SELF-CONSISTENT DEVELOPMENT OF FAST MAGNETIC RECONNECTION WITH ANOMALOUS PLASMA RESISTIVITY

Transcription

1 Plasma Phbsics and Controlled Fusion. Vol. 26. ho. IZB. pp to Pnnted in Great Britain S3.W~.OO Q 1984 Instilure of Physics and Pergamon Press Lid. SELF-CONSISTENT DEVELOPMENT OF FAST MAGNETIC RECONNECTION WITH ANOMALOUS PLASMA RESISTIVITY M. UGAI Department of Electrical Engineering, Ehime University, Matsuyama 790, Japan (Receiced 27 February 1984; and in rerised form 20 July 1984) Abstract-The generic feature of the fast reconnection process that develops self-consistenti> with the growth of anomalous resistivity is numerically studied. It is demonstrated that the anomalous resistivity due to current-driven instability is quite favorable to the occurrence of the fast reconnection mechanism. and vice versa, which leads to the rapid magnetic energy conversion. As the growth of the anomalous resistivity becomes saturated, a magnetic island is suddenly formed and is attached to slow shocks standing in the extended region. In this respect, we argue that the Petschek's configuration involving a large-scale X-type magnetic field is unstable against resistive tearing mode in high-temperature plasmas ahere the anomalous resistivity is most effective in providing an electrical resistivity. 1. INTRODUCTION MAGNETIC field reconnection has been recognized as playing an important role in the large-scale conversion of magnetic energy observed in solar flares and geomagnetic substorms (PRIEST, 1983; NISHIDA, 1978). In a plasma of very large electrical con- ductivity magnetohydrodynamic (MHD) waves should be much more effective in releasing the stored magnetic energy than Ohmic heating. A steady configuration of magaetic reconnection involving standing slow shocks was first proposed by PETSCHEK (1964) and has been extensively studied (SOWARD and PRIEST, 1982). The reconnection process that gives rise to a rapid magnetic energy conversion under significant influence of MHD waves may hence be called the fast magnetic reconnection. UGAI and TSUDA (1977) recognized the importance of local onset of anomalous resistivity for the development of fast reconnection and first demonstrated numerically the evolution and the establishment of fast reconnection. Later, SATO and HAYASHI (1979) assumed a continuous local injection of magnetic flux to cause a local resistivity enhancement and obtained a similar evolution of fast reconnection. In high-temperature plasmas, which are most important in actual plasma systems (both space and laboratory), electrical resistivity due to Couiomb coilision is extremely small, so that anomalous resistivity should play a crucial role in the effective proceeding of magnetic reconnection. Therefore, it must be a problem of high current interest to demonstrate a possible evolution of the fast reconnection mechanism by selfconsistent growth of anomalous resistivity. The theme of the present paper is to study this problem by computer simulation. Since the general form of anomalous resistivity consistent with current-driven instability has not fully been understood to date, computations will be done for some typical functional forms q( VD) of anomalous resistivity, each of which is assumed to increase with the relative electron-ion drift velocity VD when some threshold value is exceeded. By this simulation model, the generic feature of the self-consistent development of fast reconnection that does not depend on the detailed form of q( VD) will be derived and be argued in detail; in fact, the generic feature should be most applicable to actual plasma processes. We also 1549

