The Kinetic Approach

3.1 Plasmapause Formation

The kinetic approach was also used to study the formation of the plasmapause by the mechanism of quasi-interchange instability (Lemaire 1985, 2001). This mechanism causes the peeling off of the plasmasphere as a result of sudden enhancements of the convection velocity associated with substorms in the night side. This enhanced azimuthal convection velocity leads to increased centrifugal acceleration in the outermost layers of the plasmasphere and a reduction of the total field-aligned potential barrier that ions have to overcome to reach the equatorial plane. This prompts the uplift of ions out of the underlying ionosphere. The enhancement of the field-aligned expansion velocity reduces the plasma density at high altitudes along all geomagnetic field lines traversing the zero-parallel force (ZPF) surface. This surface is the locus of points where the components of the gravitational and centrifugal acceleration balance each other in the direction parallel to the geomagnetic field lines. As a consequence of the plasma density diminution at high altitude along the field lines traversing the ZPF surface, a steep density gradient is formed into the plasmaspheric equatorial density profile. This steep plasma density gradient corresponds to a new plasmapause. It is formed where and when the azimuthal component of magnetospheric convection velocity is occasionally raised to values larger than the corotation velocity in the unperturbed core of the plasmasphere. This is how outer layers of the plasmasphere are peeled off and how plasma elements are detached according to the interchange theory for the formation of the plasmapause (Lemaire 2001).

The increased upward ionization flux which is prompted by the lowering of the field-aligned potential barrier depletes the mid-latitude ionosphere in the nightside MLT sector, where and when the eastward component of the convection electric field is suddenly enhanced. This leads to the formation of F-layer ionization troughs in the mid-latitude ionosphere as observed by Muldrew (1965) from Alouette data. As an additional result of the upward H+ and He+ ions fluxes along all geomagnetic field lines beyond those which are tangent to the ZPF surface, the concentrations of these light ions are depleted, in both conjugate ionospheric regions. This is how light ion troughs (LIT) are developing beyond the projections of the ZPF surface at low altitudes in the ionosphere (Taylor and Walsh 1972). All these well documented events and features are observed following substorm events. According to Lemaire's theory for the formation of the plasmapause, they are consequences of the field aligned plasma distribution driven unstable by enhanced centrifugal effects at the innermost edge of substorm injection clouds (Lemaire 1974, 1985).

Dynamical simulations have been developed to determine the position of the plasmapause due to the erosion of the plasmasphere by the combined influence of the interchange instability and Kp-dependent electric field distributions (Pierrard and Lemaire 2004). The results predicted by models that include interchange have been compared with success to different Image EUV observations during various periods of time including quiet periods, substorms and storms (Pierrard and Cabrera 2005, 2006; Pierrard 2006). The models predict an extended equatorial plasmasphere during prolonged quiet periods and plumes formation in the afternoon sector when the level of geomagnetic activity suddenly increases. Beside the sunward shift of the dusk-side plasmasphere during episodes of enhancement of the dawn-dusk component of magnetospheric convection electric field, an alternate mechanism of formation of such attached plasmatails or plumes was proposed by Lemaire (2000). Results of the interchange-included simulations have also been compared successfully with CIS (Cluster Ion Spectrometer) (Reme et al. 2001) and WHISPER (Waves of HIgh frequency and Sounder for Probing Electron density by Relaxation) (Decreau et al. 2001) observations onboard CLUSTER by Dandouras et al. (2005), Darrouzet (2006) and Schäfer et al. (2007). Moreover, Schäfer et al. (2008) studied the spatio-temporal structure of a poloidal Alfven wave near the dayside plasmapause based on CLUSTER observations of magnetic and electric fields.

The results of the dynamical simulations based on the inclusion of the interchange mechanism have been compared to the traditional MHD-based convection-only mechanism, in which the plasmapause location is determined dynamically under the influence of a time-dependent potential electric field. The axioms and assumptions of both mechanisms have been recalled in detail in a recent article by Lemaire and Pierrard (2008). The plasmapause positions predicted by these two alternative theories (i.e., convection only and convection plus interchange) have been determined numerically. They have been compared for different empirical electric field models described in this issue by Reinisch et al. (2008): Volland-Stern-Maynard-Chen (VSMC) model, McIlwain's E5D model, and Weimer's model. The predicted positions and overall shape of the equatorial plasmapause cross-section have been compared for these three different electric field models (Pierrard et al. 2008).

