Lecture Notes. Geometry and Dynamics of Magnetic Reconnection

Lecture Notes on Geometry and Dynamics of Magnetic Reconnection Wang, Xiaogang Institute of Plasma Physics Department of Physics Harbin Institute of Technology Email: xgwang@hit.edu.cn

Preface This lecture is for The 4 th East-sian School & Workshop on laboratory, Space, and strophysical Plasmas, July 7 to ugust 1 14, hold at Harbin Institute of Technology (HIT). It is planned for first or second year graduate students in plasma physics with an emphasis on basic geometry and ideas of magnetic field line reconnection in space, astrophysical, and laboratory plasmas. The lecture is arranged as follows. In the Introduction, we try to discuss the geometry and basis of magnetic reconnection. In the next two sections we present the analysis for free magnetic reconnection, the tearing mode. Then we further discuss forced magnetic reconnection in Section V. The final section is for recent developments in Hall MHD reconnection studies. I. INTRODUCTION 1.1 Frontiers of Plasma Physics 1. Geometry of Reconnection 1.3 Kinematics of Reconnection 1.4 Dynamics and Time-Scales II. RESISTIVE TERING MODES.1 Energetics. FKR & Rutherford Theories of Constant- Modes.3 The Non-Constant- Modes, m=1 Kink-Tearing

III. COLLISIONLESS TERING MODES 3.1 LPV Theory of Constant- Modes 3. Ion Tearing 3.3 Drake-Lee Theory with Guide Field 3.4 Sawtooth Collapses in Tokamaks IV. FORCED RECNNETION IN SLB GEOMETRY 4.1 Taylor s Model 4. Fast Reconnection 4.3 In Rotating Plasmas V. HLL MHD RECONNECTION 5.1 Hall Effects in Magnetized Plasmas 5. Hall MHD Reconnection: Reduced Model 5.3 Hall MHD Forced Reconnection with Guide Field 5.4 Hall MHD Forced Reconnection without Guide Field 5.5 Steady State Hall MHD Reconnection

I. INTRODUCTION 1.1 FRONTIERS OF PLSM PHYSICS Reconnection of magnetic field lines is an important physical process in the frontiers of space, astrophysical and laboratory plasma physics. Fast energy transfer events in space, astrophysical and laboratory plasmas For instance: Relevant solar system phenomena: Solar phenomena Flares, coronal heating, coronal mass ejection (CME), etc., leading to Space phenomena Solar wind, interplanetary magnetic field (IMF), etc., leading to Geophysical phenomena (space weather) Magnetosphere storms and substorms Disruptive phenomena in laboratory (tokamak) plasmas: Neoclassical tearing modes leading to Disruption Kink-tearing modes leading to Sawtooth collapses

Why reconnection? Ohm s Law in D: d v J, B = zˆ B zˆ dt t 1 d J= dt The Way to Dissipate Energy: 1. Diffusion. Reconnection d ~ dt, PDIF J ~ d ~ dt J ~ 1 J ~ O 1 1 PREC J ~ 1 P P REC DIF ~ ( 1) 1. Solar Flares Figure 1.1 typical solar flare event.

Giovanelli Model The concept of magnetic reconnection was first proposed by Giovanelli as magnetic field lines merging and annihilated at a magnetic null, in an effort to understand solar flares [Giovanelli, 1946]. Sweet-Parker Model Figure 1. Magnetic field line geometry of Sweet-Parker model. Giovanelli s idea and further developed by Sweet [1958] and Parker [1957] into a neutral line merging model, which is now a fundamental model of magnetic reconnection. However the Sweet-Parker reconnection rate is on the order square root of electrical resistivity, too slow to be counted on such a fast event as a solar flare. Petschek Model Figure 1.3 Magnetic field line geometry of Petschek model. With a fast driven and an X-point geometry, it was claims that a fast reconnection rate could be almost on the order of logarithm of resistivity, much faster than the

Sweet-Parker rate [Petschek, 1964]. It is nevertheless found that in high resolution simulations, the X-point geometry cannot be realized in high Lundquist number (S > 1 4, S ~ 1/) regime. nd in the low Lundquist number regime (S < 1 3 ), Petschek reconnection rate and Sweet-Parker reconnection rate are more or less on the same order.. Solar Coronal Heating Fast reconnection and current sheet heating model Proposed by Parker [197], this model claims that a fast topological reconnection driven by foot-point twisting yields singular current sheets which lead to the anomalous heating of the solar corona. Challenge: foot-point tying keeps the coronal loop from certain topologic changes. Other models The wave heating model suggests that particle resonance heating caused by shear lfvén wave (lately, kinetic lfvén wave) is the reason for the high temperature of the solar corona. Challenge: what is the mechanism to generate kinetic lfvén waves? The synthesis heating model finds that kinetic lfvén waves can be generated on the singular current structures such as current sheets caused by foot-point slipping or current filaments caused by foot-point twisting. 3. Solar Wind Merging to Magnetosphere This is a forced reconnection process first described by Dungey [1961, 1963], for southward and northward solar winds, then by Cowley [1973] and Stern [1973] for arbitrary solar wind orientations.

Figure 1.4 Dungey s reconnection model Figure 1.5 Cowley s reconnection model 4. Tearing instability of Magnetotail This is thought a free reconnection process on the nightside neutral line: Laval et al [1966], Coppi et al [1966]: Schindler [1974]: Lembege and Pellat [198]: n electron tearing mode n ion tearing mode tearing stable geometry

N EXMPLE: Solar Wind Merging to Magnetosphere What we know: Coupling to south or north-ward IMF [Dungey, 1961; 1963] to arbitrary orientation IMF [Cowley, 1973; Stern, 1973] Open questions: Where? Geometry and kinematics How fast? Solar-wind driven reconnection dynamics 1. GEOMETRY OF RECONNECTION D SEPRTRICES: 1) rms of X points: Petschek s geometry; D coronal loop geometry ) Neutral-lines Sweet-Parker geometry 3) Rational surfaces Tokamak tearing mode theory Figure 1.6 n X-point (on the left) and a neutral line (on the right)

3D SEPRTRICES: -type ( s -type) nulls, B-type ( B s -type) nulls and B lines, and B surfaces Figure 1.7 -type and B-type magnetic nulls and their -lines and -surfaces. Dungey-Cowley Model of 3D reconnection: dipolar field + a constant IMF a). Northward IMF [Fig. 1.4(a)] n -type null above the north pole, a B-type null above the south pole. N-type reconnection b). Southward IMF [Fig. 1.4(b)] X-points at both sides. S-type reconnection c). rbitrary orientation IMF [Fig. 1.5] single separator connects the -type null and B-type null. G-type reconnection Figure 1.8 Data-based calculation, theory, and numerical simulation of dayside 3D magnetic reconnection geometry.

