H.K. Moffatt- The amplification of a weak applied magnetic field by turbulence in fluids of moderate...

8/3/2019 H.K. Moffatt- The amplification of a weak applied magnetic field by turbulence in fluids of moderate conductivity

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Reprinted without change of pagination from heJournal of Fluid Mechanics,volume 11, part 4, p p . 625-635, 1961

The amplificationof a weak applied magnetic field byturbulence in fluids of moderate conductivity

By K E I T H M O F FAT TTrinity College, Cam bridge

(Received 31 May 1961)

The effect of turbulence on an applied magnetic field is considered in the casewhen the magnetic Reynolds number R, is large compared with unity but smallcompared with the ordinary Reynolds number R of the turbulence. When theapplied field is sufficiently weak, it is argued that its effect on the velocity fieldis negligible. The equation for the field is then linear and its spectrum may beobtained throughout the equlibrium range of wave-numbers. It appears tha t thespectrum increases as ki up to a wave-number kc marking the threshold of con-duction effects, and falls off as k beyond k,. The net effect of the turbulence isexpressed in terms of an eddy conductivity equal to R;B times the electricalconductivity of the fluid. The effect of magnetic forces when these are notnegligible is also tentatively considered.

1. IntroductionThe behaviour of a magnetic field in a turbulent conducting fluid is largely

determined by the relative magnitudes of the Reynolds number R and the mag-netic Reynolds number R, of the turbulence. These may be defined in terms ofthe root-mean-square velocity U' and a length L characteristic both of the energy-containing eddies and, i t may be supposed, of any large-scale magnetic fielddisturbance (or magnetic eddies) that may be present. Thus

R = u'L/v, (1.1)

and R, = ~~T,uwu'Lu'L/h, (1.2)where v is the kinematic viscosity of the fluid, and ,U, cr and h are its permeability,conductivity and magnetic diffusivity, respectively. Throughout this work, weshall suppose that R is a t least five or six orders of magnitude greater than unity.

R (i.e. h < U ) , any weak random magnetic field is intensified bythe action of the turbulence; indeed its mean-square value increases exponen-tially until magnetic stresses react back upon the velocity field (Batchelor 1950).A t the other extreme, when R, < 1, i.e. in a weakly conducting fluid, conductioneffects are dominant at all length scales, so that any random field will rapidly

decay to zero. Steady conditions are possible, however, if a large-scale magneticfield is maintained by externally applied electromotive forces. The turbulence

, will then give rise to small fluctuations in this field whose spectral properties willbe closely related to the turbulent spectrum and whose level will be controlled bythe small conductivity of the medium. Golitsyn (1960)has recently analysed

When R,

P

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62 6 Keith Moffatt

th e particular case of turbulence of a weakly cond ucting fluid in a uniform mag-netic field, and has obtained the anisotropic spectrum of the small-scale fieldfluctuations th a t are induced.

There remains the possibility, which we investigate in thi s paper, t h a t

l < R , < R . ( 1 . 3 )

This is the case of mo derate co nductivity which m ay well arise in problems ofastrophysical and geophysical interest. The condition implies thath $ v , soth at , according to Batchelor, random magnetic field perturbations decay to zeroin t h e absence of electromotive forces. The reason for th isis that conductioneffects become im porta ntat a length scale1, large compared with th e length scaleI , of th e sm allest turbulen t eddies a t which viscous dissipation begins to pre-dominate. The lines of forceof an y magnetic field disturbance on a leng th scale

larger than I, are, to a good approxim ation, carried w ith th e fluidso th at init ialintensification m ay result. B u t such a fieldis presumably distorted b y the smallestvelocity eddies and broken up into com ponents much smaller tha n1, which m ustultimately decay t o zero thro ugh th e predominan t conduction effect. A gainitappears that a steady spectrum can be maintained only if externally appliedelectromotive forces are present. We shall suppose th a t these gen erate, on thescale L, a magnetic field H,(r) which is distorted by th e turbu lence, intensifica-tion through the stretching mechanism occurring at length scales larger thanl,, with conductive decay a t length scales smaller th an1,.

