Hall fields maximize just inside but along the separatrices and at the outer plasmoid boundaries where the Hall electrons turn around. Added are confirmation measurements in the Earth's magnetotail [ 37 ]. The top panel shows the bulk velocity-field reversal typical for plasma jetting away from the X point. The bottom panel contains the magnetic out-of-plane field B y exhibiting the plus-minus sequence expected of a Hall field near the X point. One may note that in these observations the Hall field undulates around a finite stationary guide field of 6 nT amplitude.
Figure 6. Hall fields, observation and simulation. Central panel: Two-dimensional simulation of reconnection including different electron and ion dynamics data taken from Vaivads et al. Shown is only the magnetic field with background field lines drawn in thin white. Dark blue regions indicates positive Hall- B y , white regions negative Hall- B y. The Hall fields concentrate along and inside the separatrices and boundaries of the two plasmoids where the Hall electrons dashed yellow lines turn away from the X point. Hall fields twist the original magnetic field.
They can be considered as self-generated magnetic guide fields in the ion diffusion region. Such guide fields embedded into the convection electric field E y black circles may accelerate electrons out of the plane thereby strengthening the current in the ion diffusion region. Open arrows show convective inflow, yellow arrows outward jetting of plasma.
Top panel: Plasma velocity measured along the spacecraft orbit long red arrow in the central panel in Earth magnetotail during a substorm reconnection event. Flow reversal is seen during passage near the X point indicating the two plasma jets emanating from X. Note the presence of a weak 6 nT magnetic guide field in y direction. The pronounced differences in electron and ion flows in the ion inertial diffusion region has been nicely confirmed in particle-in-cell simulations [ ] as shown in a summary plot in Figure 7.
The left part of this figure shows the usual ion jetting away from the X point, also clearly indicating that only a small fraction of the ion flow is passing the X point. At the boundaries of the ion jet discontinuities develop where the ion flow suddenly turns around. These are the separatrices. The lower panels show the profiles of the jet and current velocities. Similarly, ion current velocities are decelerated in the X point region, while electron speed become completely deflected from positive to negative velocities, an effect of reconnection, such that the reconnection current in the X point region is carried almost solely by the electrons.
Pritchett [ ] also included the third dimension with open boundary conditions in order to investigate the extent of the electron diffusion layer and distortion of its two-dimensional symmetry. No such distortion was detected except for the evolution of an electric component E z that is required by pressure balance. These conclusions are valid in the absence of guide fields.
When guide fields are included see the corresponding section on guide field reconnection below the Hall fields become distorted, asymmetric and compressed. This has been demonstrated by Daughton and Karimabadi [ ] and Karimabadi et al. A thorough comparison between different theory based simulations hybrid, Hall-MDH and non-Hall hybrid, where the Hall term is removed has been undertaken by Karimabadi et al. Reconnection was found to be independent on the Hall effect even in these ion-kinetic simulations including an anomalous resistivity thereby anticipating strictly Hall-free simulation results [ 42 ].
Including the Hall effect reconnection turns becoming asymmetric, and the X line grows in the direction of electron drift with current carried by electrons. Figure 7. Top panels: Velocities near the X line. Left: Left half negative x of space for ion velocity. Right: Right half positive x of space for electron velocity. Electron inflow continues across the ion inertial region until close to the center of the current sheet mapping the electron Hall current flow.
Bottom: Jetting and currents exhibiting different ion and electron dynamics. Left: Electron and ion jetting velocities. Ion mass flow in jets barely reaches 0. Right: Current speeds indicate deceleration for the ions at the X point. Electron velocities are inflected such that the current in the center is carried about solely by electrons. Hall fields were observationally inferred first by Fujimoto et al.
They represent self-generated guide fields. Only along the separatrices they approach the electron diffusion region near the X point. For this reason electrons become accelerated in E y in the direction opposite to E y. This acceleration amplifies the current in those domains where the Hall magnetic field is remarkable, an effect that leads to current bifurcation outside the reconnection site in the ion diffusion region.
