On the reformation at quasi and exactly perpendicular shocks: Full particle in cell simulations

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2010ja015458, 2010 On the reformation at quasi and exactly perpendicular shocks: Full particle in cell simulations Takayuki Umeda, 1 Yoshitaka Kidani, 1 Masahiro Yamao, 1 Shuichi Matsukiyo, 2 and Ryo Yamazaki 3,4 Received 15 March 2010; revised 5 July 2010; accepted 23 July 2010; published 29 October [1] A full particle in cell (PIC) simulation study is carried out on the reformation at quasi and exactly perpendicular collisionless shocks with a relatively low Alfven Mach number (M A = 5). Previous self consistent one dimensional (1 D) hybrid and full PIC simulations have demonstrated that ion kinetics are essential for the nonstationarity of perpendicular collisionless shocks. These results showed that reflection of ions at the shock front is responsible for the periodic collapse and redevelopment of a new shock front on a timescale of the ion cyclotron period, which is called the shock reformation. Recent 2 D hybrid and full PIC simulations, however, suggested that the shock reformation does not take place at exactly perpendicular shocks with M A 5. By contrast, another 2 D hybrid PIC simulation showed that the shock reformation persists at quasi perpendicular shocks with M A 5. Although these two works seem to be inconsistent with each other, the reason is not well understood because of several differences in numerical simulation conditions. Thus this paper gives a direct comparison between full PIC simulations of quasi and exactly perpendicular shocks with almost the same condition. It is found that the time development of the shock magnetic field averaged over the shock tangential direction shows the transition from the reformation to no reformation phase. On the other hand, local shock magnetic field shows the evident appearance and disappearance of the shock front, and the period becomes longer in the no reformation phase than in the reformation phase. Citation: Umeda, T., Y. Kidani, M. Yamao, S. Matsukiyo, and R. Yamazaki (2010), On the reformation at quasi and exactly perpendicular shocks: Full particle in cell simulations, J. Geophys. Res., 115,, doi: /2010ja Introduction 1 Solar Terrestrial Environment Laboratory, Nagoya University, Nagoya, Japan. 2 Earth System Science and Technology, Kyushu University, Kasuga, Japan. 3 Department of Physical Science, Hiroshima University, Higashi Hiroshima, Japan. 4 Also at Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Japan. Copyright 2010 by the American Geophysical Union /10/2010JA [2] Previous kinetic particle in cell (PIC) simulations revealed that collisionless shocks in plasmas can be strongly nonstationary in both spatial and temporal scales of ions. In the direction normal to the shock surface of a quasi perpendicular collisionless shock, a new shock front periodically appears [e.g., Biskamp and Welter, 1972; Quest, 1985; Lembege and Dawson, 1987; Lembege and Savoini, 1992; Hellinger et al., 2002], which is called the cyclic reformation. [3] The shock reformation is due to bunching of ions reflected at the shock surface that involves compression of magnetic field. The upstream ions are decelerated at the shock ramp by a cross shock potential at supercritical quasiperpendicular collisionless shocks, and part of them are reflected upstream. The reflected ions are responsible for the formation of the shock foot. The shock reformation occurs due to accumulation of specularly reflected ions during their gyration when the reflection rate of ions becomes large [Hada et al., 2003; Scholer et al., 2003; Scholer and Matsukiyo, 2004]. Another mechanism for bunching of reflected ions is steepening of various plasma waves excited in the shock foot region [e.g., Krasnoselskikh et al., 2002; Scholer et al., 2003; Scholer and Matsukiyo, 2004]. [4] In the shock tangential direction, on the other hand, there appear fluctuations at the shock front on the spatial scale of ion inertial length in the direction parallel to the shockmagneticfield[winske and Quest, 1988; Lowe and Burgess, 2003], which are called the ripples. The ripples are thought to involve Alfven ion cyclotron waves excited by an ion temperature anisotropy at the shock front. This process has been confirmed in two dimensional (2 D) hybrid PIC simulations where ions are treated as particles while electrons are treated as a fluid. So far, it has been difficult to perform 2 D full PIC simulations where both ions and electrons are treated as individual particles, since current computer resources are not necessarily enough to cover a large simulation domain of several ion inertial lengths. 1of7