It will be possible to neglect the back-reactionof the magnetic field on thevelocity field provided the mean magnetic energy generated is small comparedwith th e kinetic energy of those eddies whose scale is small com pared w ithL, tselfa factor R* less th an th e tot al kinetic energy of th e turbulence (see$4). Sinceth e eq uation for th e m agnetic field is linear, th e m agnetic energy will be propor-tional to .@ (and probably larger t ha ngg in view of the initial intensification).We shall therefore suppose th atgt is sufficiently small for us to neglect th e back-reaction, although we ma y later examine, a t least qualitatively, departures fromth is condition.

Und er th e conditions outlined above, th e statistical properties of the small-scale motion, characterized by wave-numbersk large compared with 1/L, areaccording to Kolmogorovs theory, stea dy, isotropic, an d determine d solely bythe parameters v and 8, th e ra te of dissipation of kinetic energy per un it m ass. Th ejustification for these claimsis t h a t R is large compared with unity . SinceR ,is likewise large compared w ith un ity, th e statistical p roperties of the small-scalemagnetic field are,t o the same approximation, steady an d isotropic, though th eym ay d epend upon certain field param eters (e.g.A )in addition to v and 6.

A problem closely related t o t h a t under consideration was studied b y Batchelor,Howells & Townsend (1959) who obtained the spectrum, a t high wave-numbers,

of a dynam ically passive scalar solute8 und er th e combined action of convectionand diffusion a t small Pran dtl num ber.It is interesting to note t h at th e magnetic-field spectrum th a t we shall obtain in th e following sections is identical withthe spectrum of VB,although the underlying kinematical reasoning is not thesame in th e tw o cases, sinceH and VBdo not satisfy th e same equations.


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Amplification of a magnetic field by turbulence 62 7

It is perhaps worth stressing a t the outset that we are assuming t hat the levelof the magnetic spectrum can be controlled by ohmic conduction, even thoughthis predominates only a t higher wave-numbers, simply because the intensifica-tion through stretching of lines of force is associated with a decrease of scale,

i.e. the magnetic energy may be increased but is necessarily directed towardsthe ohmic sink at high wave-numbers at the same time. Some other authors(e.g. Biermann & Schluter 1951)have taken the view th at the increase of fieldintensity a t low wave-numbers may continue until equipartition with the kineticenergy a t the same length scale is established. However, the results tha t we shallderive are entirely consistent with our assumption which we therefore retainwith some confidence.

It is easy to see why ohmic conduction does not necessarily control the magneticspectrum level in the case of high conductivity when h < U. For in this case, theconduction length scale Z is much smaller than the viscous length scale I,,

sothat

the magnetic field cannot be broken up by turbulent eddiesinto small componentsa t which conduction dominates, Indeed it can only be broken down into loopsof size O(Z,) a t which the small-scale straining motion is very efficient at intensi-fying the field. But to pursue this argument is not the purpose of the presentpaper.

a

2. The magnetic energy spectrum in the range 1/L < k < (/A3)&

to be, in order of magnitude,

Similarly the conduction cut-off kc,which is small compared with k, and cannottherefore depend on U , can depend only on E and h and is therefore given by

The viscous cut-off wave-number k,, depending only on E and U , is well known

k, = z = ( 4 V 3 ) t (2 .1)

kc = z = (/h3)*.

We shall suppose that L-4 < k$ < kt, (2 .3)( 2 . 2 )

so that we may consider separately two subranges of the inertial range of wave-

and

recognizing th at if either oft he inequalities of ( 2 . 3 ) ails to obtain, the correspond-ing subrange simply shrinks to zero. Since E is given by the semi-empiricalrelation

the condition (2 .3 ) s tantamount to the condition

@ numbers, subrange A :1 / L < k < k, subrange B : , < k < k,, (2 .4)

(2 .5)= u3/L,

a more exacting requirement than we anticipated in (1 .3 ) .I n subrange A themagnetic energy spectrum is in a st ate of dynamic balance under the influenceof the convection and consequent stretching of lines of force alone; we shallproceed to determine its form in this section leaving until $ 3 the s tudy of sub-range B in which conduotion plays an important part.


