J. Plasma Physics (1990), vol. 44, part 2, pp. 337-360 337 Printed in Great Britain Nonlinear magnetic reconnection models with separatrix jets By E. R. PRIESTf AND L. C. LEE Geophysical Institute, University of Alaska, Fairbanks, Alaska 99775-0800, U.S.A. (Received 26 February 1990 and in revised form 29 June 1990) A new theory for fast steady-state magnetic reconnection is proposed that includes many features of recent numerical experiments. The inflow region differs from that in the classical model of Petschek (1964) and the unified linear solutions of Priest & Forbes (1986) in possessing highly curved magnetic field lines rather than ones that are almost straight. A separatrix jet of plasma is ejected from the central diffusion region along the magnetic separatrix. Two types of outflow are studied, the simplest possessing an outflow magnetic field that is potential. The other contains weak standing shock waves attached to the ends of the diffusion region and either slowing down the flow (fast-mode shock) after it crosses the separatrix jet or speeding it up (slow-mode), depending on the downstream boundary conditions. A spike of reversed current slows down the plasma that emerges rapidly from the diffusion region into the more slowly moving downstream region, and diverts most of it along the separatrix jets. In the simplest case the outflow possesses no vorticity over most of the downstream region. The models demonstrate that both upstream and downstream boundary conditions are important in determining which regime of reconnection is produced from a wide variety of possibilities. 1. Introduction In the overall picture of a steadily reconnecting configuration (figure 1), oppositely directed magnetic fields are carried in by a flow towards a central diffusion region or current sheet OY of length L. They are broken and reconnected at an X-type magnetic neutral point within the diffusion region and are then expelled outwards. The external field strength and inflow speed at a point A at a distance L e, say, from O are denoted by B e and v e, while the corresponding inflow values just outside the diffusion region are B i and v (, and the outflow values at a point C a distance L e from O are B o and v 0. A dimensionless form of the flow speed is the Alfven Mach number M = ~, (1.1) V A where v A = B/(/ip)* is the Alfven speed. A shock wave YH (normally of slowmode type) stands in the flow and turns the inflowing plasma to the right, at the same time decreasing the magnetic field strength. In the incompressible case t Permanent address: Mathematical Sciences Department, The University, St Andrews, Fife KY16 9SS, Scotland.
338 E. R. Priest and L. C. Lee Bo, v o FIGURE 1. One quadrant of a reconnecting configuration, indicating schematically the magnetic field lines (arrowed), the central current sheet OY, the separatrix YS and the shock wave YH. that is being analysed in this paper, YH becomes an Alfve'nic discontinuity of slow- or fast-mode type, but we shall still loosely refer to it as a shock. Apart from understanding the detailed structure of the plasma flow and magnetic field, the main aims of reconnection theory have been to determine how the length L of the central current sheet varies with the external Alfven Mach number M e and to find whether there is a maximum allowable (dimensionless) reconnection rate (M e ) less than unity. The two key assumptions of the classical Petschek (1964) analysis are that (i) the inflow magnetic field is current-free (or potential), and (ii) the inflow magnetic field lines are almost straight. In other words, Petschek (1964) considered a linear potential perturbation to a uniform magnetic field (-B g x) in the inflow region. The resulting value of the field at the inflow to the sheet is B t -B. (1.2) and so, by putting B t = \B e, he estimated the maximum reconnection rate M* to be M t = fi]n 7 'r,r- (1-3) 8 log LJL For the diffusion region (of width I) to be in a steady state, the inwards advection of magnetic flux must balance its outwards diffusion, and so v t = l (1.4)
Reconnection models 339 If the plasma is expelled from the diffusion region at the inflow Alfven speed v Ai, conservation of mass in the incompressible case implies Lv i = lv Ai. (1.5) Furthermore, in a steady-state two-dimensional configuration with the magnetic field frozen to the plasma, the electric field E is uniform and equal to v x B everywhere, where v x is the plasma flow speed perpendicular to the magnetic field. Thus, evaluating this at both A and the inflow to the current sheet gives v t B t = v,b t. (1.6) Finally, elimination of I and v t between (1.4)-(1.6) gives so that the maximum reconnection rate (1.3) becomes (1.7) typically 001-01. Subsequently, Sonnerup (1970) proposed a model with an extra discontinuity in the inflow region ahead of the slow shock for which the maximum reconnection rate is of order unity. Vasyliunas (1975) pointed out that Petschek's inflow region has the character of a weak fast-mode expansion, while Sonnerup's extra discontinuity is a slow-mode expansion generated at some external point. Also, Parker (1973) and Sonnerup & Priest (1975) discovered a stagnation-point flow solution with straight field lines, while Sonnerup & Wang (1987) studied the structure of the downstream region in detail, treating it as a narrow boundary layer. More recently, Priest & Forbes (1986) presented a unified theory for linear reconnection: linear in the sense that the inflow magnetic field is a linear perturbation to a uniform state (assumption (ii) above). They did this by generalizing assumption (i) and allowing the inflow configuration to possess significant pressure gradients. The result is a continuous sequence of reconnection regimes, of which Petschek's solution and a Sonnerup-like solution with a diffuse slow-mode expansion are particular members. It is the inflow boundary conditions along the