A qualitative study of the reconnection between the Earth ' s magnetic field and an interplanetary field of arbitrary orientation

To the present date only the reconnection process for exactly antiparallel fields has been discussed in detail. In magnetospheric terms this restricts us to the consideration only of southward interplanetary fields. The qualitative study presented here shows how reconnection takes place between arbitrarily oriented fields in infinite and finite geometries. The process may best be thought of as a continuous exchange of field-line partners during the time a field line maps into the diffusion region. In a finite geometry the diffusion regions lie on field lines which connect neutral points of the field configuration, and along which a potential drop is imposed. Although the discussion here centers principally on the magnetosphere, the described reconnection process and field topology should also be applicable to other astrophysical problems.


INTRODUCTION
Reconnection between the Earth's magnetic field and the interplanetary field is thought to be the basic driving mechanism of plasma convection within the magnetosphere [Dungey, 1961].However, only the special case of reconnection between exactly antiparallel fields has to date been discussed, i.e., reconnection between the Earth's field and an interplanetary field directed exactly southward.The purpose of this paper is to obtain a qualitative understanding of the reconnection process which occurs between nonantiparallel fields and to determine how this can be applied to the magnetosphere.This objective is achieved by a two-fold approach.First, the reconnection of nonantiparallel magnetic fields is considered in a system which is infinite along the X line, by a simple modification of the solutions of Yeh and Axiord [1970] and Sonnerup [1970].However, the qualitative results we obtain should not be dependent on the particular solution we choose to.consider.Second, the expected topology of the open magnetosphere in the general case is discussed, following Dungey [1963].We repeat this discussion here in the interests of obtaining a self-contained paper.Our method, as in Dungey [1963], is to Copyright (•) 1973 by the American Geophysical Union.
superpose linearly a uniform magnetic field of arbitrary orientation with a dipole field.
We then combine these two approaches to obtain a consistent qualitative picture of the general reconnection process for arbitrary orientations of the interplanetary magnetic field and the general topology of the magnetosphere.Several authors have recently considered this process briefly, but do not seem to have been led to the picture presented here [Stern, 1972;Gonzales and Mozer, 1973].
Even though the discussion here will center specifically on the magnetosphere, it should also be applicable with suitable modification to other systems of astro.physicalinterest.

FIELDS IN AN INFINITE SYSTEM
No complete solution of the magnetic field structure and plasma flow in the vicinity o.f an X neutral line has yet been accomplished.Sonnerup [1970] and Yeh and Axiord [1970] have considered the problem within the framework of collision-dominated fluid theory (the plasma conductivity • being taken to be a scalar) and have considered separately two regimes within the total solution.These two.regions are the convection region well away from the neutral line, and the diffusion region immediately surround-ing the neutral line.In the convection region the plasma may be considered to be infinitely conducting, so that Ohm's law becomes E = --vxB/c and the field lines are tied into and convect with the plasma.This approximation breaks down in the diffusion region (v x B -• 0 at the neutral line) and finite conductivity effects become important.In this region, then, the field lines are no longer tied to the plasma and can become broken and reconnected.
However, no detailed matching between these two regions has been attempted.
Within the framework of collision-free plasmas there as yet exist no self-consistent solutions of the field and flow near an X neutral line; only neutral sheets and other simple geometries have been considered [Harris, 1962;AIfvdn, 1968;Eastwood, 1972;Cowley, 1973].We therefore consider here the extension of the fluid solutions of Sonnerup and Yeh and Axford for an X neutral geometry to the case where the field lines flowing towards the X line are no longer antiparallel.
In We now consider the addition of a uniform magnetic field B, to these solutions, the field direction being parallel to the neutral line.This does not affect the dynamic equilibrium of the system since the currents are unchanged and j x B, = 0.However, we must also introduce a new electric field component which lies in the plane of the flow lines given by E' = -v x B,,/c where v is the fluid velocity of the new (and old) systems.This field is required by Ohm's law, and is the field required to maintain E. B = 0.The condition that this field has zero curl everywhere is simply div v -0.Thus within the framework of incompressible plasma flow we may simply add a uniform magnetic field and its corresponding electric field to any solution for reconnection of antiparallel magnetic fields and hence obtain a solution for reconnection of nonantiparallel fields.Even in this case more complicated solutions are possible in which the magnetic field parallel to the neutral line is changed in strength across the shocks of the convection region leading to changes in plasma flows parallel to the X line.These will not be discussed, the simplest possible model being sufficient to obtain a qualitative understanding of the processes involved which is all we require here.
The addition of a uniform magnetic field parallel to the X line to the field in the convection region simply twists the field in opposite directions on either side of the diffusion region in an obvious manner.The projection of the field lines on planes perpendicular to the X line remains unchanged.The latter statement is also.true for the field lines in the diffusion region.However, within the diffusion region the perpendicular field components become progressively smaller as the X line is approached while the parallel field remains of constant strength.The field line twisting in the diffusion region is therefore more pronounced than in the convection region.
Figure 1 shows the field line structure near the X line, the field lines being projected onto the plane perpendicular to the X line.The perpendicular field components near the X line depend linearly on distance from the X line, i.e., Bi --oliixi where a• = OB•/Ox•l•=o since the field may be Taylor-expanded about the X point, and since the current there is nonzero.
The field lines are thus systems of hyperbo,lae.The field lines which map, to the X point in this projection will be referred to as the separatrices between the inflow and outflow regions of the system.In order to apply these qualitative results to the magnetosphere we must first consider how this system must be modified in order that field lines mapping out to infinity may be incorporated into a finite geometry.This requires a study of the general topology to be expected for an open magnetosphere and is the subject of the next section.

