Electron holes, ion waves, and anomalous resistivity in space plasmas

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi:10.1029/2004ja010482, 2006 Electron holes, ion waves, and anomalous resistivity in space plasmas Lars P. Dyrud 1 and Meers M. Oppenheim Center for Space Physics, Boston University, Boston, Massachusetts, USA Received 17 March 2004; revised 5 October 2005; accepted 1 November 2005; published 31 January 2006. [1] Phase-space electron holes are seen in simulations, laboratory plasmas, and many regions of the Earth s space environment. We present simulations of beam plasmas showing that the generation and decay of electron holes results in a reduction of electron current, implying a parallel resistivity. We show that resistivity occurs in simulations where a cold electron beam is coincident with a warmer background plasma and appears to be mediated by the generation of ion acoustic waves propagating obliquely to the magnetic field. Initially, electron holes scatter electrons in the beam direction, steepening the electron beam distribution, eventually launching ion acoustic waves that cause resistivity and strong ion heating perpendicular to ~B. These effects occur in both strongly and weakly magnetized plasmas. Given that electron holes are observed in many space plasmas, these results have important implications for a number of magnetospheric and auroral ionospheric processes. For auroral plasmas, electron hole resistivity could support parallel electric fields on the order of several mv/m, accounting for parallel potential drops from tens to hundreds of ev. For the magnetopause, simulations show effective collision rates of 0.00015 w pe, which could enhance dissipation and diffusion across the boundary. Citation: Dyrud, L. P., and M. M. Oppenheim (2006), Electron holes, ion waves, and anomalous resistivity in space plasmas, J. Geophys. Res., 111,, doi:10.1029/2004ja010482. 1. Introduction [2] Over the past decade, electron holes were observed in space plasmas by NASA satellites from Polar to GEOTAIL and Cluster but most frequently in the downward current region by FAST [Matsumoto et al., 1994; Ergun et al., 1998; Franz et al., 1998; Bale et al., 2002]. While electron holes are a common occurrence and their structure is fairly well understood, it remains unclear whether they are merely an interesting plasma phenomenon and perhaps tracers of a turbulent environment or whether their generation and existence play a critical role in the flow of energy and momentum in the magnetosphere-ionosphere system. [3] We present simulations showing that the generation and decay of electron holes creates a parallel resistivity. This anomalous resistivity could impact several important auroral and magnetospheric processes, including supporting field-aligned potential drops, enhancing diffusion, and heating of both ions and electrons. [4] Electron holes have been well studied over the past decades using theory, computer simulations, laboratory plasmas, and most recently satellite observations. Early analysis examined electron hole stability in terms of Bernstein-Green-Kruskal (BGK) modes, and nonlinear Landau 1 Now at Center for Remote Sensing Inc., Fairfax, Virginia, USA. Copyright 2006 by the American Geophysical Union. 0148-0227/06/2004JA010482$09.00 damping [Bernstein et al., 1957; Morse and Nielson, 1969; Turikov, 1984]. Simulations further contributed to our understanding of the complex nonlinear dynamics associated with electron phase space holes [Berk et al., 1970; Morse and Nielson, 1969]. The discovery of holes in the Earth s space environment has triggered a number of additional numerical studies with results applicable to holes in space [Omura et al., 1996; Mandrake et al., 2000; Oppenheim et al., 2001; Muschietti et al., 1999; Singh et al., 2000; Goldman et al., 1999]. For a review of electron and ion holes in computer simulations and laboratory experiments, see Guio et al. [2002]. [5] The results presented here are relevant to any of the space plasma regimes where electron holes are observed. However, two new studies indicate that electron hole resistivity has particular importance for the auroral downward current region and the magnetopause boundary [Ergun et al., 2001; Cattell et al., 2002]. [6] Results from FAST show a clear relationship between electron holes, VLF saucers, and upward electron beams energized by parallel potential drops [Ergun et al., 2001]. Additionally, James [1976] showed that VLF Saucers emanate from point sources that are no longer than 10 km in altitude. These combined results suggest that these phenomena are generated within a 10 km region in altitude along a downward current flux tube. We extend the results from recent simulations of magnetized holes in two and three dimensions that show them persisting for many hundreds of w e 1 before eventually developing kinks 1of12

