Anomalous resistivity and the nonlinear evolution of the ion-acoustic instability

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2004ja010793, 2006 Anomalous resistivity and the nonlinear evolution of the ion-acoustic instability P. Petkaki, 1 M. P. Freeman, 1 T. Kirk, 2 C. E. J. Watt, 3 and R. B. Horne 1 Received 17 September 2004; revised 4 November 2005; accepted 14 November 2005; published 27 January [1] Collisionless magnetic reconnection requires the violation of ideal MHD by various kinetic-scale effects whose relative importance is uncertain. Recent research has highlighted the potential importance of wave-particle interactions by showing that Vlasov simulations of unstable ion-acoustic waves predict an anomalous resistivity that can be at least an order of magnitude higher than a popular analytical quasi-linear estimate. Here, we investigate the nonlinear evolution of the ion-acoustic instability and its resulting anomalous resistivity by examining the properties of a statistical ensemble of Vlasov simulations. The simulations differ in their initial electric noise field but are otherwise identical with a Maxwellian electron-ion plasma of low number density and low electron to ion temperature ratio, appropriate to collisionless space plasmas. By studying the evolution of an ensemble of 104 Vlasov simulations with reduced mass ratio m i /m e = 25, we show that (1) the probability distribution of anomalous resistivity values produced during the linear, quasi-linear, and nonlinear evolution of the ion-acoustic instability is approximately Gaussian, (2) the ensemble mean of the ion-acoustic resistivity during the nonlinear regime is higher than estimates at quasi-linear saturation, and (3) the ensemble standard deviation is comparable to the ensemble mean. We argue that the large variability during the nonlinear phase is due to electron and ion bounce motion which is sensitive to the initial conditions. We demonstrate that the results are essentially similar for a real mass ratio simulation. Citation: Petkaki, P., M. P. Freeman, T. Kirk, C. E. J. Watt, and R. B. Horne (2006), Anomalous resistivity and the nonlinear evolution of the ion-acoustic instability, J. Geophys. Res., 111,, doi: /2004ja Introduction [2] In collisionless space plasmas, magnetic reconnection requires the breakdown of the ion and electron frozen-in condition at the magnetic null point. The breakdown of the electron frozen-in condition can be provided by electron momentum changes through ion-acoustic wave-particle interactions with an associated anomalous collisionality and hence anomalous resistivity [e.g., Sagdeev, 1967; Coroniti and Eviatar, 1977; Galeev and Sagdeev, 1984; Labelle and Treumann, 1988]. It is valuable for many numerical experiments [e.g., Ugai et al., 2003] to be able to obtain and use an expression for resistivity due to kinetic scale wave-particle interactions in terms of macroscopically measured or calculated plasma parameters that is valid for realistic conditions, such as low electron to ion temperature ratio, non-maxwellian plasmas, and nonlinearity. [3] Although recent simulations show that fast reconnection can be obtained without anomalous resistivity when the Hall current term is included in generalized Ohm s law [Birn et al., 2001], laboratory experiments detect waves 1 British Antarctic Survey, Cambridge, UK. 2 Emmanuel College, University of Cambridge, Cambridge, UK. 3 Department of Physics, University of Alberta, Edmonton, Canada. Copyright 2006 by the American Geophysical Union /06/2004JA associated with reconnection [Carter et al., 2002], and there is increasing evidence for wave-particle interactions associated with reconnection events in space plasmas, at least inside the ion diffusion region [Deng and Matsumoto, 2001; Bale et al., 2002; Farrell et al., 2002; Matsumoto et al., 2003]. The role of wave-particle interactions, for example, as a mechanism to decouple electrons from the field or as a drag force on the particles after they are accelerated by the reconnection electric field [e.g., Drake et al., 2003; Omura et al., 2003], remains an open question [e.g., Lapenta and Brackhill, 2002; Silin and Büchner, 2003; Scholer et al., 2003; Tanaka et al., 2004]. Some wave modes like lowerhybrid waves and whistler mode waves require the presence of a magnetic field, but ion-acoustic waves can exist in the heart of the current sheet, where the magnetic field becomes small. Ion-acoustic and Buneman instabilities can be excited in the weak field region and are observed in simulations [Drake et al., 2003; Petkaki et al., 2003]. [4] Most analytical estimates of resistivity due to waveparticle interactions use the quasi-linear theory approximation [Sagdeev, 1967; Davidson and Gladd, 1975; Galeev and Sagdeev, 1984; Labelle and Treumann, 1988]. Quasilinear theory gives an expression for the evolution of the spatially averaged distribution function (DF) due to interactions between the unstable linearly growing waves and the plasma. By taking moments of the quasi-linear equation, it is possible to obtain expressions for the evolution of the 1of12

