H.K. Moffatt- Turbulent dynamo action at low magnetic Reynolds number

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  • 8/3/2019 H.K. Moffatt- Turbulent dynamo action at low magnetic Reynolds number


    J . Fluid Mech . ( 1970) , vol. 41, par t 2, p p . 436-462Boeing S ym po siu m on Turbulence


    Turbulent dynamo action at low magneticReynolds numberBy H. K. MO FFA T T

    Department of Applied Mathematics and Theoretical Physics,Silver Street, Cambridge, U.K.(Received 30 November 1968 and in revised form 31 October 1969)

    The effect of turbulence on a magnetic field whose length-scale L is initiallylarge compared with the scale 1 of the turbulence is considered. There are noexternal sources for the field, and in the abseilce of turbulence it decays byohmic dissipation. It is assumed that the magnetic Reynolds number R, = u,l/h(where uo is the root-mean-square velocity and h the magnetic diffusivity)is small. It is shown that to lowest order in the small quantities ZIL and R,,isotropic turbulence has no effect on the large-scale field; but that turbulencethat lacks reflexional symmetry is capable of amplifying Fourier components ofthe field on length scales of order R;2Z and greater. I n the case of turbulencewhose statistical properties are invariant under rotation of the axes of reference,but not under reflexions in a point, it is shown that the magnetic energy densityof a magnetic field which is initially a homogeneous random function of positionwith a particularly simple spectrum ultimately increases as t-iexp (a%/2h3)where a(= O(ugZ)) is a certain linear functional of the spectrum tensor of theturbulence. An analogous result is obtained for an initially localized field.1. Introduction

    A theory that is likely to be of the greatest significance in geomagnetism andin cosmical electrodynamics has been developed recently by Steenbeck, Krause& R'adler (1966), Steenbeck & Krause (1966, 1967), Radler (1968) and Krause(1968). The theory is concerned with the effect of a turbulent velocity fieldon a magnetic field distribution in an electrically conducting fluid, it beingsupposed that there is no external source of magnetic field, the only sourcebeing the electric current distribution within the fluid itself.

    A principal conclusion of these authors is that dynamo action (i.e. systematictransfer of energy from the velocity field to the magnetic field) will occur providedonly that the statistical properties of the turbulence lack reflexional symmetry,i.e. are not invariant under a change from a right-handed to a left-handed frameof reference. The arguments can be justified with some degree of rigour only whenthe magnetic Reynolds number R, = p d u , satisfies the condition

    here, ,U is the magnetic permeability of the fluid, r~ its electrical conductivity, 1the length-scale characteristic of the energy-containing eddies and uo the r.m.8.

    Printed, in. Great Britain28-2


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    436 H . K . Moffattturbulent velocity. It seems quite likely however that a lack of reflexionalsymmetry is a sufficient condition for dynamo action irrespective of the orderof magnitude of R,.The above result is in striking contrast with previous predictions concerningturbulent dynamo action. For example, Syrovatsky (1959) argued (on the basisof ideas put forward by Biermann & Schluter 1950) that dynamo action willoccur only if R, is greater than some number of order unity; Batchelor (1950)argued on the basis of the vorticity analogy that sustained dynamo actionwill occur only if the magnetic diffusivity h = ( p a ) - l is smaller than the kine-matic viscosity of the fluid, i.e. only if R, > R where R ( S 1 ) is the turbulentReynolds number; and Saffman (1963) argued that dynamo action might notoccur under a ny circumstances due to an accelerating ohmic diffusion effectwhich occurs when the length scale of a convected magnetic field is systematicallydecreased. These authors concentrated on the tendency for magnetic energyto be swept towards higher wave-numbers, in spectral terminology, but theyunderestimated the potential importance of the possible leak back to lowerwave-numbers which can occur and which turns out to be at the heart of theSteenbeck, Krause & Rkdler mechanism. The possibility of such a leak backwas recognized but not quantitatively analyzed by Kraichnan & Nagarajan(1967) . Some of the relevant arguments were reviewed by Moffatt (1961) .A first version of the present paper was submitted for publication before Iwas aware of the existence of the above series of papers by Steenbeck et al.There is a considerable overlap between the work described in $82 and 3 and thework of these authors. The notation and the detailed method of analysis aredifferent, but the conclusions are substantially the same. The treatment givenhere is certainly more compact than that given by Steenbeck et al., and sincethe results have been by no means widely recognized or accepted, it is felt thatthis complementary approach is well justified. The physical interpretation of thehelicity effect given in $ 4has not been given previously, and the demonstrationof dynamo action given in $85 and 6 is definitely simpler and more convincingthan the arguments (based on more complicated models) given by Steenbecket al.