boudnary AD that determine which regime of reconnection is produced. The theory was developed further by Jardine & Priest (1988a-c), who extended it to include compressibility and to allow a weakly nonlinear inflow by considering the second-order terms. Numerical experiments on steady magnetic reconnection (Biskamp 1982, 1984, 1986; Forbes & Priest 1982, 1983, 1984; Lee 1986; Lee & Fu 1986; Fu & Lee 1986; Scholer 1989) have revealed four new features that are not present in the classical models: (a) the inflow is usually not a weak fast-mode expansion with slightly converging field lines and a weakening field strength; (b) the inflow magnetic field may be highly nonlinear with a strong curvature; (c) there may be strong jets of plasma along the separatrices; (d) spikes of strong reversed current may be present near the ends of the diffusion region. It was feature (a) that stimulated Priest and Forbes to develop their unified
340 E. R. Priest and L. C. Lee theory for linear reconnection, in which the inflow may have the character of a slow-mode compression, fast-mode expansion or slow-mode expansion, depending on whether the inflow is converging or diverging. Forbes & Priest (1987) have shown in detail the way in which many features of the inflow region in numerical experiments may be understood in terms of this theory. The elements of a theory for the separatrix jets has been presented by Habbal & Tuan (1979) and Soward & Priest (1986), while Jardine & Priest (1988d) have given a physical argument for the existence of the reversed currents. It is the aim of the present paper to set up a theory which incorporates features (6), (c) and (d) for the first time; 2 gives some physical considerations to suggest why the inflow magnetic field may be highly curved and why the shocks, separatrix jets and reversed currents are present; 3 gives a basic model without shocks; 4 incorporates the shocks and analyses the resulting nonpotential downstream region. The main conclusions are summarized in the final section. 2. Physical considerations Before embarking on the mathematical analysis, let us first briefly consider the following physical questions. Why is the inflow magnetic field often highly curved? Why are the shock waves present? What causes the separatrix jet? Why do the reversed current regions exist near the ends of the diffusion region? Forbes & Priest (1987) have stressed that the number of independent boundary conditions that may be imposed in a magnetohydrodynamic situation is equal to the number of characteristics that are propagating information into the region, and so it depends on the magnitude of the inflow or outflow speed. In particular, for two-dimensional coplanar incompressible ideal MHD (the case studied here), when there is an inflow slower than the normal Alfven speed, three boundary conditions may be imposed, such as the normal flow speed v n, the tangential flow speed v t and the pressure p; for faster inflows four conditions may be specified. Consider, for example, the inflow boundary AD in figure 1, and suppose that we prescribe v x 0, v y = v e (a constant), p = p e (a constant) (2.1) along AD. Then the electric field equation E + vxb = 0 (2.2) with E uniform implies that B E > is constant, while V.B = 0 (2.3) implies that B y = B y (x). Furthermore, provided dvjdy = dbjdy = 0, the x component of the equation of motion (VxB)xB (2.4) reduces to 0 = ~(» + M, ox \ 2/i j
Reconnection models 341 so that B y has a constant value, which must be zero in order to make B y vanish on the axis of symmetry (x = 0). In other words, imposing the boundary conditions (2.1) implies that the magnetic field is uniform on the boundary and parallel to it. If instead one imposes values of v x, v y and p only slightly different from those in (2.1) then a magnetic field with a slight curvature may be produced, as studied in the models of Petschek (1986) and Priest & Forbes (1986). Imposing boundary conditions greatly different from (2.1) would in general produce highly curved magnetic field lines. In other words, the feature (6) mentioned in the previous section is a direct consequence of the form of the imposed boundary conditions. Soward & Priest (1977) have described in detail the role of characteristics in the reconnection problem. The curl of (2.2) gives (v.v)b-(b.v)v = 0, (2.5) and so for an incompressible plasma (2.4) and (2.5) may be combined to give i( g) (2.6) (2.7) where v + = v + v yj> v - = v "" v^> V A B In other words, when the total pressure p+b 2 /2/i is uniform, the quantity v_ is constant along the characteristics C_ (the streamlines for the quantity v_) and v_ is constant along the characteristics C + (the streamlines for v + ). When the total pressure is non-uniform, its gradient acts as a continuous distribution of sources for the wave equations everywhere in the flow. The C + characteristics are shown schematically in figure 2 (a), where it can be seen how they propagate information from both sides in towards the particular characteristic YS which comes from the end (Y) of the central current sheet. It is because the information contained in v_ cannot propagate upstream that an Alfvenic discontinuity may in general exist along YS. In the compressible case this becomes a slow-mode shock in Petschek's model, where the shock exists physically because the inflow is super-slow-magnetosonic, since the slow magnetosonic speed is zero for propagation across the field. We shall find in practice (see e.g. figure 46) that the streamlines are highly distorted from those indicated schematically in figure 2(a), coming in to the separatrix and leaving it at small angles. In the limit when the half-length OY of the central diffusion region vanishes, there is no flow along the separatrix itself and a double jet of plasma exists with a strong inflow towards 0 upstream of the separatrix together with an outflow downstream. When OY is non-zero, the separatrix jet consists of a similar double jet together with an extra outflow along the separatrix, which essentially represents the outflow of the plasma that flows into OY and is then diverted along the separatrix. Since there is a net mass flux along the separatrix, we shall use the term 'separatrix jet' even though it has a complex structure. In numerical experiments with very high resolution, notably the excellent ones of Biskamp (1986) and Lee & Fu (1986), much of the plasma that enters
