Short large-amplitude magnetic structures and whistler wave precursors in a full-particle quasi-parallel shock simulation



[1] A one-dimensional (1-D) full-particle electromagnetic simulation code (PIC) is used to investigate the role of upstream whistler and low-frequency upstream waves during cyclic reformation of a medium Alfvén Mach number quasi-parallel collisionless shock (magnetic field - shock normal angle = 30°). The ion to electron mass ratio is assumed to be 100. Compared with previous PIC simulations, the upstream region is large enough to allow for the emergence of low-frequency upstream waves by the interaction of backstreaming ions with the solar wind via an ion/ion beam instability in the far upstream region. It is shown that the low-frequency upstream waves steepen up into pulsations (or short large-amplitude magnetic structures (SLAMS)), as has been shown earlier by hybrid simulations. As these pulsations are added to the shock and thus comprise the shock, the upstream edge radiates a phase standing whistler train. This whistler train propagates partway into the newly arriving pulsation. The nonlinear interaction of reflected ions and incoming solar wind ions in the electrostatic potential of the whistler leads to ion trapping and rapid whistler damping. This results in SLAMS consisting of two regions with different ion temperatures. The cyclic reformation is essentially due to the SLAMS being added to the shock and is of larger scale (∼10 ion inertial lengths) as compared with the whistler scale.

1. Introduction

[2] It has first been shown by Burgess [1989] on the basis of hybrid simulations that quasi-parallel collisionless shocks disrupt and reform periodically in time. Since then a number of mechanisms have been proposed in order to account for such a behavior. Already Burgess [1989] and later Scholer and Terasawa [1990] pointed out that upstream low-frequency waves steepen up as they approach the shock and eventually become the reformed shock. Onsager et al. [1991] suggested that shock reformation can be due to the interaction of specularly reflected ions with upstream waves. The beam is deflected parallel to the shock front, and the beam attains zero velocity at a position upstream which depends on the angle of the magnetic field and the shock normal direction, ΘBn. This deflection is enhanced by an upstream wave, and as shown by Scholer and Burgess [1992], the original wave amplitude grows and the wave steepens. Thus reformation can be caused by the interaction of specularly reflected ions with upstream waves. Equally important seems to be the interaction of low-frequency waves with the increasing density of diffuse ions as they approach the shock. In the simulations by Scholer [1993], reformation cycles were due to steepening and growth of upstream waves into pulsationlike structures when they are convected into the region of strongly increasing diffuse ion density immediately upstream of the shock. It has actually been suggested that the shock can be considered as an extended region containing three-dimensional large-amplitude pulsations or structures (SLAMS) which convect with the flow and grow in amplitude [Schwartz and Burgess, 1992].

[3] Winske et al. [1990] showed that artificial removal of backstreaming ions at a distance greater than 10 ion inertial lengths upstream of the shock had little influence on the shock evolution. Reformation was due to a nonuniform resonant beam instability which occurred in a region of overlap between the incident ions and a somewhat less dense component of downstream ions. This so-called interface mode was suggested to be the main dissipation mechanism during the reformation process. Lyu and Kan [1990] found in their quasi-parallel shock simulations that the incident ion beam can be decelerated and reflected at the higher-frequency whistler waves immediately upstream of the shock ramp. The incident ions are ultimately thermalized in the whistler waves, and a new shock forms upstream. All the simulation work mentioned so far has been performed with the hybrid method, where ions are followed as macroparticles, while the electrons are approximated as a massless background fluid establishing charge-neutrality. The importance of whistler waves in collisionless quasi-parallel shock physics had already been pointed out by Kan and Swift [1983]. However, the amplitude of the whistler waves in hybrid simulations seems to depend on the assumed empirical resistivity and even on the numerical method used. Furthermore, the expected whistler wavelength is of the order of the ion inertial length so that they are poorly resolved in hybrid simulations. This led Pantellini et al. [1992] to perform full-particle electromagnetic simulations of a quasi-parallel shock with an implicit code. They found that whistler waves are indeed an important element in shock dissipation and shock reformation. The whistler precursor grows in amplitude by cyclotron resonance with reflected ions, and the strong nonlinear interaction of the incident ions with the whistler precursor leads to shock reformation. Since in hybrid codes the assumed resistivity has a strong influence on the whistler saturation, such codes cannot correctly describe whistler dynamics. On the other hand, Pantellini et al. [1992] had a rather limited upstream region so that the influence of low-frequency waves and their steepening into large-amplitude pulsations could not be studied. Pantellini et al. [1992] used a box with size Lx = 68λii = ion inertial length). Their simulation frame was identical with the shock rest frame. The shock was approximately in the middle of the box, i.e., the upstream region was about 30 ion inertial lengths in distance. This did not allow upstream low frequency waves to grow, which have a wavelength of ∼40λi.

