inertial length λe = cpe (c = velocity of light, ωpe = electron plasma frequency) the maximum pickup ion energy is close to about 1 MeV/nucleon. A somewhat different analysis by Lee et al. [1996] results in maximum energies of 12 MeV/nucleon. If, however, the scale length is of the order of the ion inertial length, λi, the maximum energy is by the ratio mi/me smaller. Thus, the length scale of the cross-shock electrostatic potential determines the effectiveness of shock surfing acceleration. Furthermore, intrinsic nonstationarities of the shock may also compromise the acceleration efficiency.

[4] Direct observations of shock potential length scales at interplanetary shocks and the Earth's bow shock are rare, the reason being that it is not possible with single spacecraft observations to distinguish unambiguously between spatial and temporal structures. There are two reports on thin shock ramps: based on ISEE-1 and ISEE-2 data, Scudder et al. [1986] reported a high β (β = magnetic field to particle pressure) shock observation with a length scale of the magnetic field ramp of ∼4λe, and Newbury and Russell [1996] observed one almost perpendicular bow shock with a magnetic field ramp scale given by ∼2λe. But is not clear whether these observations are rare exceptions or a common occurrence. A determination of the magnetic field ramp of 11 quasi-perpendicular shocks measured by OGO-5 and ISEE-1/2 was made by Balikhin et al. [1995]. Four of the shocks reported had magnetic ramp scales smaller than 0.15∼4λi. In addition it is not clear how the magnetic ramp scale compares with the scale of the shock potential. Scudder [1995] derived empirically the potential profile through a quasi-perpendicular bow shock. The total potential jump occurs over a scale, which is an order of magnitude larger than that of the magnetic ramp. According to the result by Scudder [1995], the electric field within the ramp is considerably smaller than one would get by putting the whole potential drop across the magnetic ramp.

[5] In the present paper, we will determine the length scale of the potential by computer simulations of collisionless shocks. Computer simulations of shocks have been performed in the past by the hybrid method and by the particle-in-cell (PIC) method. In the hybrid method, ions are treated as macroparticles and electrons as a fluid. With a massless electron fluid, it is not possible to resolve the electron scale. Application of the hybrid method with a finite electron mass to shocks can produce serious problems: on one hand the electron inertial length has to be resolved, on the other hand the resistive length scale should not be resolved. Otherwise, the electron fluid is artificially heated in the shock ramp. Since the hybrid method has artificial and intrinsic resistivities, one should take refuge to the PIC method, where both ions and electrons are treated as particles. This will be done in the present paper.

[6] Beginning with the study of Biskamp and Welter [1972], collisionless shocks have been investigated by PIC simulations for more than three decades. It has been shown that perpendicular and quasi-perpendicular shocks are intrinsically nonstationary and considerable attention has been paid to the cyclic self-reformation [e.g., Biskamp and Welter, 1972; Lembège and Dawson, 1987; Lembège and Savoini, 1992]. This process is due to the fact that part of the upstream ions are reflected at the shock front and are responsible for the formation of the foot. The accumulation of these ions in time is responsible for the cyclic self-reformation with timescales of the order of the inverse ion gyrofrequency Ωci. We will concentrate in this paper on quasi-perpendicular shocks with ΘBn close to 90°, i.e., on shocks where the deviation of ΘBn from 90° is small so that dispersive effects are negligible. At shocks with ΘBn > Θtr where cos Θtr = (me/mi)1/2MA (MA = upstream Alfvén Mach number). a whistler wave train will trail the shock, whereas for ΘBn < Θtr a whistler precursor is emitted from the shock [e.g., Balikhin et al., 1995]. For example in the case of a shock with Alfvén Mach number MA = 5 one obtains Θtr ≈ 83° (assuming mi/me = 1840). In the case ΘBn < Θtr upstream whistlers can interact with reflected ions and steepen up, which also results eventually in shock reformation [e.g., Krasnoselskikh et al., 2002]. We will concentrate in the following on quasi-perpendicular shocks with ΘBn > Θtr. The length scales of the magnetic ramp in more oblique shocks (ΘBn = 55°) as obtained from PIC simulations has been compared on a statistical basis with the length scales of the electric field gradient by Lembège et al. [1999]. These authors found that the spatial scales over which the magnetic field changes are of the same order as those over which the potential acts, and that these are of the order of a few cpe.