2 1550 M. UGAI examine how resistive tearing mode instability influences the fast reconnection development, since the tearing mode plays an important role in distinct plasma phenomena such as formation of magnetic island. 2. SIMULATION MODEL A. Initial-boundary-value problem As basic equations, we employ the compressible MHD equations Dp/Dt = -~V.U, pdu/dt = -VP + J x B, ab/& - V x (U x B) = - V x (U J), pde/dt = -?V.U + qj2, V.B = 0, J = /&lv x B, where D/Dt =?/at + u.v, and e is the internal energy per unit mass. We have employed Ohm s law, E + U x B = 119, and the gas law, P = (7 - l)pe, is assumed (the specific heat ratio 3 = 2 will be specified throughout). These equations are restricted to two dimensions so that all the quantities depend on x and y co-ordinates only and that the electric field E = 10, 0, E(x, y)] and the current density J = [Q, 0, J(x, y)] have z components only. The basic equations are further transformed to a conservation form, and the two-step Lax-Wendroff method is employed for numerical ca!culation (UGAI and TSUDA, 1977). In actual computations, ali tne quantities are replaced by the ones normalized on the basis of the initial field configuration that will soon be shown. Distances x and y, etc., are normalized by a, the half-width of the current sheet; magnetic field B and fluid velocity U by BO and VAO( = Bo/(~o~o) ~), the AlfvCn velocity measured ir? the antiparallel-field region; therefore, time t is normalized by the Alfven travelling time cx/vao. Also, electric field E, current density J, energy density, and magnetic flux are respectively normalized by VAOBO, Bo~(wo), B;/(~PO), and JBO. In order to see the overall dynamical process of magnetic reconnection, we assume an isolated current-sheet system as an initial static field configuration (UGAI, 1981, 1982). The associated magnetic field B = (B,(y), 0) is given by: B,(y) = y for lyl < 1 (current-sheet region), B,(y) = f 1 for 1 < ly I d 2.4 (antiparallel-field region), and B,(y) linearly decreases to zero for 2.4 < /yj (return-current region). The gas pressure P(y) is initially assumed so that the pressure balance condition, P(y) + Bx(y)2 = 1 + Po, is satisfied, where Po is the ratio of the gas pressure to the magnetic pressure in the antiparallel-field region (Po = 0.1 is specified in the present simulation). Also, plasma density p = 1 and fluid velocity U = 0 are initially assumed. Numerical results will be obtained in a square box, 1 x 1 < 2 and 1 y I 6 3. Considering that the condition for the effective magnetic energy conversion to take place is such that the plasma is allowed to be freely ejected from the system (UGAIL, 1982), we assume the boundaries enclosing the computational region to be free boundaries: i.e. the first derivative of all the variables (except for B,) with respect to x vanishes OR the side boundaries (x = & 2), and that of all the variables (except for BY) with respect

3 Self-consistent development of fast magnetic reconnection 1551 to y vanishes on the top and bottom boundaries (y = k3); B, on x = +2 and By on y = i-3 are determined by the solenoidal condition (UGAI and TSUDA, 1977). Also, the conventional symmetry conditions are imposed on the x 2nd y axes. If the electrical resistivity is zero everywhere, any plasma flow cannot grow, and the initial field configuration does not change at all. In order to disturb the initial configuration, we impose such a finite resistivity locally near the origin in the currentsheet region that becomes reduced to zero in a few normalized times in accordance with the decrease in the current density near the origin. As was shown in detail in UGAI (1982), such a resistive disturbance causes a local magnetic reconnection, leading to a local formation of X-type field configuration near the origin. The plasma behaviour near an X-type neutral point in a plasma of infinite conductivity was first discussed by DUNGEY (1953) and was extensively studied (IMSHENNIK and SYXOVATSKI, 1967). It was suggested that the two branches of the separatrixes tend to approach each other with the current being concentrated near the neutral point. A similar plasma process may be expected to proceed in the present local X field configuration. UGAI (1982, 1983) in fact found that, if the disturbance magnitude is not so small, global plasma flow grows on a significant time scale so as to flatten the local X configuration and hence to cause the thinning of the effective width of the current sheet. The resulting increase in the current density may strongly suggest that anomalous resistivity due to current-driven instability should be likely to take place. B. Anomalous resistirjity model It has been well recognized that current-driven instabilities, such as lower-hybriddrift instability, and ion-sound and electron two-stream instabilities, give rise to anomalously high electrical resistivity (DRAKE et ai., 1981; SMITH and PRIEST, 1372; CORONITI and EVIATAR, 1977). If the relative electron-ion drift velocity becomes sufficiently large (compared with the electron thermal velocity), current-driven instability can grow and become nonlinearly saturated in a few tens growth times with the drift energy being dissipated into plasma heating (e.g. BUNEMAN, 1959). One may hence estimate the effective frequency of collision between electrons and ions through electrostatic fields in terms of the maximum growth rate, but it is very difficult to obtain the general description of the anomalous resistivity. In fact, this problem involves very complicated nonlinear interaction between particles and waves, 2nd the generic form of the anomalous resistivity has not convincingly been shown to date. Certainly, it seems difficuit to introduce the anomalous resistivity in a unified manner into the present MHD framework, since both the threshold value and the maximum growth rate of the current-driven instability depend on electron and ion temperatures. However, it may be quite relevant to assume the anomalous resistivity q to be an increasing function of the relative electron-ion drift velocity VD. A most important problem must be to derive the generic feature of the self-consistent reconnection development that should be independent of the detailed form of ~(VD). For this purpose, the present study will examine the following three typical forms of ~(VD):(A). q(r, t) = 0.05exp(--[(VD(r, t)- 11)/412} for VD < 11, and q = 0.05 for VD > 11; (B). ~(r, t) = kr(vd(r, t) - 4) for VD 2 4, and q = 0 for VD < 4; (e). q(r, t) = kr(vd(r, t) - 4)2 for VD 2 4, and q = 0 for VD < 4. Here, the drift velocity VD(r, t) is given by I J(r, t)/p(r, t) 1 at any time t and at any spatial point r. For the case (A), the resistivity is significantly enhanced when VD 27 and becomes constant