[4] In this paper we present a new one-dimensional (1-D) full-particle simulation of a quasi-parallel shock obtained with an explicit electrodynamic particle-in-cell (PIC) code. The main purpose of this work is to evaluate the importance of whistler waves for the quasi-parallel shock structure. As outlined above, both hybrid codes and PIC codes gave in the past conflicting answers: in the hybrid simulations by Swift and Kan [1983] whistler waves were the dominant mode leading to downstream thermalization. On the other hand in the Burgess [1998], Scholer and Burgess [1992], and Winske et al. [1990] hybrid simulations the upstream low-frequency waves or interface waves were dominant. Since growth and damping of whistler waves is not correctly described by hybrid codes, it is necessary to perform PIC code simulations, as was done by Pantellini et al. [1992] for a small simulation system. However, since the structure of the quasi-parallel shock also depends on large-scale structures, as upstream low-frequency waves and pulsations running into the shock, a large upstream region is necessary to describe correctly excitation and growth of these waves. The present system size is large enough so that low-frequency waves are allowed to be excited in the upstream region and that their interaction with the shock can be studied within the run time.

2. Simulation Method

[5] We use a 1-D explicit electromagnetic particle-in-cell code. The compromise between a reasonable large mass ratio, a large upstream region, and a long run time allows presently only 1-D PIC simulations of quasi-parallel shocks. It should be pointed out that 1-D and 2-D simulations are limited because in magnetic fields with one ignorable coordinate charged particles are forever tied to the same magnetic field line, i.e., the cross-field diffusion coefficient is zero [Jokipii et al., 1993; Jones et al., 1998]. 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 one-dimensional 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. Furthermore, since the downstream region is at rest, the shock develops in the downstream rest frame, which means that in the simulation frame the shock propagates upstream to the left-hand side of the numerical box. 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 = cpepe = electron plasma frequency, c = velocity of light), 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. The system size is Lx = 2000λe. As in the work of Pantellini et al. [1992], an ion to electron mass ratio mi/me = 100 has been chosen in order to reduce computational time while keeping the ion and electron time scales separated. The system size thus corresponds to 200λi, where λi is the ion inertial length. In order to stay within computer resources, the ratio of the electron plasma frequency to the electron gyrofrequency is chosen to be 101/2, which is equivalent to assuming a ratio of velocity of light to Alfvén velocity of c/vA = (ωpece)(mi/me)1/2 ≈ 32. We use ΘBn = 30° and inject the plasma with Alfvén Mach number MAO = 3.5. This leads to a MA ∼ 4.7 shock. In the run shown here the electron and ion beta (particle pressure to magnetic field pressure) in the upstream region was assumed to be βe = 0.5 and βi = 0.1, respectively. Pantellini et al. [1992] used in their simulation βi = 0.5. We have also performed a run with βi = 0.5 and the results are rather similar to the lower ion beta run. It is important in quasi-parallel shock runs that there is a large upstream region so that the interaction between the incident solar wind and the diffuse ions can excite low-frequency waves by ion/ion beam instabilities. We follow the shock up to a time t ∼ 100Ωci−1, at which there is still an upstream region of ∼80λi.

