Dynamic 3D reconnection in a separator geometry with two null points

Transcription

1 Dnamic 3D reconnection in a separator geometr with two null points D. I. Pontin 1 & I. J. D. Craig Department of Mathematics, Universit of Waikato, Private Bag 35, Hamilton, New Zealand. ABSTRACT The dnamic behaviour of disturbances in the vicinit of a pair of magneticall connected 3D null points is eamined. The aim is to investigate how nonlinear disturbances lead to strong localized currents that initiate magnetic reconnection at the separator. The problem is formulated in an incompressible clindrical geometer b superposing arbitrar disturbance fields onto a background two-null field. Two different regimes are found for the dnamic evolution, depending on the relative strengths of the background magnetic and velocit fields. In one regime, disturbance pulses split into ingoing and outgoing components, which propagate along the background field lines. In the other flu pile-up regime, a strong driving flow localises the disturbances towards the null point pair. Current structures aligned with the spines, fans and separator present in the field are found to result, and the structure of these currents and their scaling with resistivit is investigated. Subject headings: MHD plasmas Sun: flares Sun: magnetic fields 1. Introduction Magnetic reconnection is a process which is fundamental to man phenomena in astrophsical and laborator plasmas. Reconnection is the onl mechanism capable of releasing topologicall bound magnetic energ, in the form of Ohmic heating and the kinetic energ of mass motion. Most astrophsical plasmas, however, are highl conducting the dimensionless collisional resistivit is an inverse Lundquist number of order η and for reconnection to be effective, localized regions comprising huge magnetic field gradients, and 1 Present address: Space Science Center, Universit of New Hampshire, Durham NH3, USA.

2 therefore strong currents, must be present. In this paper, we will investigate the growth of such intense currents in a magnetic separator topolog. We concentrate here on magnetic reconnection in the presence of 3D null points of the magnetic field. The resulting spine and fan separatri topolog of the background field will be outlined in the following section. For the present, we emphasize that reconnection at separatri surfaces, as well as at separator lines linking the nulls, is thought to be important in both solar and stellar atmospheres, as well as closer to home in the Earth s magnetosphere. In the solar corona in particular, it is predicted that there should be present an abundance of 3D null points (e.g. Inverarit and Priest 1999; Albright 1999; Longcope et al. 3) and separators (Schrijver and Title ; Beveridge and Longcope 5). It is further predicted that such sites should provide priviliged regions for heating the corona (Longcope 199; Antiochos et al. ; Priest et al. 5). There is also observational evidence that reconnection at a 3D null point ma act as a trigger for some solar flares (Fletcher et al. 1). Our present aim is to stud separator reconnection. Despite being invoked as an eplanation for man dnamic phenomena in the corona, such as X-ra bright points (Longcope et al. 1) and solar flares (Longcope and Noonan ), the mechanisms of separator reconnection are not well understood. What is known is that separator reconnection occurs quite naturall on the Sun, for eample, when two photospheric flu sources move relative to one another in the presence of an overling field (Parnell and Galsgaard ). Observational evidence of separator reconnection in the corona has recentl been presented b Longcope et al. (5). In addition, it has been shown that flare statistics can be modelled using a superposition of separators (Wheatland ). In this paper, we develop solutions which model current growth and reconnection at a separator. Due to the etremel low value of the resistivit in astrophsical plasmas, a crucial propert of an reconnection model is the predicted reconnection rate, and whether it is sufficient to eplain observed dnamic phenomena at realistic plasma parameters. To this end, we will discuss the scaling properties of the peak current with resistivit in each of the regimes described. The model is based upon the principles first developed b Craig and Henton (1995), who proposed a stead-state, resistive, planar model for reconnection at a two-dimensional (D) null point. This work has since been generalised into three dimensions, firstl with the addition of an aial field to the D X-point, such that there is no null in the field (Craig et al. 1995), and secondl to spine and fan reconnection at single 3D nulls (Craig and Fabling 199). Further generalisations are possible to include the effects of timedependence (Craig and Fabling 199), as well as additional (non-resistive) non-ideal effects (e.g. Ji and Song 1; Craig and Watson 3). In what follows we develop a resistive, time-dependent model for reconnection occurring