[7] There is a further motivation to perform additional PIC simulations of perpendicular and quasi-perpendicular shocks, and to determine length scales in these shocks. Two important parameters enter a PIC simulation: (1) the mass ratio mi/me and (2) the ratio of electron plasma frequency to gyrofrequency ωpece. The problem of performing PIC simulations with a realistic value of mi/me is well known; not so well known is the problem arising from the second quantity. As far as mi/me is concerned there seems to exist in the literature only one quasi-perpendicular shock simulation with a realistic value. This is the work by Liewer et al. [1991], where the electron dynamics in low Mach number (MA ≈ 2.8) shocks is investigated by a 1-D PIC simulation. These subcritical shock simulations were only run to a fraction of an inverse ion gyroperiod. In most PIC simulations the second quantity, ωpece, is assumed to be of the order of 1, i.e., simulations are done for shocks in the strongly magnetized condition. Note that ωpece can be written in terms of the ratio of velocity of light c and Alfvén velocity vA:

equation image

[8] In the solar wind at the Earth's orbit this quantity is 100–200. In contrast, because of computational constraints of PIC codes, Biskamp and Welter [1972] used a value of ωpece = 5, Lembège and Dawson [1987] used 1, and Liewer et al. [1991] used 1–4. Quest [1986] has pointed out that the use small values of ωpece overemphasizes charge separation effects. He argued that separation electric fields in thin shock structures possibly destabilize the shock and could lead to artificial reformation. Recently, a 1-D simulation in the weakly magnetized condition with a value of ωpece = 20 has been presented by Shimada and Hoshino [2000]. However, because of computational constraints, these authors chose a value of mi/me = 20, which makes it difficult to differentiate between ion and electron length scales.

[9] In this paper we will present a number of 1-D shock simulation runs with different mass ratios and different ratios of the electron plasma to gyrofrequency. At present, computer resources prohibit a simulation with a realistic mass ratio and at the same time a value of ωpece appropriate for the solar wind. However, we can vary both parameters independently and are then able to infer the structure of a bow shock/termination shock or an interplanetary traveling shock. Among the runs, we will present one run with mi/me = 1840, τ = (ωpece)2 = 8, and one run with mi/me = 400, τ = 60. We will also vary the second important parameter, the upstream ion β, between βi = 0.1 and βi = 0.4. Low β values are expected to be characteristic for the heliospheric termination shock. We will restrict ourselves to nondispersive quasi-perpendicular shocks, and chose ΘBn = 87°. Less oblique collisionless shocks, which have dispersive upstream whistlers, have recently been investigated by Krasnoselskikh et al. [2002]. A compilation of the parameters used in the various runs is given in Table 1.

Table 1. Parameters of the Runs

2. Simulation

[10] The shock is produced by the so-called injection method: a high-speed plasma consisting of electrons and ions is injected from the left hand boundary of a 1-D simulation system and travels toward positive x. The plasma carries a uniform magnetic field which has a Bz and a Bx component. At the right hand boundary, the particles are specularly reflected. A shock then propagates in the −x direction, and the shock normal is the x axis. Thus, the simulations are done in the so-called normal incidence frame where the upstream bulk velocity is parallel to the shock normal. Initially there are 100 particles for each species in a computational cell, the size of which is comparable to the Debye length λD. In the following, time will be given in units of the inverse of the ion cyclotron frequency Ωci, distances in units of the electron inertial length λe, the velocity in units of the upstream Alfvén speed vA, magnetic field and the density in units of their upstream values B0 and n0, respectively. The potential eΦ is given in units of cB0e. We use ΘBn = 87° and inject the plasma with Alfvén Mach number MA0 = 3 and 4.5, respectively. This leads to MA ∼ 4.2 and MA ∼ 6.2 shocks, respectively. One exactly perpendicular shock (ΘBn = 90°) is presented for comparison. We vary the parameters mi/me, τ = (ωpece)2, and βi, βe. In the following we will discuss the various runs and start with a low β, low electron to ion mass ratio, and low τ run usually presented in the literature. We then increase the mass ratio to the realistic value, keeping τ low, and then increase τ, leaving the mass ratio low. Finally, we discuss runs with larger values of ion and electron β, respectively.

