Introduction

The Earth's intrinsic dipolar magnetic field is separated from the ambient magne-tosheath field by the magnetopause, a thin current-carrying plasma surface layer. The surface current is what is required from Ampere's law to bring about the observed net change in the field across this plasma boundary. A schematic drawing of the magnetopause surface as well as the magnetosphere and its various features is shown in Figure 8.1.

The first unambiguous observations of the magnetopause were made by the Explorer 12 spacecraft, as reported by Cahill and Amazeen (1963). Figure 8.2, taken from their paper, is an excellent illustration of the magnetic field behaviour typically seen in a dayside crossing of the magnetosphere, magnetopause, and magne-tosheath. The magnetospheric field magnitude decreases with increasing distance from Earth, but less rapidly than a dipole field in a vacuum. Indeed, the surface current induces a magnetic field that adds to the geomagnetic field inward of the magnetopause, while it prevents the Earth's field lines from penetrating into the

1 Belgian Institute for Space Aeronomy, Brussels, Belgium

2Space Science Division, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, UK

3Mullard Space Science Laboratory, Holmbury St. Mary, Dorking, Surrey, UK

4Thayer School of Engineering, Dartmouth College, Hanover, NH, USA

5Max-Planck-Institut für extraterrestrische Physik, Garching, Germany

6Swedish Institute of Space Physics, Uppsala, Sweden

'International Space Science Institute, Bern, Switzerland

8Swedish Institute of Space Physics, Kiruna, Sweden

9CETP/IPSL/UPMC, Velizy, France

Space Science Reviews 118: 231-320, 2005. DOI: 10.1007/s11214-005-3834-1

© Springer 2005

Figure 8.1. Three-dimensional cutaway view of the magnetosphere. The light blue outer surface is the magnetopause, its boundary layers are shown in darker blue. Magnetic field lines are shown in blue, electric currents in yellow. The polar region where the magnetic field lines converge is the polar cusp. The bow shock has been omitted for clarity. (Adapted from Kivelson and Russell, 1995).

Figure 8.1. Three-dimensional cutaway view of the magnetosphere. The light blue outer surface is the magnetopause, its boundary layers are shown in darker blue. Magnetic field lines are shown in blue, electric currents in yellow. The polar region where the magnetic field lines converge is the polar cusp. The bow shock has been omitted for clarity. (Adapted from Kivelson and Russell, 1995).

magnetosheath, where the field is relatively weak. Just before the magnetopause is encountered, at about 8.2 RE, the measured field is about twice that of the dipole (see e.g., Chapman and Bartels, 1940). In the magnetosheath, the field intensity is usually lower and has larger variability, reflecting the fluctuations in the interplanetary magnetic field. The increase in the field elevation angle, a, just earthward of the magnetopause suggests the presence of a plasma boundary layer, presumably connected to the dayside ionosphere via field-aligned (Region 1) currents. Due to lack of plasma measurements, the actual presence of such a layer in the Explorer 12 pass shown in Figure 8.2 remains uncertain. But measurements by later spacecraft have established the frequent presence of a plasma boundary layer immediately Earthward of the magnetopause. It is now called the 'low-latitude boundary layer' or LLBL (e.g., Eastman et al., 1976; Sckopke et al., 1981).

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Figure 8.2. Magnetic field measured by Explorer 12 on September 13, 1963. Measured (dashed curve) and dipole (solid curve) field magnitudes are shown, along with measured field elevation angles, a, and longitude angles y. (From Cahill and Amazeen, 1963).

Figure 8.2. Magnetic field measured by Explorer 12 on September 13, 1963. Measured (dashed curve) and dipole (solid curve) field magnitudes are shown, along with measured field elevation angles, a, and longitude angles y. (From Cahill and Amazeen, 1963).