3 3 in a magnetic field in clindrical geometr, where either one or two 3D null points are present. This etends and generalises the work of Watson and Craig (), who made a preliminar investigation of similar configurations in a stead-state regime (see also Tassi et al. 3), and provides the possibilit of reconnection at curved current sheets and also separator current sheets. Although we consider here onl resistive non-ideal effects, we do not epect that the introduction of further non-ideal effects would alter the qualitative results significantl. Hall currents, for eample, are capable of having a profound effect on the detailed microphsics of the reconnection region. Their effect depends, however, both on the smmetries of the reconnection problem, and on the presence of background guide fields (Craig and Watson 3, 5). For the present, we simplif our discussion b concentrating on the buildup of strong field gradients in the vicinit of the reconnection region. In Section we discuss the topolog of the magnetic fields considered. In Section 3 we introduce the governing MHD equations and describe the method of analsis, emphasising in particular the wave properties of the solution, which we believe provide generic signatures for all transient magnetic merging solutions. In Sections and 5 the results of numerical simulations of the equations are discussed and their scalings with resistivit are summarized. We present our conclusions in Section.. Topolog of connected null point pairs We begin b summarizing the magnetic field topolog associated with 3D nulls. The field lines which asmptoticall approach 3D null points provide a local skeleton of the magnetic field. In the case of a single 3D null the skeleton comprises a spine line and a fan surface. The spine is a pair of field lines which approach (recede from) the null in opposite directions, while a famil of field lines radiate out from (into) the null in the fan plane (e.g. Fukao et al. 1975; Lau and Finn 199; Parnell et al. 199). B considering a Talor epansion of B about the null point, it can be seen that the orientation of the spine and fan are determined b the eigenvectors of the matri B. This matri is traceless, and the eigenvectors corresponding to the eigenvalues of like sign (or whose real parts have like sign) determine the orientation of the fan, while the other defines the spine. In general, the field strength in the fan will not be isotropic, and this anisotrop is determined b the fan eigenvalues. When two null points are present a separator a field line which directl connects the nulls ma eist. Kinematic considerations for fields containing multiple nulls (Lau and Finn 199; Priest and Titov 199) suggest that spines, fans and separators provide special sites at which magnetic reconnection can occur, and this view has been well supported b stead state reconnection solutions (Craig et al. 1999).

4 In what follows we consider a two-null field whose separator is formed b the intersecting fan planes of the component nulls. This structure provides a generic separator geometr in the sense that the separator line is topologicall stable, being robust to arbitrar, small perturbations in the background field. Note that, although a separator could be formed b the coeistence of two null spines or one spine and one fan field line, neither of these structures is topologicall stable, due to the uniqueness of the spine lines. This is one advantage of the present stud over that of Craig et al. (1999), who consider a general class of fields containing multiple nulls in a stead-state regime. In this case spine-fan separators are present, which can onl be maintained b the high degree of smmetr present in the field. We believe the geometr of the present stud also provides a significant improvement on man previous studies of separator reconnection, which model the separator simpl using a planar X-point threaded b an arbitrar aial field (e.g. Heerikhuisen and Craig ). As we will show, the influence of the nulls themselves cannot be discounted when considering separator reconnection. 3. The MHD equations 3.1. Form of the solution We assume the equations of collisional resistive MHD, based on an incompressible plasma in an open (unbounded) geometr. The problem is scaled according to the reference coronal values B c = G, l c = 9.5 cm, n c = 9 cm 3, v A = 9 cm s 1. Time is now measured in units of the Alfvén time τ A = l c /v A, which is tpicall a few seconds in coronal applications. In this formulation the plasma resistivit is an inverse Lundquist number η 1. The simplifing assumption will be made that the viscosit ν is isotropic. In the simulations described later, ν is chosen to scale linearl with η this choice has been shown to reproduce the pure resistive scalings of the current laer (Craig and Watson 5). The dimensionless induction equation is given b B t = (V B) + η B, (1) while taking the curl of the viscous momentum equation we obtain Ω t = (J B) (Ω V) + ν Ω, ()

5 5 where J = B is the current, Ω = V is the vorticit, and η and ν are assumed constant. We must also impose the constraint equations B =, V =. (3) Once the magnetic and velocit fields have been found, the plasma pressure ma be obtained from the uncurled form of the momentum equation. The assumption of an incompressible plasma ( V = ) is made in order to facilitate our method of solution, since in this form the equations displa a high degree of smmetr between B and V (Craig and Henton 1995). Furthermore, once the current sheet has formed, its behaviour is epected to be largel incompressible, as the global timescale for merging is tpicall much longer than the timescale for fast mode propagation across the current sheet. It has been shown that finite compressibilit has onl a weak effect on the scaling properties of flu pile-up current sheets, and in fact acts to marginall speed up the resultant reconnection (see Litvinenko and Craig 3, and references therein). In order to construct reconnection solutions, we use the superposition technique of Craig and Henton (1995), and let B = βp() + b(, t), V = αp() + v(, t), () where P is a stead-state potential background field, and b and v are disturbance fields of arbitrar amplitude. The method of solution is based on choosing a form of reduced dimensionalit for the disturbance fields such that Equations (1) and () ma be reduced to a sstem of ordinar differential equations. In the stead-state regime, this corresponds to automaticall satisfing the momentum equation, leaving onl the induction equation to be solved. In this case, v() = (β/α)b(), and in order for the momentum equation to be automaticall satisfied we require (( b) b) =. (5) In previous models based on Cartesian geometr b has taken either the form b = f(, )ẑ or b = f()ŷ +g()ẑ (or cclic permutations thereof). It is interesting to note that these forms are in fact over-restrictive for this method, in that the satisf ( b) b =, whereas this quantit need onl be curl-free. In clindrical coordinates we find that there is a reduced choice of such low-dimensionalit disturbances. Of those forms which satisf the divergence condition, the onl two which additionall satisf (5) are b 1 = b(r, θ, t)ẑ, b = a(r, t)ˆθ + b(r, t)ẑ. ()