[11] Figure 1a shows magnetic field profiles (Bz component) stacked in time for a run (1) with the following parameters: mi/me = 400, τ = 10, βi = 0.1, βe = 0.2. As has been shown previously by many authors, the shock consists of a foot with reflected ions and a ramp. The reflected ions accumulate in the foot and the shock self-reforms on a timescale of 1–2 Ωci−1, with the foot becoming the new shock ramp. Figure 1a exhibits two of these reformation cycles from run 1, which demonstrates that this is not an initial artifact but a repetitive event. For run 2 we have assumed a realistic mass ratio mi/me = 1840 and have used τ = 4 (we have repeated the low mass ratio run 1 with τ = 4 and found virtually no differences in the profiles of the various parameters at identical times). Figure 1b shows the stacked magnetic field profiles (Bz component) in the same format as in Figure 1a for the time period 5.6–7.3 Ωci−1. It can be seen that the magnetic field magnitude in the foot increases with time, and a new shock ramp emerges out of the foot. Thus, reformation is not an artifact of the use of an unrealistic low ion to electron mass ratio. However, in the realistic mass ratio case (run 2), the pronounced dip between the newly emerging shock ramp and the former ramp disappears. This has consequences for the identification of shock reformation from situ spacecraft data. In addition, once the magnetic field in the foot is large enough, i.e., about twice the upstream magnetic field, low-frequency waves appear in the foot region. We will return to these low-frequency waves below.

Figure 1.

(a) Stacked profiles of the magnetic field Bz component for a shock simulation (run 1) with upstream parameters ΘBn = 87°, βi = 0.1, βe = 0.2. Mass ratio is assumed to be mi/me = 400, (ωpece)2 = 10. (b) Stacked profiles of the magnetic field Bz component for a shock simulation (run 2) with upstream parameters ΘBn = 87°, βi = 0.1, βe = 0.2. Mass ratio is assumed to be mi/me = 1840, (ωpece)2 = 4.

[12] At the beginning of a reformation cycle the magnetic ramp develops a sharp profile. Figure 2 shows the magnetic field Bz component and the cross-shock potential Φ at two different times during a reformation cycle for run 1 (upper four panels) and run 2 (lower four panels), respectively. The corresponding profiles are marked in Figure 1 by arrows on the y axis (time axis). Here and in the following cross-shock potential always refers to the potential jump between far upstream and a position x given by the electric field Ex normal to the shock in the normal incidence frame. The scale in x for the two runs varies by a factor 2 so that in units of the ion scale due to the different mass ratios the profiles of both runs are comparable. Let us first discuss results from run 1 (upper four panels) At tΩci = 6.7 the magnetic field increases in the foot over 20λe. The cross-shock potential has the same length scale, i.e., it extends over about 1 ion inertial length. As the shock reforms (tΩci = 7.3) a sharp ramp appears which leads to a large electric field spike over ∼4λe (not shown here), but a large part of the potential drop occurs already in the foot region, so that at tΩci = 7.3 only 60% of the cross-shock potential exists across the main ramp. The scale of the foot is determined by the gyroradius so that the whole potential drop occurs over a scale given by the gyroradius. The magnetic field and potential profiles of the mi/me = 1840 run are similar. At tΩci = 6.4 the magnetic field strength in the foot has increased to twice the upstream value, and small wavelength waves are superimposed on the profile of the foot. After reformation, i.e., by tΩci = 7.1, the ramp exhibits again a sharp profile and the small wavelength waves have disappeared. Table 2 summarizes the length scales of the foot Lf and the ramp Lr and the corresponding cross-shock potential drops during two time periods. During period 1 (tΩci = 7.1) a new reformation cycle just begins, whereas during period 2 (tΩci = 6.4) the foot has increased in magnetic field amplitude, and the whole cross-shock potential drop occurs in the foot region.

