Current sheets in MHD turbulence

a.__

-_ I!3

PHYSICS

ELSEVIER

REPORTS

Physics Reports 283 (1997) 227-2.51

Current sheets in MHD turbulence S.C. Cowley”,

D.W. Longcopeb>‘, R.N. Sudanc

Unit’ersity ofCali$ornia at Los Angeles, Los Angeles, CA 90024, USA bSpace Sciences Laboratory, Unicersity of California Berkeley, CA 94720, USA ‘Laboratory of Plasma Studies, Cornell University, Ithaca, NY 14853, USA

aPhysics Department.

Abstract Current sheets may be the origin of fast dissipation rates in many MHD systems. The structure, formation and destruction of current sheets in force free MHD turbulence is reviewed. Special attention is given to the topological constraints that limit the possible sites of current sheets. The importance of field lines that form closed loops, including null-null loops, and the current sheets that form on them is discussed. In many situations the MHD approximation describes the plasma away from the current sheets but not in the immediate vicinity of the sheets. The plasma processes operating in the sheets are considered briefly. PACS:

52.30.Bt; 52.30%;

Keywords:

96.60.Pb

Magnetohydrodynamics,

Current sheets

1. Introduction In steady-state turbulence the energy input must be balanced by the energy sink - often the sink is the conversion of turbulent energy to heat via dissipation. In MHD turbulence, the turbulent energy is part kinetic energy, which is dissipated by viscosity, and part magnetic energy which is dissipated by resistance. Viscosity and resistance like most dissipative processes are diffusive processes. Diffusive processes dissipate energy at a rate proportional to ZF2, where 1 is the scale length of the turbulent energy being dissipated. In many turbulent systems the energy input is on large scales where dissipation is negligible. In order to dissipate the energy the turbulence must therefore transfer turbulent energy to small scales. There are two ways in which this transfer can be achieved. The first transfer mechanism is the cascade process where energy is passed successively from motions of a certain scale to motions of the next smaller scale. The energy is thus passed down to the dissipative scale via all intermediate scales. Kolmogorov [l] proposed the cascade mechanism to explain the dissipation of energy in incompressible hydrodynamic turbulence. Kolmogorov’s theory does indeed ’ Present mailing address: Physics Department, 0370-1573/97/$32.00 Copyright PZISO370-1573(96)00064-6

Montana State Univ., Bozeman,

0 1997 Elsevier Science B.V. All rights reserved

MT 59717, USA

228

S. C. Co~~le_vet ul. I Phpics

Reports 283 (1997) 227-251

describe many of the features of incompressible turbulence particularly the turbulent spectrum at small scales. The second transfer mechanism is the spontaneous formation of isolated small-scale structures in the turbulence. These small-scale structures would be singularities in the absence of dissipation. This mechanism can be distinguished from the first by the absence of energy in scales between the large and dissipative scales. Shocks formed in compressible turbulence and caviton formation in Langmuir turbulence [2] are examples of the second mechanism. Both mechanism can operate simultaneously. Indeed numerical simulations of incompressible hydrodynamic turbulence show strongly intermittent regions of localized vorticity. It is known that singularities cannot form in finite time in 2D hydrodynamic flows. Pumir and Sigia [3] have conjectured that singularities in the vorticity occur in 3D incompressible turbulence. Recent evidence for the formation of such a singularity is given by Boratav and Pelz [4]. In this paper we will discuss the role of current sheet singularity formation in turbulent MHD systems. We treat 2D and 3D cases separately although they have many similarities. The ideal dissipationless magnetohydrodynamic fluid admits discontinuous solutions. In general, these discontinuities have been classified as follows (see e.g. [5]): ( 1) Contact discontinuities: /)2)”= [u,] = [p] = 0 ;

[&I = 0 ; (2) Tangential PUn=O,

B”#O. discontinuities:

[vl#O,

l&=0=

[PIZO,

p+g [

(3) Rotational

[PI # 0 7

;

[Btl

# 0.

1

discontinuities:

[Pl=o=bl?

i.e. Btchanges direction but not its magnitude. (4) Shock waves: PU”#O,

[pl=O;

B,,B2 and ii are coplanar.

Here p, pv and B are the fluid density, pressure, velocity and magnetic field, respectively, ri is the unit normal across the discontinuity front in the direction from side 1 to side 2, subscripts n and t denote normal and tangential components, [ ] is the jump across the discontinuity (i.e. [A] = A2 - A 1 and subscripts 1 and 2 label quantities on the two sides of the discontinuity. At flow velocities well below the sound c, and Alfien velocity CA the production of shock waves is eliminated. We also were not concerned with contact discontinuities (1) between two media. In a low /I MHD system where p = cz/ci Q 1, the plasma pressure may be neglected with respect to magnetic pressure and the discontinuities of type (2) become one which we denote a current

S.C. Cowle~~ et al. I Physics Reports 283 (1997)

sheet that has an associated

rix

227-251

229

singular current K,

[B,]=K.

We shall devote most of our attention in this paper to current sheets in low p systems. Thus, we exclude discussion of systems which have high beta or large flows. The possible appearance of such current singularities in MHD system is of great importance in our understanding of the physics of solar and stellar coronae, planetary magnetospheres as well as magnetic configurations for plasma confinement. Normally, the spatial scale length I in astrophysical systems are so large that the Lundquist number S E rR/rA 9 1, it exceeds lOI for the solar corona; rR = 4rr012/c2 is the magnetic flux diffusion over a distance 1 and rA is the Alfven transit time over the same distance. In this limit the plasma is almost ideal and the nominal rate of dissipation, is exceedingly slow. There are, however, many situations in which the observed dissipative TR-‘, rate is much higher than the nominal rate r;‘. Perhaps the most dramatic examples are solar flares whose impulsive phases, during which a large amount of the energy is released, last - 100 s in a plasma where rR 2 10” s (see e.g. [6]). Another example is steady-state coronal heating which far exceeds the nominal resistive heating [7,8]. There is also evidence that efficient reconnection occurs in the interstellar medium. Measurements indicate that the galactic magnetic field has fluctuations 6B of magnitude comparable to the mean field B [9]. Theoretical arguments suggest, however, that without enhanced magnetic dissipation of magnetic flux the strong interstellar turbulence would rapidly generate 6B 9 B [lo]. One way to account for the observations is to postulate that actual scale length responsible for dissipation I, is orders of magnitude less than the nominal scale length 1. For an ideal current sheet 1,/Z + 0 and S, + 0 and thus the formation of current sheets may be the key to our understanding of fast dissipative phenomena in large-scale highly conducting plasmas. In actual fact I, would, of course, not be vanishingly small, but be of finite extent (S, + 1 N 102) determined by dissipative effects and the breakdown of the MHD approximation just as in the structure of MHD shock fronts. In addition to energy dissipation current sheets can effect a change in the magnetic field topology [l l-131. In stable equilibrium a magnetic field has the minimum energy permitted by its topology. By changing topology, and removing this constraint, reconnection gives the field access to still lower energy states. Relaxation to these states effectively liberates magnetic energy which can then take the forms of bulk plasma motion, nonthermal particles or radiation. Therefore driven systems with very large S would presumably evolve without significant dissipation, conserving their initial magnetic topology, if no current sheets were to form. If fluid motions lead to the formation of tangential/rotational discontinuities then finite dissipation rates with changes in global topology could be expected. Thus a driven, low /3, large S system would be strongly dissipative on time scales orders of magnitude less than ra and much of the observed phenomena, such as solar flares could perhaps be accounted for. Current sheets can be formed in stationary equilibrium plasmas and in plasmas with large flows. Many of the situations of interest (for example, the solar corona) are systems evolving slowly compared to the Alfven time and fast compared to the nominal resistive rate z;‘; we concentrate on such systems here. For these cases, one may assume the plasma is almost in a stationary equilibrium at all times and that the field line are “frozen in” during the evolution. The evolution of the system

