Global role of E,, in magnetopause reconnection: An explicit demonstration

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. A7, PAGES 13,015-13,022, JULY 1, 2001 Global role of E,, in magnetopause reconnection: An explicit demonstration G. L. S iscoe and G. M. Erickson Center for Space Physics, Boston University, Boston, Massachusetts B. U. Sonnerup Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire. N. C. Maynard, K. D. Siebert, D. R. Weimer, and W. W. White Mission Research Corporation, Nashua, New Hamshire. Abstract. We use a global MHD simulation to compute the distribution of E,, on the face of the magnetopause as represented by the last closed field line surface. In MHD codes, E,, is a proxy for magnetic reconnection. Integrating E,, along the topological separator line between open and closed magnetic field lines gives the global reconnection rate at the magnetopause. In the case studied here, where the interplanetary magnetic field (IMF) is precisely duskward, we find the global reconnection rate to be ~49 kv, comparable to potentials inferred from measurements made in the polar cap. The exercise demonstrates an application of a general reconnection theorem that, in effect, equates reconnection with E,,. It prepares the way for MHD imaging of reconnection in terms of contours of E,, on the magnetopause. The result also illustrates a property of parallel potentials in the global contexthat is not generally recognized. Nearly the full magnetopause reconnection voltage exists on some closed field lines between the northern and southern polar caps, so that they leave the dawn, southern hemisphere with a sizable positive polarity and enter the dusk, northern hemisphere with a sizable negative polarity. An unexpected finding is a substantial parallel potential (between 10 and 15 kv) between the magnetopause and the ionosphere in northern dawn and southern dusk sectors. (Interchange"dawn" and "dusk" for dawnward IMF.) This potential has the polarity that accelerates electrons into the ionosphere in the dusk sector and, so, might be the origin of the "hot spot" seen there in precipitating electrons. 1. Magnetic Reconnection at the Magnetopause "Reconnection occurs when an electric field component E, is present along a magnetic separator (or X line). The reconnection rate is proportional to E,,." The source of this quote, Sonnerup [1988, p. 143], is one of five papers published in 1988 that emphasized the role that E, plays in magnetic reconnection. Papers before 1988 had similarly remarked on the role of E,, starting already with Petschek [1964] and Dungey [1978]. Vasyliunas [ 1984] opened the discussion of E, in a global context, leading to the five papers of 1988, which serve as the point of departure for the study presented here. In the first of the five papers ('first' in a nonchronological order suited for exposition), Sonnerup [ 1988] emphasized local conditions for reconnection. In the second and third, Schindler et al. [1988] and Hesse and Schindler [1988] stated and proved a theorem on general magnetic reconnection that identified reconnection with E, that could be applied globally. In the fourth paper, Greene [ 1988] showed that E, is needed to eliminate electric field singularities at magnetic nulls that arise in global reconnection topology. The contributions of Schin- Copyright 2001 by the American Geophysical Union. Paper number 2000JA /01/2000J A dler's group and Greene are general and apply to arbitrary situations. In the fifth paper, Siscoe [ 1988] showed how they apply in the context of global reconnection at the magnetopause. Here we use a global MHD simulation to exploit these earlier findings by computing the distribution of E., (as a proxy for magnetic reconnection) on the face of the magnetopause as represented by the last closed field line surface. Figure 1 (adapted from Figure 15 of Siscoe [1988], which is similar to Figure 10 of Sonnerup [ 1979], which, in turn, derives from Figure 9 of Cowley [ 1973]) gives the relevant magnetic topology with the crucial null points. Here, for the case in which a uniform field is superposed on a dipole field, are surfaces that separate volumes in which field lines are disconnected, open, and closed. These are topological separatrix surfaces. Separatrix surfaces are field line surfaces, and only field lines lying on these surfaces are shown. The outer, cylindrically shaped surface separates disconnected (interplanetary(imf)) field lines from open field lines. The inner, torus-shaped surface separates open field lines from closed field lines. The separator line is the intersection of these two surfaces. Two magnetic null points (shown as dots) occur on the separator line. The null points can be said to generate the separatrix surfaces out of an infinite set of magnetic field lines that radiate out from them or converge onto them. The dusk null point generates the northern half of the cylinder and the southern half of the torus, while the dawn null point generates the southern half of the cylinder and the northern half of the 13,015

