J. Plasma Fusion Res. SERIES, Yol.2 (1999) 54-57

Energy Release and Plasma Heating by Reconnective
Magnetic Relaxation
JAIN Rekha and VEKSTEIN Grigory
University of Manchester Institute of Science and Technology,Manchester, M60 lQD, United Kingdom(Received: 4 January 1999 I Accepted: 13 May 1999)

Abstract
A brief review of recent progress in the theory of magnetic energy release and dissipation via
magnetic reconnection is presented. The relaxation aspect of this process is emphasized, which is ofinterest for astrophysical plasmas such as the solar corona.

Keywords:
current sheets, forced magnetic reconnection, magnetic relaxation

heating by forced magnetic reconnection [6,7]. The

1. lntroduction
It is well known that magnetic reconnection, alatter means that reconnection is triggered by the current
sheet formed inside initially smooth magnetic field due
to its external perturbation. Such forced reconnection

phenomenon associated with breaking and mending of

the magnetic field lines, plays an important role inseems to be a viable mechanism for solar coronalheating, because current sheets are easily formed in thecorona by photospheric shuffling of magnetic footpoints

laboratory and space plasmas. In fusion-oriented devices

reconnection results in changes in the topology ofmagnetic flux surfaces which can destroy plasma
confinement. For astrophysical plasma objects theenergy aspect of this process is essential, sincet8,el.

reconnection allows fast release of the excess magnetic

2. Magnetic Reconnection and Energy

energy which maintains the hot solar corona Il] andenergizes extragalactic jets [2]. Usually, interaction of
the plasma with magnetic fields has a quite complicatedturbulent character. Therefore significant progress in its
understanding has been achieved not starting from thefirst principles but by applying more general ideas ofrelaxation in complex systems [3]. This theory, based on

ReleaseWe are interested in a low-B plasma, hence weconsider a sheared force-free magnetic field

B, = {0, Bo sin osx, Bo cos 06.r} ,(l)embedded in a planar slab of a highly conducting

plasma with walls at x = ,t) = to. The initial
equilibrium is perturbed by deforming the right-handmagnetic reconnection as the relaxation mechanism, wasvery successful in predicting magnetic configurations
sustained in various laboratory plasma systems. It wasalso applied to the problem of plasma heating in thecorona and jets, though the related studies [4,5] wereboundary.rl*) as

only semi-phenomenological. This report presents a selfconsistent derivation of the energy release and plasma

dimensional: periodic in the y direction but invariant*{) = o - 6s cos lcy e- i'ta)

thus making the magnetic configuration two

C o r r e s p o ndin g autho r' s e - mai I : rj ain @ s urai. p hy. umi s t. ac. uk

54@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchJainR. et al., Energy Release and Plasma Heating by Reconnective Magnetic Relaxation

along the z-axis. For quasistatic evolution (aru<< l; ru
is Alfven time-scale), a new equilibrium can beformed at the resonant surface. Obviously, such atransformation cannot be realized in a perfectly

represented with the deformed magnetic field (function

conducting medium for which magnetic reconnection isprohibited. Therefore the solution for this case, denotedof r and y) as
B (x, y) = IY V (x,y)

xi ] + B, (x, y) i,

ut V|, has the additional requirement ,fl'tro) = 0, i.e.

(3)

Iand this is a force-free field if the poloidal flux function.' IvPG)=\o, -a3 x<qin",,_i,'-a>x="0(g)^I -Aasincrna'i"l:'-'.1 , xnsx<- a.

tg and toroidal field component 8. satisfy the Grad-

Shafranov equation" -|-srnK(r-xo)

B,(x,t)=9(D; Y'W+9$-=0.
(4)' "dutThis solution, though preserving the field ropology, has

In the linear approximation (do << a) it yields

a singularity which is allowed in ideal MHD: the currenrV @, y) = Vo @) + t1f 1 (x) cos lq e-i',,sheet at r = r0, where B, = dvl dx, is discontinuous. If a
finite plasma resistivity is present, it tries to destroy the9 @)=9oW),\(5)current sheet by reconnective transition from rg[0 to V/4.

where tge(x) and go\trd relate to the initial magnetic

This transition is also associated with the release ofmagnetic energy [0], which, being related ro a unitfield (1). Thus, the perturbed flux function y1(_r)
satisfies the following equation

volume, is equal tov'i+1.4-k'lv,=o.

Bi /6. \'?y1ap,(6)

Lt'=____:_v,i'G)=- Bo4 sin d6aka,x),

(9)

where

tt:;t"n;"^ sin2a,gJ-8=----:-:KalcotK(a -xo)- cot2 Kal.

.

(4-k'\:l)l2iln\a l'

The solution of equation (6) subject to boundary
condition (2) is (for k. oo)a)The dependence of / on the scale length fta of
perturbation is shown in Figure I for various values ofThis solution is regular and contains nothing
special unless at some location x - .re within the slab
B tr(rd = 0, i.e. .r - -16 is the resonant surface where k'B ;= 0. Since Vl"@d is, generally, non-zero, the newequilibrium (5) has topology different from that of (l),ArA.