4 1552 M. UCAI for VD > 11 even if the drift velocity becomes larger. For the case (B), the resistivity linearly increases with VD if a threshold value, assumed to be 4, is exceeded. Here, the two cases kr = 6 x and 1.5 x will be shown. For the case (C), the resistivity becomes enhanced much more rapidly compared with the preceding two cases when VD becomes sufficiently large. In this case, numerical results for the cases kr = 1.0 x and 0.5 x will be demonstrated. For each case, the threshold value for the occurrence of anomalous resistivity is assumed to be sufficiently large (note that at time t = 0 VD = 1 in the current-sheet region, IyI < 1). Each resistivity model will be imposed at time t = 4 when the finite resistivity initially imposed as a disturbance has disappeared completely and there is no effective resistivity everywhere. It should be noted that the resistivity q(r, t) has been normalized by (poctv~o), so that q-l defines the conventional magnetic Reynolds number Rm(r, t), which may now be a function of r and t. 3. RESULTS A. General remarks In general, large-scale X-type field configuration is fundamental for the fast reconnection to proceed effectively (or for slow shocks to stand steadily), so that the sufficiently large By field component should be retained in the overall configuration. The y component of the third equation of (1) on the x axis (i.e. at y = 0 where B, = up = 0 by symmetry) readily leads to The first term on the right-hand side indicates the magnetic field diffusion due to finite resistivity, and the second term the magnetic field convection by which the B, field is ejected away from near an X-type neutral point. Therefore, in order for a large-scale X configuration to develop and to be retained, IqJI must have a peak value at the X point along the x axis, since B,(x, y = 0) should result only from magnetic reconnection through the term a(qj)/ax. (More generally, it is the necessary condition for magnetic reconnection to take place that qj has a finite gradient near an X point.) UGAI and TSUDA (1977) found that, if localized enhancement of resistivity is retained near an X point, global plasma flow eventually grows so as to concentrate the current density J near an X point and hence to make a sharp gradient in q.l near the X point, leading to the establishment of the fast reconnection mechanism. The most important physics of magnetic reconnection apparently lies in the nonlinear hydromagnetic interaction between magnetic field diffusion near an X point and global plasma flow in the surrounding region. Resistive tearing mode grows in a way such that perturbations grow so as to enhance the current density J near an X point and hence to give rise to magnetic reconnection. For the later purpose, it may be instructive to show the simplest version of the classical tearing mode theory (FURTH et al., 1963). In a slab geometry without sheared field (Bz = O), magnetic field Et 0, 0) may be such that B,(y) = B, tanh(y/a*). The field configuration is unstable for perturbations proportional to exp(ytt + ikx). Denote the perturbed quantities by subscript 1, and make an X point be located at the origin (x = y = 0). Integration of (2) in terms of the