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628 Keith Moffatt

Let us start from the observation, first made by Batchelor, that ina Auidfor which h = v , th e equation s for th e rates of change of magnetic fieldH andvorticity w = V A U are form ally identical, nam ely

and

Th us, if H = CO a t any t ime, whereC is a suitably small dimensional co nstant,so th a t th e neglect of th e L orentz-force ter m in(2 .8 )is justified, th en H willremain equal to Cw a t all subsequent times. I n this case therefore all statisticalproperties of H and w will be identical, an d in particular their spectra will havethe same dependence on wave-numbers fork 9 L-l. If the turbulence arisesfrom some instability of a large-scale flow whose vorticity coincides with anapplied m agnetic field, th enit is clear th a t th e small-scale vorticity an d magnetic

field th at are generated m ust also coincide.Now if it is tr u e th a t th e statistical properties of th e small-scale motion an d

field are independent of the large-scale specification, then under statisticallystea dy conditions, fork 9 L-l, an d in a fluid for whichh = v, he statistical pro-perties of an y magnetic-field distribution are presumably th e same as those ofth e particular field th a t coincides with th e vorticity field. H ence t he spectrumr H ( k )of H , defined for a n isotropic, solenoidal field by th e e quations

aH/at-+ U .

VH=

H .V U+hV2H,aw/at +U. w = w . vu +V V 2 0 , (2.7)(2 .8 )

H,(x) j(x r ) = /I?,$(k)eiker d k ,

has the dependence, like the vorticity sp ectrum ,

r H ( k )K k ) for L-l < k < k,. (2 .10)If now th e ratioh/v s increased,80 t h a t k, decreasesrelative tok,, the magnetic-

field spectrum w ill be modified, b u t only over th a t pa rt of wave-num ber space inwhich conduc tion effects are relev ant, i.e.k > k,. The relation (2 .10)is unlikelyto be altered throughout its sta ted range, although we might an ticipatea fairlyrapid cut-off for k > k,, due t o th e rapid smoothing o ut of small-scale variationsby conduction.

Since the foregoing argum ent m ay not carry complete conviction,it may beas well t o supplement it with another which leans less heavily on th e analoguewith vorticity. The following argument employs the vector potentialA(r, )defined by V A A= H , V. A = 0. (2 .11)

It is readily shown, from M axwells equations a n d O hms Law th a tA satisfies

(2 .12)

where q5 is th e electrostatic potential. T he curl of this equation gives equation(2 .7 )fo r H . If we now multiply (2 .12)scalarly b y A,, average over ensembles,and use th e property of homogeneity (b y which the divergence of an y q ua nt ityvanishes on averaging), we obtain w ithou t difficulty

d P l a t = - ~ ,, (au,laxj)- 24m. (2 .13)


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Amplification of a magnetic field by turbulence 629

It is easily shown that the spectrum of A, r , ( k ) , defined by a pair of equationssimilar to (2 .9) ,is by virtue of (2 .11)related to r,(k) by

(2 .14)

It is reasonable to suppose that r,(k) varies as some power n of k in subrange A .We shall prove tha t if n lies between - 1 and 1, then it necessarily has the value a.

For in these circumstances,

of P,(k) in the neighbourhood of k = kc,whereas

r , ( k ) dk is determined largely by values

r , ( k ) d k s determined

by values of r,(k) n the neighbourhood of k = L-l since r , ( k ) by hypothesisfalls off more rapidly than k-l for k >> L-I. In this sense, it is true to say that thewave-number ranges determining A2 and do not overlap, provided R, islarge enough. Let us denote by x the total rate of generation of contributionsto 5 or p-stuff ) including generation by electromotive forces (not repre-sented in equation (2 .13))and by interaction with the turbulence, represented

(2 .15)y the term

of equation (2 .13) .Under steady conditions, equation (2 .13)then gives

0-G{A} = -2AiAj(8ui/8xj)

x = 2h(VAo2. (2 .16)

Thus, ,@-stuff is generated at a rate x a t wave-numbers of order L-l, and isdestroyed a t a rate x at wave-numbers larger than k,. It is therefore transferredat a rate x through the spectrum which is therefore determined in subrange Asolely by the parameters x and e. The dependence of FA(,%)on x must be mereproportionality because of the linearity of the equation for A, and dimensionalanalysis now gives r,(k)m ( x / e j ) -8, so that I?,@) m (x/$)3.

