H.K. Moffatt and R.E. Hunt- A model for magnetic reconnection

8/3/2019 H.K. Moffatt and R.E. Hunt- A model for magnetic reconnection

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A model for magnetic reconnectionH.K. MOFFATT & R.E. H U N T

Isaac Newton Instz tute for Mathematzcal Sczences,20 Clarkso n Road. Cambrzdge CB3 SOE. UKhkrnZ@arntp arn.ac.uk

I f youre not to o dreadfully weary,You may ask: what s the poin t of th i s theory?Just please recollectTh a t B- l i n e s r ec onne ct ;

I hope that wil l answer your query!1. Introduction

M agnetic reconnection is a diffusive process whereby th e topologyof a magnetic field imbedded in a highly conducting fluid may changew ith time. T h e process is frequently represented by th e simple diagramshown in figure 1 in which th e opp ositely dire cted lines of force AB, CDreconnect to the configuration AC, BD. A huge l i teratu re emanatingfrom th e early pap ers of Sweet (1958). Park er (1957) a n d Petschek (1964)has evolved: for a recent thoro ugh exposition including discussion of th eimportant application of the theory in solar. magnetospheric and othercontexts. see Priest & Forbes (2000) .Despite the prolonged research on this topic, there seems to be asyet no simple model which demonstrates unambiguously the necessarilytime-dependent process indicated in figure 1 which takes account of thecurvature of the field lines before and after reconnection. It is the pur-pose of this short paper to provide such a model. We restrict attentionhere to two-dim ensional evolution i n a n incompressible fluid; generalisa-tion to the more challenging problem of three-dimensional skewed fluxtubes may however also be possible.2. The initial configurationp erbolaI t is natural to represent the curves AB. C D in figure 1 by the hy-

y2 - 2 2 = y2 (1)where k a n d Y are constants. We first construct a magnetic field

125K. Bajer and H.K. Moffatt (eds.). Tubes, Sheets and Singularities in Fluid Dynam ics, 125-1380 002 Kluwer Aca demic Publishers. Printed in the Netherlands.

www.moffatt.tc


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126 H.K. Mof fa t t and R. E . H u n tY

A BB

\ C DD

Figure 1. Change of topology associated with magnetic reconnection.

B = V AA which is confined to two oppositely directed flux sheets cen-tred on this hyperbola. With A = (O.O,Ao(z,y)),his is achieved bythe choice

(2)o(z ,y)= ;@ tanh{(y2 - Y 2- k 2x 2 ) / 2 } ,where S is the transverse scale of the flux sheet. We shall adopt non-dimensional variables using S as the unit of length: so from here on wemay take 6 = 1. We shall assume further that Y > 1 (so that the twoflux sheets are 'separated' at t = 0 , well separated if Y >> 1): and weshall assume that the curvature parameter k is < 1.

The field components associated with (2) are= @ysech2(y2 Y 2- k2z2)= @k2zsech2(y2 Y2 k2x2).J dA0

(3)(4)

dA0B, =-y - d x

With Y >> 1. the flux (per unit distance in the x-direction) in the uppersheet is

and the flux in the lower sheet is similarly -@ .irrotational straining flow

In order to induce rapid reconnection, we subject this field to the

U = (ax; a y , O ) (6)with Q > 0 (Figure 2) . We may adopt a-' as the unit of time, sothat henceforth Q = 1. The Lorentz force j A B with j = &'VAB willobviously perturb this flow; but we may adopt (6) as an initial conditionfor the velocity field, and as an outer boundary condition as y + fxfor all t > 0. The initial streamfunction associated with (6) (with nowQ = 1) is

$0(z: 9)= X Y (7)


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Magnetic Reconnection 127

Fzgure 2. The initial field configuration subjected t o imposed irrotational strain3. T he evolution equations

Consider first the magnetic induction equation

where q is the (dimensionless) magnetic diffusivity parameter. For thetwo-dimensional situation. we take B = V A with

A = (O.O.A(z ,y , t ) ) . V . A = 0 . (9 )Note that U A B is in the x-direction and V (U A B) = 0. There is noapplied electric field in the x-direction. In these circumstances. (8) maybe uncurled to give

Equivalently, with U = ( a $ / a y ; -a$/axL.! ) and $ = $(x? : t ) ?

The Navier-Stokes equation including the Lorentz force is. in standardnotation. dU 1 1- u.Vu - - V p + -jnB + u V 2 u .at P P (12)With vorticity w = V A = (0,O; -V2$) . the vorticity equation is


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128or equivalently

dV2$a t

Equations (11)

H.K. Moffatt and R . E . Hunt

and (14); coupled with the initial conditions A =A o ( z ,y ) . $ = z y at t = 0 and the outer boundary conditionsA + O . $ - z y as y + * m (15)

provide a complete specification of the problem. Note the symmetries. Aeven in z nd y. $ odd in z nd y which are respected by the evolutionequations.4. Boundary-layer approximation

As the straining flow sweeps the two sheets towards each other. it isobvious that the field gradient in the y-direction will increase, while thatin the x-direction will diminish. In these circumstances. a boundary-layer approach is legitimate. We expand A and $ in the form

$ = zf (y . t ) + ;x3s(y, t ) + 0 ( ~ 5 )A = ; a ju (z ,y . t ) = ;aj[g(Y.t)+ ; ~ ( ~ . t )0 ( ~ 4 ) 1

~ 2 $ x ( f Y Y s ) + 0 ( ~ 3 ) .V 2 u= (gYY h )+ 0 ( x 2 ) .

