H.K. Moffatt- Transport effects associated with turbulence with particular attention to the...

8/3/2019 H.K. Moffatt- Transport effects associated with turbulence with particular attention to the influence of helicity

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Rep . Prog. Phys., Vol. 46, pp 621-664, 1983. Printed in Great Britain

Transport effects associated with turbulencewith particular attention to the influence of helicity

H K MoffattDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street,Cambridge CB3 9EW, UK

AbstractThe action of turbulence on a passive convected scalar field (e.g. temperature) orvector field (e.g. the magnetic field in an electrically conducting fluid) is reviewed,with particular a ttent ion paid to anom alous effects that can arise through the influenceof Corio lis forces in a rotating system on the statistics of th e turbulenc e. T he simplestsuch effect (which corresponds to a breaking of the Onsager symmetry relations) is askew-diffusion effect, i.e. th e a pp ear an ce of a component of turbulent heat fluxperpendicular t o the local mea n temper ature gradient. Th e fam ous a effect ofmagnetohydrodynamic dynamo theory is also in this category, as is the more subtleRadler effect (the appearance of a mean electromotive force perpendicular to themean current in a plasma). These effects are all associated with the helicity of aturbulent flow, i.e. the correlation between the velocity field u (x , t ) and the vorticityfield w(x, t )= curl U.Sections 1-4 are introductory in nature, and discuss the problem of heat spotdispersion, t he interaction of molecular and turb ulen t convective effects, the spectraldescription of rand om scalar an d vector fields, and t he spectral pro perti es of a passivescalar field which is subject t o bot h influences. In 0 5 , the m ean-field (or double-lengthscale) approach is presented for the scalar field problem, and the general theory ofed dy diffusivity and skew diffusivity is develop ed. Sections 6 and 7 are devoted tothe Lagrangian approach and the first-order smoothing approach-complementaryapp roac hes which have som e interesting points of contact and which provide som euseful insights. Sections 8-10 describe the corresponding theory for the vector fieldproblem. H er e the general theory of flux expulsion is shown to em erge in a naturalway from th e mean-field app roac h, provided effects of inhomogeneity of the turbulencear e included; the associated flux tra ns po rt velocity is invariably down the g radient oflocal turbulence intensity.Section 11 extends th e double-length scale app roach to incorpo rate many lengthscales; the effect of successive averaging over increasing scales leads to equations (inthe vector field conte xt) which a re invariant in form , and which permit the evaluationof the dynamo coefficient a and the turbulent diffusivity /3 in the astrophysicallyinteresting limit of large magnetic Reynolds n um ber R,.@ 1983 T he Ins titu te of Physics 621

www.moffatt.tc


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622 H K MoffattFinally, in g 12 , a model problem is analysed, showing how, when the moleculardiffusivity q is extremely small, an intermittent structure can develop in a passiveconvected field. In this situation, the simple concep t of an eddy diffusivity isinade qua te, and attem pts to rela te the statistical properties of the convected field to

low-order spectral properties of the convecting velocity field are unlikely to succeed.This review w as received in A ugust 1982.


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Transport effects associated with turbulence

Contents

623

1. Introduction2. He at spot dispersion

2.1. Lagrangian analysis of particle dispersion2.2. Effects of weak molecular diffusivity

3. Spectral terminology; energy and helicity spec trum functions4. Spectral theory for convected scalar field5 . Do ubl e-len gth scale ap pro ach applied to th e scalar field problem6. Lagrangian analysis of the fluctuation field

6.1. T he antisymmetric part of Dij6.2. Limitations of the Lagrangian analysis7 . First-order sm oothing theory8. Double-scale analysis for the convected vector field9. Lagrangian analysis for the convected vector field (77 = 0)9.1. Effects associated with mean-field gradient

10. First-order smoothing theory for a convected vector field11. Multiple-scale (or renormalisation gro up) determ inat ion of12. De velo pm ent of intermittency in convected fields

transport coefficientsAcknowledgmentsReferences

Page625628628630632635638642645646646650652654655657660663664


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Transport effects associated with turbulence 62 5