Now n cannot be less than - ; for if so , we could apply the above argument toboth I?,@) and F A @ ) ,proving that both these spectra have the dependencek-8, contrary to equation (2 .14) .Moreover, it is extremely unlikely tha t n shouldbe greater than 1; no physical argument can be found to support such a rapidincrease with k. Hence we are again led to the result (2 .10) .

It is interesting to note the resemblance between equation (2 .13) and the quation for rate of change of @ derived from equation (2 .7) ,namelyd I F / d t= G{H} - h(VH,)2, (2 .17)

where G{H} = 2 H , H j ( 8 ~ , / 8 ~ ~ ) .

The term G{H}, representing the generation of magnetic energy by the turbulence,is of vital importance in determining the spectral properties of H. This is becauseconsiderable vorticity is associated with the wave-number region near k, (sincek, 8 L-l) in which the magnetic energy is concentrated. The same cannot besaid of the term - G{A} in relation to the A-spectrum, since there is very littlevorticity associated with wave numbers of order L-l a t which F, ( k ) is maximal,a comment best expressed by the inequality

G{A)/G{H} < @/m. (2 .18)


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3. The magnetic energy spectrum in the range k, 6 E 4 ,I n this section we shall follow the method of Batchelor et al. (1959) o determine

the spectrum in subrange B. The Fourier transforms of the fields U and H maybe defined by the equations

u(r) = [ p ( k ) --ikardk, H(r) = [q(k) e i k e r dk, (3.1)

where p ( k ) nd q ( k ) have the solenoidal property

k . p ( k ) = k . q ( k ) = 0. ( 3 4

The Fourier coefficients p , ( k ) nd q i ( k ) re related to the kinetic-energy spectrum

E ( k )and the magnetic-energy spectrum r ( k ) dropping the suffix H )by the equa-

where the star indicates a complex conjugate.I n terms of the Fourier coefficients, equation (2 .7)may be written

a4,(k)+i k;p, (k- k)qj (k)dk i k;q,(k-k)pj(k)dk-hk2qj(k). (3.4)at s s

I n the integral on the right-hand side, it is expedient t o change the variable ofintegration by writing k = k - k , dk = - k. Using the fact that k;q,(k) = 0,equation (3.4) ecomes

at = - i S [ l c l p , ( k - k ) q j ( k ) + k , p , ( k - k )q , ( k ) ~ d k - h k e q l (k ) . 3.5)

The immediate aim is to convert this equation to one relating r ( k )and E ( k )by means of equations (3 .3 ) .The result of this manipulation will be found inequation (3.10).

Let us focus attention on values of k in subrange B, i.e. those satisfying

k, < k < k,. ( 3 . 6 )The integral in (3.5) s over all wave-number space; but since we expect thatqi(k) will decrease rapidly as k ncreases beyond E,,because of the predominatinginfluence of conduction, the integral will be dominated by the contribution fromthe range k < E,, that is, using (3 .6 ) )from values of k satisfying E 4 , orequivalently

k < ( k - k l . (3 .7 )This argument was given by Batchelor e t al., only the first summand of theintegral in (3.5) ppearing in their context. They moreover gave arguments whichalso carry over to the present case to show th at the time derivative in (3.5)just balances the small contribution to the integral from values of k near k ,and that both may therefore be neglected.


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Amplificatim of a magnetic field by turbulence 631

It then follows from (3.5)that

h2k4qj() qF(k) = fj{ki k;Epi( - k) pE(k - ) j ( k )qT(k)

+ k ; k k pi(k- k)pf(k-k)qj(k)&(k)+ki k$ pj( k - )p z(k - k) q,(k) @ (k)+ ki k k pj( k - k pf( - ) i( k q$(k)} k dk. (3.8)

This double integral is dominated by contributions from the range for which(k ,k) < (Ik - l, Ik - kl), and in this range the statistical connexion betweentheps and qs of (3 .8)is slight. Hence, for example,

pi( k - k) p:( k - ) j( k) $(k) + pi( k - k) pE(k - ) qj(k) qj(k).The orthogonality of the coefficients,

pi(k - )p$( - k) = p i (k - k) p$(k - k) 6(k - k),* now allows a trivial integration throughout the k-space. Further, since k < k,andpi(k) decreases slowly compared with qi(k) n the range (3 .6) ,we may replacep,(k - )p*(k - ) byp,(k)p$(k), whichmaynow bebroughtoutsidetheintegral.These simplifications reduce (3 .8)to the form

h2k4qj(k) @(k) = pi(k)p$(k) k;k;q,(k)qf(k)dks

+ kdkkp$i@jiij /qi(k) q*(k)dk. (3 .9)