COMPLEX 3D RECONNECTION GEOMETRY: MSPMSH MSHMSP MSPMSP MSHMSH Figure 1.9 Complexity of 3D reconnection geometry. Issue: Magnetic reconnection on magnetopause: nti-parallel or Component?(Cowley, 1976; Crooker et al., 1979; Pu et al., 5) Figure 1.1 Cluster + Double-Star observation of dayside reconnection line.

SEPRTOR RECONNECTION: separator is the intersection of two separatrices, connecting two anti-polarity nulls. From the 3D view point, component reconnection is only a D approximation of 3D separator reconnection when the reconnection site is away from the nulls. Figure 1.11 Cluster observations of separator reconnection events; left: the component-like, right: the anti-parallel-like. (Li et al, JGR 13) GUIDE-FIELD RECONNECTION IN 3D GEOMETRY: Magnetic reconnection in tokamaks. The toroidal magnetic field is very strong. No real null in tokamaks. Then magnetic reconnection occurs only on rational surfaces which are also magnetic separatrix. Figure 1.1 Tokamak configuration

1.3 KINEMTICS OF RECONNECTION [J. M. Greene, 1993] In a fluid with an embedded magnetic field B c E, (1.1) t V is called the velocity of the magnetic lines, if B ( V B ); (1.) t or, for E = V B / c E R, (1.3) if we can find V and so that the reconnection electrical field E R =. Thus if B E B (1.4) has a decent solution, we have the force line velocity CONCLUSION: c V = E B. (1.5) B Reconnection occurs at the place where 1. Eq. (1.4) does not have a decent solution (tokamak rational surfaces);. Eq. (1.5) gives a very large or singular V discrepancy between fluid flow and field line flow (arms of X-points, neutral lines, etc.). n Example: N--B--LINE can been approximately modeled as B z sin ky x By cos ky k Bz x z cos ky where parameters and are positive.

1. The field has the following features: i. n B null line at the intersection of x = and z =, i.e., the y-axis; ii. -type nulls at x =, ( 1/ ) / y n k, z =; B-type nulls at x =, y (n 1/ ) / k, z =; where n 1,, iii. Separatrix surfaces: z=-x plane as and z= x plane as B ; iv. way from x =, reduced to the IMF B Bzˆ. z. Field Lines: / z x ky tan C z x 4 z x cos ky C 1 s Lau & Finn s [199],Greene [1993],we add a time-varying vector potential te 1 ˆ y, to solve Solution: B BE y dl dl ByE B E E y - y ( x, y,z) B y 3. t the separatrix: The field line velocity y By z x arctan C1, y k z x k E C1 B V= c xb ˆ ˆ z zbx B k1 C1 has a power law singularity, as C zc x z x C1 xc1 z z x 1 1,

1.4 DYNMICS ND TIME-SCLES Magnetic reconnection dynamics: How fast? Sweet-Parker model: 1/, ~ SP R Petschek model: ln / 1/, ~ ln Pk B SP Wang-Ma-Bhattarjee model (1996):, 1/5 1/5 3/5 N R SP ~ 1/5 Types of reconnection (i). Free (or spontaneous) reconnection is caused by the spontaneous occurrence of a tearing instability; Due to release of free energy already stored in the outer region; (ii). Forced (or driven) reconnection is driven by external source through changing the boundary conditions of stable equilibrium. Because of externally imposed driven energy. To distinguish the spontaneous and forced from each other, we then need to discuss the typical time-scales: (i). characteristic driven time-scale av /, where a the typical length-scale, and V the typical boundary driven velocity; (ii). The magnetic field in between the boundaries responds to the driven on an lfve n time-scale a/ V,

where V the lfve n speed; Then, R where R the resistive diffusion time; (iii). On the other hand, the presence of a small but finite resistivity causes free reconnection on a time-scale S for linear non-constant /3 1/3 1 R regime 1/ S for nonlinear non-constant SP R regime S for linear constant regime /5 3/5 R then S R. Forced reconnection ; S R i.e., the external driven time scale should be shorter than the tearing mode growth time. Free reconnection ; S R i.e., the driven time scale (if any) should be longer than the tearing mode growth time. Magnetic reconnection: Magnetic field lines in various topological domains separated by a separatrix merge, cut, and reconnect to each other to form new field lines of new geometry different from their original topology. Magnetic reconnection is a fast process, in comparison with diffusion, to release magnetic energy stored or input in plasmas and transfer it to other forms of energy.

REFERENCE I Coppi, B. G. Laval, and R. Pellat, 1966: Phys. Rev. Lett. 16, 17. Cowley, S. W. H., 1973: Radio Sci. 8, 93. Dungey, J. W., 1961: Phys. Rev. Lett. 6, 47. Dungey, J. W., 1963: in Geophysics, The Earth s Environment, Gordon & Breach, New York. Giovanelli, R. G., 1946: Nature 158, 81. Greene, J. M., 1993: Phys. Fluids B5, 355. Lau, Y. T., and J. M. Finn, 199: strophys. J. 35, 67. Laval, G., R. Pellat, and R. Vuillemin, 1966: in Plasma physics and Controlled Nuclear Fusion Research (IE), 59. Lembege, B., and R. Pellat, 198: Phys. Fluids 5, 1495. Parker, E. N., 1957: J. Geophys. Res. 6, 57. Parker, E. N., 197: strophys. J. 174, 499. Petschek, H. E., 1964: in S/NS symposium on the Physics of Solar Flares, edited by W. N. Hess, 45. Schindler, K., 1974: J. Geophys. Res. 79, 83. Stern, D. P., 1973: J. Geophys. Res. 78, 79. Sweet, P.., 1958: in Electromagnetic Phenomena in Cosmical Physics, edited by B. Lehnert, Cambridge University Press, New York, 13. Wang, X., Z. W. Ma, and. Bhattacharjee, 1996: Phys. Plasmas 3, 19. Wang, X., and. Bhattacharjee, 1996: J. Geophys. Res. 11, 641.