The predictions of both alternative kinds of simulations have been compared to whistler-derived densities and global images obtained by EUV, during several storm and substorm events. These simulations confirmed that the presumed plasmapause positions and shapes depend on the variation of the geomagnetic activity level during the preceding days and also on the magnetospheric convection electric field model. When the modeled plasmapause is determined by the combined influence of instability and convection, it is formed slightly closer to the Earth than with the convection only scenario. Plumes are formed in both scenarios and for all electric field models considered in Pierrard et al. (2008) study. Nevertheless, different features are obtained for the plasmasphere structure depending on the simulation scenario and electric field models which had been adopted.

Figure 5 illustrates the differences obtained for the equatorial position of the plasmapause with the different mechanisms (ideal MHD in upper panels, interchange in the bottom left panel) and with for all three electric field models at 21:00 UT, after the magnetic storm of 17 April 2002. The E5D, VSMC and Weimer are used in the three upper panels; only E5D is used in the bottom left panel. The Image EUV observation of the equatorial plasmapause position at 21:07 UT is illustrated in the bottom right panel. VSMC and Weimer models produce a plasmapause closer to the Earth than E5D. In the midnight sector, VSMC and Weimer electric field models lead to a plasmapause position too close to the Earth compared to the EUV observations, while the plasmapause position obtained with E5D corresponds to the EUV observations in this midnight sector. In the noon sector, E5D reproduces EUV observations when the interchange is taken into account.

3.2 Plasmaspheric Wind

The distribution of cold plasma in the Earth's inner magnetosphere depends on the interplay between the corotation electric field, the convection electric field, and plasma instabilities. The corotation electric field is fairly stable but the convection electric field is unsteady and, therefore, the plasma distribution may vary substantially both in space and time during active periods. When there are disturbances in the solar wind, flux tubes outside the corotation region drain their plasma toward the magnetopause under the effect of the duskward convection electric field and a well-defined and sharp density gradient, the plasmapause, is observed. On the contrary, when magnetospheric convection is very weak for a prolonged period, the effect of the plasmasphere corotation with the Earth dominates up to large radial

Fig. 5 The equatorial cross section of the plasmasphere after the magnetic storm of 17 April 2002 at 21:00 UT. (Upper panels) MHD simulations with E5D (left), VSMC (middle), Weimer (right) electric field models. The Z component of the interplanetary magnetic field (IMF) (Bz ), the disturbance storm-time index (Dst) and the geomagnetic activity index Kp from 16 April 2002 0:00 UT up to 18 April 2002 24:00 UT are illustrated below the upper left panel. (Bottompanels) Interchange simulations with E5D (left) and observations of IMAGE EUV at 21:07 UT (right). The white circles correspond respectively to L = 2, 4, 6 and 8. (Adapted from Pierrard et al. 2008)

Fig. 5 The equatorial cross section of the plasmasphere after the magnetic storm of 17 April 2002 at 21:00 UT. (Upper panels) MHD simulations with E5D (left), VSMC (middle), Weimer (right) electric field models. The Z component of the interplanetary magnetic field (IMF) (Bz ), the disturbance storm-time index (Dst) and the geomagnetic activity index Kp from 16 April 2002 0:00 UT up to 18 April 2002 24:00 UT are illustrated below the upper left panel. (Bottompanels) Interchange simulations with E5D (left) and observations of IMAGE EUV at 21:07 UT (right). The white circles correspond respectively to L = 2, 4, 6 and 8. (Adapted from Pierrard et al. 2008)

distances (<7 RE). The magnetic flux tubes inside the corotation region are supplied with plasma continually flowing up out of the ionosphere, and building up a smooth electron density transition from plasmasphere to the outer subauroral regions as observed RPI onboard the Image spacecraft (Tu et al. 2007). Therefore, the plasmasphere is not always bounded by a steep density gradient as commonly believed.

Noting the systematic differences between theoretical hydrostatic models and the observed density distribution in the plasmasphere, Lemaire and Schunk (1992,1994) suggested the conceptual existence of continual losses of plasma from the plasmasphere, a plasmas-pheric wind, driven by interchange motions. They postulate the existence of a slow and permanent transport of plasma across the magnetic field from the inner to the outer regions of the plasmasphere, even during prolonged periods of quiet geomagnetic conditions when substorm disturbances are absent. This plasmaspheric wind is rather similar to that of the subsonic expansion of the equatorial solar corona. Such a radial plasma transport implies in deed that plasma streamlines are not closed, and, therefore, that cold plasma elements slowly drift outward from the inner plasmasphere to the plasmapause along winding up spiral drift paths.

3.2.1 The Role of Quasi-Interchange Modes

From a theoretical point of view, the presence of a plasmaspheric wind has been considered to result from a plasma interchange motion driven by an imbalance between gravitational, centrifugal and pressure gradient forces.