Bifurcation was observed first in [ 66 ] and [ , ] in Earth's magnetotail current sheet. Though several different mechanisms have been proposed to produce current bifurcation cf. Asymmetric reconnection effects. Most cases of reconnection occur in interaction of plasmas with unlike properties. Such reconnection is non-symmetric. A famous example is reconnection between the interplanetary solar wind magnetic field and the geomagnetic at Earth's magnetopause.
In contrast symmetric reconnection like that in the tail of the magnetosphere is a rare case. Theoretically one expects that asymmetric reconnection affects mainly the weaker magnetic field side than the high field side. This was demonstrated in asymmetric simulations under conditions prevalent at the magnetopause [ ]. Figure 8 gives an impression on the non-forced asymmetric case with no guide field imposed. The most interesting effect is probably that the strong-field magnetosphere remains well separated from the distorted region by a slightly deformed but stable magnetopause which itself is adjacent to two legs of the separatrix system.
The two newly formed plasmoids and the X point lie entirely on the weak field side. The second lower Hall dipole is completely suppressed as there is no electron inflow from below. Electrons and ions flow in from the top and become diverted into jets along the magnetopause. This is shown in the lower part of Figure 8.
Figure 8. Non-forced non-guide field particle-in-cell simulations of asymmetric magnetopause reconnection with Top Hall field B H in color coding overlaid on background field simulation data after Pritchett [ ] courtesy American Geophysical Union. The fat green line is the magnetopause which separates the magnetosheath from the undistorted magnetosphere.
The magnetopause coincides approximately with the inner separatrix. Only two large Hall field vortices develop on the magnetosheath side. Bottom panels: Ion left and electron right bulk flows on the magnetosheath side develop only half spaces are shown. Jetting of electrons is restricted to a narrow domain only along the magnetopause.
Partial support to these simulations has been given by magnetopause observations of reconnection events [ 38 , ]. Observations should show jetting and Hall field signatures only outside though close to the magnetopause. The zero-field crossing should be asymmetrical if in agreement with the non-guide field simulation. In this point the observations disagree since the magnetic field is close to symmetry and one leg of the Hall field seems to lie inside the magnetosphere. This discrepancy with the simulations might possibly have been caused by not having recognized the presence of a finite guide field which should be subtracted from the data.
Recently Malakit et al. Structure and extension of the electron diffusion region has attracted interest since this point had been made first by Daughton et al. They have become available only recently [ ]. The width of the electron diffusion region equaling the electron skin depth applies to the direction perpendicular to the current sheet. Outside, this component re-magnetizes the electrons. Hence, in plasmoids the length of the electron diffusion region is a fraction of the extension of the plasmoid along the current sheet, i.
It is not identical to the extension of the Joule dissipation region defined earlier in Equation 13 [ 54 ] which is the inner region in [ ], the region of unrecoverable Joule heating. It also includes the length of the electron jetting or exhaust which depends only weakly on the presence of the non-magnetic ions but includes some energy transfer from electrons to ions resulting in ion heating. This is shown in observations of electrons at the geotail reconnection site and in accompanying simulations in Figure 9 , both referring to symmetric reconnection conditions without or with only weak guide fields.
Figure 9. Determination of the energy dissipated in the magnetospheric tail current sheet during ongoing reconnection from observations of Geotail observations and simulation data taken from Zenitani et al. Top: Geotail data of electron energy W vs. The gray histogram is the dissipated electron energy W ed determined from the measurements.
W c is the convection energy. The vertical red line shows the time of crossing the location of the X point as estimated from the magnetic and plasma profiles when passing the current sheet. The electron dissipation region is indicated by gray shading. The dotted histogram shows for comparison the convective energy term in Ohm's law. Bottom: Accompanying particle-in-cell simulations for the observational conditions. Shown are the mean electron blue and ion red velocities as well as the simulated dissipated and convection energies W ed gray shading and W c almost unaffected black curve along a cut through the X point in the center of the current sheet.