6 In contrast to the Cartesian case, these forms do not satisf the more restrictive condition ( b) b =. The disturbance form of most interest to us is b 1, since the second form describes disturbance field structures which var onl in the radial direction. Disturbances based on b 1 provide a richer structure to the resulting currents, and can in an case be used to model a single component of the form b b turning off the θ-dependence. Guided b the above considerations, we take B = βp + b(r, θ, t)ẑ, V = αp + v(r, θ, t)ẑ. (7) B analog with the Cartesian case, we look for a potential field P = (P 1 (r), P (r, θ), P 3 (r, z)) which is linear in θ and z. The onl potential field of this form is ( { γln(r/l) P = + κr } γθ, (r/l) l (r/l), κz ), () l where γ, κ and l are constants. We note that the field is singular at the origin, and nonperiodic in θ, and so the solution ma not be considered as global, but rather must be considered valid onl in some restricted annular domain which ecludes the origin and the negative -ais (θ = ±π). The background field P ma contain one or two null points, depending on the values of γ and κ, while l is a scale factor which determines the distance of the null point pair from the origin. If γ and κ are of the same sign, then onl a single null is present, however if γ and κ are of opposite sign and γ > κ e then two nulls are present. The skeleton of the field in each case is shown in Figure 1. Here we will concentrate primaril on the case where two nulls are present in the field. As shown in the figure, the nulls are joined b a separator (along θ = z = ), formed b the intersection of their fan planes. 3.. Equations for the disturbance fields Substituting the above epressions for B and V (Eqs. (7) and ()), into Equations (1) and (), we find that ( κ ) ( κ ) b t = α l L b β l L v + η b (9) ( κ ) ( κ ) v t = α l + L v + β l + L b + ν v + g(t) () where L = P 1 r + P { γl ln(r/l) r θ = + κr } r + γlθ r l r θ (11)

7 7 is the directional derivative along P in the rθ-plane, and a = (1/r) (ra r ) r + (1/r )a θθ, (1) and where subscripted letters denote partial derivatives. Equation () is obtained b integrating either component of () emploing integration b parts, with the arbitrar function of time g(t) being the constant of integration. Note that g simpl gives a time-varing but spatiall uniform z-component of V, that is a uniform shift of the flow structure in the z-direction. Thus, in order to maintain the co-spatial nature of the null points of the background magnetic and flow fields, we hereafter set g =. A procedure analogous to that described above can be used to derive the corresponding equations when the disturbance field takes the form b in Equation (), as described in Appendi A Wave properties of the solution In the majorit of astrophsical plasmas, non-ideal effects onl become important when ver small length scales develop, due to the fact that the resistive and viscous coefficients (η and ν) are so small. It is therefore natural to eamine the properties of our sstem in the ideal limit, when η = ν =. This problem is simplified b invoking the Elsasser variables M = b v and N = b + v, which from (9) and () satisf M t = (α β) κ N (α + β)lm, (13) l N t = (α + β) κ M (α β)ln. (1) l To simplif further, observe that the operator L must be epressible as a total derivative since it defines the directional derivative along the background planar field. More formall, we can change from the (r, θ) coordinates to a sstem (ψ, χ) based on the (planar projected) field lines dr = rdθ, (15) P 1 P which are labelled b ψ. It follows that χ is a coordinate running along the field lines, such that ψ χ. B a suitable choice we can make the directional derivative L χ. The further change to a comoving frame τ = t, s = χ αt reduces sstem (13,1) to M τ = (α β) κ l N βm s, (1) N τ = (α + β) κ l M + βn s. (17)

8 It follows from this that both M and N, and thus b and v, satisf the generalised Klein- Gordon equation b ττ = β b ss + κ l (α β )b. (1) This equation for the disturbance field highlights the wave-like nature of the problem. First note that Equation (1) has two characteristics, C ± = s ± βτ = χ (α β)τ. (19) Obviousl, if α > β, then both characteristics correspond to waves propagating along the field lines in the positive χ direction, of different speeds. B contrast, if α < β, then the characteristics correspond to two waves which propagate in opposite directions (see also Craig and Fabling 199). Thus, there are two different possible regimes for the solution, depending on the relative sizes of α and β. The waves are Alfvén waves, which represent the incompressible limit of (compressive) fast-mode Alfvénic disturbances. A further ke propert is the possibilit of growth in the solution. Obviousl growth, as opposed to oscillator behaviour, can occur onl if the source term of (1) is positive, that is, α > β. Note that Equation (1) is a generalization of the Cartesian fan equation analzed b Craig and Fabling (199) using Fourier transform methods. Appling their analsis to (1) suggests that the condition for growth derived above ma be sufficient, even when small dissipation coefficients are accounted for. In an case, growth depends not onl on a sufficientl strong driving flow, but also on the sign of the parameter κ of the background field. In order to obtain growth, we require in addition that the background flow has the capacit to stretch and amplif the disturbance field, bẑ. For this reason, we will concentrate on the case κ >. 3.. Numerical simulations We now consider numerical simulations of Equations (9) and () based on a purpose built, predictor-corrector scheme on a clindrical mesh. In all cases the potential background fields are perturbed b imposing a single initial magnetic field pulse (b) at some location within the numerical domain. We first concentrate on two-null oscillator β > α solutions, before going on to discuss flu pile-up models. In all of the simulations we take ν = η, since a linear scaling of ν with η is known to preserve the purel resistive scalings (Craig and Watson 5). Note also that we find that the chosen value of ν has little effect on the results discussed.