and decaying while emitting electrostatic whistler waves propagating obliquely to the ambient magnetic field, ~B [Oppenheim et al., 1999]. These waves have been identified as the plasma mode responsible for auroral observations of VLF saucers [Vetoulis and Oppenheim, 2000; Newman et al., 2001]. Our results indicate that e-holes may play a role, not only in VLF saucer generation, but contribute to ion conics, and generate resistivity that supports substantial field aligned potential drops, even over short distances. [7] Weakly magnetized simulations also show the development of electron holes and strong resistivity. Recent Polar observations of electron holes near the magnetopause indicate the electron holes could provide dissipation and diffusion across the boundary [Cattell et al., 2002]. Our simulations show that effective collision frequencies caused by electron holes are as large or larger than many of the anomalous resistivities proposed by other researchers studying magnetopause diffusion. [8] We present studies of anomalous resistivity resulting from a current driven through a plasma. These results show that beam-driven electron holes behave similarly to the current-driven ion acoustic instability. Several theoretical studies have calculated effective resistivity from currentdriven instabilities, or CDI. These results focused on resolving current sheet thickness problems in the magnetosphere and substorm onset and have included ion acoustic waves, lower hybrid, modified two stream, and ion cyclotron instabilities [Perraut et al., 2000; Watt et al., 2002]. This paper effectively adds a new current-driven instability (CDI) to the above list. 2. Electron Hole Simulations [9] Our kinetic simulations use a massively parallel, electrostatic, particle-in-cell (PIC) algorithm capable of modeling both a finite and an infinite magnetic field in one, two, or three dimensions [Birdsall and Langdon, 1985]. This code applies periodic boundary conditions to an initial-value problem and uses quiet-start algorithms to minimize particle noise. For this research we conducted approximately 50 one-dimensional (1-D), 2-D, and 3-D simulations of electron beam type instabilities. Running each simulation on an IBM SP supercomputer with 16 CPUs takes approximately 4 5 hours. All of the simulations were run in normalized units with the electronic charge, e, the permittivity of free space, o, and the electron mass m e equal to 1. For further information on this simulator, see Oppenheim et al. [2001]. [10] We initiate these simulations with three different populations of particles, one background population of electrons in thermal equilibrium with the background ions, and a third population of beam electrons. The electron beam begins with a velocity between 1.0 and 25.0 times the background thermal speed and has a density of 0.1 0.5 that of the total electron density. The beam temperature was also varied from equal with the background to 1/10 of the background temperature. A magnetic field, ~B 0, parallels the ^x axis with an amplitude such that the ratio of the electron cyclotron, W e, to the electron plasma frequency, w e, is between 0.5 and 5. The time step varies from Dt = 0.1 to 0.25 w 1 e, which just resolves W e, and w e results from the combined density of the electrons. The grid resolves 2048 512 cells and spans 800 250 initial Debye lengths, l D0. Since the two stream instability leads to substantial electron heating parallel to ~B 0, the parallel grid span about 400 l D after a few tens of w e 1. We note here that for the remainder of the paper we use units of time referenced in inverse electron frequency w e 1. Additionally, halving the grid spacing and time step or doubling the simulation box does not change the results. [11] Ion populations measured in the auroral region and magnetosphere have a range of masses, temperatures, and mean velocities. To illustrate the simplest physical processes, we simulate a uniform H + population with a mass ratio of m i /m e = 1832 and a temperature equal to that of the background electron population. 3. Simulation Results [12] To demonstrate our results, we compare two simulations run with the same parameters, except for the initial electron beam temperature. Both simulations show electron hole formation and electrostatic whistler wave generation. However, a simulation initialized with equal temperature electron populations shows no anomalous resistivity, while a run with a cooler electron beam shows intense resistivity. These simulations are hereafter referred to as the nonresistive and resistive simulations. We present the nonresistive simulation first as a baseline case and to tie these results in with previous electron hole simulations. 3.1. Baseline/Nonresistive Simulation [13] Figure 1 shows the evolution during the nonresistive simulation. The parameters for this simulation are W e /w e =4, n beam = n background, T beam = T background, and v beam = 4.0 v thermal. This figure shows the electrostatic energy, E 2, and spectra of E 2 as a function of simulation position at four different times. The first panel shows E 2 at t = 96 w 1 e into the simulation. The electron holes appear as spaghetti-like tubes aligned primarily perpendicular to the magnetic field. The next panel shows that by t = 336 the holes have merged, generating larger phase space holes. The third and fourth panels display merged holes persisting and launching electrostatic whistler waves. The k? versus k k spectra show two features, the electrostatic whistler waves, which have narrow spectra and appear as waves with ~ k aligned nearly perpendicular to ~B, and the electron hole spectra which show power over a broad range of ~ k values. For a more detailed description of the evolution of this type of simulation, see Oppenheim et al. [1999], and for discussions on how electron holes launch electrostatic whistlers, see Vetoulis and Oppenheim [2000] and Newman et al. [2001]. [14] Since our simulations utilize periodic boundary conditions, we are unable to directly simulate the development of zeroth-order potential drops. In order to find anomalous resistivity, we employ the common technique used when simulating and analyzing current driven instabilities [Perraut et al., 2000; Watt et al., 2002]. That is, we treat the electron beam as an initialized current within the simulation and look for a reduction in current as a function of time. Then, from the reduction in current we calculate an effective collision frequency due to the wave-instability effects, using dj e /dt = nnev [Davidson, 1972]. 2of12