5 Self-tonsistent development of fast magnetic reconnection 1553 perturbed quantities bctween the X (at x = y = 0) and 0 (at x = n/k and y = 0) points leads to where 0: is given by a line integral of B,l(x, y = 0) between X and 0 points and provides the total magnetic flux residing in the magnetic island, and, for simplicity, the resistivity q* is assumed to be constant in both space and time. Let 2~ denote the effective width of the tearing layer where mzgnetic field diffusion is important. If E - ~ % k2, the perturbed current density J1 in the tearing layer may be given by -plg ab,~/dy. Since B,1 may take the form Yl(y)sin(kx)exp(j,t), equation (3), together with V*B1 = 0, readily leads to yr = pi lq*yy(y = O)/Y1(0), where YY(y = 0) is the second derivative of Y ~ (y) measured at y = 0. An exact value of YY(y = O)/Y1(0) can be obtained by matching the solution in the inner region ( 1 y I 5 E) with that in the outer region ( 1 y 1 b E), but it may be possible to estimate that value only by the solution in the outer region where magnetic field is frozen into plasma flow. As is well known, in the limit of incompressible flow, the outer-region solution provides A % (Y<( +E) - Y<( --E))/Y~(O) x -2(ka* - &%*)- )/a*, if (&/a*) is sufficiently small. Since YY(y = O)/Yl(0) x A /(~E) by definition, we obtain yt z plg lq*((kx*)- - ka*)/(ca*). The condition for the tearing mode to grow (i.e. for Yt io have a real positive value) is 1. Considering that t x.*(ka*)- 2~~fr;2jrS)114, where 7; = a*/v;(v; = B,/(p0p*)li2) and 78 = p,,a*2/q*, we have the well-known growth rate where RZ = zbjzf is the magnetic Reynolds number. B. Development of fast magnetic reconnection Initiated by the resistive disturbance, the current sheet becomes thinner and the drift velocity VD becomes larger near the origin (UGAI, 1983). We in fact find that for each resistivity model VD becomes sufficiently large to cause a finite anomalous resistivity in the current-sheet region at time t z 19, when magnetic reconnection suddenly commences. The reconnection process may depend on which of the resistivity models is specified. In order to have an overall picture of the fast reconnection process, Fig. 1 shows the temporal variations of the total magnetic energy UM(~) residing in the first quadrant (0 6 x 6 2 and 0 < y < 3). In the initial time range (0 < t 64), magnetic energy is reduced by an amount sufficiently small compared with the total magnetic energy, which was employed as an initial disturbance. For each resistivity model, the stored magnetic energy is remarkably released for t 2 20, which should result from the buildup of the fast reconnection mechanism. Figure 2 also shows for different resistivity models the temporal variations of the drift velocity VD(~ = 0, t) measured at the origin. Note that the anomalous resistivity is completely determined by VD(F, t ) according to each of the resistivity models. These figures apparently indicate that, both quantitatively and qualitatively, the resistivity growth process and the magnetic energy conversion process are not seriously sensitive to the functional form ~(VD), which is surprising in view of the quite different forms of anomalous resistivity

6 1554 M. UGAI "M I I I I I L t 32 I FIG. 1.-Temporal variations of the magnetic energy content UM residing in the first quadrant for (-) the resistivity model (A), (--.--.) the resistivity model IB) ( k = ~ 6 x (------) the resistivity model (C)(kR = 1.0 x lo-:), and (...) the resistivity model (C) (kr = 0.5 x 10-3). 3 FIG. 2.-Temporal variations of the drift velocity Vn at the origin (r = 0). Each line is shown in the same manner as in Fig. 1. now at issue. Note that the location of the VD maximum indicates when the growth of znomzlous resistivity at the origin has just become saturated. The global p!asma flow resulting from the reconnection process tends to enhance VD near the origin, whereas the enhanced resistivity tends to reduce the current density and hence VD.

7 Self-consistent development of fast magnetic reconnection 1555 When these two effects become counterbalanced the growth of anomalous resistivity should be saturated and be checked, so that the quantitative differences between the results in the different cases of ~(VD) of Figs. 1 and 2 may come from the difference in dq( VD)/dVo by which the temporal growth of anomalous resistivity should largely be controlled. In particular, we find that the qualitative feature of the fast reconnection development is quite the same even for different anomalous resistivity models. In order to illustrate how magnetic reconnection proceeds in accordance with the growth of anomalous resistivity, Fig. 3 typically shows, for the resistivity model (A), magneticfield and plasma-flow configurations at different times, and in each figure the presence 3 - a 0.5 I I I I I Y -3-2 o x 2 FIG. 3(a) FIG 3 -Magnetic-field and plasma-flow configurations for the resistivity model (A) at times (a) t = 25.1, (b) t = 29.0, (c) t = 30.8, (d) t = 32 8, (e) t = 34 8, and (f) t = 400. In (a) the scale of velocity is indicated, rind In (c) the local current sheet is enclosed by dashed lines. The separatrixes are shown by bold lines and the presence of anomalous resistivity is indicated by (*) at each mesh point All the figures that will appear are shown in the same manner