2 13,016 SISCO ET AL: GLOBAL ROLE OF E, IN MAGNETOPAUSE RECONNECTION North North \ Stemline Dusk Null Point i Figure 1. Magnetic topology associated with magnetic reconnection at the magnetopause, showing the separatrix surfaces (topologically a cylinder and a toms) that result from superposing a uniform field and a dipole field. (Adapted from Siscoe [1988].) toms. There is also a single field line from the dawn null point potentials of say 0, 10, 20, 30, 40, 50, and 60 kv, corresponding into the southern dawn ionosphere and a similar line from the to the solar wind flowing across the open field lines. To prevent dusk null point into the northern ionosphere. We call these sin- an infinite electric field, however, the potentials of all field lines gular field lines 'stemlines'. This topology is general in that it must be equal at the null point. The differences in the potentials represents all IMF orientations except the singular cases in which that exist far from the null point mustherefore be cancelled by the IMF is parallel or antiparallel to the dipole axis. Although parallel potential drops along each of the field lines before they Figure 1 depicts the topology of superposed fields in a vacuum, reach the null point. Note thathis implies thathe full presumed which omits fields from magnetopause currents, simulations 60 kv, which is the global reconnection rate in this illustration, prior to the one presented here have shown that this vacuum to- must exist as a parallel potential drop along the separator line pology holds even in MHD contexts [Crooker et al ; White that connects the two null points. This general observation about et al., 1998]. the requirement for parallel potential drops says nothing about The role of E, in global magnetopause reconnection can be where along the field lines the drop occurs. It could be localized stated in reference to Figure 1. If reconnection were to occur in or distributed or even patchy. Nor is our sketch necessarily a the absence of parallel potentials, then the electric field would be good representation of the actual shape of the field lines on the infinite at the two magnetic nulls. This is because, as Vasyliunas separatrix surface. They could get nearer to the separator line be- [ 1975] pointed out, reconnection is equivalent to a condition of fore veering off to reach the null point. Then the parallel potennonzero potential across the cylindrical separatrix surface. All tials could be confined to the vicinity of the separator line rather field lines on the Northern Hemisphere surface (for definiteness), than be spread over a wide area, as Figure 1 suggests. (Figure 10 with their separate potentials, come together at the dusk null of Sonnerup [1979] depictsuch a situation, as does Figure 2 of point. Thus, in the absence of parallel potentials to equalize the Vasyliunas [ 1984].) separate potentials, a finite potential would exist 'across' a point, Global MHD simulations of the interaction between the solar giving an infinite electric field. As Greene [1988, p. 8588] ele- wind and the geomagnetic field exhibit magnetic reconnection at gantly puts it: "Since many lines of force converge on the null the magnetopause and so should exhibithe properties just depoints and each line can be assigned its own potential, regularity scribed. They can therefore be used to demonstrate quantitaof the electric field at nulls can only be achieved if it has a par- tively the principle that equates reconnection with parallel elecallel component." Because Stem [ 1973] first pointed out the un- tric fields. This paper treats the case of steady solar wind condiphysical situation that occurs in the absence of parallel potentials tions with duskward IMF (pure B, no B or B..). By mapping the [see also Lyons, 1985], the singular point is called the Stem sin- location of the parallel electric fields, these simulations can also gularity. provide pictures that show where (in the simulation) reconnec- Again, parallel electric fields remove the Stem singularity by tion occurs on the magnetopause. Pictures of the areal distribuequalizing the potentials at the null points on all field lines that tion of E, on the magnetopause will be given in a separate paper converge there or radiate from there. For example, Figure 1 in which results for different IMF orientations will be compared. shows seven of the infinite set of field lines that generate the cy- Here we are concerned with the magnitude and distribution that lindrical separatrix surface in the Northern Hemisphere. Far the simulation gives for E, along the separator line and along the from the null point, these seven field lines might have electrical field lines connecting the null points with the ionosphere.

