Particle-in-cell simulations of current shear-driven instabilities and the generation of broadband ELF fluctuations

Authors


Abstract

[1] Three-dimensional electromagnetic particle-in-cell (PIC) simulations are performed to study the properties of current shear-driven (CSD) instabilities, which are driven by the free energy stored in the inhomogeneous transverse magnetic field and associated with the transverse gradient of the field-aligned current. Current shear-driven instabilities are unstable in both a static field-aligned current equilibrium and the sheared field-aligned current sheet embedded in a transversely finite Alfvén wave. The simulation results demonstrate that the electromagnetic fluctuations generated by CSD instabilities have characteristics similar to the broadband ELF (BBELF) fluctuations observed in the topside auroral region and at higher altitudes. It is also shown that CSD instabilities have the capacity to accelerate ions transversely. Comparison of the PIC simulations with the satellite observations and three-dimensional, two-fluid MHD simulations supports CSD instabilities as potential candidates for the generation of BBELF fluctuations and the correlated transverse acceleration of ions.

1. Introduction

[2] A sheared field-aligned current sheet with a transverse scale near the electron inertial length λe = equation image has been shown to be unstable to the single tearing (ST) mode, the double tearing (DT) mode, and the current-advective shear-driven interchange (CASDI) mode [Seyler and Wu, 2001]. These instabilities relax the equilibrium to a lower magnetic energy state through the advection of the current or by allowing slippage between the flow and the magnetic field due to dispersion. The ST and DT modes are global modes whose scales are determined by the scale of the current sheet. The CASDI mode is a local mode for which the smallest scales are on the order of the ion gyroradius ρi = equation image/eB0. All modes are driven by the free energy associated with the transverse gradient of the field-aligned current. We refer to these modes as current shear-driven (CSD) instabilities. The DT mode and the CASDI mode have very similar properties. The distinction between the DT mode and the CASDI mode is subtle and deserves some explanation. The DT mode is a magnetic tearing-type instability in which two points having k · B = 0, called singular points, exist in the equilibrium profile [Schnack and Killeen, 1978; Pritchett et al., 1980; Seyler and Wu, 2001]. In the collisionless case, when these points are within a few λe, the perturbation fields and flows about these points are strongly coupled. The coordinated motion results in a displacement of the region between the singular points relative to the regions outside. This slippage is called tearing and is associated with spontaneous magnetic reconnection at the singular points. In this process the system relaxes to a lower magnetic energy state. When the two singular points are sufficiently well separated, the plasma motion in the vicinity of the singular points becomes decoupled and the double tearing mode degenerates into two interchange-type perturbations. Interchange modes are transverse fluid displacements that are strongly aligned with the equilibrium magnetic field. The magnetic shear causes the perturbations to be localized about the singular points. Thus in the limit of strongly localized interchange perturbations that do not couple to another singular point, the eigenfunctions are distinct, and we refer to the instability as the current-advective shear-driven interchange (CASDI) mode. This acronym is derived from the J advection term that appears in the parallel component of the generalized Ohm's law [see Seyler and Wu, 2001, equation (2)]. The mechanism of CSD instabilities is not restricted to a static field-aligned current equilibrium. The instabilities also occur in the sheared field-aligned current sheet associated with a transversely finite Alfvén wave. Although the transverse sheared flow due to E × B drift in the Alfvén wave reduces the growth rate of the CASDI mode, it is not totally stabilizing. The CASDI instability still exists, even in the presence of the flow shear [Wu and Seyler, 2003]. The results from three-dimensional, two-fluid MHD simulations show that the characteristics of the CASDI mode are similar to satellite observations of broadband ELF (BBELF) fluctuations [Seyler and Xu, 2003]. Our three-dimensional electromagnetic particle-in-cell (PIC) simulation results confirm most of these previous results and demonstrate the energy conversion from magnetic energy to transverse ion kinetic energy, which may at least partly account for the observed transverse acceleration of ions (TAI).

1.1. BBELF Observations

[3] BBELF fluctuations is the descriptive phrase for a phenomenon frequently found in the topside auroral F region and at higher altitudes. Several mV/m or larger, these fluctuations in the electric field have a power law spectrum in the observer reference frame extending from well below the local O+ ion cyclotron frequency to well above the H+ cyclotron frequency, which is a few hertz to a few kilohertz [Kintner et al., 2000]. This phenomenon has been observed by numerous satellites (Freja [Louarn et al., 1994; Stasiewicz et al., 2000a], FAST [Carlson et al., 1998; Chaston et al., 1999, 2003], CLUSTER [Wahlund et al., 2003]) and sounding rockets (AMICIST [Bonnell et al., 1996], SCIFER [Kintner et al., 1996], SIERRA [Klatt et al., 2005]). At lower frequencies (below the O+ cyclotron frequency), a magnetic component to the fluctuations is present [Louarn et al., 1994]. The magnetic component amplitude typically decreases with frequency faster than the electric field amplitude [André et al., 1998]. Attempts to measure the wavelength of BBELF fluctuations at higher frequencies have led to the conclusion that it is characterized by short wavelengths, in some cases the order of the O+ gyroradius [Bonnell et al., 1996; Kintner et al., 2000]. No structure at the cyclotron frequencies is observed in the electric field power spectrum [Kintner et al., 1996]. However, the substantial Doppler shift and consequent frequency spreading make this an ambiguous measurement. BBELF fluctuations are generally correlated with TAI [Lynch et al., 1996; Carlson et al., 1998; Chaston et al., 1999, 2003]. The SCIFER sounding rocket project has demonstrated a one-to-one correlation between TAI, BBELF electric fields, and reduced plasma density at 1400 km altitude in the prenoon (1000 MLT) cleft [Kintner et al., 1996]. Noting that the TAI is closely associated with field-aligned currents up to 20 μA/m2, Kintner et al. [1996] suggested that a current-driven instability like the electrostatic ion cyclotron (EIC) instability [Kindel and Kennel, 1971] might be operating. However, André et al. [1998] showed that the observed field-aligned currents were generally too weak to trigger the EIC mode. The observation that BBELF fluctuations are frequently found in regions of Alfvénic activity [Wahlund et al., 1998] has motivated studies seeking connections between BBELF and Alfvén waves. Stasiewicz et al. [2000a] showed that BBELF fluctuations observed by Freja had dispersive inertial Alfvénic features and were generally close to zero-frequency (spatial) frozen-in turbulence, Doppler-shifted by the satellite velocity. CLUSTER observed similar emissions at distances around 4–5 RE with the frequency dependence of the δE/δB ratio consistent with inertial Alfvén waves [Wahlund et al., 2003]. These results have led some to conclude that BBELF fluctuations are composed of small-scale inertial Alfvén waves that have been Doppler-shifted by spacecraft motion. However, SIERRA sounding rocket observations at about 700 km altitude in the Alfvénic auroral region recently showed that the small-scale inertial Alfvén wave interpretation was only partially correct [Klatt et al., 2005]. The spectral analysis of the phase difference between δE and δB revealed that for scale sizes near to or less than λe, the observed field fluctuations were electromagnetic and nonpropagating in the reference frame of the plasma. All of these general characteristics of observed BBELF fluctuations should be taken into account when interpreting the phenomenon.