2 plasma DF moments including the electron momentum associated with resistivity, assuming the fastest-growing waves dominate [see, e.g., Davidson and Gladd, 1975]. This technique is valid when the range of resonant phase velocities is comparable to the electron thermal speed and when nonlinear wave-wave interactions and higher order correlations are unimportant. For the current-driven ionacoustic waves (CDIAW) that we are interested in, further approximations for the linear growth rate and wave number of the fastest-growing mode are introduced, which are only valid for T e T i (T e is the electron temperature, T i is the proton temperature) [i.e., Galeev and Sagdeev, 1984; Labelle and Treumann, 1988]. It should also be noted that the electrostatic dispersion relation is modified in the presence of wave-particle interactions. Further analytical work on anomalous resistivity due to ion-acoustic turbulence includes work by Horton [1985], where renormalized turbulence theory is used to extend the validity of quasilinear theory. Horton [1985] obtain a scaling law for the effective collision frequency, and thus the resistivity, that have the same dependencies on the electron drift velocity and the electron to ion temperature ratio as the quasi-linear theory of Sagdeev [1967]. [5] In view of the above approximations in deriving an expression for resistivity due to wave-particle interactions by analytical calculations, it seems appropriate to use simulations to obtain such an expression [Biskamp et al., 1972]. Recent simulations have shown that the analytical calculations of ion-acoustic anomalous resistivity by Labelle and Treumann [1988] have underestimated its level by at least an order of magnitude because the details in the way the particle DFs are changing are not taken into account [Watt et al., 2002; Hellinger et al., 2004]. Petkaki et al. [2003] have further shown that a Lorentzian DF enables significant anomalous resistivity for absolute drift velocity conditions where it would be negligible for a Maxwellian DF with the same moments and that the anomalous resistivity at or near wave saturation can be an order of magnitude higher, even when the drift velocity and current density for the Maxwellian plasma are larger. The results of Watt [2001] and Petkaki et al. [2003] also indicate that the dependence of the ionacoustic resistivity on electron drift velocity is stronger than the linear relationship by quasi-linear theory [Sagdeev, 1967] or it extensions [Horton, 1985]. [6] In the papers by Watt et al. [2002] and Petkaki et al. [2003], the ion-acoustic anomalous resistivity values are measured at quasi-linear saturation as defined by the saturation of the fastest-growing wave mode, in order to compare with the results of quasi-linear theory. However, a plasma is unlikely to remain in this state. Although not discussed in detail in these papers, there is an evolution of the ion-acoustic wave modes after the quasi-linear saturation point. In fact the behavior of CDIAW close to the quasi-linear saturation is already nonlinear because there is significant power in growing waves outside the linearly resonant region. The nonlinearity develops further and it is this nonlinear state that is likely most appropriate to the common state of the real space environment. [7] As we shall see, the anomalous resistivity behavior close to and after the quasi-linear saturation is sensitive to the initial noise electric field conditions of the simulation. Thus in order to quantify the behavior of the ion-acoustic anomalous resistivity at and beyond quasi-linear saturation, we must use a statistical ensemble approach. We performed 104 Vlasov simulations with identical initial conditions except for the initial white noise electric field. The results are presented here. [8] The layout of the present work is as follows. In section 2, we briefly describe the one-dimensional (1-D) electrostatic Vlasov code that we use. We present the time integration method, the way we set up the simulation grid, and the initial conditions of the simulation. In section 3 we present the results of an ensemble of 104 Vlasov simulations. The anomalous resistivities due to CDIAW resulting from each simulation are compared and discussed in sections 3 and Method [9] The simulations presented in this paper study the evolution of unstable ion-acoustic waves and associated anomalous resistivity using a finite difference approximation to the Vlasov-Maxwell equations. Here we describe the details of the time-integration method, the simulation grid, and the initial conditions Time Integration Method [10] We use a 1-D electrostatic Vlasov simulation code with periodic boundary conditions [Horne and Freeman, 2001; Watt, 2001]. Electrons and protons are modeled using their distribution functions, f a (a 2 {e, i}), and the Vlasov equation is integrated forward in time to describe their evolution. For the study of electrostatic waves, the only force acting on the plasma is that due to an electric field, and the 1-D Vlasov equation þ a þ q a E z ¼ 0; m z where it is assumed that the magnetic field is zero or aligned along z. [11] The electric field is integrated forward