3 SISCOE ET AL: GLOBAL ROLE OF E, IN MAGNETOPAUSE RECONNECTION 13, Integrated Space Weather Prediction Model, E = -VxB + lj + E o (1) Parameters for Duskward IMF Run, where V is flow velocity, B is magnetic field, rl is explicit elec- Methods for Calculating E, trical resistance, J is current density, and Epo M is a contribution The Integrated Space Weather Prediction Model (ISM) operates within a cylindrical computational domain with its origin at the center of the Earth and extending 40 R sunward, 300 R anfrom the PDM algorithm. Here E om is an electric field that represents a diffusive flux of magnetic field introduced by the numerical algorithm. It is generated to prevent development of tisunward, and 60 R radially from the Edith-Sun line. The do- spurious extrema in the numerical advance of Faraday's law. main has an interior pherical boundary at the approximate bot- Since it causes magnetic flux to move across grid cells, it can be tom of the E layer (100 km in simulations described here). expressed as an equivalent electric field. The curious fact is that The ISM is based on standard MHD equations augmented all three terms on the right-hand side can contribute to E. with hydrodynamic equations for a collisionally coupled neutral thermosphere [White et al., 2000]. The combined equations go continuously from pure MHD for plasma in the solar wind and magnetosphere proper ionospheric/thermospheric equations at low altitudes. For purposes of the simulations reported here, Though not surprising for the rij and Ep, terms, this statement seems to be a mistake for the VxB term. In many global MHD codes, however, the constrainthat the magnetic field have zero divergence is satisfied through the use of a staggered mesh that centers different MHD variables at different spatial locations. specific selections of parameters and simplifying approximations That is, the components that make up a vector are not spatially have been made. The finite difference grid resolution varies from a few hundred kilometers in the ionosphere to several R at coincident; for example, B, B., and B: are centered on cell faces to which they are normal. Similarly, components of the com- E the outer boundary of the computational domain. At the magne- puted vector cross product, VxB, are evaluated at cell edges to topause, resolution ranges between 0.2 and 0.8 R. Explicit vis- which they are parallel. Consequently, the vector product cosity in the plasma momentum equation has be n seto zero. VxBeB is a hybrid spatial composite, which is not guaranteed to To approximate nonlinear aspects of magnetic reconnection be zero. Nonetheless, it is nearly zero almost everywhere in the within the context of a finite-difference grid, the coefficient for computational volume, and almost everywhere the cross product explicit resistivity in Ohm's law is zero when the current density thus computed is closely orthogonal to the crossed vectors. 10 perpendicular to B is less than 3.16x 10 '3!xA m '2 and is 2x 10 m ~ Where V or B changes appreciably over the size of a grid cell, s" otherwise. In practice, this pammeterization leads to nonzero however, an appreciable parallel component of the cross product explicit resistivity primarily in the subsolaregion of the dayside magnetopause, and then typically only for strongly southward or dawn/dusk directed IMFs. The dissipation for numerical stability is based on a form of the partial donor-cell method (PDM) developed by Hain [1987]. The solar wind inflow boundary conditions for the simulation used typical values: speed of 350 km can result. Thus VxB acquires a parallel component precisely where one expects the parallel components of rij and E o to be big. A parallel componento VxB is not physically meaningless. Vasyliunas [1988] points out that when V and B fluctuate on small enough scales (in this application on the scale size of the s 'l, density of 5 protons cm" ; temperature of 20 ev; IMF strength differencing grid) any coherence between fluctuations will add a of 5 nt. For this simulation, thermospheric hydrodynamics and explicit chemistry between ionospheric and thermospheric speterm to the grid-averaged cross product, which can have a parallel componento the grid-averaged V and B. Since, as emphacies have been disabled. The ionospheric Pedersen conductance sized earlier, a component of VxB parallel to B is equivalent to is uniform in latitude and longitude with a value of 6 Siemens; reconnection, this constitutes a form of turbulent dissipation. no Hall conductance was used. An analysis of boundary layer transport properties in the ISM simulations indicates the effective kinematic viscosity due to artificial numerical dissipation in the Thus the numerical differencing procedure for computing VxB introduces, / la Vasyliunas [1988], numerical dissipation that, analogous to turbulent dissipation, can cause reconnection. It ISM's finite difference approximations to be of the order of 109 combines with rij and E o,to provide dissipation steep gradim 2 s". This level of dissipation consistent with estimates of kinematic viscosity in the magnetospheric low-latitude boundary layer as determined by satellite data [e.g., Sckopket al., 1981 ]. In this simulation the Earth's dipole is perpendicular to the solar wind flow and the IMF. Results are presented in the geocentric solar ecliptic coordinate system (GSE), which in this case is the same as the geocentric magnetospheric coordinate system ents. In the case studied here, the VxB contribution to the total magnetopause reconnection rate is of the order of 30%. We used three methods to calculate the parallel potential. Method 1 simply integrates the parallel component