1.2. Theoretical Explanations of BBELF

[4] Understanding the generation mechanism of BBELF fluctuations and its correlation to TAI has been an important problem in auroral physics. BBELF fluctuations are significant to auroral physics because they are the wave phenomenon most closely correlated with and presumably most responsible for TAI in the auroral ionosphere [André et al., 1998; Knudsen et al., 1998; Norqvist et al., 1998]. The generation mechanism of BBELF fluctuations is still not well understood, although there have been a number of theories: electrostatic ion cyclotron instability [Kindel and Kennel, 1971], plasma shear flow instability [Ganguli et al., 1985], field-aligned plasma flow instability [Gavrishchaka et al., 1998], drift Alfvén waves [Maggs and Morales, 1996], wave emission resulting from nonlinear one-dimensional steepening [Seyler and Wahlund, 1996], and CSD instabilities driven by the transverse shear in the field-aligned current [Seyler and Wu, 2001; Wu and Seyler, 2003]. All of these theories have merit and are motivated by one or more observational characteristics. However, conclusive evidence in support of one has not been found. In this article we present evidence in support of the CSD instabilities as potential candidates for the generation of BBELF fluctuations and the correlated TAI.

1.3. Organization of the Paper

[5] Section 2 describes the simulation model we use. Section 3 presents two cases of simulation results. In section 4 we discuss the comparison between simulation results and observations. The conclusions are summarized in section 5.

2. Methodology

[6] The simulations are performed using a three-dimensional electromagnetic PIC simulation model with periodic boundary conditions in all three dimensions. Readers may refer to Birdsall and Langdon [1985] for a detailed discussion of this approach. For the purpose of simulating CSD instabilities, we nondimensionalize the system, as in the work of Seyler and Xu [2003]. Time is normalized to the ion cyclotron time Ωi−1, and length is normalized to ion inertial length λi = equation image. The magnetic field is normalized to the z component of the background magnetic field B0, and the plasma density is normalized to the background density n0. Mass, charge, and velocity are normalized to ion mass mi, electron charge e, and Alfvén velocity VA, respectively. The equations after nondimensionalization are Faraday's law:

equation image

Ampere-Maxwell's law:

equation image

Newton-Lorentz equations of motion for ions:

equation image
equation image

and for electrons:

equation image
equation image

A similar initial equilibrium condition is then applied:

equation image
equation image

for 0 ≤ x < Lx.

[7] For simplicity, uniform density and temperature are imposed for both ions and electrons. Thus to maintain static equilibrium:

equation image

we initialize Bz using

equation image

which is produced by Jy:

equation image

Both Jz and Jy are carried by electrons. Therefore both ions and electrons are initially force-free from the two-fluid model perspective:

equation image

where j = i, e stands for ions and electrons, respectively.

[8] The quiet start technique [Byers, 1970] is used to initialize velocities and positions of the particles with a low thermal fluctuation level. We use a first-order weighting method to calculate current and charge densities and to interpolate electric and magnetic fields at particle positions. To suppress short wavelength numerical noise, a Hamming filter [Hamming, 1977] is applied three times when we calculate current and charge densities every time step. This procedure has the equivalent effect of using a higher-order weighting method but is computationally faster. To achieve better numerical performance, a staggered grid mesh system known as a Yee lattice [Yee, 1966] is employed, which enables all of the spatial derivatives to be calculated using the central finite differences with second-order accuracy. The time advance is accomplished using a leap-frog method, which allows time centering. The leap-frog method is both simple (easy to understand and with minimum storage) and second-order accurate. The particle accelerations are computed using a nonrelativistic version of the Boris scheme [Boris, 1970]. Furthermore, the code has been parallelized using MPI (Message Passing Interface) with domain decomposition in the x direction to run on the parallel computer cluster of the Cornell Theory Center.

[9] When different energy components of the system are calculated for diagnostic purposes, a common factor is neglected in order to achieve higher computation efficiency. The various expressions employed for different energy components are the kinetic energy of ions:

equation image

the kinetic energy of electrons:

equation image

electric field energy:

equation image

and magnetic field energy:

equation image

The meaning of symbols can be found in Table 1. Note that all of these components should be multiplied with the common coefficient NcmiVA2/2 for comparison with real energy units.