in time using Ampère s law given ¼ m 0c 2 ðj int þ J ext Þ; ð2þ where J is the current density in the plasma. The subscripts int and ext in equation (2) indicate the internal and external components of the current density. The internal component is given by the plasma current density which can be calculated using the plasma DFs. The moments of the plasma DFs are calculated using the in-pairs method [Horne and Freeman, 2001]. The external component of the current is assumed to be given by the gradients of an external magnetic field J ext = rb ext. We assume that J ext balances the spatially averaged internal current such 0 /@t =0 always, where E 0 is the spatially averaged electric field [see Omura et al., 1996]. [12] Equations (1) and (2) are integrated forward in time using the MacCormack method, which is an explicit finite difference method using a predictor-corrector algorithm and first-order derivatives to calculate second-order accurate solutions [Anderson, 1992, pp ]. We ensure the 2of12

3 Table 1. Summary of Simulation Parameters Parameters All 104 Runs m i /m e 25 n = n i = n e m 3a T e /T i (T i =1) 2 DF Maxwellian v de 1.2 q m e = ms 1 l min m L z = l max m Dz = l min / m v ph,min ms 1 v ph,max ms 1 v cut,e ms 1 v cut,i ms 1 Dv e ms 1 Dv i ms 1 N v,e 891 N v,i 289 N z 642 a Here l De = 3.97 m and w pe = rad s 1. stability of the integration algorithm, by imposing constraints on the timestep of the numerical integration. These are the same as in section 3.1 of Petkaki et al. [2003]. [13] The resistivity h is calculated at each timestep in the simulation using the following definition [Davidson and Gladd, 1975]: h ¼ m e n e q 2 n ¼ m e e n e q 2 e e p ¼ m e n e q 2 e ; ð3þ R where n is the effective collision frequency and p e = m e vz f e0 dv z is the electron momentum (measured in the rest frame of the ions, with ion drift velocity zero) Simulation Grid [14] The dimensions and resolution of the ion and electron phase space grids are calculated by the simulation program to suit each individual set of initial parameters [Watt, 2001]. A full description of how we set up the simulations is presented by Petkaki et al. [2003]. In Table 1 we present the simulation parameters. Here l min, l max are the smallest and largest wavelengths of linearly unstable ion-acoustic wave modes calculated by solving numerically the linear dispersion relation. L z = l max is the length of the simulation box. Dz = l min /12 is the spatial grid resolution and N z = L z /Dz = 642 is the number of spatial grid points. Here v cut,e is the largest electron velocity and v cut,i is the largest ion velocity in the grid. Here v ph,min is the smallest phase velocity of the unstable wave modes and v ph,max is the largest phase velocity of the unstable wave modes calculated from the solutions to the linear dispersion relation. The resolution of each of the velocity grids (Dv e and Dv i ) is controlled by the phase velocities of the growing wave modes, such that there are three velocity points in the linear unstable region (v ph,min v v ph,max ) so that interaction with the unstable wave modes is possible. The number of velocity grid points for electrons is N v,e = 891 and for ions is N v,i = Initial Conditions [15] The Vlasov simulation runs presented here were performed for a temperature ratio T e /T i = 2, ion temperature T i = 1 ev, number density n = n i = n e = m 3 (see Table 1). The mass ratio is m i /m e = 25. The reduced mass ratio is used in order to keep the total run time of the simulations at a reasonable level. [16] CDIAW are generated by giving the electrons a uniform drift velocity (v de ) relative to the ions such that v de > v crit, where v crit is the threshold for the ion-acoustic instability. The drift velocity between electrons and ions is v de =1.2 q e m = ms 1, where q e m is the thermal velocity of the electrons. The wave vector is assumed to be parallel to B ext and to v de, k k = k z. The CDIAW grow out of a white noise electric field applied at t =0, E 1 ðz; 0 Þ ¼ XNmax n¼1 E tf sinðk n z þ f n Þ; ð4þ where f n is a random phase and N max = N z /2 is equal to the number of modes permitted in the simulation box. The initial amplitude of the electric field is set at the thermal fluctuation level E tf of the plasma: E tf ¼! 2k 1=2 BT e 0 l 3 ; 1 n N cut De E tf ¼ 0; N cut < n < N max ; ð5þ where l De = ( 0 k B T e /n e e 2 ) 1/2 is the Debye length of the plasma [Treumann and Baumjohann, 1997]. At saturation, the amplitudes of the growing waves are much greater than the thermal level [see Watt et al., 2002; Petkaki et al., 2003]. To minimize numerical aliasing, N cut = l max /l min is chosen to be less than the maximum number of k-modes N max = L z /(2Dz) =l max /(24l min ) allowed in the simulation box. 3. Results [17] We performed Vlasov simulations for 104 different initial white noise electric fields applied at t = 0 (see equation (4)). For each of the 104 Vlasov simulations the phases of the initial white noise field were randomly picked out of the uniform distribution, otherwise the simulations are identical. [18] In Figure 1a we see the time evolution of anomalous resistivity