KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT: SPINE-ALIGNED CURRENT

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1 Geophsical and Astrophsical Fluid Dnamics Vol. 98, No. 5, October, pp. 7 8 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT: SPINE-ALIGNED CURRENT D.I. PONTIN a, *, G. HORNIG b and E.R. PRIEST a a School of Mathematics and Statistics, Universit of St. Andrews, St. Andrews, Fife, KY6 9SS, Scotland, UK b Theoretische Phsik IV, Ruhr-Universiteit, D-78, Bochum, German (Received 6 Februar ; In final form 7 Ma ) Magnetic reconnection at a three-dimensional null point is the natural etension of the familiar two-dimensional X-point reconnection. A model is set up here for reconnection at a spiral null point, b solving the kinematic, stead, resistive magnetohdrodnamic equations in its vicinit. A stead magnetic field is assumed, as well as the eistence of a localised diffusion region surrounding the null point. Outside the diffusion region the plasma and magnetic field move ideall. Particular attention is focussed on the wa that the magnetic flu changes its connections as a result of the reconnection. The resultant plasma flows are found to be rotational in nature, as is the change in connections of the magnetic field lines. Kewords: Magnetic reconnection; Magnetohdrodnamics; Magnetic flu; Magnetic null points INTRODUCTION Magnetic reconnection is a fundamental process in man areas of plasma phsics, whereb the magnetic field, B, becomes restructured. When null points are present the global topolog of the field changes. Our ideas on how this restructuring occurs come mostl from the well-studied case of reconnection in two dimensions. In two dimensions, reconnection occurs at hperbolic null points of the magnetic field (see, e.g. Priest and Forbes,, for a review), commonl known as X-points (see Fig. ). A plasma flow transports magnetic flu towards the X-point, where the reconnection takes place, and the flow then transports the reconnected magnetic flu awa from the X-point. In terms of magnetic field lines, the process of reconnection involves a pair of field lines being brought in from two quadrants on opposite sides of the null. At the X-point each of these field lines breaks, when the lie along the separatrices of the field. The are then rejoined and move out in the other two quadrants of B. The result is that the field line footpoints are pair-wise differentl connected when the leave the reconnection region. *Corresponding author. davidp@mcs.st-and.ac.uk ISSN 9-99 print: ISSN 9-9 online ß Talor & Francis Ltd DOI:.8/997

2 8 D.I. PONTIN et al. FIGURE Two-dimensional reconnection at an X-point. The thin lines are magnetic field lines and the bold arrows indicate the direction of the plasma flow. Magnetic reconnection is a fundamentall non-ideal process; in an ideal plasma, magnetic field lines maintain their identit for all time, and are said to be frozeninto the plasma. The non-idealness ma be the result, for eample, of a non-zero resistivit,, in which case, assuming no other non-ideal effects are important, the process satisfies Ohm s law in the form E þ vtb ¼ J: ðþ In order to investigate the evolution of magnetic flu it is useful to define a flu transporting velocit w (Hornig and Schindler, 996; Hornig and Priest, ) which satisfies E þ wtb ¼, ðþ which is possible in two dimensions since the electric field, E, is alwas perpendicular to B. Note that for reconnection to take place, the flu transporting velocit w must become singular at the null point, which is a signature of the breaking of the field lines (Hornig, ) and the subsequent discontinuit in the mapping of their endpoints. B comparison with an ideal Ohm s law, we can consider w to be a flow within which the magnetic flu is frozen. The component of w perpendicular to B can be found from w? ¼ ETB=B : ðþ If we assume that the non-ideal term on the right-hand side of Eq. () is localised within a finite region, D, about the null point, then outside D, w coincides with the plasma velocit perpendicular to B, v?. The situation in three dimensions is much more complicated. In general, for reconnection in three dimensions, EEB (or JEB) is non-zero within a finite region D, and hence in general no unique flu-conserving velocit eists, or in other words there is no unique field line velocit for field lines which thread D (for a proof, see Hornig and Schindler, 996; Priest et al., ). Nonetheless, it is still possible to stud the evolution of magnetic flu and field lines under certain circumstances. Consider the case of a resistive non-ideal term J, which is localised within some diffusion region, D. We consider a finite region D as the generic situation for astrophsical plasmas since