Table 1. Simulation Parameters for Case I
QuantityValue
System length in the x directionLx = 1λi
System length in the y directionLy = 1λi
System length in the z directionLz = 6.25λi
Number of grid points in the x directionNx = 64
Number of grid points in the y directionNy = 64
Number of grid points in the z directionNz = 32
Number of particles per cellNc = 100
Number of particles in the whole systemN = 13, 107, 200
Ion-to-electron mass ratiomi/me = 100
Light speed to Alfvén speed ratioc/VA = 5
Plasma constant β = μ0nkBT/B2β = 0.000625
Ion-to-electron temperature ratioTi/Te = 1
The amplitude factor of ByBy0 = 0.12
The width factor of the initial current Jzκ = 1.3
Time step lengthΔt = 0.00125Ωi−1

3. Simulation Results

[10] For many years the electrostatic ion cyclotron (EIC) mode has been the leading contender for generating BBELF fluctuations and the correlated TAI. It has the lowest threshold of all parallel electron drift instabilities and, through ion cyclotron damping, it transfers parallel electron drift energy into transverse ion kinetic energy [Kindel and Kennel, 1971; Okuda et al., 1981]. However, its likelihood has fallen into question in recent years. André et al. [1998] showed that the observed field-aligned currents were generally too weak to trigger the EIC mode. Although drifts exceeding the EIC threshold have been found [Kintner et al., 1996], they are relatively rare. No structure at the cyclotron frequencies is observed in the electric field power spectrum of BBELF fluctuations [Kintner et al., 1996; Chaston et al., 2006]. Perhaps most importantly, in the TAI regions of highly elevated transverse ion temperatures, the drift required for the EIC instability becomes too large. In any case, even when EIC is initially unstable, it would stabilize at transverse ion temperatures well below the observed values. The situation for CSD instabilities is very different. There is no obvious current instability threshold [Seyler and Wu, 2001]. Current shear-driven instabilities can still develop in the regions of highly elevated transverse ion temperatures only if there are transverse shears in the field-aligned current. We present two simulation cases in this paper, although we have performed numerous simulations. In case I, we carefully choose the simulation parameters to excite both EIC and CSD instabilities. EIC mode grows faster and dominates in the early stage. The consequent parallel electron drift damping and transverse ion heating reduce its growth rate. Current shear-driven instabilities catch up and dominate the fluctuations in the late stage. Dissipation of the free energy associated with the transverse gradient of the field-aligned current continuously heats ions. To better study the properties of CSD modes, case II simulates pure CSD instabilities with plasma parameters selected to resemble the plasma in TAI regions.

3.1. Case I: Combination of EIC and CSD Instabilities

[11] The parameters for case I are given in Table 1. There are 100 ions and 100 electrons distributed in each cell. These parameters satisfy the Courant condition (Δx, Δy, Δz > cΔt, where c is the light speed and Δx, Δy, Δz is the grid size in the x, y, z direction, respectively) and ensure that the system is in the inertial plasma regime (β < me/mi). As with most PIC simulations, both a small ion-to-electron mass ratio and a small light speed to Alfvén speed ratio are used to suppress the numerical noise and enable a longer time step. This should be kept in mind when interpreting the results.

[12] Figure 1 shows the initial sheared field-aligned current Jz and the corresponding magnetic field By. The transverse spatial scale is on the order of the electron inertial length λe. The peak current is 0.55 and corresponds to a peak parallel electron drift velocity at 0.55 in units of VA. For Ti = Te, the threshold for EIC mode is about 15cs, where cs = equation image is the ion acoustic speed [Kindel and Kennel, 1971]. In case I, cs = 0.018 and 15cs = 0.27. So the peak parallel electron drift velocity is well above the EIC instability threshold and thus the current sheet is unstable for the EIC mode near the current peak.

Figure 1.

Case I: The initial sheared field-aligned current Jz and the corresponding magnetic field By. (The solid line stands for Jz and the dashed line is By.)

[13] The unrealistic electron mass introduces a trivial factor into the simulation. The electron thermal velocity ve = equation image = 0.25 < 15cs makes the ion acoustic mode even more unstable than the EIC mode. However, since the simulation has k/k ≪ 1, this suppresses the ion acoustic mode artificially. For a realistic mass ratio, the electron thermal velocity will be much larger than the parallel electron drift velocity, and the ion acoustic mode is stable.

[14] Figure 2 displays a snapshot of the contours of the y component of electric field Ey, the x component of magnetic field Bx, electron density ne, and the z component of current Jz in the x-y cross section at nz = 16 and t = 10,000Δt when the EIC instability is growing. Vortical structures are established and are apparent in the contours of Ey, Bx, and ne. As expected, they are confined to the region aligned along the peak of the parallel electron drift since this is where the EIC mode is most unstable. In the later stages of the run, CSD instabilities dominate the fluctuations in the system. Another snapshot at t = 50,000Δt is shown in Figure 3 when CSD instabilities begin to dominate. Comparing Figure 3 to Figure 1, it is clear that these vortices are most prominent near the current shear extrema, as is expected for CSD instabilities since they are driven by the transverse gradient in the parallel current. To confirm that these vortices are the result of CSD instabilities, a snapshot in the y-z cross section at nx = 24 (the same time) is given by Figure 4, which shows the vortical structures are aligned parallel to the local magnetic field. The field-aligned nature is a hallmark of CSD instabilities. The same snapshot of the EIC instability at t = 10,000Δt does not have the field-aligned property.

Figure 2.

Case I: Snapshot of the contours of Ey, Bx, ne, and Jz in the x-y cross section at nz = 16 and t = 10,000Δt.

Figure 3.

Case I: Snapshot of the contours of Ey, Bx, ne, and Jz in the x-y cross section at nz = 16 and t = 50,000Δt.

Figure 4.

Case I: Snapshot of the contours of Ey, Bx, ne, and Jz in the y-z cross section at nx = 24 and t = 50,000Δt.