for three simulations (runs, 1, 9, and 80) superposed. The time axis is normalized by the inverse plasma frequency (w pe = rad s 1 ). We notice that the qualitative evolution of the individual simulations is similar but the exact values of the anomalous resistivity differ from one simulation to the next. The anomalous resistivity is time variable with oscillations that are sensitive to the relative phase of the initial electric field fluctuations. We identify three regimes in the evolution of the resistivity that can be explained by comparing with Figure 1b where the fastestgrowing mode of one run (Run 9) is plotted. [19] The first regime is the linear regime, which starts at w pe t 50 and continues to w pe t 200, where the fastestgrowing mode grows exponentially in time, as predicted by linear theory. (The initial fluctuations for w pe t 50 are due to the ballistic free streaming solutions to the Vlasov equation [Krall and Trivelpiece, 1973]). The evolution of 3of12

4 Figure 1. (a) Anomalous resistivity versus time from three simulations, runs 1, 9, and 80. (b) Fastest linearly growing electric field mode for run 9. (c) Spatially averaged electron and ion distribution functions for run 9. each simulation is very similar to the rest of the simulations as would be expected for a linear system. Throughout the linear regime, the anomalous resistivity remains close to zero. [20] The second regime is the quasi-linear regime, which starts at w pe t 200 and ends at w pe t 240, where the fastest-growing mode of each simulation starts deviating from exponential growth and saturates. In Figure 1c we have plotted the spatially averaged distribution functions of ions (centered at v i /q m e = 0) and electrons (centered at v de = v e /q m e = 1.2) for Run 9. Overplotted using dashed vertical black lines are the limits of the resonant velocities as predicted by linear theory. The two distributions are shown at the beginning of the simulation (t = 0) using a full black line and near quasi-linear saturation w pe t = 220 using a dotted line. The arrival at quasi-linear saturation can also be identified by the plateau formation at the range of resonant velocities in the spatially averaged electron distribution function [see also Petkaki et al., 2003, section 3.2]. In the quasi-linear regime, the time evolutions of the simulations start deviating from each other and the resistivity rises rapidly. [21] The third regime is the nonlinear regime, which starts at w pe t 240. Looking at the distribution functions at time w pe t = 300 in Figure 1c, the plateau in the spatially averaged electron distribution function has spread to lower and higher velocities outside the linear resonant region. The spatially averaged ion distribution function shows a high-energy tail, which suggests that momentum exchange has taken place between electrons and ions via the ion-acoustic waves. The time evolutions of the simulations are increasingly different but the resistivity is relatively stable, oscillating about the resistivity level reached at the end of the quasi-linear regime. [22] The development of the full wave spectrum during the linear, quasi-linear, and nonlinear regimes of our simulation is shown in Figure 2 where we have shown the absolute value of the fast Fourier transform (FFT) of the electric field E k for simulation run 5. Similar behavior is seen in the k-space electric field spectrum of all 104 simulations. The spectrum is shown at the beginning of the simulation at t = 0 in Figure 2a (see section 2.3). The electric field spectrum during the linear regime is shown in Figure 2b at time w pe t = 130, where we see that the wave power is above the initial noise field. We compared the growing modes of our simulation with the linear resonant modes calculated by solving the linear dispersion relation. The fastest linear growing modes, with growth rates g/w pi 0.05, were calculated to lie in the range k 0.2. Thus in the simulation, the wave power mainly is put into the fastest-growing modes in the linear resonance region, although in Figure 2b we also see that by w pe t = 130 some wave power is already present at higher k modes. In Figure 3 we plotted the growth rates calculated from linear theory by solving the dispersion relation of unstable ion-acoustic waves. Overplotted on the same graph (triangles) are the growth rates of the linearly growing modes calculated from our Vlasov simulations. Our simulation reproduces accurately the linear growth rates and the linear phase of the ion-acoustic instability. [23] Figure 2c shows the electric field spectrum in the quasi-linear regime at w pe t = 220. Wave energy has 4of12