Acceleration of electrons into formation of two quasi-symmetric electron jets in opposite x -directions is indicated by the blue curve. Note that the electron jets extend substantially out from the dissipation region until becoming braked and assuming the same speed as the ions. Here the quasineutral plasma jets are born.
Weak braking starts already soon after the electron jets have left the dissipation region indicating interaction with the ion fluid already here. Energy transfer from electrons to ions is indicated by negative dissipation energies W ed. Since no effect is seen in the ion bulk flow the dissipated electron jet energy goes into ion heating. This, however, is a highly variable quantity, because B z increases with distance from the X point until ultimately becoming comparable to the external field B 0.
Moreover, electrons are strongly heated during reconnection. Both effects increase the electron gyroradius in the B z field. The pure electron jet ceases once ions become involved. Then ion inertia retards the jetting electrons, and a quasi-neutral reconnection plasma jet forms. This is then also the theoretical one-sided maximum length of the diffusion region. This value can be larger since the electron density in the diffusion region decreases due to outflow. Shay et al. Their estimate is based on the competition between the Lorentz force v x B z of the electrons in the B z field and the reconnection electric field E rec.
At the point where these two become of equal magnitude, the electron current vanishes and reverses sign outside. The electron exhaust can be longer. Such lengths are in overall agreement with observations in the magnetosheath [ ]. Le et al. The trapping is mediated by the finite reconnection electric potential field along the field lines, not by magnetic mirroring.
This potential, as was shown above, is the result of reconnection but is, at the same time, generated by the particle anisotropy which leads to non-diagonal terms in the pressure tensor. Preliminary confirmation of the pressure anisotropy was provided via numerical simulations [ ] and made complete by the kinetic derivation of the pressure anisotropy [ ] accompanied by extended simulations.
Earlier simulations [ 57 ] had already shown the various contributions to the reconnection electric field being localized in the electron diffusion region as a consequence of the pressure anisotropy produced by the reconnection process. The analytical asymptotic expressions for the pressure in the high-density low-field case applicable to reconnection are similar to the heat-flow suppressed CGL expressions.
These expressions hold in any local frame along the magnetic field. Transforming them into the X point frame generates the wanted non-diagonal pressure tensor elements. Hence, in the electron diffusion layer both effects are closely related. Electron trapping in the reconnection electric field causes pressure anisotropy, and pressure anisotropy causes the non-diagonal elements responsible for reconnection.
It should, however, be noted that the pressure anisotropy depends on the presence of a guide magnetic field [ ]. It disappears if the guide field become too weak. In the simulations of Pritchett [ 57 ] the pressure anisotropy was given. Hence non-guide field reconnection requires an initial anisotropy that is not generated by the reconnection process. In this case, however, an initial weak normal magnetic field will do the same service as a guide field. Moreover, the Hall field in the ion diffusion region which plays the role of a weak guide field [ 40 ] may generate an anisotropy outside the reconnection site by similar processes; once this anisotropy is produced, it will be transported by the convective flow into the X point.
A recent detailed analysis of three-dimensional reconnection [ 55 ] in an asymmetric setting with weak field on one side and strong field on the other side of the current layer, a scenario that applies to the conditions at the magnetopause, partially disqualifies the generality of the dissipation measure of Zenitani et al. It is of interest at the separatrices where wave fluctuations maximize. This conclusion is also strongly supported by the recent symmetric three-dimensional simulations of Liu et al. Recent high-resolution simulations in three spatial dimensions [ ] following [ ] and [ ] have been used to further investigate the structure of the electron diffusion region.
The interesting new feature found was that the current J y in the electron diffusion region, where it is carried mainly by electrons, splits into two or more thin tilted current layers. This splitting is different from the Hall induced one proposed above as it restricts to the electron diffusion region.
