H.K. Moffatt and H. Kamkar- The time-scale associated with flux expulsion

8/3/2019 H.K. Moffatt and H. Kamkar- The time-scale associated with flux expulsion

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HIG H C ONDUC T IVIT Y D YNAMOS AND F L UX E XP UL S ION 912.3Th e t ime-scale associated with fluxexpu sionH.K. MOFFATTtandH. KAM KAR ttSchool of Mathematics, Bristol University(Received February 1 I , 1981; in rev ised form March 20, 1981)

A simple model problem is solved in order to show that the time-scale associated with theprocess of flux expulsion ist~~ = R,3 to

where to is a time-scale characterising the flow (for example, the eddy turnover time, orinverse shear rate) and R , is the magnetic Reynolds number. This estimate is in agreementwith that of Weiss (1966) based on numerical experiments. By decomposing the vectorpotential into a product of a rapidly varying part (in space) and a slowly varying part, i t isshown how numerical work can be extended to much higher values of R , than has beenachieved hitherto .

1. IntroductionWhen a steady two-dimensional motion u(x) with closed streamlines actsupon a magnetic field in the plane of the motion, it is well known that, ifR, 9 1, the field is eventually expelled from regions of closed streamlines,and is ultimately concentrated in layers of thickness O(R;*) at theboundaries of these regions.

The process is described by the equation for the vector potentialA ( x , y , t ) k of the magnetic field, viz

Present address: Department of Applied Mathematics and Theoretical Physics, Silver Present address: Berkeley Nuclear Labo rato ries, Berkeley, Gloucestershire, U K .

Street , Cambridge CB3 9EW, U K ./ A

www.moffatt.tc


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92 S T E L L A R A N D P L A N E T A R Y M A G N E T IS MD uri ng a n initial pha se, diffusion is negligible, an d

where x(a , t ) is the position at time t of the fluid particle initially at a .D urin g this pha se, th e m agnetic field is disto rted into a tight d oub le spiralwithin each eddy, and the magnetic energy increases essentially like t 2 .Obviously the field gradient increases during this process, and so after atime, say ffe, diffusion must b ecom e imp orta nt. This is the stage a t whichclosed field loops form (Parker, 1966), and the process of flux expulsioncommences. T he magnetic energy within any eddy reaches a maximum a tt = fe and then falls of f, ultimately to a value of ord er R;'.The computational study of Weiss (1966) suggested that t f . - R,"3fo,and that in consequence, B L , - R,2'3. Th e purpose of this note is toprovide a simple theoretical explanation for this scaling, and to explainwhy the alternative scaling tfe- R,"'to, B',,, - R , suggested by Moffatt(1978, 03.8) is in fa ct incorrect.2. The action of uniform shear on a space-periodic

magnetic fieldFlux expulsion from an eddy occurs essentially because, at t = 0 , u . Bvaries (and indeed changes sign) on each closed streamline within theeddy. A much simpler flow and field configuration, with a similar pro pe rty ,is sketched in Figur e 1.W e suppose tha t U = (ay,0, 0), and tha t a t t = 0 ,

. Y

(a) (b)F I G U R E 1 Effect of uniform shear on a unidirectional space-pe riodic magnetic field.


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HIGH C ONDUC T IVITY DYN AMOS AND F L UX E XP UL SION 93B (x,0 ) = (0 , Bo COS k G , 0). (2.1)

Correspondingly,A ( x , y , 0) = - k; Bo Im { e i kG } ,

and th e solution of (1.1) has the formA ( x , y , t) = -k; Bo Im { a ( t) eik(') "},

where~ ( 0 ) 1 , k ( 0 ) = (ko, 0, 0 ) . (2.4)

It is easily show n tha t

so that the wave-fronts of B are progressively tilted as indicated inFigure l b , and thatda ld t = -q k2a,

so thata( t ) = e x p { - / q k 2 d t } = exp{-qk : ( t+ l /3a2 t? )} . ( 2 . 6 )

T he effect of t he sh ea r is repr ese nte d in the te rm '/3a2t3. If (Y = 0 , thetime-scale of decay of B is the usual diffusion time-scale t, = (qk;)-l. Ifa # 0, an d m or e particularly if a 9 qk:, the n t he time-scale of decay is