3. Simulation Results

[6] Figure 1 presents an overview of various parameters at a specific time during the run. Shown from top to bottom are the magnetic field Bz component (note that the magnetic field is initially in the x-z plane), the electric potential in the shock normal direction x, the ion bulk velocity vi in the shock normal direction, and the ion density ni. The shock is at this time (tΩci = 85) at about x = 1100λe. Note that only part of the whole simulation box is shown. The magnetic field Bz profile exhibits two large downstream pulsations indicated by 1 and 2, which are the remnants from previous upstream waves. Previous pulsation 2 radiates a whistler wave from the leading edge in the upstream direction. Further upstream, one can see pulsation 3 with a steepening edge, at which the solar wind is decelerated, and a low-frequency upstream wave 4 far upstream. The main shock transition can also be recognized in the potential jump coincident with the velocity and density jump. In the following we will discuss the evolution of the steepening pulsation 3 and the upstream wave 4 into new large-amplitude pulses as they are added to the shock and eventually comprise the shock.

Figure 1.

(top to bottom) Magnetic field Bz component, shock normal potential Φ, ion bulk velocity vi in the shock normal direction, and ion density.

[7] Figure 2 and Figure 3 show on an enlarged scale the development of the magnetic field from tΩci = 82.5 till tΩci = 105. The Bz component is stacked in time; time runs from bottom to top. The shock ramp is at time tΩci = 82.5 located at ∼1140λe and radiates a phase standing whistler upstream. Owing to the upstream directed group velocity consecutively more and more wave trains appear further upstream until they merge with the approaching pulsation 3 and disappear within ∼2Ωci. By tΩci = 92.5 the large-amplitude pulsation 3 takes over the role of the new shock ramp. The occurrence of the dispersive whistler train at the leading edge of SLAMS 3, the steepening of the low-frequency wave 4, the merging with the whistler and the rapid disappearance of the whistler train repeat itself. At tΩci = 97.5 the whistler wave length is λw ∼ 22λe = 2.2λi. The wave number k0 for which the phase velocity is the same as the shock velocity is given by [e.g., Tidman and Krall, 1971]

equation image
Figure 2.

Time sequence of profiles of Bz. Results are shown in the simulation frame in which the shock propagates to the left. The spacing between profiles is Ωcit = 2.5.

Figure 3.

Same as Figure 2 for later time period.

[8] The whistler which is stationary with respect to the shock forms the standing precursor. By tΩci = 97.5, the upstream pulsation has slowed down the solar wind to ∼2.8vA and the ramp drifts with a velocity of about 0.3vA to the left. Thus equation (1) predicts λw ≃ 2λi, which is in close to the observed value. We note that most hybrid simulations used a resolution of Δx = 0.5λi and were thus not able to properly resolve the upstream whistler waves.

[9] Figure 4 shows the Bz component, the cross-shock potential Φ, and vx-x ion phase space shortly before the disappearance of the whistler train. From the ion phase space plot it can be seen that the solar wind is considerably decelerated at the upstream edge of the pulsation. In the whistler train both reflected ions and the incoming solar wind ions are trapped, resulting in phase space vortices. Such ion trapping was also observed in the full particle simulation by Pantellini et al. [1992]. In the Pantellini et al. simulation the loops led to ion heating and reformation cycles. We will discuss ion heating in more detail below. Here we first note that the whistler is rapidly damped and disappears as the upstream pulsation steepens up. The whistler waves do not introduce reformation cycles by themselves, the reformation is rather due to the addition of a new large-amplitude pulsation to the shock. Pantellini et al. [1992] explained the trapping of the solar wind ions in terms of a resonant process. They concluded that the ions are decelerated and accelerated in the electric field oscillations of the whistler precursor so that many ions have a vx velocity close to the phase velocity of the wave (which is zero in the shock frame). This supposedly leads to strong m = 1 resonance in the cyclotron resonance condition