2 [5] Recently, Hellinger et al. [2007] performed 2 D hybrid and full PIC simulations of exactly perpendicular shocks and claimed that the shock reformation is suppressed by large amplitude whistler waves excited in the foot region. Lembege et al. [2009] conducted further full PIC simulations of a supercritical (M A 5) exactly perpendicular shock and reported that whistler waves are excited in an earlier phase of their simulation run with a high ion toelectron mass ratio (m i /m e = 400) and that there is no evident shock reformation. On the other hand, in their simulation run with a low mass ratio (m i /m e = 42), the shock reformation takes place in an earlier phase but becomes less evident by excitation of whistler waves in a later phase. This transition from the reformation phase to a no reformation phase has been confirmed by another 2 D full PIC simulation with similar parameters but a much lower mass ratio (m i /m e = 25) [Umeda et al., 2010]. In contrast to these results, Yuan et al. [2009] has confirmed the periodic appearance and disappearance of the shock front at a supercritical (M A 5) quasi perpendicular shock in their 2 D hybrid simulations, suggesting that the existence of the shock reformation. Although the shock reformation period is changed by the excitation of whistler waves, the shock reformation itself is dominated by ion reflection. [6] Here, the former results [Hellinger et al., 2007; Lembege et al., 2009; Umeda et al., 2010] and the latter result [Yuan et al., 2009] seem to be inconsistent with each other, which raises one big question: Is the shock reformation suppressed at exactly perpendicular shocks but persistent at quasi perpendicular shocks? However, this question is not answered because of several differences in the former results [e.g., Lembege et al., 2009] and the latter results [Yuan et al. [2009]: First, Lembege et al. [2009] performed full PIC simulations while Yuan et al. [2009] performed hybrid PIC simulations. Second, Lembege et al. [2009] analyzed the data of the shock magnetic field averaged over the shock tangential direction, while Yuan et al. [2009] analyzed local magnetic field data. [7] The purpose of the present study is to answer the above question by performing a 2 D full PIC simulation of a quasi perpendicular shock with a shock normal angle Bn = 80. By using the similar physical parameters as a recent 2 D full PIC simulation of an exactly perpendicular shock [Umeda et al., 2010], it is possible to make a direct comparison between the results of exactly and quasi perpendicular shocks. It should be noted that we define the periodic appearance and disappearance of the shock front as the shockreformationinthepresentstudy,whichiscommon with the previous work [Yuan et al., 2009]. The paper is organized as follows. Section 2 describes the model and the parameters of full PIC simulations. Section 3 shows the development of shock magnetic field found in the PIC simulations and gives the direct comparison with the results of exactly perpendicular shock [Umeda et al., 2010]. Finally, section 4 gives conclusion and discussion of the present study. 2. Simulation Setup [8] We use a 2 D electromagnetic full PIC code [Umeda et al., 2008, 2009], in which the full set of Maxwell s equations and the relativistic equation of motion for individual electrons and ions are solved in a self consistent manner. The continuity equation for charge is also solved to compute the exact current density given by the motion of charged particles [Umeda et al., 2003]. [9] In the present simulation, the simulation domain is takenintheshock rest frame [e.g., Umeda and Yamazaki, 2006; Umeda et al., 2008, 2009, 2010]. The shock rest frame model is achieved by using the relaxation method [e.g., Leroy et al., 1981, 1982], in which a collisionless shockisexcitedbyaninteractionbetweenasupersonic plasma flow and a subsonic plasma flow moving in the same direction. [10] The initial state consists of two uniform regions separated by a discontinuity. In the upstream region that is taken in the left hand side of the simulation domain, electrons and ions are distributed uniformly in space and are given random velocities (v x, v y, v z ) to approximate shifted