MAGNETOSPHERE
The expected topology of the open magnetosphere in the general case is here investigated by linearly adding a uniform magnetic field o.f arbitrary orientation to a dipole field.This procedure and its results have previously been described by Dungey [1963] but will be repeated here in the interests of selfcontainment.
The superposition of two fields in this manner is valid only for media with zero conductivity, so that the resultant magnetic fields we produce are not expected to be valid magnetospheric models.However, we expect the field to be topologically correct, particularly if we can show them to be consistent with the reconnection picture discussed above.Such proves to be the case.
The conventional picture of the reconnection of the Earth's field with a purely southward interplanetary field may be represented within this framework by the addition of a uniform field parallel to the dipole moment vector as sketched in Figure 6.An X neutral ring is generated in the equatorial plane at a certain radial distance which corresponds to the dayside and nightside X-type neutral lines.

The field lines which map from this neutral ring are the separatrix field lines of the X configuration and form the boundary surfaces between .the closed, o,pen, and interplanetary field lines. Surface ,4 is the interface between closed and open lines generating in three dimensions a doughnut-shaped boundary, while surface B forms the interface between open and interplanetary lines and generates two cylindrical regions connected to the north and south polar caps (the tail lobes).
This situation is rather a special case; for any other orientation of the uniform field a neutral ring is not generated, rather two neutral points are produced.These points lie in the plane containing the imposed magnetic field direction and the dipole magnetic moment vector.Only in this plane does the imposed field have no azimuthal component; the dipole field has no azimuthal component in all planes containing the magnetic moment vector.In order to discuss the three-dimensional topology we must therefore consider the magnetic field configuration near a neutral point.Close to the null the field may be expanded in a Taylor series so, to lowest order, we can write a matrix expression for the field components of the form

IN THE MAGNETOSPHERE
Having made the above identification of the fieldline ring with the center of the diffusion regions on the dayside and nightside, we can now use the results of the second section to describe qualitatively the reconnection process for the general case.In order to keep the figures as simple and consistent as possible we describe the process in terms of the previous fieldline structures of Figures 9, 10, 11, and 12, and we will not attempt to represent the sweeping back of the reconnected field to produce an antisolar tail, the day-night asymmetry of the magnetosphere, etc.These are the MHD aspects of the.problem and will not change the fundamentals of the processes discussed here provided that the.topology we have arrived at and its interpretation in terms of the second section are correct.In addition we discuss only dayside reconnection; the nightside process is the same but reversed in time-sequence to that which we discuss here.
In Figure 14 this limit the field lines and plasma parameters between the shocks are uniform and the system is amenable to reasonably simple analytic calculation.
Fig. 1.The field line structure near the X line in the diffusion region: the proiection onto a plane perpendicular to the X line.

FigFig. 7 .Fig. 8 .
Fig. 5. Showing in plan projection the two inflow convection region field line A, B on opposite sides of the diffusion region which are ultimately connected together, and the resultant reconnected field lines, C and D.

Figure 13 .
Figure13.An X-type configuration is generated, but the center of the X is not a neutral point since there exists a magnetic field component perpendicular to the plane of the projection.The field lines along the separatrices are the field lines in the boundary surfaces between closed, open, and interplanetary lines, and map from either the Earth or the interplanetary medium into one or another of the neutral points.The equivalence of this situation and that discussed above for an infinite X-type region should now be clear.In the infinite system the field lines of the separatrices are extended indefinitely along the line running down the center of the X; in a finite system they do not extend to infinity but map into neutral points.The field-line ring joining the two neutral points is simply the field line which runs along the center of the X in the infinite system, and thus defines the centers of the day and night diffusion regions.For the finite system, therefore, the diffusion regions lie between and are terminated at neutral points.

FigureFig. 14 .
Fig. 13.Projection of the fi'eld lines near the ring onto a plane perpendicular to the ring at noon.