Figure 1. Electrostatic energy and spectra at four different times for a simulation with equal beam densities, T beam = T background, and v beam =4.0v thermal. The magnetic field is in the ~x direction. [15] Figure 2 shows our method of analyzing this simulation for resistive effects. The top panel shows the total electron current in the simulation as a function of time, normalized to the magnitude of the initialized current density. It is important to note that we are not merely plotting the beam flux but the combined electron current of the background and beam populations. This panel shows that the initial formation of the electron holes has quite the opposite effect of resistivity, i.e., we find that the holes, which traverse the simulation at slightly less than beam velocities, preferentially scatter electrons in the forward direction, thus increasing the current. We find this effect in every simulation that generates electron holes, including one-, two-, and three-dimensional simulations. The increase in electron current is small, from 1 to 5%, but occurs rapidly, representing a large time derivative in the electron momentum. This effect acts to steepen the electron beam distribution, which as shown in the next simulation, drives marginally stable electron distributions unstable and acts to continually feed energy to instability. 3.2. Electron Hole Antiresistivity [16] The increase in current is quite unexpected and, while not the main focus of this paper, we have spent some time examining the processes that drive it. Several candidate processes have been examined and some excluded. Figure 3 plots the parallel electron distribution function just after the the formation of electron holes and again after 1000 w e 1. This simulation started with an initial beam at v b = 4.0 and the holes form and traverse the simulation with an average velocity of about v h 3.5, which is closer to the beam velocity due to the cold beam. The steepened beam distribution peaks at about v = 4.25 and is associated with a net increase in electron flux. The following paragraph discusses candidate physical mechanisms for this increase in flux. [17] A process that we examined is that electrons are accelerated via a Landau damping type process by interacting with the passing electron hole potentials. This process is effectively the same as demonstrated by Chen [1984, section 7.5.1]. However, this process is quickly excluded by the evidence that the holes traverse the simulation at slightly less than beam velocities, the increase in current is due to a reduction in thermal velocity electrons and an increase in beam velocity particles. This effect appears quite clearly in a plot of the electron distribution function as a function of time between t = 100! 500 w e 1 in Figure 11. In every beam driven electron hole simulation we have conducted, the increase in phase-space density occurs at velocities faster than the electron hole velocity. This is entirely inconsistent with a simple Landau resonance process between thermal electrons and passing electron hole potentials. [18] The exclusion of a Landau resonance leaves few mechanisms remaining. Every beam-driven electron hole simulation we conducted launches ion acoustic waves with ~ k k ~B of varying amplitudes. Additionally, while electrons are accelerated to larger velocities than the holes, the accelerated velocity appears to be near the sum of the hole velocity and that of a trapped hole electron near the phase space separatrix. This revelation leaves us with the following interpretation: passing ion acoustic waves modify the hole potentials to generate asymmetric potential wells. 3of12

Figure 2. The top panel plots simulation electron current, normalized to the initial current. The second panel plots energy densities, average electron parallel kinetic energy (solid line), and electrostatic wave energy multiplied by 10 (dashed line). The third panel shows the expected electric field and effective collision frequency associated with a change in current. All traces are shown as a function of time, measured in inverse w 1 e. Electric field calculated from d~v e = e dt m ~ E using auroral parameters: w pe = 10 5 s 1 l D = 100 m. These asymmetric potential structures free trapped electrons and accelerate passing electrons. These accelerated and newly liberated electrons will have velocities slightly larger than the original hole velocity. [19] It is important to note that these simulations conserve momentum to within floating point accuracy. Therefore in order for the electrons to be accelerated in the beam direction, ions must be accelerated in the opposite direction. For example, simulations with stationary ions do not develop antiresistivity physics. Figure 4 demonstrates this effect on the ion population. Figure 4 shows that ions absorb momentum through more complicated wave processes. We intend on looking at this electron hole antiresistivity process more closely in the future. While interesting, it is not the main focus of this paper. 