    2. The averaged effect of the turbulence on the magnetic fieldWe consider a fluid of infinite extent in a state of homogeneous turbulentmotion with zero mean velocity. It will be further supposed that the mean pro-perties of the turbulence (e.g. correlation tensors) do not change with time, i.e.

    the turbulence is stationary as well as homogeneous; departures from theseidealized conditions can be incorporated at a later stage. The theory that followsis essentially a kinematical one, in which all statistical properties of the tur-bulence (determined by dynamical processes) are assumed known, and the evolu-tion of a passive vector field, which is both convected and diffused, is investigated.An electric current distribution J(x, ) in the fluid will give rise to a magneticfield distribution B(x, ) satisfying

    V A B =k J, V * B= 0, (2 .1)

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    Turbulent dynamo action at low magnetic Reynolds number

    aB/at = v A (U A B )+ P B ,437

    and this develops according to the equation(2 . 2 )

    where u(x, ) is the turbulent velocity field. The condition R, < 1 ensuresthat magnetic fluctuations on scales of order 1and less will tend to be rapidlysuppressed. Suppose however that, at some initial instant t = 0, we have amagnetic field B ( x , ) whose characteristic length scale L satisfies

    L 9 1. (2 .3)

    FIUVRE. Schematic picture of the velocity field u(x, ) on the scale I and the initialmagnetic field B(x, 0) on the scaleL. he box V, is large enough to contain a large numberof turbulent eddies, but small enough for the field B(x,0) t o be approximately uniforminside it.Figure 1 shows the sort of picture that is envisaged. The light line represents aninstantaneous streamline of the field u(x , ) . The heavy lines represent the linesof force of B ( x , ))with mean curvature O ( L - l ) . The field B ( x , ) may be alocalized field decreasing to zero outside some finite region, or it may itself be arandom function of position, possibly homogeneous. I n the absence of any tur-bulence, such a field would decay according to the diffusion equation

    asjat = A V ~ B , (2 . 4 )(2 .5 )n a time of order tdwhere td=P / h .

    I n the presence of the turbulence, the governing equation becomes (2 . 2 ) ) heterm V A (U A B ) epresenting the inductive effect of flow across the magneticfield. This term will undoubtedly generate magnetic fluctuations on the scale I

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    438 H . K . Moffatt(and smaller). It is then appropriate to treat the total field B(x, ) as the sum ofa field B,(x, ) on the scale L and the generated field b(x, ) on scales of order 1:

    B(x,t )= B0(x, )+b(x, ) . (2.6)If equation (2.2) s averaged throughout a box V ,of side a satisfying

    l < a < L , (2.7)and the average is denoted by a n overbar, then assuming that U = 0 (with the

    (2 .8)implication 6 = o ) , aB,/at = V A (U A b)+hV2B,,and this is the appropriate modification of (2.4) as an equation describing theevolution of the large-scale field B,(x, ) .

    The fluctuation field b(x, )may be calculated within the box V , o the lowestorder in the small parameter e = 1/Lby neglecting variation of B, throughoutV,, i.e. by treating B, as locally uniform. The equation for b is then7iibliiii = V A [U A (B,+b)]+hV2b. (2.9)

    The term V A (U A B,) has the character of a forcing term in this equation andit is responsible for the generation of the fluctuations b on scales comparablewith the scale 1 of U .

    Equation (2.9)is in general difficult to solve due to the presence of the randomcoefficient in one of the terms linear in b, viz. V A (U A b). However, under thecondition R, < 1, (2.10)(as may easily be verified a posteriori),so that V A (U A b) may be neglected in(2.9)in comparison with V A (U A B,). We are then left with the more tractableequation, ab/at= V A (U A B,)+hV2b. (2.11)

    The first objective is to solve equation (2..11)for b in terms of U; and then toevaluate the term V I\ (U A b ) in equation (2.8). It will be supposed that com-pressibility effects are unimportant, so that V e U = 0, and, neglecting (for themoment) spatial variation of B,, (2.11)takes the form

    ab/at = B, * VU+hV2b. (2.12)If the forcing term B,.Vu were steady, then, after the disappearance of

    any irrelevant transients, the term abjat in (2.12)would be identically zero. Infact the term B, * Vu is unsteady, partly due to variation of U on a time-scale(the turnover time) to = l/u,, (2.13)and partly due to variation of B, on some time-scale t , (as yet undetermined). Itwill be assumed a t th is stage that t, is not less th at to (actually, it will appear later,see (5.13), hat t, 9 to), o that the effective time during which the term B,*Vu

    t Strictly, a term -V A (U b) should be included, but this is obviously small com-pared with V A (U b) and may be omitted.

