EXPANSION ONSETS
A.T.Y. LUI
The Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA
Abstract.Space plasmas present intriguing and challenging puzzles to the space community. Energy accessible to excite instabilities exists in a variety of forms, particularly for the magnetospheric environment prior to substorm expansion onsets. A general consensus of the pre-expansion mag- netosphere is the development of a thin current sheet in the near-Earth magnetosphere. This review starts with a short account of the two major substorm paradigms. Highlights of some observations pertaining to the consideration of potential plasma instabilities for substorm expansion are given.
Since a common thread of these paradigms is the development of a thin current sheet, several efforts to model analytically a thin current sheet configuration are described. This leads to a review on the instability analyses of several prominent candidates for the physical process responsible for substorm expansion onset. The potential instabilities expounded in this review include the cross-field current, lower-hybrid-drift, drift kink/sausage, current driven Alfvénic, Kelvin-Helmholtz, tearing, and entropy anti-diffusion instabilities. Some recent results from plasma simulations relevant to the investigation of these plasma instabilities are shown. Although some of these instabilities are generally conceived to be excited in spatially localized regions in the magnetosphere, their potentials in yielding global consequences are also explored.
Table of Contents 1. Introduction
2. Two Major Substorm Paradigms 2.1 Near-Earth Initiation 2.2 Mid-tail Initiation
2.3 Comparison on the Paradigm Predictions
3. Observations Guiding Plasma Instability Development 3.1 Near-Earth Substorm Observations
3.2 Mid-tail Substorm Observations 4. Analytic Current Sheet Models
4.1 Harris Current Sheet 4.2 Yoon-Lui Current Sheet 4.3 Kan Current Sheet
4.4 Schindler-Birn Current Sheet 4.5 Cowley Current Sheet
4.6 Lembége-Pellat Current Sheet 4.7 Forced Current Sheet
5. Plasma Instabilities
Space Science Reviews 113: 127–206, 2004.
© 2004Kluwer Academic Publishers. Printed in the Netherlands.
5.1 Cross-field Current Instability 5.2 Lower-hybrid-drift Instability 5.3 Drift Kink/Sausage Instability 5.4 Current-driven Alfvénic Instability 5.5 Kelvin-Helmholtz Instability 5.6 Tearing Instability
5.7 Entropy Anti-Diffusion Instability 5.8 Other Instabilities
6. Numerical Simulations
6.1 Cross-field Current Instability
6.2 Lower-hybrid-drift, Drift Kink/Sausage Instabilities 6.3 Entropy Anti-Diffusion Instability
7. Role of Plasma Instabilities in Global Substorm Dynamics 7.1 Equilibrium Transition from Current Disruption 7.2 Substorm Current System
8. Summary and Concluding Remarks
1. Introduction
The importance of plasma physics in understanding space phenomena can hardly be overemphasized in view of the fact that over 99.9% of space consists of plas- mas in various states and parameter regimes. Many impulsive energetic processes occur frequently in space plasmas, challenging us to decipher Nature’s secrets in expedient means to tap energies from plasma reservoirs in the universe. Space indeed constitutes an indispensable natural laboratory to probe exotic processes unfathomable on the Earth’s surface.
An outstanding question in the space plasma community is the cause of mag- netospheric substorms (Akasofu, 2004). Substorms typically exhibit a sequence of well-documented disturbances in the ionosphere and the magnetosphere. For instance, the characteristic evolution of global auroral pattern during substorms has been known since the mid-1960’s (Akasofu, 1964). In fact, the auroral behavior documented then led to the inception of the substorm concept and identified it as a distinct class of disturbances in the near-Earth space environment. Similar char- acteristic substorm changes in the magnetosphere soon followed (Akasofu, 1968).
The period of substorm disturbance was first sorted into two phases of activity, namely, expansion and recovery, which reflect the poleward advance and retreat of auroral disturbance in the polar regions. A third phase, the growth phase, was later introduced to indicate the progressive equatorward movement of the auroral oval and magnetospheric reconfiguration toward an elevated energy state prior to the sudden onset of some substorm expansions (McPherron, 1970; McPherronet al., 1973). Since substorms can occur without preceded by this reconfiguration, the
growth phase is not universally accepted as a necessary phase for a substorm (e.g., Meng and Liou, 2004; Parks, 2004).
Since these early works on substorm based primarily on ground-based observa- tions, there have been extensive investigations of substorm phenomena supplemen- ted in a major way by space-borne measurements as well. In addition, global nu- merical simulations of the magnetosphere-ionosphere system have been performed and are apparently successful to reproduce substorm-like disturbances during some specific events (e.g., Fedder et al., 1995; Winglee et al., 1998; Ashour-Abdalla et al., 1999; Raederet al., 2001; Slinker et al., 2001; El-Alaoui, 2001). Numer- ical simulations have also been used to test and substantiate a proposed substorm mechanism (Leeet al., 1995, 1998).
At this stage of substorm research, it is reasonable to state that ionospheric and magnetospheric disturbances considered to be key substorm phenomena have been identified with general consensus (Akasofu, 2004; Meng and Liou, 2004), although there are still confusion in the proper identification of the storage/release aspect of this magnetospheric disturbance as recently discussed by Rostoker (2002) and on the exact causal sequence of substorm phenomena (Meng and Liou, 2004).
Given the key substorm phenomena are now identified, it is timely to review some prominent substorm onset mechanisms.
The main objective of this article is to review some major plasma instabilit- ies proposed for the onset of substorm expansion. Magnetic reconnection is not included here because it is not a plasma instability although its occurrence may vitally depend on one or more plasma instabilities. In other words, magnetic recon- nection is a physical process that may result from the development of some plasma instabilities. For the background leading to plasma instability discussion, we shall elaborate on the two primary substorm onset scenarios in Section 2. Observations pertaining to these two scenarios are highlighted in Section 3 with emphasis on those guiding the formulation of potential plasma instabilities. A common feature of these two scenarios is the development of a thin current sheet in the magneto- tail prior to the sudden onset of expansion activity. Theoretical consideration of a thin current sheet configuration, a prerequisite for plasma instability analysis, is thus discussed in Section 4. A brief review of several candidates of plasma instabilities for substorm expansion onset is presented in Section 5. This section intentionally excludes the ballooning instability that is covered by Cheng (2004) in the accompanying article of this issue. Section 6 presents numerical simulations on the evolution of some potential instabilities. While most plasma instabilities discussed in this article are kinetic in nature and are thus spatially localized, their onsets could lead to global consequences, as explored in Section 7.