8.1.1 Orientation, motion, and thickness

The orientation of the magnetopause surface is of importance, not only for studying its geometry (e.g., surface wave phenomena), but also for the purpose of establishing whether there is a magnetic connection and associated plasma flow across this boundary. Under certain assumptions, the orientation can be deduced from single-spacecraft measurements of the magnetic field alone. If one believes that no normal field and flow components were present, that is, if the magnetopause was a tangential discontinuity (TD), one may obtain the vector normal to the magnetopause simply as the cross product of the field vectors on opposite sides of the layer. If one wants to allow for the possible presence of a normal field component, then one can adopt a one-dimensional model of the local magnetopause structure where the orientation of the current layer is also unchanged during the crossing. Under these assumptions, the condition V • B = 0 requires the normal field component to be strictly constant during the crossing. The orientation can then be found as that direction in space along which the component of the magnetic field has minimum variance (Sonnerup and Cahill, 1967; Sonnerup and Scheible, 1998). Provided the 1D assumption holds and the variance matrix is not near degeneracy, this method, referred to as MVAB (Minimum Variance Analysis of the magnetic field, B), can in principle show whether or not a nonzero normal field component was present during a crossing. With a few exceptions, it was found that this component is so small that it is hidden within its uncertainty estimate; plasma flow across the magnetopause is even more difficult to establish and requires precise knowledge of magnetopause orientation and plasma velocities.

In a time-independent picture, the magnetospheric field magnitude should decrease monotonically with increasing distance from Earth up to the magnetopause. This is not what is seen in Figure 8.2, where magnetic field magnitude fluctuations are found from 6.7 RE outward, followed by a steep decrease to a more or less constant plateau value that persists out to the magnetopause. These features could be spatial, temporal, or both. In particular, they could be the result of magnetopause motion. From a single-spacecraft magnetic field record alone, one cannot tell for sure, but frequent observations of multiple magnetopause traversals during a single outward or inward pass, by Explorer 12 and later single-spacecraft missions ever since, have provided convincing evidence that the magnetopause is almost always in rapidly changing inward-outward motion with typical speeds of tens of km/s. The magnetopause does not have much mass, so its speed can change abruptly, and by large amounts, in response to even minor changes in the magnetosheath pressure. The amplitude of the radial motion of the magnetopause can be several Earth radii. Surface waves can also lead to multiple crossings of the boundary.

An important parameter is the thickness of the magnetopause, which determines the intensity of the currents in the layer and the steepness of the density gradients and velocities across it. Given the orientation of the magnetopause, its thickness can be inferred from the boundary speed and the crossing duration. The results, however, require that one makes certain assumptions (e.g., planarity of the layer), and they critically depend on a precise determination of the magnetopause speed. Reliable determinations of the magnetopause speed and, from it, the thickness, became possible in the late seventies. The time difference between the passage of the magnetopause over the ISEE-1 and ISEE-2 spacecraft led to speeds ranging from near zero to more than 200 km s^1, with an average around 40 km s^1. The crossing durations then implied thicknesses ranging from 200 km to 1800 km, with an average of about 800 km (Berchem and Russell, 1982). Reliable measurements of the lowest-order plasma moments also became available from ISEE and, later on, from the Ampte spacecraft. This led to the development of single-spacecraft methods for determination of the magnetopause speed. The earliest of these methods is referred to as MVAB/HT. It is based on the existence of a deHoffmann-Teller (HT) frame, a concept first used by deHoffmann and Teller (1950) in the study of shocks and applied to the magnetopause by Aggson et al. (1983); for a modern version, see (Khrabrov and Sonnerup, 1998a). The HT frame moves with velocity VHT relative to the spacecraft, in such a manner that, when transformed to this frame, the measured plasma flow becomes as nearly field aligned as the data permit. The component of VHT along the magnetopause normal then represents the inward/outward motion of the magnetopause, while the tangential components describe motion of field structures along the magnetopause, past the observing spacecraft. Later on, a method referred to as Minimum Faraday Residue (MFR) analysis was developed (Terasawa et al., 1996; Khrabrov and Sonnerup, 1998b). Faraday's law requires the electric-field component tangential to a one-dimensional layer, of fixed structure and moving at constant speed, to be constant. This property, along with V • B = 0, is used in MFR to produce a prediction of both the normal vector and the speed of the magnetopause. Tests of these methods against the timing of magnetopause passages over the two closely spaced spacecraft Ampte/IRM and Ampte/UKS revealed substantial uncertainties in the single-spacecraft predictions as well as difficulties in the timing results, the latter caused by difficulties in identifying features in the magnetic field profiles that are needed for an unambiguous determination of the time delays (Bauer et al., 2000).