Maxwellian momentum distributions with the drift velocity ~u 1, number density n 1 0 m e w pe1 2 /e 2, and isotropic temperatures T e1 m e v te1 2 and T i1 m i v ti1 2, where m, e, w p, and v t are the mass, charge, plasma frequency, and thermal velocity, respectively. Subscripts 1 and 2 denote upstream and downstream, respectively. The upstream magnetic field ~B 01 with a magnitude of m e w ce1 /e is also assumed to be uniform, where w c is the cyclotron frequency (with sign included). The downstream region taken in the right hand side of the simulation domain is prepared similarly with the drift velocity~u 2, density n 2, isotropic temperatures T e2 and T i2, and magnetic field ~B 02. [11] We take the simulation domain in the x y plane and assume an in plane shock magnetic field (B x0 and B y0 ). The shock normal magnetic field B x0 is set to be uniform in both upstream and downstream regions. The shock normal angle is defined as Bn = arctan(b y01 /B x0 ), which is set as 80 and 90 in the present study. As a motional electric field, a uniform external electric field E z0 = u x1 B y01 u y1 B x0 = u x2 B y02 u y2 B x0 is applied in both upstream and downstream regions, so both electrons and ions drift in the x y plane. At the left boundary of the simulation domain in the x direction, we inject plasmas with the same quantities as those in the upstream region, while plasmas with the same quantities as those in the downstream region are also injected from the right boundary in the x direction. We adopt absorbing boundaries to suppress nonphysical reflection of electromagnetic waves at both ends of the simulation domain in the x direction [Umeda et al., 2001], while the periodic boundaries are imposed in the y direction. [12] In the relaxation method, the initial condition is given by solving the shock jump conditions (Rankine Hugoniot conditions) for a magnetized two fluid isotropic plasma consisting of electrons and ions [Hudson, 1970]. In order to determine a unique initial downstream state, we need given upstream quantities u x1, w pe1, w ce1, v te1, v ti1 and an additional parameter. Here, we assume a low beta and weakly magnetized plasma as shown in Table 1, which gives the light speed c/v te1 = 40.0 and the bulk flow velocity of the upstream plasma u x1 /v te1 = 4.0. The ion to electron temperature ratio in the upstream region is given as T i1 /T e1 = 1.0. In this study, downstream ion to electron temperature ratio T i2 /T e2 = 8.0 is also assumed as another initial parameter to obtain the unique downstream quantities. 2of7

3 Table 1. Simulation Parameters Used by Different Authors Authors (code) M A Bn b i b e m i /m e w pe /w ce Present (Full PIC) Umeda et al. [2010] (Full PIC) Lembege et al. [2009] (Full PIC) Lembege et al. [2009] (Hybrid PIC) Hellinger et al. [2007] (Hybrid PIC) Yuan et al. [2009] (Hybrid PIC) [13] In this study, we use N x N y = cells for the upstream region and N x N y = cells for the downstream region, respectively. The grid spacing and time step of the present simulation are Dx/l De1 = Dy/l De1 = 1.0 and w pe1 Dt = , respectively. Here l De1 is the electron Debye length upstream. Thus the total size of the simulation domain is 10.24l i 5.12l i, which is long enough to include the ion scale rippled structures, where l i = c/w pi1 ( = 200l De1 ) is the ion inertial length. We used 16 pairs of electrons and ions per cell in the upstream region and 64 pairs of electrons and ions per cell in the downstream region, respectively, at the initial state. 3. Results [14] In the present study, we focus on the shock reformation at a quasi perpendicular shock with a relatively low but supercritical Alfven Mach number (M A = 5). As shown in Table 1, the present simulation parameters are similar to those used in the previous works. [15] Figure 1 shows the tangential component of the magnetic field B y as a function of position x and time t. The position and time are renormalized by the ion inertial length l i1 = c/w ci1 and the ion cyclotron angular period 1/w ci1, respectively. The magnitude is normalized by the initial upstream magnetic field B 01. In Figure 1a, the tangential magnetic fields B y are averaged over the y direction, which means that fluctuations in the shock tangential direction are smoothed out. In Figures 1b and 1c, the local tangential magnetic field at two different positions (y/l i1 = 1.28 and 3.84) are plotted. [16] In the present shock rest frame model, a shock wave is excited