230

S. C. Cowlry rt ul. I Ph_ysics Reports 283 (1997) 227-251

through a series of equilibrium states may take it from a state without current sheets to one with such sheets. Although this topic has received some study the key questions are only partially answered. Let us first outline these key questions since they provide the backbone of this paper. (i) What are the properties of these current sheets? Essentially, these properties are dictated by the jump conditions. This question is addressed in Section 2. (ii) Where do th e current sheets form? In particular, if we start with a smooth equilibrium and slowly change the boundary conditions, on which field lines do current sheets form? Current sheets cun and do form along boundaries separating domains of topologically distinct field lines. In Section 3.1 we discuss this in the context of two-dimensional fields with and without boundaries. The prototypical example of an equilibrium current sheet in two dimensions was given by Syrovatskii [ 141. In Section 3.2 these arguments are generalized to show the possibility of current sheet formation on topological boundaries in three dimensions. Current sheets can also form on isolated closed field lines in three dimensions (Section 3.2.2) as is well known in fusion research [15, 161. There has been considerable debate over the possibility of current sheet formation in threedimensional magnetic fields without topological boundaries or closed field lines. This possibility was originally raised by Parker [7, 171 and the context in which it is most often addressed is referred to as “the Parker problem”. This consists of a uniform straight magnetic field B = l&i anchored in parallel conducting planes at z = 0 and z = L (the solar photosphere). These planes are then deformed in an arbitrary but finite and spatially continuous manner. The initial field is free of topological boundary and the continuous motion does not introduce one. The question is whether such a field could ever relax to a discontinuous equilibrium. It was suggested by several subsequent authors that it could not [ 18,191. This argument and its implications are discussed in Section 3.2.1. Nevertheless, it was shown that extremely narrow intense layers of current were possible in the Parker problem [22] under fairly general conditions. Such layers, called pseudo-current sheets or pseudo-rotational discontinuities, which would have many of the physical properties of genuine current sheets, are discussed in Section 3.3. We shall assume that the topological structures of the field that allow current sheet formation (e.g. closed loops) are formed at a point when the plasma was much more resistive. They were then frozen into the plasma. This, for instance, applies to the solar corona where the basic topology was established deep inside the sun and rose buoyantly to the surface; similarly in tokamaks the closed field lines on rational surfaces are formed during the cold early phases of the discharge. We do, however, ignore important subtle effects where nulls and loops may arise during resistive evolution ~ an interesting example of this is described in a recent paper by Lau and Finn [23]. The topological structures (generically boundary mapping discontinuities see Section 3.2) cannot be formed by ideal flux frozen motion with continuous boundary conditions. (iii) How long does it take to form a current sheet? Formally, we seek the time 5F for the sheet to narrow to the dissipative scale. In some situations this time scales with S and thus would be very large in astrophysical plasmas. Thus for current sheets to be of interest their formation time must be either independent of S (finite time singularities) or proportional to In S. Formation times are discussed in Section 3.4. (iv) Are current sheets stable? Macroscopic instabilities may make the reconnection highly unsteady. This issue is presented in Section 5.1. Microscopic instabilities would presumably enhance dissipation in the sheet (see Section 5.2). In Section 5.3 we show how runaway electrons can be made in sheets.

S. C. Cm:ley rt ul. I Physics

Reports 283 (1997)

227-251

231

(v) What is the rate of dissipation and reconnection in the sheets? Magnetic reconnection of a current sheet of finite thickness (once resistivity is allowed) was treated first by Parker [ 121 and Sweet [ 1 I]. In the Sweet-Parker reconnection model the fluid flows towards the current sheet from both sides but the normal velocity 2;” is diverted to tangential velocity u, in the current sheet itself. The resulting rate of reconnection scales as S-I:‘. This rate is sufficiently slow that away from the immediate vicinity of the sheet the plasma is approximately in equilibrium and well described by Syrovatskii’s current sheet equilibrium solutions [ 141. A modification was introduced by Petschek [ 131, who introduced slow shocks to deflect the incoming fluid flow away from the current sheet. In Petscheck’s solution the system is not close to the current sheet equilibrium anywhere. The length of the current sheet in Petscheck’s solution is postulated to be short. This modification allows for greater flow velocities 0, resulting in the so-called “fast” reconnection where the rate scales like I/ In S. Recent numerical simulations by Biskamp [24] have failed to support the Petschek model. In fact, the structure of the sheet in Biskamp’s simulations is very similar to Syrovatskii’s current sheet. This has led Waelbroek [ 161 to suggest that the solution is well approximated by the Syrovatskii solution everywhere but in the narrow sheet. There continues to be debate about the possibility of fast reconnection. We cannot give a detailed review of this debate here. It must be appreciated that if reconnection is indeed fast then current sheets barely form before they are dissipated and equilibrium current sheet solutions have little relevance. (vi) Is the fin a 1 state stable? Does it lead to a fast Alfvenic motion and the possible formation of shocks? Almost nothing is known on this topic and we shall mention it only briefly in the conclusions. We have not resolved many of the questions raised in this paper. Our treatment is necessarily incomplete and biased towards parts of the problem that we feel most comfortable with. Extensive reviews of current sheet physics exist, for instance, the fine review by Priest [25]. These reviews generally cover different material than this paper and can be considered complimentary.

2. Jump conditions and the structure of current sheets in low p plasmas Current sheets in low /? equilibrium

satisfy the jump conditions

[B’] = 0

(1)

ix[B]=K

(2)

and the condition

on both sides of the sheet that

B, .ri=B2.1i=0. Here again fi is the unit vector normal to the sheet directed from side 1 to side 2 and K is the current per unit length in the sheet. Note both ri and K can vary in the sheet. Eq. ( 1) ensures that the normal forces on the sheet vanish. Amperes law applied to the current sheet as a whole yields Eq. (2) and Eq. (3) makes the tangential force on the sheet zero. Clearly from Eq. (3) the field lines on both sides of the sheet are tangential to the sheet. It is also clear from Eq. (1) that B changes its direction but not magnitude across the current sheet. Let us define the mean of the field on both

232

S.C. Codey

et al. /Physics

Reports 283 (1997)

227-251

sides of the current sheet (4) Then it is easy to show that K=c&

(5)

where r = 4(i x (B,

B2) +

ii

.B, &)2

x

=

[B] 82

B .

(6)

Note that the c( expression in Eq. (5) is hard to interpret where IsI = 0, however it is clear from Eq. (2) that at such points K = 2(n x B2). In two-dimensional cases where B is in a plane it is evident from Eq. (2) that K is normal to the plane and from Eq. (1) that B, = - B2 (B = 0) on the current sheet. In three dimensions B may vanish along particular lines on the sheet [26]. Now consider the regions not on the current sheet. The plasma equilibrium at low B satisfies the force free equation J x I? = 0, thus J is parallel to B. Therefore, since B, and B2 are tangential to the current sheet (Eq. (3)) there is no current flowing onto or off the sheet. Applying Gauss’s theorem to a thin volume which spans the surface yields K.iixdl=O

(7)

.i

where dl is a line element in the surface. This is a mathematical statement of the fact that the current on the sheet stays on the sheet. If the sheet is finite in extent and simply connected then the current path on the sheet must close. On a surface which is topologically a toroidal surface or infinite in extent the current path need not close.