5 Figure 2. Electric field spectrum in k-space (E k (V m 1 ) k(m 1 )) at four different times. (a) E k kat time t = 0, initial setup of the electric field spectrum. (b) E k k at time t = 130 w 1 pe, electric field spectrum during the linear regime. (c) E k k at time t = 220 w 1 pe, electric field spectrum during the quasi-linear regime. (d) E k k at time t = 300 w 1 pe, electric field spectrum during the nonlinear regime. cascaded into wave modes outside the linear resonant region, both at harmonics of the linear modes and not, and generally all modes have increased in power. In Figure 2d we have plotted the electric field spectrum at time w pe t = 300 which is during the nonlinear regime. The electric field spectrum evolves toward a fully developed forward (and possibly also inverse) cascade with a power law inertial range from the injection frequency of the fastest-growing linear wave mode to the highest (lowest) k-mode. At w pe t = 350 we stop the simulation because we begin to observe aliasing of the highest k-modes to lower k-modes. The development of the power law electric field spectrum is evidence of nonlinear wave-wave coupling. [24] Figure 4 shows the whole ensemble evolution of the ion-acoustic anomalous resistivity and wave power. In Figure 4a we have overplotted the time evolution of all 104 anomalous resistivities, each one represented by a different color. This representation shows the similarity and diversity in the behavior of the resistivity. In Figure 4b we plot the mean value of the resistivity (continuous black line) calculated by averaging the value of the 104 resistivity values at each timestep. On the same graph plus/minus three standard deviations from the mean are plotted (continuous red lines). Comparing Figures 4a and 4b we see that the range of resistivity values is well confined in ±3s of the mean, as would be expected by a Gaussian distribution. Figure 3. Linear growth rates g/w pi versus k (m 1 )ofthe ion-acoustic instability for the parameters of our simulations. The full line represents the growth rates calculated from solving the dispersion relation of unstable ion-acoustic waves. The triangles are the growth rates calculated from our simulations. 5of12

6 Figure 4. (a) Overplotted time evolutions of ion-acoustic resistivity from the ensemble of 104 Vlasov simulations. (b) Mean value (hðþ) t (black line) and ±3s curves (red lines) of the ensemble ion-acoustic resistivities. [25] To investigate this further, in Figure 5 we present histograms of the normalized probability distribution (PD) of resistivity values. The x-axis represents the standardized value of h at time t, (h(t) hðþ)/s t where hðþ t is the ensemble mean value of h at time t, and s is the standard deviation of h at time t. On each plot we have plotted the Gaussian of mean zero and standard deviation of one in order to compare it with the probability distribution of the standardized resistivity values. In Figure 5 we present histograms for three time periods, coming from the three different phases in the evolution of the instability. Each histogram comprises the sum of probability distributions from 11 consecutive timesteps where the standardization of Figure 5. Probability distribution of standardized resistivity values (h(t) hðþ)/s) t during three time periods of the ensemble evolution, (a) linear regime, (b) quasi-linear regime, and (c) nonlinear regime. 6of12

7 Figure 6. Time evolution of (a) the skewness and (b) the kurtosis of the probability distribution of ensemble resistivity values. h was performed individually for each of the 11 timesteps before summation, using the sample mean and standard deviation of each timestep. In Figure 5a we show the PD during the linear phase of the instability, for time period w pe t = The distribution of resistivity values appears to fit a Gaussian reasonably well. In Figure 5b we show the PD during the quasi-linear phase, for w pe t = , which seems to deviate from a Gaussian, being sharply peaked and with a left tail longer than the right. In Figure 5c we show the PD during the nonlinear phase, for time period w pe t = At this time period the distribution appears to be symmetrical and approximately Gaussian. [26] We have performed a chi-square test for each time timestep, where we tested the goodness of fit to a Gaussian distribution of mean zero and unit standard deviation of each one of the probability distributions of standardized resistivity values at the 0.05 and 0.01 significance level. Overall, the test fails in 8.4% of the timesteps at the 0.05 level and at 1.8% at the 0.01 level, which suggests that the distributions of resistivity values may not fit a Gaussian at all times. Figure 6 shows the times when the chi-square test fails at the 0.05 level (dotted lines) relative to the time evolution of the skewness and kurtosis of the resistivity distributions. The skewness of the resistivity distributions fluctuates but remains for most of the time evolution close to the value zero, implying that the distributions of resistivity values are mostly symmetrical. The kurtosis of the resistivity distributions has a value close to three for most of the time evolution, consistent with a Gaussian. However, on several timesteps deviations from the expected Gaussian values for skewness and kurtosis are observed. This is most apparent as we approach quasi-linear saturation and beyond where the amplitude of the fluctuations in the values of skewness and kurtosis increases. At quasi-linear saturation (see also Figure 5b) skewness is negative, which indicates that the left tail is longer. At the same time the kurtosis is well above three indicating a more sharply peaked distribution than a Gaussian. 4. Discussion [27] In this paper we have presented results of an ensemble of D electrostatic Vlasov simulations of the CDIAW performed for Maxwellian distribution functions under identical plasma initial conditions, except for a difference in the initial white noise electric fields. This 7of12

