Space Plasma Physics: The Study of Solar-System Plasmas (1978)

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Magnetic Field Reconnection Bengt U. 0. Sonnerup Radiophysics Laboratory Dartmouth College Hanover, New Hampshire 03755 879

880 1. Introduction Magnetic-field reconnection has been proposed as a basic energy-con- version process which may occur in many parts of the universe. Its primary function in the cosmic scheme is to prevent the build up of excessive amounts of magnetic energy in association with intense electric current sheets formed in highly conducting plasmas. The reconnection process is thought to cause a relaxation of such configurations, either partially or completely, and either continuously or sporadically, toward their lowest energy (current-free) state. The magnetic energy released during reconnection is converted into kinetic and internal energy of the plasma. The process causes the transfer of magnetic flux and plasma from topological cells with excessive flux to cells deficient in flux. This fact provides the basis for a precise definition of reconnection to be given in Section 3.4. Reconnection is also often referred to as magnetic field merging or magnetic field annihilation but, as will be seen, the three terms should not be used synonymously. Figures 1-5 show examples of cosmic current sheets where reconnection • may occur. Figure 1 represents the field produced by two photospheric dipoles 94 which gradually move toward each other . In the absence of reconnection, a current sheet of increasing length forms between the dipoles in the highly conducting solar atmosphere above them. If reconnection suddenly sets in, the magnetic field may relax toward a potential one, as indicated in the last picture of the sequence. This represents a possible, perhaps even plausible, 57 91* mechanism for a solar flare ' ". Figure 2 illustrates current-sheet formation caused by the stretching of magnetic loops on the sun during rapid plasma 14 ejection . Figures shows current sheets* separating interplanetary magnetic 123 •sectors with different polarity ' . Figure 4 shows the magnetopause current stretching of magnetic loops on the sun during rapid plasma ejection

881 layer, formed as the solar wind presses the interplanetary magnetic field against the terrestrial field, as well as the tail current sheet, resulting directly from tangential stresses exerted by the solar wind on the magnetic field in the two tail lobes. The topology shown in the figure was first proposed by Dungey32. Figure 5 shows the magnetic field configuration expected for a rapidly spinning planetary magnetosphere such as that of Jupiter45'57. All of the above examples, and many possible other ones, such as 70 7R R7 supernova remnants , accretion disks , and galactic dynamos , illustrate cosmic situations in which magnetic field reconnection may occur. However, we do not know with certainty that the process does in fact take place in any or all of these geometries. And if it does take place, we still do not know much in detail about its dynamics. Are both continuous and sporadic re- connection possible, and if so, what are the plasma parameters and geometries in which these two modes are to be expected? What are the conditions for onset of reconnection? What is the energy conversion rate? In spite of twenty years of theoretical effort, recently summarized in a brilliant manner by Vasyliunas , as well as several laboratory experiments^' ' ' ' and computer experiments 3~ ' f no universal agreement exists concerning the answers to most of these basic questions. Even in the most recent liter- ature, opinions about the cosmic occurrence of the process range from full 1 1 ft P ? acceptance to outright rejection * . On the other hand, there is conclusive evidence that reconnection occurs in tokamaks and other fusion ^9? ^9 devices as an end product of the resistive tearing-mode instability ,*« One of the difficulties with the cosmic reconnection research effort to date is that to a large extent it has lacked the detailed integration of theoretical and experimental work essential to the effective advancement of our knowledge concerning the process. On the one hand, an extensive but

882 118 rather abstract body of theoretical work exists , concerned primarily with the steady-state process and utilizing the fluid description. The latter is likely to be inadequate for the analysis of certain critical aspects of the process. On the other hand, laboratory experiments J indicate the importance of sporadic reconnection. However, the plasma param- eters in these experiments are sufficiently different from those prevailing in most cosmic applications so as to pose serious difficulties in the appli- cation of the laboratory results in cosmos. A wealth of indirect observational evidence in the terrestrial magnetosphere, both at the magnetopause and in the tail, suggests that if the process occurs, it is likely to do so sporadi- cally rather than continuously. In current observational magnetospheric work, the reconnection process is often invoked to account for a great variety of observations but with little effort to check theoretical predictions in detail or to consider alternate interpretations. The result is that the observational case for the occurrence of the process in the magnetosphere is not as solid as it might be. For other astrophysical applications, the situa- tion is even worse. On balance, our best opportunity for learning about reconnection as a viable cosmic energy conversion process is likely to be in the earth's magneto- sphere. It is difficult to account for the overall dynamic behavior of the magnetosphere without invoking time-dependent transfer of magnetic flux from closed to open field lines and vice versa. And such transfer is one of the principal features of the reconnection process. The magnetosphere offers the unique advantage of permitting in situ plasma and field observations with probes that are much smaller than relevant plasma length scales. Thus an intense magnetospheric observational program with a focus on reconnection, coupled with a theoretical effort aimed at the geometries and plasma parameters

883 prevailing at the magnetopause and in the magnetotail would seem to have high potential for success. What is learned about reconnection in the magneto- sphere may then be applied to other cosmic systems which do not permit in situ observations. It is seen that a research effort focused on magnetospheric reconnection is likely to lead to significant advances in our understanding of many other astrophysical and cosmic problems. It is the purpose of this paper to provide a concise qualitative summary of the present state of reconnection theory and observations, with special reference to the earth's magnetosphere, and to bring into focus a number of specific problems and questions concerning the reconnection process in its magnetospheric application which should be studied both theoretically and observationally. The organization of the paper is as follows. First, a number of basic concepts are introduced via a qualitative discussion of steady two- dimensional reconnection in Section 2, and of possible nonsteady and/or three- dimensional configurations in Section 3. With this background, the more de- tailed technical discussion in subsequent sections can be presented in a com- pact fashion. Specifically, Section 4 deals with the external flow region, which is usually described in terms of the fluid approximation. Section 5 discusses one-fluid and two-fluid approaches to the.plasma dynamics in the diffusion region, which is the site of the field reconnection process itself, and in which plasma microinstabilities are likely to be important. Section 6 discusses possible mechanisms for the generation of finite resistivity in the diffusion region and for the onset of reconnection. Section 7 contains a brief summary of present observational evidence for or against magnetospheric reconnection. Finally, Section 8 provides a summary of outstanding problems along with certain recommendations concerning the organization of future re- connection studies.

884 Three comments should be made about the scope of the paper. First, it does not attempt to provide a historical perspective. Rather it is organized to elucidate basic physical principles and recent significant approaches to the development of adequate theories of cosmic reconnection. Second, the paper does not attempt to cover all direct and indirect evidence for or against reconnection in the magnetosphere, on the sun, or elsewhere in cosmos. Third, the paper does not deal with applications in tokamaks and other labo- ratory devices where the physical boundary conditions are such that spatially periodic behavior results. It should be stressed, however, that vigorous interaction between fusion plasma physicists and cosmic physicists on the problem of reconnection is likely to be of substantial benefit to both groups.

885 2. Plane Steady-State Reconnection; A Qualitative Picture In order-to develop an understanding-of certain basic features of magnetic field reconnection, it is desirable first to examine the simplest possible qualitative model of the process. To this end, consider the two-dimensional time-independent electromagnetic field configuration shown in Figure 6. The magnetic field B is confined to the xy plane and has a hyperbolic (/if-type) A null point at the origin. An electric field of the form E_ = E z is present along the direction perpendicular to the plane of the figure. Since V x £ = o in a steady state, and since partial derivatives with respect to z are assumed to be zero, it follows that E is independent of x and y, i.e., the electric field is uniform. This electromagnetic field is imagined to be imbedded in an electrically conducting fluid or plasma. In the following subsections we examine several aspects of this physical model: flux transport, external plasma dynamics, nature of the region around the magnetic null point, and electro- magnetic energy conversion. The discussion is qualitative. More detailed discussion of existing analyses is presented in later sections of the paper. 2.1 Flux Transport 0 O7 It is well known J that E_ • B_ = 0 is a sufficient condition for the flux transport velocity u_ = E x B/B2 to move points which are on a —& given magnetic field line at one instant in such a way that they remain linked by a field line at all later times. For example, points which at a certain instant are located on field lines C\Ci and D1D2 in Figure 6 will move in such a way that at a later time they are located on field lines €{€2 and D{D2, res- pectively. Thus, a set of points, originally located on a field line and subsequently moving with v.,, may be thought of as representing a "moving field • line". This fact explains the use of the term flux transport velocity for u_. ~

886 Note that the reconnection process may be discussed entirely without reference to moving field lines and that indeed the latter concept might become invalid if substantial electric fields parallel to the magnetic field should develop. However, in the present simple model no such parallel fields occur except in the region very near the magnetic null. The use of the concept of moving field lines is then just another way of referring to the electric field E. In this model, the use of the term "reconnection" to describe the process is best understood in terms of moving field lines. As the lines CiC2 and DiD2 move with v^ toward each other through positions C{C-L and D{D2 they ultimately reach locations Ci"C2" and Di"D2" where the lines meet at the origin. The sur- faces through these lines and perpendicular to the plane of the figure are called separatrices, because they separate families of field lines of different topological origin. When the lines have reached this critical position, they appear to be cut and reconnected so that at still later times they are connected as C["D{" and C2'D2", as shown in the figure. It is evident that the reconnec- tion may be thought of as leading to a transport of magnetic flux from flux cells QT) and (2j across the separatrices into cells Q3) and 9 2.2 External Plasma Dynamics Up to this point the description of the reconnection process has con- tained no reference to plasma dynamics. Indeed, the process may well have been imagined to occur in a vacuum. In such an instance, or if the field configu- ration is imbedded in a weakly'conducting plasma, few restrictions exist on the magnitude of E , i.e., on the magnitude of « . And the magnetic field will O —fa be equal to, or nearly equal to, a vacuum configuration with an angle a of nearly -n/2 between the intersecting separatrices at the origin. The coupling between the electromagnetic field and the plasma is weak or absent. But in

887 virtually all cosmic applications of interest, the field configuration in Figure 6 would be imbedded in a plasma of high electrical conductivity. Indeed, in many cases Coulomb collisions may be considered entirely absent and the con- ductivity, if such a term is to be used, is associated with plasma turbulence and/or inertia and gyro effects, occurring near the magnetic null. Away from that point, the coupling between the B field and the plasma is strong and the plasma dynamics of the process will have dramatic effects in determining the detailed magnetic field configuration and perhaps in limiting the magnitude of the electric field E . We now outline some basic features of the plasma dynamics of the reconnection process. First, it is observed that in a collision-free plasma the guiding centers of charged particles move with some velocity y_ under the influence of the electromagnetic field in Figure 6. In the drift approximation, which is expected to be valid, except in the immediate vicinity of the origin, the component of v^ parallel to the xy plane and perpendicular to B is identical with the flux transport velocity v,,. Thus, in that plane, and as long as ~ti F • B = 0, the magnetic field lines may be thought of as moving with the plasma or vice versa. We note that the simplified magnetohydrodynamic des- cription also yields this result in the limit of an infinite electrical con- ductivity. The region away from the magnetic null in which plasma and fields move together is referred to as the convection region. Qualitatively the plasma motion is the one shown by the velocity arrows in Figure 6. Plasma approaches the origin along the positive and negative x axes and leaves along the positive and negative y axes. The motion may be the result of an external electric field E applied between capacitor plates at z = ± h. Alternatively, E may be a polarization field created by an impressed plasma flow, specified in terms of a prescribed inflow rate at large \x\ values, say. The details

888 of the overall flow and field configuration will depend on these and other boundary conditions in a manner discussed in Section 4.1. However, all MHD models are expected to have in common the occurrence of large-amplitude standing waves in which the plasma is accelerated into the exit flow along the ± y direc- tion, as shown in Figure 7. In incompressible analyses, these waves are Alfve"n waves; in compressible flow they are slow shocks approaching the switch-off limit. The occurrence of these standing wave patterns is related to the fact that the propagation speed of these modes is very small in directions nearly perpendicular to the magnetic field. Thus, by arranging the angle between the wave normal and the upstream magnetic field to be sufficiently near 90°, the wave front can remain stationary even for very small inflow speeds along the ± x direction. The set of waves divides the flow field into two inflow regions and two outflow regions. These regions do not coincide exactly with the four flux cells in Figure 6. Because the separatrices are located upstream of the standing waves, parts of cells (3) and (4) overlap the inflow regions. The standing waves contain concentrated electric currents, directed along the z axis as shown in Figure 7. The j_ * B_ force associated with these currents serves two purposes: it balances the difference in perpendicular momentum and in pressure of the plasma across the shock, and it accelerates the plasma in a direction tangential to the shock. It should be emphasized that currents are by no means confined to flowing only in the wave fronts. Distributed currents j, may occur throughout the flow field. In particular, Z as will be shown in Section 4.1, the current distribution in the inflow region may influence the reconnection process in a crucial way. An approximate balance of the magnetic shear stress at the shock and the exit momentum flow* yields *In this calculation it is assumed that the plasma has a negligible velocity component along the y direction as it enters the shock. This assumption is not always valid. See Sections 4.1 and 4.2.