3 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT 9 these plasmas have etremel high magnetic Renolds numbers, and dissipation is enhanced in well localised regions, for eample, when micro instabilities form in a thin current sheet. If no closed magnetic field lines eist within D, then we can still follow the motion of individual field lines from each end, since we know that in the ideal region on either side of D the must remain attached to the same plasma elements for all time. Suppose the surface of D is split into two parts, through one of which magnetic flu enters D, and through the other of which it leaves. Since each field line is anchored twice in the ideal environment, once on either side of the non-ideal region, one can follow the motion of the field lines b tracing them through space from either of these two sets of footpoints. B reference to the ideal environment then, it is possible to define a velocit with which the field lines passing into D move, sa w in, and another velocit w out at which the field lines passing out of D move. In the stationar case, these two velocities can be calculated throughout space b mapping the electric potential from each set of footpoints along the field lines. This leads us to the mathematical epressions given in Eqs. (6) and (7). These two (pseudo-)field line velocities must each be identical to v? on the relevant section of the surface of D, but in contrast to the two-dimensional case, the are not identical to each other inside D, nor are the identical to v? on their continuations through D. This is a manifestation of the non-eistence of a unique field line velocit, w, as stated above. The result is that, following field lines with w in and w out, respectivel, the field lines seem to split as soon as the are transported into the diffusion region, and inside D the continuall change their connections. Reconnection can occur in three dimensions either at a null point or in the absence of a null point (Schindler et al., 988; Lau and Finn, 99; Priest and Forbes, ). The nature of the field line splitting, and subsequent restructuring of the magnetic flu, for reconnection in a simple magnetic structure with no null point has been discussed in general b Priest et al. (), and in more detail b Horing and Priest (). Here we stud instead the kinematic problem at a three-dimensional null point. We find that the induction equation implies that such reconnection is profoundl different from two-dimensional X-point reconnection. The implications for further stud are addressed in the conclusion. In Section, we describe the structure of three-dimensional null points as well as describing ideal field line behaviour in their vicinit. In Section, the equations solved in our model are given, as well as the assumptions made, and we detail our method of solution. The elementar solution is given in Section, before we describe the result of adding a phsicall relevant ideal flow in Section 5. The eistence of perfectl reconnecting flu tubes is discussed in Section 6, where the effect of adding a simple time-dependence to the model is described. We give our conclusions in Section 7. STRUCTURE OF D NULL POINTS AND IDEAL BEHAVIOUR The nature of reconnection at a three-dimensional null point is of particular interest since it is the three-dimensional generalisation of an X-point. Three-dimensional null points are also of crucial importance in the topolog and interaction of comple fields on the Sun. The are found in abundance in the solar corona (see e.g. Brown and Priest, ; Longcope et al., ), where their associated separatrices

4 D.I. PONTIN et al. z spine fan FIGURE Basic structure of a three-dimensional null point. and separators are thought to be likel candidates for sites of coronal heating (Longcope, 996; Antiochos et al., ; Priest et al., ). There is also evidence that null point reconnection ma act as a trigger for at least some solar flares (Fletcher et al., ). The local structure of the magnetic field around a three-dimensional null point is shown in Fig.. The skeleton of the null point is made up of a pair of field lines directed into (or out of ) the null from opposite directions, known as the spine, and a famil of field lines which are directed out of (or into) the null ling in a surface, known as the fan plane (Priest and Titov, 996). A general mathematical formalism is given b Parnell et al. (996), who classif nulls depending, amongst other things, on the size of the current, J, and its direction with respect to the spine ais and fan plane. If the current is zero then the null point is known as potential. When the current is directed onl parallel to the spine (J ¼ J k ), the fan and spine are perpendicular, and the field lines in the fan form a spiral structure (Parnell et al., 996). When the current is directed onl perpendicular to the spine (J ¼ J? ), the spine and fan are no longer perpendicular. In general, when the current has components in both directions, both of these effects are present, as well as further characteristics, depending on the relative magnitudes of J k and J?. The kinematics of stead reconnection at a three-dimensional null point have been studied previousl b Priest and Titov (996). The start off b discussing the ideal behaviour in the vicinit of the simple potential magnetic null given b B ¼ ð,, zþ: ðþ The reconnection is classified as one of two tpes near an isolated null point, termed spine reconnection and fan reconnection. Reconnection associated with a separator, a special field line which joins two null points, is also discussed. Due to the fact that the configuration considered is ideal, to achieve reconnection, the plasma velocit is necessaril singular at the null, and the field line mapping is discontinuous. During spine reconnection a flow is imposed across the fan (z ¼ ) resulting in singularities in E and v at the spine ( ¼ ¼ ). In fan reconnection, the flow is imposed across

5 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT the spine resulting in singularities in E and v in the fan plane. The effect of adding nonpotential and diffusive terms is then considered in a preliminar manner. We aim here to investigate the structure of reconnection when the null point is nonpotential, with a current parallel to the spine, where a localised diffusion region is included in order to get a realistic model with no singularities in an phsical quantities. It is instructive to seek insight into the structure of the highl comple process of three-dimensional reconnection b considering solutions to a reduced set of the MHD equations. We follow the method of Hornig and Priest () in adopting the kinematic approimation, b solving the induction equation and Mawell s equations, though not solving the equation of motion. In a dnamical situation we would epect to start from an initial field, which then forms a current sheet (diffusion region) at which reconnection takes place. The wa in which a current sheet ma form b the collapse of such null points is discussed b Parnell et al. (997) and Mellor et al. () as well as b Klapper et al. (996) and Bulanov and Sakai (997). It should be noted, however, that as in two dimensions, reconnection ma be initiated in man different was (see e.g. Priest and Forbes, ), such as b driving from boundaries, or as a result of instabilities of man different tpes, as well as b null point collapse (following response to eternal boundar motions). Our investigation here is independent of the initiation mechanism. THE MODEL We seek a solution of the kinematic, stead, resistive MHD equations given b E þ vtb ¼ J, JTE ¼, ð5; 6Þ JEB ¼, JTB ¼ J: ð7; 8Þ The non-ideal term on the right-hand side of Eq. (5) is assumed to be localised, for the reasons described in Section. We investigate the behaviour in the vicinint of a simple spiral null point, where J lies parallel to the spine. In the future, we will go on to consider the case where J is perpendicular to the spine. In general the field in the vicinit of our spiral null point can be written (Parnell et al., 996) as B ¼ B j, þ j, z, ð9þ where B and j are constants, such that J ¼ðB j= Þ^z (from Eq. (8)) is directed along the z-ais, which is coincident with the spine. Due to the clindrical smmetr of the field, it simplifies matters to work in clindrical polar coordinates ðr,, zþ, in which B has the form B ¼ B R, jr, z : ðþ It is crucial for the following analsis to be performed analticall that we are able to find analtical equations describing the field lines in terms of some initial starting