342 E. R. Priest and L. C. Lee Fast-mode FIGURE 2. (a) Schematic representation of the magnetic field lines ( > ), streamlines ( ) and C + characteristics ( >- ). (b) Notation at a point P on the shock that is inclined at an angle 6 to the x axis: >, downstream field line (B 2 ) for a fast-mode shock; «, downstream flow streamline (v 2 ) for a fast-mode shock; >, B 2 for a slow-mode shock;, v, for a slow-mode shock. the diffusion region is not observed to flow outwards along the direction of the diffusion region, as in Petschek's model. Instead it escapes in strong jets along the separatrices, the magnetic field lines that intersect at the X-type neutral point in the core of the diffusion region (figure 3). With hindsight, this is not surprising since, if there is a partial magnetic blockage at the ends of the sheet (associated with a reversed current), outflow along the separatrix will be unimpeded magnetically since there is no magnetic force along it. When the outflow is sub-alfvenic with respect to the transverse magnetic field, at the outflow boundary DH in figure 1, two conditions may be imposed, such as v n and v t ; but when it is super-alfvenic, one boundary condition may still be prescribed, such as v n (Forbes & Priest 1987). Thus, in the nonlinear regime, there is a whole new continuum of possible solutions depending on what
Reconnection models 343 FIGURE 3. (a) Overall mass flux into and out of the diffusion region. (6) Streamlines, showing the central ones, which pass out of the ends, and the outer ones (shaded region), which are deflected along the jet and then by the shock (dashed). is imposed on the outflow boundary. In the Petschek solution and the linear unified solutions of Priest and Forbes, there is no such freedom since the outflow speed is the upstream Alfven speed. However, in the weakly nonlinear solutions (Jardine & Priest 19886) the freedom to impose the outflow speed at next order is present. Reversed currents near the ends of the diffusion region have been observed in many numerical experiments (Podgorny & Syrovatsky 1981; Forbes & Priest 1982; Forbes 1986; Biskamp 1986; Lee & Fu 1986). In an extreme case where the outflow speed is zero, so that there is a magnetic obstacle downstream, Forbes (1986) performed a detailed analysis of the numerical data and found that the reversed current consists of a fast-mode shock, which slows down the plasma flowing out of the diffusion region, followed by a deflection current, which deflects it sideways. The reversed currents seem to be produced because of a mismatch (Jardine & Priest 1988a!) between the prescribed (often slow) speed at the outflow boundary and the Alfven speed with which the plasma is being squeezed out of the diffusion region. The Lorentz force associated with the reversed current slows down the plasma and may deflect it along the separatrices. Thus one would expect the combination of a long diffusion region (into which a lot of mass is flowing) and a slow imposed outflow speed to produce large reversed currents and strong separatrix jets. The magnitudes of the reversed currents and separatrix-jet mass flux may be expected to increase as the diffusion-region length increases and the outflow speed is reduced. 3. Basic model Our aim is to construct a reconnection model with highly curved (i.e. nonlinear) inflow magnetic fields, separatrix jets and reversed currents. In particular, it is of interest to determine the relation L = L(M e ) between the diffusion-region half-length L and the external inflow Alfven Mach number M e. We need to calculate in turn the magnetic field upstream of the shock, the
344 E. R. Priest and L. C. Lee upstream flow, the diffusion region parameters, the shock relations, and the downstream flow and field. In the process, the values of the external inflow. speed v e, field B e at a distance L e and the outflow speed v 0 will be prescribed. 3.1. Upstream magnetic and velocity fields The basic equations to be solved for a steady-state two-dimensional flow (v x (x,y),v y (x,y)), magnetic field (B x (x,y),b y (x,y)) and pressure p(x,y) are the electric field equation (2.2) with E = Ei. uniform, the magnetic field constraint (2.3), the equation of motion (2.4) with density p uniform, and the mass- continuity equation V.v = 0. (3.1) For simplicity, we shall assume that the inflow is highly sub-alfvenic and the plasma beta much less than unity, so that (2.4) reduces to VxB = 0, (3.2) which, together with (2.3), implies that the magnetic field is potential everywhere except at the current sheet of half-length L. The solution for the upstream magnetic field that we shall adopt outside the current sheet is j (3.3) where Z x + iy is a complex variable and there is a cut in the complex plane from Z = L to Z = L. B t is the value of B x at the origin just above the current sheet, i.e. at the inflow to the diffusion region. Equating real and imaginary parts of (3.3) after squaring yields expressions for B x, B y and B: B\ = ^{-{x 2 -L 2 -y*) + [{x 2 -L 2 -y 2 ) 2 + ±x 2 y 2 t, (3.4) ^ % (3-5) B 2 = [y* + 2y 2 (x 2 +L i ) + (x 2 -L 2 ) a ]i (3.6) Evaluating (3.3) at the point A(0,L e ) in figure 1 gives an expression for B t in terms of the external field B e : Furthermore, the flux function A such that da B * = -dy-' can be found from (3.4) and (3.5) to be B y = da -c% where K*+ >'+ te+^j (38) \ \ lx]*, X = x 2 -y 2 -L\ Y = 2xy.