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    Turbu len t dynamo ac t ion a t low magnetic Reynolds num ber 439varies significantly is of order to.The term ab/at arises only through the time-variation of the forcing term on this time-scale; hence

    (abjatl= O(bu,/Z), ( 2 . 1 4 )whereas IhV2bl = O(hb/12) . (2 . 15)Since the ratio is O(R,), it is consistent to neglect ab /a t in ( 2 . 1 2 ) giving the

    (2 .16)quationwherein B, is to be treated as uniform.It is now convenient to use the Fourier-Stieltjes representation (Batchelor

    u(x, ) = d Z ( k , ) ikeX, b(x , ) = d Y ( k , ) ikVx, (2 . 17)1953 , 52 . 5 )

    hV2b= -B, ' V U ,

    s sin terms of which (2 . 16) becomes

    d Z ( k , ).B, * khk2Y ( k , t ) =-We are now in a position to calculate U A b. Evidentlyso that , using (2 .18) ,

    ur\b = / d Z * ( k , t ) A d Y ( k , ) ,(U A b), = h-lAijB,,,

    Aij= i P [ d Z * ( k , ) A dZ(k, ) l i k j .shere

    (2 . 18)

    (2 . 19)(2 . 20)(2 . 21)

    Aij is a tensor determined by the statistics of the turbulence, and is uniformand steady only in so far as the turbulence is homogeneous and stationary.

    Returning now to ( 2 . 8 ) , and recognizing that B,(x) does vary on scales oforder L, e have

    (2 . 22)and this is the equation that replaces (2 .4) when the effects of the turbulenceare taken into consideration. The tensor coefficient A,, is placed under theoperator ajax, o allow for the possibility of inhomogeneity of the turbulenceon scales much greater than a ; in the derivation of ( 2 . 2 2 ) it was necessary onlythat the statistical properties of the turbulence should have negligible variationthroughout the box V,. The effects of the turbulence on the large-scale field arewholly summarizedin the first term on the right-hand side of ( 2 . 2 2 ) .The importantdifference between ( 2 . 2 2 ) and ( 2 . 2 ) is of course that the quantity A,, is not arandom quantity (like u(x, ) ) but a smoothed-out average property of theturbulence. I n the particular case of homogeneous, stationary turbulence,on which attention will be focussed in the following sections, A,, is independentof x and of t ; and, dropping the suffix zero on the field B,, (2 .22) becomes

    (2 . 23)

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    440 H . K . MoSfatt3. Properties of the tensor A,,of the turbulence b y th e equationThe Fourier-Stieltjes transform d2, is related to t h e spectrum tensor @$,(k)

    (3 .1 )d@(k, ) dZj(k, )d3k$,(k) limdsk-+O( 3 . 2 )Hence (2 .21) gives A,, = ieikl /k-2k,@kl(k)d3k.J

    The spec trum tensor satisfies th e condition of Herm itian sy m me try,@ k l W = @&(k), (3 .3 )

    AT. = - i ~ i k l k-'kj(DZk(k) d3k = - egk k -2k j@k l ( k )d3k= A,,, (3 .4 )s s

    s ss

    so t h a ti.e. Aijis real (as is required b y th e relation (2 .20) ) .Moreover QkZ(k)atisfies t h eincompressibility conditionsso th at , for example, kk@kl(k)= 0, k,@kZ(k)= 0, (3.5)

    A,, -A,, = i k 2 k 2 ( 23 - 3 2 ) d3k- k2l(Q3, - 1 3 ) d3k= i1k-2(k2@23+ k, QI3)-k2k, 3 2+ k, @3,)] d3k= i k2- , CD,, +k3@33] d3k = 0,

    and it follows th a t Aiiis symm etric. It is therefore possible t o diagonalize Ai,b y su itable choice of axes, so th a t say,

    where cc, /3, y are real parameters determined b y th e statistical struc ture of theturbulence.Suppose first that the turbulence is isotropic, i.e. its statistical propertiesare invariant w ith respect to ro tation of th e axes of reference a n d with respect toreflexions in th e origin of reference. Then 0$,(k) akes th e simple form,

    and it is easy to see from ( 3 . 2 ) h at in this caseA,j= 0. (3 .8 )

    The conclusion is th a t t o th e lowest order in the small parameters e and R,,isotropic turbulence ha s no effect on th e decay of a (large-scale) magnetic fielddistribution.

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    Turbulent dynamo action at low magnetic Reynolds number 441If we relax the condition of reflexional symmetry, then (Batchelor 1953, $ 3 . 3 )

    we can have an additional term in the expression for the spectrum tensor, viz,(3 .9 )F ( k ) e k@,,(k)= @,(k)+ 2 jk k,

    where, by virtue of ( 3 . 3 ) )F(k) s real. The tensor (3 .9 ) is still invariant withrespect to rotation of the axes, but the second term changes sign on reflexionin the origin, i.e. on change from a right-handed to a left-handed co-ordinatesystem. Turbulence for which F(k) + 0 may be described as pseudo-isotropic.Substitution of ( 3 . 9 ) n (3 .2 ) gives

    Aij = CC$,, (3 .10 )where a! = -$Iorn2 F ( k ) k . (3 .11 )

    The lack of reflexional symmetry, represented by the function F ( k ) s associatedwith the degree of right-handedness or left-handedness (or helicity ) of theturbulence; the appropriate measure of this quantity for homogeneous tur-bulence (cf. Moffatt 1968) is

    u . 0 = i dZ*(k, t ) .k dZ(k, t )= - i ( 3 . 1 2 )- sand, with ai, iven by ( 3 . 9 ) ,

    (3 .13)so that F(k)dkmay be regarded as the contribution to mean helicity of theturbulence from the element of wave-number space between the spheres ofradii k and k:+dk .