[15] In case I, λi = 10λe = 56.6ρi. Only the electron inertial length λe is well resolved in the transverse directions. The transverse dimension of the structures appearing in Figure 3 is comparable to the equilibrium scale. Therefore we believe the structures are the result of the double tearing mode. The even larger scale-enveloping vortices in the Bx panel indicate an emergence of the single tearing mode, which is clearer at later times (not shown).

[16] To clearly demonstrate the transition from an EIC-dominating stage to a CSD-dominating stage, a spectral analysis of Ey in different stages has been performed. The results are shown in Figure 5. The time series of Ey are taken at ny = 32 and nz = 16 for different nx. Then a spectral analysis is applied in two stages: from t = 0Δt to t = 50,000Δt and from t = 50,000Δt to t = 100,000Δt separately. The same scale is used to make the comparison straightforward. The top panel shows the early stage and the bottom panel displays the late stage. In the early stage, the spectrum exhibits the excitation of the first and second cyclotron harmonics in the central high electron drift region and the coexistence of low-frequency fluctuations generated by CSD instabilities in the high current shear regions. In the late stage the EIC fluctuations have abated and the CSD fluctuations dominate.

Figure 5.

Case I: Ey spectra for different stages: t = 0Δt to t = 50,000Δt (top) and t = 50,000Δt to t = 100,000Δt (bottom). The time series of Ey are taken at ny = 32 and nz = 16 for different nx.

[17] The energy evolution of the system is shown in Figure 6. Energy transfers from By mainly to the transverse kinetic energy of ions and partially to the electric field and magnetic field fluctuations (shown as EE and MEx in Figure 6). During the entire process, the transverse kinetic energy of ions increases by 25%, clearly indicating TAI. The total energy of the system is reasonably well conserved during the evolution. The increase of electron kinetic energy vanishes in case II when we reduce the electron temperature. The essential point is that both EIC and CSD instabilities have the capacity to heat ions transversely. The abrupt rise in the transverse ion kinetic energy around t = 10,000Δt is due to the EIC mode. The slower but steady increase thereafter is mainly due to the CSD instabilities. Another notable point is the abrupt increase of the electric field energy around t = 10,000Δt in the bottom left-hand panel (EE) of Figure 6. There is no obvious increase in the magnetic fluctuation energy associated with it, as shown in the top right-hand panel (MEx) of Figure 6, indicating that the fluctuations in the early stage are mainly electric field fluctuations with relatively negligible magnetic components, which agrees with the electrostatic nature of the EIC mode.

Figure 6.

Case I: The energy evolution of the system. KEi, KEe, KE, EE, MEx, MEy, MEz, ME, and TE represent the kinetic energy of ions, the kinetic energy of electrons, the total kinetic energy, the electric energy, the magnetic energy of Bx, the magnetic energy of By, the magnetic energy of Bz, the total magnetic energy, and the total energy of the system, respectively. In some panels, different colors are used to symbolize different components (red-x, green-y, blue-z, and black-total).

[18] Figure 7 displays Jz(x) profiles at various times. The current profiles are taken at nz = 16 and averaged over different ny. Figure 7 shows that the nature of the EIC instability at early times (from t = 0Δt to t = 20,000Δt) is to reduce the electron drift speed. At later times (after t = 20,000Δt) the CSD instabilities have the effect of broadening the current profile, which also lowers the equilibrium transverse magnetic energy in By, as shown in Figure 6. The transverse magnetic energy is the dominant source of energy for driving both the EIC and the CSD instabilities. To our knowledge, this is the first PIC simulation to allow the EIC mode to extract energy from the transverse magnetic field. In an electrostatic picture the energy source for the EIC instability is solely the parallel electron drift kinetic energy. The parallel electron drift kinetic energy is transferred to the electrostatic waves through inverse Landau damping which, in turn, is absorbed by the ions via cyclotron damping [Okuda et al., 1981]. In an electromagnetic picture, the transverse magnetic field and the field-aligned current are closely coupled. Even if we assume that the instability directly extracts energy from electron drift energy, the reduction of the electron drift energy, and consequently the current, will lower the magnetic energy. Both the parallel electron drift kinetic energy and the transverse magnetic energy are ultimately transferred to the instability. However, the parallel electron drift kinetic energy is relatively insignificant due to the small electron mass. The transverse magnetic energy is the dominant energy source. In the present simulation, the slow decay of By induces a parallel electric field that maintains the parallel current, thereby driving the EIC instability until either Jz has decreased below the threshold or Ti has increased to raise the EIC drift threshold.

Figure 7.

Case I: Jz(x) profiles at various times. The current profiles are taken at nz = 16 and averaged over different ny.

3.2. Case II: Pure CSD Instabilities

[19] Case II is set up to simulate pure CSD instabilities. Comparing with case I, we use a larger ion-to-electron temperature ratio, Ti/Te = 30, β = 0.005625, By0 = 0.152, and a smaller ion-to-electron mass ratio, mi/me = 25. High Ti/Te suppresses the EIC mode and is similar to the plasma in hot-ion BBELF regions. This arrangement allows us to demonstrate the characteristics of the CSD modes without EIC mode fluctuations and thus to directly compare the results with BBELF observations. For these parameters, λi = 5λe = 13.6ρi. Both the electron inertial length and the ion gyroradius are well resolved in the transverse directions and are marginally distinguishable. Colder electrons, along with a larger electron mass, help to reduce thermal noise and suppress the artificial heating of electrons in case I. Furthermore, larger me/mi and By0 increase the growth rate of CSD instabilities according to the growth rate formula given by Seyler and Wu [2001, equation (10)].

[20] Figure 8 shows a snapshot of the contours of Ey, Bx, ne, and Jz in the x-y cross section at nz = 16 and t = 6000Δt when the CSD instabilities are beginning to develop. Vortical structures emerge in the contours of Ey, Bx, and ne. The scale of these vortices is on the order of ion gyroradius ρi. We classify these vortices as the CASDI mode, since the structures are small scale and are consistent with the advection minimization of the current gradients. There are both density depletions and enhancements in the electron density perturbations, which are up to about 10% in this stage and even larger in later stages. The depletion amplitude is slightly larger than the enhancement amplitude, which is a common feature in most stages of the simulation. Once again, it is a natural consequence of the CASDI mode for these vortices to be most prominent near the current shear extrema.