6 D.I. PONTIN et al. coordinates,. For B defined as in Eq. (), this can be done b integrating dxðsþ ds ¼ BðXðsÞÞ ðþ to find X ð, sþ, given b Eq. (), as R ¼ R ep ðb sþ, ¼ þ B jrs, z ¼ z ep ð B sþ: ðþ Note that the parameter s does not denote the distance, dl, along a field line, which is instead given b dl ¼jBjds: ðþ From Eq. (6) we can see that E can be written as the gradient of some scalar, sa, so that Eq. (5) becomes J þ vtb ¼ J: ðþ A solution to this equation for given B ma be found as follows. Firstl, since we require our non-ideal term to be localised and since J is constant, we prescribe a localised resistivit,. The component of Eq. () parallel to B is ðjþ k ¼ J k, which ma be integrated along field lines to give Z Z ¼ J k dl þ ¼ JEBds þ : ð5þ Thus, if we can prescribe some function ¼ ð, sþ in such a wa that, having substituted Eq. () into B and J to find B ð, sþ and J ð, sþ, Eq. (5) is integrable, then we can deduce ð, sþ. From here it is possible to substitute the inverse of Eq. () to find ðþ. Then the electric field, E, and the component of the plasma velocit perpendicular to B, v?, can be found from E ¼ J ð6þ and v? ¼ ðe JÞTB=B : ð7þ Prescribing a phsicall reasonable profile for ðþ we can obtain an epression for in terms of the field line coordinates ð, sþ with the help of the coordinate transformations (Eq. ()). This epression can then be used for the integration (Eq. (5)). In order to have a unique coordinate transformation!ð, sþ, we need to choose the initial points on surfaces such that each field line passes through these surfaces eactl once. Two obvious choices which we will use in the following discussions are the planes z ¼, or a clindrical surface such as R ¼.

7 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT It is now necessar to choose ðþ in such a wa that the integral in Eq. (5) can be performed analticall. One analticall integrable form for ðþ is a piecewise polnomial function, such as ( ðr=aþ ¼ ðz=bþ R < a, z < b ; ; otherwise; ð8þ where, a and b are constant and is the value of the resistivit,, at the null point. This form of gives a clindrical diffusion region, of radius R ¼ a, sa, etending in the z-direction to b. We are choosing this shape for simplicit, and aim to consider other shapes in the future. is continuous and smooth everwhere. To proceed we need to perform the integration given b Eq. (5). In order to do this we must choose in which direction along the magnetic field lines to integrate. That is, we ma choose either to set s ¼ on some surface of R ¼ R (where R is a constant, R >a) and integrate in towards the null point and thus through D, or we ma integrate down and up towards the null from z ¼ z and z ¼ z, respectivel (constant z >b), setting s ¼ on these surfaces for each half-space. In the latter case the starting potentials ( )at z ¼z must be chosen such that is continuous and smooth at the fan plane (z ¼ ). However, the choice between these two directions of integration is irrelevant provided we consider our model with respect to some chosen eternal/boundar conditions, equivalent to imposing some on one boundar. The results from each case then are equivalent, and so we choose here to describe the result of setting s ¼ on z ¼z, constant, and point out an significant differences for the other case when the occur. B the method described above, E ð Þcan be found. However, it turns out that for the choice of given in Eq. (8), E is singular in the fan (z ¼ ). The reason is that is not sufficientl flat near the null point, where the diverging field lines make the integration of ver sensitive to an variations of. To obtain a smooth (and therefore phsicall acceptable) E, we find that the power of the variables in Eq. (8) must be greater than, and so take, for simplicit, ( ðr=aþ ¼ 6 ðz=bþ 6 R < a, z < b ; otherwise; ð9þ where, a and b have the same phsical meanings as before. This form for has essentiall the same structure as that given b Eq. (8). It is now possible, via the method described at the beginning of this section, to calculate E and v?. The analtical epressions for these are too length to present here, but can be calculated using a smbolic computation program. The resulting solution, described in detail in the following section, gives, in a sense, a local rearrangement of the flu within an envelope of field lines enclosing the diffusion region, D, see Fig.. The rearrangement is local in the sense that onl changes on field lines which pass through D (see Eq. (5)), and thus in a region of space threaded eclusivel b field lines which have not passed through D, is constant, and Eqs. (6) and (7) impl that E and v? are zero.

8 D.I. PONTIN et al..5.5 z.5.5 FIGURE Field lines on the boundar of the envelope enclosing the diffusion region (clinder), showing the region of influence of the local solution. Now, for a given magnetic field, Ohm s law (Eq. ()) ma be decomposed into an ideal component (Eq. ()) and a non-ideal component (Eq. ()) as follows J non-id þ v non-idtb ¼ J; J id þ v id TB ¼ : ðþ ðþ We have the freedom to add an ideal flow to the non-ideal solution since we do not solve the momentum balance equation here, which would otherwise determine the ideal part of the flow. We can thus add an ideal flow to our local solution b taking id to be our in Eq. (5). Note that id should be constant with respect to the integration in Eq. (5), i.e. id ¼ id ð Þ. The perpendicular component of the ideal flow v id? ma then be calculated, b taking the cross-product of Eq. () with B, as v id? ¼ J id TB=B : ðþ We ma thus add an ideal flow to transport magnetic flu into and out of the local envelope shown in Fig., allowing us to see the global effect of the restructuring of the magnetic flu b the reconnection process. This will be discussed in Section 5, but first let us consider the elementar solution. ELEMENTARY SOLUTION. Induced Flows We eamine here the nature of the solution with id ¼, i.e. just the local behaviour of the flu envelope enclosing D with no etra ideal flow. All phsical quantities are completel smmetric about z ¼, and we will therefore describe the results onl for z>. Choosing to integrate Eq. (5) from z ¼ z we automaticall start with constant for z>b. Hence the electric field and plasma velocit are zero for z>b. The velocit for