3.3. Resistive Simulation [20] Figure 5 shows the evolution of electric field energy during the resistive simulation in the same format as Figure 1. Running a simulation with identical parameters as the nonresistive simulation except for the electron beam temperature leads to strong anomalous resistivity. For this run the electron beam is cooler than the background ions and electrons with T beam /T background = 1/6. The first panel in this figure shows essentially the same behavior shown in the the previous simulation: the formation of electron holes which appear as spaghetti-like tubes. However, the next panel at t = 336 departs from the previous simulation. The tubes have merged as before but are losing coherence perpendicular to ~B. This loss of coherence is due to interference with electrostatic whistler waves, which again have ~ k just off perpendicular ~B, and a new mode not seen in the nonresistive simulation. This new wave has ~ k at more oblique angles to ~B which is of critical importance and discussed below. The final two panels show the waves becoming much stronger and masking the remnant hole structures, which become weaker and circular-shaped. [21] Again we look at the electron current to examine the simulation for anomalous resistivity. Figures 6 and 7 shows the resistive simulation results as a function of time in a Figure 3. Change in parallel to ~B electron distribution function for the nonresistive simulation. The solid line shows the distribution function just after electron hole formation, while the dotted line is after 1000 w 1 e. The plot shows that electron hole ion acoustic resonance reduces the background population while steepening the bump. 4of12

Figure 4. Change in the ion distribution function for a 1-D simulation with the same parameters as the nonresistive simulation. The plot shows the ion distribution function is perturbed nonuniformly as a result of the antiresistivity. format identical to Figure 2. The top panel shows that again the electron current does initially increase but that the effect is short lived. The corresponding simulation times from Figure 5 are marked on this plot by the letters a d. First, a shows that the increase in current corresponds to the immediate formation of electron holes (antiresistivity); second, b marks the maximum rate of increase in current which coincides with the appearance of the waves. Line c marks the middle of the decreasing current period. This effect coincides with wave saturation, as evidenced by the simulation electrostatic energy shown in the second panel. The time marker d shows that current is no longer decreasing and corresponds to the waves weakening and the electron distribution function flattening and stabilizing (shown in Figure 11). The second panel of Figure 6 plots both the parallel kinetic energy and the total electrostatic energy as a function of time. The resistive effects reduce the parallel energy by 25%. This shows that the loss of parallel energy is approximately 10 times that of the total electrostatic energy at any given time. Comparing E 2 with the nonresistive simulation shows the resistive simulation has 5 times the peak electrostatic wave energy. [22] The final panel estimates the effect this resistivity would have in the auroral downward current region and magnetosphere. We relate the loss of electron momentum to an electric field by a simple electron momentum equation that assumes current continuity, E resist = 1/e dhmv ek i/dt where hmv ek i is the parallel momentum averaged over all electrons. We use the resulting field, E resist, as an estimate of the resistive effects in the auroral ionosphere. Using auroral parameters suitable for 2000 km altitude, w pe =10 5 s 1 and l D = 100 m, our simulation yields parallel electric fields, E resist, of approximately 2 mv/m. [23] We use these simulations to estimate the total potential drop that may be incurred due to this resistivity. While the periodic boundary conditions of the simulation make direct calculations of this impossible, we may still estimate potential drop in the following manner. Assuming that the process of electron production in the auroral zone remains constant for at least a few thousand plasma periods, we can estimate the total potential drop over the entire region where Figure 5. Electrostatic energy and spectra in the simulations at four different times. Equal beam densities, T beam /T background = 1/6, and v beam =4.0v thermal. 5of12

Figure 6. Same format as Figure 2 but for the simulation shown in Figure 5. The electrostatic energy has been multiplied by 10 such that it appears on the same axis. Again, electric field calculated from d~v e = e dt m ~ E using auroral parameters: w pe =10 5 s 1 l D = 100 m. anomalous resistivity occurs. Since the resistive behavior lasted for approximately 1500 w 1 pe in our initial value simulation and the average electron hole velocity, h v hole i, is 3.5 v th, the resistive region extends for about 5250 debye lengths or 525 km. Our simulation tells us that the resistive electric field along this length is nearly 2 mv/m, making the total potential reach 1000 V. We note here that we chose the hole velocity as the coordinate transformation velocity from time to space, for the following reason. If the presence of the holes and associated physics is responsible for the initiation of resistivity, following