H.K. Moffatt- The oxymoronic role of molecular diffusivity in the dynamo process

8/3/2019 H.K. Moffatt- The oxymoronic role of molecular diffusivity in the dynamo process

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WHO1-78-67

NOTES ON THE 1978 SUMMER STUDY PROGRAMON DYNAMO MODELS OF GEOMAGNETISMI N

GEOPHYSICAL FLUID DYNAMICSAT

THE WOODS H O L E OCEANOGRAPHIC INSTITUTION

Willem V. R. Malkus, DirectorMary Thayer, Editor

and

WOODS HOLE OCEANOGRAPHIC INSTITUTIONWoods Hole, Massachusetts 02543

November 1978

T E C H N I C A L REPORTPrepared for the Office of Naval Researchunder Contract NO001 4- 78-G-0072.Reproduction i n whoZe or i n part i s permit tedfor any purpose of the United States Government.This report should be cited as:Oceanographic Institution TechnicaZ ReportWoods HoleWHOI-78-67.Approved for pub l i e re lease; dis tr ibut ionun2 m i e . (ZfLJ? b. , / i ,pproved f o r Dis t r ibu t io n: Robert W. Morse

Dean o f Graduate Studies

www.moffatt.tc


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The oxymoronic role of molecular diffusivity in the dynamo process

H.K. Hoffatt

Ab tract

The delicate question concerning the behaviour of the regenerationcoefficient Q and the turbulent diffusivity B in the limit ofvanishing molecular diffusivity (n + 0) in helical turbulence isdiscussed, in the light of an exact result of Bondi & Gold (1950)viz. when q = 0 the external dipole moment of a currentdistribution in a sphere is permanently bounded.


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1. The oxymoron is a figure of speech which embodies an apparentcontradiction; e.g. creative destruction, relaxed tension, devastatingtriviality, etc. The oxymoronic role of molecular diffusivity rl(= (p0o)-') is this:responsible for the natural ohmic processes of dissipation and decay, it

that while non-zero diffusivity (r l > 0) is directly

is also indirectly responsible for the means of regeneration of themagnetic field; the dynamo process may be described as a process of'regenerative decay', or perhaps better 'reinvigorating dissipation'.

2 . Consider the dipole moment p(t) associated with a current-distribution d(x,t) = po-lV ,, in a conducting sphere V:r < a.-This is given by various alternative expressions:

where S is the surface r = a ; and its rate of change is given by

With = - A B + r lV , and n.u = 0 on S, this gives5 -

SThe first term on the right describes the mechanism identified by Bondi &Gold (1950) for increase of the dipole moment: field sweeping towards themagnetic poles (defined by the instantaneous direction of the vector p )can increase 1p I , ut, as emphasised by Bondi & Gold, this mechanism isstrictly limited when rl = 0, since Ipl then attains a finite maximumwhen all the flux of is concentrated at opposite ends of a diameterof the sphere (as in an elementary bar magnet). To see this explicitly

from the above equations, .let S, denote those parts of S on which-n.! > or < 0 respectively, and let


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so that y = !+ + F - We then have

3 .

where @ = 1 (g.J3)dS = - I (g.l3)dS.s+ s-

Now, when rl = 0, is constant, since flux through every closed materialcircuit is conserved, and so

(7)Ill1 5 I y + l + 1F-l i [email protected] maximum being attained only when the flux is entirely concentratedat the poles, as mentioned above.

There can therefore be no doubt that, when 17 = 0, exponentialincrease of the dipole moment is impossible, no matter what thecomplexity (laminar or turbulent) of the velocity field in V may be.The situation is transformed if r) > 0, because then diffusive increase

in the dipole moment (represented by the second term of (3) is possible,provided the velocity field is such as to maintain a field with asuitably negative gradient near the boundary r=a.

4 . The impossibility of sustained dynamo action (in the sense of anexponentially increasing external dipole moment) applied equally to suchbasic systems as the homopolar disc dynamo. If the disc conductivity isinfinite, then the magnetic flux across it cannot change with time, andexponential growth of the magnetic field associated with the device isimpossible no matter how fast we rotate the disc or how ingeniously wetwist the wire, and whatever conventional wisdom may tell us to thecontrary. In terms of growth rate, if, in general, B Q. ept, then pmust depend on the disc Reynolds numberfigure 1.behave in this manner.

Rm in the manner indicated inIt is reasonable to conjecture that fluid dynamos also must


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for homopolar disc dynamo.(a): wire,resistance zero; (b): wire resistance non-zero.In either case, p + 0 as Rm + -.

Rmigure 1. Possible dependence of p on

S . Consider now the situation in mean-field electrodynamics, in which,in conventional notation,

= is the large-scale (mean) field, andb = B - B . Under first-order smoothing theory (Moffatt 1978 - hereafterreferred to as M-chap.7) we have the results- -0

wherethe random u-field. If

F(k,w), E(k,w) are the helicity and energy spectrum functions of-F(k,u) = O(w2) , E(k,o) O(u2 ) as w + 0, (11)

cc a. rl , B 5 ~ ~ ' r l s n + 0, (12)then clearly

f

where oo' and i3,' are in general non-zero constants ( B 0 ' > 0). Thisis clearly t h e situation when the y-field is a field of random waves withno zero-frequency ingredients. In this case, the regenerative processnormally associated with the pseudo-scalar a vanishes as rl * 0,


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con sis ten t with th e remarks of 81.of Bragniskii (M , chap.8) gives an expre ssion .for th e regenerat ivecoef f i c i en t ve ry s imi l a r to (9), and aga in with the property a = O ( Q )as Q -F 0.