Figure 2.

Comparison of magnetic field and shock potential profiles at two different times between run 1 (top two panels) and run 2 (bottom two panels).

Table 2. Characteristic Lengths of Foot (Lf) and Ramp (Lr) and Corresponding Potential Drops in Percent of the Whole Cross-Shock Potential for Run 2

[13] We return to the high-frequency waves seen in the mi/mee = 1840 run during the reformation process. Figure 3 shows an enlarged view of the foot and ramp region at tΩci = 6.4 for the mi/me = 1840, low β run 2. Shown are in the top two panels the magnetic field profile and the normal electric field component. The two bottom panels show an electron vezx phase space plot and an ion vixx phase space plot, respectively. Note that vz is almost parallel to the magnetic field; however, because the shock is propagating at an angle ΘBn to the upstream magnetic field the electric field has a component parallel to the magnetic field. The electrons are accelerated along the field lines and can get trapped by the potential of the high-frequency waves. Acceleration and trapping is evident from the vezx phase space plot in Figure 3. The trapping leads to preheating of the electrons in the foot region.

Figure 3.

From top to bottom: Magnetic field component Bz, electric field Ex, and electron vezx phase space plot and ion vixx phase space plot in the foot and in ramp of the shock for the mi/me = 1840 run (run 2) during reformation.

[14] When the ions are reflected the incident solar wind electrons are decelerated so as to cancel the zeroth-order current. The relative velocity between the incident electrons and the incident ions can lead to an electromagnetic modified two-stream instability [Wu et al., 1983]: in the quasi-perpendicular case ΘBn < 90° an interaction between the ion beam mode of the incoming solar wind ions and the electron whistler wave can take place. We have determined the growth rate of this instability on the basis of three-fluid theory where we have assumed that the ions are unmagnetized and the electrons are magnetized. The values for the densities and the velocities are taken from simulation run 2 and are average values in the foot of the shock. For the analysis, it has been assumed that the k vector is parallel to the incident ion, reflected ion, and electron beam, respectively, and that the magnetic field is inclined by 87° to the beam direction. It should be noted that this analysis neglects kinetic effects and is only qualitative. The resulting growth rate γ in units of the ion gyrofrequency as a function of the wave vector k in units of the inverse of the electron inertial length is shown in Figure 4. The growth rate is shown for three different values of the mass ratio mi/me, and for each mass ratio we present the result for two different values of τ: τ = 4 (solid line) and τ = 60 (dashed line), respectively. First, it can be seen that the wave vectors of maximum growth for the mi/me = 1840 case is close to what is seen in run 2 (see Figure 3). Second, the growth rate has a strong dependence on mi/me. For mi/me = 400, the growth rate is of the order of the ion gyrofrequency so that in low mass ratio cases it is not expected that the instability is excited within one reformation cycle. This may explain why the instability has not been seen in previous low mass ratio PIC simulations. Third, the instability is only weakly dependent on τ. As can be seen from the bottom panel of Figure 3 the incident solar wind ions interact strongly with the whistler waves, which leads to a deceleration in the individual whistler trains and to vortices of the incident ions in vixx phase space. Since the instability results from the interaction between the ion beam mode of the incoming solar wind ions and the electron whistler the reflected ions are not much affected, and these vortices do not result in individual small-scale reformation events.

Figure 4.

Results from linear analysis: growth rate of the (electromagnetic) two-stream instability between incident electrons and (unmagnetized) ions for different mass ratios and two different values of τ. Parameters from the foot region of run 2 have been used.