9 9. Wave-like regime ( β > α ) According to the previous analsis, when the background magnetic field dominates the driving flow the evolution of the disturbance field is characterised b inward and outward propagating waves. Thus the initial disturbance is epected to spread along the background field lines, within an envelope bounded b peaks travelling in and moving out at speeds α ± β, as determined b (19). Figures (-) show that this behaviour is reproduced in the simulations (these figures show frames from animations, which can be viewed at math97/cl anim.html). We see that the magnetic disturbance develops two separate peaks, which travel along the field lines of P toward and awa from the null point pair. However, the geometr of the background field lines, and the location of the initial disturbance with respect to the background field structure, are found to influence, quite strongl, the localisation of the ingoing pulse. In the present simulations the direction of the magnetic field and plasma flow are quite arbitrar: the flow ma be in either direction, and although this influences the relative speeds of the peaks of the disturbance, the localisation and current growth properties remain unaffected. The background field structure is the two-null structure of Figure 1(b). There is one null, hereafter named null 1, whose fan lies in the rz-plane and whose curved spine lies in the rθ-plane; and another ( null ) whose fan lies in the rθ-plane and whose spine lies in the z-direction. The spine of each null bounds the fan surface of the other. If the initial pulse is located to the negative- side of the spine of null 1, or if it disturbs this spine, then the disturbance spreads out and localises to the fan of null 1, as shown Figure. However, if the disturbance is initiated in the fan of null (i.e. to the positive- side of the spine of null 1), then in general the ingoing pulse is squeezed in towards null along P, where it localises towards the spine, as in Figure 3. In addition, a third tpe of behaviour ma occur. If the disturbance is initiated in the fan of null, but sufficientl close to the spine of null 1, then the ingoing pulse will be transported inwards towards the separator joining the two nulls, as in Figure, the disturbance being aligned with the fan plane of null 1. The relative size of the region of space in which this separator localisation occurs depends on the geometr of the field lines in the fan of null, or equivalentl the isotrop of this null, since the disturbances propagate along P. In each of the three cases described above, the outgoing peak moves outwards along P, spreading due to diffusion and the increasing plasma velocit awa from the nulls. Of central interest, in each of the above cases, is whether a growth in the current densit occurs, and to what etent this depends on the resistivit η. As epected, the magnitude of both disturbance peaks decas in time, as there is no flu pile-up occurring. The current associated with the outgoing pulse does indeed deca, but the ingoing pulse is

10 alwas associated with current growth. The growth is stalled when resistive effects begin to dominate. Independent results for the advection of magnetic pulses suggest that, in the absence of flu pile-up, the peak current is controlled b the width η of the current laer (e.g. Craig and Fabling 199). This corresponds to a slow Sweet-Parker dissipation rate. We find here that, while this scaling is not adhered to strictl in the present more comple field structure, it is approimatel followed. Figure 5 shows a logarithmic plot of peak current versus resistivit for each of the spine, fan and separator cases. Thus if J ma follows a power-law dependence of the form J ma η µ J, () then the gradient determines µ J. Also plotted, for comparison, are the results for a similar run in which a localisation towards the curved fan plane of the single null field shown in Figure 1(a) was considered. The results are displaed in the top portion of Table 1. Note that while the fan current scaling is similar in the two cases considered, the spine and separator currents scale somewhat more slowl. These descrepancies can perhaps be attributed to the lack of highl localized current structures in the present clindrical geometr. In an case, since the scaling of the peak current with resistivit is such that the resulting reconnection rate is unlikel to be energeticall significant at realistic values of the resistivit, it seems more profitable to consider possible enhancements in the flu pile-up regime with α > β. 5. Flu pile-up regime ( α > β ) 5.1. Localisation phase We now eamine the evolution of our sstem in the regime α > β. Consider first the ideal limit with η = ν =. Since we require that our dnamic models must be able to reproduce the stead state solutions, in which v = (β/α)b, we introduce the function f = v (β/α)b to measure departures from the stead state. Now, rewriting Equations (9) and () in terms of b and f, we have in the ideal limit (η = ν = ) that b t + α Lb = κ ( κ ) l α b β l L f, (1) f t + α + Lf = α κ l f β α α ( κ l L ) b, () where α + = α + β α, α = α β. α