by the relaxation of the two plasmas with different quantities. Since the initial state is given by the shock jump conditions for a two fluid plasma consisting of electrons and ions, the kinetic effect is excluded in the initial state. Thus the excited shock becomes almost at rest in the simulation domain. The shock front appears and disappears on a timescale of the ion cyclotron period, which corresponds to the cyclic shock reformation. However, the reformation seems to be less evident after w ci1 t 8 in terms of the averaged magnetic field (Figure 1). The present simulation result is in agreement with the previous works [Hellinger et al., 2007; Lembege et al., 2009; Umeda et al., 2010] in which the averaged magnetic field showed the transition from the reformation to no reformation phase. [17] By contrast, the evolution of local magnetic fields (Figures 1b and 1c) shows the shock reformation, which is in agreement with the other previous work [Yuan et al., 2009]. The reformation period is 1.6/w ci1 in the early phase (w ci1 t < 8). However, in the later phase where there is not evident shock reformation in the averaged field (Figure 1c), the reformation period is changed to 2.6/w ci1. As seen in Figure 1, the phase of the reformation is opposite in Figures 1b and 1c. That is, the magnetic field broadens upstream at w ci1 t 10 in Figure 1b, while the magnetic field broadens upstream at w ci1 t 11 in Figure 1c. Thus one can understand that the reformation is canceled out Figure 1. Tangential magnetic field B y as a function of position x and time t at Bn = 80. The position and time are normalized by l i1 and 1/w ci1, respectively. The magnitude is normalized by the initial upstream magnetic field B 01. (a) The averaged magnetic field over the y direction. (b) A local magnetic field at y/l i1 = (c) A local magnetic field at y/l i1 = of7

4 Figure 2. Tangential magnetic field B y as a function of position x and time t at Bn = 90. The position and time are normalized by l i1 and 1/w ci1, respectively. The magnitude is normalized by the initial upstream magnetic field B 01. (a) The averaged magnetic field over the y direction. (b) A local magnetic field at y/l i1 = (c) A local magnetic field at y/l i1 = when the magnetic field is averaged over the shocktangential direction. Note that the phase of reformation depends on the position. In other words, the phase of reformation changes depending on the timescale of shock reformation and the phase velocity of ripples at the shock front. Thus a local magnetic field sometimes shows no evident shock reformation at a specific position. [18] It is noted that the existence of the shock reformation in a local magnetic field and the change in the reformation period are also pointed out by Umeda et al. [2010], but no figures are shown. Thus in the present study, we replot the evolution of local magnetic fields with the averaged one at an exactly perpendicular shock [Umeda et al., 2010]. [19] Figure 2 shows the tangential component of magnetic field B y as a function of position x and time t in a simulation run with the parameters used by Umeda et al. [2010] ( Bn = 90 ) with the same format as Figure 1. We found the transition from the reformation to no reformation phase in the averaged field (Figure 2a) but also recognized evident reformation in the local fields (Figures 2b and 2c). The phase transition takes place at w ci1 t 7.5. The reformation period is 1.6/w ci1 in the early phase (w ci1 t < 7.5) and is 2.5/w ci1 in the later phase (w ci1 t > 7.5). The simulation result at Bn = 90 is essentially the same as that at Bn = 80. [20] The present simulation results suggest that the shock reformation takes place locally at perpendicular shocks and that the shock reformation seems to be less evident when the spatial structure in the shock tangential direction is averaged. The present simulation results are in agreement with both previous 2 D full PIC simulation of exactly perpendicular shocks [Hellinger et al., 2007; Lembege et al., 2009] and 2 D hybrid PIC simulation of quasi perpendicular shocks [Yuan et al., 2009]. It should be noted, however, that Hellinger et al. [2007] and Lembege et al. [2009] did not show time evolution of local magnetic field data, and Yuan et al. [2009] did not show time evolution of averaged field data. [21] The transition from the reformation phase to the noreformation phase is reported only by several recent 2D full PIC simulations, in which the