Reconnection models 345 Having obtained the magnetic field from (3.4) and (3.5) with B i given by (3.7), we must next calculate the velocity from (2.2) and (3.1) with E = -v e B e. (3.9) First of all, we can evalute (2.2) at the points 0 and C (where the velocity and magnetic field are perpendicular) to give v ( in terms of B i and v 0 in terms of B o as v o = V -^. (3.11) Indeed, in general (2.2) implies that the flow speed perpendicular to the magnetic field is with B given by (3.6), and then (3.1) determines the flow speed parallel to the magnetic field. A useful way to determine v is to write it in terms of a stream function T such that so as to satisfy (3.1) automatically. Then (2.2) reduces to x dy' y dx' with solution Y = v e B e, (3.14) where the integration is along a magnetic field line. For field lines above the separatrix YS in figure 1, one integrates from a point on the y axis where *F = 0 since the y axis is a streamline (*F = 0). For field lines below the separatrix, one integrates from a point on the lower boundary (YC) where *F = 0. The separatrix itself becomes a vortex-current sheet, whose structure Soward & Priest (1986) have discussed. Across the sheet, the flow velocity abruptly changes direction, but in practice, just the like the central diffusion region, such a sheet is broadened to a finite thickness by viscosity and magnetic diffusion. 3.2. The diffusion region On y = 0 just above the current sheet the magnetic field strength is and so the inflow speed into the sheet is B x = l -(L 2 x 2 )*, (3.15) Vy = v J$JL = z i L 2i, (3-16) where v { = v e B e /B i. Thus the mass flux into the half-sheet from above is fjo pv y dx = \nplv i. (3.17)
346 E. R. Priest and, L. C. Lee (a) L = 005 L,. L = 0-2 L e xll c 1 0 x/z.,, FIGURE 4. For caption see facing page. In order of magnitude, if a fraction / of the incoming mass flows out of the end of the sheet at the Alfven speed v M and a fraction 1 / goes out along the separatrix then mass continuity implies = v M l, (3.18) where v t = v e B e /B f and v Ai = v Ae BJB e. Also, steady diffusion implies ««= -. (3.19) and so elimination of I between (3.18) and (3.19) gives < - 2 B* t (3.20) in terms of the external magnetic Reynolds number R me and Alfven Mach number M e, where BJB e is given by (3.7). If L/L e is known then (3.20) determines the fraction/, whereas if/is found from some as yet unknown properties of the current sheet and the reversed
Reconnection models 347 L = 005., (b) L = 0-2 L e L = 0-4 L e L = 0-6 L e FIGURE 4. (a) Magnetic field lines and (b) streamlines for shockless reconnection. Four values of the current sheet half-length L are shown (0'05L e, 0-2L e, 0-4Xi e, 0-6L e ), with the same external field B e and flow v e. current region then (3.20) determines L/L e. We shall use (3.20) in the following sections, but it is also of interest to note that the variation of the width of the sheet is given by _V _V (L 2 -x 2 )* l{x) = = -, v y v t L and so the sheet narrows towards its end and l(x) decreases as the inflow speed increases. At the same time, the variation in the flow along the sheet may be found from mass continuity, v x l(x) = Vydx = V ( Lsin 1, so that Thus the flow speed increases with distance. In practice, there will be a cut-off before x reaches L, and therefore the sheet will possess a minimum width and a maximum outflow speed. Such a narrowing of the sheet to a neck may be seen in some of the numerical experiments.