To a large extent, this review is based on the author’s exposure to the subject and may thus unjustifiably emphasize on plasma instabilities investigated by the author and his collaborators. Nevertheless, other plasma instabilities are also covered and the associated references are cited so that the reader can pursue further examination of these other plasma instabilities as well.
2. Two Major Substorm Paradigms
Although there are many proposed candidates for the substorm expansion onset process, they can be generally classified into two major scenarios based on the magnetospheric location of the process responsible for the initiation of substorm expansion. The first scenario elaborated in Section 2.1 may be referred to as the near-Earth initiation scenario with the process located in the downstream distance of ∼6-15 RE. The other scenario elaborated in Section 2.2 may be referred to as the mid-tail initiation, with the process located in the downstream distance of
∼15–30 RE. These two scenarios have both similarities and differences on the temporal evolution and characteristics of substorm phenomena. A comparison on the predictions of these two scenarios is provided in Section 2.3 to highlight these similarities and differences so as to guide future works in distinguishing these two alternatives. It must be borne in mind that there is a possibility that both scenarios can occur (e.g., Lui, 1992) and which scenario is appropriate for a given substorm could depend on the external solar wind condition as well as the internal state of the magnetosphere just prior to the time of substorm expansion. The latter factor may well represent the hysteresis of the system encompassing active components of the solar wind, the magnetosphere, and the ionosphere. This is in line with the synthesis viewpoint expressed by several substorm researchers (e.g., Lui, 1991;
Erickson, 1995; Akasofu, 2004).
2.1. NEAR-EARTHINITIATION
This scenario, elaborated as the synthesis substorm model (Lui, 1991; Erickson, 1995), invokes some physical process acting close to the Earth, as depicted in Figure 1. Several potential mechanisms are proposed in the literature under this scenario and most of them deal with disrupting the cross-tail current in the near- Earth region to set up the substorm current wedge, a feature considered to be essential for any credible substorm mechanism. (This perception turns out to be not entirely correct, as elaborated by Akasofu (2003, 2004) and in Section 7.2).
The substorm expansion phase in this scenario commences with a plasma process initiated on magnetic field lines linked typically to the most equatorward auroral arc (Akasofu, 1964; Rouxet al., 1991; Lui and Murphree, 1998; Perrautet al., 2003).
Plasma instabilities suggested for this onset process will be further discussed in Section 5.
A major effect of this process is to cause the stretched magnetic field lines in the disturbance region to relax. As a consequence, the magnetic field configur- ation earthward of the disturbance site becomes dipolar-like, accompanied sim- ultaneously by plasma sheet expansion. Tailward of the activity site, the current disruption process instigates further current disruption in adjacent locations of the magnetotail by momentarily thinning the plasma sheet and enhancing the cross-tail current density nearby. This causes current disruption and dipolarization to proceed
Figure 1.A schematic diagram to illustrate near-Earth substorm initiation scenario. The substorm onset process is visualized to start current disruption in the near-Earth region, which then instigates other regions in the tail for more current disruption and eventual onset of magnetic reconnection.
Signal on occurrence of current disruption is conveyed to other magnetospheric regions via rarefac- tion waves set off by plasma depletion in the current disruption site due to plasma flows generated by force imbalance at the site.
progressively down the magnetotail (Jacquey et al., 1991, 1993; Ohtani et al., 1992a; Perrautet al., 2003), corresponding to the poleward advance of the auroral bulge in the ionosphere (Liou et al., 2002). This tailward progression of current disruption leads to the development of the substorm current system appearing in the ionosphere at progressively higher latitude as observations indicate. One of the current disruption sites may eventually evolve to enable magnetic reconnection in the mid-tail region to occur, producing a significant plasma sheet expansion.
There are some global magnetohydrodynamics (MHD) simulations that yield res- ults consistent with this scenario. For example, Tanaka (2000) simulated substorm onset by using a highly precise numerical scheme with a state-of-the-art ionosphere model and found near-Earth dipolarization rather than mid-tail magnetic recon- nection onset to be the cause of substorms. El-Alaoui (2001) simulated an actual substorm event on November 24, 1996 with a global MHD code and found current disruption in the near-Earth region to trigger the subsequent substorm disturbances.
Furthermore, Luet al.(1999) have modeled the magnetic field changes during three substorms using observed values at multi-points within the magnetosphere. Their result indicates the development of a thin current sheet in a narrow region around X∼ −7.5REat the end of the growth phase, consistent with this scenario.
2.2. MID-TAILINITIATION
The scenario for mid-tail initiation stems essentially from the idea that magnetic reconnection is the key process responsible for substorm onset. The early idea of putting the reconnection site in the near-Earth region (e.g., Nishida and Hones, 1974) has been abandoned due to the lack of expected reconnection signatures in that region (Luiet al., 1977, 1999; Maynardet al., 1996; Erickson et al., 2000).
Figure 2.A schematic diagram to illustrate mid-tail substorm initiation scenario. Magnetic reconnec- tion starts the whole sequence of events, from generating high-speed plasma flows in the magnetotail, to braking of the flows in the near-Earth region to set up the substorm current wedge.
The latest version of this idea, shown in Figure 2, is to have reconnection take place typically at the downstream distance of 20–30RE, a revision to modify the model to be consistent with recent Geotail and Cluster observations (Nagaiet al., 1998; Nakamuraet al., 2002; Runovet al., 2003).
In this scenario, the link of mid-tail activity to near-Earth disturbances is made through fast plasma flow resulting from magnetic reconnection (Haerendel, 1992, 2000; Shiokawaet al., 1997, 1998). The plasma flow carrying magnetic flux from the mid-tail is slowed down by the strong magnetic field and high plasma pres- sure in the near-Earth region, producing an eastward inertial current to drive the substorm current wedge. Later MHD simulation shows that the braking scenario accounts for only the initial buildup of the substorm current wedge. A sustained buildup requires an enhancement of the north-south pressure gradient from the accumulated effect of magnetic flux and plasma pileup in the central plasma sheet (Birnet al., 1999). Several global MHD simulations yield results consistent with this scenario, which requires mid-tail magnetic reconnection occurring a few minutes earlier than the substorm onset time on the ground to allow for the time delay in propagating disturbances from the mid-tail to the near-Earth region (Baker et al., 1996).