1. IntroductionFluid flow in large-scale geophysical an d astrophysical c ontexts is generally tu rb ule nt,i.e. ther e is a ran dom velocity fluctuation sup erpo sed on the mean velocity, and thisrand om fluctuation plays an imp orta nt part, not only in the ov erall dynamics of theflow field, but also in th e tra nspo rt of any fluid pro pert y (e.g. tem pe rat ure ) or of anycon tam inan t (e.g. radioactive waste in the oceans). T he rand om convective action ofthe tu rbu lent eddies is roughly analogous to th e molecular agitation th at is responsiblefor molecular diffusion effects, and the action of turbulence may in the simplestcircumstances be represented by turbulent diffusivities (sometimes known as eddydiffusivities), e.g. turb ul en t viscosity vT, turb ulen t the rm al diffusivity K T , etc, quantitieswhich are generally much larger than their molecular counterparts Y, K , . , . . Thesetur bu len t diffusivities all have th e physical dim ensions of velocity x length, and a crudeestimate of th eir mag nitude is generally given as

where u o s the roo t m ean squ are of t he velocity fluctuation an d I is (in some sense)a dom inant scale of the turbulence. M ore generally, the tran spor t coefficients vT , K T ,etc, are determined in principle by the statistical properties of the turbulence; andturbulent transport theory is concerned with establishing the precise relationshipbetween the tran spor t coefficients and these statistical pro per ties.In large-scale geophysical and astrophysical contexts, Coriolis forces associatedwith global rota tion have an effect on th e statistical prope rties of the t urb ulen ce andhen ce on th e tran spor t coefficients. Firstly, it is obvious that the global rotati on (e.g.rotation of the Earth in meteorological and oceanographic contexts, rotation of theSun in the solar convection zone context) introduces a preferred direction in anyturbulen t region, so that th e turbul enc e may be severely anisotropic. Secondly, andm or e subtly, a combination of energy flux parallel to the rot ation vector and Corioliseffects leads to a correlation between turb ulen t velocity U and turbulent vorticity w,known as the m ean helicity of t he turbulence %= U * 0).T h e existence of this m eanhelicity not only modifies the values of the standard transport coefficients, but alsointroduces new effects, qualitatively distinct from the traditional transport effects,which might well be described as pseud o-tra nsp orteffects, in view of th e pseudo -scala rcharacter of the quantity %. T h e most fa m ou s of these new effects is th e a effect(Steenbeck et a1 1966) which is responsib le for th e gen era tio n of mag netic field(spo ntan eou s dyn am o action) in an electrically conducting fluid in turbu lent m otion .Th er e is, however, an analogous effect in th e ther m al diffusion con text; the prese nceof helicity can lead to a component of heat flux which is perpendicular to the localmean temp erature gradient (no matter what the direction of the latter may be!) (Thiseffect, which may be described as skew diffusion, will be analysed by means of thedouble-length scale pro ced ure in 5 below.)In this review, we shall consider the action of turb ulen ce on dynamically passiveconvected fields which ar e also subject t o th e influence of a molecular diffusivity,which in all cases we shall denote by 77 , The convected field may be either a scalar


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626 H K Moffattfield 0 (x , t ) satisfying th e transport equation

a e l a t + U vo = 7 7 ~ ~ 0where u(x, t ) is th e tu rb ul en t velocity field, or a vector field B ( x , ) satisfying V 4 B = 0and aB/a t = v A ( U A B ) + 7 ~ ~ . (1.2)In either case, we shall suppose, unless otherwise stated, that U is solenoidal, i.e.V U = 0, and that U is a stationary random function of x and of t with zero me an ,i .e . ( U )= 0. (T he angular brackets (. . .) represent an ensemble average.)The function 0 in (1.1) may represent either temperature fluctuations (or thetemperature excess over some standard temperature), 77 then being the thermaldiffusivity, or th e concen tration of a convected solu te, e.g. salt in water. W e shalladopt thermal terminology, but the wider applications may be borne in mind. Theequation may equally be written

aolat = -V - w h e r e q = u @ - q V @ . ( 1 . 3 ( ~ ,) )q represents the flux of temperature due to convection by the velocity field anddiffusion down the gradient of 0. If lo is the scale of the temperature fluctuationsand u o is th e root mean squ are velocity ( u o= ( u ~ ) ~ ~ ) ,he ratio of th ese two contribu -tions has ord er of magn itude