II. RESISTIVE TERING MODES.1 ENERGITICS OF TERING MODES 1. The Energy Integral From the linearized Maxwell s equations, it follows that 1 c dx B1 E1 J1 E1 d 1 1 t 8 a E B (.1) 4 where all perturbed quantities are signified by the subscript 1. The linearized resistive MHD equations are V J B1 J1B p 1 (.) t c c 1 V (.3) t V B E1 J 1 (.4) c and 1 B1 4 E1 B1 J 1 (.5) c t c where n1 ne 1 ni 1 by quasi-neutrality, and V =V 1 in the linear approximation. We then get J1 B V J1 E1 J 1 (.6) c The first term on the right-hand-side (RHS) of (.6) can be calculated from the momentum equation (.) that gives J1 B V 1 J B1 V p1v p1v V. (.7) c t c For a D slab equilibrium model as a Harris sheet, z B = yˆ ˆ ˆ B tanh x B F( z) x, (.8) L B= ŷ, (.9) 1 1 with F tanh z / L, we obtain

V F J1 E1 1 J1 p1v p1 V, (.1) t 8 F where the non-adiabatic pressure perturbation J 1 p1 p1 p1 p1. (.11) c Eq. (.1) can then be cast in a form c t 4 d a E B 1 1 p V 1, (.1) where da the differential inner-outer interface W W Q K (.13) f c with the free energy of the magnetic field and W 1 F d 8 x B F, (.14) f 1 1 W t d p c 1 x V (.15) due to the plasma compression. Note that the dissipation Q t d 1 x J (.16) and the fluid kinetic energy 1 K d x V (.17) are positively defined quantities. If the boundary in (.1) is chosen in such a way that the surface term vanishes, then a sufficient condition for stability is W W W. (.18) f c. The Energy Flux For an incompressible plasma,. Then, there are three terms in the energy W c integral (.13) among which K is the kinetic energy and always positively defined. It is clear that a tearing instability can occur if and only if there is magnetic

free energy available, i.e., W f. However simultaneously, there also exists a mechanism of dissipation causing We rewrite (.14) in the form Q (which is always positively defined too). 1 F Wf d k 16 x F, (.19) where the perturbed flux z t inertia and the dissipation, we get 1, cos kx. In the outer region, neglecting the k ( F / F). (.) If the first term in the integral (.19) was integrated by parts and (.) was used, W f reduces to a xy z axy Wf (.1) z 16 16 where for instability as dx a dz with axy xy dxdy, and the parameter shows a condition '( ) '( ) ' (1 kl). (.) () kl On the interface between the inner and outer regions, the Poynting flux is then c c da E1 B1 dydx dydxzˆ E1 B 1 4 4 z z axy W 16 t t f, outer (.3) Clearly, the condition for instability shows that there is a net energy flux into the inner region, and then absorbed by dissipation and/or transferring to kinetic energy in the inner region. Dissipation, such as resistivity effect of J ~, is usually very small due to low collisionality in space and tokamak plasmas. However, in the inner region, the spatial scale is very small, then the dissipation can be significant. Thus the free energy

absorption can be mainly due to dissipation, particularly in resistive MHD model.. FKR & RUTHERFORD THEORIES OF CONSTNT- MODES 1. The outer region: Ignoring the dissipation and the inertia terms in the outer region, the linearized equation for the flux is (.), with an exact solution of z kz tanh z / L e 1 kl (.4) which gives the jump discontinuity (.).. The inner region: Since the resistive effect in this region can be substantial, then the inner equations for incompressible plasmas can be written kb z L t L R Bz k 4 t L with the incompressible condition ŷ1 k / t, BL and z / L z, kl k, s with V, z t t, f z, t f z 1, sin kx. Making, (.5) the lfve n time, we have (.5) reduced to s z / R s k z (.5 ) In a constant- approximation ( x ~ 1) 1 inner if L ~1 or smaller, since the size of the inner region / L 1. Following Furth, Killeen and Rosenbluth [1963], we can solve (.5) and match,

the solution to the outer region by dz z inner to get / k 4/5.55. (.6) /5 3/5 R Or, ~ ~S 3/5 3/5. This is a slow process for most plasmas since resistivity is usually very low, as 6 1 S ~ 1 1. In this rate, the field line is broken and reconnected into a new closed geometry called magnetic island. Figure.1 Magnetic field lines are reconnected into magnetic islands. The Rutherford nonlinear theory s the magnetic island grows out of the inner region, magnetic reconnection goes into the nonlinear regime. The detail of the theory is in Rutherford [1973]. We can however present the result of the theory in a simple way [Wang & Bhattacharjee, 1997]. For the constant- tearing mode, the nonlinear evolution can be calculated as follows. From Ohm s law, we obtain d L L B dt w R R where w is the island width, B is the magnetic field jump cross the island. In the

constant- regime, one can calculate w [ () / B, and B ( ), we ] 1/ have then B 4 L B, w 4 dw w dt R dw dt L R w L ~ t t. (.7) R It is not only algebraically slow, but also scaled as, just like a resistive diffusion..3 THE NON-CONSTNT MODE,m=1 KINK-TERING 1. The linear theory The non-constant- regime is for a perturbation with an extremely big, or say, L 1, therefore in the relation ( x ~ 1) 1 inner ~ 1. From (.), this mode can either be a long-wavelength perturbation, or a zero initial as m=1 kink modes or forced reconnection. The linear growth rates of non-constant- tearing instabilities, both long-wavelength modes and kink-tearing modes, ~ ~ /3 1/3 1/3 R, (.8) are much faster than those constant- modes with ~ ~ /5 3/5 3/5 R in (.6). The linear m=1 kink-tearing mode was discussed elsewhere, such as in ra et al [1978]. We here give a brief discussion of long-wavelength modes. The outer region solution and the inner region equations of these modes are the same as those of constant- modes in (.4) and (.5). The only difficulty is that we cannot use the constant approximation. ssuming the size of the inner region is so z/ z,, and

s L, under the new spatial scale, we have z kz (.9) with, indicating a growth rate of (.8) /3 1/3 L R. The nonlinear theory The first analytic nonlinear study of the resistive m=1 kink-tearing mode was given by Hazeltine et al [1986]. They claimed that the mode continuously grew exponentially in the nonlinear stage but with a slower growth rate reduced by a factor of.7 if one ignores the current sheet effect. This result is then found a late nonlinear growth when the current sheet is indeed resolved and can be obtained in a simple way shown as in the end of the section [Wang & Bhattacharjee, 1999]. However, taking the current sheet effect, first derived by Rosenbluth et al [1973], into account, Waelbroeck [1989] found that the early nonlinear growth of the mode was slow down to an algebraic growth as w~ (.3) t with a Sweet-Parker reconnection time-scale. This time scale was also gotten by Kadomtsev [1975] in discussing the sawtooth oscillations caused by the m=1 modes. Biskamp [1991] then analytically extended Waelbroeck s estimate in a cylindrical geometry. This work then has been completed by Wang and Bhattacharjee [1995] recently. We here give a brief of this theory. Under the reduced MHD assumption,a tokamak magnetic field: B B BT B p p T B T B const (.31) ˆ B B B B ˆ qs p * r qr s m n where