equation image

In the solar wind frame ω ≫ Ωci. Thus for the m = 1 resonance to occur, the whistler waves have to have a phase velocity close to the thermal velocity of the incident ions, a condition which is rather difficult to fulfill in the present situation. A different approach has been taken by Biskamp and Welter [1972] in order to explain the strong interaction between incident ions and reflected ions in their oblique shock simulations. They have proposed that the interaction is not caused by a linear instability but is due to a nonlinear two-ion beam instability excited by the electric potential oscillations coupled to the whistler precursor. Biskamp and Welter [1972] have shown that a two-ion beam situation is nonlinearly stable but that the presence of potential oscillations in the region of the two-ion beams lead to a nonlinear instability and a strong coupling of the beams. The middle panel in Figure 4 shows the electric potential, and the oscillations due to the whistler wave train coincident with the ion trapping can very well be seen. The instability is easier excited the smaller the velocity difference between the two ion beams is. The decrease of the incident solar wind ions at the steepening upstream edge of the pulsation is thus favorable for the excitation of the instability.

Figure 4.

(top to bottom) Magnetic field Bz component, shock normal potential Φ, and ion vx-x phase space plot.

[10] Figure 5 shows ion distribution functions f(vx) versus vx in a log versus linear representation within the SLAMS 3 at tΩci = 90. The top part shows the distribution in the part of the SLAMS where the solar wind is decelerated but where the whistler precursor does not exist. The bottom part shows the distribution function within the area where the whistler precursor exists. The dashed curve shows for reference the ion distribution far upstream. The diffuse ions have been eliminated for the dashed distribution. At this time the SLAMS consists of two regions with different ions distributions: in the upstream part the distribution is decelerated and adiabatically heated, and in the whistler train the distribution has a high-energy tail. In the plasma frame, specularly reflected ions have a velocity of vb ∼ 2vA–4vA. From the cold whistler dispersion relation one obtains for waves with a wavelength of ∼2λi a frequency of ωw ∼ 7.5Ωi. Thus for a not too cold beam the m = 1 cyclotron resonance can be fulfilled, and beam ions can get strongly accelerated. Figure 6 shows the ion distribution functions f(vx) within the pulsation after the whistler wave train is completely damped by the trapped ions (tΩci = 95). The pulsation exhibits a two-temperature structure: a distribution with a high-energy tail in the leading part and a much hotter distribution in the part of the damped whistler train.

Figure 5.

Ion velocity distribution function log [f(vx)] in arbitrary units integrated over two intervals within a large-amplitude pulsation at tΩci = 90.

Figure 6.

Ion velocity distribution function log [f(vx)] in arbitrary units integrated over two intervals within a large-amplitude pulsation at tΩci = 90.

[11] Pantellini et al. [1992] have discussed the interaction of electrons with the whistler precursor and have concluded that electron Landau damping is rather inefficient. For higher-order resonances, on the other hand, the resonance condition cannot be fulfilled. In Figure 7 we show integrated electron distribution functions log [f(vx)] at tΩci = 92.5, i.e., after the whistler precursor is damped. Also shown by dashed curves is the upstream electron distribution function. First, the asymmetry in the far upstream distribution shows the existence of an electron heat flux directed upstream. This heat flux has been seen in early simulations by Quest et al. [1983] and in the Pantellini et al. [1992] simulations. The electron distributions in the two parts of the pulsation, the leading steepened up part and the trailing part where the whistler precursor has been damped, are rather similar. As in the Pantellini et al. simulations the whistler precursor accelerates some electrons to suprathermal velocities.

Figure 7.

Electron velocity distribution function log [f(vx)] in arbitrary units integrated over two intervals within a large-amplitude pulsation at tΩci = 92.5.

[12] After a cycle is completed and a new whistler precursor just builds up, the electron distributions exhibit strongly asymmetric profiles in the vz direction downstream of the ramp. Figure 8 shows electron distributions log [f(vz)] versus vz at the beginning of a new cycle averaged in the region downstream of the previous pulsation. At these times the electric potential exhibits a sharp increase immediately at the ramp, which we define here as the leading edge of the previous pulsation. Since the magnetic field downstream of the ramp is mainly in the vz direction the asymmetry is a consequence of the cross-shock potential which accelerates the electrons along the magnetic field lines. This has also been reported by Pantellini et al. [1992]. However, in a three-dimensional setup, where electrons are allowed to cross field lines, the direction of the heat flux might change.