3. Formation of current sheets In this section we discuss the sites where current form. 3. I. One- and two-dimensional

sheets form and the time the sheets take to

planar fields

As we have seen in the previous section when the field is entirely in one plane the jump conditions imply that the field reverses sign across the sheet i.e. B, = - B2. We argue from continuity in time (as we shall several times in this paper) that as the system is evolving from a smooth state to a state with a current sheet it passes through a state where the current sheet is very small in magnitude. At this point using Eq. (2) we find that B, and B2 are very small. Thus in the planar case the sheet must form on lines B = 0. A simple, almost trivial, planar case with a current sheet is the one-dimensional situation where Bf =&(x)9 and Bf(x) = B0 for x > 0 as Bf(x) = - B0 for x < 0. One might think of forming such a current sheet by cooling a plasma with pressure [28]. Suppose at t = 0, B = 2B0(x/L)$ for -L < x < L and p(x) = ~~(0) - 2Btx2/L2 so that the plasma is in equilibrium (p + B*/2 = ~~(0)). Let us cool

S. C. Codey

et al. I Physics Reports 283 (1997)

227-251

233

the plasma so that p + 0 everywhere; then conservation of flux (in the regions 0 -=cx < L and 0 > x > - L) and the equilibrium condition, Bf/2 = constant, yields the simple current sheet solution given above as the final state. The essential features of the two-dimensional planar case are contained in the solutions of Syrovatskii [14]. These solutions refer to forming a current sheet at a hyperbolic zero of B - an “x point” - in the smooth equilibrium. In fact they can also be applied to forming a current sheet at a particular elliptic zero of the smooth B, i.e. at an “0 point”. We noted that in the planar case where J is perpendicular to the plane, J x B = 0 implies that where B # 0, J = 0. Take B=V$xi. Everywhere

(8)

but on the current sheet, we have

-V2+Jz=0.

(9)

On the current sheet we must have $ = constant to satisfy Eq. (3). O$ must change sign across the current sheet. From elementary theory of functions of a complex variable we know that if F(z) = $(x, v) + ix&

Y) ,

(10)

where z =x + iy then $ satisfies Eq. (9). Syrovatskii wrote down a function F(z) which yields a $ with a current sheet on x = 0 between y = - b and y = + b. Syrovatskii’s F(z) is

(11) L

J

where the branch cut lies on the current sheet, i.e. at x = 0 between y = - b and y = + b. It is easy to verify that $ = 0 along the current sheet and O$ = &//ax = - By changes sign across the sheet. The total current in the sheet is I and the current distribution is K(y)

=

2((G’n) + W2/2>-

NY=)

pq7

.

(12)

If -ab2/2 < 1/27c < rb2/2 then the current is zero at c(y* = (1/27r) + ab2/2. We distinguish the following solution types: (i) & > $

c 1ockwise enclosed flux Fig. l(a),

(ii) A!._= Clb2 “Y solution” Fig. l(b), 27t 2 (iii) -$

< k

(iv) & < - $

< $

positive and negative current Fig. l(c),

anti-clockwise

enclosed flux Fig. l(d).

Note that in type (iii) sheets the current density is zero and B = 0 where the separatrix crosses the sheet. With no initial enclosed flux, i.e. a simple x point, only types (ii) and (iii) are topologically accessible in the limit of zero resistance in the plasma. The enclosed flux current sheets demonstrate current sheets at o points.

S. C. Colvley et al. I Physics Reports 283 (1997) 227-251

234

b

-2.5

2.5

-2.5

0.0

b

2.5

0.0

b

2.5

(b)

64 2.5

-2.5 -2.5 (c)

0.0

b

2.5

-2.5 -2.5 (d)

Fig. 1. Syrovatskii’s current sheet for 2 = h = 1 (see Eq. (11)). (a) Clockwise enclosed flux with I = 5. (b) The “T solution” with I = 71. (c) Positive and negative current on the sheet with I = 0.5. (d) Anticlockwise enclosed flux withZ= -5.

The obvious generalization of the Syrovatskii’s case is to remain two dimensional, but include a magnetic field in the z direction. In this case, the fields away from current layer need not be current free and obtaining solutions in any generality is difficult. A trivial special case is obtained by simply adding a constant z magnetic field to Syrovatskii’s solution. This solution yields identical B,, By and current to Syrovatskii’s solution (see Fig. 1). Force-free fields which are not current-free can also develop current sheets at X-points in the initial configuration. Longcope and Strauss [20] found one example whose initial magnetic field was a doubly periodic array of square magnetic islands. They showed that this equilibrium was linearly unstable and that other equilibria existed consisting of pentagonal islands (irregular pentagons) with current sheets separating pairs of them. Since a pentagon has a shorter circumference than a square,

S. C. Cm&y

et al. I Physics Reports 283 (1997) 227-251

235

for a given area, the former has shorter field lines and therefore lower magnetic energy. The relative difference between the two states was 2.7% of the non-potential energy. In addition, the continuous field aligned current inside each island was slightly lower in the pentagonal state. Pentagonal equilibria existed corresponding with each of categories (i)-(iv) above, and once again only (ii) and (iii) were accessible without magnetic reconnection. In addition, Longcope and Strauss showed that of the dynamically accessible solutions the Y-point solution had the lowest magnetic energy. Two-dimensional magnetic equilibria relevant to various configurations of the solar corona can be constructed by the introduction of a conducting boundary surface at x = 0. As in Syrovatskii’s model the field is taken to be current-free everywhere except the current sheet and expressed using a harmonic potential with a branch cut (the current sheet). Analytic functions have been found which model interacting dipoles [30] (see Figs. 3(a) and (b)) and a pair of neighboring arcades [31] (see Figs. 3(c) and (d)). In the case of interacting dipoles the current sheet forms where a null point had previously been, exactly as in Syrovatskii’s case. The neighboring arcades are an example of a current sheet forming where no X-point had been. This case uses a field similar to the Priest and Raadu field, but with the null point placed “below the photosphere” where it is not relevant. In this case, however, the manner in which field lines map from the conductor back to itself is discontinuous across a separatrix field line, which “grazes” the photosphere. The current sheet forms at this discontinuity in the mapping. Indeed, one feature of an X-point above the photosphere is that it also introduces a discontinuity in the field line mapping along its own separatrices. This mapping discontinuity is an important pre-condition for current sheet formation in two dimensions. This condition is discussed in the next section for three-dimensional fields. 3.2. Current sheets in three-dimensional jields Topological boundaries are also logical candidates for the location of current sheets in threedimensional fields. By topological boundaries we refer to those places where the field line mapping is discontinuous. In cases where there is no boundary we consider \rl --f 30 to serve as a boundary on which the magnetic field induces a mapping (i.e. a mapping of asymptotes). In two dimensions this generalized definition of a topological boundary (i.e. separatrix) includes all of the cases we have discussed, nulls and grazing field lines. The topology of three-dimensional magnetic fields is more complex and, in addition, to magnetic nulls we must include closed field lines among those features which introduce mapping discontinuities. Indeed, the problem is challenging enough that it is worthwhile considering first, the case without topological boundaries. In particular, the Parker problem, whose field lines all map continuously from z = 0 to z = L, serves as the prototypical setting for theories of such topology-free current sheet formation. 3.2.1. Parker’s model This simple problem was proposed by Parker to demonstrate a mechanism for heating the solar corona by current sheets. Begin at t = 0 with a uniform field B = &z^ embedded in a low-density conducting plasma with high-density conducting fluid boundaries at z = 0 and z=L. Suppose that these boundaries have smooth prescribed velocities in the x, y planes at z = 0 and L. Further suppose that the field lines are frozen to the low-density plasma and the boundary motions - the boundary velocity therefore prescribes the motion of the ends of the field line. The boundary conditions model the photospheric boundary of the corona. For boundary motions slow compared to the coronal Alfven

236

S. C. Cowley et ~1.I Physics Reports 283 (1997) 227-251

z=o

Fig. 2. Two field lines on either side of a current sheet for the “Parker Problem”. of the sheet and line 2, the dashed line, is underneath the sheet.