Figure 8.

Case II: Snapshot of the contours of Ey, Bx, ne, and Jz in the x-y cross section at nz = 16 and t = 6000Δt.

[21] To visualize the three-dimensional structures of these vortices, Figure 9 displays the results in the y-z cross section at nx = 40 while Figure 10 shows a three-dimensional isosurface plot of the electron density at the same time. The light tubes correspond to density enhancements ne = 1.02 and the dark tubes correspond to density depletions ne = 0.98. The shading is chosen to enhance the three-dimensional perspective. It is obvious that the vortical structures are aligned parallel to the local magnetic field, indicative of a localized interchange instability in which the equilibrium magnetic field lines are minimally bent.

Figure 9.

Case II: Snapshot of the contours of Ey, Bx, ne, and Jz in the y-z cross section at nx = 40 and t = 6000Δt.

Figure 10.

Case II: Three-dimensional iso-surface plot of ne at t = 6000Δt. The light tubes correspond to density enhancements ne = 1.02 and the dark tubes correspond to density depletions ne = 0.98. The shading is chosen to enhance the three-dimensional perspective.

[22] The spectral ratio of the electric field fluctuations to the magnetic field fluctuations (∣δE/δB∣) as a function of kyλe in the nonlinear stage at t = 10,000Δt is shown in Figure 11, where ky is the wave number in the y direction. The simulation results agree with the theoretical curve for homogeneous inertial Alfvén waves derived from reduced MHD [Seyler and Xu, 2003] (after nondimensionalization):

equation image

where we neglect the (1 + k2ρs2) term since ρs = equation image/eB0 and TeTi. Equation (17) is the same as the result for dispersive Alfvén waves presented by Stasiewicz et al. [2000a]. Figure 12 displays the spectral ratio of the electric field fluctuations to the plasma density fluctuations (∣δE/δn∣) as a function of kyλe at the same time. The electron density ne is used as the plasma density n in the calculation because nne. The results also agree with the theoretical curve based upon the linear response relation derived from reduced MHD [Seyler and Xu, 2003] (after nondimensionalization):

equation image

Equation (18) reveals a clear ion polarization-ion Boltzmann response of the plasma density fluctuations to the electric field fluctuations. At wave numbers i < 1, the response is the usual Alfvénic ion polarization response. For i > 1, the response is ion Boltzmann.

Figure 11.

Case II: The log-log plot of the spectral ratio ∣δE/δB∣ versus kyλe at t = 10,000Δt. The solid line and dashed line are the theoretical curves with and without thermal correction (ρi = 0 in equation (17)), respectively. The dots are the simulation results.

Figure 12.

Case II: The log-log plot of the spectral ratio ∣δE/δn∣ versus kyλe at t = 10,000Δt. The solid line is the theoretical curve given by equation (18). The dots are the simulation results.

[23] Figure 13 presents the energy evolution of the system. Energy transfers from By and parallel kinetic energy of electrons, which are related through Ampere's law since the electrons carry the parallel current, mainly to the transverse kinetic energy of ions and partially to the electric field and magnetic field fluctuations (shown as EE and MEx in Figure 13). It is important to note that the fluctuations continue to transversely heat ions in spite of the very high initial ion-to-electron temperature ratio. During the entire process the transverse kinetic energy of ions increases by 5%. The artificial electron heating has disappeared in case II. The change in the total energy is negligible compared to the change in the transverse kinetic energy of ions, which suggests the validity of the simulation results. Similar features of the CASDI mode have been reported in a previous three-dimensional, two-fluid MHD simulation [Seyler and Xu, 2003], although the fluid approach cannot effectively investigate the kinetic properties of the CASDI mode. This agreement further supports the validity of the PIC simulation results.

Figure 13.

Case II: The energy evolution of the system. KEi, KEe, KE, EE, MEx, MEy, MEz, ME, and TE represent the kinetic energy of ions, the kinetic energy of electrons, the total kinetic energy, the electric energy, the magnetic energy of Bx, the magnetic energy of By, the magnetic energy of Bz, the total magnetic energy, and the total energy of the system, respectively. In some panels, different colors are used to symbolize different components (red-x, green-y, blue-z, and black-total).

4. Discussion

[24] No structure at the cyclotron frequencies is observed in the electric field power spectrum of BBELF fluctuations [Kintner et al., 1996; Chaston et al., 2006]. Studies by André et al. [1998] indicate that the observed field-aligned currents are generally too weak to excite the EIC mode. In some rare situations the field-aligned current observed in BBELF regions may be above the threshold of the EIC instability. The consequent damping of parallel electron drift and transverse ion heating will arrest its development, as demonstrated in case I. In strong BBELF regions where Ti is significantly elevated over Te [Bonnell et al., 1996; Knudsen and Wahlund, 1998], the threshold for EIC is likely to be too high for instability. Stasiewicz et al. [2000a] pointed out that the ion acoustic mode should also be excluded because of the high ion-to-electron temperature ratio in these regions. On the other hand, there is no current threshold for CSD instabilities [Seyler and Wu, 2001]. Current shear-driven instabilities can still develop in the regions of highly elevated transverse ion temperatures if there is transverse shear in the field-aligned current. This makes CSD modes strong candidates for the generation of BBELF fluctuations.