B * ˆ then J c ˆ B J( ) qr 4 s J( ). Due to incompressibility, u ˆ. (.3) Within a cylindrical geometry ( m=1, n=1 radial displacement r,, r,, rcos / R ), the kink-tearing modes are induced by the (.33) where R the major radius of the tokamak. The singular (rational) surface is give by q r s rb RB r s 1. In the resistive MHD, the Ohm s law is p s c u J (.34) t 4 HELICITY CONSERVTION ND CURRENT SHEET Following Rosenbluth et al. [1973] and Waelbroeck [1989], the nonlinear evolution of m=1 islands can be described through a series of neighboring equilibria ( x, ) ( x ) ( x ( x, )) (.35) with Then x r rs, / R, and 1 ( x ) x, rs( k B ) s B P( rs) q( r. s) 1 ( x ( x)cos ). (.36) For the linear stage or the thin-island stage, 1 x x ( x)cos. (.37) nd the eigenfunction of the internal kink-tearing mode can be asymptotically written

x ( x) x (.38) From, the condition for nulls of the auxiliary field, we have x cos sin, (.39) that gives an O-point at x, =, and an X-point at x, =. It can be shown that, however, this X-point is stretched into a current sheet, if helicity is an invariant. Obviously, since reconnection is symmetric to the rational surface, it can be expected that two flux surfaces labeled by x X will reconnect and form an island with a width w 4X, shown in Fig..1. In the other words, the outer branch of the separatrix is given by and the inner branch is which gives w x X, x X, (.4) 4 w x cos X cos, (.41) 4 w. (.4) Figure. The reconnection geometry for the m = 1 mode: (a) initial state; (b) intermediate state after reconnection begins.

However, in the context of helicity conservation [see, for example, Kadomtsev, 1976; and Bhattacharjee, et al, 198], S S S f 1 (where S f, the reconnected flux; S 1 and S, the fluxes before reconnection) we have w ( rs X ) ( rs X ) rs rdrd 4 inner where inner 1 w w w rdrd d rs cos rs w 4 4 8 Then 4r X s w rsw, or 8 X w w w (.43) 4 36r 4 s It means that, for reconnecting fluxes, in the constraint of helicity conservation (.43), the outermost reconnected flux is X i w w w Xe (.44) 4 36r 4 s where X e is the innermost unreconnected flux; and all flux surfaces in the region X x X are squeezed into a narrow strip along the separatrix and a jump of the i e magnetic field is yielded as the order of ( ) X X, which gives a current sheet all around the edge of the island. e i Y-POINT STRUCTRE The current sheet geometry, sustained by the helicity-conserving reconnection process can be determined by the functions f( x ) and g( ), defined in Rosenbluth et al [1973] and Waelbroeck [1989]. For reference, these functions are defined by the relation

1 1 F( ) G( ) f ( ) g( ) r (.45) which has a form of E( x) K U( ) mv / U( ). For the thin islands discussed in Waelbroeck [1989], we have, near the X-point, f ( x ) x g( ) cos. (.46) Figure.3 Geometry of the nonlinear reconnection for the m=1 kink-tearing mode nd near the O-point, g( ). The flux surface x satisfying the inequality f x g m ( ) ( ) will be trapped (reconnected) between the mirror points m ( in Fig..3). Therefore, for the separatrix x w/4, we obtain the mirror points m, from 3 w w 4 cos m, (.47) which is the same as a numerical solution of the nonlinear m=1 equations [Zakharov et al, 1993]. We have then a ribbon in between two Y-points 1/ 1 Y sin It is clearly a result of helicity conservation. w. (.48)

ERLY NONLINER GROWTH OF RESISTIVE MODES On the current sheet between the Y-points, the Ohm s law and incompressibility d dx u B dt r B s * * dt R, (.49) with uxrsd u ( ) V (.5) V B w w V, (4 ) 4( ) r * 1/ 1/ u cos x u and. Then we have u s dx dt 1 4 V dw dt r (4 ) s 1/ dx dt ( /sin ) 1/ X, (.51) K where, growth X w/r s, and 1/ K SP ( R ). For fixed, we have the algebraic X ( t/ K ). (.5) sin It is indeed the relation of (.3). LTE NONLINER GROWTH OF RESISTIVE MODES s the island size grows out of the current sheet, the thin current sheet with a Y-point structure is resolved and the angle should be determined by [see, Figures.. and.3] w w d( ) cos wsin, (.53) 4 4 where d ( ) is the local island width at. Integrating Eq. (.49) over in the range from to, we obtain r s ( SX 1/ / ). Then

1/ 1 sin 1/ 3 w, (.54) S X and dx dt X. (.55) /3 1/ 3 R Recalling the linear m=1 growth rate ( k /3 L1 r s ) /3 1/3 R, we have the late /3 nonlinear growth rate N1 ( k r s ) L1, close to the rate of N1. 71 L1 in Hazeltine et al [1986]. The saturation of the island growth can also be calculated. The details of the derivation can be found in Wang & Bhattacharjee, 1999.

REFERENCES II ra, G., B. Basu, B. Coppi, G. Laval, M. N. Rosenbluth, and B. V. Waddell, 1978: nn. Phys. 11, 443. Bhattacharjee,., R. L. Dawer, and D.. Monticello, 198: Phys. Rev. Lett. 45, 347; 117. Biskamp, D., 1991: Phys. Fluids B3, 3353. Furth, H. P., 1963: in Propagation and Instabilities in Plasmas, edited by W. I. Futterman, Stanford University Press, Stanford, C, US. Furth, H. P., J. Killeen, and M. N. Rosenbluth, 1963: Phys. Fluids 6, 459. Hazeltine, R. D., J. D. Meiss, and P. J. Morrison, 1986: Phys. Fluids 9, 1633. Kadomtsev, B. B., 1975: Sov. J. Plasma Phys. 1, 389. Rosenbluth, M. N., R. Y. Dagazian, and P. H. Rutherford, 1973: Phys. Fluids 16, 1894. Rutherford, P. H., 1973: Phys. Fluids 16, 193 Waelbroeck, F. L., 1989: Phys. Fluids B1, 37 Wang, X., and. Bhattacharjee, 1993: Phys. Rev. Lett. 7,167. Wang, X., and. Bhattacharjee, 1995: Phys. Plasmas, 171. Wang, X., and. Bhattacharjee, 1997: Phys. Plasmas 4, 748. Wang, X., and. Bhattacharjee, 1999: Phys. Plasmas 6, 1674. Zakharov, L. E., B. Rogers, and S. Migliulo, 1993: Phys. Fluids B5, 498.