2.3. COMPARISON ON THEPARADIGMPREDICTIONS
There are significant similarities and differences between the above two major paradigms that are useful to consider for their evaluation. Both paradigms have magnetic reconnection occurring during substorms so that the appearance of mag- netic reconnection signatures during substorms is not a distinguishing element.
Tailward progression of dipolarization in multiple activity sites is also a shared feature of these two paradigms. In fact, a lot of substorm phenomena (e.g., sub- storm injection, particle acceleration, substorm current wedge, dipolarization) can equally be interpreted in terms of either paradigm. Therefore, it will be inaccurate to claim one scenario can account for more substorm phenomena than the other.
There are some differences between these two scenarios that may be used to distinguish these two possibilities. One obvious difference is on the onset time of magnetic reconnection with respect to substorm expansion onset time. The near- Earth initiation paradigm suggests magnetic reconnection occurring after substorm onset while the mid-tail initiation paradigm suggests it to commence a few minutes before substorm onset. Based on this difference in prediction, some studies repor- ted observations favoring the latter paradigm. For example, Baker et al. (2002) reported a case study of a substorm with Cluster, IMAGE, and geostationary satel- lites. They surmised that Cluster observations indicated magnetic reconnection started ∼7 min before the ground signature of substorm expansion onset. Miy- ashitaet al. (2000) conducted statistical study of timing plasma flow onset with substorm expansion onset to show that plasma flows in the mid-tail occurred prior to substorm expansion onset. Ohtaniet al. (1999) showed a case study of magnetic reconnection signature at the downstream distance of∼30REcommencing∼3 min prior to substorm onset. On the other hand, Lui et al. (1998) examined plasma flow onsets in the magnetotail based on timing the substorm onset with global auroral images. They found the results to be consistent with the near-Earth initi- ation. Furthermore, Liouet al. (1999, 2000, 2002) conducted extensive systematic studies of several substorm onset identifiers and found the results to favor the near- Earth initiation scenario, as discussed in more detail by Meng and Liou (2004).
Therefore, there are observational supports for both paradigms based on the timing approach.
The timing of activity onset to differentiate these two paradigms turns out to be difficult in practice because the timing difference is small (a few minutes at most) and is plagued by the localized nature of activity (Luiet al., 1992, 1998; Angelo- pouloset al., 1996; Ohtaniet al., 1998; Franket al., 2001a,b) and the occurrence of pseudobreakups that may produce substorm-like signatures (Parkset al., 2001;
Fillingim et al., 2001), causing confusion in matching activity onset in the tail with substorm expansion onset. Mid-tail activity occurring a few minutes prior to ground substorm onset as reported by Ohtaniet al. (1999) and Bakeret al. (2002) may well correspond to pseudo-breakup activity prior to substorm expansion onset.
Only by extremely meticulous studies, such as those by Liouet al. (1999, 2000, 2002), could the subtle timing difference be evaluated confidently (Meng and Liou, 2004).
There are also other ways of differentiating the two paradigms that one can ex- plore. For instance, if dipolarization is caused by magnetic flux pileup from plasma flow, then the flow must be convective and satisfies the frozen-in-field approxima- tion, i.e.,E= −v×Bduring dipolarization. Furthermore, since dipolarization can be seen as deep as the geostationary orbit, earthward plasma flow should precede the dipolarization at distances beyond that location for the earthward propagation of magnetic flux accumulation. Case studies of some dipolarization events by Lui et al. (1999) showed absence of plasma flow, tailward flow, and non-compliance
with the frozen-in-field approximation, contradicting the expectation from the mid- tail initiation scenario.
Another means is to examine activity poleward of the substorm onset arc. No activity prior to initial arc brightening is expected for near-Earth initiation scenario while activity associated with midtail magnetic reconnection onset is expected for the other scenario. Most recent IMAGE results by Mende et al. (2003a,b) from far ultraviolet (FUV) images of aurora indicated the onset location to be separated from the closed/open field line boundary by an extended region of closed field lines.
No clear signature of activity in this extended poleward region was seen prior to onset (see also Lui, 2000a). Furthermore, from a comparison between FUV and high energy neutral atom data, Mendeet al. (2002) showed indications of a pre- onset increase in the plasma pressure of the inner magnetosphere. Both these results are consistent with the near-Earth initiation scenario.
Another important difference is on the identification of substorm onset pro- cess. The near-Earth scenario includes a number of substorm onset processes (see, e.g., Lui, 1991) with testable predictions. On the other hand, the mid-tail scenario has not identified what physical process is responsible for the creation of mag- netic field configuration required by magnetic reconnection. The tearing instability, which was assumed to play that role, has not been demonstrated theoretically to be tangible (see discussion in Section 5.6). One possible connection between near- Earth and mid-tail substorm activities is through the launching of rarefaction waves (Chaoet al., 1977). Consistent with the existence of rarefaction waves are the oc- currence of plasma sheet thinning in the mid-tail region during substorm expansion – a prediction by Chaoet al. (1977) and the findings that equatorial plasma pressure in the mid-tail region is reduced during substorm expansion phase (Petrukovich et al., 1999). More comparisons between these scenarios are discussed by Ohtani (2004) and Cheng (2004) in this issue.
Another point that needs to be made concerns the claim of identifying recon- nection location at substorm onset that differs from both paradigms (Zelenyiet al., 2004). This claim arises from the single event study of Petrukovichet al.(1998) in which substorm onset timing and location were inferred based on plasma flow measurements from two spacecraft (Geotail and Interball) closely aligned along the tail axis. The substorm onset location was estimated to be atX ≈ −15.5REfor a weak substorm. This location does not contradict the near-Earth initiation scenario since the substorm was weak, implying the magnetosphere not to be highly stressed such that the inner edge of the cross-tail current (i.e. the transition region between dipolar and tail-like field configurations) would be at a further downstream distance than normal. Furthermore, while Geotail detected tailward plasma flow soon after substorm onset, the plasma flow reversal seen a few minutes later did not have the expected signatures of an X-line for magnetic reconnection – the magnetic field had a significant northward component during the later part of tailward flow and no particle energization was found at the time of plasma flow reversal, which should correspond to the encounter of anX-line for magnetic reconnection. Fur-
thermore, large magnetic fluctuations were seen by Interball atX ≈ −11.5 RE
associated with earthward plasma flow, a feature consistent with current disruption onset rather than a region 4RE earthward of the assumed X-line location, espe- cially for a weak substorm. The interpretation of reconnection pulse for this event was based solely on the oppositely directed plasma flows at the two spacecraft.