simulation domain in the shock tangential direction is taken to be much longer than the upstream ion inertial length (N y Dy/l i1 4byLembege et al. [2009], N y Dy/l l1 5byUmeda et al. [2010] and present). We performed another run with the same physical parameters but N y Dy/l l1 = However, there is no rippled structure and the structures of the shock becomes very similar to those in a 1 D simulation. [22] It is worth showing the 2 D structures of the shock front in the run with N y Dy/l l1 = Figure 3 shows the time development of the spatial profiles of the shocktangential magnetic field B y and the ion density n i at the quasi perpendicular shock. At w ci1 t = 4 and 5, when there is the shock reformation, there exist fluctuations in both B y and n i with a wavelength of 1.7l i1 at the shock overshoot. These fluctuations propagate along the shock surface, which is almost parallel to the ambient magnetic field. Note that the local ion inertial length at the shock overshoot is 0.4l i1. This result is consistent with a fact that the linear dispersion relation for an ion temperature anisotropy gives the maximum growth rate at the wavelength close to the ion inertial length (k k l i 1). On the other hand, for w ci1 t > 8 (when there is no shock reformation), the wavelength of the fluctuations at the shock overshoot becomes much longer than the ion inertial length ( 5.1l i1 ). Thus there exist MHD scale fluctuations at the shock overshoot. Although the mechanism of 4of7

5 Figure 3. The spatial profiles of the shock tangential magnetic field B y and the ion density n i at different times. The solid lines represent magnetic field lines. this inverse cascade is not clear, it is expected that the inverse cascade would play a role in the transition from the reformation phase to the no reformation phase. [23] Finally, it should be noted that there is no clear and general definition of the shock reformation in multidimensions. The concept of the shock reformation came from 1 D simulations where a new shock front is periodically formed upstream of the shock ramp [e.g., Biskamp and Welter, 1972; Quest, 1985; Lembege and Dawson, 1987]. Meanwhile, we defined the shock reformation as the periodic appearance and disappearance of the shoch front [e.g., Yuan et al., 2009]. Thus it is worth analyzing how the structures of the shock front develop in time by showing time variation of physical quantities, especially ion density and shock magnetic field. Figure 4 shows the time development of the spatial profiles of the magnetic field B y, the ion density n i, and the ion x v x phase space density with the width of 16Dy = 0.08l i1 at a local point (y/l i1 = 3.84). From Figure 1c, there appears to be strong compression of the shock magnetic field at w ci1 t 8.5 and We see strong reflection of ions from the shock ramp at these times in Figure 4, which may result in the formation of new shock front upstream of the shock ramp. At w ci1 t 10.0, on the other hand, the ion reflection becomes weaker due to the lack of ion density at the shock overshoot, and the period of reformation is changed. It is noted that the spatial profiles of B y and n i are not correlated with each other, because of the existence of the MHD waves at the shock overshoot. This feature is different from the shock dynamics obtained by 1 D simulations, and it is not simple to identify the formation of new shock front upstream of the shock ramp. 4. Conclusion and Discussion [24] Recent 2 D hybrid and full PIC simulations of perpendicular shocks obtained a contradictory result: The full PIC simulations showed suppression of the shock reformation at exactly perpendicular shocks [Hellinger et al., 2007; Lembege et al., 2009], while the hybrid PIC simulations showed persistence of the shock reformation at quasi perpendicular shocks with Bn = 85 [Yuan et al., 2009]. Although these two sets of authors used the very similar simulation parameters (see Table 1), the obtained results seems to differ substantially. It was not clear whether the different results are due to the difference in the numerical approach or the shock normal angle. In the present study, we performed a 2 D full PIC simulation of a quasi perpendicular ( Bn = 80 ) and an exactly perpendicular shock to make a direct comparison between them. [25] We confirmed that both simulations ( Bn = 80 and 90 ) give similar results. The shock magnetic field averaged over the shock tangential direction shows that the shock reformation takes place in an early phase but is suppressed in a later phase in which the shock front ripples are strongly enhanced [Lembege et al., 2009; Umeda et al., 2010]. On the other hand, local shock magnetic fields show evident 5of7