348 E. R. Priest and L. C. Lee 3.3. Shockless reconnection Consider first for simplicity the case when there is no shock wave present or it is very weak, so that the magnetic field is given by (3.3) everywhere outside the current sheet. The stream function follows from (3.14), where the integral for streamlines below the separatrix is performed by integrating from a point on YC where *F = 0, so that one has here given up the ability to prescribe the stream function on SC. Plots of magnetic field lines and streamlines are shown in figure 4 for four particular values of L. The effect of increasing B e is simply to introduce more field lines without changing the direction of the magnetic field. Also, increasing v e does not change the shapes of the streamlines, but simply scales up the magnitude of the flow everywhere. In order to be able to follow the streamlines through the diffusion region and along the separatrix, the stream function has been smoothed over small regions of width 8A = 001B e L e at the diffusion region and separatrix, where (3.14) with (3.3) leads to discontinuities. The outflow field strength B o is given by evaluating (3.5) at (L e, 0) with B t given by (3.7), and so B The outflow speed v 0 follows from (3.11) as Also, by substituting forb ( /B e from (3.7), the diffusion region relation becomes If one insists that none of the inflow to the diffusion region flows along the jet and so puts /= 1 then (3.23) determines L/L e to be a monotonically increasing function of M e, with the maximum reconnection rate M* being given by *?=-A and so having a Sweet-Parker scaling. If instead one allows a separatrix jet and imposes the value of v 0 then (3.22) determines the sheet half-length L/L e as (3.24) Thus, as v e increases from zero (with v 0 held constant) up to a maximum value of v 0, so the sheet half-length L decreases from L e to zero (figure 5a). (For example, if v 0 is held equal to the local Alfven speed v Ao = v Ae B 0 /B e then L?/L\ = (1 M e )/(i +M e ), and L decreases from L e to zero as M e increases from 0 to 1.) At the same time, the fraction / of plasma-diffusion-region plasma that does not flow out along the separatrix jet is given from (3.23) and (3.24) by me v\. v Ae irtiv\/vl
Reconnection models 349 I0 2 v 10 5? ' 10-' io- 2 0-5 v,/v o 10 10 FIGURE 5. For shockless reconnection with' a constant outflow speed v 0, the variation with external inflow speed v t of (a) diffusion-region length L; (6) the fraction / of diffusion region plasma not escaping along the separatrix jet; (c) diffusion-region width; (d) magnetic-energy conversion, where W t is the external inflow of magnetic energy and W o the outflow. as plotted in figure 5(b). Thus, unless v e is extremely small, the presence of the factor R me (p 1) means that/^ 1 and most of the plasma flows out along the jet. For example, for v e «v Ae,f «R^e. The sheet width is given from (3.7), (3.10), (3.19) and (3.22) as As shown in figure 5(c), it decreases from rj/(2*v e ) to zero as v e increases from zero to v 0. The inflow of magnetic energy through AD in figure 1 is W e = v e B e [ L 'B x (x,l e )dx, Jo where B x is given by (3.4), B { by (3.7) and L by (3.24). Also, the outflow of magnetic energy through DC is f J
350 E. R. Priest and L. C. Lee 10 0-4- ^ ^ 0 2^^=^ L/L e = 005 0-6 -l 0? 0-5 - - 0 1 i 0-5 xll e 10 2-0 10 04 ^ = = : 0-2 - - (c) i UL e = 0-6 i 0-5 005^ "" - 20 0 - id) I LIL e = / 0-61 J - -20 I 10 0-5 0-4 / 0-6 / 0-4 yll, FIGURE 6. (a, b) Variation with distance along the inflow boundary A of (a) the flow speed v y normal to the magnetic field and (b) the flow speed t^ parallel to the field, (c, d) The variation with distance along the outflow boundary of (c) v ± and (d) v f. where B y is given by (3.5). The rate of magnetic-energy conversion as a fraction of the inflow of magnetic energy is therefore (W e W o )/W o and is plotted as a function oiv e /v 0 in figure 5(d). It declines from a maximum value of about 06 at v e = 0 to zero when v e = v 0. For given values of v 0 and B e, the absolute rate of magnetic-energy conversion is proportional to v e (W e W o ) and so increases with v e up to a maximum value and then declines to zero as v e approaches v 0. The way in which the flow velocity components vary with distance along the inflow and outflow boundaries is shown in figure 6. Generally, the flow speed v ± perpendicular to the magnetic field decreases with distance from the axes. It increases with L on both the inflow and outflow boundaries. The flow v^ along the field on the inflow increases in magnitude with distance from the axis and decreases with L. On the outflow boundary the parallel flow speed increases up to the separatrix jet and increases with L. 4. Reconnection with shocks The characteristic W+A = constant which comes from the end (Y) of the current sheet may support an Alfvenic discontinuity (a shock in the compressible case), since a disturbance generated at Y may propagate out along \ -1 /- 305" 0-2 10
Reconnection models 351 0 x/l e 1 0 xll, 1 FIGURE 7. Characteristics * + A = constant for L = 0-2L e, with the inflow Alfven Mach numberitf, = 0-05 (a), 0-2 (b), 0-4 (c) and 05 (d). it (i.e. YH). The position of the characteristic depends on the magnitude of the flow (figure 7), since a very slow flow produces characteristics that are inclined only slightly to the field lines (except the separatrix), and, as the flow speed increases, so the angle of inclination increases and the spacing between the separatrix and the shock widens. For the upstream region ahead of the shock, the magnetic field and flow are still given by (3.3) and (3.14), but the shock modifies the region downstream of the shock. Having found the shock position, we therefore need to apply the shock relations and then determine the downstream flow, bearing in mind that we are free to specify one condition at the outflow boundary HC (see 2). 4.1. Shock relations In our incompressible regime, the condition that W+A = constant, or equivalently that the normal flow v n be related to the normal Alfven speed v An ^ v +v = 0, (4-1) determines the shape and position of the shock YS at each point P along its length starting from the end Y of the current sheet. In other words, it determines the angle 6 of inclination of the shock at P. 12 PLA 44