9 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT 5 z<b is a rotation within the flu envelope. (For integration from R ¼ R, the plasma velocit, v, vanishes for R >a, and we have a rotation within the remainder of the flu envelope.) For the purposes of illustration of the results, it is at this stage convenient to add a component of v parallel to B such that v z ¼. This is achieved b defining v ¼ v? ðv?z =B z ÞB: ðþ Our freedom to do this comes from the fact that Eq. () determines onl the perpendicular component of v, since it is the part that affects the behaviour of the magnetic flu, and so for our purposes the parallel component is arbitrar. The resulting flow pattern of v in a plane of constant z is shown in Fig. a. The radial component, as well as b definition the z-component, of v is zero, so we have purel rotational flow, and the wa that this varies with radius is shown in Fig. b. Note that here the angular plasma velocit is strongest in a ring centred on the spine. Note also that in the z ¼ plane the fan field lines rotate like a solid bod, as does the plasma outside the diffusion region. The source of the rotational flows can be eplained as follows. We consider the case where is integrated from z ¼ z, and the argument can be easil adapted to the reverse integration. Consider the potential drop along sections of the loop illustrated in Fig. 5. L andl are radial lines in planes z ¼ b and z < b, respectivel, while L is a field line on the surface of the envelope of flu threading D, and L lies along (a).5.5 (b).5.5 R FIGURE (a) Vectors of the plasma flow v, along with a projection of the magnetic field lines in the plane z ¼ :, for the parameters B ¼, a ¼, b ¼, ¼, j ¼. The corresponding radial variation of the plasma velocit v (black line) and field line velocit w out (gre). L L L L FIGURE 5 Closed loop made up of line sections. L and L are radial lines in planes z ¼ b and < z < b respectivel. L is a field line ling on the surface of the flu envelope enclosing D, and L lies along the spine (R ¼ ).

10 6 D.I. PONTIN et al. the spine of the null. The potential drop around an closed loop must be zero. The potential drop along lines L andl must be zero, as L lies at z b and L lies on the boundar of the envelope, and so ¼ all along them. Thus or L þ L ¼, L ¼ ð spine Þ 6¼, ðþ where spine is the value of at the verte of L andl, which must be different from since E is non-zero along the spine (EEB ¼ JEB 6¼ on spine). Since there is a potential drop along L there must also be a non-zero electric field along it. This electric field induces a plasma flow perpendicular to such a radial line, i.e. a rotational flow. This rotation has the same sense for z> and z<, and has maimum magnitude in the z ¼ plane. Note that this argument is completel independent of the particular profile of within D.. Reconnection Rate It is possible to calculate a reconnection rate for the flu in this reconnection process, although it is important to note that this has a ver different phsical meaning from the concept of a reconnection rate in two-dimensional reconnection. In two dimensions, flu is cut and rejoined at the null point, and the reconnection rate gives a measure of the amount of flu that undergoes this process in a given time. However, in three dimensions we need a different definition since we have a localised flu envelope within which all of the flu continuall changes its connections, so during an arbitraril short length of time ever field line is reconnected (ecept, for smmetr reasons, the spine field line). Considering just the flu reconnected in the half-space z >, we define, b close analog to Hornig and Priest (), the reconnection rate, F, as the integral over the parallel electric field along the spine ais, F ¼ Z E k R¼ z¼ dl ¼ ðr ¼, z ¼ bþ ð R ¼, z ¼ Þ ¼ 7 9 B j b: ð5þ To obtain some idea of the meaning of this quantit, consider the difference between the velocities of the field lines anchored in the surfaces, z ¼b and R ¼ a, ofd through which flu passes in and out. Since the velocities w in=out must match the plasma velocit on the relevant boundar of D, the are given b w in=out ¼ r in=out B=B : ð6þ Assuming B >, we can now identif the surface through which flu enters D as z ¼ b, and the surface through which flu leaves it as R ¼ a, namel, w in ¼ J z¼b TB=B, w out ¼ J R¼a TB=B : ð7þ