8 1556 M. UGAI Y O % 2 FIG. 3(b and c), of anomalous resistivity is also indicated by (*) at each mesh point. Except for Fig. 3(a), the configurations are shown only near the x axis, where most novel phenomena take place. The generic feature of the fast reconnection development can be summarized as follows. (1) As the anomalous resistivity grows in the field-reversal region, magnetic reconnection commences and grows, and an X-type neutral point is formed at the origin (r = 0). The reconnection process further enhances the drift velocity VD near the X point, which is consistent with the recent result of UGAI (1983), so that the resistivity grows and becomes localized near the X point. The resulting localized enhancement of anomalous resistivity allows a large-scale X-type field configuration to be set up in the overall region, as was suggested by UGAI and TSUDA (1977). In this way, the fast-reconnection configuration eventually builds up where the outflow velocity Iuxl in the field-reversal region attains an AlfvCn velocity [Fig. 3(a)]. (2) As the fast reconnection grows further, the resistivity becomes larger and extremely localized near the X point [Fig. 3(b)], so that the By field becomes larger from near the X point, since magnetic reconnection rapidly proceeds because of the term a(qj)/ax which has a large value [equation (2)]. When the growth of the anomalous resistivity is eventually saturated, the separatrixes become suddenly flattened from near the X point, forming a local current sheet, which is enclosed by dashed lines in Fig. 3(c). (3) The local current sheet becomes elongated with time, and as its effective iength exceeds the effective width, the local current sheet suddenly splits into a pair of X

9 Self-consistent development of fast magnetic reconnection 1557 FIG. 3(d-f). points and an 0 points [Fig. 3(d)]. A magnetic island is accordingly formed and is attached to the large-scale X field configuration existing in the extended region. As the island swells further, the X points move in the directions oppositely to each other; simultaneously, the anomalous resistivity decays near the 0 point but is still enhanced locally near the X points, which enables the effcctive proceeding of magnetic reconnection [Fig. 3(e)] as well as the effective magnetic energy conversion (Fig. 1). For each resistivity model, the fast reconnection process is found to terminate when the antiparallel-field region becomes reduced to zero and the plasma p becomes

10 1558 M. UGAI larger (p 2 1) outside the field reversal region [Fig. 3(f)]. This may readily be understood from the recent result of UGAI (1984) who shows that, when plasma /? exceeds unity outside the field-reversal region, the drift velocity VD cannot effectively be enhanced, so that no effective anomalous resistivity nor magnetic reconnection result as is shown in Fig. 3(f). In general, the anomalous resistivity due to current-driven instability is quite favorable to the occurrence of fast magnetic reconnection and hence to the rapid magnetic energy conversion involved. One of the most distinct phenomena may be the formation of a magnetic island. In order to see the resulting configuration in more detail, Fig. 4 shows the current density distribution corresponding to the field configuration shown in Fig. 3(e). There is a current core in the island where plasma density and pressure are also found to be notably enhanced. In the extended region attached to the island, there are two pairs of thin layers of remarkably large current density which are certainly identified with standing slow shocks (UGAI and TSUDA, 1977; UGAI, 1982). We hence conclude that the fast reconnection mechanism, consistent with the anomalous resistivity due to current-driven instability, is eventually combined with a swelling magnetic island. It should also be noted that, even quantitatively, the magnetic energy conversion process is not significantly FIG. 4.-Current density distribution for the resistivity model (A) at time t = influenced by the different choice of q(vd). We in fact find that for various parameter values of kr in the resistivity models (B) and (C) the corresponding energy conversion rates do not change so much; for instance, Fig. 1 shows that the magnetic energy conversion processes for the two cases of kr = 1.0 x and 0.5 x for the resistivity model (C) are quantitatively quite similar to each other. Such a quantitative feature of the self-consistent fast reconnection process should, of course, be quite fundamental in applications to actual plasma systems. C. Formation of magnetic island It has geceraily bee:: discussed that a magnetic island is eventually formed during the fast reconnection development consistent with the anomalous resistivity that is supposed to result from current-driven instability. For instance, Figs. 5 and 6 further