Now - -H aH, 1 a

kLqj(k)qz(k)dk= H i d = H.- = --HiHi = 0, by homogeneity.f ax, jax, 2axkHence the second term of (3.9)vanishes. Also

am.f axi axklkiqj(k)@(k)dk = - 2 3 .S(VHj)26ik,aby the isotropy of the small-scale magnetic field, and

Using these results and equationb (3 .3) , the relation between E ( k ) and r ( k )follows in the form

m 4 r ( k ) = g ~ ( k )vH,)2++ k 2 ~ ( k )q. (3.10)The interpretation of this equation is as follows. Part of the magnetic energy

in the wave-number range (k , +d k ) is derived from the simple interaction ofvelocity components of wave-number k with an effectively uniform field gradient{(VHj)2}i. his is exactly as for the scalar spectrum, and corresponds to a transfer


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632 Keith Mo8att

of magnetic energy down the spectrum. The other contribution to r ( k ) s a directtransfer from kinetic energy of wave-number k through the action of velocitygradients (i.e. rates of strain) on an effectively uniform field of magnitude

{(H,2)P.Now we may estimate the value of (vH3)2in terms of @ as follows.

+(OH,)2= som21?(k) k = k21?(k) k+1k21?(k) k.In the first integral the integrand is increasing throughout most, if not all, of therange; in the second it is decreasing throughout the range. The integrals convergea t the origin and a t infinity, respectively. They are therefore determined by the

FIGTJRE . Wave-number dependence of magnetic energy spectra (on logarithmic scale),(a) hen 1 4 R, 4 R, and ( b )when R, c 1. The slope is represented by the letter S. Thedotted curve in (a ) represents the cut-off of the spectrum of the applied magnetic fieldH m .

value of the integrand at E , and may both be approximated by the expression

r(k)k may beapproximated by k,l?(k,). Hence (vH,)2w k : q , so that the second term on theright of (3 .10)is the larger for k B k, . Hence, neglecting the first term, and usingthe Kolmogorov expression for E ( k ) ,valid in the range considered,

E ( k )w dk-9 , (3 .11)

P(kc) ,neglecting numerical constants. Similarly 3 q = s

we find, for the magnetic spectrum,

l?(k)w q e ) h - 2 k - Y for k, < k < k, . (3 .12)This spectrum does fall off rapidly compared with the energy spectrum E ( k )in the same range as we were led to presuppose.


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Amplifimtion of a magnetic field by turbulence 633

It is illuminating to compare this result with the work of Golitsyn (1960)who foun d a similark* law, modified by a n anisotropic factor, for the spectrumof small-scale field variations induced b y turbulent mo tion in a uniform appliedfield. He used a pertur bation me thod, supposing th a t the field fluctuations are

always small compared with the applied field,a situation th a t would seem topersist only when R, < 1. I n Golitsyn's case, th e lines of force a t an y inst an twould be approximately straig ht with small fluctuations. I n the caseR, 4 ,the lines of force will be randomly oriented, and approximately uniform withsmall fluctuations through each region of fluid of dimension1, = (A3/s)*.Thespectrum of the small-scale fluctuations within each such region could bedetermined by Golitsyn's method, introducing an anisotropic factor for eachregion. W hen we average over all the regions, the anisotropic factor disappears,an d we are left with th e isotropic law(3.12) h a t we have already determined by

a n independent method. The m agnetic-field spectra in th e two cases are sketchedin figure 1.

4. Conclusions

an d since the dimensionless con stant m ust be chosenso t h a tSince the s pectra (2.10)and (3.12)m ust agree in order of magnitude a tk = k,,

they may be writ ten in th e formr(k)= ~ A F c - M (L-1 @ k 4 k,), (4.2)r(k)= g ~ - z @ b k - + (k, @ k e k ~ . (4.3)

It ma y fairly be supposed th a tr ( k )does not behave too erratically fork < L-1,a n d t h a t it falls off very rapidly for k > k,.