889 MO 2-1 where pi is the plasma density in the inflow. Further, v\t B\t and y2, B2, are inflow and outflow speeds and magnetic fields, respectively. They are related via E - yIB1 = y2B2 2-2 o If v1 is eliminated between equations (1) and (2), we find y, * — = v. 2-3 and M = — = — 2-4 Thus it appears that, regardless of the inflow speed, the exit speed y2 is always of the order of the Alfve"n speed y , based on inflow conditions. Also, for fixed Bt, the magnitude of the magnetic field B2 in the exit flow increases with increasing Alfve"n number M in the inflow. When M = 0, B2 = 0 and A 1 A 1 the configuration reduces to a current sheet. When M = 1 the two fields " 1 are approximately equal, i.e., B2 = BI. In steady-state reconnection models, the inflow A1fve*n number M is 1 commonly used as a measure of the reconnection rate. For very small values of M , and in a collision-less plasma, the plasma " 1 ejection along the ± y axis, postulated in the model in Figure 6, may become gradually replaced by an ejection at z = -h and z = +h, respectively, of electrons and positive ions meandering in the current layer, as suggested by 7 0 ^ Alfven and discussed further by Cowley . The charge separation effects in that case lead to an electric field E which is a function of the 2 coordinate 2. This limit will not be dealt with in the present paper.

890 2.3 Region Near the Magnetic Null The preceding discussion has dealt with plasma motion away from the magnetic neutral point at the origin in Figure 6. Let us now briefly consider the region immediately adjacent to that point. As the origin is approached, the flux transport velocity v., tends to infinity. Thus it is — D evident that the plasma can no longer move with v,, in the xy plane. In fact, ~~Ci as the plasma approaches the origin from both sides it must be brought to rest for symmetry reasons. In hydrodynamic terms, the magnetic neutral point is also a double stagnation point. The region in which the plasma velocity deviates significantly from y., is referred to as the diffusion region; its — c dimensions are denoted by 2x* and 2y* as indicated in Figure 6. In this region finite conductivity effects of some type must come into play, allowing the current density to remain finite at the null point for E f 0. Three main possibilities exist. (i) In a collisional plasma with large but finite electrical conductivity a, the half width x* of the diffusion region is expected to adjust itself in such a way that a balance is established between the rate of magnetic flux con- vected into the diffusion region and the rate of diffusion of that flux through the semistagnant plasma in the diffusion region. The ratio of these two transport rates is measured by the magnetic Reynolds number R = \i0avix*. Thus we expect R = 1, i.e., x* is of the order of the resistive length: x* = (uooyi) L 2-5 We note that x* decreases with increasing conductivity and increasing i?i . •Since vi = VE = £0/5i' Bi being the magnetic field at (x = ± x* , y = 0), increasing vi corresponds to increasing £0 , assuming B\ to remain fixed.

891 (ii) In a collision-free plasma one might expect the value of z* to be determined instead by the scale of the particle orbits near the null point. Four such scales may be of relevance: the electron and ion gyroradii and the electron and ion inertial lengths. Further discussion of these scales is presented in Section 5. An equivalent electrical conductivity may be imagined in this case, such that the effective residence time of a particle (an electron or an ion) replaces the usual collision time T in the expression o = «e2T/m (m = particle mass). This residence time is found to be inversely proportional to yi so that x* - (uooyi)~l becomes independent of Ui and hence of E0 for fixed Bi. For further discussion, see section 6.1. (iii) In each of the above two cases, the current density or the gradients in the diffusion region may become sufficiently large to cause plasma micro- instabilities. The resulting plasma turbulence will lead to a reduction in the effective conductivity, as discussed in section 6.2. Whether the plasma dynamics in the diffusion region is described in a continuum fashion, i.e., by use of an effective conductivity, or in terms of individual particle orbits near the magnetic null point, it is easy to see that the net current I in the diffusion region will be along the positive z axis so that E_ • I > 0. Thus the diffusion region, along with the entire shock system, acts as a dissipator of electromagnetic energy. We note that the overall conservation of mass in the diffusion region yields 2-6 which may be combined with 2-2 and 2-4 to yield Bj p2 -* _ = __ _ _ _ v ' Bl ~ p! y* " i 2-7

892 Assuming the density ratio p2/Pi to vary only moderately with M , we see ^ i that the diffusion region is very elongated along the y axis for small M. " i values. Additionally, in a collisional plasma the thickness x* increases with decreasing M. , as shown by equation 2-5: "• i x* = (y av. }~l(M. )~* 2-8 Combining equations 2-7 and 2-8 it appears that x* ~ M' ~ , y* - M in a collisional plasma (case i) while x* ~ const., y* ~ M. ~1 in a collision- A i free case dominated by inertial resistivity (case ii). Finally, we estimate the separatrix angle a in the outflow (see Figure 6). Near the magnetic null point we may write . B = ay "\ x I > 2-9 B = bx I y J where a and b are positive constants, and the angle a = 2 tan~Va/Z>. Esti- mating ay* = B2 and bx* = BI we find by use of equation (7) -l fB2x* , /•_+ /rrv , / /JT™ \ a = 2 tan 7^-y = 2 tan'1/5* AV 2 tan / P— M. } 2-10 indicating that the range of Alfvdn numbers M from zero to /Pa/Pi corresponds «i to an a range of zero to ir/2. The latter value corresponds to b = a, i.e., to a current-free state because j = (b - a)/u0. 2 2.4 Energy Conversion The reconnection model described in this section serves as a steady- state converter of electromagnetic energy into plasma kinetic and internal energy. For example, the rate of electromagnetic energy flow into and out of the diffusion region may be estimated as follows:

893 Inflow = By*h E Outflow = 8x*h - E where the diffusion region has been taken to be a rectangular box with sides 2y*t 2x* and 2h. Thus the net rate of inflow of electromagnetic energy is - f If) which upon use of equations 2-2 and 2-7 may be written 2— 2-11 It is evident from this approximate expression that the energy conversion rate has a maximum at some value of the reconnection rate M. intermediate ^1 between 0 and a maximum value, which in the present approximate set of rela- tions appears to be M. = /p2/p1. Note that */„,,, = 0 both for M, = 0 and A 1 . kM A 1 for M. = /p2/P1. For the latter value of MA , the configuration near the AI A\ null is current-free and symmetric (b = a; a = ir/2). In such circumstances one may expect /p2/P1 - 1. Thus M. = 1 appears as a theoretical upper limit A 1 for the reconnection rate (based on plasma conditions at x = x* y = 0). It is, however, by no means assured that boundary conditions at large distances or plasma processes in the diffusion region will always permit this upper limit to be reached. The net rate of increase of kinetic energy of the plasma may be expressed as follows and conservation of energy requires the difference V™, - w to be the rate tJM KJL of increase of the internal energy of the plasma, V '. This latter rate may include thermal as well as nonthermal parts, for example- in the form of run-away s * electrons.

894 The analysis given above applies to the diffusion region. But usually only a minute part of the total energy conversion occurs there, the main part taking place in the shocks. In approximate terms, the formulas 2-11 and 2-12 may be modified to be valid for the entire reconnection geometry by replacing y* by L, where 2L is the height of the total configuration, as shown in Figure 6. Also, all quantities bearing the subscript 1 (which are evaluated at x = x* y = 0) should be replaced by quantities bearing the subscript «, i.e., they should be evaluated at x » x*, y = 0. Depending on the nature of the boundary conditions, the inflow may be such that M. A<*> differs significantly from M. (see Sections 4.1 and 4.2). Ai The phrase magnetic field annihilation has been used to describe the reconnection process. In the light of the preceding discussion, this term appears appropriate only in the limit of small M. values where the magnetic "• i field B2 in the exit flow is small (or absent as in AlfveVs model, mentioned earlier*J23 ). Henceforth, annihilation will refer to situations where M is sufficiently small so that the diffusion region occupies the entire A l length of the current sheet, i.e., y* >_ L. By combination of equations 2-7 and 2-8 this is seen to occur for 0 < M < /(p2/Pi)/(y<>ay- L). «i " i In reconnection, energy conversion occurs on a time scale comparable to the Alfve"n wave time T = L/v (assuming the inflow regions to extend to /t "1 \x\ - L), while in annihilation the scale is /T.T_, T_ being the time for purely resistive decay of a current sheet i.e,, T_ = \i0oL2. T is enormous in most cosmic applications, so that reconnection rather than annihilation is required to account for the rapid energy release in solar flares, geomag- netic substorms, etc.

895 3. Flux Transfer in Time-Dependent and Three Dimensional Configurations The two-dimensional steadyreconnection model outlined in Section 2 is useful as a vehicle for introducing certain basic aspects of reconnection. But it appears likely that in any real cosmic applications of the process, three-dimensional and temporal effects are important, perhaps even dominant. For this reason it is useful to consider briefly a few reconnection configu- rations which incorporate these effects. To date, the plasma dynamics asso- ciated with such geometries has not been dealt with in a substantial way, so that the discussion is confined mainly to the electromagnetic field topology and flux transfer aspects of the process. In the following subsections we describe the two-dimensional but time-dependent double inverse pinch configu- ration, a simplified steady-state three-dimensional magnetopause topology and a possible three-dimensional time-dependent magnetotail configuration. Finally, in Section 3.4, a general definition of reconnection is given. 3.1 Plane Time-Dependent Geometry A plane vacuum magnetic field geometry associated with the double inverse pinch laboratory experiments is shown in Figure 8. The X type magnetic null point is again located at the origin. The magnetic field is maintained by the currents I in the two metal rods at the center of flux cells fn and \2j, and a return current ZI3flowing in the plasma along an \ outer envelope, which coincides with the outermost field lines in flux cell In the experiments, the current I increases with time so that magnetic flux is generated continually at the two rods, i.e., in cells fn and If we assume for a moment that no plasma is present, the flux in cell in- creases proportionately so that magnetic flux may be thought of as being t transported from the rods into cells (T) and (F) and from there across the

896 separatrix into cell (J). It is of interest to calculate the electric field responsible for this flux transport. The vector potential for the vacuum configuration is given by where the rod separation is 2c, the minor diameter of the return-current en- velope is 2d, and the radii ri and r2 are measured from the two rods as shown in Figure 8. Note that A_ - 0 at the envelope. In the experiment, the current I and the envelope diameter both increase with time; in a more general case, the rod separation might be imagined to depend upon time also. But for our purposes it suffices to consider the time variation of the current I and the diameter d. Then, the electric field is £ = _ M. = ~ E°£ £„ Q2+<i2 . " .Po-T 2dd —n— — 2ir 2dd Since at each instant >1 remains constant on a magnetic field line, the in- 2 stantaneous electric field has the same value on a given field line but its value changes from one line to another. In particular, on the separatrix it has the value yo-Z" 0 (, . dz Uo-Z" 2dd E = __ inn + _r- -r- -=- z 2it \ C*J ZTT 02+d2 • Thus for increasing current J and diameter d, E is positive as required for flux transport into cell In the presence of a .plasma, the field configuration is modified as follows, The electric field now drives plasma currents in the vicinity of the magnetic null line, causing a field deformation of the type shown by the dashed lines in Figure 8. The separatrix intersection angle a falls below its vacuum value

897 of TT/2. These effects imply an excess of magnetic flux in cells (7) and (?), a deficiency in cell (§), compared to the vacuum configuration, which is the lowest energy state. Thus, a certain amount of free magnetic energy is stored in the system. However, at the same time a considerable amount of flux trans- port into cell (5) takes place. That is, reconnection occurs continuously*. The principal difference between the present case and the steady-state model in Section 2 is the spatial nonuniformity of the instantaneous electric field. This effect occurs because in the nonsteady case some of the flux transported in the xy plane is being deposited locally,causing a field magnitude increase at each point. Associated with this flux accumulation, a plasma compression must also occur. But this would appear to be a relatively minor effect so that the steady model in Section 2 may provide an adequate instantaneous des- cription of the flow away from the rods and the return envelope. Thus the essential qualitative features of the reconnection flow may be obtained by examination of a sequence of steady-state configurations. Impulsive flux transfer events are observed in the double inverse pinch experiments. It appears that,as the magnetic field and associated plasma cur- rents near the null point grow, anomalous resistivity associated with ion sound turbulence sets in abruptly with an associated rapid increase of elec- tric field and decrease of currents at the null point. The net result is a much more rapid flux transfer into cell (T) and an associated relaxation of the entire magnetic field configuration toward its potential form with the separatrix intersection angle a increasing toward ir/2. Evidently the stored free magnetic energy described in the previous paragraph is being rapidly con- verted into plasma energy. These events occur on a time scale much shorter .than that associated with I. Hence it is unlikely that they may be described, *By contrast, Ref. 25 analyzes a hyperbolic-field collapse, where a decreases from ii/2 to 0, without any reconnection.