the holes should follow the onset of resistivity. This choice of velocity is however an approximation, and arguments could be made for other frame of reference choices. [24] To provide a more precise calculation of potential drop, we integrate the calculated electric field shown in Figure 6 over the entire resistive time. Using the same electron hole velocity, plasma period, and debye length yields a total potential drop integrated to 717 V. While such parameters yield an extremely large potential drop they do so with a large beam velocity of 40,000 km/s. If we choose the same w pe =1 10 5 s 1 but reduce l D =20m,the resulting beam velocity for the resistive simulation would translate to 8000 km/s and result in an integrated potential drop to be 28 V. [25] For a simulation with the same parameters but with a slightly denser and faster beam of n beam =.65n total,v beam =5, and v hole = 4.4 yields stronger anomalous resistivity. Using the method applied above and w pe =1 10 5 s 1 and l D = 20 m, we calculate that such a system generates a much larger total potential drop of 221 V with a beam velocity of 10,000 km/s. These values demonstrate that electron hole resistivity can extend from a moderate to a large effect on the system, depending on the local plasma conditions and Figure 7. Ion distribution functions at t = 0 and t = 1344. 6of12

Figure 8. Spectra of electric potential and ion density from the two 2-D simulations during the resistive period between t = 600 and t = 2000. These spectra vary the phase velocity and angle of the k-vector with respect to the magnetic field for a fixed k = 0.1. The dispersion relation for warm ion acoustic and electrostatic whistler waves has been plotted over the spectra as a white line. Note the most dramatic difference between the two simulations at ion acoustic phase velocities near v ph = 0.1. beam strength. Finally, for more useful comparisons with magnetospheric plasmas, the scale on the right side of Figure 6 shows the effective collision frequency per w 1 pe. The effects of this anomalous parallel resistivity on space plasmas is discussed in more detail in section 4. 3.4. Summary of Simulations [26] We have conducted nearly 50 1-D, 2-D, and 3-D simulations with a range of initial conditions and find the above example is fairly representative of the evolution of the simulations for a wide range of parameters, with the following variations. Resistivity only occurs when the electron beam is cooler than the background electrons by a temperature ratio of at least T beam /Tbackground = 1/2. The effective collision frequency increases linearly with increasing temperature ratio. The threshold velocity for resistivity also varies with beam temperature, but for equal density beams, electron holes and resistivity occur for v beam > 1.75 v th. Additionally, once above threshold, the effective collision frequency increases linearly with v beam suggesting, that for a fixed beam temperature and density, there is a standard linear dependence between j and E. We have also conducted simulations with a range of magnetic field strengths and find that the development of resistivity has a fairly weak dependence on the magnitude of B. Ion temperature plays an important role, large ion temperatures cause ion landau damping of the ion acoustic waves, which diminishes the effects of resistivity. Finally, simulations with the same parameters, but in higher dimensions (i.e., 3-D versus 2-D, and 2-D versus 1-D) show show stronger resistivity by about 20%. 4. Discussion [27] In this section we discuss the physical processes associated with the the electron hole resistivity and the numerous implications for space plasmas. We examine several diagnostics of these simulations in order to understand the physics of this anomalous resistivity. [28] Comparing spectra between the two simulations shows that the resistive simulation has an additional plasma mode present, waves ~ k oriented at 45 with respect to the magnetic field. This wave was appears in the lower panels of E 2 ( ~ k) in Figure 5 of the resistive simulation, but was absent in the nonresistive simulation shown in Figure 1. Since they are only present in the resistive simulation, these oblique waves may play a role in the resistivity, but we therefore examine the various modes present in the two simulations here. [29] Figure 8 plots the power in the electric potential and ion density for both simulations as a function of phase velocity and the angle of the wave vector ~ k with the magnetic field. We have superimposed the dispersion relation for warm ion electrostatic whistler waves, w pi 1 = + w2 pe w 2 1 T k 2 w 2 cos 2 (q), which shows both an m w 2 ion acoustic solution and the more common w = w p cos(q). For these plots we chose the wavelength of k = 0.1 which can be compared with the spatial spectra seen in Figure 5 and 1. This plot shows the most dramatic difference between the two simulations occurs at ion acoustic phase velocities near 0.1. While significant power exists at these frequencies in the resistive simulation, it is much weaker in the nonresistive simulation. We see both simulations show power at ion acoustic phase velocities at a range of angles. However, the resistive simulation shows power at ion acoustic phase velocities (near 0.1) at a broad range of angles, including at angles parallel to the magnetic field. In the nonresistive simulation the ion acoustic power is weaker; between 3 and 10 times weaker for most wavelengths. Figure 8 also reveals that the power in the ion acoustic waves is enhanced at angles near the electrostatic whistler waves. This suggests a further relationship between electron holes, whistler waves, and acoustic waves, and demonstrates that under some conditions, warm ions should be considered in calculating whistler wave properties. This plot shows the most dramatic difference between the two simulations occurs at ion acoustic phase velocities near 0.1. While significant power exists at these frequencies in the resistive simulation, it is much weaker in the nonresistive simulation. 7of12