11 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT 7.5 z.5 FIGURE 6 Flu surfaces of constant radius (for given z), within which field lines periodicall eactl reconnect with themselves. For eample, the highlighted (black) field line will split in the rotational flow, with the top half remaining fied, while the bottom half reconnects with each of the gre field lines until eventuall being reconnected again into the configuration shown after performing a rotation of radians. Now, ¼ on z ¼ b, and hence J z¼b ¼ and w in ¼, so field lines passing in through the top of D remain fied and stationar there. Due to the rotational nature of v, w out is also rotational, leading to a rotational mismatching between w in and w out. The value of F, given b Eq. (5), provides an idea of the maimum difference between the rate of rotation of the inward flu bundle and outward flu bundle, or the maimum rate of relative slippage (see Hornig and Priest,, for a further pedagogic eample). From Fig. b it can be seen that the outward bundle of field lines rotates approimatel like a solid bod close to the spine, with the rotation falling off to zero at the edge of the flu envelope. As mentioned above, w ¼ w out w in, the rate of slippage between the inward and outward bundles of field lines, is purel rotational, so that it has a -component onl. Thus field lines are continuousl reconnected in shells of constant radius (for a given value of z) onl. These shells have the same three-dimensional shape as, and are concentric with, our flu envelope, see Fig. 6. An initial field line which splits into two will be periodicall eactl reconnected with itself, ever time the half embedded in R ¼ a performs a full rotation of radians. Note, however, that this period is different for each shell, since it is not a rigid rotation. For this reason, if we consider a flu tube within the envelope with a finite radial etent, there will be no periodic return to the initial state. It is important to note the implications the rotational slippage has for helicit production. If we imagine that the small flu tube within our flu envelope is closed somewhere far from D, then we can see that the relative rotation of the two ends of the tube would act to twist up the tube and thus act as a source of self-helicit with respect to an initiall untwisted tube. 5 COMPOSITE SOLUTIONS As discussed in Section, we ma impose an ideal flow upon the solution described in the previous section, as shown in Eqs. () and (). We would like to choose an

12 8 D.I. PONTIN et al. ideal flow which shows the global effect of the local rotational slippage behaviour b transporting magnetic flu into and out of the local flu envelope. For this reason, we choose to impose a stagnation-tpe ideal flow. Stagnation flows are phsicall relevant flows to choose as the ma perform the localisation of the quantities contained in the non-ideal term in Eq. (5) (Priest and Forbes, ; Hornig and Priest, ), and are common in reconnection solutions, such as Craig and Fabling (996). The stagnation flow is chosen in such a wa as to satisf Eq. () b imposing a suitable id ð, Þ on the surface z ¼ z, and then substituting the inverse of the field line equations () to obtain id ð,, zþ. The equivalents of Eq. (6) and (7) can then be used to find E id and v id?, respectivel. Again, for ease of analsis, we choose to set ðv id? Þ z ¼, via Eq. (), to obtain v id. We choose to take id ð, Þ ¼, ð8þ where is a constant. The resulting flow, v id, takes the form of a stagnation-point flow, with separatrices coincident with the - and -aes in the plane z ¼ z. For z 6¼ z, the flow separatrices are rotated with respect to this configuration, due to the spiralling of the magnetic field lines. The total flow, obtained b adding this ideal flow to the elementar solution, and given b v þ v id, has the same stagnation-tpe structure outside the enclosing flu envelope, but inside we have a superposition of X-tpe and O-tpe flows. Whichever of these flows dominates is dictated b the nature of near the spine, R ¼. If we consider all other parameters fied, there is a critical value of, namel ¼ crit, at which the transition from O-tpe to X-tpe flow takes place. This can be determined b eamining the total at small R, which (in cartesian coordinates) takes the form ¼ þ þ þ þ þ O,, ð9þ where ¼ B j 9b z 7z þ 6z 6 b 6 9b, ¼ 9B j 5a z, z ¼ cos sin, z ¼ j ln z ; z ¼ cos sin z : ðþ In a plane of constant z, changes from a local maimum to a saddle point, and thus the flow changes from O-tpe to X-tpe, when z ¼ :

13 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT 9 Substituting in the above epressions for, and and rearranging for we find that crit ¼ 8B j z 5a : ðþ For > crit, the flow is of X-tpe at the spine ais, whereas for < crit it is O-tpe. Note that crit is independent of z, so the flow tpe is the same for all z for a given value of. 5. The Case u >u crit For > crit, the stagnation flow dominates the rotational flow of the elementar solution everwhere, and hence v, w in and w out all have a basic X-tpe structure. In order to analse the effect the reconnection process has on rearranging the flu, we consider the mismatching of the two flu velocities w in and w out. The flow lines of these velocities are coincident with the contours of constant in and out, respectivel. For clarit we eamine the mismatching of these contours, and hence the flu velocities, in the plane z ¼ b, without loss of generalit. The two sets of contours are shown in Fig. 7. Note that the flow w in is eactl the ideal plasma velocit, v id, since the associated elementar solution from the previous section is zero on z ¼ b. Note also that in the plane z ¼ b, the boundar of the flu envelope associated with the elementar solution coincides with the boundar of the diffusion region, at R ¼ a. The superposition of the flow lines of w in and w out is shown in Fig. 8a, which ma be understood as follows. Consider a magnetic field line in an inflow region outside the diffusion region, whose intersection with the chosen plane, z ¼ b, lies on one of the sketched flow lines. Outside D, the two sets of flow lines coincide (with each other and with v), and the field line moves ideall. Once the field line is transported into D, the flu velocities w in and w out are no longer the same, and so we ma choose to follow either the field line anchored in the ideal region above D, which moves at w in, or follow the field line anchored in w out. Plotting both motions the field line seems to split, as shown b the splitting of the flow lines. There are three distinct tpes of behaviour of the magnetic flu, which occur in different regions of the flows. These regions are separated b the separatrices of the w-flows, as well as those flow lines which just touch (but do not enter) D, all of (a).5.5 (b).5.5 FIGURE 7 Flow lines in the plane z ¼ of the field line velocities w in (a) and w out (b). The adopted parameter values are B ¼, a ¼, b ¼, ¼, j ¼, ¼ > crit.