[15] Since the instability discussed above is on the whistler mode branch it should disappear for exact perpendicular shocks. In Figure 5 the results from high mass ratio two mi/me = 1840 runs with ΘBn = 87° and ΘBn = 90° are compared. Instead comparing run 2 with the equivalent ΘBn = 90° run, we have performed two additional realistic mass ratio runs with higher Alfvén Mach number and lower values of the ion and electron β: run 3 with ΘBn = 87°, MA = 6.2, βe = 0.1, βi = 0.05, and run 4 with ΘBn = 90° and the same values for MA and the ion and electron β, respectively. Figure 5 compares the magnetic field Bz profile and the profile of the electric field Ex component between these two runs at the same time. As can be seen from Figure 5 the short wavelength waves in the foot region indeed disappear in the ΘBn = 90° run. Furthermore, no other electrostatic wave component is observed in the foot in the ΘBn = 90° case. How does this compare with the results of Shimada and Hoshino [2000]? These authors found in PIC simulations of a ΘBn = 90° shock excitation of the Buneman instability in the foot region. The instability is triggered by the interaction between the incident electrons and the reflected ions and is on the upper hybrid mode branch with exact perpendicular propagation. Shimada and Hoshino [2000] used a mass ratio of 20 and an Alfvén Mach number of 10.5. This instability requires the velocity difference between the electrons and the reflected ions to be larger than the thermal velocity of the electrons. As the mass ratio becomes large the electron thermal velocity vthe goes up as vthe2 = (βe/2)(mi/me)v2A. At the high mass ratio and medium Mach number of run 4 the reflected ion beam cannot interact with the upper hybrid mode and the Buneman instability is not excited. A detailed investigation of the electromagnetic modified two-stream instability and of the associated electron heating is beyond the scope of the present paper. Here, it is important to note that for the details of the shock structure (and for a comparison of simulations with in situ observations) simulations with a realistic mass ratio are paramount.

Figure 5.

Comparison of magnetic field component Bz and electric field Ex profiles between two runs (3 and 4) with mass ratio mi/me = 1840 but different values of ΘBn.

[16] We have increased in run 5 the parameter τ, leaving all other parameters as in run 1. Stacked magnetic field profiles (Bz component) of run 5 with τ = 60 are shown in Figure 6. This higher τ run exhibits also the reformation phenomenon. At such large values of τ = (ωpece)2 charge separation effects should not play any role, so that we conclude that the reformation process is not a breakup of the ramp due to artificially large charge separation electric fields. From Figure 6 it can be seen that small amplitude, small wavelength fluctuations are superimposed to the magnetic field Bz profile of the foot, and larger amplitude fluctuations are present downstream. Such fluctuations can not be seen in the equivalent smaller τ = 10 run 1. Since according to linear analysis the growth rate of the electromagnetic modified two-stream instability for τ = 60 is about twice as large as for τ = 10 (see Figure 4) it is most likely the small amplitude waves seen in the lower mass ratio, high τ run 5 in the foot region are also due to the electromagnetic modified two-stream instability. However, larger amplitude fluctuations in the magnetic field and in the density are generated in the ramp and can be subsequently found downstream. Figure 7 shows from top to bottom Bz, Φ, the ion bulk speed V, and n at one particular time (tΩci = 4.5). The result is similar as in the τ = 10 run 1: the extent of the foot determines essentially the scale of the whole potential drop, but ∼40% of the potential drop can exist over a few electron length scales across the ramp. The large amplitude fluctuations immediately behind the ramp are due to vortices in vxx ion phase: the density and magnetic field minima are related to the vortices, while the maxima are related to the regions in between the vortices. Such vortices have not been seen in the lower τ simulations, and their generation process is currently under detailed investigation.

Figure 6.

Same as Figure 1a, for run 1 but with (ωpece)2 = 60 (run 5).

Figure 7.

Various profiles for run 5 at tΩci = 4.5. From top to bottom: magnetic field Bz normalized to the upstream B0 field, electrostatic potential Φ, ion bulk velocity Vx, and ion density n.