3. The action of non-uniform shear on a space-periodicmagnetic field

Suppose now tha t U = (U y ) ,0, 0), so that


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94 S T E LL AR AN D P L AN E T AR Y MAGNE T ISMdAldt + u ( y ) dA ldx = q V2 A . (3.1)

Figure 2 a , b show s th e effect of such a velocity field on a magnetic fieldgiven initially by (2.1). Flux expulsion occurs from the region in whichlduldy 1 B q k z , th e field topology changing throu gh th e diffusion process.T he solution of (3.1) now has the form

A ( x , y , ) = -k{' Bo Im { a ( y ) ei k (y , f , , 1 , ( 3 4where

anda ( y , t )- xp { -qkz ( t + '/3 (du/dy)* 3 ) } , (3.4)

This solution describes flux expulsion on the time-scale (2.7) where nowa = Idu/dyl max.

4. Flux expulsion from a single eddy w it h circular streamlinesSuppose now that

U = (O,sw(s),O)

(a ) 1.0 (b) t %F I G U R E 2 Effect of non-un iform sh ea r on the same initial field as in Figure 1. Acceleratedohmic diffusion in the region of maximum shear leads to apparent flux expulsion in a timetfe of order I 1 du/dy ,,,&I.


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HIGH CONDU CTIVITY DYNAMOS AND FLUX EXPULSION 95in cylindrical polar coo rdinates (s, 8, z ) , and that

A ( s , 8 , 0 ) = B o s s i n 8 , (4.2)corresponding t o a uniform field B,, parallel to 8 = 0. Then

A(s ,8, t ) = Bo Im {f(s, t) eie} , (4.3)where

aflat + i o ( s ) f = (a%* + s-'a/as-s-2)f, (4.4)with f ( s , 0) = s.th e solution of (4.4) isT he results of 2 an d 3 now suggest the best way to proceed. If q = 0 ,

f ( s , t ) = f(s, 0)e-iWf= se-'"'. (4.5)Th e function e-'"' is a rapidly varying function of s, when t is large. W henq # 0, let

f(s, t) = e-'"'g(s, t ) , g(s,O) = s , (4.6)so that

af/as = (-ito'g + ag/a s) e-'"', (4.7)a2f/ds2 = (- t2~ '~g-"o ' 'g -22to 'dg/ds + a2g/as2)e-iwr. (4.8)Substitution in (4.4), and retaining only the term on the right whichincreases like ?,we have

agiat = - q ( d 2 t 2 g + . . .), (4.9)giving

g(s , t )- e-'/q""P,which again desc ribes flux expulsion o n the time-sca le

(4.10)

tfe = R,1'3 (s CO')-', (4.11)


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96 STELLAR AN D PLANETARY MAGNETISMin agr eem en t with the results of Weiss (1966).do ub le limiting processThe asymptotic symbol - n (4.10) needs interpretation in terms of a

l s o t l 1 and I q ? d e , (4 .12)the latter arising from back-substitution of the solution (4.10) in (4 .8) tosee at what stage the term a2g/ds2becomes com parable with ?o2g. T h esolution (4.10) is thus valid for

1 < o o t


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HIGH C ONDUC T IVIT Y DYNAMOS AN D F L UX E XP UL S ION 97then evaluate the neglected term to see whether it was indeed negligibleis her e un reliable an d gives a misleading result!

AcknowledgementsW e ar e grateful to D r Nigel Weiss, whose disbelief in the RA* estimate led to a continuinginvestigation of the problem, and to Dr John Chapman, who contributed to some of ourearlier discussions.

ReferencesKamkar, H., Kinematic and dynamic aspects of flux expulsion in magnetohydrodynamics,Moffat t , H.K. , Magnetic field generation in electrically conducting fluids, Cambridge:Parker , R.L., Reconnexion of lines of force in rotating s phe res and cylinders, Proc. Roy.

Ph D thesis, Bristol University (1981).University Press (1978).Soc. A291.60-72 (1966).Weiss, N.O., Th e e xpu lsion of m agnetic flux by eddies , Proc. Roy. Soc. A293, 310-328(1966).

Reprinted from:s t e l l a r & Planetary Magnetism(ea . A . M . Soward)Gordon & Breach, London, 1983

H.K. Moffatt and H. Kamkar- The time-scale associated with flux expulsion

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