Its provisional explanation is that oblique tearing modes evolve in three dimensions which may form chains as predicted by Galeev [ 95 ] in the kinetic case. Such narrow layers indicate the evolution of turbulent reconnection structures. Figure 10 shows a combination of the two-dimensional [ 57 ] and three-dimensional [ ] simulation results on the electron diffusion layer. The three-dimensional case shows in addition the splitting of the single electron current layer into three distinct layers.
Figure Structure of the electron diffusion region. Top: Contributions of the electron pressure tensor left and inertial terms right to the generation of the reconnection electric field and dissipation centered in the X point frame after Pritchett [ 57 ] courtesy American Geophysical Union. The pressure tensor elements contribute mostly away from the X point near the plasmoids. Note the spiral form of the region.
The non-linear inertial term contributes near the X point above and below the symmetry line all relative units given in the colour bar. Bottom: Three-dimensional simulation data from Liu et al. White lines are three-dimensional magnetic field contours in two-dimensional cut exhibiting the complicated flux rope structure generated in three-dimensional reconnection. These observations are not in contradiction to the simulational results [ ] of asymmetric reconnection.
An important question is which plasma waves do and to what degree do they participate in reconnection. Reconnection is an instability, whether forced or not; it grows out of some initial state and thus is accompanied by waves. In the non-guide field, reconnection has been brought into relation to the generation of whistlers that are localized at the reconnection site.
There is a simple reason for this difference which is barely mentioned in the literature. In no-guide field simulations electrons are magnetized in the non-magnetic ion diffusion region. These may form large amplitude quasi-localized waves which structure the electron outflow region and also radiate along the separatrices. In a sufficiently strong guide field as discussed in [ ] ions become involved by re-magnetization. Transferring energy from the drifting electrons is the source feeding the waves. The transverse wave-electric field can, in the ion diffusion region, stochastically heat ions in the perpendicular direction.
The parallel wave-electric field component accelerates or heats the electrons parallel to the field.
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The latter process is of particular interest. The normalized spectral energy density that is available per electron for heating is obtained by integration with respect to time over the wave frequency. These waves propagate in oblique direction causing density ripples above and below the X point. In the pair plasma, the waves are the equivalent of lower-hybrid-drift waves in the current density gradient in electron-proton plasmas and have a similar effect in scattering the plasma particles.
Non-linearly they evolve to large amplitudes and cause structuring of the current, generating of anomalous resistance, and accelerating particles along the magnetic field into field-aligned beams. These beams are similar to the electron fluxes which close the Hall current system along the separatrices in electron-proton reconnection, in pair plasma they are of different nature, i.
The waves do not necessarily cause field-aligned current flows as no Hall current system requires any closure. Moreover, though wave activity was remarkable, its visible heating effect on the distribution was small and did not have any profound effect on the reconnection process itself. Lower-hybrid drift waves are the main mode which is known to be generated in the ion diffusion region plasma gradient.
At short wavelength it is purely electrostatic and related to whistlers, at long wavelength it is electromagnetic. These waves have been predicted long time ago to be of importance in reconnection [ 80 , , ]. However, as has been discussed earlier, observations show [ 83 , 84 ] that they are of very weak amplitude in the vicinity of reconnection sites and in simulations do barely appear in the electron diffusion region, in spite of the relatively strong density gradients at its boundaries.
Presumably they play little role in reconnection other than in accelerating electrons outside the diffusion region near the separatrices. However, importance has been attributed to their non-linear evolution of being capable of causing current bifurcation see Figure 11 [ 63 , 69 , ] and, by modulating the electric field and flow, affecting the reconnection rate. Bifurcation of a thin current layer caused by the lower-hybrid-drift instability simulation data taken from Daughton et al. Top: Sheet current strength color coded highest intensity red, lowest green.
After evolution of the lower-hybrid drift instability it has split into two current layers separated by about one initial layer half-width d indicating lower-hybri-ddrift instability mediated current bifurcation. The bifurcated current is structured in x by the lower-hybrid wavelength. The anisotropy is positive and concentrated mainly on the bifurcated current. A weak anisotropy is attributed to the original current center.