[17] For run 6 we have increased the ion and electron β: βi = βe = 0.4. The other parameters are as in run 5, mi/me = 400, τ = 60. From the stacked magnetic field Bz profiles shown in Figure 8a it can be seen that over a time period of 2.8 Ωci−1 no reformation occurs. The lower three panels in Figure 8 show profiles of various parameters at tΩci = 4.8. No substructures in the electric field occur, and the potential has the same scale as the magnetic field ramp. Direct comparison between Figures 6 and 8a proves that self-reformation is a low β process and disappears in higher β plasmas. The question arises whether the ion or electron β is the parameter responsible for the disappearance of reformation. In Figure 9, we compare results from the low ion and electron β run 1 with results from a run 7 where all parameters, in particular the electron β, are identical, but where the ion β has been increased from 0.1 to 0.4. Clearly, low ion β is responsible for reformation.

Figure 8.

(a) Stacked profiles of the magnetic field Bz from a run (6) with βi = 0.4, βe = 0.4. All other parameters are as in run 1 (Figure 1a). (b)–(d) Magnetic field B, electrostatic potential Φ, and ion density n profiles for run 6 at tΩi = 4.8.

Figure 9.

Comparison of magnetic field profiles (Bz component) from run 1 (upper panel) with a run (7) (bottom panel) where the electron β is the same but the ion β has been increased by a factor 4 to βi = 0.4.

[18] Figure 10 demonstrates the process by which reformation occurs. The top panel shows vixx phase space plots for the βi = 0.1 run 1 (top panel) and for the βi = 0.4 run 7 (bottom panel) at two specific times. Superimposed onto the phase space plots are the respective magnetic field profiles. As can be seen from the top panel, the large velocity difference between the cold incoming solar wind distribution and the cold specularly reflected ions results in a large-scale vortex between the ramp and the position upstream where the reflected ions are turned around perpendicular to the shock normal and to the magnetic field, and accumulate. Since B/n is essentially constant in a compressible plasma this results in an upstream magnetic field hump. The solar wind is slowed down, and the solar wind density increases, causing a further magnetic field increase. Eventually the hump takes over the role of the reformed shock. This behavior of the ion phase space density is typical for all reformation cycles. As discussed above, in the realistic mass ratio run 2 the solar wind ions interact with the whistler waves in the foot, and additional smaller-scale vortices in ion phase space can appear.

Figure 10.

Comparison of vixx phase space from the low ion β run 1 (upper panel) with vixx phase space from the higher ion β run 7 (bottom panel) at two instants of time.

[19] In contrast, in the higher β case due to the large thermal spread of the solar wind and of the specularly reflected ions relative to the upstream bulk speed the incoming and reflected ions mix already immediately upstream of the ramp. This leads to a smooth foot in front of the ramp. In the βi = 0.1 run 1 the ratio between the upstream bulk velocity and the ion thermal velocity is ∼12, whereas it is a factor 2 lower for the high β case 7. This suggests that the upstream bulk speed has to be an order of magnitude larger than the thermal speed of the ions in order for reformation to occur.

[20] A realistic mass ratio does not change this conclusion. Figure 11 shows results from a high β run 8 with realistic mass ratio (mi/me = 1840). Because of computational constraints we reduced τ for this run to τ = 8. Figure 11a shows stacked magnetic field profiles. At early times the magnetic field profile exhibits fluctuations in the foot. However, these fluctuations are most probably remnants from the initial reflection of the incoming solar wind at the right hand side and die out later in the run (note that the lowest profiles are obtained very early in the run). As in the higher ion β runs 6 and 7 with mass ratio mi/me = 400 there is no evidence for reformation. The shock ramp exhibits occasionally a somewhat steeper profile than in the lower mass ratio runs. The lower two panels of Figure 11 shows the magnetic field and the potential on an expanded scale during one particular instant. The main potential drop occurs over several ion scales in the foot, but 25% of the potential occurs over ∼7λe in the steepened up ramp.

Figure 11.

(a) Stacked magnetic field profiles (Bz component) for a high β, realistic mass ratio run (8) (βi = 0.4, βe = 0.4, mi/me = 1840, (ωpece)2 = 8). (b) and (c) Magnetic field and electrostatic potential profiles for run 8 at tΩi = 5.8.