8 Figure 7. Mean of the wave energy (solid line) and standard deviation of the resistivity (dotted line) for the ensemble of 104 Vlasov simulations versus time. ensemble simulation is designed to examine the nonlinear evolution of the ion-acoustic instability and the associated resistivity, extending the results of analytical theory and the recent simulations of Watt et al. [2002] and Petkaki et al. [2003], in which the ion-acoustic anomalous resistivity values were measured at quasi-linear saturation. [28] The main result of the ensemble simulation is that the anomalous resistivity during the nonlinear phase is highly variable in time and sensitive to the initial noise field, such that the resistivity at any given time varies considerably from simulation to simulation. Consequently, a single simulation may not adequately describe the anomalous resistivity during the nonlinear phase. However, the distribution of resistivity is found to be approximately Gaussian such that the resistivity could be fully represented by the mean and standard deviation estimated from a relatively small ensemble. [29] The variation of the ensemble mean was shown in Figure 4. In the previous simulations of Watt et al. [2002] and Petkaki et al. [2003], the ion-acoustic anomalous resistivity values were measured at quasi-linear saturation, defined by the saturation of the fastest-growing wave mode. This is estimated to be at about w pe t = 220 in the simulations presented here. At quasi-linear saturation w pe t = 220, the mean value of resistivity is 74 W m and is consequently an underestimation of the anomalous resistivity since the resistivity continues to grow. An alternative point in the time evolution to take the measurement of anomalous resistivity is the point of saturation of the ensemble mean value of the ion-acoustic resistivity at time w pe t = 253. At this time, the mean value of resistivity is W m. The saturation of mean value of the ensemble of resistivities is possibly a convenient alternative measurement point of the anomalous resistivity in the nonlinear regime. However, the mean resistivity continues to evolve but it is always higher than the ensemble mean at quasi-linear saturation. [30] The variability of the resistivity is represented by the standard deviation of the ensemble. Referring to Figure 4b, the standard deviation of resistivity continues to grow monotonically (with minor fluctuations) throughout the whole evolution. At quasi-linear saturation w pe t = 220, the standard deviation s =12W m, which is 16% of the mean. At saturation of mean value of the resistivity at time w pe t = 253, the standard deviation s = 38.7 W m, which is 20% of the mean. By the end of our simulation at time w pe t = 350, the mean value of resistivity is 198 W m and the standard deviation s = 105 W m, which is 53% of the mean value of resistivity. [31] The large standard deviation can be explained as a combination of two effects: (1) The large variation with time of the anomalous resistivity in the nonlinear regime and (2) the sensitivity of the nonlinear evolution to the initial conditions such that the anomalous resistivity variations of the simulations are not in phase. [32] The chaotic sensitivity is likely attributable to the increasing wave energy of an increasing number of nonlinearly interacting wave modes as the power-law electric field spectrum develops. This is shown in Figure 7, which compares the evolution of total wave energy and the standard deviation of the anomalous resistivity. It can be seen that the two quantities grow similarly in the quasilinear and nonlinear phases. The development of the wave spectrum beyond the linearly resonant region is due to a change in the wave dispersion relation, and hence resonance, caused by the change in the Maxwellian distribution function [Hellinger et al., 2004] and by nonlinear wavewave interactions. Considering equation (2), we see that a spatial frequency present in the particle distribution function will also be present in the electric field. In equation (1), the electric field and particle distribution function interact through the E a /@v z term. Consequently, power in f a at an initial wave number k in the linearly resonant region will produce power in f a and E z at spatial frequencies equal to zero and 2k. Subsequently, the k and 2k waves will nonlinearly interact with themselves and each other to produce power at k, 0,k, 2k, 3k, and 4k, and so on. Thus the initial wave power growing in the linearly resonant region (Figure 2b) is transferred to harmonics of this frequency band (Figure 2c) and eventually cascades to all frequencies (Figure 2d). [33] The large variation with time of the anomalous resistivity can be explained as due to the bounce motion of electrons and ions in the growing