Lengths are measured in halfwidths d. Roytershteyn et al. This case is, anyway, not realized at the magnetopause and also not in the tail. The waves had the expected lower-hybrid polarization with electric field highly oblique to the magnetic field. They formed very-large amplitude localized patches along the separatrix. Lower-hybrid waves in magnetopause crossings related to reconnection had been occasionally observed at their high frequency whistler tail [ 84 ] and on the low-frequency branch [ 83 ].
Mozer et al. The latter conclusion is true since the waves do not occur in the electron diffusion region near the X point as confirmed by the simulations. At the separatrix the amplitudes and the effect of waves may be underestimated due to spacecraft resolution and smearing out the localized wave patches. Much earlier investigations [ 81 , 82 ] had already ruled out their importance in generating anomalous diffusion at the magnetopause, even when localized.
Finally, the Weibel instability [ 41 ] is another mode that could be of importance in contributing to the spectrum of waves near the reconnection site. This was mentioned first in [ ]. If it can be excited, it contributes to the formation of seed X points in the current layer which serve as initial disturbances in causing reconnection.
A possible scenario refers to the acceleration of electrons along a guide field [ ] producing the required temperature anisotropy of the electrons for excitation of the Weibel mode in the current sheet. Guide fields do arise naturally in interaction of asymmetric plasmas as suggested in [ 71 ] for only the exactly anti-parallel field components do merge and annihilate each other. Tilted magnetic fields therefore carry guide fields. Eastwood et al. Including a guide field leads to profound changes of the reconnection process. A guide magnetic field parallel or anti-parallel to the sheet current acts stabilizing on reconnection once the guide field becomes strong.
This is a consequence of the magnetization of the electrons by the guide field which enforces the frozen-in condition. In addition, guide fields cause a tilt of the X line against the z axis, i. The upright X becomes an italic X. They also tilt the Hall field geometry and favor one leg of the Hall current. This was demonstrated first by Karimabadi et al. An example is shown in Figure Guide field reconnection. Top: Ion velocity arrows overlaid by the out-of-plane magnetic component of the magnetic field [simulation data after , courtesy American Geophysical Union].
This field includes the guide field B yg and the self-consistent Hall field. The guide field causes an asymmetry in the field and ion flows. The diffusion region becomes tilted. Bottom: The highly asymmetric electron flow pattern overlaid by the out-of-plane electron current J ey which is opposite to the out-of-plane electron flow.
Yellow-to-red colors indicate positive, blue-to green colors negative values. It is highest along the yellow-red diagonal, indicating that it has been accelerated along the reconnection amplified convection electric field. Evolution of reconnection depends strongly on the strength of the imposed guide field. This has been investigated by Daughton and Karimabadi [ ] and Karimabadi et al.
In the weak regime the effect of the guide field is small. In this regime the tearing mode becomes the drift-tearing mode being modified by the diamagnetic drift of electrons magnetized in the guide field. This mode has finite frequency comparable to the electron drift frequency. Moreover, the presence of a guide field has an important effect on the quadrupolar structure of the Hall magnetic field [ 46 , ]. It not only distorts the Hall field, introducing an asymmetry, but compresses the spatial range of the Hall field down to the electron gyro-scale. This Hall field structure survives into the non-linear regime of the tearing mode.
In absence of guide fields anti-parallel case electrons become non-gyrotropic and anisotropic. This process dominates over saturation at singular layer thickness due to electron trapping. The latter becomes important only when Weibel-modes or turbulence are excited causing pitch angle scattering. Guide fields allow for a number of other important effects: due to the magnetization of electrons in guide fields, they introduce the whole spectrum of magnetized electron plasma modes, if only conditions can be generated which drive one or the other mode unstable. Guide fields thus open up the gate to excitation of electron whistlers and electrostatic electron cyclotron waves if the electron pressure becomes anisotropic, a situation which is quite realistic under various conditions like forcing of the current sheet and has been observed in simulations see below.