Figure 8.

Electron velocity distribution function log [f(vz)] in arbitrary units downstream of the ramp after the completion of two consecutive reformation cycles.

4. Conclusions

[13] We have presented in this paper time full particle electromagnetic simulations of a quasi-parallel shock in a system which is large enough to allow for the occurrence of low-frequency upstream waves. The system is allowed to evolve up to more than 100Ωci−1, at which time hybrid simulations have resulted in steepening up of the upstream waves into large magnetic pulsations. We have essentially recovered in the present simulation both features described on the basis of full particle simulations in a short simulation system by Pantellini et al. [1992] and on the basis of hybrid simulations in a large simulation system by Scholer [1993]. The results can be summarized as follows.

[14] At the more oblique quasi-parallel shock studied here, reformation cycles are due to steepening of upstream waves when they are convected into shock. Steepening is due the increase of diffuse upstream ion density toward the shock. After a reformation cycle is completed and the solar wind is completely thermalized within the pulsation, the upstream edge, or shock ramp, radiates dispersively a phase standing whistler wave. Because of the large group velocity of these whistlers they appear eventually on the downstream side of the next approaching pulsation and run into the pulsation. The pulsation is then divided into two parts: a trailing part with a whistler train and a leading, steepened up part. A nonlinear instability between the specularly reflected ions and the solar wind ions is excited by the electric potential oscillations of the whistler. This instability results in trapping of the solar wind ions and the specularly reflected ions. At the same time the solar wind is decelerated at the upstream edge of the pulsation. This makes the trapping easier and eventually results in fast damping of the whistler. The SLAMS then consists of two parts with different ion distributions: a hot distribution with a high-energy tail in the region near the trailing edge and a colder distribution in the part near the leading edge of the pulse. No significant electron damping is observed in the region of the whistler precursor. The most significant aspect of the electron distribution is the asymmetry parallel to the magnetic field due to the cross shock potential immediately after the cycle is completed and sharp shock ramp has emerged.

[15] Although we have recovered most of the aspects described by Pantellini et al. [1992] concerning the whistler occurrence and the ion and electron distributions in the whistler, we conclude from the present simulations that reformation cycles are large scale (∼10λi) and due to the upstream low frequency waves. The pulsations or SLAMS comprise eventually the shock as in the Schwartz and Burgess [1992] quasi-parallel shock scenario. The Schwartz and Burgess [1992] scenario is three-dimensional and predicts a patchwork; this can of course not be verified by the present one-dimensional simulation. The whistler precursor has the effect to divide the SLAMS into two regions with a hotter and colder ion distribution, respectively. The steepened upstream pulsation limits the whistler precursor in the upstream direction, i.e., the whistler does not result in more than a few wave trains after it has run into the slowed-down solar wind behind the upstream edge of the pulsation.

[16] Pantellini et al. [1992] proposed that the whistler is driven by a cyclotron resonance of the reflected ions, a mechanism proposed first by Cippola et al. [1977]. In the simulation the whistler train does not occur without reflected ions and vice versa: reflected ions always occur after a cycle is completed and the shock has a sharp ramp with a well defined potential jump. A sharp ramp on the other hand can radiate whistlers dispersively. The density of reflected ions is of the order of 15% of the solar wind ion density. With such a density one obtains from Cippola et al. [1977] for the cyclotron resonance instability a wavelength at maximum growth of about 8 ion inertial lengths, which is larger than the wavelength obtained in the simulation. This suggests that the whistler train is dispersive. Additional enhancement of the whistler train by cyclotron resonance is possible. Actually, an indication for such an enhancement is the fact that the amplitude of the whistler is larger than the amplitude of the magnetic field in the shock ramp. A final comment concerns the interface instability proposed by Winske et al. [1990]. By eliminating backstreaming ions some distance upstream of the shock, these authors obtained reformation cycles in hybrid simulation purely by the interaction of the solar wind ions with part of the heated downstream distribution. Scholer et al. [1997] concluded from hybrid simulations that the waves produced by the interface instability actually dominate the downstream state. However, the Scholer et al. [1997] results were for almost parallel shocks, whereas the Pantellini et al. [1992], Scholer [1993], and the present results are for a more oblique situation (ΘBn = 30°). It will be interesting to see whether the result of interface waves being dominant in almost parallel shocks also holds for full particle simulations.