    Suppose now that we weaken the symmetry conditions further, and supposethat the turbulence is axisymrnetric (but without reflexional symmetry) aboutan axis in the direction of a unit vector A . The tensor QJj(k)and so the tensor4,) s then invariant with respect to rotations of the axes of reference aboutthis axis. Hence A must coincide withone of the principal axes of A,,, sayA = ( 1 , 0 , 0 ) .The general form for CD,,(k) atisfying the conditions (3 .3 ) and (3 .5 )is

    + i M [ ( kA A),kj-(k A h)jk4]+iAT[(kA),hj-(k A (3 .14)where A , B, . ,N are real functions of k .k and k . , i.e. of E and k,, related by

    @,j(k) Ak,k j+ Bh,hj+CSij+ D(k%Aj+kj&)i G ~ i j k h k + i H ~ $ j k k k

    k2A+C+k,D= 0,k ,B +k2D= 0 ,

    - G + k 2M + k , N = 0.Substitution of (3 .14 ) in ( 3 . 2 ) and simplification using (3 .15 ) leads to

    (3 .15)

    (3 .16)

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    442 H . K . MoffattIt is evident from this expression that (as we know already) A , , = AS3 , ut thatA,, may be quite different in magnitude (and possibly in sign). Once again,the only contributions to A,, come from terms of QLj(k)hich change sign onreflexion in the origin.

    It is not necessary to go into similar details for the general case of homogeneousturbulence with no directional symmetry, for which the most general form ofspectrum tensor consists of 31 terms of which 21 change sign on reflexion in theorigin (Batchelor 1953, Q 3 .3 ) . These 21 terms give (in general) unequal contribu-tions to the principal values a, , y of A$,.Comparison of (3 .2 ) with the equation

    uiu,_ = Oij(k)d3k (3 .17)suggests that A i j is determined by the structure of the energy-containing eddies(with perhaps a little weighting a t the low-k end of the spectrum), and so, on

    (3 .18)imensional grounds,where a,,POand yo are dimensionless constants (positive or negative) of orderunity.It is evident that the effect of turbulence on a large-scale magnetic fielddepends critically on whether the turbulence has keflexional symmetry in apoint, and it may be as well to consider briefly whether a lack of reflexionalsymmetry is a likely state of affairs in geophysical and astrophysical turbulence.In general such turbulence arises as a result of an instability of a mean flow, theinstability being frequently driven by buoyancy forces. It seems likely thatthis turbulence can lack reflexional symmetry only if the mean fields (velocity,temperature, etc.) themselves exhibit some lack of reflexional symmetry. Thesimplest, and most frequent, example arises in the case of turbulent thermalconvection in a rotating fluid. The rotation vector and the direction of meanheat flux are together sufficient to give a definite right-handedness or left-handedness to the system, and it seems IikeIy that a corresponding property willbe represented in the statistics of the turbulence.

    (a, ,Y) (a02 P O , 7 0 ) iuk

    4. The helicity effect; physical interpretationIn the pseudo-isotropic case, Akl= a&,, nd equation ( 2 . 2 2 )becomes

    aB, a- = - V A B, +hV2B,.at AThe term (a /A)VA B, in (4 .1 ) represents a tendency to generate magnetic fieldin the direction opposite (if a < 0) to that of the large-scale current J = uV A B,,and it is of interest to examine the physical mechanism underlying this effect.Suppose that we choose axes Oxyx a t a point in the fluid so th at , locally,

    Jo = (090, ,), B, = -,uJo(Y, 0, ), (4 .2 )

    (4 .3 )and consider the action of a typical 'helicity wave'

    U = uo(O, in(kx- d),os(kx- d)) Reiei@"-ut),

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    Turbulent dynamo action at low magnetic Reynolds number 443where fi = uo(O,-i, l ) , and, to be definite, k > 0, cr > 0. For this motion,

    0 = ku, u.w = ku; > 0) (4.4)(the average being with respect to x or t) . The streamlines (and vortex lines)of the motion are straight, but the particle paths (when g + 0) are circular.If we imagine the vortex lines closed at large values of IyI and 1x1, then they arelinked in the manner that gives positive helicity (Moffatt 1968).