The density gradient also causes excitation of electron drift waves. Some of these waves are known to produce anomalous resistivity and may thus contribute to plasma heating, dissipation weakly supporting reconnection. If electrons are accelerated along the guide fields, which is expected in the presence of convection electric fields, the Buneman instability [ 75 ] can be excited either leading to anomalous resistivity and plasma heating, or allowing for the growth of chains of electron holes found in simulations cf.
The latter family of holes indicates hole generation away from the electron diffusion region near the separatrices. It may be related to the non-linear lower-hybrid wave evolution reported in [ 69 ].
Without proof Che et al. Electron holes are local concentrations of electric potential drops along the guide magnetic field due to charge separation between trapped and passing electrons. Their occurrence in relation to substorms and reconnection was supported by observations in the magnetosphere [ ], in the laboratory [ ], and in one-dimensional Vlasov simulations [ ] of electron hole formation based on electron distributions obtained from the symmetric numerical simulations of reconnection in presence of a magnetic guide field [ 57 ]. The latter simulation demonstrated that the simulated electron distribution functions inside the electron diffusion layer and exhaust region are consistent with electron hole formation during reconnection.
They thereby confirm the assumption made earlier that electron acceleration by the reconnection electric field in the electron diffusion region is strong enough to exceed the Buneman threshold for instability and non-linear evolution of the Buneman instability which results in the formation of electron holes. Electron holes give rise to further violent local acceleration; by cooling the passing electron component, electron holes may generate bursts of fast cool electron beams which result in the excitation of Langmuir waves and non-fluid plasma turbulence. For all these reasons, recent simulation research favors guide-field reconnection in two or three dimensions.
So far we considered the case of spontaneous reconnection when a sufficiently thin current sheet spontaneously pinches and decays into a number of current braids located in the centers of a chain of plasmoids separated by magnetic X points of weak magnetic fields. We noted that in this case the Parker idea applies that the plasma is sucked in into the X point reconnection site.
The philosophy changes when reconnection is driven by a continuous inflow of plasma into the current layer. This case is analytically almost intractable and requires a numerical treatment which will be discussed below. Continuous plasma inflow forces the current sheet to digest the excess plasma and magnetic fields. This can happen only by violent reconnection and plasma ejection from the X points. One of the first full-particle kinetic investigation of forced reconnection was reported by Birn et al. Such an inflow is produced by imposing a cross-magnetic electric field here for a limited time.
It results in a compression of the initial current sheet causing current thinning. It seemed to demonstrate that all kinds of simulations would lead to about the same reconnection rates, except for the MHD codes where the rates were much less. But the amount of reconnected flux at the final state was about the same. This has by now changed profoundly having been superseded by more sophisticated two-dimensional forced simulation [ 52 ] in which an external cross-magnetic electric field was imposed at the boundaries above and below a symmetric electron-proton Harris current sheet for a temporally limited time.
It causes a spatially varying inflow of magnetic flux that compresses the initially assumed thick Harris current sheet. Initiating of reconnection is not done artificially. Instead it is waited until it sets on by itself which happens due to both thinning of the current layer due to inflow and numerical noise.
This causes significant delay of reconnection, however. These effects lead to reconnection based on non-diagonal pressure tensor elements in thin current layers, significantly different from the case of an isotropic Harris layer. Of course, quadrupolar Hall- B y are as well generated. Mass ratio effects come into play at the stage when plasmoid formation takes place slowing down further reconnection. Thus, forced reconnection proceeds through two stages: fast initial and slow plasmoid formation.
The early kinetic theory [ 90 , ] stressed the importance of the presence of a residual normal magnetic field component B z in reconnection. Such a component is natural in magnetospheres where the magnetic fields are anchored in the central object, a planet like Earth or Jupiter or a Star. Open boundary conditions were imposed in x. Forcing of reconnection was achieved by switching on a uniform external cross- B electric field for a brief period of the order of one ion-gyro period to both sides of the Harris current layer.