Pe is th e PCclet num ber, an d it is generally large in situatio ns of practical im portan cewhen Z o is taken to b e of the sam e order as the scale of th e most energetic ingredientsof th e turbu lence (th e energy-containing eddies-see Ba tchelor (1953) ). Th is m eansthat the turbulence causes a very severe distortion of the isoscalar surfaces 0 =constant and high gradients of 0 develop, before molecular diffusivity effects aresignificant. It is this strong disto rtion of t he te m pe ra tur e field which constitutes thema in difficulty in th e analysis of eq ua tio n (1.1).Th e function B in (1.2) may rep resent th e magn etic field in an electrically con duct-ing fluid; an d indeed this is perhaps th e only example of any imp ortance. (T he vorticityfield in a non-conducting fluid satisfies (1.2) also, but in general it can hardly bedescribed as dynamically passive, since th e velocity field instantaneously adapts itselfto satisfy V A U =U, V * U = 0.) In t he m agnetic context, (1.2) is known as the inductionequation, and it is obtained by eliminating E and J from th e equationsa B / a t = - V A E F O J = V A B V * B = O (1.5)

andj = a ( E + u AB).

Here , E is the electric field, J is th e electric current and U is th e electrical conductivityof the fluid (77 = (poa)- ) , Displacem ent current is omitted from th e second of (1.5)(the magnetohydrodynamic approximation). Eq uation (1.6) is Ohms law ad apted totake account of th e motion of th e medium ; it may be written equally in the formE = - U A B + T V A B

(cf 1.3(b)). If Zo is now the scale of variation of B , th e ratio of the two contributions


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Transport effect sassoc iated with turbulence 627to E is

(1.7)as in (1.4). In this context, th e ratio u o l o / ~s known as the m agnetic Reynolds num berR,. Th e problem of convection and d istortion of B by turbulence is of ac ut e interestin astrophysical contexts, particularly in th e context of th e ma gneto hydr odyn am ics ofturb ule nt convection zon es of stars like the Sun, and in these contexts R, is gene rallylarge also (typically of t he or de r of 106 or gr eate r). Again this implies that a magn eticfield will be very strongly distor ted by turb ulence before th e weak effects of mo leculardiffusivity can have any significant influence o n th e field evolution.If we simply put 7 = 0 in ( l . l ) , then th e Lagrangian derivative of 0 vanishes:

and so 0 s constant for any fluid elemen t. Let x = x(a, ) represent the path of thefluid particle which passes through a at t ime t = 0 (figure l ( a ) ) ; hen the solution of(1.8) is simply@(x, t )= @(a, ) . (1.9)

Similarly, if we put 77 = 0 in (1.2) and expan d th e right-hand side, using V U = 0 ,we obtain(1.10)

T he solution of (1.10) analogous to (1.9) isB i ( x , ) =&(a, O)8xi/aai . (1.11)

Equ ation (1.10) implies that magnetic lines of force are frozen in the fluid, the fluxof B across any material element being conserved. Equation (1.11)expresses thisconservation in a different way: the tensor axi/aai is the strain tensor for the fluidelement initially at a ; its antisymmetric part represents the rotation of the element,and the symmetric part represents the stretching, and the initial field threading thefluid elem ent a t a is subject to bo th rotation a nd stretching (figure l( 6 ) ).v l a . t lf l t I O f l

a,t:O

lal lb )Figure 1. ( a ) Lagrangian path of a fluid particle. ( b ) Distortion of a fluid element; an initially sphericalelement becomes ellipsoidal under the action of irrotational strain plus rigid body rotation.