of equation (1) along a field line. Method 2 computes the electrical potential everywhere in the ISM domain by integrating from the domain's upwind boundary the electric field vector given by (1). Method (GSM). The ISM code was run for 2.5 hours of magnetosphere 3 traces points where the potential is desired back along flow time; by then, global-scale parameters related to magnetopause streamlines to the solar wind, where the electric field is uniform, reconnection had become steady. We must discuss here subtleties of calculating E,, with a numerical MHD code (in this case the ISM code, but in this respect it is like other codes.). E, is not computedirectly by the code. and measures the potential there. Method 1 is most straightforward, and we show results based on it. Method 2 assumes that steady state applies everywhere. Although this is a good assumption upwind from the magnetopause, it becomes less good Instead, it is a 'postprocessing' variable. Discussion is necessary within the magnetosphere, and it becomes bad in the plasma because different postprocessing procedures for calculating E, or, equivalently, the parallel potential give answers that differ by 10-20%. In most cases, such differences are unimportant, but in one or two cases they might matter. The electric field that the code uses to advance the magnetic field according to Faraday's induction law is sheet in the magnetotail. In principle, methods 1 and 2 should give identical results, but differences in numerical implementations of the methods lead to differences in results up to 20%. Method 3, which is based on streamlines being equipotential in ideal MHD, works only for points connected by streamlines to the solar wind. It assumes that ideal MHD applies right up to the

4 13,018 SISCO ET AL: GLOBAL ROLE OF E, IN MAGNETOPAUSE RECONNECTION Separator Line Front View I Stemlines Figure 2. 'Top' and 'front' views of a magnetic field line that lies very close to the true separator line that connects the two magnetic null points, which the dots labeled A and B represent. We use this separator line between the tips A and B to approximate the true separator line. The stemlines are singular field lines connecting A and B to the ionosphere. The 'Earth' gives the scale. The inset shows a front view of field lines that lie on the lastclosed-field-line surface. The separator line is one of these lines. point where the nonideal parallel potential is desired. This is an parallel potential (by method 1) as a function of distance along unrealistic assumption, and the method is used mainly as a sanity the separator line of Figure 2. The plots start at A and end at B. check on the results. It gives a lower bound on nonideal MHD effects, such as parallel potentials. The top panel gives the logarithm of magnetic field strength. The minima in magnetic field magnitude at A and B mark where the line comes nearesthe null points. Field strength reaches a local maximum of ~40 nt at the halfway point, which corre- 3. Parallel Potential Along the Separator Line and sponds to the stagnation point in the solar wind flow. The midthe Global Closed-to-Open Reconnection Rate dle panel gives E,, along the field line in mv/m. E, reaches its highest values of around 0.4 mv/m in two broad maxima near Figure 2 shows, in the inset, a suite of field lines seen from the sun (the 'front') labeled 'Last Closed Field Lines'. This set of 20 ø north and south latitudes. It has a dip where the separator crosses the subsolar point, and it has secondary peaks at the lines comes as close as our search procedure allows to defining minima in magnetic field strength associated with the null points. the separatrix surface that the (true) dawn null point generates. (One sees this at B.) We say more in section 5 abouthe shape of Figure 2 also shows two views of one of these field lines, the one the distribution of E,, along the field line. The bottom panel gives that comes closest to touching the magnetic nulls. This special the parallel electrical potential along the field line in kv. We field line is seen here projected onto an xy plane (i.e., top view) place the zero of potential on the x axis, which is appropriate to and a yz plane (front view) in standard GSE coordinates. The the symmetry of the problem. The total potential along the field middle part of this line, labeled "separator line," is very close to line from A to B is ~49 kv. This is the answer that method 1 the actual separator line that connects the two magnetic nulls in (integrating E,, along the field line) gives. Method 2 (integrating the reconnection topology shown in Figure 1. (Figure 3 validates E in from the domain boundary to find the electric potential evethis claim.) The points labeled A and B sho where the depicted rywhere) gives ~58 kv, and method 3 (tracing the end points A separator line comes closest to the null points. (In sections 4 and and B out into the solar wind along streamlines) gives ~39 kv, 5 we say more about the distinction between the depicted sepa- which, as expected, is the lowest of the three. Method 2 shows rator line and the true separator line and about where the depicted the same general distribution of potential along the separator line A and B are located relative to the true null positions.) The portions of the field line that connect the depicted nulls A and B to the Earth represent our closest approaches to the stemlines. Figure 3 shows magnetic field strength, E, (from(1)), and as the bottom panel of Figure 3; that is, it also shows that the electric field strengt has a dip across the equator. As described next, method l's result (~49 kv) seems most consistent with evidence from the ionosphere.