6 Figure 4. The time development for the spatial profiles of the shock tangential magnetic field B y the ion density n i, and the corresponding x v x phase space for ions with the width of 16Dy = 0.08l i1 at y/l i1 = shock reformation. The period of the shock reformation in the early phase corresponds to several ion cyclotron periods, but the reformation period is modified by the excitation of strong ripples in the later phase [Yuan et al., 2009; Umeda et al., 2010]. Hence, we reached a simple conclusion that both previous works obtained similar results but analyzed them from different points of view. That is, Hellinger et al. [2007] and Lembege et al. [2009] analyzed the averaged shock magnetic field and found the transition from the reformation phase to the no reformation phase, while Yuan et al. [2009] analyzed local shock magnetic fields and found the modification of the reformation period during the transition. The present simulation results of both quasi perpendicular ( Bn = 80 ) and exactly perpendicular ( Bn = 90 ) shocks are in agreement with these previous results. [26] Finally, problems of the present study are listed below. First, the differences between hybrid and full PIC simulations should be clarified. The ratio of plasma tocyclotron frequency is not defined in hybrid PIC simulations (see Table 1). However, this parameter can change the type of microinstabilities in the shock foot region [e.g., Matsukiyo and Scholer, 2003]. Second, the influence of the ion toelectron mass ratio should be discussed. Lembege et al. [2009] has shown that the timescale of the transition from the reformation to no reformation phase shortens as the mass ratio becomes increases. The mass ratio also changes the type of microinstabilities. With higher mass ratio, whistler mode waves propagating obliquely to a magnetic field become unstable due to modified two stream instability (MTSI) [Matsukiyo and Scholer, 2003, 2006], which is absent in the present study with the small mass ratio. It is noted, however, that MTSI is also absent in hybrid PIC simulations. The influence of microinstabilities to the shock reformation is an important issue, and large scale full PIC simulations including full kinetics of both electrons and ions are quite essential to reveal physics of collisionless shocks. The simulation runs with different frequency ratios and mass ratios, however, are left as future works. [27] Acknowledgments. The computer simulations were performed on the Fujitsu FX1 and HX600 systems at the Information Technology Center, Nagoya University, and in part on the NEC SX8 system at Yukawa Institute of Theoretical Physics, Kyoto University. This work was carried out as a computational joint research program at the Solar Terrestrial Environment Laboratory, Nagoya University. This work was supported by agrant in Aid for Scientific Research on Innovative Areas (T.U.), a Grant in Aid for Scientific Research on Priority Areas (R.Y.), and a Grant in Aid for Young Scientists (B) (S.M.) and (R.Y.) from MEXT of Japan. [28] Philippa Browning thanks Martin Lee and another reviewer for their assistance in evaluating this paper. References Biskamp, D., and H. Welter (1972), Numerical studies of magnetosonic collisionless shock waves, Nucl. Fusion, 12, Hada, T., M. Oonishi, B. Lembege, and P. Savoini (2003), Shock front nonstationarity of supercritical perpendicular shocks, J. Geophys. Res., 108(A6), 1233, doi: /2002ja Hellinger, P., P. Travnicek, and H. Matsumoto (2002), Reformation of perpendicular shocks: Hybrid simulations, Geophys. Res. Lett., 29(24), 2234, doi: /2002gl Hellinger, P., P. M. Travnicek, B. Lembege, and P. Savoini (2007), Emission of nonlinear whistler waves at the front of perpendicular supercritical shocks: hybrid versus particle simulations, Geophys. Res. Lett., 34, L14109, doi: /2007gl Hudson, P. D. (1970), Discontinuities in an anisotropic plasma and their identification in the solar wind, Planet. Space Sci., 18, Krasnoselskikh, V. V., B. Lembege, P. Savoini, and V. V. Lobzin (2002), Nonstationarity of strong collisionless quasiperpendicular shocks: Theory and full particle numerical simulations, Phys. Plasmas, 9, of7

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