I t may be noted t h a t t he theor y

86 . Di ff ic ul t ie s a r i s e however i f the ?-f ield has non-zero sp ec tr aldensi ty a t w = 0, as i s the case f o r conventional turbulence. Thezero-frequency ingre dient s of t he turbulence ar e pr ec is el y those th a tare responsib le for the d ispers ion of p ar t ic le s i n a tu rbulen t f low,

*

and they are of v i t a l importance a l s o i n the f i e ld - l ine -s t r e t ch ingcontext. I t must be noted however that results such as Q 2 D t2-fo r the re l a t iv e d ispers ion of two pa r t i c l es separated by vectord i s t ance S ( t ) i s ultim ately lim ite d by th e physi cal dimensions ofth e fl u i d domain; and ca re may then be needed i n carrying overasymptotic re su lt s from s t r i c t l y homogeneous turbulence t o turbulencei n a f i n i t e domain, par t ic u la r ly when these re su l t s are s e n s i t i v e t othe l imi t ing ( t + =) behaviour.

7 . When rl = 0, there i s an al te rn at iv e approach t o the determinat ionof the coeff ic ie n ts a and 8 using Lagrangian averages. I f a t somei n s t an t t = O , the U and b f i e ld s a re unco r re l a ted , t hen a and Bare funct ions of t (which cl ea rl y vanish a t t = O ) . The Lagrangian

- -procedure (M, g7.10) leads to th e expressions

t

0 0

where v(t) i s the veloci ty of t he f l u i d pa r t i c le i n i t i a l l y a t


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p o s i t i o n a.behave f o r a t y pi c al f i e l d of homogeneous turbulence asKraichan ( 1 9 7 6 a,b) has argued that , i n t h e case of tu rbu lence wi thnon-zero hel ic i ty ,

The d i f f i c u l t y h e r e i s to determine how th es e expr essio nst + a.

a ( t ) 't a. y B ( t ) ,l 4 B, as t + - , (15)t h e a p p a r en t p o s i t i v e d iv er g e nc e i n t h e s ec on d term of (14) beingcance l led by an equa l nega t ive d ivergence i n the th i r d t e rm, whichinvolves th e awkward t r i p l e Lagrangian co rr e l a t io ns . Kraichnan 'sarguments r e s t i n p a r t o n co mp ar is on w i th t h e r e s u l t s o f f i r s t - o r d e rsmoothing theory i n s i t ua t i on s where both approaches ( f i r s t o r d e rsmoothing and Lagrangian) may be expected t o be v al id , and i n p a r t onnumer ica l eva lua t ion o f a ( t ) and B ( t ) f o r v e l o ci t y f i e l d s w i thpresc r ibed Eule r ian s t a t i s t i c s .i s needed however, b efo re th e r e s u l t s (15) can be regarded asa b s o lu t e ly and d e f i n i t i v e ly e s t a b l i s h e d .(15), and pursue the consequences i n the context of a2- and au-dynamo mdels.

Fur the r numerical exper imentat ion

Let u s n e v e r th e l e s s a c c e p t

88 . For an a2-dynam i n a sphere r < a (M. chap.9), th e growthra tes have the form

a " + @ , and"ehereR = I a [ a /qe ,a

and dynamo action occurs whenfor the s imples t mode of dipole symmetry when

F(R,) > 0. This genera l ly occurs

wherepr ec is e assumption made about any large -scal e va r i a t io n ofthroughout the sphere . L e t us suppose that , as q + 0, t h e r e l e v a n t

Rac i s a po si t i ve number of ord er un i t y which depends on t h ea


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behaviour of a and B Xcf 15) is

9.

B % 8 , as T + O .0 ,% aThen (16) becomes

B - = bob2P L * G a L Raa 6 0is certainly satisfied if a is large enough,he condition R -and then p tends to a strictly positive value as T- -f 0, implying

a exponential increase of the mean field, and in particular of theexternal dipole moment..with the Bondi & Gold result (7), which applies when rl = 0 whatever

This appears to be in fundamental conflict

the complications of the velocity field, and whether laminar orturbulent.

The conflict does not arise under the alternative limitingbehaviour (12). In this case,

and the dipole mement does not grow exponentially in the limit rl = 0.

For dynanros of ao-type, growth rates are generally given by

where is a measure of the shear associated with differentialrotation. The condition for dynamo action is now of the form

XXC v (23)where Xc is model-dependent, but generally of order unity. Againunder the behaviour (19), as rl -t 0,

and we encounter the same fundamental conflict with the Bondi & Goldresult.

Under the alternative behaviour (12),


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To determine the behaviour of p as q + 0, e need to know thebehaviour of F(X) as X -+ 00. If F(X) = o(X) as X * 00, thenp -+ 0 as r) -+ 0, and conflict with Bondi & Gold is avoided. Theasymptotic behaviour of F(X) as X + Q) does not appear to have beeninvestigated for aw-dynamos in a spherical geometry. A clue ishowever provided by the results for angeometry (modelling the galactic disc).solved completely (M, 9.9),

aw-dynamo in a CartesianFor this case, which can be

f(X) log X as X + 00and so p * 0 as rl + 0 as required.

lO. It is hard to escape the conclusion that the result (19) cannot becorrect, or that, if it is correct in homogeneous turbulence, it is, forsome deep reason, not applicable when the turbulence is confined to afinite region (see the remarks of $6).

ReferencesBondi H. & Gold T. (1950) Mon.Not.R.Astr.Soc. 110, 607-11.Kraichnan R.H. (1976a,b) J. Fluid Mech. 75, 657-76 and 77, 753-68.Mof f t H.K. Magnetic field generation in electrically

conducting fluids (C.U.P.)

- - -- - - -

(1978)

AcknowledgementI am grateful to John Chapman who helped me to sort out the argument of 52.

H.K. Moffatt- The oxymoronic role of molecular diffusivity in the dynamo process

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