    According to (2.12), this motion generates a perturbation field

    AIf cr g hlc2 (as is the case if cr w o1, k x 1-1, R, < 1 ) then h w (iB,/hk)fi, nd so(4.6)B uhlcb = - O O(o,cos(kx-at) , -sin(kx-crt)).

    The lines of force of the perturbed field B, +b are left-handed helices, all withthe same curvature (figure 2 ) . t The total field in the region between the linesAC, AC' receives contributions from all the lines of force which penetrate thatregion; and it is evident that the unequal cancellation of contributions to thez-component of the generated field from neighbouring lines of force can lead toan average (negative) component of B in the x-direction.This generation of B, is assured, however, only if the direction of U (indicatedby the small arrows) is such as to reinforce the convection of the field into theregion between AC, A'C'. This is so in the caseg hk2considered here, for which

    U A b = - B,u;/hk)(O, ,1 ) , (4.7)V A (U A b) = (a/h)VA B,, a = - u t / k .AConsider, however, the other limit cr $ hk2, for which b w - kB,/cr)fi,and

    b = - kB,/cr)u,U A b = 0. The lines of force of the perturbed field B,+bare still helices, but the motion d does not reinforce the distortion, and so anaverage B, does not materialize. It is evident th at U A b is non-zero only if Uand b are out of phase, and this arises only if the process of generation of b hasa dissipative character.

    Likewise, a fully turbulent motion with non-zero helicity (say positive) willdistort lines of force into 'random helices' with negative mean screw. In the caseR, < 1 treated in this paper, the expression for U A b contains contributions ofthe form (4.7) or each k represented in the Fourier decomposition of the u-field,together with terms periodic in x which vanish when the average U A b is taken.The cancellation effect between neighbouring lines of force will be incompletewhen the initial field is non-uniform, and the generation of a mean componentin the direction of - V A B, will result. This effect is so intimately related to the

    t Note that( B , + b ) . V A ( B , + b )= b. (VAb)= B:u,2/kha> 0

    which might suggest (wrongly) a right-handed helical structure.

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    444 H . K . Moffatthelicity of the background turbulence that the term helicity effect would seemappropriate.?change of B are

    The orders of magnitude of the two terms in (4.1) ontributing to the rate ofI :VAB~=O(%), I A V ~ B IO

    so thatHence, if L >> h2/a, the development of the field will be dominated by the tur-bulence term, a t any rate as long as the scale of the field remains of order L, orgreater. t t

    z z(4 (4FICTJRE. Distortion of initially straight lines of force by the helicity wave (4.3).Theinitial gradient of B is indicated by the different thickness of the lines of force. The velocityis indicated by the symbols ?. 0 J. 0 n (a ) ,where 0 means into the paper and 0 meansout of the paper. The lines of force are distorted into left-handed helices, and considerationof all the contributions to B from within the region in ( b ) between the lines AC, ACindicates a net generation of B, in the negative z-direction) due to unequal cancellationof contributions from neighbouring lines of force.

    The equation for magnetic energy density, from (4.1),s- + B ~ = - B . V A B + ~aat h

    In the case of a random homogeneous field, this gives- (&B2)= a(B.(V A B))-h((%)?.dt h (4.9)

    (where the brackets (... signify a space average$ over scales much greater thanL),o that the magnetic energy density is affected by the turbulence only throughthe appearance of a mean helicity (B. (V A B)) of the field B. That such a meanhelicity must appear is evident from th e equation,

    d 2a- B. (V A B)) =dt ((V A B)2)+h(B.V2(V B)+ V A B).V2B), (4.10)t The effect has been described as the a-effect by Steenbeck et al. (1966).$ O r equivalently, an average over an ensemble of realizations of the B-field.

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    Turbulent dynamo action at low magnetic Reynolds number 445derivable from (4.1) together with its curl. The f i s t term on the right re-presents a production of mean helicity with the same sign as a, .e. opposite tothat of the mean helicity u.0 of the background turbulence. The case of alocalized blob of magnetic field (on the scaleL) s similar, the brackets (. .. inequations (4.9) and (4.10)being then replaced by ...dv, the integral beingthroughout the whole extent of the blob.5. Dynamo action in the pseudo-isotropic case

    (i)Evolution of a single Pourier component of B(x, )As a preliminary t o the study of the evolution either of a homogeneous randomB-field or of a localized B-field, it is appropriate to examine the evolution of asingle Fourier component of the field; equation (4.1)evidently admits solutionof the form BeUteiKmX, K.B = 0,A A (5.1)where A iahw + h K 2 ) B = K A 6 ,(The symbol K will be used for the wave-vector of any Fourier component of theB-field, with the implicit understanding that IKI < Ikl where k is any wave-vectorhat wFch tkere is significant contribution to the background turbulence),With 3 = B,+ iB,, the real and imaginary parts of (5.2)areA A