electric fields. The bounce frequency is defined as w ba = p kq a ke k /m a k [Chen, 1984] where a is the species. Referring to Figure 2c, we estimate k 0.1 m 1 (the largest amplitude wave mode), for which E k 2mVm 1. This gives an electron bounce period of 25w 1 pe and an ion bounce period of 125w 1 pe at time w pe t = 220, decreasing to lower periods thereafter due to the increasing E k (Figure 2d). By inspection of Figure 1, we see that the electron bounce period corresponds to the fine scale structure of the ion-acoustic resistivity variation. The ion bounce period corresponds to the slower variation that gives rise to the two peaks in the ensemble mean (Figure 4). [34] The Vlasov simulations presented so far were performed for reduced mass ratio m i /m e = 25. It is impractical to perform 104 ensemble runs for much higher mass ratio. However, we have performed one run with the real mass ratio m i /m e = , in order to demonstrate that the main 8of12

9 Table 2. Summary of Real Mass Ratio Simulation Parameters Parameters m i /m e n = n i = n e m 3a T e /T i (T i =1) 2 DF Maxwellian v de 1.2 q m e = ms 1 l min 28.8 m L z = l max m Dz = l min / m v ph,min ms 1 v ph,max ms 1 v cut,e ms 1 v cut,i ms 1 Dv e ms 1 Dv i ms 1 N v, e 3729 N v, i 307 N z 529 a Here l De = 3.97 m and w pe = rad s 1. physical processes are not changed. The parameters of the simulations are written out in Table 2 and the results are summarized in Figure 8, which can be compared with Figure 1. Qualitatively, the evolution of the anomalous resistivity, the growing modes, and the distribution functions seen in the simulations using m i /m e = 25 are reproduced by the real mass ratio simulation. Quantitatively, the timescale of the evolution is longer (e.g., Figure 8a). For example, saturation of the fastest-growing modes is reached later because the growth rate is smaller for m i /m e = (Figure 8b). In Figure 8c we show the evolution of the spatially averaged electron and ion distribution functions at w pe t = 0, 1200, 1485 and Saturation of the fastestgrowing mode at w pe t 1200 coincides with the formation of a plateau in the resonance region of the electron distribution function. [35] As with the reduced mass ratio simulations, the real mass ratio simulation shows considerable variation of the anomalous resistivity in the nonlinear regime. To demonstrate that this is also due to electron and ion bounce motion, we show in Figure 9 the full distribution function of electrons and ions. The top two panels show the full electron distribution function at w pe t = 990 and w pe t = 1485 for the full length of the simulation box in x and from 2 q e m to 4 q e m in the velocity space (q e m is the thermal velocity of the electrons). The lower two panels show the full ion distribution function at w pe t = 990 and w pe t = 1485 for the full length of the simulation box in x and from 0.1 q e m to 0.1 q e m in the velocity space. We have zoomed in velocity space in order to see the details of the ion distribution function. At w pe t = 990 a very small trapping region has started to form at the resonant electron velocity region. By w pe t = 1485 the trapping region has expanded in velocity space and several trapping islands are present in phase space diagram. Coalescence of some islands is visible too. Using either the average wavelength of the island structures seen in Figure 9 or the wave number of the largest amplitude wave mode, the characteristic electron trapping wave number is k m 1.Atw pe t = 990 the ion distribution function is undisturbed yet by the presence of the growing electric field modes. By w pe t = 1485 ion trapping regions are beginning to form with a similar wave number. Figure 8. (a)anomalous resistivity versus time for real mass ratio m i /m e = (b) Fastest linearly growing electric field mode. (c) Spatially averaged electron and ion distribution functions. 9of12

10 [36] In Figures 10a and 10b we plot the electric field spectrum around this wave number at w pe t = 990 and 1485, and in Figure 10c we calculate the corresponding bounce frequencies for electrons normalized to the electron plasma frequency at w pe t = 990, 1485, and At k 0.01 m 1, the electron bounce frequencies vary from about 0.003w pe at w pe t = 990 to about 0.05 w pe at w pe t = 1980, corresponding to periods of 300 to 20 w 1 pe, respectively. Correspondingly, the ion bounce periods vary from 13,000 to 860 w 1 pe. Looking at Figure 8a we can see that the anomalous resistivity fluctuates with a period of 100w 1 pe around w pe t = 1000 which decreases to 10w 1 pe toward the end of the simulation. Thus these timescales and their evolution are roughly consistent with the timescales of the inverse electron bounce frequencies. It should also be borne in mind that there is not only one mode present at any one time but several are growing simultaneously each with a different frequency and amplitude. As a result, beating effects are also present in the time evolution of the anomalous resistivity. The ion bounce frequencies are much smaller so their effect would not be noticed in this simulation. [37] Our results on the ion-acoustic anomalous