14 D.I. PONTIN et al III (a).5.5 (b) I II.5.5 II I FIGURE 8 (a) The superposition of the two sets of flow lines shown in Fig. 7, highlighting the mismatching, and (b) the regions of different reconnective behaviour, distinguished b separatrices of w in (gre) and w out (black), as well as flow lines just touching the surface of D (dashed). which are shown in Fig. 8b. Regions I show ideal behaviour everwhere. Here w in and w out coincide, and so field lines alwas remain connected and frozen to the plasma. In regions II, the flow lines split inside D, and so an initiall unique field line splits, with each footpoint becoming differentl connected. Notice, however, that although an two flow lines separate when the enter D, the same two flow lines alwas come back together when the leave D. We can therefore think of these regions as slippage regions, since, although in general the two halves of the initial field line will take different times to pass through D, and will hence not rejoin upon leave D, the do move awa from the reconnection region along the same flow line. Hence after both halves of the field line have left D, their separation remains constant in time. Regions III are quite different. Here again there is a splitting of flow lines, and thus field lines, at the edge of D. In contrast to regions II, however, the flow lines in this case never rejoin, and in fact head off in opposite directions. Hence after we leave D each footpoint of our initial field line is now connected to a field line from a topologicall distinct region of magnetic flu. Regions III, then, can be thought of as more of a classical tpe of reconnection region, with the crucial characteristic being that initiall joined field line footpoints continue to move awa from each other for all time after leaving D. As, and hence the strength of the ideal flow, is decreased, these classical reconnection regions grow in width, until we reach ¼ crit. 5. The Case u <u crit When < crit, the rotational flow dominates w out near the spine, whereas w in retains the X-tpe structure, as it still coincides with v id. The two sets of flow lines and their superposition are shown in Figs. 9 and a. Again the different tpes of behaviour can be most clearl distinguished b studing the separatrices of these flows, which are shown in Fig. b. Regions I, II and III have essentiall the same characteristic behaviours as before. Notice, however, that regions III where the classical flu separation occurs are now wider than when > crit (if all other parameters are fied). In addition we now have a region, marked IV in Fig. b, where the nature of the flu

15 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT (a).5.5 (b).5.5 FIGURE 9 Flow lines in the plane z ¼ of the field line velocities w in (a) and w out (b). The chosen parameters are B ¼, a ¼, b ¼, ¼, j ¼, ¼ < crit..5.5 IV.5.5 (a).5.5 (b) I II III II III.5.5 FIGURE (a) The superposition of the two sets of flow lines shown in Fig. 9, highlighting the mismatching, and (b) regions of different reconnective behaviour, distinguished b separatrices of w in (gre) and w out (black), as well as flow lines just touching the surface of D (dashed). reconnection is different again. In this region the flow lines of w out are closed, and so following a field line anchored in this flow, we find that it simpl rotates round and round, never leaving the region or the local flu envelope. So we have a continuousl rotating flu bundle moving at w out, which reconnects with a succession of different field lines, which are swept into and then out of the reconnection region at w in. Note that the reconnection rate for both of the composite solutions described in this section is eactl the same as that given in Eq. (5) for the elementar solution, since id ¼ when R ¼, and so the calculation is unaffected. This reconnection rate now, however, summarises the sum of man different effects. Lastl, note that the above analsis would follow through in eactl the same wa had we chosen the option initiall of integrating from R ¼ R. The onl difference would be that flow lines of w in would become flow lines of w out and vice-versa. 5. Reconnection of Flu Tubes To visualise the effect the mismatching of the field line velocities has on the reconnection of flu tubes, we now describe two eamples. In each case, field lines are integrated from four arbitrar cross-sections chosen smmetricall about the spine such that initiall we have two unique flu tubes on opposite sides of the diffusion region. The

16 D.I. PONTIN et al. cross-sections are chosen so that the never pass into the diffusion region, and hence we can identif a set of field lines which remains frozen into these cross-sections for all time, defining our flu tubes. The resulting motion of the flu tubes is plotted, although we stop the field line integration at a chosen point to see more clearl what is happening. When the flu tube cross-sections are chosen to lie in region II, the resulting flu tube behaviour is a slippage, as described in Section 5., and shown in Fig.. We see that the flu tubes slip apart as the enter D, but the two sections of each initial flu tube leave the vicinit of the reconnection process in the same direction. The degree of slippage depends on the initial positions of the cross-sections, and the size of. The evolution of flu tubes whose cross-sections lie in regions III is shown in Fig.. Now the two sections of each flu tube flip around the spine in opposite directions after splitting. The then leave the diffusion region in opposite directions, as described in Section 5.. Notice, however, that the four flu tubes formed during the splitting process never rejoin cross-wise to form two unique flu tubes again as the do in two-dimensional reconnection. The ma share a few common field lines, although this need not necessaril be the case, depending on eactl where in region III the initial cross-sections are chosen. Z Z X X Y Y Z Z X X Y Y FIGURE Evolution of a pair of flu tubes integrated from cross-sections ling in region II of the w-flows. As the enter the diffusion region each tube splits into two parts, which slip apart, but on leaving D the move off in the same direction.