5 SISCO ET AL: GLOBAL ROLE OF E, IN MAGNETOPAUSE RECONNECTION 13, South _ North 1.2E Dawn _ _ Dusk.o 0.6 _ 0.4 A... =-,, E 0.30 B ' 0.20,,', 0. o Distance Along Separator Line (RE) Figure 3. Magnetic field strength, E,, and parallel potential (obtained by method 1' see text) as a function of distance along the separator line of Figure 2 from the souther null A to the northern null B. 4. Relation Between Global Magnetopause equatorward of the separatrix surface. As we show below, the Reconnection Rate and the Transpolar Potential proximate cause of this displacement a significant potential drop along field lines. The electrical potential in the ionosphere is found by inte- Consider next the polar cap contour shown in Figure 4. As grating the ISM-computed VxB in the ionosphere, putting the mentioned, this contour connects the points where field lines on zero of potential at equatorial noon, as mandated by the symme- the last-closed-field-line surface of Figure 2 intersect the ionotry of the problem. Figure 4 shows the result for the Northern sphere. Four of these lines are shown. The line located at Hemisphere. It also shows the contour of the open polar cap as ~17:15 local time is the singular Northem Hemisphere stemline determined by the intersection of the last closed field line surface that comes from the tip of the field line at B in Figure 2. It carwith the ionosphere. (The contour's noncircularity and irregu- ries the duskside part of the reconnection potential down to the larities are a snapshot of a constant agitation that pervades the ionosphere. The corresponding field line that carries the dawngeometry, as if Kelvin-Helmholtz waves and other deformation side reconnection potential down to the ionosphere from A is lowaves wash along the boundary and are recorded in the bound- cated at ~9:30 local time. To find it, we could not use the stemary's image in the ionosphere. Movies of the contour show these line from A, for it goes to the Southern Hemisphere. Instead, we irregularities migrating generally tailward.) The lines in Figure 4 launched a suite of field lines from the vicinity of A, followed are identified by their labels. This figure contains much infor- those that went into the northern ionosphere, and picked the one mation, which we will extract in steps starting with the potential that ended up at the highest potential. This is the "Dropline from pattem. A" in Figure 4. We consider nexthe electrical potentials along Here we see the standard two-cell convection pattern with variou segments of the field lines that Figure 4 shows. This "round-and-crescent" asymmetry characteristic of a dominant, leads to the discovery of a substantial potential drop along the positive IMF B [Crooker, 1979; Reiff and Burch, 1985; Heppner dropline from A. and Maynard, 1987; Weimer, 1995]. The total transpolar poten- First, however, note that Figure 4 allows us to determine that tial between the centers of the two cells is ~44 kv, which is 5 kv the total potential along the entire field line in Figure 2 is ~46 less than the 49 kv separator-line reconnection potential found kv. This is twice the ~23 kv (negative) that Figure 4 gives for above by method 1. We return below to discuss the meaning of the point where the stemline touches the ionosphere. Since, as this discrepancy, but first we continue the description of the io- Figure 2 shows, this field line is diagonally symmetric, it must nospheric potential pattern. The peaks of the two cells are dis- leave the southern ionosphere at a positive 23 kv, which we have placed from the separatrix surface by 5 or 6 kv. On the duskside confirmedirectly. Thus the full potential is 46 kv, which is ~3 the round cell's center lies poleward of the separatrix surface. kv less than the ~49 kv reconnection potential along the sepa- This is the well-known lobe cell owing perhaps to reconnection rator portion of the field line. A symmetrical ~1.5 kv parallel on open field lines with resulting overdraped geometry [Crooker potential drop along the stemlines each hemisphere probably et al., 1998]. On the dawnside the crescent cell's center lies makes up this 3 kv difference. The sign of the required potential