Line 1, the continuous

line, is on top

speed the light plasma will remain in force free equilibrium; this is called quasi-static evolution. Parker conjectured that the light plasma would form current sheets once the boundary motions had tangled the field lines with sufficient complexity. Van Ballegooijen [ 181, however, showed that the continuous line tying provides severe constraints that prevent the formation of a simple current sheet. Let us review a version of Van Ballegooijen’s argument. Assume, for the purpose of showing a contradiction, that a current sheet forms. Let this sheet be represented by the plane of the page in Fig. 2, of course, in reality it is not a plane but for the purpose of visualization we represent it by a plane. The field on top of the plane is B, and on the bottom, &. Consider a field line labeled 0 in Fig. 2 that lies immediately on top of the current sheet and another line (labeled @ in Fig. 2) which is on the bottom side of the sheet and starts from z = 0 infinitesimally close to 0. Since the boundary motions are smooth these two lines must have been infinitesimally close at t = 0 at z = 0 and z = L. Hence (again because of the smoothness of the boundary motions) at subsequent times Q and @ are infinitesimally close at z = L, see Fig. 2. Now we use the important fact, discussed in Section 2, that no current leaves the sheet. Thus, ,$ B. df = 0 for any loop on top or on the bottom of the sheet. Note we will not use any loops that cross the sheet. First, we use a loop on top of the sheet that starts at z = 0 and follows Q to z = L (path @ ) then returns to z = 0 following @ (path 0). It is important to note that we follow @ even though the field on the top of the sheet on path 0, B,, is not parallel to the direction of the path. For this first loop, we have (13) where dl is the incremental distance along the loop. The second loop we consider is on the bottom of the sheet following @ from z = 0 to z = L and @ from z = L back to z = 0. The magnetic field

237

S. C. Cowley et al. I Physics Reports 283 (1997) 227-251 AY

.!.x -a

a

(a)

!Gjiijl

4

3 2

1

Yo -1

(4

x

Fig. 3. Examples of current sheets formed on two-dimensional mapping discontinuities. (a) Two interacting dipoles without a current sheet. (b) Current sheet formed by pushing dipoles towards each other. (c) Arcades without current sheet. (d) Current sheet formed on “grazing” field line by moving arcades towards each other.

on the bottom of the sheet of path A is denoted &(I).

For this second loop, we have (14)

Note that since [B*] = 0 then II.?21= 1BI1= B(I). Let 0 denote the angle between top and bottom field lines so that B2 B1= B* cos 0 and llyl = B sin 8. Adding Eq. (13) and Eq. (14) we get

J B(I)(

1 - cos 0) dl

= 0

a+@

since the left-hand side is positive definite we must have cos 0 = 1 everywhere, the current sheet has zero current, i.e. does not exist.

(15) which means that

238

S. C. Cot&y

et al. IPhysics

Reports 283 (1997)

227-251

This proves the nonexistence of simple current sheets in this particular configuration. The proof has relied on the construction loops out of paths @ and @ which we cannot prove rigorously is possible for current sheets of arbitrarily complicated geometries. Within the limits described in Section 2, however, we find it unlikely that such pathological geometries are possible. Thus, it would seem that the continuous nature of the field line mapping prevents the formation of an internal current sheet in the Parker problem. As in the two-dimensional case [30], a discontinuity in the mapping would have defined a separatrix invalidating the argument, see Fig. 3 (see also [32]). In summary, we believe that a discontinuous boundary mapping is a necessary condition for the formation of a force-jiee current sheet. As we noted in the introduction such discontinuities can arise spontaneously during resistive evolution but not during the ideal evolution considered here. 3.2.2. Periodic systems Periodic boundary conditions where B(x, y,z) = B(n -t L,, y, z) = B(x, y + L,,z) = B(x, y,z + L,) are also of interest, especially in fusion problems. Here we shall consider the special case where all field lines that leave the surface z = 0 cross the surface z = Lj, i.e. they do not return to the surface z = 0. It is common in such problems to represent the mapping of the field lines from z = 0 to z = L3 by a Poincare surface of section - the intersection of the field lines with the x-y plane z = 0. Generically, [43] there are three kinds of field lines; closed jield lines that map onto themselves after a finite number of map iterations, field lines that lie on a surface (a line in the x, y plane) and ergodic field lines that fill an area of the x, y plane. The ergodic field lines map to a different point in the X, y plane on each iteration. We imagine, as before, changing the field in a flux frozen manner so that a current sheet just forms. Since the jump in the field across the sheet is infinitesimal, the current vector K is essentially in the direction of one of the undisturbed field lines just prior to the formation of the sheet. Current sheets in the form of a ribbon can form on closed field lines - we shall discuss such loops shortly. Flux surfaces - surfaces on which field lines lie - are of two types: “islands” where the surface forms a closed line in the Poincare section and surfaces that span the periodic box. On both kinds of flux surface, one can define a winding number for the field. To do this for the nonisland surfaces we follow the field line on the surface and count the number of times it crosses z = 0, N, and the number of times it crosses x = 0, M. The island surface winding number is similar but we define M as the number of times the field line goes around the island. The winding number is the ratio M/N in the limit that we go an infinite distance along the field line. If the winding number is rational then the field line is closed. If it is irrational then one field line essentially covers the whole surface _ the surface is then called an irrational surface. Flux frozen motions cannot change the winding number. Current sheets cannot form on field lines on irrational surfaces, this can be seen from the following argument. Suppose (for contradiction) there is a current ribbon following a line on an irrational surface. Since the current is unidirectional on the ribbon and the line covers the surface the whole surface must be covered with a unidirectional current. From the jump condition Eq. (I), it is clear that the winding numbers on top and on bottom of this sheet are then different. Such a jump is impossible since the winding number is conserved by flux frozen motions and prior to the formation of the sheet no such jump existed. Thus, the sheet formation is impossible on irrational surfaces. On rational surfaces the B lines are closed and so the K lines are also closed. Thus, it is possible to have positive and negative currents in sheets on these surfaces so that there is no jump in the

S. C. Codey

et al. I Physics Reports 283 (1997) 227-251

239

A Type Null

B

D B Type Null Fig. 4. A simple and null-null

loop.

winding number across the sheet. Thus current sheets can form on rational surfaces - this is just a restatement of the previous statement about closed field lines in the situation where the field lines lie on surfaces. Current sheets cannot, it seems, form on an ergodic field line. To show this imagine forming a ribbon on such a field line, the ribbon contains a small constant current but infinite current density. Iterating the map infinitely many times puts infinite current density everywhere in the ergodic region. An infinite total current is surely absurd and we therefore conclude that no such current ribbon can be formed. 3.2.3. Closed field lines We now focus our attention on closed field lines since, in general, we expect these to be sites of current sheet formation. Indeed current sheets that are simply connected, finite in extent and do not intersect any boundaries must form on closed field lines. This can be seen by again considering the instant at which the sheet just forms. At this time K is parallel to the field on either side of the sheet. Therefore, since K lines must form loops on the sheet (see Section 2) so must B lines. In general there are two types of closed field line loops: “simple loops” in which B # 0 anywhere on the loop and “null-null loops” (field nulls where B = 0 are discussed in [41, 421) where two field lines (called separators) emanating from a type B null terminate in a type A null, see Fig. 4. In a vacuum field only null-null loops exist since we must have $ B . dl = 0 around the loop. The current sheet formation on null-null loops is discussed in Longcope and Cowley [26]. Current sheets on simple loops are well known in tokamaks where the nonlinear evolution of the internal kink gives rise to a sheet on the q = 1 rational surface [27]. First let us examine the field line structure on loops. We illustrate current sheets on simple loops in Figs. 5(a) and (b) and null-null loops in Figs. 6(a) and (b). The ribbon-like sheets are flattened to the plane of the paper and the field lines on top (the continuous lines with arrows) and underneath