III. COLLISIONLESS TERING MODES 3.1 LPV THEORY OF CONSTNT - MODES Even if a plasmas is collisionless, without any dissipation, one can still be able to find certain nonideal mechanism such as inversed Landau damping to break field lines. Collisionless reconnection is important due to the fact that the resistivity in most space and solar plasmas is extremely low. Furthermore there are certain observed reconnection events much faster than any resistive reconnection time-scale. We here only present a brief physical heuristics. D slab equilibrium as a Harris sheet without the guiding field was employed by Laval et al (LVP) [1966] and Coppi et al [1966] to model the collisionless tearing instabilities in magnetotail plasmas: ˆ B y xb ˆ tanh z / L. (3.1) Writing the perturbed flux as 1 ~ γ ( z) e t coskx, we have from the momentum equation for electrons m V ee e / c, which yields e ey y pe J y () -nev ey = (), 4 c where [4 n e / m ] pe 1/ e. On the other hand, integrating J y c 4 z cross the inner region gives us c c J y() (), 4 z 4 or c, (3.) k pe with k 1/ de, and de c/ pe is the electron skin depth. Let us now use (3.) to estimate the growth rate of the mode: Since the collisionless damping is caused by the wave-particle interaction, the

growth rate of the mode should satisfies the resonant condition kv e, / V T m. (3.3) e e e Making use of (.) and (3.), we get 3/ e e ce k L i k L e V V T (1- ) 1 (1- ), (3.4) k L L L T where one of the equilibrium conditions B /8 n ( T T ) is used and the size of e i the inner is estimated as [see, for example, Galeev, 1984], /. ce L V e ce The growth rate (3.4) is the same as derived in the LPV s kinetic theory. 3. The Ion Tearing Clearly the mode growing as (3.4) is caused by the electron dynamics as we discussed above. In fact the dispersion relation in LPV theory is 1 ( k) ( k), (3.5) e where ( k ) is given by the RHS of (3.4) and e i 3/ Vi ci T e i ( k) 1 (1 k L ) L L Ti. (3.6) In general, Te Ti, e( k) i( k). Even if Te ~ T i, we still have 1/ e( k) m e 1 ~ 1 i( k) mi 43. Therefore e( k ), which is (3.4). LPV theory was proposed for the magnetotail plasmas. It is a good approximation for the tail region far away from the earth. Nevertheless, because in the near Earth region the field is more depolarized, there should be a significant equilibrium Bz B component. It can be shown that with n this so called normal field B n, the tearing mode is stabilized since the plasma is now compressible.

Figure 3.1 Near earth magnetosphere with a "thick" tail To make our point clear, we can calculate W c of (.15) in the energy integral a B (), (3.7) xy Wc k 16 Bn which is due to plasma compressibility, including not only electrons but also ions [Wang and Bhattacharjee, 1993a]. In fact, if B n is non-zero, the electrons are magnetized everywhere and bouncing between mirror points along a field-line. Furthermore, there has to be a pressure gradient along z= to balance J B n. Therefore to perturb the field, one has to move the B n field line in a way of a sound wave pressing a fluid, which does a work to move the magnetized electrons and dissipates free energy. Since typically Bn ~.1B, the magnitude of W c is much bigger than that of the free energy of the outer region W a 16. xy f However, this stabilizing effect was called electron compressibility initially, given the fact that it is due to the magnetized electrons moving with the compressed field lines. This is very misleading, eventually leading to a controversy in the literature for years. Since only the electron was thought stabilized, a new and much faster mode was proposed as ion tearing. Simply removing the electron term from (3.5) since its stabilizing effect, a new growth rate was then obtained for the collisionless tearing mode as

3/ Vi ci T e i ( k) 1 (1 k L ) L L Ti, (3.8) which as we knew was more than 4 times bigger than the initial electron tearing, much closer to the observed substorm onset time scale. Lembege and Pellat [198] raised issues about this ion tearing instability. Our further work [Wang and Bhattacharjee, 1993a] showed a clear discussion on this issue. We found that one couldn't just simply throw away the electron term since it had to be calculated as (3.7). ny efforts to restore the last ion term of (3.5) for an instability was not possible since this so-called ion tearing term did not exist at all. In fact, from the process to calculate (3.7), anyone can find that the effect of ion compressibility, which comes into play due to quasi-neutrality constraint n n 1i 1e, is much larger than the effect of electron compressibility. Not only is the energy sink term W c from the work done to compress electrons but mostly to compress ions which have to move with electron to keep the charge-neutrality. Since the ion inertia is much bigger than the electron, in fact it is the ion that stabilizes the mode. We should call it ion stabilizing rather than ion tearing. The deep reason is lied on the fact that as soon as there is a B n field, the topological separatrix z= is destroyed. s we pointed in the Introduction, no separatrix, no tearing! 3.3 DRKE-LEE THEORY WITH-GUIDE-FIELD If there is a guiding field B y in the equilibrium (3.1), the plasma then is also magnetized. However, topologically the z= has to be a separatrix, with certain constrain such as the rational surface, and physically, B y dose not induce a pressure gradient on the z= surface since J B. Therefore, we can still expect an y unstable tearing mode. The theory for such a mode was developed first by Drake and Lee [1977]. From the same discussion in between (3.1) and (3.), and by the resonant condition for a

magnetized plasma kv e where k k B / B kbx / By k / L, ( By / B ), we get the D-L growth rate kve kv, (3.9) L k L e s where Ls L is the shear length and the relation (3.) is applied. Both (3.4) and (3.9) are the same as the result from exactly kinetic calculations, except for a factor of that is a kinetic effect from the integral of plasma dispersion function. Clearly the condition for the instability is the same as (.) and (3.4) but the growth is slower than the unmagnetized case. 3.4 Tokamak Sawtooth Collapse The tokamak sawtooth collapses was explained by Kadomtsev [1976] as a nonlinear growth of resistive m 1 kink-tearing modes. However he did not give a time dependent solution. Lately, experiments found: 1. The resistive m 1 growth of a Sweet-Parker time scale was too slow (~ 1 ms ) to count for the fast ( 1 s) sawtooth crash.. s described in Section.3, the nonlinear resistive m 1 mode was then found to initially grow algebraically as time. This time behavior is not in any agreement to the suddenness of the sawtooth onset. 3. The most fatal challenge to the theory is that the sawtooth crash is said to stop after a full reconnection when the safety factor on the magnetic axis q() raised above unity; while experiments always see an almost unchanged q().7 through the entire process. Therefore the sawtooth problem is a great challenge to fusion plasma theory. To solve the first difficulty, collisionless m 1 modes are proposed: electron inertia effect [Wesson et al, 1991], hyperresistivity effect [ydemir, 1989; Drake and Kleva, 1991] and etc. ll those give a faster time-scale, but not time evolution.