6 13,020 SISCOE ET AL: GLOBAL ROLE OF E, IN MAGNETOPAUSE RECONNECTION Stemline fro B 16..:;--- *'--,...,,, 8 Dropline :...,-;' from A I:' /..',/.' I.: / -"...':...'x x / /'//?\... ;;'",. (",.:,, +/ /,,, I W:! I, /!!,, :. / Polar Cap Boundary '. ": True Stemline 0... '..,,-... '" '...:.. '"' ' "" _ True Dropline " '......' LT Northern Hemisphere Ionosphere Figure 4. A polar view of the northern ionosphere showing equipotentials and a contour obtained by connecting the "footpoints" of the field lines that define the last-closed-field-line surface. (Irregularities in this contoureflect the presence of tailward-propagating waves on the boundary.) Contour levels are separated by 5 kv. The first solid contour is 0 kv. The numbers ( and 15.94) give the peak potentials of the two cells. The la- beled field lines are discussed in the text. drop is consistent with a resistive potential associated with region methods, as already mentioned. A 14.5 kv or 18 kv parallel 1 currents. We note that this small potential drop makes the re- potential along a segment of a field line not involved with reconsult of method 1 (discussed in section 3) the most reasonable of nection along the separator line is surprisingly big, though a the three methods we have used. Given, however, the uncer- qualitatively similar behavior was found in an analytical treattainty in the estimates of potential as indicated by the differences ment of the coupling of the low-latitude boundary layer with the between the values returned by the three methods, we should not ionosphere [Siscoe et al., 1991 ]. (The full 18 kv cannot be the attribute too much to potential differences less than ~5 kv. result of time variability, for it changes by <3 kv during the next Method 2 gives potential drops along the stemlines of ~6 kv also half hour.) By contrast, recall that the corresponding parallel in the sense of a resistive electric field generated by region 1 cur- potential along the stemline is only ~1.5 kv (or 6 kv by method rents. Method 3 gives ~3.5 kv but in the unphysical sense of 2). A direct application of method 2 to the dropline from A gives being opposite to a resistiv electric field generated by region 1 the same answer as seen in Figure 4. Even method 3 gives ~10 currents. (Note: we did not use Method 1, i.e., direct integration kv. Though we have isolated one field line for attention (the of E,,, to obtain the 1.5 kv potential along the stemline. The nu- dropline), a similar potential must exist along field lines in its merics pertaining to the PDM algorithm in the strong magnetic neighborhood. This result is the basis of the statement made earfield region close to Earth make the contribution of the PDM lier regarding the displacement of the crescent cell from the sepaterm to E, exceed that of the resistive term.) ratrix surface. The parallel potential along the dropline reduces As we now show, the ~1.5 kv potential along the stemline the ionospheric potential, thus shifting the peak that would oththat in Figure 2 touches down in the round cell is considerably erwise be there equatorward. less than the potential along field lines in Figure 2 that touch It is interesting to relate the parallel potential along the drodown in the crescent cell. (This statement applies even for the ~6 pline to the difference in the potentials of the round and crescent kv found by method 2.) To see the presence of a substantial par- cells. Denote the potential drop along the dropline from A to the allel potential on the dawnside, simply note that from Figure 3 dawn northern ionosphere by V d and the potential drop along the the potential at A is 24.5 kv, whereas the potential in the iono- stemline from the dusk northern ionosphere to B by V. Also, sphere at the foot of the field line from A is ~ 10 kv. This gives denote the reconnection voltage along separator line from A to B ~14.5 kv along the dropline from A. This inference can be by Vr,and let the potential be zero on the x axis. Then the footchecked using method I and simply integrating E, from A to the point in the ionosphere of the dropline from A is at potential ionosphere. The result is ~18 kv, which is ~3.5 kv bigger than (Vr/2-- Vd), while the footpoint of the stemline from B is at po kv obtained by subtraction, but perhaps not bigger than one tential (--Vr/2 + V,)' The sum of these values is (-V + V ). On might be allowed to expecto result from the uncertainties of the the other hand, if the center of the crescent cell is at potential V

7 SISCOE ET AL: GLOBAL ROLE OF E,, IN MAGNETOPAUSE RECONNECTION 13,021 and the center of the round cell is at potential-v 2, then the sum of these values is (V, - V2). The two sums should differ only to the extent that the footpoint voltage of the dropline and stemline are displaced by unequal amounts from their corresponding cell centers. Neglecting this latter correction (which is pretty small in this case), we then find v) = (v,.