[21] The late time magnetic field profiles in the high β case exhibit a rather smooth foot and ramp. The question arises why in the high ion and electron β run 8 with mi/me = 1840 the modified two-stream instability in the foot of the shock is not excited. We have determined the percentage of reflected ions in run 2 and run 8, respectively. In the low β run 2 more than ∼50% of the incident ions are reflected at the beginning of a reformation cycle. Assuming that to lowest order the current in the foot region is zero results in a velocity difference between the incident ion population and the solar wind electrons of about half the shock velocity. In contrast, in the high ion β case (8) the reflection rate is ∼25%, resulting in a considerably smaller velocity difference between incident ions and solar wind electrons. However, linear analysis results in a growth rate that is still sufficiently high so that the waves should be seen. From the linear dispersion relation for the two ion beam plus electron system we have determined the phase velocity of the whistler wave. Since in the high ion β case (8) the phase velocity of the wave is of the order of the ion thermal velocity, it is expected that ion Landau damping becomes important and does suppress the wave activity in the foot. The kinetic behavior of the instability will be investigated in detail in future work.

3. Summary and Discussion

[22] We have performed a number of 1-D PIC simulations of quasi-perpendicular shocks with a medium, but supercritical Alfvén Mach number. ΘBn has been chosen to be 87° so that the results are only valid for shocks that do not have an upstream whistler precursor. One exactly perpendicular shock run ΘBn = 90° is shown for comparison. Since we are still not able with present computer resources to run a simulation with a ratio of ωpece as present in the solar wind and at the same time with a realistic mass ratio of protons to electrons we have varied both parameters independently. In addition we have investigated the dependence of the shock reformation process on the value of the ion and electron β. The results can be summarized as follows.

  1. Self-reformation is not a computational artifact due to a low ωpece value used in many PIC simulations of previous works but occurs also in a (ωpece)2 ≫ 1, low β (βi = 0.1) simulation. The present simulations recover that the time for self-reformation is determined by the gyroperiod of specularly reflected ions in the magnetic field of the ramp.
  2. We have shown for the first time that self-reformation also occurs at low ion β for a realistic mass ratio. However, details of the shock profile during buildup of the foot depend considerably on the mass ratio used. In particular, in low β shocks with a realistic mass ratio an electromagnetic modified two-stream instability is excited in the foot, which leads to waves on the whistler mode branch with wavelengths of the order of a few electron inertial lengths. These waves result in parallel electron acceleration and trapping.
  3. After a reformation cycle the length scale of the shock potential is of the order of the ion scale. During the build up of the foot due to reflection of ions the ramp steepens up. Part of the potential occurs over the ion scale of the foot, and part of the potential (∼40%) occurs over a few electron inertial lengths in the steepened up ramp.
  4. Self-reformation is a low ion β process and disappears for a MA∼ 4.5 shock at or above βi ≈ 0.4. This holds also when the realistic mass ratio mi/me = 1840 is used. It is argued that for reformation to occur the difference between the speed of the incoming solar wind and of the specularly reflected ions has to be so large that upstream of the ramp a vortex in vixx phase space can build up. According to the simulations this requires that the upstream bulk speed (shock velocity) is an order of magnitude larger than the thermal velocity of the ions.
  5. There is no indication for a two-stream instability in the foot of the shock in the higher β, realistic mass ratio (mi/me = 1840) run at late times. This is attributed to kinetic effects at high solar wind temperature.
  6. In the higher β nonreforming shocks potential drops of the order of 25% of the total potential can exist over scales of 5–10 λe.

[23] The present simulations confirm some of the earlier findings [e.g., Biskamp and Welter, 1972; Lembège and Dawson, 1987; Lembège and Savoini, 1992]. Self-reformation is a genuine process and not a feature due to the small (ωpece)2 values used in many PIC simulations because of computational constraints. The similarity between the magnetic field scale and the potential scale has been stressed in the statistical investigation by Lembège et al. [1999]. We found that also in the realistic mass ratio case the spatial scales of the electrostatic potential and of the magnetic field in the foot as well as in the ramp are comparable. However, although similar scales occur for various parameters, the gradients of different parameters differ: for instance, as can be seen from Figure 2, the relative potential drop in the foot is considerably larger than the relative magnetic field change. This is due to the additional contribution by the gyration of upstream reflected ions in the foot to the electric field Ex, and is well known from hybrid simulations of shocks [e.g., Leroy et al., 1981, 1982]. Vice versa, the relative potential drop through the ramp is smaller than the relative increase in B through the ramp.