Data from Cluster taken during the passage of all four spacecraft through the magnetopause, together with certain assumptions, in principle permit an unambiguous determination of magnetopause speed, thickness, and orientation. All methods utilised in this book assume the magnetopause surface to be flat and to maintain a fixed orientation in the entire time interval encompassing the four crossings. In the so-called Discontinuity Analyser, or DA for short (e.g., Dunlop and Woodward, 1998), the magnetopause orientation is taken from a single-spacecraft method such as MVAB. From the time delays between the crossings and the spacecraft separation vectors, the magnetopause speed and its variations can then be determined and, from the duration of each crossing, the corresponding magnetopause thickness can be found. In another method, called the Constant Velocity Approach, or CVA (Russell et al., 1983; Schwartz, 1998), the magnetopause speed is assumed constant during all four crossings. The magnetopause orientation, speed, and a thickness for each crossing, can then be determined from the timing and the crossing durations. In the more recent Constant Thickness Approach (CTA - see Haaland et al., 2004a), the magnetopause thickness is assumed to be the same for all four crossings. This method allows determination of magnetopause orientation and thickness, as well as the speed during each of the four crossings. A combination of CVA and CTA, called Minimum Thickness Variation, or MTV, has also been developed (Paschmann et al., 2004). In this method, the orientation of the magnetopause is a weighted average of the results from CVA and CTA. The magnetopause velocity and thickness are allowed to vary from crossing to crossing but the variation of the latter is minimised. Comparisons of results from various single-spacecraft methods with those from these multi-spacecraft methods are presented in Sections 8.2 and 8.3. Included in these comparisons are two new methods, first tested with Cluster data, that allow determination of the magnetopause normal and speed, either from the conservation of mass, or from the conservation of charge. The former method, referred to as Minimum Massflux Residue (MMR) analysis (Sonnerup et al., 2004a), requires measurements of plasma velocity and density from a single spacecraft. It was made practical, for the first time, by the precision of the Cluster plasma measurements. The latter method, called Minimum Variance of Current (MVAJ) (Haaland et al., 2004b), utilises the curlometer capability of Cluster, i.e., the capability to determine the electric current density from measured magnetic fields on the four spacecraft by use of Ampere's law (also a Cluster 'first', see Section 8.3).

8.1.2 Magnetopause substructure

Over the years, evidence has been mounting that the dominant transport of plasma across the magnetopause is caused by magnetic field reconnection. This process is discussed in detail in Phan et al. (2005). It implies an X-line geometry at the magnetopause. The magnetopause then has a small, non-zero normal magnetic field component with opposite sign on either side of an X-line or reconnection site. Some process must decouple electrons and ions from the magnetic field, but this process needs only be present in the immediate vicinity of the reconnection site. Elsewhere on the magnetopause, it is the normal magnetic field component accompanying reconnection that creates a direct magnetic coupling across the magnetopause and that allows solar-wind plasma to flow into the LLBL. The magnetopause then has a compound structure, with a rotational discontinuity (RD) as its principal ingredient; in a magnetohydrodynamic description, this rotational discontinuity is a large-amplitude standing Alfven wave. Direct measurements of plasma flow across magnetopause are difficult, as noted earlier. An indirect approach is to perform the so-called Walen test, which is, in effect, a test of the tangential stress balance at the magnetopause (Paschmann et al., 1979). In its modern form, the Walen test consists of plotting the velocity components measured during a magnetopause crossing, and transformed to the HT frame, against the corresponding measured components of the Alfven velocities. A regression slope of + 1 indicates Alfvenic flow parallel to the magnetic field while a slope of — 1 signifies antiparallel flow at the Alfven speed. Assuming the flow is directed earthward, the former case indicates an inward, the latter case an outward directed normal magnetic field component. Thus the sign of the slope indicates on which side of the reconnection site the observations were taken. The above results are based on a one-dimensional description of the magnetopause structure during reconnection. Strong 2D effects, such as magnetic islands embedded in the current layer, or, perforce, 3D structures, can lead to Walen slopes that deviate significantly from unity. Here, Cluster can help to identify and assess the existence of additional terms in the tangential stress balance in order to understand such deviations (Section 8.4.1).

It is therefore important to assess the geometry of the current layer. The single-spacecraft methods for the determination of magnetopause orientation and motion are based on the assumption that the current layer is a one-dimensional, time-independent structure with a fixed orientation and moving at constant speed during the crossing. If these assumptions are well satisfied, the one-dimensional field and plasma structure of the layer can be easily reconstructed. If the orientation and 1-D structure are both time-stationary but the velocity is not, then the structure can nevertheless be reconstructed, in the simplest case by allowing for a constant acceleration of the HT frame or, for more complicated motions of the HT frame, by doing the HT analysis with a sliding data window. Local reconstructions of magnetopause structures, based on integration of the Grad-Shafranov equation, have been performed, starting with work by Sonnerup and Guo (1996), and followed by several more recent studies. In this technique, measurements along a single spacecraft trajectory are used as spatial initial values for the integration. Validation of the technique by use of Cluster data is discussed in Section 8.4.2, along with a technique for ingesting data from all four Cluster spacecraft into the reconstruction.