240

S. C. Cowley et al. I Physics Reports 283 (1997) 227-251

Fig. 5. Current sheets on a simple loop (called simple current sheets). Continuous/dashed lines with arrows are field lines on the top/bottom of the sheet. (a) A Y-type simple current sheet. (b) A simple current sheet with positive and negative current. The dotted line is the K = 0 line. See text for details.

Bl

Fig. 6. Current sheets on a null-null loop (called null-null current sheets). Continuous/dashed field lines on the top/bottom of the sheet. The dotted line is the K = 0 line. (a) A Y-type (b) A null-null current sheet with positive and negative current. See text for details.

lines with arrows are null-null current sheet.

S. C. Cowkey et al. I Physics Reports 283 11997) 227-251

241

(the dashed lines with arrows) are drawn. Fig. 5(a) illustrates a case with current unidirectional on the sheet; in cross section (i.e. in a plane perpendicular to the singular current direction), the field structure around the sheet resembles Syrovatskii’s “Y-type” solution illustrated in Fig. l(b). Fig. 5(b) illustrates a sheet with positive and negative current which, in cross section, resembles Fig. l(c). The lines on which K = 0 are dotted. A null-null current sheet with unidirectional current is illustrated in Fig. 6(a). The A- and B-type nulls are split into two A-type nulls, Al and A2, and two B-type nulls, Bl and B2. The sheet in Fig. 6(a) resembles the “Y-type” sheet of Syrovatskii (Fig. l(b)) in cross section (except at the nulls). Finally, Fig. 6(b) illustrates the null-null sheet with positive and negative current - again the K = 0 line is dotted. The field structure of Fig. 6(b) in cross section resembles Fig. l(c) except at the nulls. Some discussion of the global structure of the null-null loop fields is given in Longcope and Cowley [26]. We have not shown &en current sheets form on loops but we have shown that they could. An evolution of the external field would, in general, attempt to alter the flux through a given loop [26]. To prevent this, and obey constraints of frozen-flux, a current flows along the closed loop. This current generates a self-flux through the loop which cancels the externally imposed change. The relation between the current and its self-flux is a self-inductance. Longcope and Cowley find that current sheets on closed loops have a nonlinear self-inductance (i.e. the inductance depends on current). This problem can be posed in the solar corona, where the potential component of the field reflects the locations of magnetic features (i.e. active regions) on the photosphere [33]. This field also contains separators which are the only truly closed field lines in the problem. As the magnetic features move current is induced along these separators [34]. This current stores energy (see Section 4) and are likely candidates for reconnection. Reconnection occurring at these specific sites is qualitatively and quantitatively suggestive of observed features of coronal evolution such as transient loop brightenings [35,34] X-ray bright points [36] and solar flares [37]. In a turbulent plasma, a distribution both simple loops and null-null loops can be expected. This distribution will presumably dictate the distribution of current sheets in the evolving plasma. Thus to understand the physics of a turbulent magnetized plasma it is necessary to understand the statistics of loops; the distribution of sizes, fluxes, self-inductances and mutual inductances. Such a complex study involves analyzing the statistical topology of a random vector field. Clearly, we give no proof that the sites of current sheet formation must be loops in all cases. Further work is needed to classify the kinds of sheets that are possible in general. Of course in plasmas with finite p we can escape the restriction imposed by the conservation of current on the sheet. This allows greater freedom in the possible sites of sheets.

3.3. Pseudo-current sheets In the Parker problem current sheets were ruled out because the field line was tied to boundaries in a continuous manner. It is worth asking “how continuous” a typical mapping would really be. To answer this question quantitatively we define the mapping by following each field line from z = 0 to z = L. The field line itself, (x, v) = xl(z), satisfies the equation

(16)

242

S. C. Cowley et al. I Physics Reports 283 (1997) 227-251

Integrating

this from a point on the z = 0 plane, x(z = 0) = xy, to z = L gives the mapping

Xl(Z = r,) = x;
(17)

Before boundary motions (t = motions of the boundary fluid, To simplify things we will plane, with a velocity ul. L The dx$ = dt

0), the field is straight and the mapping is trivial: xi(xj) = xj. The however, cause this to change. consider holding fixed the z = 0 plane and moving only the z =L mapping will then evolve according to

uj(xj).

(18)

The inadmissibility of current sheets follows because the gradient of this mapping By gradient we mean the Jacobian matrix of derivatives

is bounded.

1

axL/axO axL/ayO &L(4)

=

Differentiating

4%

-

dt

=

i ay”/axO

ay”/ay”

Eq. (18) in an analogous CL

XL

(19)

.

fashion leads to an equation for the evolution

of 2

(20)

>

where CL is the rate-of-strain

tensor for the boundary

motion

av,L/axL av,;/ayL CL(Xj>

=

[ a$IaxL

av$layL 1

’

(21)

The mapping will become most distorted, and its gradients the largest, at a point where the flow is locally hyperbolic. A constant velocity will not affect the rate-of-strain, so we use “hyperbolic” to refer to the local behavior relative to the fluid element; the point need not be a stationary point of the flow. Indeed, this designation applies simultaneously to whole domains of the flow. We focus our analysis about that point at which the rate of strain is largest. At such a point the rate-of-strain tensor takes the generic form

(22) where A is the Lyapunov exponent, and the principle axes of the hyperbolic point have been aligned with the coordinate axes. Based on dimensional arguments we expect 3, N 27c/r.,, where r, is average the eddy turnover time of the boundary flow. Assuming the axes of the hyperbolic point remain stationary Eq. (20) can be integrated in time (23) As expected, there is one component of the derivative along the direction along which the flow converges.

matrix which grows rapidly in time; this is

S. C. Cowley et al. I Physics Reports 283 (1997)

227-251

243

To ascertain the influence of this motion on currents we must express yL in terms of magnetic field lines. For simplicity, we make the reduced MHD approximation - that B, is large and L is long. Differentiating (18) gives [22]

dA = Y(z) . jz

(24)

)

dz

where yZ is Jacobi matrix of the partial mapping to z and Y(z)

is (25)

Integrating this from z = 0, where f0 = 3, to z = L gives an expression terms of the magnetic field. At a hyperbolic point in the field

0 y=a 0-a 1’ [

for the mapping gradient in

(26)

where scaling implies IX- Jz/B,. If the field line hyperbolic point coincides with the fluid hyperbolic point then analogous integration would yield CCL= At. In terms of currents this means J, N (t/ze)B,/L so the current grows only linearly in time. In a case analyzed in detail by Longcope and Strauss [21] this turned out to be the case for a known (symmetric) equilibrium. This equilibrium was linearly unstable and numerical simulation suggested a second equilibrium containing something like a current sheet near what had been the hyperbolic field lines. Since genuine current sheets are forbidden in this problem they hypothesized that the final equilibrium might contain a layer of intense, but finite current density J,. Near such a layer they showed that the matrix Y took the form

F(z)=.@“.