(16)(17)

where f is odd in y, and g and h are even in y . Note that then,(18)(19)

In each case. the leading-order term will be dominated by the part in-volving the y-derivatives, i.e.

V2$ N X f Y Y , V 2 uWith this boundary-layer approximation,

- YY .we have


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Magnetic Reconnection 129and so, to leading order, (11)yields

ht = -2h f y + f hy + qhvy .f y y t = f f y y y - y f y y - M 2 hgyyy - h,ygy) + V f y y y y >

(25)

(26)and (14) yields

where M = $ ( p ~ p ) - ~ / @ ,measure of the strength of the initial field.This equation integrates with respect to y (as expected within the boundary-layer approximation) to give

f y t = 1+ f f y y - f,- @(hgyy - hygy) + fug/?/ * ( 2 7 )where the constant of integration ( + l ) s fixed by the boundary condi-tions

f - y . g + l , h+O as y + * m . (28)We have thus reduced the problem to three nonlinear coupled evo-

lution equations (24) . (25) . (27) . Clearly the initial condition on f isjust

f (Y. ) = Y . (29)To obtain initial conditions on g and h, we expand (2) in the form

A o ( x , y ) = ;@ [tanh(y2- Y 2 ) k2x2sech2(y2 Y 2 ) 0(z4)] (30)so that, from (4 .2 ) . these initial conditions are

g ( y , 0) = tanh(y2- Y 2 )h ( y .0) = -2k2sech2(y2- Y 2 )

(31)(32)

5 . Weak field limitpassive vector field. We then have f = y . and (24) . ( 2 5 ) becomeIf M


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H.K. Mof fa t t and R.E. Hunt30and so g ( y ; t )= tanh(y2e2t- Y2) (36)

(37)-2 t 2 2 2 t -h ( y , t )= -2k e sech (y e Y 2 ) .We then have(38)g,, - h,g, = -4k2sech 4 (y 2 t - Y2) .

Thus, although the magnetic field strength increases exponentially (likeet), the relevant part of the Lorentz force in (27) ( that part not compen-sated by the pressure field) remains O ( M 2 ) :his is because the curvatureof the field lines near 2 = 0 decreases in proportion to e- t . There maybe a weak cumulative effect of the Lorentz force over a long time; butthe perturbation velocity field driven by this Lorentz force certainly doesnot increase exponentially, and remains O ( M 2 ) elative to the imposedstrain field.

When q > 1, we have

Near the origin, the B-lines A = cst. are hyperbolic and given byho(t)z2+ g 2 ( t ) y 2 = cst. (45)


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Magnetic Reconnection 131The separatrices through 0 are the lines

Y/Z = +ho(t)/g2(t))1'2 ' (46)Note that at t = O! although the field is exponentially weak near 0 ,these separatrices are well-defined as y / x = fk. Under the frozen-fieldevolution (36): (37), they move with the fluid to position

(47)ty l x = +ke-i.e. the angle between the separatrices decreases exponentially.is

At time t = 0, the flux crossing the positive y-axis is a; at time t , it

= ;a [l- o ( t ) ] . (48)(49)

The difference,is the flux that has reconnected and is associated with the field in theregion

a - a&) = ;a (1 + g o @ ) ) jo < y < k e - 2 t x , (50)

(see figure 3). Clearly, this reconnection process occurs in an E-neighbour-Y'r

Figure 3. Field configurat ion dur ing reconnect ion.hood of the saddle point of the field B where E = O ( q 1 / 2 ) , nd cannotoccur in the absence of such a saddle point. Figure 4 shows the computedevolution of g o ( t ) based on equations (24), (25). (27), for the followingparamet er values:

M = 1 , u = q = O . O l , Y = 1 . 5 . I c = O . 5 . (51)


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132 H.K. Mof fa t t and R . E . H u n t

I

- 1iFigure 4. Evolution of the function gO( t ) based on numerical solution of (24) . (25)and (27 ) ; note (i) the initial .frozen-flux period of constancy of g O ( t ) . and (ii) thesubsequent monotonic increase from -1 to 1 (so tha t, from (48). a Z ( t )ecreases fromto zero).

As expected, gO( t ) remains constant at near -1 for small t , while diffu-sion is still negligible; it then rises quite rapidly and tends to $1, th ereconnection process being 99% complete by t = 8.Contrary to received opinion. there is no indication here that theLorentz force has a dom inating effect on thi s process. T h e time-scaleof the process is 0(1), .e . O(cr-) when we return to dimensional vari-ables; this is because, however small the value of 7. the transverse scaledecreases to O(7/cr)1/2before the onset of diffusion, an d on dimensionalgrounds the reconnection time-scale is necessarily O ( a - l ) . Of coursethis conclusion may need modification if the magnetic field parameterM is greatly increased. Nu me rical exp erim ents over th e space of theparameters M . v a n d 7 are in progress.ReferencesP A R K E R . . N . 1937 Sweets mechanism for merging magnetic fields in conductingPETSCHEK.. E. 1964 Magnetic field annihilation. In Physics of Solar Flares (Ed.PRIEST,. R . &L FORBES,. 2000 Magnetrc Reconnectron Camb. Univ. Press.SWEET.P. A . 1958 The neutral point theory of solar flares. I n Electromagnetic Phe-nomena in Cosmical Physics. IAU Symp. 6 , (Ed. B. Lehnert) Camb. Univ. Press,

fluids. J . Geophys. Res . 62, 09-520.W . N . Hess). NASA SP-30. Washington, DC. 425-439.

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H.K. Moffatt and R.E. Hunt- A model for magnetic reconnection

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