898 even approximately, by a sequence of steady-state configurations. But the conditions for onset of such an event may perhaps be identified by examination of such a sequence. 3.2 Steady Three-Dimensional Geometry A three-dimensional magnetic-field configuration of interest for steady-state magnetopause reconnection is obtained by the superposition of a dipole and a uniform field of arbitrary direction. This topology, shown in one cross section in Figure 9, has been discussed extensively in the litera- ture 24j32j . Two hyperbolic magnetic null points Xl and X2 are formed in the plane containing the dipole moment vector and the uniform field vector. A basic topological property of such a null point is that many field lines enter it forming a separatrix surface and two single field lines leave it along directions out of that surface, or vice versa. The separatrix surfaces associated with ATX and X2 intersect along a circular ring located in a plane through the two points and perpendicular to the plane of Figure 9. This ring is referred to alternatively as a singular line, a reconnection or merging line, a critical line, an X line, or a separator line. At a chosen point on the ring the magnetic field does not vanish in general, but it is directed along the ring. Only at Xl and X2 is the field intensity zero. If the uni- form field is exactly antiparallel to the dipole field a degenerate situation arises in which the magnetic field vanishes at each point on the ring. A schematic picture of the two separatrix surfaces is shown in Figure 10, in a configuration that may be appropriate for magnetopause reconnection. The upper part of the figure shows a view in the antisolar direction of field lines on the separatrix surface associated with the null point X2; the lower part shows the same view of the Xl separatrix. The total picture is an overlay

899 of the two diagrams with the reconnection line connecting *i and X2. Part of the solar-wind electric field E is impressed across the configuration and must be sustained along the reconnection line. Thus, in the vicinity of that line a strong electric field component is present along the magnetic field. Unless special circumstances exist, such parallel electric fields do not arise in highly conducting plasmas. However, it is believed that the field lines on the separatrix and its immediate vicinity bend to become nearly parallel to the reconnection line extremely close to that line, as shown in Figure 10. Thus parallel electric fields occur only within the diffusion region which surrounds the reconnection line and in which finite resistivity effects permit their presence. Figure 10 suggests that it may be possible to study reconnection in this geometry by use of a locally two-dimensional model which is then applied to each short segment of the reconnection line. Such a model will be similar to that discussed in Section 2, but with an added magnetic field component B (x,y). Thus the reconnection of fields that are 2 not antiparallel is obtained. Further discussion of such geometries is given in Section 4.4. The dynamics of the motion near the points Xl and X2 has not been studied to date. It may well be that these points mark the end points of a reconnection line segment on the front lobe of the magnetopause surface. Referring to Figure 5, which represents a cut through the earth's magneto- sphere in the noon midnight meridional plane, it is seen that reconnection at the magnetopause, as described above, serves to transport magnetic flux from the interplanetary cell (T) and from the front-lobe magnetospheric cell into the polar cap cells (5) and (7). 3.3 Time-Dependent Three-Dimensional Geometry • ; As a final example of reconnection geometries of cosmic interest,

900 consider the magnetic-field topology associated with the formation of a re- connection bubble in the geomagnetic tail. The evolution of the field geom- etry in the noon-midnight meridional plane is shown in Figure 11. Note the formation of an X type and an 0 type neutral point. The bubble originally has a very small longitudinal dimension. As it grows in size in the noon- midnight plane, it also occupies an increasing longitude sector. The actual three-dimensional magnetic field topology of such a bubble is not known, but it may be represented schematically by an X type and an 0 type null line as in Figure 12. The points A, X, B and 0 in that figure all emerge at the same place at the time of onset of reconnection. Subsequently they move apart as the reconnection process continues and the bubble grows. An electric field exists along the reconnection line AXB but none, or almost none, along the 0 line AOB. This field presumably has an inductive and an electrostatic part which tend to cancel along AOB while adding along AXB. - 3.4 Definitions On the basis of the preceding discussion we now formalize the defi- nition of several terms, used in the magnetic-field reconnection literature: (i) A separatrix is a surface in space which separates magnetic field lines belonging to different topological families. By necessity the separatrix is everywhere tangential to the magnetic field. The field lines constituting the surface originate at a hyperbolic neutral point in the field, (ii) A separator is the line of intersection between two separatrices or the line of intersection of one separatrix with itself. The separator is also called reconnection line, merging line, or X line. The terms neutral line, singular line, or critical line should be avoided, since they may refer to the 0-type topology as well.

901 (iii) The diffusion region is a plasma channel, surrounding the separator, in which resistive diffusion, caused by collisional processes, turbulence, or inertial effects, is important. In a highly conducting plasma, the dif- fusion region is imbedded in a much larger convection region, in which magnetized plasma moves toward and away from the separator, in the inflow and outflow regions, respectively, and in which dissipative effects are confined to shocks. (iv) Magnetic-field reconnection is said to occur when an electric-field component E (induced or electrostatic) is present along a separator or a macroscopic portion thereof. It is proposed that the term magnetic-field annihilation be reserved for the case where the separator has degenerated (for dynamic purposes*) to a surface (e.g., the surface separating two half spaces containing antiparallel uni-directional fields). The term magnetic-field merging may be taken to encompass both reconnection and annihilation. (v) The local instantaneous reconnection rate at a chosen point on a separator is measured by the instantaneous magnitude of the electric-field component E along that line. It is desirable to express this rate in a nondimensional form by dividing the electric field by the product of a characteristic velocity and a characteristic magnetic field. The latter two quantities may be taken to be the Alfve"n speed i> and magnetic field B at a chosen reference point, denoted r by the subscript r, in the inflow, such as (x = x*, y = 0) or (x = L, y* - 0). Since E /B represents a characteristic flow speed, the dimensionless reconnec- tion rate takes the form of an Alfve"n number: E /B ' n f> M = ° r ° \ In steady, two-dimensional (B = 0, a/32 = 0), models the electric field E is z o constant throughout the xy plane so that £\ = E . With the reference point in *see comments in sections 2.4 and 6.1.

902 the convection region, and on the X axis where B = 0, E /B is then the plasma x o r flew speed toward the separator at the reference point and M is the local Alfve"n number, M - M. . In nonsteady flow, the electric field at the refer- o Af ence point, E , in general differs from E , and M ? M. . v 228 Vasyliunas has defined magnetic merging as "the process whereby plasma flows across a surface that separates regions containing topologically different magnetic field lines"; he takes the magnitude of that flow as a meas- ure of the merging rate. For reconnection in a highly conducting plasma, such that R = vQavL > > 1, the two definitions are essentially equivalent. How- ever, the one adopted here, in terms of an electric field component along the separator works also for flows at arbitrary R . It corresponds to m the occurrence of flux rather than plasma transport across the separatrix, be- cause flux transport is but an alternate way of referring to the electric field*. Note also that for the degenerate case of magnetic field annihilation there is no plasma flow across a separatrix. There is, however, an electric field and a corresponding magnetic flux transport. 119 *This equivalence is seen most clearly " by casting Faraday's law into the form of a conservation equation, viz., in subscript notation, 85 ./3t + (c..,E.) = 0, where e.., is the antisymmetric (Levi-Civita) unit tensor t-JK K. IJ K

, 903 4. The Convection Region The plasma dynamics in the regions away from the immediate neighborhood of the reconnection line usually is described by use of continuum equations. Nonsteady solutions have not been found to date, which describe rapid con- figuration changes such as might be associated with impulsive flux transfer events in the double inverse pinch experiment (for a circuit model, see Bratenahl and Baum, ). Three-dimensional solutions also have not been obtained. Hence the discussion in the present section is confined to steady- state plane reconnection. The incompressible assumption corresponds to the limit B-,•«i, where 8 is the ratio of plasma pressure to magnetic pressure. While this limit is invalid in most cosmic applications, it has the advantage of yielding simple analysis. Thus it provides an opportunity to study certain basic features of the reconnection flow without undue mathematical complications. We first des- cribe two incompressible reconnection flows with fundamentally different be- havior. Certain compressibility effects are considered in the second subsection. The third subsection discusses asymmetric reconnection configurations, perhaps applicable to the magnetopause. The fourth subsection deals with the recon- nection of magnetic fields that are not antiparallel, a common situation at the magnetopause. Finally, a partial single-particle model is discussed briefly. 4.1 Two Incompressible Symmetric Flow Models Figure 13, reproduced from Vasyliunas^5, shows a field and flow 90 map for a reconnection model initially analyzed by Petschek and sub- sequently considerably refined and improved by Vasyliunas. The model contains a set of four Alfve"n discontinuities which in compressible flow may be identified

904 as slow-mode shocks and across which the plasma is accelerated into the exit flow regions. Note that the plasma flow converges toward the x axis in the inflow and that the magnetic field intensity decreases on that axis for de- creasing \x\ values. As pointed out by Vasyliunas, this behavior is charac- teristic of fast-mode expansion of the plasma as it approaches the reconnection line. Because the fast-mode propagation speed is infinite in the incompressible limit, such expansion is by necessity an elliptic effect, that is, no standing expansion wavelets are possible. The maximum reconnection rate in this model corresponds to an Alfve"n number M of about one in the inflow just adjacent " 1 to the diffusion region. But because of the increase in flow speed and decrease in magnetic field associated with the fast-mode expansion, the A1fve"n number, M. , at large distances upstream is considerably less than unity. Values in 00 the range .05 < M < .2 for the maximum rate are obtained (see Ref. 218 Fig. <» 12). Recently, Soward and Priest have reexamined Petschek's reconnection geometry by use of an asymptotic approach, valid away from the reconnection line. Their analysis in all essential respects supports the conclusions sum- marized above. Figure 14 also taken from Ref. 118, shows a flow and field map for a different model , which is the sole nonsingular member of the similarity solutions derived by Yeh and Axford . This model contains a second set of Alfve"n discontinuities located upstream of the slow shocks and originating at external corners in the flow, as shown in the figure. These discontinuities represent the incompressible limit of slow-mode expansion fans centered at the external corners. They cause a large deflection of the plasma flow away from the x axis and a substantial increase in field magnitude. It is now generally agreed that these discontinuities will not occur in any real situation. Rather • they represent a suitable mathematical lumping of slow-mo'de expansion effects

905 in the inflow. The maximum reconnection rate in this model* is M. = (1 + /2). «i On the x axis this value remains constant, independent of |x|. However, this is a result of the lumping of the slow-mode effects. In a model where these effects are spread over the inflow region the value of M on the x axis will decrease with decreasing \x\ in association with a decrease in plasma velocity and an increase in magnetic field. Thus, in reality it is unlikely that the inflow into the diffusion region can occur at M. as high as (1 + /2~); more "i likely that value corresponds to the maximum M. at large |a;| values. Fur- oo ther discussion of this point is given in Sections 4.2 and 5.1. The two models discussed above represent two extreme sets of conditions in the inflow: pure fast-mode and pure slow-mode expansion. In any real ap- plication both effects may be present. Vasyliunas has pointed out that from a mathematical viewpoint the difference between the two models is related to the boundary conditions at large distances from the reconnection line. Far upstream, the fast-mode model is essentially current free and has a nearly uniform flow and magnetic field, while the slow-mode model contains substantial currents which bend the magnetic field lines and cause a deflection of the flow away from the x axis. Vasyliunas has further suggested that the former state of affairs may obtain when a demand for magnetic flux originates at the current sheet itself (the xy plane) or in the exit flow, as may be the case in the geo- magnetic tail, while the latter set of conditions may correspond to externally forced inflow such as at the magnetopause. In this context, it is worth noting 132 that slow-mode expansion effects have been argued ° to be present outside *The estimates given in Section 2, viz., v2 - v. and (v\) - v. assumed a -' AI l'max Ai negligible flow component along the y axis as the plasma enters the shock. Such a component is present in this model, the result being that the exit " flow speed v2 and the maximum inflow speed (yi) both exceed v. by a factor (1 + /2).