The plasma flows were not sorted into perpendicular and parallel components.
Also, the plasma flow observations were not substantiated by simultaneous electric field measurements available at Geotail. As will be discussed later in Section 7.1, oppositely directed plasma flows at two separate tail locations can be produced by current disruption as well. Thus, the interpretation given by Petrukovichet al.
(1998) cannot be judged as unambiguous.
3. Observations Guiding Plasma Instability Development
As discussed in the previous section, there are mainly two regions in the mag- netosphere in which the substorm process is perceived to take place. One is the near-Earth magnetosphere near the geostationary altitude, probably in the inner edge of the plasma sheet where the magnetic field changes from dipolar to tail-like configurations. The other is the mid-tail region where the magnetic field normal to the neutral sheet is weak enough to allow possibly the occurrence of tearing instability to set up the magnetic geometry for reconnection. This section focuses on observations in these two regions that provide guidance in the formulation of potential plasma instabilities responsible for substorm onset. More detail observa- tions on substorm phenomena are reported by Ohtani (2004) and Meng and Liou (2004).
3.1. NEAR-EARTHSUBSTORMOBSERVATIONS
One key substorm feature is the injection and energization of plasma sheet particles into the geostationary altitude (McIlwain, 1971). This substorm injection is ac- companied by the magnetic field becoming a more dipolar orientation, consistent with the interpretation that the near-Earth cross-tail current is reduced. A quantit- ative estimate of current change was made by Lui (1978) who modeled the cross- tail current sheet by a slab with current density decreasing in the tailward direc- tion. Assuming the current density decreases linearly with downstream distance x, i.e., j (x) = j0(1−x/L), whereL is the length of the tail current sheet, he noted that the magnetic field changes inx- andz-directions based onBiot-Savart law are
Bx = −µ0j0
2π z L
0
(1−u/L)
(u−x)2+z2du≈ −µ0j0
2π π
2 −tan−1x z
, (1)
Figure 3.Analysis on the change in the magnetic field orientation at∼11REdownstream observed by IMP-6 satellite during a contracted oval substorm on November 17, 1971. The magnetic field orientation is plotted along the satellite trajectory to show dipolarization occurring simultaneous withByperturbation signaling the generation of field-aligned currents (afterLui, 1987).
Bz= −µ0j0
2π L
0
(u−x)(1−u/L)
(u−x)2+z2 du≈ −µ0j0
2π
ln(L/
x2+z2)−1
.(2) These expressions can be inverted to obtainj0 andx by fitting the observed field changes:
j0≈ −Bz/(ln(L/
x2+z2)−1)µ0/2π, (3)
x≈zcot(2π Bx/µ0j0). (4)
When this technique is applied to a well-documented substorm event of di- polarization observed by IMP-6 (shown in Figure 3), it is found that the field changes are consistent with a tailward retreat of the near-Earth cross-tail current
Figure 4.An example of current disruption in the magnetosphere during which magnetic field fluctu- ates wildly and energetic particles increase tremendously in intensity. The schematic diagram of the magnetosphere indicates the close proximity to the Earth for this kind of event.
sheet by 0.4–3REwith a current decrease of about 5–25%. Current reduction is not monotonic but interrupted momentarily by partial recoveries. The field changes are also accompanied by deflections of theBy component indicative of simultaneous generation of field-aligned currents.
The above preliminary analysis was extended with the inclusion of propagation of current reduction by Jacquey et al. (1991, 1993) and Ohtaniet al. (1992a) as substorm expansion activity develops. Jacquey et al. (1991) showed that if the current density enhancement during the substorm growth phase is represented by a current slab of length L with its earthward edge at X0, and current reduction propagates at a uniform speedVtailward, then the magnetic field components at a satellite locationXsat,Zsatat timetare given by
Bx =µ0j{tan−1[(X0−Xsat+V t)/Zsat] +tan−1[(Xsat+L−X0)/Zsat]}/2π,
(5) Bz =µ0j{ln[(X0−Xsat+V t)2+Z2sat]
−ln[(Xsat+L−X0)2+Zsat2 ]}/4π.
(6) Jacquey et al. (1991, 1993) used these expressions to perform best fitting to single- and multiple-point magnetic field measurements. They concluded that cur-
Figure 5.Time evolution of plasma pressure components from protons, electrons and oxygen ions as well as plasma beta during the current disruption relative to the disruption onset time at T=0. The plasma beta was large and the proton pressure was dominant among the particle species shown prior to the current disruption onset (after Luiet al., 1992).
rent disruption starts at 6–9 RE in the tail and propagates at a tailward speed of 150–250 km/s over tens ofRE during substorm expansion. A similar procedure was carried out by Ohtaniet al. (1992a) who found from a statistical study of 13 substorm events that current reduction starts usually in the near-Earth region and its tailward propagation speed is estimated to be∼200 km/s. These speeds should not be equated to the speed at which the rarefaction wave front in the near-Earth initiation scenario propagates because the rarefaction waves only lead to plasma sheet thinning. A finite time delay is expected between the start of plasma sheet thinning and the onset of current disruption for at least two reasons. First, it takes time for the plasma sheet to be thinned to the extent that the thinned current sheet becomes favorable for the onset of a plasma process for current disruption. Second, there should be a delay required for a plasma process to develop sufficiently to cause current disruption because any plasma instability necessarily requires a finite time to reach from its onset to its nonlinear saturation stage.
When the point of observation is close to the neutral sheet, current reduction is often observed to have a short duration of high-level magnetic fluctuations before reconfiguration of the magnetic field is established. The left side of Figure 4 illus- trates the proximity of current disruption site to the Earth. The right side shows the magnetic fluctuations of a current disruption event first reported by Takahashiet al.
(1987). An important point worth noting is that the magnetic field component nor- mal to the current sheet (BH) is strongly northward at∼5 nT prior to the occurrence of current disruption. This strong northward component, as will be discussed in Section 5 later, suggests that the magnetic fluctuations are not initiated by magnetic reconnection or tearing instability occurring locally because the normal magnetic field component is too strong for either process to commence.