Later work such as Zakharov et al [1993] showed a nonlinear exponential growth for isothermal plasmas. But neither the mode growth, nor the onset is exponential. Numerical simulation suggested either electron inertia [Ottaviani and Porcelli, 1994] or the electron pressure (temperature) gradient term [ydemir, 199] in the Ohm s law might be responsible for the fast time scale. The effects are then discussed thoroughly in the theory [Wang and Bhattacharjee, 1993b; 1995], which also gives a theoretical analysis of the suddenness and the q() problem. Now let us see the outline of the theory. In the same geometry as discussed in Section.3 and following the same procedures but replacing the resistivity term in the Ohm s law by other collisionless terms in a fast time scale process such as the sawtooth, we have the island evolution equation as dx dt d X X r sin sin e (3.1) s i where the (electron inertia term) gives the initial nonlinear exponential growth rate of de c (3.11) r sin Integrating Eq. (3.1), we get with s X() exp( ct) Xt (), (3.1) 1 [ / ] X [exp( t) 1] c c (3.13) sin i to measure the finite-time explosion. However taking the large island effect into account, we have the island growth equation as 1/ 1/ 3/ de i dx 1 X (1 X ) X (1 X ) d i rs (3.14) where τ ω t. This equation is integrated to show a good agreement between the

theory, numerical simulation and the observation data, shown in Figs. 3. and 3.3. Figure 3. (a) The nonlinear growth rate predicted analytically. (b) The nonlinear growth rate obtained numerically in ydemir, 199. In the derivation for above results (3.1-3.14), the relation (.53) is applied as S d( ) 4X sin, with in this case sin ( / 1/ i ). Then following length parameters are obtained: the width of the reconnection layer wsin ( / ) X scaled by S / d / r, where di c / pi V / i ; and the length of the reconnection i i s layer r r s 1/ 1/ s ( / i ) ( dirs ). In fact, it is the first time to obtain the scaling of Hall-type reconnection. The scaling is lately confirmed in our recent PRL paper for Hall MHD reconnection without the guide field [Wang, Ma, and Bhattacharjee, 1]. Figure 3.3 The plasma displacement (a) predicted analytically (b) measured in JET

The q problem is solved as the argument follows. In all sawtooth crash models, outside of the island, including the hot plasma core surrounded by the cold island, is assumed ideal MHD as for a constant density. p p ( ), or T ( ) e Te From experiments, it is known that, as the island grows, the inner edge of the island is deformed [McGuire et al (TFTR team), 1991]. This deformation is also seen in the numerical simulation [ydemir, et al, 1989; ydemir, 199], and theoretical model [Wang and Bhattacharjee, 1993b; 1995]. If the mode is a internal kink, the inner island edge is badly distorted on the side opposite to the X-point when it is close to the plasma center shown in Fig. 3.3. Figure 3.4 Geometrical deformation of the island as the size of the island increases from (a)-(d), the shapes (c) and (d) are not physically realizable. Figure 3.5 (a) The temperature (pressure) jump formed on island edge, (b) The pedestal which causes ELM.

s the flux is deformed, the edge current is built up rapidly. Since the electron temperature inside of the island drops sharply, there is an electron temperature jump on the edges of island (see Fig. 3.5(a)). This temperature jump is very similar to the pedestal on the edge separatrix during a H-mode operation which causes edge localized modes (ELMs) (see, Fig. 3.5(b)). We can estimate the jump as T nte n e JB ~ c It is also the current singularity discussed in Section.3 * (3.15) J nct B ~ e * (3.16) as, and similar to the bootstrap current in the pedestal region, that keeps the force balance. Similar to the pedestal that induces peeling-ballooning instability to cause ELM collapses, the edge singularity can also generate instability, crash the temperature jump, and then stop the fast reconnection and flatten the current profile. We discuss this process in three phases. 1. The ELM-like Phase This phase starts with the local crash of the edge current and temperature jump. Since the loss of the force balance of (3.15), the momentum equation can be reduced to, if the island temperature is much lower than core temperature T e, Du nt p e, (3.17) Dt which leads to the transported energy 1 u nt e, or an out-flow speed T e cs In the transport equation u m, (3.18) T t e i u T T T, (3.19) e e e

we can see that Te s u Te u ite Te - T e (3.) where the time scale of the heat mixing or convection can be estimated as hm d e i -1 ~ i ~.1sec., (3.1) and the time scale of the diffusive heat pulse ~ 1 ~ 1 sec., (3.) hc 7 i 3 r s is much slower [see ydemir et al., 1989]. It is clear that the break of the edge current triggers an extremely fast heat flow out of the outermost flux surfaces. This local temperature loss on those surfaces will lead to a parallel heat transport phase.. The Parallel Transport Phase The last term in (3.3) is the parallel heat transport with a time scale r, s ~1 ~.1 sec which is even faster than the first heat convection term. Therefore the temperature on a flux surface is flattened almost immediately, i.e., the local temperature loss due to the local current sheet dissolution becomes global within no time. Then we can take the temperature as a flux function always, i.e., T T (, t), and make a flux average to (3.3) to eliminate the parallel term. e e 3. The Domino Phase s soon as the outermost fluxes lose their temperature, they become a part of the cold sink as the island did. We then see the temperature jump moves into interface between those fluxes and the fluxes right next to them. Since there is no more current sheet in this interface, a global heat convection flow with the speed (3.18) will immediately flatten the temperature on these next fluxes in a time scale of (3.1). This process then continues like dominoes until the total temperature peak is flattened.