906 the subsolar magnetopause regardless of whether or not reconnection occurs there. 4.2 Compressible Symmetric Models A detailed compressible analysis of the external region of Petschek's reconnection geometry is not available at present. On the other hand, the slow-mode expansion model has been extended to include compressibility effects. An isothermal analysis was given by Yeh and Dryer . But the isothermal assumption leads to unacceptable entropy variations with decreasing entropy across the shocks and increasing entropy across the expansion waves. More 725 recently, an analysis has been performed by Yang and Sonnerup " » which assumes isentropic flow in the inflow and uses the ordinary jump relations for slow shocks. It is found that the expansion-wave discontinuities in the in- compressible solution do indeed dissolve into slow expansion fans centered at the external corners in the flow (see Figure 15). It might be thought that the reflection of these fans in the x axis, and the subsequent interaction of the reflected waves with the shocks, shown schematically in Figure 16, may be treated exactly by the method of characteristics. However, it is found that the flow from region (T) in the figure, across the last expansion wavelet and the innermost portion of the shock, cannot be dealt with without the inclusion of elliptic (fast-mode) effects. This is extremely difficult to do. Thus, in the main part of their work, Yang and Sonnerup, after calculating the isentropic plasma and field changes across the fans, considered them to be lumped into a single discontinuity, i.e., they ignored the reflection altogether. While such a procedure is perhaps justified in a first attempt to study com- pressibility effects in the external flow, it nevertheless seriously limits the usefulness of the resulting solutions. The width of the slow expansion fans in the inflow increases dramatically with increasing compressibility,

907 i.e., with decreasing values of B1 = 2u0p1/fl12, so that for B1 - 1 the lumping of the fan into a single discontinuity is difficult to defend. Furthermore, except perhaps for very large B values, conditions immediately outside the diffusion region are not adequately represented so that the solution may not be used to provide external boundary conditions for compressible matched dif- fusion-region analyses. However, the analysis is valid at large distances from the originjand it is of interest to examine its predictions concerning flow and plasma conditions in the exit regions. When conditions typical of geomagnetic tail reconnection are substituted, flow speeds in the range of 1000 km/s are calculated, in rough agreement*with observed proton speeds in the tail during energy-release events35'53. The analysis also predicts exit flow speeds considerably greater than the fast-mode propagation speed so that standing transverse fast shocks may be present in the two exit flow regions, causing a decrease in flow speed and an associated increase in plasma density, temperature, and in the exit magnetic field. •j o r Yang and Sonnerup also calculated the change in plasma and flow prop- erties along the x axis in Figure 16, caused by the reflection of the slow expansion fan, but ignoring the elliptic effects mentioned earlier. The solid curve in Figure 17 shows the resulting relationship between the Alfven numbers M and M , in regions (T) and (~) of Figure 16, respectively. For comparison, the corresponding relationship for the fast-mode model, developed by Soward and Priest , is shown by the dashed curves. It is evident that the dif- ferent distant boundary conditions for the fast-mode and the slow-mode models may lead to profoundly different inflow conditions into the diffusion region for the two models. .*The agreement is however not sufficiently detailed to support this particular reconnection configuration over others.

908 4.3 Asymmetric Models A qualitative reconnection model for the asymmetric flow and field 72 91 conditions existing at the magnetopause was first described by Levy et al. J . In this -model, shown in Figure 18, the magnetosheath plasma is assumed to carry with it a compressed interplanetary magnetic field which is due south so that a field reversal results at the magnetopause (see Figure 4). In impinging on the earth's field, the plasma encounters a wave system consisting of an intermediate wave (rotational discontinuity; large amplitude Alfve"n wave) followed by a narrow slow expansion fan. The intermediate wave, which marks the magnetopause, accomplishes the field direction reversal and an associated plasma acceleration parallel to the magnetopause and away from the reconnection line. The magnetic-field magnitude remains constant across this wave but it then increases to its higher magnetospheric value in the slow expansion fan across which the plasma pressure also is reduced to match the pressure in the magnetosphere, taken to be equal to zero in the model. The quantitative details of this model have not been worked out for fast-mode ex- pansion in the inflow. But for the incompressible slow-mode expansion model, various types of asymmetries in the flow and field have been analyzed 26 ft? TOR - , L...,)i/o Certain compressible counterparts of these geometries have been studied by Yang by use of the procedure of lumping slow-mode effects in the inflow, discussed in Section 4.2. In particular, the case of vacuum 126 conditions in the magnetosphere has been reported in detail . A typical resulting field and flow configuration is as shown in Figure 18. The model pre- dicts the following features of the magnetic field: (i) a small magnetic • field component normal to the magnetopause; (ii) rotational behavior of the magnetic-field component tangential to the magnetopause; (iii) a gradual in- 9 crease in magnetic field intensity inside the magnetopause. In terms of plasma

909 flow, the model predicts: (i) flow of magnetosheath plasma normal to and across the magnetopause with speed equal to the AlfvSn speed based on the normal magnetic field component; (ii) tangential acceleration of the plasma to speeds of the order of 500-750 km/s as it crosses the magnetopause; (iii) no change in plasma density or temperature as it crosses the magnetopause; (iv) an isentropic decrease in density and temperature and an associated velocity in- crease as the plasma expands in the slow expansion fan inside the magnetopause. To date, the predicted plasma behavior has not been observed. At various times 1 08 some but usually not all of the predicted magnetic signatures have been seen . An example is shown in Figure 19. 4.4 Reconnecting Fields Forming an Arbitrary Angle 90 In his original paper on reconnection, Petschek observed that in incompressible two-dimensional reconnection flow, a constant magnetic field component B could be added to any solution without changing the flow or mag- 2 netic field configuration in the xy plane (refer to Figure 6). Thus, it is possible to generate solutions that describe the reconnection of fields that form an arbitrary angle. This provides a link between the traditional cosmic reconnection models and the problems of reconnection in nearly force-free field configurations, such as the tokamak120^21^22. This procedure has provided the basis of a number of attempts to describe the dependence of the cross- magnetospheric electric potential difference on magnetosheath field direction, assuming the former to be caused by magnetopause reconnection^ 5^55, ^07. The result of these geometrical analyses is that the potential should have a principal angle dependence given by the function* (B IE. - cose)2 „ . *In Ref. 49 the functional dependence is (/(£>))2

910 where 9 denotes the angle between the magnetic field B. in the magnetosphere y and B in the magnetosheath. For coaQ >, B /B. no reconnection occurs* and /(6) = 0. Observations indicate a low energy injection rate from the solar wind into the magnetospheric ring current system when 6 < ir/2 , a result which appears compatible with Equation 4-1. Observations^ of the magneto- spheric cusp location as a function of 6 also are in qualitative agree- ment with this equation. 25 28 Recently, Cowley J L has pointed out that incompressible solutions i exist in which B is a function of x and y. Thus the assumption underlying 2 Equation 4-1 , of one and the same value of B on the magnetospheric and the 2 magnetosheath side of a typical magnetopause reconnection model, may be invalid. It is noted that this assumption corresponds to a situation where the net current in the magnetopause (and in the diffusion region) is parallel to the separator. In the incompressible MHD approximation the equations describing the flow and field in thexy plane are completely uncoupled from the differential equations 25 for the velocity component y and forB . However, as pointed out by Cowley , . 2 2 an indirect coupling exists via the requirement that the line integral $E* dl = 0 for any closed loop which encircles the diffusion region. This requirement poses an additional constraint on the shape of the diffusion region, a constraint that does not arise when B = 0, or B = const., and that does not appear to be 2 2 automatically satisfied by the diffusion region of plane reconnection geometries. Thus it is not clear at the present time whether Cowley's criticism of Equation 4-1 hasafirm foundation in incompressible MHD theory. But even ifitdoesn't, the use of a constant B in the real compressible magnetopause flow situation to 2 construct reconnection geometries for arbitrary 6 values remains hypothetical. • Further theoretical study of this problem requires detailed analysis of *If the magnetosheath field magnitude exceeds the magnetosphere field, the subscriptso and i are to be interchanged.

911 compressible external and diffusion-region flows and appropriate matching of these flows at the edge of the diffusion region. . 4.5 Collisionless Model No complete or nearly complete collisionless models for the exter- nal reconnection region have been developed to date. But certain partial . results have been obtained by Hill . He suggests that for small 6 values the principal field reversal is accomplished by a current sheet located on the y axis, as shown in Figure 20. with only a small amount of field-line bending at the slow shocks. Hill does not treat the flow and field configu- rations in the inflow or in these weak shocks. Rather he assumes that, away from the magnetic null point at the origin, the field lines form an angle x with the current sheet. He then proceeds to discuss the properties of the sheet. One-dimensional self-consistent Vlasov equilibria of such sheets have 36 37 been obtained numerically by Eastwood ' ; an approximate analytic theory using the adiabatic invariant J J associated with the meandering par- 38 tide orbits in the sheet has also been given . However, the result pri- marily used by Hill is obtained directly from the overall stress balance at the sheet, without reference to the sheet structure: in a frame of reference sliding along the y axis (see Figure 20) with a speed such that the reconnection electric field E vanishes, the usual firehose limit must apply. Assuming the magnetic moment of individual particles to be preserved, Hill shows that for inflow along the x axis* this condition may be expressed as a local re- connection rate V 4-2 *Hill's analysis also includes an unspecified velocity component y of the incoming particles, which we have set equal to zero. ^

912 where v. = B//u0p and B are the Alfve*n speed and magnetic field, respectively, in the region adjacent to the current sheet. Further, the anisotropy factor a of the incident particles is defined by Note that a = 0 corresponds to isotropic pressure, a = 1 to firehose conditions (taking account of the fact that the total plasma density p is twice the density, p. , of the incident particles). fprl The preceding result is incomplete in that the angle x must be a function of the reconnection rate. Also, the rate should refer to conditions on the x axis in the inflow region. Equation 4-2 is nevertheless interesting because it suggests that pressure anisotropy in the incoming plasma may be an important factor. In particular, it appears that reconnection may cease al- together for a=l. Using the same approach, Hill has also considered the case of reconnec- tion of fields of unequal magnitude and with a constant B component present. 2 He obtains the formula 4-1 for the angular dependence of the reconnection potential . The particle energization in a model of Hill's type is seen to be the direct result of inertia and gradient drifts in the current sheet, moving posi- tive ions in the direction of the reconnection electric field, electrons in the opposite direction. It is also important to note that the energized plasma will be streaming out nearly parallel to the y axis, i.e., for small angles x» nearly parallel to the magnetic field on both sides of the current sheet. By contrast, the symmetric fluid dynamical models yield an exit plasma flow that is perpendicular to the weak magnetic field in the two exit flow regions.

913 5. Fluid Description of the Diffusion Region A complete theoretical treatment of the reconnection problem requires not only a self-consistent solution for the external flow, but also an internal, or diffusion-region, solution which describes the essential dis- sipative physical processes operating in that region, and which joins smoothly to the external solution. In the present section we review attempts to describe the diffusion region in terms of continuum equations which incor- porate effects of plasma microinstabilities, if any, by means of an effective conductivity a. A brief discussion of one-fluid theories is given in section 5.1. In magnetospheric applications of reconnection, the collisional resistive length (u^yj)~1 is much smaller than relevant inner plasma scales, such as the electron inertial length. Thus one-fluid theory is directly applicable only if turbulent processes generate an effective resistive length which exceeds these inner scales. But even if that condition is not met, one-fluid theory serves the important purpose of providing an opportunity for a careful mathe- matical treatment in one region of plasma-parameter space. The two-fluid description of the diffusion region is dealt with in sections 5.2 and 5.3. The former discusses the importance of the electron inertial length in determining the width 2x* of the diffusion region when no collisional or turbulent resistivity is present. In the latter section, the importance of Hall currents and of the ion-inertial length and gyroradius are discussed. 5.1 One-Fluid Models Detailed studies of the flow and field configuration in the dif- • * fusion region, using one-fluid magnetohydrodynamics, have utilized two

914 approaches: series expansion around the magnetic null point, and development of integral or "lumped" equations for the entire diffusion region. Addition- ally, certain exact solutions exist. 93 27 Series expansions have been given by Priest and Cowley and by Cowley . They found that 1n incompressible flow, and assuming analytic behavior at the null point, a plasma velocity of the form v = -k\x, v = k\y to lowest order yields a magnetic field behavior of the form B = k2y3 B = k^x, i.e., x y the field-line configuration at the neutral point is chi-like (J^) rather than ex-like ()(). The latter type of configuration may however be obtained by assuming a third-order, rather than a first-order zero in v (x) at x = 0. x 128 Furthermore, Yeh ' has shown that the flow and field behavior near the null point need not be analytic. Logarithmic terms are possible in the expansion. Whether or not such terms are in fact present can be determined only by match- ing of the series expansion solution to an appropriate external solution, which has not yet been done. It also appears that the inclusion of compres- sibility effects will invalidate the above-mentioned result of Priest and 27 Cowley. Finally, Cowley has shown that series expansions yielding a non-constant field B (x,y) are possible. But the question of whether such 2 solutions may be matched to corresponding external solutions with non-constant B (see Ref. 25) has not been resolved. 2 The first lumped analysis of the diffusion region was performed by fifi 77^ Parker with application to Sweet's resistive current-sheet model 'l of a solar flare. The analysis yielded the following expression for the reconnection rate in incompressible flow )~^ 5-1-

915 This result is obtained directly from equations 2-5 and 2-7 with P2 = PI. Originally, the formula was used with y* replaced by L, the half-length of the current sheet. It then describes field annihilation (see section 2.4) and yields a value of MA that is far too small to account for an energy-re- lease time of the order of 103 sec in a solar flare, or for the observed potential difference of 20-100 kV across the terrestrial magnetosphere. Later Petschek used the formula 5-1, now with 2y* representing the height of the diffusion region and with y* « L, to describe the diffusion-region flow in his model. The basic point made by Petschek is that equation 5-1 deter- mines, not the reconnection rate, but the height y* of the diffusion region. In agreement with the discussion in section 2.3, y* is then seen to be pro- portional to M. ~2. Ai A more detailed lumped analysis was performed by Sonnerup in an attempt to develop a diffusion region solution for the slow-mode reconnection geometry in figure 14. The treatment is incomplete because it does not take account of the momentum balance. Additional criticism has been offered by 118 Vasyliunas on the basis that the model implicitly assumes an abrupt switch from finite to infinite electrical conductivity at the outer edge of the diffusion region. Considering the extreme simplification of the external flow in this model, with slow expansion effects lumped into a single discon- tinuity (see section 4.1 and figure 14), this latter criticism is probably not of serious consequence. The following qualitative results of the analysis are likely to remain valid for the slow-mode reconnection model in figure 14: (i) A relationship exists between M and the magnetic Reynolds number « l R = u0oy y* which is of the form given by Parker, equation 5-1, for small i/ " 1 v values of M , and which yields R =0 for M = (1+/Z). Thus, the dimensions *» i y "i of the diffusion region shrink toward zero as M approaches its maximum value.