Figure 5 presents the variations of plasma pressure and beta when the CCE spacecraft captured the entire interval of current disruption near the neutral sheet on June 1, 1985. Pressure contributions from three major particle species, namely, protons, electrons, and oxygen ions, are shown. Plasma pressure, dominated by
Figure 6.Observation of ion anisotropy during current disruption. Strong duskward ion streaming was seen just prior to current disruption onset, with a subsequent development of an ion pressure anisotropy perpendicular to the magnetic field (afterLui, 1996).
protons, increased by∼15% and was rather isotropic prior to current disruption onset at∼2314 UT. After onset, the perpendicular pressure became significantly larger than the parallel pressure. Plasma beta exceeded 20 well before onset, reach- ing to about 70 just before onset due to an increase in plasma pressure accompanied by a decrease in the magnetic field strength, and finally decreasing to less than 5 afterwards.
The ion behavior during this event is further illustrated in Figure 6. The top three panels in the figure give the magnitude and elevation angle of the magnetic field together with the ion anisotropy of 31–43 keV energetic ions. The ion thermal energy at this time is∼12 keV. The ion velocity distribution at three time snapshots
Figure 7.Observation of electron anisotropy during current disruption. The electron pitch angle distribution changed from a trapped distribution well before onset to a field-aligned distribution afterwards.
is given in the bottom row of panels. The ion distribution appeared to be isotropic well before onset. Later, the energetic ions exhibited a strong overall duskward drift, which is not a bulk flow of the entire population. More examples of a nonzero net drift not representing a bulk flow of the entire population are given in Parks et al. (1998, 2001) and Parks (2004). The skew in the distribution is consistent with the expectation of energetic ions undergoing Speiser orbits in the thin current sheet just before onset. Eventually, the ion distribution was anisotropic with the perpendicular temperature higher than the parallel.
During this event, the electrons behaved differently from and evolved in a much shorter time scale than the ions, as illustrated in Figure 7. The top panels show
Figure 8.Frequency and time decomposition of wave activity by wavelet analysis during current disruption. Several components in different frequency regimes were excited at current disruption onset and during its progress (after Lui and Najmi, 1997).
the electron pressure, measured pitch angle, and the magnetic field magnitude.
The bottom rows are three time snapshots of the electron pitch angle distribution (PAD). Before onset, the electrons had a pancake distribution, peaking at 90◦pitch angle. This indicates a trapped population. During current disruption, the electrons exhibited a rather isotropic distribution. The trapped electrons were sufficiently perturbed by the magnetic fluctuations that they lose their trapped signature in the PAD. At the end, the electron PAD was cigar-shaped with fluxes peaked along the magnetic field. This PAD is probably related to the nonlinear consequence of the physical process activated during current disruption.
There were several significant wave components associated with current disrup- tion as revealed in Figure 8 by wavelet analysis. The magnetic field componentBv
is shown at the top and the wavelet decomposition of the signal is shown at the bot- tom. Superposed in the bottom panel is the trace of local ion cyclotron frequency.
About 1 min before onset, a low frequency wave at∼10–30 mHz was seen. At on- set, a broadband (10 mHz–1 Hz) wave activity was seen, with frequency reaching well above the ion cyclotron frequency. Later, waves at intermediate frequencies at 40–90 mHz were seen. Intermittent bursts of wave activity after onset were seen
at later times as well, e.g., at∼2315:45 UT. This wavelet analysis indicates that multiple frequency components are excited during current disruption, suggesting possibly the onset of multiple plasma instabilities.
3.2. MID-TAILSUBSTORMOBSERVATIONS
Most of substorm observations in the mid-tail region were concentrated on the detection of magnetic reconnection signatures. Since magnetic reconnection is not a plasma instability, we shall confine here to discussion on some observations that may reveal the role of plasma instabilities in the mid-tail substorm phenomena.
The magnetic field geometry in the non-substorm plasma sheet is perceived to have a northward magnetic field normal to the neutral sheet in the mid-tail region.
Southward magnetic field near the neutral sheet is typically infrequent but does occur, especially during substorms.
Nagaiet al.(1998, 2001) showed detailed plasma condition during occurrence of some southward magnetic field intervals from Geotail spacecraft. Figure 9 is such a case on January 27, 1996 when Geotail was at about 29RE downstream.
During this interval, the Bx component occasionally was small, indicating Geo- tail being close to the neutral sheet. The magnetic field had mostly a southward component in the interval∼1400–1407 UT, in which theBzcomponent was large (∼10 nT) intermittently, so was theBycomponent. Coincident with this southward field interval was strong tailward plasma flow with a substantial flow component perpendicular to the magnetic field. In the early part of this interval, ions showed a single population flowing approximately tailward. Electrons had a nearly iso- tropic distribution. At a later time when Geotail was almost right at the neutral sheet whereBx∼0 nT, the ion distribution exhibited two peaks. These two peaks represent counter-streaming of ions along the magnetic field with a common very strong convection of∼2500 km/s. Electrons showed a broader velocity distribution compared with that in early part of the interval, suggesting that the electrons were heated.
The ion velocity distribution was different when Geotail was outside the neutral sheet region, as shown in Figure 10. The simultaneousBxandBycomponents were 9 nT and−12 nT, respectively. Here, there were two ion components. There was a warm ion population flowing tailward almost antiparallel to the magnetic field dir- ection. Its temperature was similar to that found near the neutral sheet region. This population most likely originated within the neutral sheet. This was accompanied by another cold ion population showing significant convection directed tailward and toward the neutral sheet. This cold component probably originated in the tail lobe.
The electrons at this time had a rather isotropic velocity distribution. However, there were other times in which the electrons showed quite a different velocity distribution than that shown in Figure 10. This is illustrated in Figure 11 for a later time at ∼1406 UT. There were apparently two peaks in the cut of the electron
Figure 9.Intervals of strong southward magnetic field near the neutral sheet detected by Geotail during a substorm on January 27, 1996. These signatures are consistent with occurrence of magnetic reconnection near the spacecraft (after Nagaiet al., 1998). In the bottom three panels, the components of the total and perpendicular plasma flow are shown in thin and thick lines, respectively.
velocity distributions shown. One high-energy (∼10 keV) component was stream- ing tailward along the magnetic field while another component at medium-energy (∼3 keV) was moving approximately earthward and also field-aligned. At this time, ions had a huge field-aligned tailward flow exceeding 2800 km/s with an insignificant convection. More cases of this bi-directional electron streaming along the magnetic field were presented by Nagaiet al.(2001).