3. Plasma Heating by Magnetic Relaxation
In this section we discuss the possibility of plasma
heating that can be provided if this forced reconnection
is continuously generated by the external perturbation

with magnetic islands of width A = 4(lrlr([\Gdl/a&ilrt2(2). In a plasma with finite resistivity a generalequilibrium configuration can be represented as a linear

a.6=3n/8

superposition of the two equilibria derived in theprevious section
IVt=AV?+G-A)Wf\(*.=o)(10)

Since the solution r4ft) is singular, its presence in (10)means that the resulting equilibrium acquires a current
a,a=n/4

sheet whose strength can be characterized by theparameter A'- thejump in the logarithmic derivative oft/r at the resonance surface x = xo. It follows from (7)o.2o.40.60.8and (8) that

1.0ka

A'- 2aFig. 1 The factor f {see Equation (9) plotted as a functionof ka for three different values of aoa i.e. aoa = rl
8, nl4 and 3nl8 lx6 = 0). Note that the magnitudeof f increases with increasing aoa but decreases
with increasing ka.

(1-A)

cottca(1 1)(here -16 = 0 is assumed for simplicity). The particularvalue of /'and hence of A is determined by the internal

structure of the current sheet where both inertial and55

lainR. et al., Energy Release and Plasma Heating by Reconnective Magnetic RelaxationA(ar.ft| @+ib)='

resistive effects are essential. Therefore the situation issimilar to usual tearing stability analysis [1 l], although

in the tearing mode /'is prescribed by the global

ta+ib)=

magnetic configuration (the external solution), and thenthe current sheet (the internal solution) determines theinstability growth rate whereas herc the perturbation(ax,fr*-e=,(#)'^,

(r3)

(14)

y(y=7,+i)(2dp.

t ll' * t

(18)

Q=* 9@tt,1; g@)r,)--ax,A2 (a1,), (19)(15)

where Ad is the excess magnetic energy given by (9).

The function $(ar,), plotted in Figure 2, has its
maximum at oq - I and tends to zero as (rm,)eta for oJt,<< I, and as (al,)-tta for ar, >> 1. In general, !}(aq)shows features typical for a relaxation process when

Knowing the stream function @, one can use the secondof equations (12) to derive the current density inside thecurrent sheet, which yields ry'i =0". Therefore

ff

dissipation is most effective for the external driver'stime-scale comparable to the internal relaxation time. It
also demonstrates explicitly that the faster reconnection,i.e. shorter 2., does not necessarily result in an increased
heating. Though this is true for a)q > l, the dissipation
rate becomes decreasing with shorter z, when @q < l,simply because too fast reconnection does not allowbuild up of the excess magnetic energy (the amplitude Aa'tl | {*=r/r(0) kvt lo) J x=?:I(*j"'[_tot(klao\"'1ar;1r-)energy dissipation rate per unit volume results inACtt')''oo,=J]

delay that determines a continuous dissipation ofmagnetic energy. A straightforward calculation of

external solution is| v'*,,.

8l . t,This means a phase shift (temporal delay) between theexternal driver and internal equilibrium, and it is thisThe appropriate solution of (14) which is an oddfunction of ( and behaves as ll ( at ( -+ - to match theur'}tnla'*

o1o'+equilibrium y[t) with strong current sheet is present. In
general case, when both r4ft) and r4[') contribute to theequilibrium state, the amplitude A is a complex quantity.

equations (12) can be reduced toX" -i('X,+i(=g .

+ 1on,f* ,r".f'It shows that magnetic equilibrium is governed by theparameter alr. which is the ratio of the internal
reconnection time z. to the time scale of the external
driver (2). In the limit of fast internal reconnection,
when rrr4 << l, A -r 0, i.e. the system is very close to thereconnected equilibrium rd". In the opposite limit ofslow reconnection, ar, >> l, A -+ I and the ideal\U4=#1,'".0{-(l -\#tlaapproximation Il] and using new variables:*=t4(a2+(r2)Bolas and Q by -ialfi. Adopting the constant-V/.

I{[,the resistive time (tu<<cn) with.x scaled by asr. V/1 by\anik- J| aotantCz-. .- l''A= A t*iAt =where T" =.fpolBoaois the Alfven time, zn = 14a2of is(=4-:-l'.*,(17)shear parameter d0, so that T, ) * when the initialconfiguration (l) approaches the tearing instability
threshold Ka = 7t/2. The solution of (16) isut'i("' 'ir r< |tf,ts rfll5
lr;*^t,l is the effective_internal
reconnection time. It has usual tearing mode scaling [11]with respect to fu and Tn, but it also depends on thethis cunent sheet. The plasma motion is incompressible
inside the current sheet i.e. v = IY Q @,y,t) x 2], where 0= @(r) sin ky s-i'' is the stream function. Then themagnetic induction equation and equation of motion canbe written there in the following dimensionless form:

Itl| ?

Here r, -frequency al is fixed by (2), and so is the internalsolution; thus the external equilibrium (10) which isspecified by yet unknown amplitude A should adjust to-i(DVt=-ikxQ+ atr:- a,10" + l<xV't =0,t-A ,l** y" o(=., 2ttf Ql4) y .

o6)Equating (16) and (ll), the following equation
which determines the amplitude A is found:of the current sheet is small).
56

JainR. et al., Energy Release and Plasma Heating by Reconnective Magnetic Relaxatron

4. Conclusions

Acknowledgements
This work was supported by the UK Particle
Physics and Astronomy Research Council. One of us

In this work, energy release and plasma heating byforced magnetic reconnection is studied selfconsistently. The most interesting regime of(G.V.) is grateful to Prof. T. Sato, Prof. T. Hayashi andDr. T. Watanabe for their kind hospitality andreconnective relaxation occurs when the system is
initially close to marginal tearing stability i.e. when asapproaches (ao).. (= trl2a for the above model). In thiscase both Lt and Q become very large (and it follows
from (9), (16) and (18) that they scale as AC- [(ai",aol4,Q - [(ai", - aof-ttt ).
From a general physics point of view this plasma

discussions of this work during his visit to the National
Institute for Fusion Science.

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57