916 (1i) When the slow-mode expansion in the inflow is concentrated into dis- continuities as in figure 14, the increase in magnetic field and decrease in flow speed, described in sections 4.1 and 4.2 must occur in the outer portions of the diffusion region. Thus, as the origin is approached along the ± x axis, the magnetic field first increases then decreases to zero while |vx| decreases monotonically to zero. Thus, in the outer portion of the diffusion region the field behaves almost in a frozen-in manner. This may account for the large diffusion region width x* found in the analysis ' Two exact solutions exist which describe flow near the magnetic null 7P9 point. Yeh obtained shock-free similarity solutions by assuming resis- tivity and viscosity to increase linearly with distance from the origin. It is not clear how such assumptions can be reconciled with an exterior solution in which dissipative effects are confined to shocks. A different type of exact solution has been discussed by Priest and Sonnerup55J^9. It describes an incompressible two or three-dimensional MHD resistive stagnation- point flow at a current sheet. The field lines are straight and parallel to the current sheet. Thus, purely resistive magnetic field annihilation without reconnection occurs, as illustrated in Figure 21. These solutions represent 88 a generalization of a case studied by Parker . The resulting magnetic field profiles are shown in Figure 22. Three features are of interest. First, the characteristic width of the current layer is of the order of the resistive length as expected. Second, the frozen field condition applies at large |x| values and leads to a gradual increase in the field magnitude \B \ y as \x\ and |y I decrease. : As resistivity effects become increasingly impor- •w tant |s | reaches a maximum and then decreases to zero at x = 0. This is 3 precisely the behavior described in (ii) above. Third,-a nonconstant value of B is possible.

917 5.2 Two- Fluid Effects; Electron Scale Lengths In the two-fluid description of an electron-proton plasma, the ordinary Ohm's law is replaced by a generalized form (see, e.g., Rossi and Olbert5*). + **- - 5-2 - — V • P + — j x B ne — =e ne *- — As pointed out by Vasyliunas, the terms on the right-hand side may introduce a variety of plasma scale length's into the problem. The first term yields the resistive length X = (unovi)'l; the second set of terms yields the r electron inertial length X = (m /yone2) , the off -diagonal terms of the & & electron stress tensor term P yield the electron gyroradius; the last term, ^^p describing the Hall effect, yields (vjvi)\. where X. = (m./yone2)1/2 is the . A % 1*1? ion inertial length. The importance of the diagonal terms in P has not "Z3& been studied; with isotropic pressure and isentropic flow, these terms are cancelled identically by an electrostatic field; To illustrate the effects of the electron inertia terms we now gener- alize the stagnation-point flows discussed by Sonnerup and Priest to include two-fluid effects. Assuming incompressible flow and diagonal stress tensors for ions and electrons, the flow and fields are of the form £ = - xkix + 5-3 where the quantities ki , k2 and k3 are constants such that ki = k2 + These assumptions lead to a Bernoulli-type pressure integral P ~ Po - 7 P (k2x2 + kly2 + klz2) - (2u0)~l (B 2 , i y

918 of the momentum equation, and to the following components of the induction equation (the curl of equation 5-2). k x B- + A - B - *XB - *B - It is seen that only the electron-inertial length and the resistive length appear in equations 5-4. The Hall current term in 5-2 is curl -free and is cancelled exactly by a Hall electric field E (x) . Thus the ion inertial *v length does not appear. The solutions of equations 5-4 in the resistive limit are illustrated in Figure 22. In the inertial limit, the odd solutions are shown in Figure 23 for various values of a E fe2Ai . As expected, the width of the magnetic-field reversal region is now of the order of the elec- tron inertial length regardless of the flow rate. Further, it is observed that for a = 1, i.e., for plane flow, the current density is logarithmically 09 infinite at x = 0, a conclusion also drawn by Coroniti and Eviatar . Thus some form of plasma microinstability or other effect by necessity must be present to reduce the current density to a finite value. The off-diagonal terms in the electron pressure tensor will provide a finite electron gyroradius correction to the preceding results. When the electron gyroradius greatly exceeds X , it will replace the electron inertial 6 775 length as the minimum width of the layer ' . However, a detailed calcula- tion of these effects is difficult because the appropriate form of the off- diagonal pressure tensor terms is not known in a field reversal region of width comparable to the orbit scale. 7 IB Vasyliunas *" pioneered the study of electron inertial effects in the diffusion region. In an approximate lumped analysis which neglected compres- oo sibility (later included by Coroniti and Eviatar ), off-diagonal stress

919 tensor elements, and Hall -current effects, he showed that the diffusion region in the inertial limit (I.e., neglecting resistivity) 1s hyperbolic in shape with width 5-5 The exact analysis given here 1n all essential respects confirms VasyliunasI results for small values of M. 2 y2/\2. It also provides magnetic field pro- f* i & files B (x) whereas Vasyliunas neglected B within the diffusion region. •J • y Note that the formula (22) yields x*(0) = A ; y* = \ /M. , in agreement with & & ni the behavior quoted in section 2.3 (between equations 2-8 and 2-9). 5.3 Two-Fluid Effects; Ion Scale Lengths It is not clear how a diffusion region of the small physical dimen- sions implied by equation 5-5 can be joined to an external solution with slow 20 shocks of thickness comparable to, or greater than, the ion inertial length. Thus we are led to ask why the ion inertial length and the ion gyroradius did not appear in the previous analysis. The former is introduced via the Hall current term j_ x B/ne in equation 5-2. It may be cancelled by a Hall electric field only when it is curl free, which was the case for the stag- nation-point flow in section 5.2. The ion gyroradius is introduced via the off-diagonal terms in the ion pressure tensor. We now demonstrate that in plane reconnection flow these effects imply the presence of Hall -current components j and j as well as a macroscopic flow v (x,y) and a field x y 2 component B (x,y). Assuming 8/3z = 0 and omitting electron inertia and 2 pressure terms for brevity* the 2 components of the momentum and induction equations are, respectively, 3y, 8y, a a i 3B, 3B, ' 5~6 „ . *

920 3B 3B > - «» + " « „ + • > 5~7 where the particle density n and the conductivity a have been assumed constant. In the second equation, the Hall term (the last term on the right) is seen to be of the form (B • Vj }/ne, i.e., it vanishes only when the current j" •" " ~ 2 Z remains constant along a field line. Otherwise it becomes a driving term in the second equation forcing values of v and B different from zero. Similar- 2 Z ly, in the first equation it appears that the stress terms will usually force values of y and B different from zero. These terms are expected to introduce Z Z the ion gyroradius as a characteristic length into the problem. However, it is difficult to discuss these effects in detail, because the form of the stress terms in a thin field-reversal region is not known. Thus, we confine attention to the Hall current term, (5 • Vj )/n . While this term vanishes -~* ~~" Z & in the stagnation-point flow discussed in the previous section (and indeed along the x axis of any configuration), it cannot vanish throughout the dif- fusion region. We may estimate the magnitude of B by approximately equating Z the first term on the left and the second term on the right in equation 5-7: x 3x ne x 3x With y - yls B ~ B2 - Bitf ,j - B!/UOX* and 3/3x ~ 1/x* we then find «Jj 'Zj M l 2 B./B! - X ./x* where X. E= (m./yo"^2)l^2 is the ion inertial length. Thus it appears that values of x* much less than X. would give rise to unacceptably high values of BZ. A similar comparison between the terms B 3y /3x and (ne)~l B 3j'2/8x yields y_/y,, ~ X./x*. Again, x* « X. leads to an unacceptable result. Z A i 1s 1* Vasyliunas has pointed out that off-diagonal electron pressure tensor

921 terms, not shown in equation 5-7, may possibly cancel the Hall term. However, there seems to be no obvious physical reason to expect such a cancellation. And problems with the off-diagonal stress tensor terms in equation 5-6 would still remain. Detailed analysis of the effects described above is not available at present. However, a nonvanishing field component B (x3y) would imply the z presence of Hall currents j = yo~l 3B /3y and j = - yo'l 9B /3x in the dif- x z y z fusion region. The expected current flow and field pattern is shown schemat- ically in Figure 24. The behavior of the B and B components indicated in y z the figure should be easy to identify in magnetic-field vector measurements from a satellite which crosses the diffusion region. The reason for the appearance of the Hall current component j with the sc direction shown in figure 24 may be understood by noting that for a = °° the generalized Ohm's law (equation 5-2)implies that apart from electron-inertia and gyroradius effects the magnetic field is frozen into the electron compo- nent of the plasma. Thus the electrons flowing toward x = 0 are brought to rest over a distance of the order of the electron inertial length or the electron gyroradius. If the ions are similarly brought to rest over a dis- tance comparable to the ion inertial length or the ion gyroradius, a relative motion of electrons and ions results, leading to currents j in the direction 3£ shown in the figure. Charge conservation then implies the presence of j y as shown. The z component of the force £ * B associated with the Hall currents also leads to an acceleration of the plasma in the ±z direction. This effect is caused by drift and meandering motion of the ions in the current sheet. It ,may be interpreted as an ion current in the layer. Indeed, if x* ~ X., the Ir principal current component jz is carried by the ions; if x* - A it is car- ried by the electrons.

922 It is evident that in the absence of plasma resistivity (classical or turbulent) the electron length scales must play a significant role in the diffusion region structure. But from the preceding discussion it appears possible that the ion length scales may determine the overall width 2x* of the diffusion region while the electron length scales give the size of the detailed structures of j 3 j 3 and j near x = 0. From the preceding discus- x y z sion, it is concluded that, even without plasma turbulence, the electromag- netic structure of the diffusion region may be far more complicated than previously assumed.