W e m ay deduce a rough rule for computing the n et effect of the turbulence onth e magnetic field as follows. I n he absence of the turbulenc e, only the large-scalemagnetic field distribu tionH,(r) would be prese nt, with a m ean-square value of

(4.4)'@ approximately Q = S o L - ' r ( k ) d k= L-ir(L-1).

If we suppose th a t th e spectral law(4.2) is valid, a t least in orderof magnitude,r ight up to k = L-l, then, using equations(1.2)and (2.5) ,equation (4.4) becomes-

H 2 = $R,@,

indicating th e e xte nt to which turbulenceat large magnetic Reynolds numb erincreases th e m ean-square field intensity (provided always th a tR, < R).

The e xte nt to w hich t he dissipation of energy by ohmic heating is increasedm ay also be readily calculated. Thu sif


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634 Keith Noffatt

is the dissipation in the absence of turbulence and

D = AIOmT ( k ) d k= Ak;I'(kc) (4.7)is the increased dissipation under turbulent conditions; then

D = RkD,. (4.8)

This result may be interpreted in terms of an eddy conductivity r e , efined sothat the total dissipation, by analogy with (4.6), is given by

D = ( 4 r p g e ~ 3 ) - 1 (L -1 ) . (4.9)Then evidently be may be expressed in the equivalent forms

ge = RE# = (~T,uu'L)-*(T-Q, (4.10)

indicating, first, the strong dependence of r e n R,, and secondly, the anomalouseffect, already noticed for a scalar spectrum, that increase of g , eeping otherparameters constant, induces a decrease in ge, simply because the conductioncut-off k, is raised so th at more thorough mixing of the magnetic field is possible.

As we stated at the outset, we have assumed throughout tha t the mean mag-netic energy per unit mass ppH2/4rp must be small compared with the kineticenergy per unit mass of the small-scale motion. By dimensional reasoning, thislatter quantityis of order of magnitude ( s v ) ~ ,hich, by equations (1.1) and ( 2 . 5 ) ,is of order R-4 times + d 2 , he total kinetic energy per unit mass, as stated in theintroduction. Using (4.5), this places the following restriction on the magnitude

p.@/4rp < da /R ,R4 . (4.11)It is not difficult, however, to visualize how our results must be modified ifthis condition is violated. The field, approximately uniform over a regionofdimension k ; l , will tend to extinguish any eddies of smaller scale whose energyit exceeds (cf. the experiments of Murgatroyd 1953). Now the energy per unitmass of Fourier components of the motion of wave-number greater than k isapproximately d k - 3 . Let us define a wave-number kf by the stipulation thatthe motion consisting of eddies of wave-number greater than kf should haveapproximately the same energy density as the total magnetic field. If kf k,,we have the situation already described wherein the applied field is very weak. Atthe other extreme, if kf < L-l, we have the case of a field strong enough tosuppress he turbulence completely. I n the intermediate case, when L - l < kf < k, ,

pUH'ij4~p= d k F % , (4.12)o that kf is given by

the effect of the field will be to suppress eddies of wave-numbers large comparedwith kf. I f kf > k,, both the kinetic and therefore magnetic energy spectra may

thus be expected to fall off very rapidly for k > k f , I f kf < k,, equipartition ofenergy between kinetic and magnetic modes will probably be established a t allwave-numbers greater than k f , both spectra falling off rapidly in the range ofohmic dissipation, k > k,. These observations are sketched schematically infigure 2.

of @,


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Amplification of a magnetic field byturbubnce 635

FIGURE . Schematic illustration of the possible effect of magnetic stabilizing forces onkinetic an d magnetic spectra in th e two cases: ( a ) kf > k, , ( b )kf c k,.

It is a pleasure to record my thanks to Dr G. K. Batchelor for the stimulating

discussions that I have enjoyed with him on the subject of this paper.

R E F E R E N C E S

BATCHELOR, . K. 1950 Proc. Boy. Soc. A, 01, 405.BATCHELOR, . K., OWELLS, . D. & TOWNSEND, . A. 1959 J . Fluid Mech. 5, 134.BIERMANN, . & SCHLUTER, A. 1951 Phys. Rev. 82, 63.GOLITSYN,G. . 1960 Soviet Phys. Doklady, 5, 536.MTJRGATROYD,W. 1963 Phil. Mag. 44, 1348.

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