%,

(27)

where BZ is the unitary matrix which rotates the sheet’s normal into f in the plane z. The middle factor in this expression is typical of a sheared magnetic field in contrast to Eq. (26) which typifies hyperbolic behavior. Using this in Eq. (24) gives [22]

. 80.

(28)

As a result of the sheared rather than hyperbolic field structure the matrix depends on J, linearly rather than exponentially. Setting this expression equal to Eq. (23) shows that the current layer must rotate by 90” from z = 0 to z = L and also gives an expression for J, J, = eAt:

N exp(2rct/z,)

.

(29)

In conjunction with this exponentially large current density there is a correspondingly small layer thickness Sideal- e-“. This thickness follows from ideal considerations; in particular, it reflects the nature of the field line mapping. While the continuity of the mapping prevents a genuine discontinuity

244

S. C. COIL&~ ct (II. I Physics

Reports

283 (1997)

227-251

(bidear> 0), the mapping is only “mildly continuous” so exponentially narrow layers are permitted. The specific case considered by Longcope and Strauss [22] has a two-dimensional analog in which genuine current sheets do occur [20]. The two-dimensional result can be recovered in the above analysis by taking L + q which gives a value Sideal= 0. The above argument assumes that fluid trajectories and equilibrium magnetic field lines behave differently. We assumed that the fluid motion at the boundary was characterized, at least at certain points, as hyperbolic in a frame moving with the fluid element. This will generally be true of isotropic fluid motion, even unsteady motion. If the magnetic field behaved similarly (i.e. Eq. (26)) the current density would never become very large. Conversely, if the field lines tended to become locally anisotropic, developing sheared rather than hyperbolic nature, then the current densities can become exponentially large. The growth to large densities occurs within several turnover times of the fluid, so its long time behavior is irrelevant. In the case studied by Longcope and Strauss [2 1,221 the initial magnetic field does have a hyperbolic nature, but when /It exceeds a threshold for instability it undergoes nonlinear evolution which gives rise to pseudo-current sheets. Numerical simulation of random isotropic end-plate motions [38] suggests that pseudo-current sheets are generic features. 3.4. Formation times The time to form a current sheet is defined as the time to for the current sheet width shrink to the dissipative scale. Let us denote the dissipative scale by 6. The timescale for formation is crucially dependent on the compressibility of the plasma. We start by examining the simple one-dimensional cooling model [28] discussed in Section 3.1. They assume a simple exponential cooling law with a cooling time r5,. The time for the current sheet to form tf is tf cc - yz, In 6 ,

(30)

where g is the ratio of specific heats. The one-dimensional model is special because the plasma cannot be removed from the sheet by flows along the sheet. The removal of the plasma from the sheet in two and three dimensions slows the formation just as it slows reconnection in the Sweet-Parker models. Shocks are well known to form in a finite time independent of dissipation, however current singularities of the type we are examining may not. Intuitively, one suspects that current sheets involve the kl, = 0 shear Alfven mode which has zero frequency. Thus, it is likely that current sheet form in a time that goes to infinity as dissipation goes to zero. In the situation where the boundaries are being distorted slowly and the plasma is almost everywhere in equilibrium the plasma around the current sheet will then remain somewhat out of equilibrium. However, macroscopically we expect the current sheet to resemble the ideal solutions [ 161. Brandenberg and Zweibel [40] have shown how plasma can be removed from the layer by recombination in partially ionized plasmas. This allows them to obtain very fast current sheet formation.

4. Energetic consequences of current sheets Until this point we have considered purely ideal equations which, in certain cases, lead to discontinuous magnetic equilibria: current sheets. The inclusion of resistivity will resolve this discontinuity into a layer of thickness 6. This thickness presumably depends on resistivity such that S(q) * 0 as

S. C. Cowle~y et ul. I Physics Reports 283 (1997)

227-251

245

y + 0 returning a genuine discontinuity in the ideal limit. ’ Pseudo-current sheets are presumably very similar to genuine current sheets when the resistive thickness 6 >> 8ideal the thickness for the ideal field. The large-scale properties of the current sheet, its transverse scale d (which is called b in the Syrovatskii solutions in Section 3.1) and net current I, are affected only slightly by the presence of resistivity. 4.1. Ohmic dissipation In the Sweet-Parker limit the thickness 6 is determined from a balance between resistive current diffusion (outward) and advection (inward) by plasma being drawn into the current sheet. Conserving plasma mass, and restricting the outflow velocity to the local Alfvkn speed gives the well-known Sweet-Parker scaling (5N

A/P4(y/lrl)‘,‘2

(31)

)

where p is the plasma mass density. Here we have used Ampere’s law to combine the magnetic field strength B_L outside the current sheet (i.e. the part that is discontinuous) with the sheet’s breadth: I- BLA. The Ohmic power dissipated by this sheet is P,, = q

.I’

J2d3x N

(32)

Thus, for a given magnetic configuration the Ohmic power scales as yl’.‘. In a magnetized plasma driven from the boundaries, however, the magnetic configuration will become increasingly stressed until resistivity can compensate [38]. The work done by boundary motions can be expressed an external change in flux, Y though each current sheet [26,34]. For a collection of N current sheets this power (equivalent to a Poynting flux) takes the form P,-N@I.

(33)

Over time the current in each sheet will increase until P,, is sufficient to balance this input power, and this will be the statistical steady state. This occurs for an average value of current-sheet current

(34) Replacing this in Eq. (32) and dropping constant factors results in an Ohmic power P,! N q-1.3 which increuses with decreasing resistivity [38]. The physical mechanism underlying this paradoxic behavior is the following: Lowering resistivity lowers the efficiency with which field can be untangled, thus allowing more stress to be accumulated. Realistically this trend cannot continue to arbitrarily small resistivity, as that would yield current sheet currents 111+ 0~. It is tempting to believe that current sheets are not stable above some current. * This assumption is a rather dubious one, as seen in the case of hydrodynamics is known to differ from the inviscid system described by the Euler equations.

where the limit of vanishing

viscosity

246

S. C. Codey

et al. I Physics Reports 283 (I 997) 227-251

4.2. Global energy liberation Local analysis of steady-state Sweet-Parker reconnection shows that magnetic energy is also lost to the accelerate the plasma in the outflows. The power lost through this process is several times the Ohmic power, but this factor does not vary with VI.This local, steady-state analysis, however, misses the most important energetic contribution of a real current sheet. By eliminating field-line constraints the process of reconnection gives the plasma access to magnetic equilibria whose global energy is lower than it otherwise could be. In their example of doubly periodic two-dimensional equilibria Longcope and Strauss [20] found that reconnection would release nearly 50% of the nonpotential magnetic field energy. This energy is a global one and cannot be estimated from local analysis of the current sheet. Conversely, knowing the difference between equilibrium energies before and after reconnection does not specify how the liberated magnetic energy will be dissipated. There will certainly be global plasma motions as the large-scale field is rearranged to its new equilibrium. Presumably these motions will dissipate through eddy-cascade and viscous dissipation. In the presence of the magnetic field this process may resemble nonlinear interaction of Alfven waves more than eddy-cascade, but the ultimate fate of the energy is the same. Recent theories of three-dimensional current sheets has given a much clearer picture of how this global energy release would work in turbulent low /? plasmas [26,34]. Current sheets on closed magnetic field lines can be treated as nonlinear inductive elements [26]. The ring of current generates a self-flux through its core Y N YZ, where T(1) N -In 111is the self-inductance of the current sheet. This self-flux occurs to cancel an imposed change in flux which has been externally applied, e.g. through boundary motions [34]. In addition to preventing a change in flux, the current in the sheet stores additional magnetic energy, as any inductor would I

E=

.I’ n

5f(I’)l’ dI’ - $I*

.