923 6. Non-Fluid Effects in the Diffusion Region In magnetospheric and interplanetary applications of the reconnection process, collisional resistive effects in the diffusion region are negligible. Thus an effective resistivity in that region must derive either from inertia effects or from plasma turbulence. The former effects were dealt with in sections 5.2 and 5.3 from a fluid point of view. However, to develop a phys- ical understanding of inertial phenomena in the diffusion region, it is useful to obtain an approximate expression for the effective inertial conductivity. This is done in section 6.1. Section 6.2 examines plasma instabilities which may generate steady-state turbulence in the diffusion region, but with details provided only for the ion-acoustic instability. Section 6.3 discusses several threshold effects that might be of importance for the onset of reconnection and for the identification of situations in which reconnection may not occur. Particle acceleration in nonsteady reconnection is discussed in section 6.4. It will become quickly apparent that most of the material in this section is speculative in nature. Different processes may occur in different applica- tions. It appears that no systematic effort has been made to sort out which mechanisms dominate in different parts of plasma parameter space. In reading the present section, it will be useful to refer to the typical values of several physical parameters given in Table 1. 6.1 Inertial Resistivity The concept of inertial resistivity was first discussed, in the 1 08 context of magnetic field reconnection, by Speiser • The basic idea is that a particle spends only a finite amount of time in the diffusion region and thus can pick up only a finite amount of energy from the electric field E0z_. The inertial conductivity is written as ° inert ~ m

924 0) u V c c o u OJ 5- o V) J_ 0) 4-i 01 « «o a ra u 0) >> 1n 'i 1 10 CO r— 1- l~ O O o O 0 0 Q. O rO r— ' r— • ,— . r— 1n S- 4J -O 1n O O) O C X X pH X X X X •4-> QJ 3 CO C V) O vn 1n 1n 0 ai •* CO* •- CM 1 (O 1n o 1 1 U) CO 1n 4J r— 0 O O O O o> ••- <0 c^ r~~ ^— !-—• r~~ r— C ra o O ^^ X in X X X X • * ra r~ 2 m CM 1£r co CO •- * 1n 1n CM £ « 1 1 (O m 4-> a1 o o 0 CD o o <u 10 «D t— 1 t— «— ~ r" C 3 O 1n CO Ol ra X • X X X X X ra Q- CM co 0 CO 1n CM ,- f— 1 CM •— 1n m 10 — 1 1 1 ^•% o o O 0) 1n CM •— r^ *^ r— CM 4-i t— f P4 u» 1 01 (U ra «— • O O o X X X • S. 4J * r— 1n o f* f^ ra VI 4_ ai CO CM •- ra •— r— 1+- CM a CO O 1 | \ 1 t/i a> r-l O ^i 0 o CD CD S- 1 — f CM * r— r-~ ^^ CM r— i— •O. O O r~ X • • • X X — r— 1n •*: co >> 0 M 1 CM - eg 1 I O C ra O O o O CD o 1 * /-\ CZ ^^ *w- ' d 1 ra Ol c/1 J. J- J- ro X CD X X X X X 0 <£ r— r—~ • p .a — a. LO CM CO CTl CO CM <a • • • • • • — ' •~ 1n l^- Lf) ^*" •— / -^ -s OJ / IM-H ^ J^- J^- -^ "^ .,_ / ^*. 1 1 •>~ >C GO / 0 CM t^"1 *^ *~^ •S- -S- / at •r~ 1 i 'g "^ r-i rQ <M 01 / >^ $- ra •o jj r-i B QJ Q) / 4-> •r- 3 S- •— S- o O t-H / ra S< QJ ^. QJ QJ CM G O ^ c / 1/I -^ •r- O O O / ^i. Ct <•* S- 0 S- -r* «f. QJ o 0 3. ri a> I QJ ^, 3 E^ 3: II 3- 3. *^^ / ^K^ •o E CL 4- J ^s^ Q) *t^ / '£ & ~ QJ -rS fO — J_ Q) ^J ^ ^E / c J- "• OJ E-1 4J ^ II II / *° 0l C a. 01 CT] ea / 3 e C r i 5^ n n E C 0 Ol O ^ 3 _^ QJ re o Q) LM «£• f ^ r -^ __ I CM 1O C o IO E p-' IS O D. CT a1 "«». >> OQ J3 <U > II •^ Ol ~ co II) N Q. I/I - u* H \ to c 1n f^ 3 ra •i Ol O X tn «o M 0) .a at IS) o r- O <U 4-> O CM

925 where n is the average particle density in the diffusion region and T is the effective time available for acceleration in the electric field. The formula 6-1 may be used with either the electron mass m - m or the ion mass m = m. & t' depending on whether the diffusion region current is principally an electron current or an ion current. Both cases may be treated the same way so that the particle mass m will be left without a subscript. The average displacement, AF, along the electric field £0, of a particle in the diffusion region may be obtained from a simple mass balance over a box of dimensions 2x* x 2y* x AF: = nv 2x*2y* = 2n2v22x*Ez 6-2 where v is the average particle speed along the electric field E0z, and 3 v = Az/T. As before, the subscripts 1 and 2 refer to conditions at the 2 points (a;*,0) and (0,i/*), respectively. From the first equality in equa- tion 6-2 we thus obtain r: ~ « ** 1 m yi 6~3 When this expression for T is substituted into equation 6-1 there results s inert Thus the inertial conductivity is very high for low reconnection speeds, vi, i.e., when the configuration approaches a current sheet. This is the be- havior referred to in section 2.3. Expressing the basic balance of field convection and diffusion as u0a. tyiz* = 1, which is valid for small reconnection rates*, we find «i x* = — \! 6-5 : " where AI is the inertial length Xi = (m/\i0nie2)1^ . For m = m this result & is in agreement with VasyliunasI formula (equation 5-5). See also Coroniti *with inertial resistivity, pure field annihilation is found to occur for 0 < MA - *l/L (comPare section 2.4).

926 and Eviatar . But the calculation gives no clue as to whether the elec- tron or the ion inertial length is to be used. It should be realized that the value of x* given by equation 6-5 repre- sents a lower limit. The calculation assumed the time 7 available for free acceleration in the electric field to be equal to the residence time of a particle within the control box. In reality T must always be less than the residence time because the magnetic field does not vanish within the entire box. Thus the inertial conductivity is less than that given by equation 6-4 and x* is correspondingly larger than in equation 6-5.. When diamagnetic currents become important, i.e., for Bi = 2u0pi/Bi > 1, it may be shown that x* gradually approaches a magnitude of the order of the gyroradius instead. The previous estimate of x* applies only for small values of the recon- nection rate. To understand this fact, we note that the expression u0ayix* = l, which was used in obtaining equation 6-5, derives directly from Ohm's law in the approximate form j = oE0 = cryiBi with J - \i0~ldB /3x = BI/\i0X*. For z z y large reconnection rates it becomes important to incorporate the term £ x B_ in the electric field, as well as the curvature term 3B/ty in the expression for j . The latter effect leads to a multiplicative factor (1 - M/.2pi/p2) on the right-hand side of equation 6-5 (see equation 6-7 below) so that the size of the diffusion region decreases toward zero as the reconnection rate approaches its maximum value. Thus, for large reconnection rates, the requir- ed width of the diffusion region may be substantially less than the relevant plasma scale (the inertial length or gyroradius, depending on Bi). It is dif- ficult to reconcile such a situation with the nature of the particle orbits in that region. Therefore, it is conceivable that steady-state reconnection with inertial resistivity as the dominant effect in the diffusion region is not pos- sible for large reconnection rates.

927 The average electrostatic particle energization in the diffusion region may be obtained directly from the first equality in 6-2: E » e*z ffo - «*V ^ 6-6 But we also have nevz - J2 - Pol (35 /8x - SB^/ty). Approximating 8B /3z by y y *, 3B/3y by B2/y*, noting that B2/Bi = vi/v2 - Pi^Vpij/* we find Pi = nevz = (B^x*) (} - — MA*) 6-7 and from 6-6 c *—±- (1 - — M* ) 6-8 C yo«i Pz ^i yo«i Pz where the relations £0 = vi^i and Vi/v2 - VI/VA = M have been used. This AI A i formula agrees with equation 2-11. Maximum acceleration occurs for small reconnection rates and densities: formagnetospheric conditions £ ~ 1 - 10 keV. It is emphasized that equation 6-8 represents the average energy gain. A small number of particles moving nearly along the reconnection line may gain larger amounts of energy in the electric field £„. 6.2 Plasma Turbulence A variety of plasma instabilities may serve to generate plasma turbulence in the diffusion region and an associated turbulent conductivity at L. We now discuss such effects in an assumed quasi-steady state of re- connection. Onset effects are dealt with in section 6.3. The tearing instability, either in its collisional resistive version ' J or in the collision-free electron-inertial version ' * * fi 7 fi Q ' , has been studied intensely in the context of reconnection. It gen- erates a pattern of alternating X type and 0 type magnetic neutral points in •a current sheet. But most analyses of this instability pertain to current- sheets with a vanishing magnetic-field component B perpendicular to the

928 sheet. In other words, in the present application either the reconnection rate M. is very small or the magnetic field behavior in the diffusion region "i is chi-like (Y) rather than ex-like (Y)' as discussed in section 5.1. Schindler55 has pointed out that for BX ? 0 the collision-less tearing instability may still proceed as long as the gyro-period T = 2-m/eB of a g •«• particle in the field BX exceeds the instability growth time TO = (x*/vth) (x*/R )fi where v., is the thermal speed and R the gyroradius. This condi- tion may be applied either to electrons (electron tearing) or ions (ion tear- ing). In rough terms, non gyrotropic behavior of the particles is required for these instabilities to be possible. While the nature or existence of steady-state tearing turbulence does not appear to have been establish- ed, one cannot exclude the possibility that such turbulence could be of importance in the diffusion region;Z5,2:Zj4'7. 89 Parker has suggested that interchange instability may serve to en- hance the flow rates in Sweet's current sheet model. In the geomagnetic tail, the instability would be driven, not in the diffusion region itself, but rather by the pressure gradient and field curvature in the near-earth section of the tail plasma sheet (see figure 4). A detailed analy- sis, including the impeding effects of the ionosphere, has been given recently fifi by Kan and Chao . It indicates growth times of the order of a few hours with ionospheric coupling, a few minutes without such coupling. The situation relative to the level of steady-state turbulence is not clear. Huba et al '" have proposed that the lower-hybrid-drift instability may provide anomalous resistivity in the diffusion region. It appears that the threshold for this instability is sufficiently low to permit the diffusion region width to be of the order of the ion inertial length. Haerendel has discussed the possibility that the electron-cyclotron

929 drift instability, which has a current threshold somewhat less than that of the ion-acoustic instability, may generate turbulence in a diffusion region of width 2x* equal to a few electron-inertial lengths. However, its impor- 22 tance has been questioned by Coroniti and Eviatar on the basis that the gyrocoherence required by the instability may not be available in the diffusion region. They also note that when the electron drift speed exceeds the thresh- old for the ion-acoustic instability the electron-cyclotron drift mode goes over nonlinearly to the ion-acoustic one. The ion-acoustic instability has been proposed J ' J as a likely agent for the generation of turbulence in the diffusion region. It will be dealt with in some detail in the remainder of this subsection. This is done for illustrative purposes and not as an indicator of a universal pre- ference for this particular mechanism. On the other hand, the ion-acoustic 9 85 instability appears in fact to occur in laboratory reconnection experiments ' But it is probably not relevant to magnetospheric reconnection. For a current-driven instability such as the ion-acoustic one to occur, the current density in the diffusion region must exceed a certain minimum value, corresponding to a critical current velocity v , i.e., j > nev . If G G Hall currents are present, as discussed in section 5.3, the total current must be considered. Here we shall confine attention to the component j . 2 According to equation 6-7 we then find 6-9 where, for the ion-acoustic instability /kfl v = /— -/f(T /T.) 6-10 o m J e ^ Combination of equations 6-9 and 6-10 yields

930 where B- E 2^nkT./B\. The function f(T/T.) is shown in Figure 25. It is *r ^ v •' seen that /Yr /T J is of the order unity for T - T. so that for small w. , e i e i ^i and for B. of order unity or less, the critical diffusion region width is of ^ the order of the electron inertial length. For large values of B ., x* must tr be considerably less than A suggesting that only a subportion of the dif- fusion region may contain ion-acoustic turbulence (compare the logarithmic 22 singularity in j at x = 0, discussed in section 5.2).Coroniti and Eviatar 2 indicate B < 5 as a condition for their analysis to remain valid. For greater B values, the size of the turbulent region approaches the wave-length of the ion-acoustic turbulence. For M. of order unity the diffusion region •"i must also be very small for ion-acoustic turbulence to occur. For high temperature ratios T/T., the instability may occur for x* Gr & considerably larger than A but probably not as large as A. (see figure 25). & "l. A large temperature ratio may perhaps be generated temporarily by electron run-away in a current sheet at the onset of reconnection (see next subsection) For example, in the double inverse pinch experiment the collisional resistive length considerably exceeds A (see Table 1), so that run-away must occur to & initiate the ion-acoustic instability. But it appears unlikely that a large temperature ratio T/T. could be sustained on a steady basis in a diffusion & "Z*i region of width much greater than X since most parts of such a region the & run-away would have to occur transverse to a strong magnetic field. We con- clude that steady-state ion-acoustic turbulence, driven by the current com- ponent j , is unlikely to be important unless the diffusion region width, 2x*3 z is .of the order of the electron inertial length. At the same time it is ob- served that the Hall current component j discussed in section 5.3 (figure 24) y may be sufficiently intense to drive the instability in parts of a diffusion

931 region of total width comparable to the ion inertial length. 22 Coroniti and Eviatar . have examined the question of the turbulent saturation of the ion acoustic instability in detail. They conclude that the current velocity will remain close to the threshold value given by equation 6-10. The resulting weak steady-state turbulence is adjusted to give the value of turbulent conductivity required to satisfy uoa. rbvix* - 1 Wltn x* given by the equality in equation 6-11. On the other hand, common estimates of the effective electrical conductivity associated with the ion-acoustic instability, in a state of turbulent saturation, such as (see, e.g., refer- ences 40» and US). FT. a = — f' — — — J — ) 6-12 a VJ /kT./me — — — — turb me ; (j/ne) where u) = (~ne2/c0m jV2, give a much too low value of the conductivity, even P& & at the critical current velocity j/ne = VQ. In other words, with reasonable reconnection speeds and with x* satisfying 6-11, one finds u0a. b vlx* « 1 , which is impossible in a steady state. In Table 1, this fact is manifested by the inequality A. . » A , where A. . is the turbulent resistive length. 6.3 Onset of Rapid Reconnection There is ample observational evidence relating to solar flares, to the earth's magnetotail, and to the double inverse pinch experiment, to in- t dicate that occasionally rapid reconnection is switched on in an abrupt, al- most explosive manner. At the earth's magnetopause, if reconnection actually occurs there, the switch-on appears more gentle and may be a direct conse- quence of the interplanetary field turning southward so that the angle between the reconnecting fields exceeds some critical value (compare section 4.4). .It is natural to assume that the more explosive events might be associated with a plasma instability and/or an abrupt decrease in the effective con-