Figure 9. Image of ion density and electric potential f as a function of time and space for a 1-D version of the resistive simulation. The potential image shows the fast electron holes (steep slope) decaying into ion waves with a slow velocity, while the ion density shows only these strong ion acoustic waves, with perturbations near 30% moving both forward and backward in the simulation. [30] To more simply demonstrate how ion acoustic waves propagating in the beam direction grow in the resistive simulation but not in the nonresistive one, we present plots from 1-D versions of the 2-D simulations. In this case, the ion acoustic waves cannot propagate obliquely but can propagate only with or against the beam. Plots of ion density and electric potential, f, are shown in Figures 9 and 10. The two simulations demonstrate similar features, such as the moving hole potentials which appear as diagonal lines with a weak slope in the plots of f. Additionally, both simulations show ion acoustic waves propagating backwards. However, the resistive simulation has strong ion acoustic waves with perturbations near 30% propagating in both directions, while the nonresistive simulation shows only weak ion-acoustic waves with ion density perturbations of only 0.5% propagating backward. It is these strong, beam direction propagating ion waves that interact with the electron beam and transfer electron momentum to the ions. [31] After establishing that ion acoustic waves are present in the resistive simulation, two questions remain. Why are the waves not damped, and what is the association with electron holes or the oblique whistler waves? The answer to the first question is that the waves draw free energy through a type of current-driven ion acoustic instability. This instability requires a positive slope in the electron distribution near the ion acoustic phase velocity [Goldston and Rutherford, 1995]. Figure 11 shows that the increase in high-speed electrons associated with antiresistivity are actually responsible for this positive slope. As described in section 3.2, the interaction of parallel ion acoustic waves and electron holes preferentially scatter electrons to beam velocities. Figure 11 shows the effect electron hole formation has on the electron distribution function and how it relates to the evolution within the simulation. The figure plots f(v x ) versus time and shows the beam distribution continually steepening and accelerating to higher velocities. This effect continues until the emergence of oblique ion acoustic waves, which flatten the beam. The timing of the growth of oblique ion waves can be seen in the spectra from Figure 5, and their emergence in the ion density plots of Figure 9 near t = 700. We conclude by noting that the electron distribution function early in the simulation, approximately 100 plasma periods after two-stream thermalization, is similar to those distributions shown by Figure 3 and 4 in the work of Newman et al. [2002], in 1-D Vlasov simulations of electron beams emerging from a strong double layer in the auroral ionosphere. The implications of this are discussed later. [32] The reason the formation of ion acoustic waves requires a cooler initial electron beam is twofold. The first is that the cool beam suffers less flattening by the twostream instability. The second, and more important, is that the cool beam generates much stronger electron holes with larger peak potentials, which are far more efficient at perturbing the electron distribution function. The electron distribution functions in the nonresistive simulation do not have a positive slope at the ion sound speed, while the resistive simulations do. A positive slope in the electron distribution provides free energy to the ion acoustic waves via wave-particle interaction. [33] Throughout the course of this discussion we have pointed toward ion acoustic waves as the mediator of the 8of12

Figure 10. A figure in the same format as Figure 9 except for the nonresistive simulation, which shows only the generation of comparatively weaker backward going ion acoustic waves, with perturbations of only 5% that do not remove momentum from the electron beam. resistivity. While the circumstantial evidence seems clear, that ion acoustic waves play a role, the specific mechanism is not entirely straightforward. It may be the oblique waves that appear in the resistive simulation or may simply be the aligned with ~B waves seen in the the spectra as a function of angle and the 1-D simulations. In 1-D, the ion acoustic waves appear to result from decay of electron hole potentials, as was shown by potential panel of Figure 9. This panel shows that after several hundred plasma periods the electron hole potentials seed the formation of ion acoustic Figure 11. Electron distribution function in the x or B k direction, as a function of time. This plot shows that the two-stream instability flattens the initial beam distribution within a few w 1 e. Over the next few hundred w 1 e the electron holes are shown to steepen the beam distribution and push it toward higher velocities. This effect continues until about t = 1000 when ion acoustic waves are launched which weaken the beam distribution. These two processes reach a steady state at t = 2200. 9of12