[17] When comparing the simulation with observations it has to be kept in mind that the simulations can only be run for short real times. It can not be expected that during these times a upstream region with a significant diffuse ion density has fully developed yet. Thus real SLAMS may have a considerably more complicated structure and their modeling requires long-time multidimensional simulations. Furthermore, we have only presented results for a medium Alfvén Mach number, low beta quasi-parallel shock. For instance, previous results from hybrid simulations and preliminary results from PIC simulations have shown that the upstream region of quasi-parallel shocks becomes more violent as the Mach number increases. Thus detailed comparison of observations with simulations requires that the simulations are extended over a wider parameter range. We note in this respect that observations upstream of a high Mach number, high beta bow shock shows that isolated SLAMS have a whistler train attached at their upstream edge [Schwartz et al., 1992], at variance with the present simulation for a medium Mach number shock. Also the turbulent wave activity between isolated SLAMS in association with ion beams observed by Wilkinson et al. [1993] is not seen in our simulation. This activity is of much lower intensity than the dispersive whistler found here.

[18] Nevertheless, we can draw a few conclusions from the present simulation concerning observations of SLAMS by Cluster 2. The simulations are performed in the average downstream rest frame. A spacecraft standing with the shock would be on a trajectory in t-x space, in for instance Figure 2, which moves to smaller x values. The SLAMS is rapidly developing in time with a time scale of the order of the inverse ion frequency (which corresponds to ∼1 s in the solar wind at 1 AU). Unless the shock, together with the SLAMS, is rapidly moving, a spacecraft would enter the SLAMS at early time in its development and leave it at a late time in its development: the spatial structure at one instant in time can only be inferred if the shock moves with a velocity larger than the velocity given by the time τ for change of the structure and the length scale L of the structure. With τ ∼ 2Ωci−1 (damping time of the whistler) and L ∼ 10λi as found in the simulation one obtains a speed of ∼5vA, which is of the order of the solar wind velocity itself. Realistic bow shock speeds (in the magnetosphere frame of reference) are considerably smaller. If the motion of the SLAMS over the spacecraft is due to motion of the bow shock, the spacecraft may measure at entry and exit of the SLAMS the structure at rather different states of its temporal development. Two spacecrafts appropriately spaced in the shock normal direction, i.e., with an interspacecraft distance of the length of the SLAMS, can be used to yield the spatial structure of the SLAMS at one time. For a SLAMS with a typical dimension of L ∼ 10λi this results in a spacecraft separation of ∼1200 km. If the spacing is smaller, the same problem arises as before with a singe spacecraft; if the spacing is much larger, each spacecraft will again sample the SLAMS at different stages of the development. These considerations assume that the SLAMS is essentially standing in the shock frame, as found in the late state of its development. In the early state of the SLAMS development it is essentially convected with the solar wind over the spacecraft. However, the solar wind is considerably slowed down in the SLAMS so that again during entry and exit in a SLAMS a spacecraft may measure different states of its development.

[19] The present 1-D PIC simulation shows that hybrid simulations with low numerical/artificial resistivity, which resolve the whistler scale, may be adequate to describe the reformation process of quasi-parallel shocks. Large-scale three-dimensional hybrid simulations are then the next step to gain more insight into the structure of the large-amplitude pulsations and to compare directly with observations. However, high-resolution hybrid simulations do not correctly describe whistler mode damping or growth so that in addition 1-D PIC simulations have to be performed as an additional test.


[20] We are grateful to D. Biskamp, M. A. Lee, and S. Matsukiyo for helpful discussions. This work was supported in part by NASA under grants NAS5-30613 and NAG5-10131.

[21] Shadia Rifai Habbal thanks David Burgess and Joe Giacalone for their assistance in evaluating this paper.