Gold (1959) was the first to introduce the concept of interchange of magnetic flux tubes in the magnetospheric context. His so-called strict interchange assumes a one-to-one interchange between magnetic flux tubes enclosing the same magnetic flux and thus leaving the shape of the magnetic field lines unchanged as well as the magnetic energy of the system unperturbed. Cheng (1989) pointed out that this model is at odds with the requirement of total pressure balance, and that a realistic flux tube interchange must be accompanied by a change in field magnitude. The so-called generalized interchange model of South-wood and Kivelson (1987) still assumes that the interchanging flux tubes preserve everywhere the direction of the local magnetic field, but they relax the condition that the energy density of the magnetic field is unperturbed by the interchange. Both models are in fact unphysical, insofar as true interchange motions of plasma elements generally also entail distortions of the magnetic field that preserves the total pressure (plasma plus magnetic pressure). The presence of stratification of the plasmaspheric pressure distribution and of non-electromagnetic forces leads in fact to the destabilizing of a broader category of modes driven by buoyancy forces, known as quasi-interchange modes that trigger transverse as well as translational plasma motions (Newcomb 1961; Ferrière et al. 1999; André 2003; Ferrière and André 2003).

These modes can become unstable in the limit of small parallel wave vector, and fall into two types in the limit of zero parallel wave vector. The type 1 quasi-interchange mode or transverse interchange mode corresponds to plasma motions which are predominantly perpendicular to magnetic field lines and results in the exchange of plasma elements across magnetic field lines. The type 2 quasi-interchange mode or translational mode corresponds to motions of the plasma predominantly along flux tubes.

3.2.2 Testing the Instability Criteria of Quasi-Interchange Modes in the Plasmasphere

André and Lemaire (2006) tested the local stability of quasi-interchange modes for various diffusive and exospheric hydrostatic field-aligned density distributions expected to be representative of the equatorial regions of a saturated plasmasphere under very quiet geomagnetic conditions. When the only effect of the centrifugal force due to corotation of the plasma with the angular velocity of the Earth is taken into account, the corotating plasma appears convectively stable inside geosynchronous orbit. However, when the magnetic curvature of the magnetic field lines is properly taken into account, the conclusions obtained in the case of straight field lines are significantly altered since the magnetic curvature is found to have a much larger influence than the effective gravity (including the effect of the centrifugal force) in the Earth magnetosphere. The thermal plasma confined in the Earth's dipole magnetic field cannot remain in magnetostatic equilibrium but becomes convectively unstable much deeper inside the equatorial plasmasphere, at R = 2 RE and beyond. The type 2 quasi-interchange or translational mode appears to play a more important role than the type 1 quasi-interchange or transverse mode. The type 2 mode appears unstable inside the geosynchronous orbit for R < 6.6 RE both in the case of a diffusive equilibrium (DE) model (Lemaire 1999) and in the case of Pierrard and Lemaire (2001) exospheric model for R > 2.3 Re . Since the latter model fits the empirical equatorial density distribution of Carpenter and Anderson (1992), the later conclusion holds also in that case. Similar conclusions are obtained with other empirical models characterized by larger density gradients (e.g., Reinisch et al. 2004). The existence of a static equatorial plasmasphere seems therefore to be questionable, even in a saturated stage following a long period of quiet geomagnetic conditions.

3.2.3 Implications of a Plasmaspheric Wind and Future Refinements

Although the type 2 quasi-interchange or translational mode is considered primarily to lead to plasma motions parallel to the magnetic field line, it should be pointed out that it does not imply strictly parallel motions: The motion necessarily acquires also a transverse component (Ferrière et al. 1999). In that sense, this would be compatible with the concept of plasmaspheric wind introduced by Lemaire and Schunk (1992), consisting of a slow and permanent cross- B transport of plasma from the inner to the outer equatorial regions of the plasmasphere accompanied by a field-aligned upward ionization flow.

Both the diffusive and exospheric models used by André and Lemaire (2006) are oversimplified models, since they assume the plasma distribution to be stationary, i.e., time-independent and with no net mass flow along magnetic field lines, whereas for example asymmetries in the geomagnetic field line geometry and in the boundary conditions at the feet of the flux tube are expected to lead to dynamic inter-hemispheric plasma flows. Their application suffers from large uncertainties due to the use of various simplifying assumptions but interesting conclusions can be drawn from their simplified formulation. Further refinements of their application will have to include, in particular, the coupling of low-energy and high-energy plasma in the plasmasphere, the ionospheric effects arising at the foot of the flux tubes, as well as the observed plasma corotation lag (Burch et al. 2004).