waves. This decay process appears to resonantly drive the ion acoustic waves at a particular ~ k. In the 2-D simulations we see an enhancement in ion acoustic energy at the electrostatic whistler angles. We will continue examining these complex driving mechanisms for the ion acoustic waves in electron hole simulations and expect to provide a more detailed description in the future. [34] Regardless of their role in mediating resistivity, the oblique ion waves have a strong effect on ion perpendicular temperature. Figure 7 shows the ion distribution functions at both the initialization and after the resistive phase for the resistive simulation. Taking the moments of the distribution functions shows that the ion T? increased by a factor of 2 and occurred in concert with the current reduction. [35] We summarize the processes leading to electron hole resistivity as follows. Beam simulations of electron hole evolution, in the presence of an ion background, show a dramatic reduction in electron current. The reduction in current occurs for cold beams but not equal temperature beams and is stronger in higher-dimension simulations. In both resistive and nonresistive simulations electron holes form and then continue to perturb the electron distribution function by preferentially scattering electrons in the beam direction. This antiresistivity in all simulations is mediated by weak forward propagating and strictly parallel ion acoustic waves. When the electron distribution forms a positive slope at the ion sound speed, the distributions become unstable to strong oblique ion acoustic waves. In 1-D these waves appear to be resonantly driven by the electron hole potentials; in 2-D they show enhancements at electrostatic whistler wave angles. These waves transfer electron beam momentum to the ions, reducing the current, and heating the ions in the perpendicular direction. 4.1. Electron Hole Resistivity in Space Plasmas [36] Since electron holes are observed in nearly every boundary region of the magnetosphere, the finding that electron holes are associated with a resistive environment may have important implications for understanding the physics of these regions. The following paragraphs discuss these implications for the auroral region, magnetopause, and magnetotail plasmasheet. [37] These results may have particular importance in the downward current region of the auroral ionosphere. Ergun et al. [2001] shows simultaneous FAST observations of electron holes, VLF saucers, and upward going electron beams which were energized by a parallel potential drop. When combined with the results from James [1976], these observations suggest that electron holes, parallel potential drops, and VLF saucers are generated within a short (10 km in altitude) region along a downward current flux tube. The simulation shown in Figures 5 and 6 show the self-consistent development of electron holes, electrostatic whistler waves, resistivity which could lead to the formation of hundreds of V potential drops, and finally significant perpendicular ion heating which, when combined with a parallel potential drop, could generate or contribute to the well known ion conic distributions. In the very least, this perpendicular ion heating will significantly obscure the interpretation of ion conic distributions as a pure result of parallel electric fields. The phenomena of wave emission and resistivity occur rapidly over about 1000 w 1 e. If we assume an auroral plasma frequency of w pe =10 5 s 1, then these events occur in 10 ms and for a beam velocity of 1000 10,000 km/s yields distances of 10 100 km. These phenomena could be generated simultaneously, within a short distance, and self-consistently by a cold downward electron current that encounters a warmer background electron population. Finally, the varying threshold conditions for this instability is consistent with the nonlinear voltage-current relationship observed by FAST in the downward current region [Elphic et al., 1998]. [38] Several papers discussing the relationship between parallel potential drops, electron holes, and VLF saucers have been recently published, with most focusing on the role of strong double layers [Ergun et al., 2003; Goldman et al., 2003; Andersson et al., 2002; Newman et al., 2002]. We conclude by noting that the electron distribution function early in the simulation shown in Figure 11, just after 100 plasma periods which is after two-stream thermalization, is similar to those distributions shown in Figures 3 and 4 in the work of Newman et al. [2002], of 1-D Vlasov simulations of electron beams emerging from a strong double layer in the auroral ionosphere. We suggest that cold electron beams from double layers should result in the same resistivity physics simulated here, which should modify and increase double layer potential drops. [39] Electron hole resistivity may also play a role in the auroral upward current region. While much less common, FAST has observed electron holes along auroral acceleration flux tubes [Ergun et al., 1998]. However, Mandrake