8.1.3 Physical properties

Although much progress has been made since the Explorer 12 epoch, our knowledge of what physical processes operate in the magnetopause current layer and determine its structure remains incomplete. One of the oldest unanswered questions is whether, and in what circumstances, an intrinsic electric field along the magnetopause normal remains after the external electric fields have been removed by, in effect, making a transformation into the HT frame. Ferraro (1952) concluded that, in a TD, i.e., in the absence of a magnetic field component along the magnetopause normal, such an electric field , directed outward from the magnetosphere, is required because of the vastly smaller gyro radii of impinging solar wind electrons compared to those of the impinging ions. This field would bring about an equal penetration depth of the electrons and ions. But such an intrinsic field is not a necessity: particles trapped in the magnetopause layer can in principle nullify the effect. If the magnetopause has a nonzero normal magnetic field component, both solar wind electrons and ions can flow across the magnetopause. An intrinsic electric field may nevertheless be required to assure that the electric current has no component along the magnetopause normal. Prior to Cluster, an intrinsic electric field along the magnetopause normal had not been seen. But in Section 8.5.2, new Cluster observations are presented that provide convincing evidence that such a field is sometimes present.

In an ideal magnetohydrodynamic (MHD) description, which is realistic to lowest order since the plasma is collisionless, the magnetopause is a tangential discontinuity. This means that there should be no particle transfer through the boundary. This is almost true, but not completely: A fraction of the solar wind flux does penetrate the magnetopause. Therefore, some deviation from the tangential discontinuity model should be present: the plasma behaves in a non-ideal way. Diffusive processes based on wave-particle interactions could be responsible for such behaviour. One possibility is that this type of diffusion permits reconnection to occur, so that the boundary becomes a rotational discontinuity with associated direct plasma entry by flow along the normal-field component. Alternatively, if wave-particle interactions are strong enough, they could lead to diffusive plasma entry over large portions of the magnetopause surface. When magnetic field measurements with high time resolution became available in the late sixties and early seventies, it quickly became apparent that the magnetic fluctuation level is high in the magnetopause. Later on, results from ISEE established and quantified the persistent presence of enhanced magnetic and electric field turbulence over a wide range of frequencies in the magnetopause (Gurnett et al., 1979). The nature of the waves and instabilities causing the fluctuations has been studied extensively, but their importance, if any, in producing macro-scale diffusive fluxes of mass, momentum, and energy from the solar wind across the magnetopause into a low-latitude boundary layer remains in doubt. Cluster results concerning these fluctuations, including their origin, relationship to free energy sources such as current density and density gradients, and their effectiveness in producing diffusion in large regions of the magnetopause surface, are presented in Section 8.5.4.

8.1.4 Meso-scale phenomena

On a somewhat larger scale, the magnetopause and the adjacent low-latitude boundary layer are usually not planar. With single-point measurements, one always has to make assumptions about the broader geometrical context. Cluster, by design, produces a picture of the local environment, as it is able to resolve at least part of the spatio-temporal ambiguity. For instance, bipolar signatures in the component of the magnetic field normal to the magnetopause surface were believed by some to be 'flux transfer events' (FTEs), caused by brief bursts of magnetic reconnection. Others interpreted these events simply as kinks in the magnetopause surface, travelling past the observing spacecraft. Simultaneous observations by Cluster spacecraft located on the two sides of the magnetopause have unambiguously shown that such FTEs represent 'bulges' or local swellings of the magnetopause, rather than magnetopause ripples (Section 8.6). Another example is surface waves (Section 8.7) that travel downtail along the magnetopause surface. Yet another example is the discovery of magnetosheath-like plasma structures inward of the low-latitude boundary layer, but spatially detached from it (Section 8.8). 'Empirical reconstruction' techniques, originally developed for single-spacecraft data (Paschmann et al., 1990; De Keyser et al., 2002), can be used in many situations to combine the four-spacecraft data from all instruments into a consistent picture of the topology of the magnetopause and the low-latitude boundary layer (see Sections 8.7 and 8.8).

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