    (U+hK 2 )B, - a/h)K A B,, (5.3)and ( w + h K 2 ) , = (a/h)K A B,,so that (5.4)and, for a non-trivial solution,

    w = - K 2f aK/h)= wl, w2, say. (5 .5 )From (5.3)we then have & = $ & A ( 5 . 6 )

    B(X, t ) = [+(B,+ fi A B,) cult+ $(B,- t A B,) ewat]e a - x .A Awhere fi = K / K , so that

    corresponding to the initial condition B(x,0) = BOeiKqxs= 16,,1and B4.B, = 0, for each mode. The solution(5.7)

    It is evident from ( 5 . 5 ) hat ifK < a / h 2= K,,say, then the eigenvalue w1 is positive, and the corresponding solution (5.1)

    grows exponentially in time. The parabolaswl, w2= hK(fK,-K) (5.9)

    are shown in figure 3 . The maximum rate of amplification occurs at K = +K,)(5.10)nd a t this value = U,,, = $hKZ = a2/4hs.

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    446 H . K. MoffattNote that , with a = O(ugZ),K, = O(ugl/h2),o that

    K,l= O(RL). (5 .11)Hence amplification will occur (in general) only on length scales of order Rz21and greater.

    (Jma x

    FIU~JRE. Variation of w1 and w 2 as functions of K . The decay rate U,, = -hK2 thatwould occur in the absence of turbulence is shown also for comparison.The assumption following ( 2 . 1 3 ) concerning the time-scale of the large-scale

    field may now be vindicated. A minimum estimate for this time-scale (i.e. assum-ing maximum rate of change of B ) s

    tl=o(-+o()max U212 * (5.12)Hence tilt,= O(RZ3) , (5 .13)and this is certainly large (as assumed) when R, < 1 . Note that waves for whichK 9 K , have N - hK2, and are negligibly affected by the turbulence.

    (ii)Evolution of a random B-jieldSuppose now that B ( x , ) (andso B(x , ) ) s itself a homogeneous random function

    B ( x , ) = dW(K, ) e i K e X , (5 .14)of x . Let swherein dW(K, ) evolves according to sub-section (i)above; then

    Bi(x, ) = &,,(K, t )d Y ( K ,0)eiKmx, (5 .15)(5 .16)here Q{,(K, t ) = +(ewit+ ewzt)Jij-+ieijkfZk(eWit w2 t ) ,

    and w,(K), w2(K)are as given in (5.5)or (5.9). Note that Qi3(K, ) satisfies thecondition, necessary for the reality of B ( x , ) ,

    &ij(-K, t ) =&t@,) . (5 .17)Clearly the development of Bi(x,) is ultimately dominated by contributions

    to the integral (5 .15) from the region K < K, of K-space for which w1 > 0,and since w 2 < 0, we may take (in that region)

    Q,Lj(K,) - +(aij- ei jkfZk)e(Jl t . (5 .18)


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    Turbulent dynamo action at low magnetic Reynolds number 447The spectrum tensor of the field B(x, ) s given by

    It will be sufficient to consider the case in which B(x,O) is (statistically)FkZ(K,0 )= ~ ( K ) ( 4 A 3 z )i W w , i m & n . (5.20)

    Substitution of (5 .16) and ( 5 . 2 0 ) in ( 5 . 1 9 ) leads, after some simplification, to

    pseudo-isotropic, so that

    Ftj(K, ) = &[M(K)(e2"l t+2"at )+N(K)(e2"lt- e2"at ) I (4j-G@-+ i [ N ( K ) ( e 2 @ +2"at)+M(K) (e 2" l t - Zwat)]etj&, (5 .21)

    and, for K < K , and umaxt9 1, this givesI & ( K , t )N + [ M ( K ) N ( K ) ] 6,,-BtB,- i~~kBk)e2Wlt . 5.22)The magnetic energy density is given by

    ( 5 . 2 3 )so that , ultimately, restricting the integration to the region K < K, which pro-vides the growing contribution,

    +(B2) N +IgcM ( K ) N ( K ) ] 2"i(mt4;rrK2dK. (5 .24)It may reasonably be assumed that M ( K ) nd N ( K )vary continuously in the

    range (0 ,K,). The function 4;rrK2M(K)s the initial magnetic energy spectrumfunction, and it may be expected to have a maximum at some wave-numberof order L-l, since the initial field has typical length scale L. The functionN ( K )will be identically zero if the initial field is strictly isotropic; if not, it maybe expected to have the same qualitative behaviour as M ( K ) . Withw,(K) = h K ( K , - K ) , it isevident from ( 5 . 2 4 ) hat thereisapreferentialamplifica-tion of different contributions in the range ( 0 ,K,) and that ultimately the integralis dominated by contributions from the neighbourhood of K = +K,. Asymptoticevaluation of the integral gives, for'umax t 9 1,


    B(B2) N 2 n ( 3 K c ) 2[N(+K,) N ( & K , ) ]

    To be quite specific, suppose that

    corresponding to an initially isotropic random B-field with a simple spectrumfunction having a single maximum at a wave-number of order L-l. Then

    (5 .27)+(Bz) $(&i-)g CK: exp ( -BPK,")

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    448 H . K . MoffattWith a = O(u;l),h e dou bling tim e for th e magnetic energy is ultimately of orderR;3 towhere to= l/u,and th e characteristic length scale of th e field t h a t ultim atelydom ina tes is of order K;l, i.e. of order Rz21. This m ay be large or sm all com paredwith its initial scale L = O(e-ll).)