et al. [2000] used simulations to show that electron holes form in upward auroral currents but become unstable as they impinge on increasing density gradients. Therefore electron holes in the upward current region could launch the ion waves shown here and contribute to local acceleration and smaller-scale auroral structure. However, the resistive processes do operate a ion timescales, so further work is required to test these ideas. [40] The resistivity in the simulations discussed above, which used auroral region type parameters (i.e., W e /w e = 2.5), also appears in simulations with weaker magnetic fields that mimic magnetospheric plasmas. A simulation with the same parameters shown in Figure 5, except that W e /w e = 0.5, shows essentially the same evolution and resistivity. The effective collision frequency as a function of time was also shown in Figure 6. The reduction in current for this simulation is equivalent to an effective collision rate of 0.00015 w e or 0.00015 collisions per w 1 e. To compare these results with recent simulations of current driven ion acoustic (CDIAW), we rewrite the Sagdeev formula for n resulting from CDIAW [Watt et al., 2002], n ¼ 0:01 w pi v beam T e : w e w pe c s T i [41] Applying this formula to the simulation parameters yields n = 0.01w e, which is over 10 times larger than simulated. However, this formula assumes T e T i, while our simulation has a T e varying from only 2 4 times larger than T i, and our simulation only has 1/2 the electrons traveling at v beam which means that only 1/2 the current exists. Further, the two-stream instability diverts much of the beam s initial free energy into electron hole formation ð1þ 10 of 12

long before ion acoustic waves appear. A more realistic comparison can be made with the recent simulations of Watt et al. [2002] for CDIAW in the magnetopause. For similar parameters but with only 1/2 the current, we achieve a collision rate about 30% as high as they published. In their simulations, the acoustic waves were driven unstable by a nonlinear evolution of the electron distribution. In our electron hole simulations, the holes themselves provide an additional and stronger perturbation to the distribution function, by steepening the electron beam. The authors would like to point out at this time that our simulations show that this electron hole resistivity smoothly yields to the Buneman instability as T i becomes sufficiently high or n beam exceeds approximately 80% of the total electron density. We expect to explore this aspect further, as there are interesting implications for magnetospheric plasmas. [42] The POLAR satellite has observed electron holes in both the cusp and magnetopause, and these simulations provide evidence that an associated anomalous resistivity could play an important role in magnetopause diffusion and contribute to cusp turbulence [Cattell et al., 2002; Franz et al., 1998]. Specifically, Drake et al. [2003] has published particle simulations of reconnection showing the formation of electron holes. Those simulations showed electron hole scatter and acceleration that are different from the organized resistivity due to ion acoustic waves shown in our simulations. Our results can be further extended to the tail plasmasheet. GEOTAIL was the first satellite to observe electron holes in the Earth s environment [Matsumoto et al., 1994; Miyake et al., 1998]. Our results indicate that that the substantial resistivity caused by electron holes is comparable to or larger than other CDI s already proposed for current sheet disruption and diffusion. 5. Summary [43] Phase-space electron holes are seen in simulations, laboratory plasmas, and many regions of the Earth s space environment. This paper presents plasma simulations and analysis of beam plasmas showing that the generation and decay of electron holes results in a reduction of electron current, implying a parallel resistivity. We shows that resistivity occurs in simulations where a cold electron beam is coincident with a warmer background plasma and is mediated by the generation of ion acoustic waves, which propagate obliquely to the magnetic field. The evolution of the resistive plasma simulations unfolds in the following manner. Initially, electron holes scatter electrons in the beam direction, steepening the electron beam distribution, providing free energy for ion acoustic waves that cause resistivity and strong ion heating perpendicular to ~B. While electron hole resistivity may play a role anywhere electron holes are observed, for auroral plasmas this resistivity can potentially support parallel electric fields as large as 2 mv/m. These results may help explain FAST satellite data that show simultaneous electron holes, parallel electric fields, and VLF saucers in the auroral downward current region. Since this effect also occurs in simulations with low magnetization, electron hole resistivity may lead to anomalous diffusion and transport at the magnetopause. These low magnetization simulations show effective collision rates of 0.0008 w pe, which is on par or larger than other instabilities that may operate at the magnetopause. [44] Acknowledgments. The authors would like to thank Licia Ray for editing many of the figures and Hans Pecseli, Andy Clark, and Harry Petschek for their helpful comments and suggestions. This research was supported by NASA (S00-GSRP-092) and NSF (ATM 9988976). 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