- v,). (2) this is not a big difference, the near agreement appears to result from partially offsetting effects. Overdraping increases the strength of the round cell while, parallel potentials decrease the strength of the crescent cell. The role of parallel electric fields in magnetic reconnection has often been noted. A demonstration of its role in a global, magnetopause setting as given here is nonetheless helpful since parallel potentials are normally associated only with regions closer to Earth where electron acceleration associated with auro- In this example the formula gives ras occurs. Parallel potentials associated with auroral acceleration are typically 1-10 kv. Here the parallel potential is the en- (V d - V,) = kv. (3) tire global reconnection rate, typically 60 kv, as inferred from measurements of the transpolar potential. We can check this result, for here we have independently V d kv and V kv; thus the agreement is quite good. In summary, the potential drop along the dropline exceeds the drop along the stemline by an amounthat is approximately equal to the magnitude of the round-cell potential minus the magnitude of the crescent-cell potential. This result could be used together with Weimer [ 1995] or other empirical ionospheric potential distributions to estimate what (Vd- V ) is in the actual system. Weimer, for example, gives a potential difference between 10 Consequences of such large potentials along field lines in the polar regions have been discussed by Crooker et al. [1998]. One significant consequence is that the separator line (and its ionospheric extensions as stemlines) and field lines around it leave the ionosphere in one hemisphere near the center of a positive convection cell (say) and enter the ionosphere in the other hemisphere near the center of a negative cell. This result explains a familiar but nonetheless puzzling observation. The polarity (positive or negative) of the larger of the usual pair of polar conand 20 kv between the round and crescent cells for the IMF B. vection cells changes between hemispheres, being positive in one case. Interestingly, and in apparent conflict with "common sense," the voltages in the dawn and dusk cells (to repeat) are difand negative in the other. This implies that the two "ends" of some field lines must have opposite polarities. We now see this ferent, the difference being opposite in opposite hemispheres. puzzling property to be a necessary consequence of magne- This is caused primarily by field-aligned voltages, which modify the magnetospheric voltage differently in the two hemispheres. To conclude this section, we quantify the difference between the ionospheric potentials associated with the separator and stemlines of Figure 2 and true separator and stemlines. We had to resort to using approximate separator, stemlines, and droplines because finding the exact lines is difficult. The line in Figure 2 is as close as we could reasonably come to the real one. The true null points, however, are relatively easy to find. The dawn null point lies at (x, y, z) -- (-14.3, -19.2, -8.4) in units of Earth radii. The dusk null point is diagonally opposite. From here it is easy to drop lines down to the ionosphere and pick as the real stemline and dropline the ones that reach the maximum potential. These lines are identified in Figure 4 as "True Dropline" and "True Stemline." The question is, by how much do they differ in ionospheric potential from the lines from A and B? That is, how big is the error resulting from using the lines from A and B instead of the true lines? Figure 4 answers the question. The potentials at the ionospheric ends of the two stemlines differ hardly at all. The potentials at the ionospheric ends of the drop lines differ by --,3 kv. But this is small compared to the potential we found along the dropline from A. Moreover, a direct application of method 2 to determine the potential difference along the --,14 R between the true null point null point A shows that the true null is 1.8 kv more positive than A. The potential along the true dropline is therefore within,--1 kv of the potential along the dropline from A. Thus the conclusions we have reached using approximate lines hold also for true lines. topause magnetic reconnection. A third point of significance is that, in this IMF By simulation, dayside reconnection is more component-like than antiparallellike. Most of the reconnection voltage accumulates along the separator line at low and middle latitudes on the sunward face of the magnetopause. Nonetheless, some reconnection occurs at high latitudes along the separator line, as the peaks in E, at the positions marked A and B in Figure 3 show. We return here to the discussion deferred in section 3 concerning the shape of the distribution of E, along the separator line. E,, in Figure 3 maximizes at midlatitudes and dips somewhat where the line crosses the subsolar point, which is also the equator. The strength of E, at the magnetopause ranges from essentially zero up to 0.4 mv/m, which agrees with the range measured by spacecraft [e.g., Lindqvist and Mozer, 1990]. The subsolar dip in E, hints at a systematic variation in E, distribution along the separator line as a function of IMF orientation in the yz plane. In a later paper that compares two-dimensional patterns of E, on the magnetopause surface for different IMF orientations, we show that the subsolar dip increases when the IMF tilts northward and fills in when the IMF tilts southward. In effect, magnetopause reconnection