resistivity differ from those of Hellinger et al. [2004], since the evolution of ion-acoustic anomalous resistivity does not show the fluctuating behavior in those simulations. This difference is due to the trapping effects that we see in our simulations (see Figure 9). The trapping effects are not seen in the simulations of Hellinger et al. [2004] (P. Hellinger, private communication, 2005). Figure 9. Full electron distribution function at (a) w pe t = 990 and (b) w pe t = Full ion distribution function at (c) w pe t = 990 and (d) w pe t = Real mass ratio m i /m e = was used in these simulations. The linear resonant region is 0.015q e m to 0.025q e m. 5. Conclusions [38] The results of the analysis of the ensemble of 104 Vlasov simulations with identical initial conditions except for the initial white noise electric fields can be summarized as follows: [39] 1. We have shown that the evolution of the ionacoustic anomalous resistivity in the nonlinear regime is variable in time and sensitive to the phase of the initial noise electric field. Because of this, estimations of the anomalous resistivity values from a single simulation could lead to errors in obtaining the correct value of the resistivity and an ensemble simulation is a more reliable estimator. [40] 2. We have shown that estimation of the ion-acoustic anomalous resistivity at quasi-linear saturation provides an underestimation of the resistivity in the nonlinear regime. The mean resistivity of the ensemble saturates at 2.5 times higher level than at quasi-linear saturation. This result is qualitatively confirmed by the real mass ratio simulation, where also the ion-acoustic anomalous resistivity at the saturation of the fastest-growing mode is much smaller than later in the time evolution. [41] 3. The standard deviation s of the ensemble anomalous resistivity and the electric field mean wave energy increase monotonically with time throughout the evolution. [42] 4. On the basis of the results of the chi-square test, the distributions of resistivity values are approximately Gaussian in the linear, quasi-linear, and nonlinear regimes of the resistivity evolution. However, the skewness and kurtosis of the distributions deviate from the Gaussian distribution close to and at quasi-linear saturation and also at instances during the nonlinear evolution. 10 of 12

11 Figure 10. Electric field spectrum at (a) w pe t = 990 and (b) w pe t = (c) Bounce frequencies for electrons at w pe t = 990 (black), w pe t = 1485 (green) and w pe t = 1980 (red). (d) Bounce frequencies for ions at w pe t = 990 (black), w pe t = 1485 (green) and w pe t = 1980 (red). [43] 5. A natural plasma that supports the ion-acoustic instability may be expected to exist predominantly in the nonlinear regime of the instability. Our calculations show that the ensemble mean of the ion-acoustic resistivity during the nonlinear regime is higher than estimates at quasi-linear saturation from both analytic [Labelle and Treumann, 1988] and simulation results [Watt et al., 2002; Petkaki et al., 2003] and that the ensemble standard deviation is comparable to the ensemble mean, providing a natural source of localized resistivity for collisionless reconnection. [44] Acknowledgment. Shadia Rifai Habbal thanks James F. Drake and two other referees for their assistance in evaluating this paper. References Anderson, J. D., Jr. (1992), Discretization of partial differential equations, in An Introduction to Computational Fluid Dynamics, edited by J. F. Wendt, pp , Springer, New York. Bale, S. D., F. S. Mozer, and T. Phan (2002), Observation of lower hybrid drift instability in the diffusion region at a reconnecting magnetopause, Geophys. Res. Lett., 29(24), 2180, doi: / 2002GL Birn, J., et al. (2001), Geospace Environment Modeling (GEM) magnetic reconnection challenge, J. Geophys. Res., 106, Biskamp, D., K. U. von Hagenow, and H. Welter (1972), Computer studies of current-driven ion-sound turbulence in three dimensions, Phys. Lett. A, 39, Carter, T. A., H. Ji, F. Trintchouk, M. Yamada, and R. M. Kulsrud (2002), Measurement of lower-hybrid drift turbulence in a reconnecting current sheet, Phys. Rev. Lett., 88, doi: /physrevlett Chen, F. F. (1984), Introduction to Plasma Physics and Controlled Fusion, vol. 1, Plasma Physics, pp , chap. 8, Springer, New York. Coroniti, F. V., and A. Eviatar (1977), Magnetic field reconnection in a collisionless plasma, Astroph. J. Suppl. Ser., 33, Davidson, R. C., and N. T. Gladd (1975), Anomalous transport properties associated with the lower-hybrid-drift instability, Phys. Fluids, 18, Deng, X. H., and H. Matsumoto (2001), Rapid magnetic reconnection in the earth s magnetosphere mediated by whistler waves, Nature, 410, 557. Drake, J. F., M. Swisdak, C. Cattell, M. A. Shay, B. N. Rodgers, and A. Zeiler (2003), Formation of electron holes and particle energization during magnetic reconnection, Science, 299, Farrell, W. M., M. D. Desch, M. L. Kaiser, and K. Goetz (2002), The dominance of electron plasma waves near a reconnection X-line region, Geophys. Res. Lett., 29(19), 1902, doi: /2002gl of 12

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