11 Self-consistent development of fast magnetic reconnection 1559 illustrate for the resistivity models (B) (kr = 1.5 x and (C)(kR = 0.5 x respectively, how the local "current sheet" is formed and the magnetic island is subsequently formed. An important question may arise as to why the local current sheet suddenly splits to form a magnetic island. The basic mechanism of the splitting seems to be closely related to that of tearing mode, so that let us examine in more detail the island formation in terms of the linear tearing mode theory, although the basic situation involved is considerably different. For this purpose, Fig. 7 demonstrates for different resistivity models the temporal variations of the total magnetic (normalized by %BO) residing in the magnetic island (@(t) is defined as the absolute value of a line integral of B,(x, y = 0, t ) between the X and 0 points). Also, the temporal variation of ln(@(t)pd0) is indicated by dotted lines for each case, where 00 is measured when the splitting has just taken place. This figure apparently shows increases exponentially in the initial time range of the splitting, which may define the growth rate of the island formation, and then tends to increase almost linearly with time. In applying the linear tearing mode theory to the present results, it should be noted that all the numerical quantities are normalized on the basis of the initial configuration described in Section 2. It is hence convenient that the growth rate of tearing mode X FIG. 5.--Magnetic-field and plasma-flow configurations for the resistivity model (B) (kr = 1.5 x at times (a) t = 34.1 and (b) t = 36.8.

12 1560 M. UGAI -2 2 FIG. 6.--Magnetic-field and plasma-flow configurations for the resistivity model (C) ( k = ~ 0.5 x at times (a) t = 32.0 and (b) t = FIG. 7.-Temporal variations of the magnetic flux CP residing in the magnetic island and of ln(@/(do) (dotted lines) for the resistivity models (A), (B) (kr = 1.5 x and (C) (kr = 1.0 x lo- )).

13 Self-consistent development of fast magnetic reconnection 1561 [equation (4)] is rewritten in terms of the present normalized quantities as where It is normalized by (~/VAO)- ' and the normalized resistivity 1 (= R, I) should be measured near the X point where magnetic reconnection proceeds. Also, we have considered that VAO x V: for each case of the simulation models. The normalized growth rate Yt can readily be estimated on the basis of the local current sheet [Figs. 3(c), 5(a) and 6(a)]. As an example, let us consider the resistivity model (B) (kr = 1.5 x [see Fig. 5(a)]. Figure 8 shows for this case the temporal variations.of the location of the X point and of the normalized resistivity 1 measured at the X point; also, the profile of B,(x = 0.4, y) along y is shown at time t = 33.8 when the splitting has just taken place. From the figure, we may estimate (a*/a) x 0.13, q z 14 x and (kcc*) x 2/3 (note that the wavelength 3. = 2n/k should be the effective length of the local current sheet) in the initial time range of the splitting. Inserting these values into (9, we have Y, x 1:5. On the other hand, the growth rate vi (normalized by (x/ya0)-l) of the island formation may readily be estimated from the slope of ln(@(t)/@o) shown in Fig. 7. We thus obtain Ii x 1.4 for the resistivity model (B) (kr = 1.5 x when measured at t x 34. For the other cases too, the growth rates can be estimated in the similar manner, and we find: For the resistivity model (A), yt x 2.7 and yi x 2.3; for the model (B), yt x 2.4 and yi z 2.2 for kr = 6 x for the resistivity model (G), yt x 2.2 and yi x 2.1 for kr = 1.0 x IO-' and X T3 x~~ : 10 c I I i I i t FIG. 8.-Temporal variations of the location X(t) of the X point and the normalized resistivity q(x(t), y = 0) measured at the X point for the resistivity model (B) (kr = 1.5 x Also, the profile of the B, field along the line x = 0.4 is shown at time t = 33.8.