Figure 10.Ion and electron velocity distributions outside the neutral sheet region during the south- ward magnetic field interval shown in Figure 9. The logarithm of the distributions (in units of s3/m6) is shown. These distributions are presented in theBCEcoordinate system, in whichBis the magnetic field direction,Cis the convection velocity direction, andEis the assumed electric field direction.
The red arrows in these panels represent the convection velocity vectors. The scale for these vectors is given by the scales on the left, 3000 km/s for ions and 80000 km/s for electrons. The dashed line in the ionBCpanel indicates the cut (vB= −1320 km/s) at which the ion distribution in theECpanel is shown. Cuts of the electron velocity distributions at theB- andC-axis are given in the line plot panel by red and blue curves, respectively, together with the one-count level given in black. Some parts of the velocity distribution in theBC-plane are missing due to lack of plasma measurements in the spacecraft spin direction (after Nagaiet al., 1998).
Nagaiet al. (1998, 2001) organized these observations in terms of the plasma environment around a magnetic reconnection site, as depicted in Figure 12. Near the X-type neutral line, the particles have velocity distributions like that shown in Figure 10, whereas near the magnetic field line connected with the X-type neutral line, the particles have that shown in Figure 11.
There is considerably wave activity in these energetic events. The early studies focused on the broadband electrostatic noise (BEN) (Gurnettet al., 1976), which was suggested to be the signature of lower-hybrid drift instability (Hubaet al., 1977, 1981). This instability is perceived to be a mechanism to provide anomal-
Figure 11.Ion and electron velocity distributions at a later time during which electrons exhibited a two-component behavior. The format is the same as in Figure 10. The ionEC-plane cut was made at vB= −60 km/s (after Nagaiet al., 1998).
Figure 12.A schematic diagram to summarize the various features in the particle (both ion and electron) velocity distributions around a magnetic reconnection site (after Nagaiet al., 1998).
Figure 13.Wavelet analysis of magnetic and electric fluctuations during the GEM substorm challenge event. The ion gyrofrequency is plotted as a black line over the scalogram of the total magnetic field.
The excitation of waves in various frequency regimes was seen, including waves at frequencies above the ion gyrofrequency (after Sigsbeeet al., 2001).
ous resistivity to initiate the onset of collisionless magnetic reconnection. Further detailed studies of the BEN waves indicated that these waves were due to a super- position of wave modes (Cattellet al., 1986; Matsumotoet al., 1994). Recently, attention has been drawn to the solitary waves that are bipolar nonlinear spikes in the electric field component parallel to the magnetic field and are now identi- fied as electron holes (Matsumoto et al., 1999). These electron holes are present throughout the magnetosphere (Ergunet al., 1998; Tsurutani et al., 1998; Cattell et al., 1999). Some basic characteristics have been determined by Cattell et al.
(2003) who have found the mean duration of these electron holes to be ∼2 ms,
with velocities from∼1000 to>2500 km/s and scale sizes∼10’s of Debye length along the magnetic field.
A detailed study of mid-tail wave activity during the Geospace Environment Modeling (GEM) substorm challenge event was reported by Sigsbeeet al.(2001).
Geotail was at a downstream distance of ∼25 RE. As shown in Figure 13, they found from wavelet analysis on Geotail measurements that low-frequency waves from around ion-cyclotron frequency to lower hybrid frequency were intensified a few minutes prior to the ground substorm onset at 2225 UT. The characteristics of this wave activity are quite similar to those seen in the near-Earth region as shown in Figure 8. The major difference is that mid-tail observations are often complicated by the flapping motion of the plasma sheet so that measurements are a mixture of those representative of the high-latitude plasma sheet and near the neutral sheet.
On the other hand, current sheet movement is typically less in the near-Earth region so that spacecraft in that region can remain near the neutral sheet throughout the entire interval, like the case shown in Figure 6. Sigsbeeet al. (2002) noted that the low frequency waves did not have the largest amplitude at dipolarization onset and used that finding to discount the cross-field current instability (discussed later in Section 5.1) as a potential mechanism for substorm expansion onset. However, this reasoning is probably not well founded since current disruption causes episodic current density increases and decreases and therefore the excited waves are not expected to always reach the largest amplitude at activity onset.
4. Analytic Current Sheet Models
One general consensus associated with substorm onset is the development of a thin current sheet in the near-Earth region. This development is crucial in terms of setting up an appropriate plasma instability for releasing the energy associated with a thin current sheet accumulated prior to substorm expansion. There are several thin current sheet configurations that are modeled theoretically. Several theoretical works that provide analytic expressions on a thin current sheet equilibrium are discussed in the following subsections. Focus here is current sheet configuration without a pre-existing magnetic X-line. In addition to these analytical models, there are several solutions of thin current sheet configuration by numerical means without explicit analytic solutions. They are usually unsuitable for implementation in theoretical plasma instability analysis by others not engaged in the numerical work and therefore are not included below.