The time scale of the final crash can be estimated by the time of the heat flow propagates the core with the speed (3.18): r s 3 1 c ~ ~ 1 i ~ 1sec. dei, (3.3) which is roughly in agreement with the observation [Edwards, et al (JET Team)1986]. Summary: In the early stage of the sawtooth, the flux brings the temperature profile to the X-point, and then they are broken by reconnection. The hot core shrinks through the X-point by the m=1 reconnection, as first suggested by Kadomtsev. In the later heat loss stage however, the confinement is destroyed at the island edge (separatrix), and the hot core disappears in a way similar to ELM collapses. Because the extremely large perpendicular electron temperature gradient crashes by the current sheet dissolution, the energy loss is very fast. Then the hot core is melt down rapidly and the major driven of reconnection, the parallel electron temperature gradient near the X-point, no longer exists as soon as the edge confinement is broken. Therefore, we can only see a completed melt down of the hot plasma core and the central q-value is still below unity since reconnection is not completed.

REFERNCES III ydemir,.y., 199: Phys. Fluids. B4,3469. ydemir,.y., J. C. Wiley, and D. W. Ross, 1989: Phys. Fluids. B1, 774. Büchner, J., and L. M. Zelenyi, 1987: J. Geophys. Res. 9, 13456. Coppi, B., G. Laval, and R. Pellat, 1966: Phys. Rev. Lett. 16, 17. Coroniti, F. V., 198: J. Geophys. Res. 85, 6719. Drake, J. F., and Y. C. Lee, 1977: Phys. Fluids, 1341. Drake, J. F., and R. G. Kleva, 1991: Phys. Rev. Lett. 66, 1458. Edwards,.W., D. J. Campbell, W. W. Engelhardt, H. -U. Fahrbach, R. D. Gill, R. S. Granetz, S. Tsuji, B. J. Tubbing,. Weller, J. Wesson, and D. Zasche, 1986: Phys. Rev. Lett. 57, 1. Galeev,.., 1984: in Basic Plasma Physics, edited by.. Galeev and R. N. Sudan, North-Holland, New York, p35. Kadomtsev, B. B., 1976: in Plasma Physics and Controlled Nuclear Fusion Research I (IE, Vienna, 1977), p555. Kuznetsova, M. M., and L. M. Zelenyi, 1991: Geophys. Res. Lett. 18, 185. Laval, G., R. Pellat, and R. Vuillemin, 1966: in Plasma Physics and Controlled Nuclear Fusion Research (IE), p59. Lembege, B., and R. Pellat, 198: Phys. Fluids 5, 1495. McGuire, K., and the TFTR team, 199: Phys. Fluids B, 187. Ottaviani, M., and F. Porcelli, 1994: Phys. Rev. Lett. 71, 38. Schindler, K. 1974: J. Geophys. Res. 79, 83. Wang, X., and. Bhattacharjee, 1993a: J. Geophys. Res. 98, 19419. Wang, X., and. Bhattacharjee, 1993b: Phys. Rev. Lett. 7, 167. Wang, X., and. Bhattacharjee, 1995: Phys. Plasmas, 171. Wang, X., Z. W. Ma, and. Bhattacharjee, 1: Phys. Rev.Lett. 87, 653. Wesson, J..,. W. Edwards, and R. S. Granetz, 1991: Nucl. Fusion 31, 111. Zakharov, L. E., B. Rogers, and S. Migliulo, 1993: Phys. Fluids B5, 498.

IV. FORCED RECNNETION IN SLB GEOMETRY 4.1 Taylor s Model THE MODEL We change the coordinates from the magnetotail to the Taylor s model [Hahm and Kulsrud, (HK) 1985] as x y, y z, z x, and L a : B ˆB zˆ, z T B x a for / x a. (4.1) The equilibrium is clearly stable. Forced reconnection occurs under a suddenly hit then stopped boundary perturbation. The perturbed boundary in this model is then x [ a cos ky], (4.) where is a constant, and the perturbed flux function can be assumed as xcos ky. (4.3) THE OUTER REGION The outer region can be studied in quasi-static ideal MHD: x The solutions is then k. (4.4) sinh kx sinh kx x cosh kx B tanh ka sinh ka (4.5) which leads to a magnetic field jump at the separatrix x. In the ideal MHD phase, no initial reconnection occurs, then t reconnection. IN THE INNER REGION We can solve the MHD equations as:,, which leads to a non constant- 1. In the linear ideal MHD phase, we can follow HK to get and inner region current sheet growing as

J L ck B t sinh ka (4.6). In the linear resistive MHD phase, as did in the non constant- tearing mode case (see the section.), after the Laplace translations t s, and U, we have the Laplace translation of the current Z, with d Z d, R 1/3 x a, s s. /3 1/3 R L It is easy to find that it again gives us exactly the same relation as (4.6), i.e., the growth of the current sheet does not change in the entire linear phase. We always have the ideal current sheet throughout the linear forced reconnection regime. nd the growth of the reconnection flux is in the Sweet-Parker time scale L ka B t sinh ka SP (4.7) 3. It was shown in Wang and Bhattacharjee [199] that for Taylor s model the transition to the nonlinear stage occurs in a non-constant- regime with a very short time-scale, as the size of the island exceeds the width of the 3 4 1 4 c R ~ L reconnection layer. This non-constant- regime with a current sheet geometry enables us to make use of the Sweet-Parker approach to solve the Ohm s law in the nonlinear phase [Waelbroeck, 1989]. The nonlinear reconnection flux then is N 3 k Ba sinh ka t SP (4.8) The current saturates, shown in Fig. 4.1, meanwhile in the stage as J N 3 1 c k R B a 4a sinh ka (4.9)

Figure 4.1 Numerically calculated current growth in the linear growth and the nonlinear saturation phases 4. Fast Reconnection We know that the Sweet-Parker reconnection has a slow time-scale compared with the Petschek s. The latter cannot be however found in simulations with reasonable low resistivity [Ma et al, 1995]. Sato and Hayashi [(SH) 1979] proposed a current related resistivity to get a fast growth controlled by the driven but almost independent of the resistivity. We revisit the problem here. From SH boundary condition, there is a flow inward to the system at x with a form V ky 1 cos. Then the perturbed boundary can be written as a x a Vt 1 cos ky. (4.1) nd the corresponding perturbed flux function can be assumed as x x cos ky (4.11) k The outer region can also be studied in ideal MHD. From forced reconnection assumption ~ ~, we have R Bx vx t a (4.1) or x and x k k k (4.13)

for the second equation in (4.1). The solutions are then x x t x 1 B a a sinh kx sinh kx t k x k x cosh kx B tanh ka sinh ka (4.14) where V, the reconnected flux ( (), k () ) vanishes in the ideal stage and is determined in the reconnection stage, and asinh kx vx V 1 cos ky xsinh ka (4.15) which also has a jump at the separatrix 1.3 and J. M. Greene, 1993]. x [please notice the discussion in Section In the inner region, for the linear phase, we can solve the resistive MHD equations as follows. The k-component of the inner region solution is the same as HK, except that t / t leads to a 3 t growth of the reconnection flux k ka B ka t 3sinh L 3 (4.16) and an ideal t current sheet build-up J k ck B t 4 sinh ka (4.17) For the zero-wavenumber component, the k limit of (4.16) and (4.17) gives us and B ka t 3 L k 3, (4.18) J ckb t 4 a k (4.19) The transition to the nonlinear stage occurs in a non-constant- regime with a