11 11 The advection terms on the left hand side of the above equations suggest that the field f evolves more quickl than b, as the coefficients α + and α determine inverse time-scales for these processes, and α + > α. If we now consider the limit of ver strong flow, that is β/α 1, and use the variable s to compute the directional derivative, so that Lf f s, then Equation () becomes f t + α + f s + α κ l f =, with solution ( ) f = f s e α+ t e α (κ/l)t. (3) Equation (3) demonstrates the spatial localisation of f along the background field lines, for α > corresponding to inflow in the rθ-plane, as well as indicating an eponential deca in time. Therefore, for large time we ma neglect f in the evolution equation for b. Setting f = in (1), we deduce, as above, that ( ) b = b s e α t e α (κ/l)t, () which shows a slower localisation of b than f, coupled with an eponential growth of b. We conclude that the immediate effect of the magnetic perturbation is to drive the velocit disturbance field towards its quasi-stead state distribution (v (β/α)b). Once the equilization phase has occurred, the disturbance fields begin a gradual localisation phase, which is accompanied b eponential growth. This growth is arrested once length scales are sufficientl reduced so that non-ideal effects become important. 5.. Simulation results We now summarize numerical simulations performed in the regime α > β. Once again, the initial conditions are chosen such that v = at t =, and b is some non-zero disturbance pulse. The earl phase of the evolution confirms that the velocit profile v does indeed quickl grow (to v (β/α)b), in order to mirror the magnetic disturbance field profile. Once this has happened, the behaviour of v closel follows that of b, with just a small phase dela. Both disturbance fields are then advected towards the null point pair, with the magnitude of the magnetic field disturbance, as well as the current, growing as the localisation proceeds. As described in Section, the nature of this localisation depends on the relationship between the initial disturbance and the background field P, and again localisation towards the spine, fan, or separator is possible in the double null field. In each case, we require α > to ensure inflow of the disturbances.

12 Fan reconnection We eamine first the case of fan reconnection. Consider first an isolated linear 3D background null, of the form Q = (λ 1, (λ λ 1 ), λ z), where λ > λ 1 >. In this case, the fan is the plane z =, and a tpical disturbance has - and -components, which are epected to scale with resistivit as J ma η µ J, B ma η µ b, (5) where µ J = (1 + A)/, µ b = A/ and A = λ 1 /λ (for the -component) or (λ λ 1 )/λ (-component) (see Craig and Fabling 199; Heerikhuisen and Craig ). Hence, when the null point is isotropic (λ 1 = λ /), each component is equall magnified (A = 1/). However, if the null is non-isotropic, and therefore the outflow in the fan is stronger in one direction (as defined b the eigensstem of the null see Section ), then one component will be stretched and magnified more strongl than the other. Our present purpose is to test whether the above scalings persist in more complicated field configurations for both the single and double null background fields. The disturbance field in our case is alwas aligned to the z-ais, and it is the background fan field that governs the strength of the outflow in this direction. Now, in order to determine the epected scalings (as predicted b the theor for a single linear null point), it is necessar to calculate the appropriate ratio of the eigenvalues of the matri B at the null (see Section ). As discussed above, this is the ratio of the outflow eigenvalue corresponding to the direction parallel to the disturbance component to the inflow eigenvalue. For the case of the single null background field, a curved current sheet is formed in the fan plane of the null (the zθ-plane). The peak current and maimum field at time of peak current are plotted in Figure (a). B comparison with the field Q, we see that the parameter A is given in our case b P 3z /P 1r. The observed scalings for J ma and b ma for two representative runs, along with the epected values of µ J and µ b, are given in Table 1. It can be seen that the results show a ver good agreement with the predictions for the much simpler field configuration. A fan current ma also be obtained in the case of the double null background field. Using the same background field parameters and perturbation as those emploed in the run shown in Figure, the fan localisation shown in Figure (b) is obtained. In this case, the relevant degree of stretching due to the ehaust flow is computed via A = rp 3z /P θ. Observed (plotted in Figure (a)) and epected scalings are given in Table 1, which again demonstrate ver good agreement between the epected values in simple configurations and the behaviour in this more complicated topolog.

13 Spine reconnection For a general disturbance in the fan plane of null, as before, a spine-tpe current sheet results. A tpical localised disturbance is shown in Figure 7(b), and it should be noted that, due to the relativel weak driving flow in the fan close to null, this localisation occurs over man Alfvén times. The single peak (in b) disturbance corresponds to a solitar rotational tube of current centred on the spine. This is the generic current structure for a single disturbance, and a building block for the standard stead-state spine solution of Craig and Fabling (199), in which, due to the smmetries of the coordinate sstem emploed, the spine current sheet is made up of two such tubes. The epected scalings in the dnamic regime of spine current sheets are given b µ J (1 + A)/(A) and µ b 1/(A) (for a field of the form Q, see Heerikhuisen and Craig ). In this case there are two inflow directions and one outflow ( < λ 1 < λ ), and the fastest scaling is obtained when the inflow speeds are equal (λ 1 = λ /, A = 1/), giving clindrical current structures. When λ 1 λ /, the clinder flattens out, and the scaling weakens. We must therefore take A to be the smaller of λ 1 /λ and (λ λ 1 )/λ (for the field Q), or in our case, the smaller of P 3z /P 1r and rp 3z /P θ. A tpical set of results obtained from our simulations is plotted in Figure 7(a). The resultant scaling parameters are again displaed in Table 1. Although the actual values of µ J and µ b are slightl higher than the predicted ones, the basic scaling is of the same order, and still super-fast, in the sense that the reconnection rate ηj will scale as a negative power of η Separator reconnection One might epect that, as in the β > α regime, a suitabl chosen initial disturbance might localise along the separator, to a separator current sheet. This is indeed the case, although one must take care when analsing such a situation. As before, a pulse initiated in the fan of null, but close to the spine of null 1, will localise towards the separator. However, due to the strong driving flow, the localisation and current growth does not halt at this time. Rather, since there is a strong flow along the separator, the disturbance will continue to localise along this direction ( = z = ) and, assuming that the resistivit is sufficientl small, the current will continue to slowl grow until the pulse full localises at null, in a spine current sheet. This effect is shown in Figure, at an intermediate time in the slow localisation along the separator. In the present simulations, we are limited b all of the usual constraints of numerical