[25] Current shear-driven instabilities have also been studied by Farengo et al. [1985]. Farengo et al. [1985] performed an analysis in the electrostatic regime using a two-fluid model and found that the transverse gradient in the parallel current can drive unstable, low-frequency, localized modes. However, the local analysis of Farengo et al. [1985] shows a significant threshold for CSD instabilities [Farengo et al., 1985, equation (8)]. A similar result can be achieved from a more general kinetic dispersion relation (also electrostatic) given by equation (A20) of Ganguli et al. [1994]. While equation (8) of Seyler and Wu [2001] shows no sign of instability threshold, the electrostatic approximation by Seyler and Wu [2001, equation (19)] implies the same threshold as Farengo et al. [1985, equation (8)]. The discrepancy arises from the differences between the electrostatic approximation and a full electromagnetic analysis. Local CSD instability occurs at the singular points where k · B = 0 and then only if all electromagnetic terms are retained. The analysis of Seyler and Wu [2001] applies to the cold fluid limit only. A full electromagnetic and kinetic analysis is needed to unequivocally determine if there is a threshold for CSD instabilities. It is possible that hot ions, for example, may introduce such a threshold, although our limited results do not suggest that this is the case.

[26] The double tearing and CASDI modes tend to develop the largest electric, magnetic, and density fluctuations near the current shear extrema. Since the current shear is the second derivative of B, the extrema of B generally correspond to regions near the current shear extrema. Observations of Carlson et al. [1998] and Wahlund et al. [2003] showed that the BBELF (δE fluctuations and Poynting flux) were most prominent where the transverse magnetic field was at an extremum (Figure 1, first and ninth panels, of Carlson et al. [1998] and Figures 1d, 1e, and 1f between 1923 and 1927 UT of Wahlund et al. [2003]). This feature in the data is particularly important since it provides strong evidence, in our opinion, that the electric fluctuations are driven by shear in the field-aligned current. Another important feature of CSD instabilities is that they are purely growing instabilities, implying that the BBELF fluctuations they generate are spatial, nonpropagating structures. Stasiewicz et al. [2000a] found that BBELF fluctuations were generally close to zero-frequency (spatial) frozen-in turbulence, Doppler-shifted by the satellite velocity. Recent SIERRA observations confirmed that BBELF fluctuations with scales near to or less than λe were spatial perturbations embedded in the plasma [Klatt et al., 2005]. These are consistent with the purely growing nature of CSD instabilities. The other feature of the CSD instabilities revealed by the PIC simulations is that the characteristic scales of the vortical structures span from the scale of the current sheet ≥λe (case I: the double tearing mode) to ρi (case II: the CASDI mode), which also agrees with the observations [Bonnell et al., 1996; Kintner et al., 2000]. Furthermore, CSD instabilities are Alfvénic. The CASDI mode in particular has the Alfvénic δE/δB spectrum and the ion polarization-ion Boltzmann response δE/δn spectrum. This is consistent with the observations of BBELF discussed by Stasiewicz et al. [2000a], Wahlund et al. [2003], and Klatt et al. [2005]. Observations of an ion Boltzmann response in BBELF fluctuations were also reported by Wahlund et al. [1998]. Finally, as shown by the simulations (Figures 3 and 8), current filamentation is the hallmark of CSD modes. This property of BBELF fluctuations has been reported by Volwerk et al. [1996], Chaston et al. [1999], and Stasiewicz et al. [2000c].

[27] Another important point is that the mechanism of CSD instabilities is not restricted to a static field-aligned current equilibrium. The instabilities also occur in the sheared field-aligned current sheet embedded in a transversely finite Alfvén wave. Although the transverse sheared flow due to E × B drift in the Alfvén wave reduces the growth rate of the CASDI mode, it does not totally stabilize it. The CASDI instability still exists even in the presence of the flow shear [Wu and Seyler, 2003]. This may explain why Louarn et al. [1994] observed both “quasi-static magnetic fluctuations” interpreted as “the magnetic signatures of localized stationary currents, Doppler-shifted by the S/C velocity” and “strong nonlinear electromagnetic fluctuations” interpreted as nonlinear development of kinetic Alfvén waves. André et al. [1998] also showed that BBELF fluctuations were associated with Alfvén waves with frequencies of up to at least a few hertz, and with field-aligned currents. A number of other observations show that BBELF fluctuations are usually associated with Alfvén waves [Louarn et al., 1994; Wahlund et al., 1994; Seyler and Wahlund, 1996; Seyler et al., 1998], explaining BBELF fluctuations as the nonlinear development of Alfvén waves. The multiple-payload observations of SIERRA showed that there was a transition from propagating Alfvén waves of spatial scales much larger than λe to nonpropagating perturbations, embedded in the plasma, of scales near to or less than λe [Klatt et al., 2005]. These measurements lead us to the following scenario: the sheared field-aligned current sheet embedded in a propagating inertial Alfvén wave is unstable to CSD instabilities, and the development of CSD instabilities results in BBELF fluctuations. The purely growing feature of the CSD instabilities determines that BBELF fluctuations are spatial structures embedded in the plasma. This explains the transition from propagating to nonpropagating structures reported by Klatt et al. [2005]. Génot et al. [2001, 2004] performed two-dimensional electromagnetic guiding center PIC simulations to investigate the propagation of an Alfvén wave in the perpendicular density gradients that characterize the edges of the auroral density cavities. The Génot et al. [2001, 2004] simulations demonstrate a possible means of producing small-scale oblique inertial Alfvén waves that are subject to CSD instabilities. However, their results do not show clear signs of CSD instabilities and the consequent TAI. In addition, Ishiguro et al. [1997] conducted two-and-a-half-dimensional electrostatic PIC simulations using a bell-shaped electron flow. Their results showed the generation of EIC modes but no sign of CSD instabilities either. This discrepancy, we believe, occurs because CSD instabilities are three-dimensional and the third dimension is essential for instability. Furthermore, considering that observations suggest significant parallel inhomogeneity of less than an Alfvén wavelength in the auroral region, a convincing simulation to demonstrate the presented scenario must be three-dimensional and requires more realistic boundary conditions. A three-dimensional fluid simulation using a reduced MHD model to explore the relation between CSD instabilities and SIERRA observations, with emitting top boundary and absorbing bottom boundary, will be reported in a separate paper. A three-dimensional PIC simulation with more complicated boundary conditions beyond simple periodic ones would also be useful. Particle-in-cell simulations of obliquely propagating Alfvén waves with periodic boundary conditions have been done and our preliminary results confirm that propagating Alfvén waves are subject to CSD instabilities. This work will be reported in a future publication.