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628 H K Mof fa t tT h e generality of eq ua tion s (1.8) and (1.10) were discussed in an early paper ofBatchelor (19 52 ), who also derived som e of their im po rtan t consequen ces. Since anyline element tends to increase in length due to the action of turbulence, it followsthat a magnetic line of force (or B line) will also increase in len gth as it evolves

according to (1.10); he field intensity increases in prop ortio n to the length of elem ent sof the B line, and so (B) inevitably increases as a result of the action of turbulence.A t the sa m e time, however, the length scale of B decreases and ultimately moleculardiffusion must in terv en e in the sub seq ue nt evolution of B ;whether (B) hen continuesto increase or not is a question of great delicacy (Kraichnan and N agara jan 19 67 ) towhich an absolutely definitive answer is not yet available.Similarly, as argued by Batchelor (1 95 2), he are a of a material surface 0 = constantwill increase exponen tially u nd er th e action of turbulence (when q = 0), and so, sincethe volume of any fluid element is conserved when V * U = 0, the normal distancebetween adjac ent surfaces 0 = constant must dec rease as t ime increases. Th e gradientof 0 ,G = VO, therefore increases nearly everywhere, a nd (G)ill certainly increas euntil the two contributions to q in (1.3) are (in the m ean) of the sam e order ofm agnitu de. A m or e detailed discussion of the effects of m olecu lar diffusivity is givenin 2.2.2. Heat spot dispersion2.1. Lagrangian analysis of particle dispersionIf th e sole sou rce of tem pe ratu re variation is a heat spot introduced at x = 0 at t imet = 0, the n th e initial condition for (1.1) s

0(x , 0) = O,S(x). (2.1)If molecular diffusivity effects are negligible (i.e. q = 0) then the heat spot is simplycarried w ith th e fluid, i.e. it beh aves like a ma rk ed fluid particle. T h e analysis of thisprob lem (Taylor 192 1) provides the na tural start ing point for all studies of turbu lentdiffusion.L e t X ( t ) e the position of the h eat sp ot (i.e. of the m arked particle) at t ime t , sothat

( 2 . 2 )X / d t = u(X( t ) , ) = v ( t ) .u ( t ) is the Lagrangian particle velocity, and it is a stationary random function of tprovided U ( X , t ) is statistically both homogeneous in x and stationary in t t . We maythen define a Lagrangian correlation ten sor R !:)(T) by

(2 . 3 )where U: = (U). This may be expected to decrease rapidly as T +CO, and we maydefine a Lagrangian correlation time t, by

U ER I ( 7 )= (ui t ) u j ( t- ) )

(2.4)o n the re ason able assumption th at th e integral converges. This is the t ime ov er whichthe particle velocity remains strongly correlated with its initial value; for t >> t,, thei n appropriate modification for decaying homogeneous turbulence is described by Batchelor andTownsend (1956).


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Transport effects associated with turbulence 629influence of initial cond itions is forgotten, at least as far as th e particle velocity isconcerned.T he stationarity of u ( t ) mplies, from (2.3), hat

R IF 7 ) =R ( -7) ( 2 . 5 )R F(7) = R ( - 7 ) . (2.6)

and so R F)(T) s not sym me tric unless it satisfies the time-reversibility conditionThis type of condition is well know n in statistical mechanics, wh ere it is a particularconsequence of the principle of microscopic reversibility (Onsager 193l a , b).However, as recognised by Onsager, this principle is not universally valid, and inparticular it fails to hold in a rotating system when Coriolis forces have a significantdynamical effect at the m icroscopic level. In th e turb ulen ce context, th e microscopiclevel is the level of turbulent fluctuation, and it is well known (see, for example,Ibbetson and Tritton 1975) that Coriolis forces can and do affect the turbulencestatistics in a rotating system. Coriolis forces are certainly important in geophysicalcontexts (meteorological and oceanographic) on scales of th e or de r of 102km andgreater , and even more so in the larger-scale contexts of planetary interiors andconvection zones of rotating stars, towards which this review is particularly oriented.W e shall there fore give particular em phasis to situations in which th e condition (2.6)is not satisfied, and to consequ ences of this lack of sym me try.Consider now the particle displacement X ( t ) which (like t ) ( t ) ) atisfies ( X )= 0,but which is not in general a stationary random function, since the particle wandersfrom its initial position with excursions which increase on average with time. Thetensor dispersion (XiXj )atisfies

ddt- XiXj)= ( V i X j + U & )

r t

Joinvolving only the sym metric pa rt of R f f ( ~ ) .t follows fro m (2.7) hat, as t + 00,m