17 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT Z Z X X Y Y Z Z X X Y Y FIGURE Evolution of a pair of flu tubes integrated from cross-sections ling in region III of the w- flows. As the tubes enter the diffusion region the split, with the two parts of each of the initial two tubes flipping around the spine in opposite directions. Notice in the final frame that after leaving D the four new tubes do not rejoin to form two unique tubes. 6 EXISTENCE OF PERFECTLY RECONNECTING FLUX TUBES AND TIME DEPENDENCE We now ask whether it is possible to choose a pair of initial flu tubes which will perfectl rejoin after reconnection, such that we have two unique tubes before reconnection and two unique cross-wise connected tubes after reconnection. We have seen in the previous section that this is not generall the case, but perhaps with the right choice of initial cross-sections we can achieve this situation. In two dimensions, things are relativel straightforward. If we choose to pick out one field line, in an inflow region of v outside D, then it is alwas possible to find a corresponding field line, b simple smmetr through the X-point, with which it will perfectl reconnect. That is to sa, if we label the two footpoints of field line g and h, and of field line g and h, then, if the field lines are chosen such that after reconnection g is connected to g, then h will alwas be connected to h (see Fig. a). So in two-dimensions there is a simple one-to-one reconnection of field lines, and hence flu tubes. In three dimensions this is not generall the case (Priest et al., ), i.e., if g becomes connected to g after reconnection then h will not be connected to h (see Fig. b). In this sense there is

18 D.I. PONTIN et al. (a) g h BEFORE (b) g h BEFORE g h g h g AFTER h g AFTER h h g h g h h FIGURE (a) The one-to-one reconnection of field lines in D. (b) Schematic picture of reconnection of field lines in D, for eample top view of reconnection in region III of the composite solution of Section 5. no one-to-one reconnection of field lines, so a given field line has no unique counterpart with which it becomes uniquel rejoined. Nevertheless, if we can find two finite sets of field lines which reconnect with each other then we ma still have perfect, or one-to-one, reconnection of these particular flu tubes. An iteration procedure for finding sets of field lines on either side of D which reconnect with each other is described in Hornig and Priest (). The procedure involves sequentiall tracking field lines as the reconnect, moving in the flows w in and w out. In fact, in the stationar case it turns out that the sets of field lines which reconnect with each other lie in surfaces (see Fig. a), and so there are no perfectl reconnecting flu tubes. We find that, if a simple time-dependence is introduced into, as in Eq. (), then perfectl reconnecting flu tubes can be found, as follows. Let ( ½ðR=aÞ 6 Š ½ðz=bÞ 6 Š ; R < a, z < b ; ¼ fðþ t ; otherwise; ðþ where fðþis t some arbitrar function of time onl, and is thus effectivel just a multiplicative constant in the calculation of and E described in Section. Hence JTE is still zero, and no etra magnetic field is induced. This time-dependence, then, shows up in, E, v and thus w in and w out. B the same argument, a time-dependence could also be added to the ideal flow. We will consider here the case where the ideal flow is still stead, and the diffusive process is localised in time via ft ðþ¼ep ð t =T Þ, ðþ where T is a constant which controls the time-scale of the localisation. This timedependence has little qualitative effect on the results described in the previous section, other than to localise the effect of the reconnection process in time.

19 KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT (a) (b) FIGURE Schematic picture of reconnection of magnetic field lines in (a) a stationar state and (b) a time-dependent state, where there is one-to-one reconnection of flu tubes but not field lines. In order to ensure numerical stabilit of the iteration process, we choose to add to the elementar solution a slightl different ideal solution to that described in the previous section. For the ideal potential, we choose id ¼ ep þ =, ðþ so that the ideal solution is approimatel the same as before near the diffusion region, but now the ideal velocit falls off with radius far awa from the reconnection region. The flow lines and separatrices of w in and w out associated with the new composite flow are shown in Fig. 5, at t ¼. We find that the result of adding the time dependence is that the flu surfaces of the stationar case close up to form flu tubes, as shown in Fig. b. Note that although these flu tubes do perfectl reconnect, there is still no one-to-one reconnection of field lines. For eample, in the left hand tube footpoint is connected to footpoint 6 after reconnection, whereas in the right hand tube footpoint is connected to footpoint 8. The mapping of the field lines in the reconnected flu tubes shows a rotation, with respect to a one-to-one correspondence. This signifies the production of twist, and hence a finite amount of self-helicit due to the rotation, consistent with the idea of the elementar solution as a helicit source as described in Section. The result of carring out the iteration procedure is that the field line footpoints that we obtain trace out closed curves, shown in Fig. 6. These curves represent the crosssections of perfectl reconnecting flu tubes in planes of constant z. Precisel what

20 6 D.I. PONTIN et al. (a) (b) FIGURE 5 (a) Flow lines of w in (gre) and w out (black) for the modified time-dependent flow for > crit and t ¼ and (b) the separatrices of these flows. (a).5.5 (b).5.5 (c).5.5 FIGURE 6 Perfectl reconnecting cross-sections in the time-dependent situation, in the plane z ¼ b. For the parameters a ¼, b ¼, B ¼, ¼, j ¼, ¼, and (a) T ¼, (b) T ¼ and (c) T ¼ 8. The gre lines are the separatrices of w out, while the separatrices of w in coincide with the - and -aes. cross-sections are traced out depends on the starting point for the iteration procedure, with these cross-sections being concentric for fied T. The effect of varing T, i.e. the localisation of the reconnection process in time, on the perfectl reconnecting flu tube cross-sections can be seen in Fig. 6. When the process is highl localised in time (T small) the cross-sections are fairl round, but as we increase T the cross-section becomes a thinner and thinner ellipse, whose major ais collapses towards a special time-smmetric flow line. When T is so large that we obtain a state which is numericall equivalent to the stead state, the iteration points (field lines) all lie along this flow line and we return again to a flu surface. 7 CONCLUSIONS We have described here a stead solution of the kinematic resistive MHD equations, which eists in the vicinit of a spiral null point of the magnetic field, with a diffusion term localised around the null. Of course there are man other questions that will need

