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Please note that terms and conditions apply. Forced Magnetic Reconnection Received June 15, 1982; accepted August 24, 1982

It has been generally accepted that a toroidal magnetostatic equilibrium can be specified by giving its boundary, the general topology of the magnetic field, and the distribution of mass. For example, if it is assumed that the magnetic field lies on nested toroidal magnetic surfaces and the pressure is given on each surface, then a unique equilibrium with these properties exists. In actual fact this need not be the case. Grad [ l ] has shown that if such an equilibrium exists, then by changing the state of the boundary in a manner that effectively resonates with the closed lines on one of the rational surfaces and demanding identical topology, a new equilibrium is produced that has surface currents on this resonant surface. However, if the change is actually carried out in time it would take infinite time for an ideal fluid t o reach this new equilibrium. Further, the large surface current which develops leads to important resistive corrections t o the equation of motion and reconnection will set in. This will enable an equilibrium of different topology t o be reached and the resulting equilibrium will have no surface currents.

This conclusion obviously has considerable importance for the theory of general toroidal equilibrium. It is still true that equilibria which are exactly axisymmetric have nested toroidal surfaces. But it is obviously impossible t o set up such exact surfaces so it is reasonable to expect that toroidal equilibria need not have nested magnetic surfaces, even of a nonsymmetric kind. The effective boundaries of these equilibria can be thought to physically change in time from axisymmetry in a manner as Grad has supposed, so it is of considerable interest to consider such processes in detail.

Grad arrived at his conclusions from very general mathematical considerations and also from experience in numerical determination of equilibria such as in the Doublet. It was not possible for him to follow the evolution of equilibria in time because of the complexity of the general situation.

Taylor suggested a much simpler model [2] which brought out these problems in a much more explicit manner. In this paper we describe Taylor?s model and then give the explicit evolution in time of the MHD system as it passes to a new equilibrium.

Taylor?s model problem is illustrated in Fig. 1. Let us consider an incompressible plasma in a magnetic field of uniform gradient in the x-direction B = BY?xy- = BoX-aYwhere Bb =Bola and Bo and a are constant. This is clearly in static equilibrium. / II (2) (4) ( 5 ) ( 6 ) ( 7 )

Now let us perturb the boundaries x = +-a of this equilibrium, x = k ( a - 6 c o s k y ) What is the new equilibrium to first order in 6? It is determined by where j, is the unperturbed current and j, + j l , the perturbed current; both in the z-direction. Similarly B is of the first order magnetic field and is in the xy-plane. Introducing the flux function J/ by B = i x V $ we have I),= $B&/u and J/l = J / ~ ( xc)osky.

Equation (3) reduces to and the boundary condition J/ = const. on eq. (2) reduces to J/l(Q) = Bo6

(1)

ILl(x) = A cosh kx + B sinh kx,

x > 0 where by symmetry J/(x) = + J/(-x). Let us first consider the solution of eq. ( 7 ) which has the same topology as the original equilibrium (I). For this solution $(O) = 0

sinh kx +Ax> = Bo6 -s?inh ka ?

x > o (1) By has a jump at x = 0 corresponding to a surface current (I) 4nj* = (By) = 2Bok6 sinh (ka) just as Grad had supposed.

Let us now consider the second equilibrium solution that has no surface currents (11), (11) $I(x) = Bo6 cosh kx cosh ka

sinh kx Biy = Bok&

cosh ka By is continuous at x = 0 so there is no surface current. However, $(O) = Bo&/coshka so the topology has changed. Equilibrium solution (11) has islands ofwidth (2a6/cosh ka)?? x O(6??).

Which of these two solutions is the correct one and how does the equilibrium pass from the zeroth order solution to one of them as the boundary is distorted in time? To answer this, let us imagine that the distortion 6 is set up at a rate slow compared to hydromagnetic time scale rA = a/VA but fast compared to any resistive time scale. rArL-? where 0 < s < 1 and r R = 471a2/q where q is the plasma resistivity.

Then we can state that the plasma is always in magnetostatic equilibrium everywhere except near x = 0. That is must satisfy eq. (5) everywhere except near x = 0. We can write this solution with boundary condition (6) in terms of $1(0)the It is easy to see that amount of flux crossing the boundary x = 0 in a quarter period in y : @ = $(x) sin ky COShkX-sinh kx tanh ka

sinh kx + Bo6 sinh ka ( 1 2 )

A

TT Figure 2

i The result of this calculation is sketched in Fig. 2 .

Four separate regions of variation are indicated: Region A corresponds to the ideal MHD region B to MHD plus small resistive corrections, C to tearing mode analysis similar to that given by Furth Killeen and Rosenbluth (FKR) [3] for the nonconstant $ approximation, and D to the constant $ approximation. We emphasize that the equilibrium which we are consider(9) ing is stable to the tearing mode. However, a time dependent solution of the tearing mode type occurs with controlled amplitude. The bulk of the variation occurs in region D. This occurs during the tearing mode time scale TT = ~2?72?. Note that the reconnection in D overshoots the island size, growing beyond the island size of (11) and then returning to it. Equilibrium (I) never actually occurs but its form is approached during regime A. These four regimes correspond to different asymptotic regions and it can be shown that their behaviour overlaps smoothly.