352 E. R. Priest and L. C. Lee L = 005 L,. L = 0-2 L e x/l, FIGURE 8. For caption see facing page. At each point on the shock front, we shall find only four independent conditions relating the five downstream variables Vj, B x and j) x (figure 26). Such an indeterminancy is expected, since we are able to prescribe one outflow variable. The conservation relations are as follows. Conservation of mass (4.2) gives v x2 sin 6 v y2 cos 0 = v xl sin 6 v yl cos 6 and determines one of v x2 and v y2 once the other is prescribed. Conservation of magnetic flux becomes B x2 sin 6 B y2 cos 6 = B xl sind B yl cos6 Conservation of electric field (E = v x B) may be written as 'n\ v u R R = v R v R x2 JJ y2 v y2 ±J x2 "xl dj yl v y\ ±J x\'> and so (4.5) and (4.6) together determine B x2 and B y2. (4.3) (4.4) (4.5) (4.6)
// Reconnection models L = 005 L,. L = 0-2 L e j 1 1 I 1 / I / / / 1 //// oa I. _ 353 0 xll e 1 0 xll e 1 FIGURE 8. (a) Magnetic field lines and (6) streamlines for reconnection with a shock ( ). Four values of the diffusion region-half length L are shown (0-05L,, 0-2L e, 0-4L e, 0-6L e ), with the same external field B, and flow v. = 0'2«,,. Finally, total pressure balance B\ (4.7) determines the downstream pressure. It may be noted for interest that the usual tangential momentum condition (that the jump in tangential flow speed equal minus the jump in tangential Alfven speed) is not independent of the above conditions, but follows from (4.1), (4.2), (4.4) and (4.5). 4.2. Downstream Region In the downstream region, one needs to solve (2.2) and (2.4) subject to the boundary conditions that the variables be given on the downstream side of the shock by the shock relations together with one condition on the outflow boundary and symmetry conditions on the x axis. With v = V x CF4), B = V x (Az) (4.8) and E = v e B e, (2.2) may be written as (v.v)a=-v e B e. (4.9) 12-2
354 E. R. Priest and L. C. Lee Also, the curl of (2.4) becomes /o(v.v)w = (B.V)j (4.10) in terms of the vorticity u> = V 2x F and current j = We shall consider two models for the downstream flow. In the first, we shall for simplicity assume that the flow is locally super-alfvenic, so that (4.10) reduces to (v.v)w = 0. (4.11) This is a good approximation away from the reversed-current spike and the outflow boundary, where the field is weak and the flow fast. The shockless model of 3 gives an outflow speed (v 0 ) and outflow field strength B o of so that the outflow Alfven Mach number is L 2 e+l* M n =- Ll-W "e Thus, for example, if-m e = 05 then M o > 2 and so (4-11) is a good approximation even near the outflow boundary when L/L e > 3~* «0-6. One way to solve (4.9) and (4.11) is to assume that the shock switches off the y component of the flow velocity so that in the downstream region v = v x (y) x and (o = (o(y). Equation (4.11) is then satisfied identically and (4.9) determines the distortion of the magnetic field produced by the shear flow. However, the resulting vorticity has a singularity everywhere along the x axis and so we shall not adopt this approach. The simplest solution is to assume w = 0 just downstream of the shock, so that (4.11) implies that the whole downstream region is vorticity-free and we have V 2l F = 0. (4.12) The boundary conditions to solve this are as follows. First, along the shock YS, *F f(s), where/(s) is determined from the shock condition (4.2) to be the same as the stream function just upstream. Secondly, along the x axis (YC), T = 0, since it is the same streamline that comes in along the y axis. Finally, we may impose any functional form *F(y) along the outflow boundary that joins *F(0) = 0 at C to the value Y(h) at the point H where the shock reaches the boundary. For simplicity, we shall adopt a linear function, corresponding to a uniform outflow. The resulting field lines and flow that follow from solving (4.11) and (4.8) are shown in figure 8 for (approximately) the same values of the external field B e and flow v e, but with different sheet half-lengths L. The shock wave is very weak indeed and is of slow-mode expansive type, increasing the flow speed very slightly and decreasing the field strength. This is because in the shockless case the flow is faster on the x-axis, whereas we are imposing a uniform outflow, and so the shock needs to bend the streamlines downward. If we had instead imposed a much weaker outflow near the x axis then a slow-mode compressive shock would have been required to bend the streamlines upward. It should be noted that in our incompressible model here the fast-mode wave speed is infinite, and so standing fast-mode shocks are impossible. Indeed, the
Reconnection models (a) 200 LIL e = 0-6 Jft ) 1 005 / 100-1 / A \ 355 FIGURE 9. (a) Variation with x of the downstream current density along the z-axis. (b) Relation between L/L t and vjv o for different values of external inflow speed v t. L is the sheet length and v 0 the outflow speed. The shockless solution is shown dashed. discontinuities that we have loosely called 'shocks' are really Alfvenic discontinuities of slow-mode type, across which there is a change of pressure (unlike an Alfven wave) but no change in density. When compressibility is allowed, the compressible discontinuity becomes a true slow-mode shock, while the expansive discontinuity becomes a narrow slow-mode expansion fan.