(35)

Reconnection eliminates the need for flux conservation, thus permitting the current in the sheet to vanish, and its stored energy to be released. In a system where reconnection could occur rapidly and the external motions are slow this release can be considered instantaneous. The resulting evolution is one of slow storage of free energy punctuated by sudden release [26,34]. Postulating a threshold current Ith needed to trigger fast reconnection permits a general expression for the energy liberated through repeated reconnection episodes. Take the (slow) external rate of flux change for the current sheet to be !@. Averaging over the periodic releases of energy gives an expression for the dissipated magnetic energy [26]. (36) Had fast reconnection been possible for even small currents (i.e. Petcheck reconnection) we would have to take I,,, + 0 and find that y10power is lost through reconnection. This simply means that if reconnection is “too easy” the field will never develop current sheets to store energy. In actuality there will still be a Poynting flux across the boundaries, but this will scale as 111/l’.By considering quasi-static evolution we have neglected this possibility.

S.C. Cowley et al. IPhysics

Reports 283 (1997) 227-251

247

In a model of the solar corona [34], the change in flux through a separator can be related to the velocity of some foot-point motion + = dY/dx . u , where the derivative dY/dx depends only on the magnetic configuration, and not on the instantaneous foot-point velocity u. Using this in the expression for dissipated power gives P - - F . u, where

is a friction whose sign always opposes u. The resemblance to mechanical friction here is not fortuitous; the slow development and rapid dissipation of current sheets causes the energy to increase and drop in a fashion analogous to simple models of friction [39]. In order to evaluate the contribution to coronal heating it is necessary to determine the statistical distribution of separator loops along with their derivatives dY/dx.

5. Macro- and micro-stability

of current sheets and electron runaway processes

5.1. Macro-stability In cases of interest (e.g. the solar corona) the MHD approximation is good away from the current sheet itself. In the current sheet the effects of resistivity, electron inertia and the Hall terms are important. As a current sheet forms it narrows and steep gradients occur. These gradients can drive macroscopic instabilities of the current sheet. Resistive tearing modes [46] can be driven by the current gradient. These modes can tear the sheet into island structures. This instability would occur during formation if,

YFKR Z A' =

(Ll')4'5TR315T~2i5 E yo(k,)(6/60)p2

[(k,,6)p' -

k,.6]/4n: = l/47$6

)

3

(37)

is the linear growth rate and z-’ = d6/6dt is the time scale for the formation of the current sheet; TR= 47ca~?~/c’ is the resistive time scale and rA = (k),~~)-’ is the Alfven time scale. If the current sheet develops exponentially as s(t) = 60e-f!fn then

Eventually, the sheet reaches a steady thickness, perhaps the Sweet-Parker thickness, see Eq. (3 1). The tearing mode growth is substantial by the time this thickness is reached so one would not expect to form a unbroken sheet. However, one must be cautious since the strong flows make the estimate above dubious with thin sheets. Assuming that the FKR instability sets is then the nonlinear evolution of the magnetic islands [50] follows a secular evolution with time. When the width of the island becomes comparable to 6 the behavior of this mode is determined by external driving forces [53]. Enhanced energy dissipation takes place in a boundary layer near the separatrix [52] while the interior domain remains approximately adiabatic. The upshot is a more rapid dissipation of the

248

S. C. Confey et (II. IPhysics

Reports 283 (1997)

227-251

magnetic flux than in a stable current layer because of the increase in current density close to the separatrix. In the sheet the Ohm’s law must be augmented by the Hall term,

E+LB-(jXBLj,a. c n,ec J’

’

(39)

where E is the electric field and ~2, is the electron density. The instability still occurs because it is dependent on the resistive action along the field lines, i.e., because El, = j,,/a must be retained; in the direction perpendicular to B we may neglect j,/o because typically v,, Q Sz,. However, the Hall term introduces much smaller scales of the order of L,,,%& where he = C/C+ is the collisionless skin depth and CK’ is the time scale of the instability, which becomes overstable [45]. Shorter scale lengths introduced by the Hall term can lead to faster rates of dissipation and magnetic reconnection. Recent two-dimensional numerical studies by Biskamp et al. [44] which include both the Hall term and electron inertia in Ohm’s law show that decoupling of electron and ion dynamics takes place due to the Hall term on scales less than the ion inertial length ii = c/n)r,. The reconnection rate is essentially independent of the electron mass, perhaps because the electron inertial term plays less of a role than the Hall term. The reconnection rate in the current sheet that develops at the X-point between two bundles of flux associated with two neighboring parallel currents approaches the Alfven rate that depends only on the global configuration. The electron and ion flow pattern is similar far from the X-point but near the X-point these patterns differ: the electrons have a converging pattern while the ions appear to form a macroscopic Sweet-Parker-like layer of width ;li and outflow velocity CA. Thus, the ion inflow velocity is licA/L, where L is the length of the layer. The conclusion to be drawn from the studies of Biskamp et al. [44] is that a current layer of thickness 6 could break up into many magnetic islands of thickness JVi+ 6 which results in rapid local magnetic reconnection at the Alfven rate with perhaps an enhanced average rate of reconnection over the entire current layer well above the Sweet-Parker rate. The exact rate of enhancement needs further study. 5.2. Microstability Microinstabilities in the current layer will be initiated when the electron drift along the field lines exceeds the sound velocity, i.e., j,, = I/6 = cB/4&

> Fz,ec,

or 6 < 6, s 4m,ec,/B

= Ai/@,

(40)

where p = 8meT/p. The subsequent microturbulence and electron heating due to the excitation ion acoustic waves, if (40) is satisfied, would cause the electron temperature to remain at the value required by the condition for marginal stability, i.e. 6, = 6. More careful analysis by Galeev and Sagdeev [47] would indicate a factor of approximately i in front of 6. Although this turbulence is accompanied by an anomalous collision frequency v, [51] on the fast quasilinear time scale, on the

S. C. Cowley et al. I Physics Reports 283 (1997) 227-251

249

longer diffusive time scale the increase in electron temperature will cause a decrease in the Coulomb collision frequency and therefore an increase in the conductivity and thus lead to an increase in the resistive time rR. Similar considerations would apply for the excitation of lower hybrid waves. What needs further investigation is the effect of microturbulence on the evolution of the resistive tearing modes; the mere replacement of V,i by v, is perhaps a little too simplistic. 5.3. Runaway electrons in current sheets For a sufficiently strong electric field along the field lines in a resistive current layer a pronounced departure of the electron distribution from the Maxwellian occurs that manifests in a flow of runaway electrons along the main component of the magnetic field. Even if at some stage most of the current is carried out by the thermal electrons ultimately it is the runaways that will carry all the current. At this stage the electric field along the magnetic field gradually vanishes. Microinstabilities generated by the runaways in the current sheets may give rise to anomalous dissipation and intermittent behavior on very fast time scales [48]. The thickness of the runaway current sheet is much smaller than the initial resistive current sheet from which it evolves. The runaways are subject to the anomalous doppler cyclotron instability [49]. This instability leads to rapidly growing frictional force that increase the transverse temperature of the runaways. An inductive electric field is induced in the sheet to maintain the current that keeps the magnetic field constant. A group of fast runaways are accelerated by this inductive field and retain their runaway status in spite of the decelerating force. The other electrons slow down and are scattered. The detailed study of electron runaway processes is very important for understanding the copious emission of X-rays in transient solar events such as flares. The final stages in the evolution of current sheets where kinetic processes dominate is an important area for further research.