( X z X , ) - u E t l o ( R I ~ ( T ) + R ~ ~ ) ( T ) ).r (2.8)provided the integral convergeq, and in particular

( X 2 ) Dot where Do= f u ; t c (2.9)a result char acter istic of a diffusion process in th re e dim ensions with diffusivity Do.Th e probability distribution of X ( t ) hen spre ads as an ellipsoidal G aussian cloud,with principal axes determined by the symmetric tensor on the right of (2.8), andlength scale increasing like t .T he me an angular mom entu m (per unit mass) of t he particle at time t is given by

h ( t ) /I( t ) ) (2.10)and this can b e non-zero if R f ( t ) s not sym metric. In fact we find tha t, for t >>to


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630 H K Moffatt

so that the me an angular mo me ntum is asymptotically constant. Thus, if R!:'(T) sno t symm etric, du e to the dynam ical influence of C oriolis forces, the particle may beexpected to follow a random path which nevertheless exhibits a preferred sense ofturning about the direction defined by (2.11) (see figure 2).

Figure 2. Particle paths for a random velocity field u ( x , t ) with zero mean ((U)= 0) bu t preferred sense ofturning ((o(a, t ) o ( a ,0) ) 0) .

Results analogous t o (2.8) and (2.11 ) may b e obt ained similarly for higher-orderstatistical quantities; for exam ple, the triple m om ent (X iX&) is given asymptoticallyby

This triple correlation may b e expected to be small if e ith er 71 o r 72 is much greaterthan t,, an d the integral in (2.1 2) will be generally convergent. It is evident fr om(2.12 ) hat any skewness in the probability distribution of leads, as might be expected,to a corresponding skewness in the distribution of X .2.2. Effectsof weak molecular diffusivityMolecular diffusion causes the convected hea t spot to sprea d, the initial spread beingspherically symm etric an d given by

(2.14)the e lem entar y solution of the three-dimensional diffusion equ ation . This diffusedspot then interacts with t he local velocity gradient d u i / d x j at the m oving point x = X ( t ) ,


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Transport effects associated with turbulence 6 3 1

Figure 3. Combined action of convection and molecular diffusion on a heat spot released at A at timet = 0. At B, the spot is still spherical under the action of diffusion alone; at C , the local velocity gradientcombined w ith diffusion has distorted the spot t o ellipsoidal s hap e; at D, the spot has spread outside theregion of locally uniform d istortion and its shape is no longer ellipsoidal.

which distorts it into app roximately ellipsoidal sha pe (figure 3). This leads t o acceler-ated diffusion (Tow nsend 1 9 5 1 ) in the sense tha t the dispersion of the sp ot relativeto t he position x = X ( t ) s greater than it would be under the influence of diffusionalone. Ho we ver, as shown by Saffman ( 1 9 6 0 ) , he sam e interaction between the localstrain field and molecular diffusion leads to a decrease in the dispersion of the spotrelative t o its initial poin t of rele ase. Th is dispe rsion is given bys2 x 2 0 ( x , ) dx(J 0J ( 2 . 1 5 )

and Saffman showed that, for small t imes t after the instant of release of th e spo t,S2= 6( D,+ 77 )t - 5 7 7 ( d u i / d ~ j ) ~ t ~ + Ot 4 ) ( 2 . 1 6 )

the interaction between the velocity gradient dui /dx j and the molecular diffusivity 77being ap pa ren t in the form of the 0 ( t 3 ) erm . This at first sight surprising red uc tionin dispersion arises because th e me an velocity o ve r the distorted spot is generally lessin magnitude than lu (X , )l (since the velocity av erages to zer o ove r the whole fluid);the effective velocity convecting the spot away from its point of release is thereforereduced whe n its spr ead and distortion b ecom e significant.Th e same type of reduction is apparent, and perhaps more easily appreciated, ifwe con sider t he d ispers ion of a blob of dy ed fluid th at is initially large comparedwith the most ene rgetic ingredients of th e turbu len ce ; e.g . we may con sider the caseof a n initially spheric al blo b,