356 E. R. Priest and L. C. Lee L = 005 L f (a) L = 0-2 L e xll e 1 0 xll, 1 FIGURE 10. For caption see facing page. An important feature is the reversal in curvature downstream produced by the fast stream of plasma near the x axis emerging into the much slower stream at the outflow boundary. The variation with x of the resulting strong reversed current along the x axis is shown in figure 9 (a). Also, the variation with vjv o of L/L e for different values of v e is given in figure 9(fc) in comparison with the shockless solution of figure 5 (a). The outflow speed v 0 at C is faster than in the shockless case because of the fast shock and because the uniform outflow boundary condition redistributes the mass flux by speeding up the flow near C and slowing it down near S. An increase in sheet length increases v 0, as in the shockless case. The above model for the downstream flow is reasonable for much of the downstream region, and it shows how the region of reversed curvature is created. However, it is poor near the outflow boundary when a sub-alfvenic outflow is being imposed and it also fails in the reversed-current spike by overestimating its intensity. The resulting magnetic force will tend to slow down the outflowing stream more efficiently and so prevent the shear in velocity building up the magnetic gradient so much.
L = 005 L e Reconnection nodels (h) L = 0-2 L e 357 FIGURE 10. Reconnection with a shock and with the full downstream solution showing (a) magnetic field lines, (6) streamlines and (c) variation with x of the downstream current density along the x axis.
358 E. R. Priest and L. C. Lee We have therefore considered a second model by solving the following incompressible MHD equations ^, (4.13) > (4.14) to obtain the steady-state solution in the outflow region downstream of the shock. The boundary conditions imposed along the outflow boundary (x = 1) are that ^(l,z) is a linear function of z to give a uniform outflow as before, dw/dx = 0, and d 2 A/dx 2 = 0. The boundary conditions along the outflow axis (y = 0) are conditions of symmetry, namely i/r = 0, w = 0, da/dy = 0. The boundary conditions imposed along the shock front YH are that iff and A have the same values as just upstream of the shock, while the vorticity w n+1 along the shock front YH at time step n+ 1 is obtained from the stream function i/r n at time step n: w n+1 = V 2^n. The initial values for &>, A and i/r are taken to be those in the case without the shock. The values of \jr along the outflow (x = 1) are then slowly changed to the imposed value within 20 time steps. A simple forward-differencing scheme is used for the time derivatives and a centreddifferencing scheme is used for the advective terms. Finite values of rj and v (x, 0*01) are used for the resistivity and viscosity in (4.13) and (4.14) to ensure the stability of the numerical scheme. The results for the second model are shown in figure 10. The relation between L/L e and vjv o (figure 5a) is unaffected, since the new model does not change the form of i/r and therefore v y on the outflow boundary, but only modifies it in the interior of the downstream region. Figures 10 (a, 6) indicate that the plasma flows along the x axis are partially diverted upward to the region just downstream of the shock front by the magnetic force associated with the reversed current. It is also found that the magnetic field lines immediately downstream of the shock YH are greatly bent by the diverted plasma flows. The plasma flows turn downward near the outflow boundary (x = 1) owing to the imposed boundary condition along x = 1. Figure 10 (c) shows that the reversed currents along the x axis are reduced to about one-tenth of those in the first model (figure 9 a). 5. Conclusion Since the development of the classical models of fast reconnection (Petschek 1964; Sonnerup 1970), we have entered an exciting era of numerical experimentation, which has shown that some features of the previous models are correct: for example, Scholer (1989) has shown (in contrast with Biskamp 1986) that at high magnetic Reynolds numbers fast Petschek reconnection can indeed occur when the appropriate boundary conditions are imposed and the central current sheet possesses an anomalous resistivity. Also, Forbes & Priest (1983), Biskamp (1986) and Fu & Lee (1986) have found clear evidence of the slow-mode shock waves. However, the experiments have also revealed new features that are not explained by the classical models and that suggest that those models need to be generalized.