6. Conclusions In this paper we have discussed the issues of current sheet structure, formation, energetics and stability. We have concentrated on current sheets formed in force-free systems evolving through a series of equilibria. In force-free systems there are severe constraints on the sites of current sheet formation. These constraints are: that the current on the sheet stay on the sheet (see Section 2) and that at the instant of current sheet formation the current on the sheet is parallel to an existing field line. We show how these constraints make singular current formation in the Parker model (see Section 3.2.1) impossible. It has been recognized for some time that (for lines embedded in a moving conducting boundary) current sheets can form on field lines on which the field line mapping from boundary to boundary is discontinuous [30-321. By explicit construction we show that current sheet can form on closed field lines of which there are two types, simple closed loops and null-null loops. We suggest, without a general proof, that force-free current sheets only form on lines with a discontinuous boundary mapping or closed field lines. The dissipation of energy by current sheets is discussed in Section 4. With a simple critical current model we show how the energy liberated can be estimated without a detailed knowledge of the dissipative physics in the sheet.

250

S. C. Cowley et ul. I Physics Reports 283 (1997) 227-251

In Section 5, we discuss the stability of current sheets and the generation of runaways. Reconnection may involve a complicated interaction between microturbulence (whistler turbulence perhaps see [54]) and macroscopic motions. Collisionless shocks have a well-known interaction between microinstabilities and macroscopic motion that may be similar. Reconnection may lead to a final state that is ideally unstable. Presumably such a situation would give rise to fast motions. Clearly this is a very active research area and many questions are unanswered. It is hoped that this review provides a basis for further research.

Acknowledgements The authors enjoyed many stimulating discussions with the participants in the 1995 ITP Workshop on Turbulence and Intermittency in Plasmas. This work was supported in part by the National Science Foundation Grant No. PHY94-07 194.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [ 171 [ 181 [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

A.N. Kolmogorov, C.R. Acad. Sci. U.R.S.S. 30 (1994) 301. V.E. Zakharov, Zh. Eksp. Teor. Fiz. 62 (1972) 1945. A. Pumir and E.D. Siggia, Phys. Fluids 30 (1987) 1606. O.N. Boratav and R.B. Pelz, Phys. Fluids. 6 (1994) 2757. L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1960) pp. 224233. E. Tanberg-Hanssen and A.G. Emslie, The Physics of Solar Flares (Cambridge University Press, Cambridge, 1988). E.N. Parker, Astrophys. J. 174 (1972) 499. W.H. Tucker, Astrophys. J. 186 (1973) 285. C. Heiles and E. Zweibel, Nature, accepted for publication. R.M. Kulsrud and S.W. Anderson, Astrophys. J. 396 (1992) 606. P.A. Sweet, IAU Symp. 6 (1958) 123. E.N. Parker, J. Geophys. Rcs. 62 (1957) 409. H.E. Petscheck, AAS-NASA Symp. on Physics of Solar Flares (1964), p. 425. S.I. Syrovatskii, Sov. Phys. JETP 33 (1971) 933. W. Park, D.A. Monticello, R.B. White and SC. Jardin, Nucl. Fus. 20 (1980) 1181. F.H. Waelbroek, Phys. Fluids B 1 (1989) 2372. E.N. Parker, Spontaneous Current Sheets in Magnetic Fields (Oxford University Press, Oxford, 1994). A.A. Ballegooijen, Astrophys. J. 298 (1985) 421. S.K. Antiochos, Astrophys. J. 312 (1987) 886. D.W. Longcope and H.R. Strauss, Phys. Fluids B 5 (1993) 2858. D.W. Longcope and H.R. Strauss, Astrophys. J. 426 (1994) 742. D.W. Longcope and H.R. Strauss, Astrophys. J. 437 (1994) 851. Y.-T. Lau and J.M. Finn, Phys. Plasmas 3 (1996) 3983. D. Biskamp, Phys. Fluids 29 (1986) 1520. E.R. Priest, Rep. Prog. Phys. 48 (1985) 955. D.W. Longcope and S.C. Cowley, Phys. Plasmas 3 (1996) 2885. M.N. Rosenbluth, R.Y. Dagazian and P.H. Rutherford, Phys. Fluids 16 (1973) 1894. V. Dormand and R.M. Kulsrud, Astrophys. J. 298 (1995) 421. D.W. Longcope and H.R. Strauss, Phys. Fluids B 5 (1993) 2858. E.R. Priest and M.A. Raadu, Solar Phys. 43 (1975) 177.

S. C. Cowley et ul. I Physics Reports 283 (1997) 227-251 [3 l] [32] [33] [34] [3.5] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [SO] [5 I] [52] [53] [54]

251

B.C. Low, Astrophys. J. 323 (1987) 358. E. Zweibel and H.S. Li, Astrophys. J. 312 (1987) 423. P.J. Baum and A. Bratenahl, Solar Phys. 67 (1980) 245. D.W. Longcope, Solar Phys. (1996), in press. T.S. Shimizu, Tsuneta, L.W. Acton, J.R. Lemen and Y. Uchida, Proc. Astron. Sot. Japan 44 (1992) L147. C.E. Pamell, E.R. Priest and L. Golub, Solar Phys. 151 (1994) 57. P. Demoulin, L. van Driel-Gesztelyi, B. Schmieder, J.C. Henoux, G. Csepura and M.J. Hagyard, Astron. Astrophys. 271 (1993) 292. D.W. Longcope and R.N. Sudan, Astrophys. J. 437 (1994) 491. J.B. Marion and S.T. Thornton, Classical Dynamics of Particles and Systems, arcourt Brace Jovanovich (1988). A. Brandenburg and E. Zweibel, Astrophys. J. 427 (1994) L91. J.M. Greene, J. Geophys. Res. 93 (1988) 8583. Y.-T. Lau and J.M. Finn, Astrophys. J. 350 (1990) 672. A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer, Berlin, 1983). D. Biskamp, E. Schwartz and J.F. Drake, Phys. Rev. Lett. 75 (1995) 3850. J.F. Drake and Y.C. Lee, Phys. Rev. Lett. 39 (1973) 453. H.P. Furth, J. Killeen and M.N. Rosenbluth, Phys. Fluids 6 (1963) 459. A.A. Galeev and R.Z. Sagdeev, Current Instabilities and Anomalous Resistivity of Plasmas in: Handbook of Plasma Physics, Vol. I, eds. A.A. Galeev and R.N. Sudan (North-Holland, New York, 1984) p. 285. A.V. Gurevich and R.N. Sudan, Phys. Rev. Lett. 72 (1994) 645. V.V. Parail and O.P. Pogutse, Review of Plasma Physics, vol. II, ed. B.B. Kadomtsev (Consultants Bureau, New York, 1986) p. 5. P.H. Rutherford, Phys. Fluids 16 (1973) 1903. R.Z. Sagdeev, Proc. Symp. Appl. Math. 18 (1967) 281. R.B. White, Resistive Instabilities and Field Line Reconnection in: Handbook of Plasma Physics, Vol. I, eds. A.A. Galeev and R.N. Sudan (North-Holland, New York, 1984) p. 642. R.B. White, D.A. Monticello, M.N. Rosenbluth and B.V. Waddell, Phys. Fluids 20 (1977) 800. R.E. Denton and J.F. Drake, Bull. Am. Phys. Sot. paper lR-2.