[24] We will now discuss the implications of the simulation for shock surfing at the heliospheric termination shock. Zank et al. [1996] have compiled the various plasma parameters for 4 different models of the heliospheric termination shock. The Alfvén Mach number for these models ranges from 4.5 to 6.9; the present simulation is representative for the lower value. In these models neglecting pickup ions the solar wind plasma β is rather small and ranges from a few times 10−3 to 0.15. At the high β end (β = 0.15) some additional heating of the expanding solar wind has been assumed. The plasma β changes drastically when pickup ions are included: β ranges then from 1.4 to 3.3. However, if the pickup ion density is not too large the solar wind will determine the shock structure, and the low β scenario applies. As shown above, in such a case half of the total potential drop can occur over ∼5λe. In case the shock is stationary, the maximum attainable energy would be a factor 50 smaller compared to the case when the ramp width of the potential is 1λe, and would thus be of the order of ∼20 keV/nucleon. In case of a higher β the maximum attainable energy is even smaller: a width of 7λe results in a maximum energy of a few keV/nucleon. This would not even be high enough for injection into a subsequent first-order Fermi mechanism: for diffusive acceleration to be a viable process a particle must scatter many times as a field line convects through the shock. This requires in the weak scattering limit energies of the order of 1 MeV [Zank et al., 1996].

[25] An additional and probably more important constraint for shock surfing at low β shocks is the unsteadiness of the shock on the timescale of the ion gyroperiod. Clearly, what is needed is a stochastic description of shock surfing, as already stressed by Lee et al. [1996]. Before such a theory is developed pickup ions should be included, either as test particles or self-consistently, in a PIC simulation of a perpendicular shock. 1-D finite electron mass hybrid simulations of quasi-perpendicular shocks with self-consistent inclusion of pickup ions have actually been performed by Lipatov and Zank [1999], and it was found that low β shocks can strongly accelerate pickup ions by shock surfing. The Lipatov and Zank [1999] hybrid code assumes an artificial resistivity, which may be provided by some current driven instability in the foot and ramp of the shock. In a simulation with large resistive length scale (ld = 0.25λi) the ramp was not sharp enough and pickup ions were not accelerated. In the case of a rather small resistive length scales (ld = 0.001λi) the electrostatic jump at the ramp was of the order of the fluctuation level and again no acceleration occurred. It is not clear whether at intermediate resistive length scales exceeding the spatial grid resolution electron heating occurs in the ramp, which can lead to large electric field spikes, whereas at smaller resistive length scales such a heating is suppressed. In any case, it is highly desirable to confirm the Lipatov and Zank [1999] results of efficient pickup ion acceleration due to shock surfing by full particle simulations.

[26] The one dimensionality of the present simulation is an important restriction. Once the ramp has steepened up to several electron inertial lengths the strong electrical current can lead to instabilities with k vectors perpendicular to the magnetic field and the shock normal, like the lower hybrid drift instability. Lembège and Savoini [1992] have presented 2-D PIC simulations of a ΘBn = 90° shock and have shown that the rippling of the shock front is due to a lower hybrid frequency instability. However, the instability has only a weak impact on reflected ions, and the period of reformation is not changed. In any case, instabilities will result in an anomalous resistivity that broadens the shock profile. Furthermore, in three dimensions the reformation process is expected to be patchy over the shock surface. Therefore higher-dimensional effects will only further hamper the shock surfing mechanism. The conclusion concerning reformation being a low ion temperature phenomenon only holds for the perpendicular shocks considered here. In more oblique shocks the whistler precursor can be an active ingredient in the shock nonstationarity [Krasnoselskikh et al., 2002].


[27] We are grateful to Dieter Biskamp and Rudolf A. Treumann for helpful discussions.