Reconnection models 359 Numerical experiments can only sample part of a parameter regime and employ a particular set of boundary conditions, and so scaling laws deduced from numerical results may not hold in different parameter regimes or with different boundary conditions (Biskamp 1986). The role of analytical theory is to help understand numerical experiments (which have their own reality) and to help understand the physics of the reconnection phenomenon and the effects of the boundary conditions, although a theorist must of course adopt simplifying assumptions in order to make analytical progress. Clearly, a combined approach of using complementary analytical and numerical tools is essential (Forbes & Priest 1987). The first new feature of numerical experiments is that the inflow region may be quite different in nature from that envisaged by Petschek. This has inspired the generalization of Petschek's model by Priest & Forbes (1986), which accounts for many observed features of the inflow regions. The aim of the present paper has been to show that there is an even greater richness and diversity of solution than found in the unified linear Priesfr-Forbes models or than glimpsed in the few numerical experiments. In particular, one range of new solutions may be obtained by imposing different inflow boundary conditions with a substantial tangential flow and a far-from-uniform normal flow and pressure, which can make the inflow magnetic field highly curved rather than only slightly curved (as in the Petschek and Priest-Forbes models). The resulting inflow region contains strong jets of plasma along the separatrices, which take most of the plasma out of central current sheet. They occur because plasma can easily escape along the separatrix field lines and because plasma on magnetic field lines that are approaching a separatrix (and moving away from it) suffers a large expansion (and contraction). The inflow boundary conditions determine the locations of both the separatrices and also the Shockwaves that are attached to the ends of the central sheet. Another range of solutions is obtained by allowing different outflow boundary conditions. In general, they lead to the presence of reversed-current spikes in the outflow region, which produce a Lorentz force that slows down the flows exitting from the current sheet and diverts part of them along the separatrices. The details of the downstream flow (and both the strength and type of the shock waves) are influenced by the imposed condition on the outflow boundary. Generally, in the cases that we have studied the shocks are very weak. When the outflow is uniform, they are weak fast-mode shocks, but, if the flow is slow enough on the axis, slow-mode shocks would be expected. As the inflow speed v e at larger distances increases from zero through a value somewhat less than 0-75w o to a maximum value somewhat less than v 0 (the outflow speed), so the diffusion region length decreases from L e through 0-5L e to zero and the fraction of magnetic energy converted decreases from 0-6 through 0-2 to zero, while the amount of magnetic energy released increases from zero to a maximum and then decreases to zero. In future, it is planned to compare the models in detail with new numerical experiments designed to allow a greater variety of inflow and outflow boundary conditions. It is also intended to improve and generalize the models by attempting a better analysis of the separatrix jets and by seeking different inflow solutions. For example, especially when compressibility is included, the roles of the pressure boundary condition and of the effects of local pressure
360 E. R. Priest and L. C. Lee gradients and magnetic forces in creating the separatrix jets need to be studied (Fu & Lee 1986). This work has been supported by NSF Grant ATM 88-20992 and DOE Grant DE-FG06-86ER. E.R.P. is grateful to the Geophysical Research Institute and the U.K. Science and Engineering Research Council for financial support and to Li-Her Lee for careful and efficient help with the computations. He is also most thankful to Li-Her Lee, Lou Lee and Syun Akasofu for warm hospitality during his stay in Fairbanks. REFERENCES BISKAMP, D. 1982 Z. Naturforsch. 37a, 840. BISKAMP, D. 1984 Phys. Lett. 105 A, 124. BISKAMP, D. 1986 Phys. Fluids, 29, 1520. FOBBES, T. G. 1986 Astrophys. J. 305, 553. FORBES, T. G. & PRIEST, E. R. 1982 Solar Phys. 81, 303. FORBES, T. G. & PBIEST, E. R. 1983 Solar Phys. 84, 169. FORBES, T. G. & PRIEST, E. R. 1984 Solar Phys. 94, 315. FORBES, T. G. & PRIEST, E. R. 1987 Rev. Oeophys. 25, 1583. Fu, Z. F. & LEE, L. C. 1986 J. Oeophys. Res. 91, 13373. HABBAL, S. R. & TUAN, T. F. 1979 J. Plasma Phys. 21, 85. JARDINE, M. & PRIEST, E. R. 1988a J. Plasma Phys. 40, 143. JARDINE, M. & PRIEST, E. R. 19886 J. Plasma Phys. 40, 505. JARDINE, M. & PRIEST, E. R. 1988C Proceedings of Workshop on Reconnection in Space Plasmas, ESA SP-285, Vol. II, p. 45. JARDINE, M. & PRIEST, E. R. 1988<Z Geophys. Astrophys. Fluid Dyn. 42, 163. LEE, L. C. 1986 Solar Wind-Magnetospheric Coupling (ed. Y. Kamide & J. A. Slavin), p. 297. Terra Scientific. LEE, L. C. & Fu, Z. F. 1986 J. Geophys. Res. 91, 6807. PARKER, E. N. 1973 J. Plasma Phys. 9, 49. PETSCHEK, H. E. 1964 Proceedings of AAS-NASA Symposium on Physics of Solar Flares; NASA SP-50, p. 425. PODGORNY, A. I. & SYROVATSKY, S. I. 1981 Fiz. Plazmy USSR, 7, 1055. PRIEST, E. R. & FORBES, T. G. 1986 J. Geophys. Res. 91, 5579. SCHOLER, M. 1989 J. Geophys. Res. 94, 8805. SONNERUP, B. U. 0. 1970 J. Plasma Phys. 4, 161. SONNERUP, B. U. 0. & PRIEST, E. R. 1975 J. Plasma Phys. 14, 283. SONNERUP, B. U. O. & WANG, D. J. 1987 J. Geophys. Res. 92, 8621. SOWARD, A. M. & PRIEST, E. R. 1977 Phil. Trans. R. Soc. Land. A 284, 369. SOWARD, A. M. & PRIEST, E. R. 1986 J. Plasma Phys. 35, 333. VASYLIUNAS, V. M. 1975 Rev. Geophys. 13, 303.