4.1. HARRISCURRENTSHEET
The most widely used 1D current sheet was formulated by Harris (1962), which is an exactVlasovequilibrium description. For this equilibrium, the particle velocity distribution functionfα(v) (where α specifies the particle species, ifor ions and
efor electrons) with particle massmα, temperatureTα (absorbing the Boltzmann constant, i.e.,Tα = mαu2α/2, uα is the thermal velocity), bulk flow Vαy, vector potentialA=(0,Ay(z),0), magnetic fieldBx(z), current densityjy(z), and number densityn(z) are given by:
fα(v)=nα(z)(mα/2π Tα)3/2exp[−mα(v2x+(vy−Vαy)2+vz2)/2Tα], (7)
Ay(z)= −B0(L)ln[cosh(z/L)], (8)
Bx(z)=B0tanh(z/L), (9)
n(z)=n0sech2(z/L), (10)
jy(z)=B0sech2(z/L)/µ0L. (11)
Here,B02=2µ0n0(Ti+Te), and the magnetic field varies in thez-direction and the current is in they-direction. The coordinate system is that of the solar magneto- spheric coordinates, i.e., thex-axis points to the sun, the y-axis points duskward, and thez-axis points northward. This coordinate system and the symbol convention will be adopted hereafter unless specified otherwise. This solution assumes the particles to have a Maxwellian velocity distribution with an isotropic temperature and the boundary condition imposed atz =0 isBx =0. For this reference frame with no electric potential, we have
Viy/Ti= −Vey/Te. (12)
The half-thickness of the current sheet is given by
L= 1
Viy−Vey
2µ0(Ti +Te)
n0e2 . (13)
Figure 14a shows the magnetic field and current density profiles for this cur- rent sheet. The number density profile is exactly the same as the current density profile and is thus not included in the plot. Also not shown is the relative drift speed between ions and electrons since it is constant across the entire current sheet. Although this is a very elegant exact Vlasov solution to the 1D current sheet problem, there are two undesirable features. First, there is no magnetic field component normal to the current sheet. This characteristic limits its applicability to realistic magnetic field geometries in space. For example, a non-zero magnetic field component normal to the current sheet almost always exists in the Earth’s magnetotail. This magnetic field component may be small but is vitally important in the consideration of plasma instabilities in a thin current sheet as discussed later in Section 5. Second, a comparison of Equations (10) and (11) indicates that the relative drift between ions and electrons is constant throughout the current sheet.
This feature is unlikely to be compatible with the expected characteristic of the
current sheet in the magnetotail that exhibits a larger relative drift between ions and electrons around the sheet center than at its edges.
4.2. YOON-LUICURRENT SHEET
A 1D current sheet model from a fluid approach with an implementation of a ve- locity shear in mind was considered by Yoon and Lui (1996) and Luiet al. (1995).
In this 1D current sheet, the drift velocity profileVy(z) is arbitrarily specified as a Lorentzian distribution function and the profiles of other parameters resemble that of Harris current sheet:
Vy(z)=V0h2/(z2+h2), (14)
Bx(z)=B0tanh h
Ltan−1(z/ h)
, (15)
n(z)=n0sech2 h
Ltan−1(z/ h)
, (16)
jy(z)= B0
µ0L h2
z2+h2sech2 h
Ltan−1(z/ h)
. (17)
The Harris current sheet solution can be recovered with h → ∞. For this solution, the magnetic field ath → ∞isB0tanh(π h/2L). Figure 14b shows the profiles of magnetic field, current density and relative drift speed between ions and electrons for this current sheet. With the exception of the drift speed profile, the resemblance of the other profiles to that of the Harris current sheet is remarkable.
The introduction of a velocity shear is important in two aspects. One is that thin current sheets in space typically have an associated velocity shear rather than a constant value as in the Harris equilibrium. Second is that this feature plays an important role in the nonlocal analysis of the cross-field current instability and the current sheet stability to Kelvin-Helmholtz instability as will be discussed in Section 5.
4.3. KANCURRENTSHEET
An important extension of the Harris 1D current sheet to a 2D current sheet was discovered by Kan (1973) through the early work of Walker (1916). His solution, like Harris current sheet, is an exact Vlasov equilibrium. Similar to the approach of Harris, the chosen particle population for both species is a drifting Maxwellian plasma as expressed in Equation (7) and the chosen frame satisfies the zero po- tential condition as given in Equation (12). The equation for the vector potential Ay(x,z) that needs to be solved in this formulation is
∂2A∗y
∂x∗2 +∂2A∗y
∂z∗2 =exp
−2A∗y , (18)
Figure 14.Profiles for current sheet parameters for (a)Harriscurrent sheet, and (b)Yoon-Luicur- rent sheet. The Harris current sheet has a flat velocity profile while the Yoon-Lui current sheet is constructed to have a Lorentzian velocity profile.
The asterisk denotes the dimensionless quantities normalized as
A∗y = −Ay/2B0L; x∗ =x/2L; z∗=z/2L. (19) The general solution to Equation (18) is found to be
A∗y = −1 2ln
4(g2x+g2z) (1+g2+h2)2
, (20)
whereg(x∗, z∗) andh(x∗, z∗) are, respectively, the real and the imaginary parts of an arbitrary analytic functionF of a complex variableξ =x∗+iz∗, i.e.,
F (ξ )=g(x∗, z∗)+ih(x∗, z∗), (21) andgx =∂g/∂x, gz = ∂g/∂z. The normalized number densityn∗and the mag- netic field components are then
n∗(x∗, z∗)= 4(gx2+g2z)
(1+g2+h2)2, (22)
Bx∗(x∗, z∗)= ∂A∗y
∂z∗ = −1 2
∂(g2x+gz2)/∂z∗
g2x+gz2 +∂(1+g2+h2)/∂z∗
1+g2+h2 , (23) Bz∗(x∗, z∗)= −∂A∗y
∂x∗ = 1 2
∂(g2x+gz2)/∂x∗
gx2+gz2 −∂(1+g2+h2)/∂x∗
1+g2+h2 . (24) Kan (1973) picked the analytic function to beF (ξ )=exp[−iξ+iβ/(ξ−α)], subject to Re(ξ ) = α, and α,β are real adjustable parameters to be determined by the boundary conditions. He also noted that the 1D Harris solution is a special case of this 2D solution with F (ξ ) = exp(iξ ). Figure 15a shows the magnetic field geometry resulting from his solution. The absence of a velocity shear in the 1D Harris solution remains in this 2D solution. However, a velocity shear can be set up if two different populations with different temperatures for both ions and electrons are introduced.
4.4. SCHINDLER-BIRNCURRENT SHEET
Systematic studies of magnetotail configuration have been reported in a series of paper by Schindler and his colleagues (e.g., Schindler, 1972; Birn et al., 1975;
Nötzel et al., 1985; Birn and Schindler, 2002). With the tail approximation, i.e.,
∂/∂x ∂/∂z, and assumingBz contribution to the total pressureP (x) = p + B2/2µ0to be negligible, the expression for a magnetic field line in a field specified by the vector potentialAyis given by (Birn, 1991)
z(x, Ay)=
Ay
A0(x)
dA 2µ0
P (x)−p(A). (25)
The lower bound for the integration is A0(x) = Ay(x, z = 0). When the magnetopause boundary and the total pressure are known functions ofx, then the solution to theGrad-Shafranovequation forAycan be written as
Ay(p)= −
√2µ0
π p
pb
a(P )
√p−PdP . (26)
Figure 15.Magnetic field configurations: (a)Kancurrent sheet, (b)Birn-Schindlermagnetotail, (c) Cowleycurrent sheet, and (d)L´embege-Pellatcurrent sheet.