14 1 resolution. However, one might epect that if the separator is sufficientl long, or equivalentl if the disturbance is initiated sufficientl close to the separator, that a diffusive current sheet will form at some location on the separator. For this to occur, we require that the time taken for the disturbance to be advected onto the separator (in the perpendicular direction) and form a sheet (τ rec ) be less than the time taken for the flow to push the pulse along the length of the separator (τ sep ) to instead form a current sheet at the null point. In order to obtain some estimate for the length scales involved, we approimate our long separator (in Cartesian coordinates) b the familiar B s = (/l, /l, b ), where l is a magnetic field length scale across the separator, and b is constant. It can then be shown (c.f. Heerikhuisen and Craig ), that if the disturbance has length scale l, then ( α l τ rec l α ln Now, since α > β, the speed of the pulse along the separator is αb. Thus, if the length of the separator is denoted b L, then τ sep L/αb. In our sstem of normalised equations (see Section 3), we have that α, α, b, L 1, and therefore in order that τ rec τ sep we require, if η 1, that l.. This certainl does not seem out with the realms of possibilit in a realistic plasma environment. With η 3, as is tpical in our simulations, we require l.3. This, unfortunatel, is on the boundar of what we can achieve with numerical resolution, so we have been unable to test this idea properl. From the above considerations, it seems unlikel that a long-lived separator current sheet can be formed in this tpe of flu pile-up regime, in which there is alwas a strong flow along the separator itself. In the present eample, the separator current buildup has a miture of fan and spine behaviours, and what is clear is that the nulls themselves pla a crucial role in determining this behaviour. Therefore, in order to obtain a quasi-stead-state current sheet located along (at least the majorit of) the separator, it ma be necessar to consider a compressible collapse-tpe model (e.g. Longcope and Cowle (199), or c.f. Pontin and Craig (5)). η ).. Summar We have investigated the dnamic behaviour of disurbances in the vicinit of a pair of 3D magnetic null points connected b a stable fan-fan separator, in an incompressible, clindrical geometr. It has been shown that the sstem can be analsed in terms of the superposition of transient disturbances onto a background field defining the two-nulls. This result generalizes the previous Cartesian description of transient reconnecting disturbances in a 3D single null geometr (Craig and Fabling 199), as well as stead-state clindrical

15 15 merging solutions (Watson and Craig ; Tassi et al. 3). The behaviour of the sstem falls into two main regimes, depending on the relative magnitudes of the background magnetic field and plasma flow (here β and α, respectivel). If β > α then the background flow is too weak to localize the disturbance field. In this case the magnetic disturbance develops two peaks, and these move in opposite directions along the background field lines. This behaviour is unaffected b changing the sign of either β or α. The pulse that moves inwards towards the null point pair provides the reconnecting field that localises towards either a spine, fan or separator. Of course, a large-scale disturbance, smeared over the entire two-null geometr, ma contain regions which are individuall focussed towards the separator and the nulls. In each case, the localisation is accompanied b a current growth, with the peak current scaling at a slow η. η. with resistivit. This implies that the resulting reconnection is slow, and that the energ release at realistic resistivities is insufficient to account for eplosive phsical processes such as solar flares. This does not, however, preclude these mechanisms from providing background heating in solar or stellar coronae. Rather, our results provide support for magnetic dissipation theories which propose separatrices and separators as preferred sites for background coronal heating (e.g. Priest et al. 5), b demonstrating how transient disturbances are channelled b the background field lines towards such topological features. In order to speed up the reconnection rate, it is necessar to consider the flu pile-up regime associated with strong driving flows α > β. In this regime, both wave solutions for the initial disturbance field travel in the same direction. Initiall, there is a relativel fast equalisation of the magnetic and velocit disturbances to a quasi-stead-state (v (β/α)b). Assuming that the sign of α is chosen to provide inflow, disturbances are localised towards the null separatrices, and ma form spine-aligned current sheets, or fan-aligned current sheets, whose scaling properties with resistivit closel match those at isolated linear nulls in Cartesian geometries. In particular, spine currents scale at a super-fast rate J η 1.5, whereas fan current sheet reconnection models provide more modest current amplitudes of order η.5 η 1, depending on the outflow geometr in the fan. For certain initial parameters and disturbances, a strong separator current ma also result. This current, however, ma not be sustained b the flow geometr. Specificall, the strong driving flow tends to push the pulse towards the spine of the null point whose fan plane corresponds to the inflow direction. Whether the current concentration grows sufficientl strong to begin diffusive/reconnective processes at the separator, or whether this occurs once the disturbance has localised at one of the nulls, depends cruciall on the length scales parallel and perpendicular to the separator. Since the connected null point pair we consider is completel generic as far as the basic field structure is concerned, this behaviour