21 to be answered before a full understanding of these processes can be obtained, such as the role of Lorentz forces in producing the current sheet and driving the reconnection. A rotational tpe of flu reconnection is found, similar to that described in Hornig and Priest () for reconnection in the absence of a null point. The reconnection of magnetic flu takes the form of a rotational slippage within an envelope of flu enclosing the diffusion region. We note that this rotational form of slippage is independent of the size and shape of the diffusion region. In order to stud the effect of the process on the global magnetic field structure, an ideal flow was added which transports flu into and out of the local flu envelope. This results in different behaviours for field lines frozen into different regions of the ideal flow. In one region in particular, the rearrangement of the flu has some similar characteristics to two-dimensional reconnection, where the two parts of an field line (mapped from initiall joined ends) separate, and the separation distance increases for all time. However, it is found that there is no simple one-to-one correspondence of reconnecting field lines after reconnection has occurred. As a result, there is also no splitting and one-to-one rejoining of flu tubes in the stead time-independent case. The result is that the re-ordering of flu b the reconnection process is much more complicated than at a familiar two-dimensional null point. In the future, we plan to set up a numerical eperiment for the full equations, including the equation of motion, and incorporating the invaluable insights that we have gained from this article. In particular, we shall calculate the flu motions w in and w out and will investigate whether the different tpes of kinematic reconnection that we have discovered here are realisable in the dnamic regime. Acknowledgments The authors would like to acknowledge financial support from the European Communit Human Potential Program under Contract No. HPRN-CT--5, PLATON. G.H. would like to thank the VW-Foundation for finanicial support and D.I.P. is grateful to the Carnegie Trust for a PhD Scholarship. References KINEMATIC RECONNECTION AT A MAGNETIC NULL POINT 7 Antiochos, S.K., Karpen, J.T. and DeVore, C.R., Coronal magnetic field relaation b null-point reconnection, Astrophs. J. 575, (). Brown, D.S. and Priest, E.R., The topological behaviour of D null points in the Sun s corona, Astron. Astrophs. 67, 9 6 (). Bulanov, S.V. and Sakai, J., Magnetic collapse in incompressible plasma flows, J. Phs. Soc. Jpn. 66,, 77 8 (997). Craig, I.J.D. and Fabling, R.B., Eact Solutions for stead state, spine, and fan magnetic reconnection, Astrophs. J. 6, (996). Fletcher, L., Metcalf, T.R., Aleander, D., Brown, D.S. and Rder, L.A., Evidence for the flare trigger site and three-dimensional reconnection in multiwavelength observations of a solar flare, Astrophs. J. 55, 5 6 (). Hornig, G., The geometr of reconnection, In: An introduction to the geometr and topolog of fluid flows (Ed. R.L. Ricca), pp. 95, Kluwer, Dordrecht (). Hornig, G. and Priest, E.R., Evolution of magnetic flu in an isolated reconnection process, Phs. Plasmas, 7 7 (). Hornig, G. and Schindler, K., Magnetic topolog and the problem of its invariant definition, Phs. Plasmas,, (996). Klapper, I., Rado, A. and Tabor, M., A Lagrangian stud of dnamics and singularit formation at magnetic null points in ideal three-dimensional magnetohdrodnamics, Phs. Plasmas, 8 8 (996).

22 8 D.I. PONTIN et al. Lau, Y.-T. and Finn, J.M., Three-dimensional kinematic reconnection in the presence of field nulls and closed field lines, Astrophs. J. 5, (99). Longcope, D.W., Topolog and current ribbons: A model for current, reconnection and flaring in a comple, evolving corona, Solar Phs. 69, 9 (996). Longcope, D.W., Brown, D.S. and Priest, E.R., On the distribution of magnetic null points above the solar photosphere, Phs. Plasmas,, (). Mellor, C. Titov, V.S. and Priest, E.R., Linear collapse of spatiall linear, three-dimensional, potential null points, Geophs. Astrophs. Fluid Dnam. 97, (). Parnell, C.E., Smith, J.M., Neukirch, T. and Priest, E.R., The structure of three-dimensional magnetic neutral points, Phs. Plasmas, (996). Parnell, C.E., Neukirch, T., Smith, J.M. and Priest, E.R., Structure and collapse of three-dimensional magnetic neutral points, Geophs. Astrophs. Fluid Dnam. 8, 5 7 (997). Priest, E.R. and Forbes, T.G., Magnetic reconnection: MHD theor and applications, Cambridge Universit Press, Cambridge (). Priest, E.R. and Titov, V.S., Magnetic reconnection at three-dimensional null points, Phil. Trans. R. Soc. Lond. A, 5, (996). Priest, E.R., Hevaerts, J.F. and Title, A.M., A Flu-tube tectonics model for solar coronal heating driven b the magnetic carpet, Astrophs. J. 576, 5 55 (). Priest, E.R., Hornig, G. and Pontin, D.I., On the nature of three-dimensional magnetic reconnection, J. Geophs. Res. 8, A7, SSH6- (). Schindler, K., Hesse, M. and Birn, J., General magnetic reconnection, parallel electric fields, and helicit, J. Geophs. Res. 9, A6, (988).