    (iii)Evolution of a localized B-jieldSuppose now t h a t th e initial field B(x,0) is th a t d ue to a localized current dis-tribution J(x,0) , .e.

    (5 .28)Such a field is in general O(XI-^) as 1x1 +CO an d i ts Fourier transform has a nassociated directional singu larity a t K = 0. Writing

    V A B(x,0) = ~ J ( x ,) , V.B(x,0) = 0.

    (5 .29 )it is known (by analogy with the problem studied by Phillips 1956) tha t , forK+O, &K, 0 ) = ( c&-&l? j )q+O(K) , (5 .30 )where 4 s indepen dent of K.The B'ourier transform again evolves as i n (i) above, so t h a t

    B,(x, ) = Q,~(K,)&K, 0) e i = . x c l 3 ~ , (5 .31 )sand th e tota l energy of th e blob isM = 2 B2dV= - Qij(K,t)QZ(K,t)Bj(K,O)@(K,)d3K, (5 .32)'s

    so th at , ult imately,&f ie2wma.t

    x&$K,, O ) d A ( ? K C ) . S e x p i - 2 h ( K - : X , ) 2 j d K ,where K, = K c 8 , an d dA (&K,) indicates integration over th e surface of th esphere 1KI = +K, n K-space. T his simplifies t o

    where 2,( = hw ih,,) = 6 (BK,, 0).(5 .33)

    6. Dynamo action for general homogeneous turbulencewave solutions of the form (5 .1) ; substitution givesW hen th e principal values a,8, y of Aijare all different, (2 .23) still adm its plane

    ( ~ + m )&( ip) yK2&-PK3&), (6 .1)

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    Turbulent dynamo action at low magnetic Reynolds number 449and two other components given by cyclic permutation. For a non-trivial

    I F-K3 w + h K 2 ? K 1solution,


    a cubic with roots w = - h K 2 , (6.3)w I , w ~ - K2f l/A)(ByK2, yaK; + /3Kf)+. (6.4)nd

    The two roots given by (6.4) reduce to those given previously in (5.9) in the iso-tropic case a = /3 = y .

    The root (6.3) is relevant if the right-hand side of (6.1) (and the two similarequations) vanishes, i.e. if

    Since the field must satisfy &. f i = 0, this can arise only if(6.5)P B3Y%E=---, -K , K2 K3

    K2, Kg K i-+-+- = 0,. P Yand this can happen only if a)3 and y are not all of the same sign. In this case,(6.6) defines a cone in K-space, and if K lies along a generator of this cone, thenthere is no interaction with the turbulence, and the wave decays as e-AKet;this circumstance must clearly be regarded as exceptional.

    If the roots (6.4)are substituted back in the three equations for the components

    where Q = + (PyK2,+ y a K ; + apKi)h. (6.8)modes, v i z . f i ( t )= B(l)emit+B@)ewzt, (6.9)


    For these modes, K . B = 0 without restriction on the direction of K . I f K doesnot satisfy (6.6)) hen the general behaviour may be represented by a sum ofthe values of Bcl)and B(2) being obtainable in terms of fi (0), using (6.7).

    The roots (6.4) are complex (with negative real part) if Q 2 < 0, and they arereal if Q2 > 0. I f , moreover,

    Q 2 = PyK2,+yaK: +a/3Kf > h4K4, (6.10)then the root w, isreal and positive, and the associated waveis amplified exponen-tially in time. I f a, 3 and y are all of the same sign, then this behaviour is quali-tatively the same as in the isotropic case; the surface Qz = h4K4 is a closedovoid intersecting the K,, K , and K , axes in the points

    respectively, and a mode of wave-number K decays or amplifies according asthe end-point of the vector K lies outside or inside this surface (figure4).29 FLM 41

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    450 H . K. MoffattIf a, 3 and y have different signs, suppose that a > /3 > 0 > y (the case/3 < 0 being similar). The surface Q2= h4K4 intersects any plane through the

    K,-axis in a figure-of-eight,and once again the mode decays or amplifies accordingas K lies outside or inside the surface. If ( y [ > a, /3, then the surface is closelywrapped round the K,-axis. In this case, only disturbances for which K is nearlyaligned with the K,-axis (and satisfying K < (a,i3)*/h2)ill be amplified.