In a recent comparison of measured radial abundance profiles from EUV observations with predicted profiles from the Sheffield University Plasmasphere Ionosphere Model, SUPIM (Bailey et al. 1997) during a period of quiet geomagnetic activity, Sandel and Denton (2007) noted some disagreement between this model and the EUV observations beyond R = 4 Re that are possibly a signature of physical processes not accounted for in the model. A plasmaspheric wind is one possible process whose inclusion in future magnetospheric convection models might resolve the model-observation disagreement. Recent analysis of cold ion measurements obtained in the plasmasphere by CIS onboard CLUSTER may have provided the first experimental confirmation of such a plasmaspheric wind (Dandouras 2008; Darrouzet et al. 2008, this issue).

3.3 Transport of Plasmaspheric Material Caused by Ultra Low Frequency Waves

The study of transport of plasmaspheric material in plumes is not only important to understand the plasmasphere dynamics but also to understand the physics of the magnetic reconnection at the magnetopause (Borovsky and Hesse 2007). The presence of dense plasma modifies the local reconnection rate by lowering the Alfvén velocity. Some escaping particles could subsequently move toward the magnetotail and subsequently be recirculated into the inner magnetosphere during substorms. Thus the formation of plumes is of vital importance if we are to understand the mass budget in near-Earth space (Chappell et al. 1987).

The formation of a plasmaspheric plume is often discussed in terms of global plasma transport during times of large geomagnetic activity such as substorms and/or storms (Gre-bowsky 1970; Elphic et al. 1996) including cases with IMF directed southward (negative BZ) (Goldstein et al. 2002; Chen and Moore 2006). However, observations during quiet and moderate periods of cold plasma in the afternoon sector, well outside the nominal plas-masphere have remained puzzling (Chappell 1974; Carpenter et al. 1993; Matsui et al. 1999; Yoshikawa et al. 2003). Are these cold plasma regions residual density features that linger long after the recovery after storms and substorms? An alternate explanation involves ULF waves, which are often excited by the variation of the solar wind dynamic pressure (e.g., Farrugia et al. 1989) and/or by the velocity shear at the magnetopause (e.g., Engebretson et al. 1995). All current electric field models include a flow stagnation point somewhere on the duskside, where E x B drifts are weak. At this stagnation point (and possibly in a large area on the duskside during quiet conditions), the wave field might exceed the background field, and thus could dominate cold plasma transport. This idea, suggested by Carpenter and Lemaire (2004) as part of the proposed plasmasphere boundary layer (PBL) concept, could help explain the prevalence of cold plasma at large L values in the afternoon sector even during extended quiet conditions. Chen and Wolf (1972) considered time variable electric fields but with longer time scale (8 hours) than ULF waves. Grebowsky and Chen (1976) considered a case applying spatially variable electric fields to examine transport of plasmas. Recently, Adrian et al. (2004) simulated ULF waves related to the formation of radially bifurcated plasmaspheric features.

Matsui et al. (2000) considered this problem with temporally variable electric fields. In this study, a test particle simulation was performed in an idealized mathematical model to examine whether ULF wave fields could cause cold plasma to be transported to the magnetopause. The background electric potential $ in the inertial frame is given by Volland-Stern model (Volland 1973; Stern 1975) without ionospheric shielding:

where C1 is a constant to specify the size of the corotation potential, L is the value of McIlwain's parameter, C2 is a variable to specify the intensity of the dawn-dusk electric field component, and 0 is the MLT expressed in radians. In addition, the wave electric potential A$ defined by the following equation is added:

where a is the wave amplitude, b is the spatial size of the wave potential taken as 1.0 x 104 Re -2, r is the distance from the stagnation point, t is time, and T is the wave period, taken as 300 s. It is assumed that there is no magnetic field perturbation, which is true for the fundamental mode of standing Pc 5 waves. These waves have a typical period of ~300 s. The location of the stagnation point is set at L = 6 and 18:00 MLT. The particle orbit without perturbation is traced from 15:00MLT, which leads to the location L = 5.99 at 18:00MLT. The orbit is traced by a Runge-Kutta method with a time step of 0.5 s.

The orbit around the stagnation point is shown in Fig. 6a. This orbit is inside the last closed equipotential (LCE) and rotating around the Earth, although the period is 55.9 hours, longer than 1 day due to a finite convection electric field in the vicinity of the stagnation point. It should be noted that this model is not based on physical processes but mathematical because the calculation relies on the existence of a singular mathematical point. Moreover,

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.0: X SM (Re) X SM (Re)

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