The plasma pressure at the boundary is pb. The current density can be ob- tained fromjy(Ay)=dp/dAy. If one adopts the pressure function to bep(Ay)= pbexp[−2(Ay−Ab)/An], whereAnis a normalization constant, the boundary field line (assumed to be defined byp=pb) is then given byz(P )=z0(pb/P )1/2cosh−1 [(P /pb)1/2], withz0 =An/(2µ0pb)1/2. Birn and Schindler (2002) considered the solution to theGrad-Shafranovequation with perturbation to this boundary surface by a factor
1−a1/[1+((P−pm)/d)2]
. The perturbation represents a local in- dentation with an amplitude ofa1 and a width ofd at the boundary location with pressurepm. They noted that solutions cease to exist when the amplitudea1exceeds a particular threshold. Figure 15b shows an unperturbed magnetic field configur- ation (top) and another (bottom) at a critical state. The perturbed configuration exhibits a singular current sheet, i.e., locally infinite current density enhancement.
Schindler and Birn (2002) have recently extended this approach to treat a current sheet with an embedded thin current structure.
4.5. COWLEYCURRENT SHEET
Another 2D current sheet model with inclusion of pressure anisotropy was put forth by Cowley (1978). This model is constructed with the assumptions that the magnetic field varies exponentially along the tail axis and the plasma pressure is anisotropic (p = p⊥I+(p−p⊥)BB/B2)with a constant ratio α = p⊥/p at z= 0 and their absolute values scale as the square of the magnetic field strength.
Here, the subscripts ⊥and refer to perpendicular and parallel to the magnetic field, respectively. This convention will be adopted hereafter. More precisely, the vector potential componentAy(x, z)and the pressure components are in the form:
Ay(x, z)=B0Lexp(x/L)Z(z/L), (27)
p(x, z)=p0exp(2x/L)Z2(z/L)/ h, (28) p⊥(x, z)=αp0exp(2x/L)Z2(z/L)/ h2, (29) whereh(z/L)=α+(1−α)g(z/L), andg(z/L)=Z/(Z2+Z2)1/2. The prime superscript for the functionZindicates its derivative. The boundary conditions are Z(0) = 1, Z(R/L) = 0, where R represents the tail radius. The function Z is taken to be even andB0=Bz(0,0). This leads to a single second order differential equation with a relatively simple solution in the case of isotropic pressure:
Bx=ξ B0exp(x/L)sin(ξ z/L), (30)
Bz=B0exp(x/L)cos(ξ z/L), (31)
p=p⊥=χ B02exp(2x/L)cos2(ξ z/L)/µ0, (32)
jy=2χ B0exp(x/L)cos(ξ z/L)/µ0L, (33)
whereχ = µ0p0/B02,ξ = (1+2χ )1/2, andR/L = π/2ξ. The last condition implies settingR/L=1/3, corresponding to a tail-like field line geometry, giving χ∼10.6. Figure 15c shows the field line shape for this isotropic case, which has a broad current sheet thickness. The assumption of magnetic field (and thus plasma pressure) varies exponentially along the tail axis may be compatible with the mid- tail observations but definitely not so with observations over a range in the tail axis from the dipolar-field to tail-like field regions (Luiet al., 1994; Spenceet al., 1989;
Wanget al., 2001). Also, the assumption that the anisotropy ratio is constant is in disagreement with observations indicating pressure anisotropy being large (α ∼ 2) in the dipolar region (Luiet al., 1987) and becomes negligible (α ∼1) in the magnetotail (Stileset al., 1978).
4.6. LEMBÉGE-PELLAT CURRENT SHEET
A simpler current sheet model than Kan’s model was considered by Lembége and Pellat (1982) who noted that an approximate 2D current sheet equilibrium could be described by the vector potentialAy(x, z):
Ay(x, z)=B0L{ln[g(x)] −ln[cosh(g(x)z/L)]}. (34) The magnetic field components and the associated plasma pressure p(x, z) (assumed to be isotropic like the previous current sheet solutions) are then given by
Bx(x, z)= −∂Ay/∂z=B0g(x)tanh[g(x)z/L)], (35) Bz(x, z)=∂Ay/∂x=B0L[g(x)/g(x)] {1−g(x)tanh[g(x)z/L)]z/L}, (36) p(x, z)=B02exp(2Ay/B0L)/2µ0, (37) in which g(x) = exp(−x/L) is used. Forces balance in both the x- and z- directions up to the order of2. This is a fluid equilibrium and produces no ve- locity shear. The magnetic field configuration from this formulation is illustrated in Figure 15d.
4.7. FORCEDCURRENTSHEET
A large number of researchers have examined the current sheet equilibrium with a highly anisotropic pressure (Eastwood, 1972; Richet al., 1972; Hill, 1975; Cowley, 1978; Francfort and Pellat, 1976; Chenet al., 1990; Burkhartet al., 1992; Holland and Chen, 1993; Kropotkin et al., 1997; Sitnov et al., 2000a,b). The latest ana- lytic result utilizes two parameters, namely, the total particle energy and the sheet invariantIz defined as Iz = 2π1
mvzdz, which is only an approximate constant of motion. Sitnovet al. (2003) used this procedure to generalize the Harris current sheet equilibrium to a bifurcated current sheet with the presence of a small pancake ion pressure anisotropy (pi⊥ > pi) and to a thin current sheet embedded in a thicker Harris sheet with a cigar pressure anisotropy(pi⊥< pi).
5. Plasma Instabilities
In this section, we present a brief account on theoretical analyses of several plasma instabilities proposed for the substorm expansion process. This is aimed to il- lustrate the level of sophistication for various proposed plasma instabilities. The original publications should be consulted for more details. It is important to note that the onset of most plasma instabilities discussed below is dependent on the threshold level of current density and other local parameters as well. Therefore, it would be misleading to invoke the total current strength across the current sheet thickness (or equivalently the total magnetic field strength in the tail lobe) as a means to judge whether or not a plasma instability should have been excited if it plays a role in triggering a substorm. In other words, even if the tail lobe magnetic field strength is high and no substorm occurs, one cannot rule out the possibility