appears to become more antiparallel-like for northward IMF and more component-like for southward IMF. We close by mentioning what might ultimately turn out to be the point of deepest significance. This is the apparent control by global physics of things sometimes thought to be controlled by local physics. The global reconnection rate appears to be such a thing, as Axford [ 1984, 1999] and others have long argued. The concept has also been emphasized in connection with the Lyon- Fedder MHD code [Fedder and Lyon, 1987; Crooker et al., 5. Significance of the Results 1998]. For reasons other than testing this idea, we have per- The primary significance of this paper is that it demonstrates formed three IMF By runs with radically different specifications the explicit use of the integral of E, along the separator line to compute the global reconnection rate at the magnetopause in a of dissipation the magnetopause and distribution of E, along the separator line. Between these runs the transpolar potential MHD simulation. In this case we find the global reconnection differs hardly at all, in agreement with Fedder and Lyon. rate to be-*49 kv. The transpolar potential is,*44 kv. Though The present study has exposed another possible instance of

8 .. 13,022 SISCOE ET AL: GLOBAL ROLE OF E, IN MAGNETOPAUSE RECONNECTION this kind. The kv parallel potential along the dropline Lyons, L. R., A simple model for polar cap convection patterns and genseems to come from nowhere. It causes the crescent cell to be eration of O-auroras, J. Geophys. Res., 90, 156 l- 1567, Petschek, H. E., Magnetic field annihilation, in Proceedings of the AASweaker than the round cell, which is consistent with observations NASA Symposium on Physics of Solar Flares, NASA Spec. Publ. 50, [e.g., Weimer, 1995], though we acknowledge that such consis , 1964 tency could be coincidence. Also, as mentioned, the displacement of the center of the convection cell equatorward from the Reiff, P. H., and J. L. Burch, By-dependent dayside plasma flow and Birkeland currents in the dayside magnetosphere, 2, A global model polar cap boundary was found in an analytic study of the interac- for northward and southward IMF, J. Geophys. Res., 90, , tion between the low-latitude boundary layer and the ionosphere Robinson, R. M., C. R. C!auer, O. de!a Beaujardiere, J. D. Kelley, and [Sistoe et al., 1991]. The potential emerges as a requiremento D. S. Evans, in Solar Wind-Magnetosl here Coupling, edited by Y. satisfy boundary conditions of a high-order differential equation Kamide and J. A. Slavin, pp , Term Sci., Tokyo, that describes the interaction in the context of the global convec- Schindler, K., M. Hesse, and J. Bim, General magnetic reconnection, parallel electric fields, and helicity, J. Geophys. Res., 93, , tion potential. We note that the potential drop has the fight sign to accelerate electrons into the dusk auroral zones in both hemi- Sckopke, N., G. Paschmann, G. Haerendel, B. U. 15 Sonnemp, S. J. spheres (Southern Hemisphere for IMF By > 0 and Northern Bame, T. G. Forbes, E. W. Hones, Jr., and C. T. Russell, Structure of Hemisphere for IMF By < 0). This might account for the dusk- the low latitude boundary layer, J. Geophys. Res., 86, , side "hot spot" reported by Evans [ 1985], although the potential Siscoe, G. L., The magnetospheric boundary, in Physics of Space Plasfound here is overkill for the job, which might mean that the premas (1987), edited by T. Chang, G. B. Crew, and J. R. Jasperse, pp. cipitating electrons do not transit the entire parallel potential. 3-78, Scientific, Cambridge, Mass., Further support for this speculation is the IMF By dependence of Siscoe, G. L., W. Lotko, and B. U. I5. Sonnemp, A high-latitude, lowthe hot spot, which is the same as predicted here [Robinson et al., latitude boundary layer model of the convection current system, J. Geophys. Res., 96, , ]. Sonnemp, B. U. O, Magnetic field reconnection, in Solar System Plasma Physics, Vol. III, edited by C. F. Kennel, L. J. Lanzerotti, and E. N. Acknowledgements. 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Res., 93, , Hain, K, The partial donor cell method, J. Comput. Phys., 73, 131, G. M Erickson and G. L. Siscoe, Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA (siscoe@bu.edu) N. C. Maynard, K. D. Siebert, D. R. Weimer, and W. W. White, Mis- Heppner, J.P., and N. C. Maynard, Empirical high-latitudelectric field sion Research Corporation, Nashua, NH models, J. Geophys. Res., 92, , B. U. (5 Sonnerup, Thayer School of Engineering, Dartmouth Col- Hesse, M., and K. Schindler, A theoretical foundation of general mag- lege, Hanover, NH netic reconnection, J. Geophys. Res., 93, , Lindqvist, P.-A., and F. S. Mozer, The average tangential electric field at the noon magnetopause, J. Geophys. Res., 95, 17,137-17,144, (Received February 25, 2000: revised July 21, 2000: accepted July 27, 2000.)