    DecayI Kz


    (i) GL2.F>y> 0 (ii) U $8 >0 .yFIGURE. Intersection of the surface Q2 = h4K4with the K , - K 3 plane. I n the axi-symmetric case (a= p ) the surface is the surface of revolution obtained by rotating thecurve about the K,-axis. The region inside the surface is the region of K-space in whichFourier components of the B-field are amplitied.If B(x,0) is a random homogeneous function of x, t is clear that, since the

    growth rates of its constituent Fourier Components depend on the direction aswell as the magnitude of their wave-vectors, anisotropy will develop even if theinitial B-field is isotropic. The anisotropy will be particularly strong if a, andyare very different in magnitude. For example, in the case y < 0, IyI >> a,P,(case (ii) in figure 4) he decay of all Fourier components for which

    implies a systematic increase in the characteristic length scale of the field inthe x and y directions. Ultimately, the field will vary significantly only in thez-direction.

    Similar considerations apply t o the case of a localized initial field. In this caseif y < 0, and Iy J9 a,p, then the lateral extent of the blob in the x and y directionswill increase as (At)*, since the dominant process affecting Fourier componentswith wave-vectors in the corresponding directions in K-space is the conventionalohmic decay. The length scale in the x-direction on the other hand will settledown to O(A2 / l y l ) s in $ 5 .

    KDK2 2 (a/lYI)K3,P/IYl)K,

    7. DiscussionIt has been established in the foregoing sections that turbulence having a

    dominant scale I is capable of systematically amplifying a magnetic field whoseinitial length scale L s large compared with I ; it was assumed that R, < 1;and a sufficient condition for dynamo action is then that at least two of the

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    Turbulent dynamo action at low magnetic Reynolds number 45 1principal values a,p, y of the tensor Aij,defined by (3.2) in terms of the spectrumtensor of the turbulence, should be non-zero, a condition that is generally satis-fied in turbulence that lacks reflexional symmetry.

    The assumption L/I B 1 is to some extent merely a convenience in describingthe problem a t the outset. I f the initial field B(x ,0) is a homogeneous randomfunction of x and if the condition L/I B 1 is not satisfied, then any of its Fouriercomponents for which IKI = O(1-1) will decay quite rapidly (since R, < 1)leaving only those components (possibly extremely weak) for which IKI < - l .These would decay slowly in the absence of turbulence; but the effect of theturbulence is to regenerate them through a coherent interaction between thevelocity and the small-scale magnetic fluctuations.

    The characteristic scale of the amplifying magnetic field ultimately settlesdown to O(Rk21).here is an obvious difficulty in designing a laboratory experi-ment to test the theory; with R, M 10-I (in mercury, this would require a tur -bulent Reynolds number R E 105)and with I M 10-2 m, a magnetic field wouldbe amplified only on scales of order 1m and greater. The possibility of anexperiment on mercury (or any other liquid metal) on this scale is remote.

    I n situations of interest in astrophysics, the magnetic Reynolds numbermay often be of order unity, or much greater. Under these circumstances, itis possible that the low wave-number Fourier components of the B-field (i.e.those for which lKl1 < 1) still grow exponentially in time (and Roberts 1969has provided evidence in the case of velocity fields that are periodic in spaceand in time th at this is the case), but the evolution of the field is likely to bedominated by the growth of Fourier components for which IKIl = O(1) andgreater, and the approach described in this paper is not adequate to trea t suchbehaviour.

    Thecrucialstageofthis workiscontainedin $ 2 ;because, once (2.23)isaccepted,dynamo action follows as an inevitable consequence without further approxima-tion. The analysis of 8 2 has been presented in a manner t hat makes some appealto physical intuition, but minimal appeal to mathematical formalism. Theanalysis may be formalized by introducing a space variable X = Rk x, and a timevariable T = Rgt (these being suggested by the results (5.11) and (5.13)), andby then allowing B to depend (independently) on x, X, t and T . The dependenceon x, t describes the fluctuations on scales characteristic of the turbulence,and the dependence on X , T describes the long-time development of the large-scale structure of the field. To lowest order in R,, this approach leads to preciselythe equations ( 2 . 8 ) and (2.16) given in $ 2 ; and it can also yield systematic higherapproximations. If the helicity spectrum F ( k ) s identically zero, then the f i s teffect of the turbulence at low R, must be sought a t these higher levels and a tthis stage, higher order spectral tensors (than the second) of the turbulenceenter the analysis. Developments in this direction perhaps merit further study.


    I am grateful to Professor Philip Saffman and Dr Robert Kraichnan for theirpenetrating comments on the first draft of this paper, and to Dr Glyn Roberts,who drew my attention to the important series of papers by Steenbeck, Krause &Rgdler.


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    452 H . K . MoffattR E F E R E N C E S

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