[28] The transverse sheared flow due to E × B drift in the Alfvén wave can also drive a variety of instabilities and possibly contribute to the generation of BBELF fluctuations. Ganguli et al. [1985] showed that transverse sheared flow can drive EIC instability through the inhomogeneous energy-density driven (IEDD) mechanism [Gavrishchaka et al., 1996]. Peñano and Ganguli [1999, 2000, 2002a] extended the study to the electromagnetic regime and found that the transverse sheared flow can also drive unstable eigenmodes at subcyclotron frequencies. Peñano and Ganguli [2002b] further confirmed that electromagnetic ion cyclotron eigenmodes can be generated as well. It has been suggested that the resonant interaction between the particles and transverse sheared flow-associated instabilities can lead to parallel acceleration of electrons [Peñano and Ganguli, 2000] and TAI [Peñano and Ganguli, 2002b]. While previous study suggests that both CSD instabilities and transverse sheared flow-associated instabilities can develop in an inertial Alfvén wave [Wu and Seyler, 2003], further investigations are needed to better understand the relation between the two categories of instabilities. The major purpose of the present work is to show that CSD instabilities are potential candidates in generating BBELF fluctuations and the associated TAI.

[29] In addition to transverse sheared flow-associated instabilities, Ganguli et al. [2002] showed that a transverse velocity gradient in the parallel ion flow significantly lowers the threshold current for the current-driven ion acoustic instability and can give rise to a new class of ion cyclotron waves via inverse cyclotron damping. They argued that the ion flow shear dominates under the assumption that ion flow shear is equal to the electron flow shear. We agree that sheared field-aligned ion flow can be more important than electron flow if the ion flow and the shear exceed the appropriate thresholds. However, for currents arising strictly from parallel electric fields, the electrons will flow faster than the ions by a factor of equation image and the electron flow shear will clearly dominate, especially if there is no threshold for CSD instabilities. Furthermore, ion conics arising from transverse ion heating cannot be the result of sheared parallel ion flow instabilities for obvious reasons. However, sheared parallel ion flow instabilities are likely to be important in regions where parallel ion flows result from folded ion distributions, which occur at altitudes above the TAI region.

[30] The broadband fluctuations generated by CSD instabilities can accelerate ions transversely as found in the simulations. Numerous mechanisms have been suggested to explain TAI [André and Yau, 1997; André et al., 1998; Stasiewicz et al., 2000b], most of which can be classified as basic gyroresonant heating by different wave modes. However, as we have presented, the electromagnetic fluctuations generated by CSD instabilities are spatial nonpropagating structures. Thus cyclotron damping mechanisms do not apply in this low-frequency regime. Stasiewicz et al. [2000b] proposed a mechanism of stochastic ion acceleration within spatial turbulence having a threshold:

equation image

In our system of nondimensionalization the expression becomes simpler: χ = equation imagekE ≥ 1. Figure 13 shows that the system begins to produce TAI at around 6000Δt. Using the spatial scale of ρi = 1/13.6λi and averaged amplitude about 0.013 of Ey at t = 6000Δt from Figure 8, we get kE = 1.1. The “coincidence” of the onset of TAI with the stochastic ion acceleration threshold suggests stochastic acceleration may play a role here. Another possible explanation is that the vortical electric field structures can cause vortical ion flow on the order of ρi through the E × B drift. Ion flow vortices on the order of ρi may dissipate and produce TAI. The study here does not investigate the exact mechanism of TAI. This issue needs to be addressed in future studies.

[31] It is generally accepted that the bulk of ion outflows are the result of some mechanism occurring at an altitude where upwelling ions have been lifted from the lower ionosphere due to ion drag and Joule heating [Strangeway et al., 2005]. Most believe that BBELF in some form is primarily responsible for the main energization process [André et al., 1998; Kintner et al., 2000; Strangeway et al., 2005]. We propose that electromagnetic fluctuations due to CSD instabilities are an important component of BBELF and, as we have demonstrated, these fluctuations predominately energize ions in the transverse plane. We do not suggest or even believe that CSD processes are unique in this regard. Other mechanisms do exist (e.g., EIC instability, flow shear instabilities), and these are likely to be important under some plasma conditions or possibly most conditions. However, an important point that deserves reemphasis is that CSD instabilities do not have a threshold for instability, at least no one we have identified in our numerous past studies. Therefore the process appears to be universal since all that is required is a transverse current structure near the electron inertial scale to enable reasonable growth times and a sufficiently energetic source of magnetic energy to relax into transverse ion energy. On the basis of this understanding of CSD instabilities, we expect that regions having enhanced Alfvén wave Poynting flux with either structure near λe or density structure that creates structure near λe [Chaston et al., 2006] will also have BBELF and elevated ion temperatures.

[32] For CSD instabilities, the transverse magnetic energy is the dominant energy source for TAI. This implies a straightforward criterion to achieve significant TAI: Bequation image/B0equation image. Essentially, this criterion requires enough transverse magnetic energy relative to the ion thermal energy. In BBELF regions where TiTe and then ββi, the criterion is equivalent to B/B0equation image. In the case of a sheared field-aligned current sheet associated with a transversely finite Alfvén wave, we use E/BVA for simplicity. The criterion becomes

equation image

Chaston et al. [2004] determined an analogous threshold for significant stochastic ion acceleration for dispersive Alfvén waves: E/B0 > Ωi/k, where the required k was a significant fraction of 2π/ρi. However, equation (20) is not a necessary condition for significant TAI. If the region is continuously showered with Alfvén waves, more than one wave will contribute to the generation of BBELF fluctuations. Persistent Alfvénic Poynting flux and subsequent decay by a CSD mechanism could allow ions to reach a high temperature. For this reason, we also do not expect to see a precise correlation of BBELF activity with the current shear from a single particular wave, as pointed out by Klatt et al. [2005].