932 ductivity in a current sheet or in the diffusion region of a slowly reconnect- ing configuration. Five such possibilities, all speculative at present, are mentioned below: (j)_ JlLermaJJ^stability^ It has been proposed2 6>19>56 that the flash phase of a solar flare may be associated with a thermal instability. For example, explosive solutions of the electron energy equation, i.e., solu- tions which yield an infinite temperature in a finite time, are known to occur when collisional Joule dissipation dominates the equation. This instability is not relevant for magnetospheric applications or for the upper solar atmos- phere (case II, in Table 1). 7 fift Jj j_)_ Beta_tJireshpkL_ It may be hypothesized that,in a collision-free pi asma^ reconnecti on is suppressed for high 61 values but may occur for small Bi. Thus, any current sheet in which B1 decreases gradually from some initially large value may be converted to a rapidly reconnecting configuration when a critical B1 value is reached. In the geomagnetic tail, an abrupt decrease in B1 value occurs if the plasma sheet in which the tail current sheet is imbedded shrinks to a thickness equal to the current sheet thickness. On the other hand, at the subsolar magnetopause, Lees and Zwan and Wolf have described a magnetosheath plasma depletion mechanism (by escape along the magnetic field lines) which would tend to maintain a value of B1 of order unity or less. The B1 threshold is not relevant to the double-inverse pinch laboratory experiment, and probably not to solar flares because B1 is small in these applications (Table 1). __ ^ Assume that a current sheet with little or no reconnection gradually thins from an initial width of an. ion gyroradius or more toward the electron inertial length, in response to an increased external

933 total pressure, pi + B\/2\i0. In this process the current density in the sheet increases gradually. When the threshold for onset of current-driven plasma instabilities is reached, e.g., for the ion-acoustic instability, when equa- tion 6-11 is satisfied, a reduction in effective electrical conductivity takes place in the layer. If this reduction occurs sufficiently rapidly, the induct- ance of the system will allow us to consider the current density initially to remain essentially unchanged. Instead an inductive electric field E (xst) is 2 developed within the sheet to maintain the current density. The magnitude of this electric field is larger, the larger the conductivity reduction. 22 For a turbulent conductivity of the size used by Coroniti and Eviatar » E is of the size usually estimated for steady-state reconnection. This 2 electric field, which is initially confined to the current sheet, is subse- quently spread by fast-mode expansion waves propagating outward from the sheet as the configuration converts itself to one of steady or quasi-steady recon- nection. If the conductivity is reduced to the level given by equation 6-12, E 2 may be one or two orders of magnitude larger than typical steady-state values. 102 Smith has pointed out that the width 2x* of the current sheet must then increase. As pointed out in section 6.2, for reasonable flow rates we find u°aturbyiX* « 1 when x* - \ and with a given by equation 6-12. Thus an increase in x* occurs in order to bring u0ayiX* toward unity as required in a steady state. The rate -3B/dt associated with the increase in x* is the principal source of E . But the main result of the increase in x* is that Z condition 6-11 ultimately is violated so that the ion-acoustic instability 102 is quenched. Smith proposes that the process may then repeat itself. A state of pulsating reconnection is established. See also Bratenahl and Baum . While the above arguments were given in terms of the ion-acoustic

934 instability, other mechanisms may produce similar effects. £iv_)_ learjjig_ thres_hp_ld_. In a collision-free current sheet with a vanishing normal magnetic field component, electron tearing should be normally present, 28 unless it is suppressed by some agent such as pressure anisotropy or fin velocity shear . With a nonvanishing normal magnetic field component B , •C a threshold for the onset of collision-free tearing does exist, as mentioned in section 6.2. If \B \ is originally large, no tearing occurs. But as |Bx| gradually decreases it will set in when the gyroperiod in B exceeds the growth »c 99 100 time. Schindler ' has noted that this threshold may be exceeded for ions (but not electrons) in the geomagnetic tail current sheet during the thin- ning of that sheet which occurs in the expansive phase of the geomagnetic sub- storm. Since the tail at this time has free energy available for dissipation the ultimate result of the onset of ion tearing should be a large-scale re- laxation (via reconnection) of the tail towards a state of minimum free energy, rather than merely the generation of tearing turbulence in the sheet. Further development of the ion-tearing instability theory has been given by Galeev A 1 1 46>4? and Zeleny J[yj__Ijitejrchan£e_i]iS^abiVij|y_._ An abrupt onset of interchange turbulence in the geomagnetic tail^SS may occur if the ionosphere becomes decoupled from the tail plasma sheet by the development of electric fields parallel to the magnetic lines of force. 6.4 Particle Acceleration One of the most important, and at the same time most poorly under- stood, aspects of magnetic-field reconnection is its presumed ability to accelerate particles to high energies. Observations in the magnetospheric

935 tail indicate the occurrence of energetic electron and proton bursts5Je5 116 during times when reconnection may be going on. And it should be re- membered that our ability to observe reconnection on the sun and in the far reaches of cosmos depends critically on the generation of energetic particles and on the electromagnetic radiation they subsequently produce. Particle acceleration may occur either in turbulent small-scale electric fields or in the large-scale reconnection electric field E . Both types of 2 acceleration are expected to be operative principally in high-current regions: the diffusion region and the shocks. To discuss turbulent acceleration one must understand the nature of the dominant micro-processes in these regions. Since no such understanding is at hand, the discussion in this section is confined to particle acceleration in the large-scale reconnection electric field. In many, but not all, cosmic applications, the total potential difference associated with a steady reconnection electric field is sufficiently high to account in principle for observed particle energies. However, it is only in the small diffusion region that particles have the opportunity to move along the electric field for any considerable distance. And even there, most parti- cles have short residence times and undergo a correspondingly small energiza- tion, as shown by equation 6-8. Thus, steady-state reconnection does not appear to be an effective mechanism for the acceleration of particles to very high energies105. Additionally, in applications such as the geomagnetic tail it is necessary to account for particle energies which exceed the steady- state cross-tail voltage by an order of magnitude or more. One is therefore led to consider the possibility of particle acceleration during nonsteady recon- 7 7 fi .nection "" • Two possible advantages are gained. First, the inductive elec- tric fields may, in principle at least, become much stronger than the quasi-

936 static ones during steady reconnection. Second, the nonconservative nature of £ permits acceleration within more localized regions of space. For example, betatron acceleration to high energy may occur in a small region of space where the particles experience a large increase in magnetic field intensity. By contrast electrostatic acceleration requires particles to move large distances along the separator. The lack of nonsteady reconnection models prevents a detailed analysis of particle acceleration. But the simple model given below may serve as an illustration of how electron energization might occur in the diffusion region. A resistively decaying one-dimensional current sheet, perhaps generated as described in section 6.3, may be crudely described by •\/x \X\ < X* 1 6-13 where the sheet width x* is an increasing function of time and BI is the con- stant field outside the sheet. Assuming no inflow into the sheet, the associat ed electric field is -1 n /, x2\ dx* c -i * £-£2 Bl (1 ~ x£ dt The direction of this field is such that it drives the particles toward the center of the current sheet (x=0). A particle accelerating freely at x=0 in this electric field may be shown to gain an amount of energy given by 2 = me2 + (_— - ) -1} 6-15 provided it doesn't leave the system (at z = ±h) during the time it takes the current sheet to widen by Ax*. In the geomagnetic tail B0 - 2Qnt, and for Ax* = 500 kn equation 6-15 predicts a possible energy gain of 1.07 MeV for * electrons (protons would gain a similar amount of energy only if Ax* ~ 15000 kn)

937 Since an electron in this energy range transverses the entire tail in less than a second, it would appear that an unreasonably large value of dx*/dt is needed. But this is not necessarily so. If the widening current sheet is located along the separator AXB of the reconnection bubble in figure 12, elec- trons may be accelerated as they move from A to B along the separator AXB. They may then return from B to A by gradient drift in the vicinity of the 0 type neutral line BOA where the electric field vanishes. Subsequently they reenter the acceleration region at A. By cycling electrons through this loop many times the energy gains predicted by equation 6-15 may be achieved even for small values of dx*/dt. The illustrative example discussed above emphasizes that it may be neces- sary to consider three-dimensional time-dependent configurations in order to account for particle acceleration in the reconnection process. For further illustrative calculations, see refs. 56' 116.

938 7. Magnetospheric Evidence Much of the observational evidence concerning the possible occurrence of reconnection in the magnetosphere has been summarized by Burch^2. Relevant references may be found in his paper and are not, for the most part, repeated here. A large amount of evidence exists indicating a relationship of various magnetospheric activity indices to the southward component of the interplane- tary magnetic field. Also, spatial asymmetries in a variety of polar-cap processes appear to be correlated with the orientation of the interplanetary magnetic field. Such evidence is compatible with, but does not prove, the occurrence of reconnection at the magnetopause. This body of observations will not be discussed here. Instead, we focus, in section 7.1, on observa- tions relating directly to the transfer of magnetic flux from closed to open field lines, and vice versa, in the magnetosphere. If such transfer in fact occurs, reconnection of some form must take place. If not, there is no need for it. Section 7.2 contains a brief discussion of direct measurements of magnetic field and plasma in the vicinity of what may have been reconnection sites. 7.1 Flux Transfer Evidence The case for the occurrence of flux transfer in the magnetosphere from closed to open field lines is based on four sets of observations, discus- sed below: (i) Existence of open field lines in the tail. Anderson and Lin have studied the shadowing effects on solar elec- 'trons (£ > 20 kev)^ produced by the moon when it is located in the geomagnetic tail. They provide persuasive evidence that a substantial amount of magnetic

939 flux in the two tail lobes occurs on open field lines, i.e., on lines that intersect the earth's surface in one place only. But the observations do not establish how large a fraction of the tail flux is on open lines at a given distance from the earth. Thus, it is not known for a fact how large a fraction of the earth's polar-cap field lines, i.e., lines emerging at latitudes above the auroral oval, that are open. A popularly held view is that all are open. But Heikkila , questioning the soundness of this 77 78 view, has drawn attention to observations by McDiarmid et al. J which indicate the common occurrence of trapped particle pitch-angle distri- butions in the day-side cusp region as well as poleward of discrete auroral arcs. (ii) Flux erosion from the front-lobe magnetosphere. The magnetopause is observed to move closer to the earth when the inter- planetary field develops a southward component. At the same time, the day- side polar cusp moves to lower latitudes. These effects cannot be accounted 79 for by simple compression of the magnetosphere. Maezawa has estimated that flux on closed field lines is removed from the magnetosphere front lobe in an amount estimated at about 108 weber during a typical event. Either this flux is transferred to open field lines in the polar cap by dayside magneto- pause reconnection or it is moved into the tail while remaining on closed field lines. In the latter case, the flux might be added to the lobe of closed field lines in the tail or it might possibly be placed on open field lines by reconnection at the tail magnetopause. The popularly held view is that the flux is transferred to open flux by reconnection somewhere on the dayside magnetopause.