14 1562 M. UCAI yr x 1.8 and yi x 1.6 for kr = 0.5 x These results clearly demonstrate that the growth of the magnetic island is generally consistent with the tearing mode growth, so that we may conclude that the island formation results, basically, from the physical mechanism of the resistive tearing mode. The present computation employs 50 x 200 mesh points in the fi,st quadrant (0 Q x Q 2, 0 Q y < 3); hence, in the half-width of the local current sheet [Figs. 5(a) and 81 there are more than 10 mesh points in the y-direction which have been found to be sufficient to resolve the diffusion region. 4. CONCLUSION The present paper has been focused on demonstrating the possible fast magnetic reconnection process that develops self-consistently with anomalous resistivity. Computer simulations have been done for quite different forms of anomalous resistivity, in each of which the anomalous resistivity is assumed to increase as the relative electron-ion drift velocity increases in view of the current-driven instability. It has been found that the qualitative feature of the anomalous resistivity growth is quite the same even for the different choice of the anomalous resistivity model, and we thus find the generic featme of the self-consistent development of fast reconnection. One of the most important results is that, according to the proceeding of magnetic reconnection, the anomalous resistivity becomes significantly enhanced and tends to be localized near an X-type neutral point. Such a localized enhancement of resistivity is in fact found to cause eventually a large-scale X-type field configuration and hence to play a crucial role in realizing the fast reconnection mechanism, which is in good agreement with the suggestion of UGAI and T S ~ (1975). A (In this respect, SATO and HAYASHI (1979) asserted that local conditions near an X point had iittie influence on the process of fast reconnection, but their conclusion was based on the specific boundary condition not consistent with the proceeding of fast reconnection.) Another significant finding is that, as the growth of the anomalous resistivity becomes saturated, a magnetic island is suddenly formed and swells with time which is combined with the slow shocks standing in the extended region. The process of the island formation has been examined in detail and found to resuit, basically, from the tearing mode mechanism. Recently, PRIEST (1984) suggests a formation of central current sheet [similar to Fig. 3(c)] and the subsequent tearing on the basis of the general consideration of the Petschek mechanism. En actual systems too, the formation of magnetic island associated with the fast reconnection process has been observed in the geomagnetic tail during substorins as well as in laboratory experiments (STENZEL and GEKELMAN, 1981). Also, it has been shown that the fast reconnection process provides a rapid magnetic energy conversion, and that, even quantitatively, the energy conversion rate is not significantly influenced by the resistivity model. These general qualitative and quantitative features of the self-consistent fast reconnection development, presented here, should, in principle, be most applicable to solar flares as well as geomagnetic substorms, although in actual systems plasma circumstances are usually much more complicated. Acknowledgements-This work was partially funded by the Institute of Plasma Physics, Nagoya University, Nagoya, Japan. The computer program was tested and run at the Computation Centres of Kyushu and Ehime Universities.

15 Self-consistent development of fast magnetic reconnection 1563 REFERENCES BUNEMAN 0. (1959) Phys. Rev. 115, 503. CORONITI F. V. and EVIATAR A. (1977) Astrophys. J. Suppl. Ser. 33, 189. DRAKE J. F., GLADD N. T. and HUBA J. D. (1981) Physics Fluids 24, 78. DUSGEY J. W. (1953) Phil. Mag. 44, 725. FURT H. P., KILLEEN J. and ROSENBLUTH M. N. (1963) Physics Fluids 6, 459. IMSHEKNIK V. S. and SYROVATSKII S. I. (1967) Soviet Phys. JETP 25, 656. N~HIDA A. (1978) J. Geomag. Geoelect. 30, 165. PETSGHEK H. E. (1964) NASA Report No. SP-50, p PRIEST E. R. (1983) Plasma Physics 25, 161. PRIEST E. R. (1984) Chapman Conference on Reconnection (edited by E. HOXES) (to be published). SATO T. and HAYASHI T. (1979) Physics Fluids 22, SMITH D. F. and PRIEST E. R. (1972) Asrrophys. J. 176, 487. SOWARD A. M. and PRIEST E. R. (1982) J. Plasma Phys. 28,335. STENZEL R. L. and GEKELMAN (1981) J. geophys. Res. 86, 649. UGAI M. and TSUDA T. (1977) J. Plasma Phys. 17, 337. UGAI M. (1981) J. Plasma Phys. 25, 309. UGAI M. (1982) Physics Fluids 25, UGAI M. (1983) Physics Fluids 26, UGAI M. (1984) Physics Fluids 27, PP 26:lZB-H