[33] Both density depletions and enhancements with transverse scales extending from the electron inertial length to the ion gyroradius are observed in the simulations. They are up to a few tens percent and the depletion amplitude is slightly larger than the enhancement amplitude in most stages of the simulations. The cause of density cavities, a subject of considerable discussion, has been variously suggested as being nonlinear vortices [Chmyrev et al., 1988], solitary kinetic Alfvén waves [Seyler et al., 1995], Alfvén cones [Stasiewicz et al., 1997], or Alfvén wave ponderomotive force [Bellan and Stasiewicz, 1998]. In the presented simulations, the density fluctuations are the ion polarization-ion Boltzmann response of the plasma density to the electric field fluctuations generated by CSD instabilities. The response is governed by equation (18) as shown in Figure 12. At wave numbers i < 1, the response is the usual Alfvénic ion polarization response. For i > 1, the response is ion Boltzmann.

[34] Recently, Knudsen et al. [2004] suggested that the lower hybrid cavity density depletions observed in space could be explained by ion heating localized on the scale of an ion gyroradius because of heated-gyroradius-scale excursions of the ions away from the heating region. Their model generates density depletions on the order of 10% for a twofold increase in Ti over the background value. This mechanism may explain why the depletion amplitude generally is slightly larger than the enhancement amplitude in our simulations. In case II, TAI mainly happens in the regions where the CASDI instability is most prominent. Their scale is on the order of ρi in the x direction (Figure 8). The TAI regions occupy about one fifth of the total system volume. Therefore an overall 5% increase of the transverse kinetic energy of ions indicates a 25% increase of the perpendicular ion temperature in the TAI regions. This in turn can cause a several percent density reduction of the whole TAI region according to Knudsen et al. [2004]. Thus the plasma density in both density depletions and enhancements in the TAI regions decreases by several percent. The amplitude difference between density depletions and density enhancements will be twice as large as this value.

5. Summary and Conclusions

[35] A sheared field-aligned current sheet with a transverse scale near the electron inertial length λe is unstable to the single tearing mode, the double tearing mode, and the current-advective shear-driven interchange (CASDI) mode, which are closely related. These related modes can occur in an equilibrium with or without a transverse sheared flow. The single tearing and double tearing modes are global modes and their scales are determined by the scale of the current sheet. The CASDI mode is a local mode and the results indicate a scale on the order of the ion gyroradius ρi. However, a kinetic stability calculation that could definitively answer this question has not been done. The double tearing mode and the CASDI mode have similar properties and both are capable of transverse ion heating.

[36] Although the EIC instability may contribute to TAI in some situations, the consequent damping of parallel electron drift and transverse ion heating will arrest its development. In strong BBELF regions where TiTe, the EIC mode is likely to be stable. Since there is no current threshold for CSD instabilities, the CSD modes can still develop in the regions of highly elevated transverse ion temperatures, subject only to the existence of transverse shears in the field-aligned current. Thus the CSD modes are strong candidates for the generation of BBELF fluctuations.

[37] The CSD instabilities develop fastest near current shear extrema. They relax the equilibrium to a lower magnetic energy state through the advection of the current or by allowing flow slippage through the dispersive effect of λe. This leads to local magnetic field-aligned vortices in Ey, Bx, ne, and current filamentation. The evolution of the CSD instabilities leads to both density enhancements and depletions up to a few tens percent with transverse scales spanning from λe to ρi. Usually, the depletion amplitude is slightly greater than the enhancement amplitude. The CSD instabilities transfer the transverse magnetic energy mainly to the transverse kinetic energy of ions and partially to the electric field and magnetic field fluctuations. The purely growing nature of the CSD modes determines that the electromagnetic fluctuations generated by the CSD modes are mainly nonpropagating, purely growing spatial structures embedded in the plasma. The δE/δB spectrum of the CASDI mode-generated electromagnetic fluctuations is the same as homogenous dispersive Alfvén waves and agrees with BBELF observations. The δE/δn spectrum has an ion polarization response for i < 1 and an ion Boltzmann response for i > 1.

[38] The increase in the transverse kinetic energy of ions observed in the simulations may account for TAI. More importantly, the CSD instabilities are not restricted to static field-aligned current equilibria. They occur in the sheared field-aligned current sheet produced by a transversely finite Alfvén wave. This makes significant ion heating by relatively weak Alfvén waves possible if the region is subjected to Alfvénic Poynting flux for a substantially long period of time.

[39] To our knowledge, these are the first reported results of PIC simulations performed to study the properties of CSD instabilities. The results agree with the previous three-dimensional, two-fluid MHD simulation but additionally demonstrate energy conversion from magnetic energy to transverse ion kinetic energy. The similarity between the observed characteristics of BBELF fluctuations and properties of CSD instability-caused electromagnetic fluctuations, as discussed in section 4, leads us to conclude that the mechanism of CSD instabilities is a potential generation mechanism of BBELF fluctuations and the correlated TAI.

Acknowledgments

[40] The authors acknowledge useful discussions with Eric Klatt. This work was supported by National Science Foundation grant ATM-0207260 and National Aeronautics and Space Administration grant NAG5-12991. This research was also conducted using the resources of the Cornell Theory Center, which receives funding from Cornell University, New York State, federal agencies, foundations, and corporate partners.

[41] Amitava Bhattacharjee thanks Christopher C. Chaston and Gurudas Ganguli for their assistance in evaluating this paper.

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