This causes continuous thinning compression of the current layer until reconnection initiates. There are several stages in this simulation which deserve mentioning. As in the above non- B z simulation of forced reconnection [ 52 ] reconnection was not ignited. 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. 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.
The coupling between the electromagnetic field and the plasma is weak or absent. But in. 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.
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.
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Plasma approaches the origin along the positive and negative x axes and leaves along the positive and negative y axes. The details. 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. 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.
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. This assumption is not always valid. See Sections 4. 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.
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. Let us now briefly consider the region immediately adjacent to that point. As the origin is approached, the flux transport velocity v. In hydrodynamic terms, the magnetic neutral point is also a double stagnation point. The region in which the plasma velocity deviates significantly from y. 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. 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. For further discussion, see section 6. The resulting plasma turbulence will lead to a reduction in the effective conductivity, as discussed in section 6.
Whether the plasma dynamics in the diffusion region is described in a continuum fashion, i. Thus the diffusion region, along with the entire shock system, acts as a dissipator of electromagnetic energy. Finally, we estimate the separatrix angle a in the outflow see Figure 6. Near the magnetic null point we may write.
For example, the rate of electromagnetic energy flow into and out of the diffusion region may be estimated as follows:. Thus M. 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. But usually only a minute part of the total energy conversion occurs there, the main part taking place in the shocks.
Depending on the nature of the boundary conditions, the inflow may be such that M. 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. 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.
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. 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. The X type magnetic null point is again located at the origin. 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. 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. In particular, on the separatrix it has the value yo-Z" 0 ,. The separatrix intersection angle a falls below its vacuum value. These effects imply an excess of magnetic flux in cells 7 and? 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. 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.
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. But the conditions for onset of such an event may perhaps be identified by examination of such a sequence.
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. 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 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. 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.
The evolution of the field geom- etry in the noon-midnight meridional plane is shown in Figure 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 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. This field presumably has an inductive and an electrostatic part which tend to cancel along AOB while adding along AXB. 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.
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. The term magnetic-field merging may be taken to encompass both reconnection and annihilation. 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.
In nonsteady flow, the electric field at the refer- o Af ence point, E , in general differs from E , and M? How- ever, the one adopted here, in terms of an electric field component along the separator works also for flows at arbitrary R. 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. 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. 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. The model contains a set of four Alfve"n discontinuities which in compressible flow may be identified. 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. Values in 00 the range. 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. 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.
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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. However, this is a result of the lumping of the slow-mode effects. Thus, in reality it is unlikely that the inflow into the diffusion region can occur at M. Fur- oo ther discussion of this point is given in Sections 4. 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. 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. 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. 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 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. 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,. 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.
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. 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.
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?
In particular, the case of vacuum conditions in the magnetosphere has been reported in detail. A typical resulting field and flow configuration is as shown in Figure In terms of plasma. 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 Thus, it is possible to generate solutions that describe the reconnection of fields that form an arbitrary angle.
Thus the assumption underlying 2 Equation , 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 ,. Thus it is not clear at the present time whether Cowley's criticism of Equation 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.
But certain partial. 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 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.
Also, the rate should refer to conditions on the x axis in the inflow region. Equation is nevertheless interesting because it suggests that pressure anisotropy in the incoming plasma may be an important factor. 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. 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. 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. 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. 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. In the latter section, the importance of Hall currents and of the ion-inertial length and gyroradius are discussed. Addition- ally, certain exact solutions exist. 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. It then describes field annihilation see section 2. In agreement with the discussion in section 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 The treatment is incomplete because it does not take account of the momentum balance. Additional criticism has been offered by 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. Thus, in the outer portion of the diffusion region the field behaves almost in a frozen-in manner. 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. 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 These solutions represent 88 a generalization of a case studied by Parker. The resulting magnetic field profiles are shown in Figure Three features are of interest. If you want to continue to stay involved, please take our survey. When we meet with politicians and planners moving forward, we want to make sure the community vision is well documented and well understood. Your email address will not be published.