940 (lii) Flux addition in the open tail lobes. A substantial body of evidence indicates that the magnetic field inten- sity in the tail starts to increase shortly after the onset of a southward component of the interplanetary magnetic field while at the same time the asymptotic tail cross section increases5^. The observed concurrent gradual thinning of the tail plasma sheet (which is believed to contain the closed tail field lines) argues against these effects being caused by an in- crease of flux on closed field lines in the tail. Rather they indicate an increase of flux on open field lines in the two tail lobes by an amount estimated at 1 - 2.5 x 108 weber. If the auroral oval (and the dayside cleft) is associated with the separatrix between closed and open field lines, the motion of this oval to lower latitudes following the southward turning fi? of the interplanetary field " supports this interpretation. But the evidence, while strong, is not conclusive. If closed field lines occur in the tail outside (i.e., above and below) the plasma sheet, the flux on open field lines could conceivably remain unchanged. (iv) Polar cap electric fields and convection. Electric field measurements " in the polar ionosphere indicate an average voltage difference across the polar cap of the order of 65 kVolt, corresponding to a magnetic flux transport across the cap from dayside to Q f\ nightside at a rate of about 2 x TO8 weber/hour . Ion flow measure- 52 ments " over the polar cap'show flow patterns that carry particles and, unless E_ • B_ f 0, magnetic flux poleward across the day-side cleft in a narrow longitude sector. Most but not all of the flow in the cap region occurs near the equatorward edge of the cap adjacent to an abrupt flow re- versal, below which the return flow to the dayside occurs. While the exact

941 location of the separatrix between closed and open field is not known, it is difficult to locate it in such a way that these results do not imply a trans- fer of flux from closed to open field lines. In spite of the ambiguities in the interpretation of the observations listed above, their mutual consistency in terms of flux transfer rates is impressive and lends credence to the idea that flux transfer from closed to open field lines does occur in the magnetosphere. However, a far greater body of simultaneous observations by satellites at different locations in the magnetosphere needs to be examined in order to establish the validity of the idea in a conclusive manner. It is noted that on the average, any flux transfer from closed to open field lines must be balanced by a reverse trans- fer from open to closed lines. Tail reconnection, occurring sporadically in connection with the expansive phase of magnetic substorms, is thought to ac- complish this latter transfer but conclusive evidence is not available (see section 7.2). 7.2 Measurements Near Reconnection Sites In a strict sense, direct evidence for reconnection consists of in- situ observations of the hyperbolic magnetic field configuration associated with a separator and an electric field along that line. The electric field observation may convincingly be replaced by the observation of plasma energiz- ed in the reconnection process (compare sections 2.4 and 4.2). £-7 Hones et al " have reported observations of proton fluxes and of mag- netic fields in the geomagnetic tail at geocentric distances in the range of 25-32 earth radii. They have found substorm events in which tailward proton flows at speeds up to 1000 km/s and an associated southward component of the magnetic field occurred during the storm expansion phase, followed by earth- ward flow and a northward field component during recovery. Such observations

942 are consistent with a separator moving tailward past the satellite. However, evidence concerning the magnetic-field component perpendicular to the tail current sheet is not entirely convincing unless the field is measured near the center of the sheet, which was not the case. And recently Lui et al74J?5 have challenged observations purporting to show the formation of a near earth reconnection line during substorms. Observations of proton jetting in the tail ' of energetic particle bursts5J68J98, and of lunar shadow 73 patterns of electron fluxes , while generally compatible with tail recon- nection, nevertheless cannot be claimed to provide unambiguous proof of the occurrence of the process. At the dayside magnetopause, magnetic field components perpendicular to 2 08 the magnetopause have been observed 3 although not as a permanent feature not even when the magnetosheath field opposes the terrestrial one. The nar- row jets of energized plasma, predicted by magnetopause reconnection models (e.g., figure 19) and flowing nearly tangential to the magnetopause, have not been seen*, even though satellites such as HEOS 2 have had the right posi- -ecent 29,35 tion and attitude to observe them J . These facts along with recent observations of a plasma boundary layer inside the dayside magnetopause suggest that magnetopause reconnection, if it occurs, may be more sporad- ic and more localized than originally expected. Furthermore, the possibility of reconnection in the cusps and elsewhere on the magnetopause surface, rather than near the sub-solar point, needs to be examined . The absence of observations of plasma energized by dayside reconnection 54 has led Heikkila to suggest that no such reconnection occurs, i.e., that the magnetopause is an electrostatic equipotential. This suggestion is "However, a layer of energetic electrons of unknown origin has been discovered • outside the tail magnetopause J.

943 difficult to reconcile with the presence of magnetic field components perpendic- ular to the magnetopause, unless one is willing to accept potential differences of the order of 50 kvolt along field lines extending from the magnetopause into the solar wind; or unless one argues that such perpendicular components are never present over any substantial part of the dayside magnetopause.

944 8. Summary and Recommendations In this paper we have given a reasonably detailed description of the present status of our understanding of reconnection. The picture that emerges is of a process, simple in concept but extremely complicated and multifaceted in detail. Nonlinear magnetohydrodynamic processes in the external flow region, governed by distant boundary conditions, are coupled to non-linear microscopic plasma processes in the diffusion region in a manner not clearly understood. And it appears that reconnection may operate in entirely different ways for different plasma parameters and for differ- ent external boundary conditions. Steady reconnection may be allowed in some cases, forbidden in others, with intermediate situations involving impulsive or pulsative events. On the whole, our theoretical and empirical knowledge of reconnection is poor. Yet the process plays a key role in solar-flare theory as well as in our present concept of the dynamic magnetosphere. And it appears as an unwanted feature in tokamaks and other fusion configurations. These facts, along with the potential importance of reconnection in other parts of cosmos, amply justify vigorous research efforts related to reconnection in the follow- ing five areas: solar-flare and astrophysical observations, magnetospheric observations, laboratory experiments, computer simulation, and analytical model building. The first area, while extremely important, is too broad to be commented upon here. In the remaining areas the following recommendations are made: Magnetospheric observations and experiments should include: (i) A coordinated program to establish (or deny) the occurrence of flux transfer across separatrix surfaces, and to study other global consequences of reconnection.

945 (ii) Direct observations of magnetic field, plasma, energetic particles, and fluctuating as well as steady electric fields, near magnetospheric recon- nection sites. Multi-satellite missions are needed to separate spatial and temporal effects. (iii) Perhaps active experiments, such as the release of barium clouds near reconnect!'on sites. Laboratory experiments. The observation of impulsive flux transfer events and of ion-acoustic turbulence in the double inverse pinch experiment il- lustrates the importance of such experiments in shaping our understanding of reconnection. Yet, (excluding fusion devices) the double inverse pinch ap- pears to be the only operating reconnection experiment in the U.S. today. A substantially expanded laboratory program is needed with four principal goals: (i) Simulation of solar-flare reconnection. (ii) Simulation of magnetospheric reconnection. (iii) Study of basic plasma processes of importance in reconnection, such as slow-shocks and anomalous resistivity. (iv) Exploration of reconnection in plasma heating devices. Computer simulation provides a potentially very powerful tool for the study of reconnection. Magnetohydrodynamic codes, and ultimately self-consistent particle-fields codes should be developed. It is particularly important to build into such simulations the effects of in.ertial and anomalous resistivity in the diffusion region. Analytical models of reconnection should emphasize the following interrelated problems: (i) Nonsteady and three-dimensional effects. (ii) Plasma processes in the diffusion region.

946 (iii) Particle acceleration. (iv) Reconnection of fields that are not antiparallel. It is through vigorous activities in the aforementioned areas, and ef- fective interaction between scientists involved in them, that our understand- ing of the reconnection process may be most rapidly advanced. To bring about such a state of affairs, two proposals are made: (A) That a special working group be assembled with the charge of promot- ing effective research on all aspects of the reconnection problem and with membership drawn from the five research areas discussed above. (B) That NASA and other funding agencies develop coordinated programs of support for reconnection research. The importance of the reconnection concept is such that we can "ill afford the present somewhat haphazard approach to its study. Acknowledgment Research supported by the National Science Foundation under Grant ATM 74-08223 A01 and by the National Aeronautics and Space Administration under Grant NSG 7292 to Dartmouth College. The author has benefitted from comments on the paper by: P. J. Baurn, A. Bratenahl, W. J. Heikkila, F. W. Perkins, E. R. Priest, and V. M. Vasyliunas. Conversations with A. Bratenahl and W. J. Heikkila have enlight- ened the author on many aspects of reconnection and its possible application in the magnetosphere.

947 r-o f.O.Sr. FIGURE 1 Qualitative time sequence for two di- poles moving toward each other on the solar sur- face. A current sheet A-B develops during time lKKro. Rapid reconnection sets in at t = t0 and relaxes the configuration toward a potential field in the short time et0. Curr«p Curranl FIGURE 2 Current-sheet formation caused by the stretching of magnetic loop on the sun (after Carmichaeli4).

20 DEC. DEC. 2I00 FIGURE 3 Sector structure of the interplanetary magnetic field in the ecliptic plane as observed by IMP-1 in 1963. Positive and negative signs indicate the direction of the measured interplanetary magnetic field away and toward the sun, respectively (Wilcox and Ness'23). FIGURE 4 The earth's magnetosphere with mag- netosheath magnetization due south.

949 FIGURE 5 Magnetic field conf1gurations for a rapidly spinning rnagnetosphere containing low- energy plasma (Gleeson and Axford48). FIGURE 6 Basic plane reconnection configura- tion. Solid lines are magnetic field lines; dashed lines are streamlines. The shaded region at the center is the diffusion region. © V v — V - Ct Ct Ct

950 FIGURE 7 Configuration of slow MHD shocks in the reconnection geometry (after Petschek90). FIGURE 8 Field configuration in double inverse pinch experiment (after Bratenahl and Baum1'). Out«r tnvtlop* -—21

951 20 15- 10 -20 -20 FIGURE 9 Field lines in the plane of the neutral points Xl and Xt for a uniform magnetic field and a di- pole field (Cowley24). Dipole moment vector at right angles to the uniform field. FIGURE 10 Schematic of separatrix surfaces for magnetopause reconnection. Lower figure shows separatrix of the null point ,V,; upper figure that of X2. The two figures are to be superimposed.

952 FIGURE 11 Formation of reconnection bubble in the geomagnetic tail. Schematic of field config- uration in the noon-midnight meridional plane. FIGURE 12 Three-dimensional sketch of recon- nection bubble with the reconnection line along AXB and an 0-type magnetic null line along AOB.

953 FIGURE 13 Petschek's reconnection model with MA = .1. Magnetic field lines (solid lines) and streamlines (broken lines) are shown (Vasy- liunasIi8). Fast-mode expansion in the entry flow. FIGURE 14 Slow-mode reconnection model with MA ) = -5. (Vasyliunasii8). Slow-mode expansion in the inflow is concentrated to waves from four external corners.

954 FIGURE 15 One quarter of a symmetric com- pressible slow-mode magnetic-field reconnection model for MA = 0.7 and/3, = 2^0 Pi (Yang and SonnerupI25). = 5 FIGURE 16 Schematic showing "reflection" of slow mode expansion fan in the x axis (Yang and Sonnerupi25).

955 M. 0.3 -, 0.2 - O.I 1.0 20 FIGURE 17 Relationship between the inflow Alfve'n numbers MA , far upstream, and MA , adjacent to the diffusion region. Solid curves refer to slow-mode expansion model,125 dashed curves to the Soward-Priest110 analysis of the fast-mode expansion model. FIGURE 18 Upper half of compressible slow- mode model of magnetopause reconnection for MAI = .2 and/3] = 2Iu0p 1/^12 = 2. The inter- mediate wave (IW) marking the magnetopause is shown as a dashed line. The slow mode expansion fan (SEF) is shaded (Yang and Sonnerup126).

956 Maqntti- 50 B1 50 B2 7 'j y « ** foahe oth 68 03 27 i6:i8:56:34 UT 50 81 50 33 FIGURE 19 Polar plots of magnetic field at the magnetopause. Left hand figure shows the field components BI andfij tangential to the magnetopause during an OGO-5 crossing; right hand figure shows the nearly constant magnetic-field component 83 nor- mal to the magnetopause. The field is given in units of y (ly = Int). Intermediate wave or rotational discontinuity is segment A i - A 2 of the left-hand trace; the slow expansion fan is segment A 2 - A 3. The seg- ment A3 -A4 may be caused by a finite gyroradius effect not contained in the MHD model. FIGURE 20 Hill's (53) collisionless reconnection model. The magnetic-field change across the slow shock becomes weak for 0 = 2/i0 p/B2 -* 0 with the principal field reversal occurring in a current layer at X = 0. Slow Shock

957 B ^ B ••.V FIGURE 21 Magnetic field lines and streamlines for stagnation point flow,^ = (-k^x,kiy,k3z), at a current sheet. The diffusion-dominated region is shaded (Sonnerup and Priest109).

958 «=0 10 - 0-25 05 -> -I FIGURE 22 Nondimensional magnetic-field profiles /£ = By(E0 \fi!0a/k> for the configuration in Figure 21 in the resistive limit (a = ki/ki ;k3=ki -Ar2 )• Plane stagnation point flow for o = 1, axisym- metric flow for a = H. The frozen-field profile for a = 1 is shown by dashed line (Sonnerup and Priesti09). i.o - FIGURE 23 Nondimensional magnetic-field pro- files for the configuration in Figure 21 in the elec- tron-inertia! limit [a = k2/ki; k3 = ki-k^; Xe = (rnel\ifpei^h\. Plane stagnation-point flow for a = 1; axisymmetric flow for a = %. The frozen- field profile for a = 1 is shown by dashed line.

959 Hall currents FIGURE 24 Schematic picture of Hall current loops in the diffusion region. Also shown is the transverse magnetic field Bz(x,y) induced by these currents. f(Te/Tj) FIGURE 25 Function f(Te/Tt) = (kTt/me)Vl/vc where vc is the critical current velocity for onset of ion-acoustic instability (after Fredricks40). I0 i 8 6 - 4 2 0 10 15 Vr-

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