16 1 ma well be a general propert of two null geometries in strongl driven, incompressible regimes. This interpretation is consistent with the stead-state results of (Watson and Craig ), who found current on the separator onl for ver special choices of boundar conditions. Hence, when modelling energetic phsical processes occuring via reconnection at current sheets along separators, it ma be preferable to consider a compressible collapse-tpe model for the current sheet formation rather than a strongl driven flu pile-up mechanism. Notabl, in the dnamic models described here, the separator localisation contains a hbrid of fan-tpe and spine-tpe behaviours that is strongl affected b the two nulls themselves. The authors would like to acknowledge financial support from the Marsden Fund, grant no. -UOW-5 MIS, and also useful discussions with Paul Watson. A. Appendi The same reduction procedure as described in Section 3 ma be performed for the disturbance form b = a(r, t)ˆθ + b(r, t)ẑ. The resulting differential equations to be solved are a t = (P 1r + κ/l + P 1 r ) [βu αa] + η ( a rr + a r /r a/r ), b t = (P 1 r + κ/l) [βv αb] + η (b rr + b r /r), ( r ) u t = β/r P 1 (ra) r + γl (ra) r /r dr r ( min r ) α/r P 1 (ru) r + γl (ru) r /r dr + ν ( u rr + u r /r u/r ) + g(t), r min v t = (P 1 r κ/l) [βb αv] + ν (v rr + v r /r) + h(t). REFERENCES Albright, B. J. (1999). The densit and clustering of magnetic nulls in stochastic magnetic fields. Phs. Plasmas, :. Antiochos, S. K., Karpen, J. T., and DeVore, C. R. (). Coronal magnetic field relaation b null-point reconnection. Astrophs. J., 575:57 5. Beveridge, C. and Longcope, D. W. (5). On three-dimensional magnetic skeleton elements due to discrete flu sources. Solar Phs., 7:193.

17 17 Run info µ J µ b β > α n loc. A observed epected observed epected 1 F..1 ±..5 1 F.3. ±..5 1 F.5.5 ±.1.5 F..5 ±..5 Sp.51.9 ±..5 Se.5 ±.1.5 α > β 1 F.3. ± ± F..73 ±..73. ± F.7.7 ± ±..35 F..7 ±..7.1 ±.. Sp ± ±..93 Table 1: Scaling results from the simulations and predictions based on previous analtical work. Column 1 (n) gives the number of nulls in the field for that run, column denotes which topological feature the current localises towards (F=fan, Sp=spine, Se=separator), and A is the isotrop of the corresponding null. Observed results are calculated b linear regression, with errors given at 95% confidence level.

18 1 Craig, I. J. D. and Fabling, R. B. (199). Eact solutions for stead-state, spine, and fan magnetic reconnection. Astrophs. J., : Craig, I. J. D. and Fabling, R. B. (199). Dnamic magnetic reconnection in three space dimensions: Fan current solutions. Phs. Plasmas, 5:35. Craig, I. J. D., Fabling, R. B., Heerikhuisen, J., and Watson, P. G. (1999). Magnetic reconnection solutions in the presence of multiple nulls. Astrophs. J., 53:3. Craig, I. J. D., Fabling, R. B., Henton, S. M., and Rickard, G. J. (1995). An eact solution for stead state magnetic reconnection in three dimensions. Astrophs. J. Lett., 55:L197 L199. Craig, I. J. D. and Henton, S. M. (1995). Eact solutions for stead state incompressible magnetic reconnection. Astrophs. J., 5:. Craig, I. J. D. and Watson, P. G. (3). Magnetic reconnection solutions based on a generalised ohm s law. Solar Phs., 1: Craig, I. J. D. and Watson, P. G. (5). Eact models for Hall current reconnection with aial guide fields. Phs. Plasmas, 1:13. Fletcher, L., Metcalf, T. R., Aleander, D., Brown, D. S., and Rder, L. A. (1). Evidence for the flare trigger site and three-dimensional reconnection in multiwavelength observations of a solar flare. Astrophs. J., 55:51 3. Fukao, S., Ugai, M., and Tsuda, T. (1975). Topological stud of magnetic field near a neutral point. Rep. Ion. Sp. Res. Japan, 9: Heerikhuisen, J. and Craig, I. J. D. (). Magnetic reconnection in three dimensions - spine, fan and separator solutions. Solar Phs., : Inverarit, G. W. and Priest, E. R. (1999). Magnetic null points due to multiple sources of solar photospheric flu. Solar Phs., 1: Ji, H. S. and Song, M. T. (1). Three-dimensional solutions for fan and spine magnetic reconnection in partiall ionized plasmas. Astrophs. J., 55:17. Lau, Y. T. and Finn, J. M. (199). Three dimensional kinematic reconnection in the presence of field nulls and closed field lines. Astrophs. J., 35:7 91. Litvinenko, Y. E. and Craig, I. J. D. Robust scalings in compressible flu pile-up reconnection. Solar Phs., 1:

19 19 Longcope, D. W. (199). Topolog and current ribbons: A model for current, reconnection and flaring in a comple, evolving corona. Solar Phs., 19: Longcope, D. W., Brown, D. S., and Priest, E. R. (3). On the distribution of magnetic null points above the solar photosphere. Phs. Plasmas, : Longcope, D. W. and Cowle, S. C. (199). Current sheet formation along three-dimensional magnetic separators. Phs. Plasmas, 3:5 97. Longcope, D. W., Kankelborg, C. C., Nelson, J. L., and Petsov, A. A. (1). Evidence of separator reconnection in a surve of -ra bright points. Astrophs. J., 553:9 39. Longcope, D. W., McKenzie, D., Cirtain, J., and Scott, J. (5). Observations of separator reconnection to an emerging active region. Astrophs. J., 3:59. Longcope, D. W. and Noonan, E. J. (). Self-organised criticalit from separator reconnection in solar flares. Astrophs. J., 5: 99. Parnell, C. E. and Galsgaard, K. (). Elementar heating events - magnetic interactions between two flu sources. ii rates of flu reconnection. Astron. Astrophs., : Parnell, C. E., Smith, J. M., Neukirch, T., and Priest, E. R. (199). The structure of three-dimensional magnetic neutral points. Phs. Plasmas, 3(3): Pontin, D. I. and Craig, I. J. D. (5). Current singularities at finitel compressible threedimensional magnetic null points. Phs. Plasmas, 1:711. Priest, E. R., Longcope, D. W., and Hevaerts, J. F. (5). Coronal heating at separators and separatrices. Astrophs. J., : Priest, E. R. and Titov, V. S. (199). Magnetic reconnection at three-dimensional null points. Phil. Trans. R. Soc. Lond. A, 35: Schrijver, C. J. and Title, A. M. (). The topolog of a mied-polarit potential field, and inferences for the heating of the quiet solar corona. Solar Phs., 7:3. Tassi, E., Titov, V. S., and Hornig, G. (3). Eact solutions for reconnective annihilation in magnetic configurations with three sources. Phs. Plasmas, :. Watson, P. G. and Craig, I. J. D. (). Analtic magnetic reconnection solutions in clindrical geometr. Solar Phs., 7:

20 Wheatland, M. S. (). Distribution of flare energies based on independent reconnecting structures. Solar Phs., :33. This preprint was prepared with the AAS L A TEX macros v5..

21 1 (a) (b) Fig. 1. Field lines of the skeleton of the background field P, when (a) κ and γ are of the same sign and a single null is present, and (b) κ and γ are opposite signs and γ > κ e, giving two nulls (with κ > in each case). The gre lines indicate the shape of the domain and the locations of the fan planes of the nulls.

22 Fig.. Evolution of a magnetic field disturbance (black red online) in rθ, with the ingoing peak forming a current in the fan plane of null 1. The dots indicate the positions of the nulls, and the dotted line is the separator, while the gre lines (white online) show some planar-projected representative field lines of P. The shading in each image, taken at times t =, 3.,.3, 11., is scaled to the maimum in that frame, and the chosen parameters are α =., β = 1, γ = 1, κ =.33 and l = 5.

23 Fig. 3. As Fig.. The ingoing peak forms a current moving towards the spine of null. The shading in each image, taken at times t =, 7.7, 1., 1.7, is scaled to the maimum in that frame, and the chosen parameters are α =., β = 1, γ = 1, κ =.5 and l = 3.

24 Fig.. As Fig.. The ingoing peak forms a current in the vicinit of the separator. The shading in each image, taken at times t =,., 11., 19.3, is scaled to the maimum in that frame, and the chosen parameters are α =., β = 1, γ = 1, κ =.3 and l = 5.

25 5.5.5 ln ( J ma ) ln (η) Fig. 5. Scaling of the peak current with resistivit and best-fit lines to the data, in the regime β > α, for the single null with curved fan current buildup (stars), and the double null fan current (diamonds), spine current (squares), and separator current (circles) runs Fan localisation ln(j );(i) ma ln(b ma );(i) ln(j ma );(ii) ln(b ma );(ii) b ln (η) Fig.. (a) Scaling of the peak current (J ma ) and maimum field at time of peak current (b ma ) with resistivit, and best fit lines, for fan currents in the presence of one null (run (i); µ J =.73, µ b =.) and two nulls (run (ii); µ J =.7, µ b =.1). (b) Tpical disturbance field profile at time of J ma for the two null field, for parameters α =, β =., γ = 1, κ =.33, l = 5.

26 Spine localisation 7 ln(j ma ) ln(b ma ) 5 3 b ln (η) Fig. 7. (a) Scaling of the peak current (J ma ) and maimum field at time of peak current (b ma ) with resistivit, and best fit lines, for the spine current, giving µ J = 1.9, µ b =.91. (b) Tpical disturbance field profile at time of J ma. Parameters are α =, β =, γ = 1, κ =.5, l = 3.

27 7 b Fig.. Tpical disturbance field profile for a separator current, during the slow localisation phase along separator, for parameters α =, β =., γ = 1, κ =.3, l = 5. Plotted on top of the bo are the positions of the null points and separator.