We give the details of the boundary layer analysis for regimes A and D only. The remaining details will be given in another paper.

The resistive MHD equations are aB - = V x (V x B) + -V77 ZB a t 471

= (B-Vj), where w, = .?* V x v and v is in the xy plane. Let v = 2 X v@ where $J is the velocity stream function and w, = Vz@.Then eqs. (13) and (14) become

(1 7) --a2- a2 a 2

f = k2X-2(~E) ar2ax2 ax aat aax22 = -k4Bnoxa aa-x22h- (19) tTohceosncsatlaendtvs.ariables U , and 6 are identical with those of FKR up As in FKR the boundary layer solution of eq. (24) and (25) Eliminate @ from eq. (19) by means of eq. (18) and introduc- lead to a jump in the logarithmic derivative of $l(x) ing f E $ J x , where E is proportional to the x component of the fluid displacement we obtain (29) A' = (20) Making use of the transform of eq. (7), we have A' = 2kE:/;i so with eq. (12) we obtain the relation where r = t f r ~T.he solution (20) must match smoothly on to Bo6Is the outer solution (12). But in the ideal limit no reconnection 'l(O) = [cosh ka + (. A.' / 2 kI)si& kal has yet occurred so $1(0) = 0 and eq. (12) reduces t o eq. (8).

Equation (20) has a similarity solution t ( x ,t ) = f ( x t )which is (We have Laplace transformed the amplitude of distortion of easily discovered t o be the boundary 6 so that = 6/s where 6 devotes its final value).

Thus eq. (30) gives us the information from the exterior solu2 Bok6 JkXt1rA e tion t o complete the boundary condition on the inner solution. f = -7r-sinh ka 0 U du x < a (21) Now region D in Fig. 2 is the region where $ may be assumed constant in the inner solution. Therefore, we may where the constant has been adjusted to match eq. (8). The vari- employ the -constant $23 solution of FKR to obtain ation of f given by eq. (21) is confined t o the region kx = const x r A / t which decreases with t. On the other hand the A' = 12f2 (31) current

a*$ 4nj1, = - =

ax2 has the value a t - + f ax 4 d l z ( o ) = -2 -k6Bo -t

71 sinh (ka)TA increases with t and is confined to the same decreasing region $l(O) proportional to l/t. Thus, a surface current of zero thickness is slowly approached. These results are those mentioned in the introduction about the ideal MHD evolution. =

For regime D of Fig. 2 we must retain the resistive term in eq. (1 5). Laplace transform eqs. (1 5) and (16) -- - 3 (2ka)'''a

(s S r i / 5 )514 However, we interpret the solution differently, s is known and A' and $1(0) are t o be obtained. Combining eqs. (30) and (31) ( 2 2 ) we can determine Jl(0) +

Bo6b

( k a ) p 5 ~ 4 / ( 2 k a ) 3 ~ 2 where P = sr?7265 is the Laplace transform variable in tearing mode time units.

Inverting the Laplace transform we find

1 $I@, t> = - jdr 51(0) est

27ri

Bo6 1 em cosh (ka)%jc dp p(1 + )\PSI4) (32) (33) (34) (23) 7 = t/(r3R15 T2A15 ) where and A = - 3 tanh (ka)

2ji2 (ka)312 (24) and the contour C in eq. (33) is the Laplace contour. Ccan be distorted as in Fig. 3. ( 2 5 ) Because of the factor p5I4 in eq. (33) there is a branch cut along the negative real axis and the integrand of I has poles at p = 0, h-415 exp(+(47r/5)i) (A and B in Fig. 3). Thus, $l(O, t ) can be reduced to real form and evaluated numerically. This gives the curve of Fig. 2.

The asymptotic results for $,(O) are where

8 = xfea, \k = - QlafBo, U E $ / 8 and The overshooting of is related t o the damped normal modes A and B in Fig. 3.

For small r , A? is very large and the constant $ approximation is no longer valid. A different analytical calculation corresponding t o the small ka limit of FKR must be carried out. Estimating the size of the nonlinear terms we find that the linearized theory is valid only if 6 / a < ( 7 ~ 7 ~ ) ~Fo?r. larger values of 6 the theory of regime A is still valid but that of the other regimes is not. However, in this case Rutherford?s nonlinear theory [4] for the tearing mode applies just as FKR did for small 6 . However, the overshooting disappears. The time scale to reach equilibrium I1 is (6 /a)?? T R .

This work was supported by the United States Department of Energy Contract No. DE AC02-76CHO 3073.