3 4 1 4 very short time-scale, for Taylor s model. In the same procedure, c R L it can be shown that the ratio of the island width w to the reconnection layer size L 5 1 w 3 t ka 3 sinh ka L cl, L ~ a ka t (4.) with a much faster transition time-scale 3 5 1 5 1 5 c1 R. Then the transition happens at c 1 t, i.e., the reconnection enters the nonlinear stage directly in a L non constant- regime. We then can again make use of Sweet-Parker model to solve this problem. For the nonlinear phase, Ohm s law at the separatrix is d N dt y It is then easy to obtain a B ab y R N R N a ka t B 1 R N sinh ka a. (4.1) B L N By Ra 1 a where L ~ a is the length of the current sheet. The nonlinear solution of the (4.) reconnected flux is then N 3 1 5 ka a t Ba 5 sinh ka L N 1. (4.3) It can be seen that the reconnection time scale is ( ). Making 3 1/5 3/5 /5 N R SP use of the forced reconnection condition SP, we find that the time scale N is much faster than 1 1 SP R, almost independent of resistivity and controlled by the external driven as be claimed in SH simulation. For example, for typical space or solar plasmas, ~ 1, ~ 1, we have 8 R N ~ 4 1 ~. In fact, in SH, they calculated E at the reconnection point. From (4.3), we can get

E 3 1 3 3 Ba 3 5 1 5 E, N N N 1 N ka a t t 1 ~ c t sinh ka L c which has the same E and dependence as found in SH with E V B / c the boundary perturbation electrical field. The corresponding current then is J N 3 1 3 c ka a R t B 4 a sinh ka L 1. (4.4) We also run a numerical simulation of this forced reconnection model. From the theory, (4.17) and (4.19) give us J J, t /. (4.5) L nd for (4.4) with ka / in our numerical runs, is J N a R 9 L 1 J 3. (4.6) Figure 4.1 The current density vs. time t /, with 5 S 1. The fitting for the simulation curves, shown in Fig. 4., is J t Jt t t 1 R 3 Jt 7 J t t t t t which is perfectly in agreement to (4.5) and (4.6), with 1/ k L / k, or the parameter factor a L 1 (4.7) 6.3 9 / 11.. The transition point t 13 is also within the analytic range of c 1 t L. Clearly from our calculation, instead of being an example of Petschek-type

reconnection, the simulations can in fact be understood by the Sweet-Parker model. The simple model discussed in this section can be applied to many important problems with continuously external driven sources such as fast reconnection of solar coronal fields by foot-point convection or dayside reconnection on the magnetopause driven by solar wind. s be seen in the numerical simulations, once reconnection starts, its rate can be as fast as the rate controlled by the inward flow and almost independent of dissipation rate. This result will give a new base for the almost dissipationless IMF merging to the dayside magnetosphere and ideally fast solar physics phenomena such as solar flares. 4.3 In Rotating Plasmas THE BOUNDRY CONDITION In tokamak plasmas, forced reconnection on a stable rational surface can either be driven by islands of unstable modes on the neighboring rational surfaces, or by the wall error field. This kind of locking of tearing modes in a tokamak is important for designing of future generation of fusion devices [Fitzpatrick et al, 1993]. The equilibrium (4.1) can be used to approximate a resonant (rational) surface in the tokamak geometry. For the same equilibrium, if the plasma rotates periodically, and the rotating velocity on the resonant surface is V, the oscillation frequency of the applied perturbation is, then the Doppler-shift frequency is kv. i). If, in this slab approximation, we get back to the Taylor s model ii). If, in a frame moving with a velocity Vyˆ, we have a rotated model with a perturbed boundary x a cosky t. (4.8) IN THE OUTER REGION If we assume the perturbed flux x y t x ky t 1,, cos, (4.9) For the forced reconnection case, again we obtain (4.5) for the quasi-static outer

region solution sinh kx sinh kx x cosh kx B. (4.3) tanh ka tanh ka IN THE LINER PHSE We have to solve the initial value problem in this region. Only was a constant- phase solved there when t c,. n easy way to solve this /5 3/5 c R SP problem is to write the boundary perturbation as then y, t e 1 i ky t, (4.31) 1 x, y, t x e f x, t e i ky t iky x, y, t i x e ig x, t e i ky t iky The inner region linearized resistive MHD equations then are kx a g f t B. (4.3) f kb x a g f t a R. (4.33) Introducing kb g h / t, x/ a t f t xh h x, ka k, we then write Eq. (4.33) as f R. (4.34) k xf Making the Laplace translations, and matching the solution to the outer solution (4.3), we obtain 1 Bk k 4 B ka SP s i sinh ka 4ka tanh ka s s i sinh ka (4.35) which yields the reconnected flux it 4 B ka fl(, t) Le 1 it e sinh and the current sheet -it ka SP, (4.36)

JL t e ck B i -it, 1 sinh ka, (4.37) which means that for a typical perturbation (4.3), the growth of the current sheet on the separatrix has the same spatial and time pattern with a phase delay of /. THE TRNSITION TO THE NONLINER PHSE In the linear phase, t L i). If t 1, we are back to the HK case and have again a transition at t. c ii). However, if f (, t) L t 1, we have ka i B t i L, t sinh ka SP t (4.38) which is much less than the reconnected flux give by (4.7). In fact, there are hardly any fluxes reconnected in the linear stage compared to the case. It again ensures that the nonlinear phase starts in the non-constant-. IN THE NONLINER PHSE In this non-constant- Sweet-Parker nonlinear phase, following the procedure in the no-rotating case, we have the reconnected flux 3/ i3 t/ k e 3 sinh ka SP fn () i B a, (4.39) which has a phase delay of /, as well as a faster oscillation frequency 3 /, and the current sheet 3/ 1/ c k R i3 t / i3 t / N, 4a sinh ka J N B a e J e (4.4) without the phase delay of the linear stage. 4.4 Effects of Viscosity The outer region quasi-static solution is not affected by the viscosity. We focus our effort in the inner region. THE LINER PHSE