( 2 . 1 7 )Consider first what hap pen s if molecu lar diffusion is neglected. Tu rbu len t ed die s onscales 1 < a cause progressive distortion of the spherical boundary (figure 4). If th eRe yno lds nu m be r of t he turbu lence is sufficiently large for th e existence of a well-established Kolmogorov energy spectrum (Batchelor 1 9 5 3 , chap VI), then the t imescale of d istortion on a scale 1 is given by

( 2 . 1 8 )where E is the rate of energy dissipation per unit mass, so that small ripples appearfirst on the surface, then larger-scale distortions, as indicated in the figure. During

-1 /3 2 / 3t l - 1


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632 H K Moffatt

IR I I ) IclFigure 4. Distortion of an initially spherical blob of dye which is larger than the integral scale of th eturbulence. In ( b ) ,eddies on the scale I have had time to distort the surface of the blob; in ( c ) largereddies with longer characteristic time scales (see equation (2.18))have had time to take effect. As th edyed fluid moves outw ards (on average) from the centroid 0, the undyed fluid move s inwards to take its place.this process, th e volu me of dy ed fluid rem ains con stant, a conseq uen ce of incom pressi-bility. Thus the evident outward spread (or flux) of dyed fluid from its centre isaccompanied by an inward flux of undy ed fluid.W ha t the n is th e effect of w eak m olecular diffusion? Clearly this simply smo othsou t th e initial discontinuity in dy e concentration-and it do es this by transferring dyefrom the fluid which is on average moving outwards to the fluid which is on averagemoving inwards; the reduction in the dispersion of the cloud of dye is an inevitableconsequence.3. Spectral terminology; energy and helicity spectrum functionsAlth oug h we shall mak e minimal appeal to the rather heavy mathematical a ppa ratusof t urb ule nc e in th e discussion that follows, so m e simple definitions and relationshipsare unavoidable, and are gath ered tog ether in this section, for future ea se of reference.T h e rand om velocity field u ( x , ) admits the Fourier representation

u ( x , )= u^(k,w ) exp [i(k * x -w t ) ] dk dw (3.1)Iwith inverse6 ( k , w )= ( 2 ~ ) - ~(x , t ) exp [-i(k - x - w t ) ] dx dt. (3.2)

Here , 6 ( k ,w ) must be interpreted as a generalised function; the reality of U and theincompressibility condition V - U = 0 imply tha t U satisfiesI

u^(-k, - w ) = u^*(k,U ) k * u^(k,w )= 0 (3.3)for all (k, ) the s tar denotes the complex co njugate) .T he Eulerian correlation tensor Rij(r, ) s defined by

Ri i ( r , ) = ( u ~ ( x ,) u i ( x + r , +. r )> (3 .4)a function only of the separat ions r and T provided U is homogeneous and stationary.


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Transport effects associated with turbulence 633Th e spect rum tensor Q i j ( k ,w ) s defined as the Fo urie r transform of R,, .e .

Qij(k,U )= Ri j ( r ,T ) ex p [-i(k * r - w 7 ) ] d r dT (3 .5)Ian d this satisfies the well-known relation(u^k, w ) a j ( k ,w) )=@ij (k , ) S ( k k ) S ( w - U ) . (3 . 6 )Th e condit ions (3 .3) may then be used to demonstrate that Qii(k,w ) satisfies theHe rmitia n symmetry conditions

@.r(k,w )= @ j j ( 4 , -0) @ji(k,w ) (3 . 7 )

kj@jj(k,0) 0 kj@jj(k,w )= 0 (3 .8)and the incompressibility c onditions

for all (k, ).with instan taneo us correlationsIn the context of homogeneous decaying turbu lence , i t is customary to de al only

Ri j ( r , )= ( u i ( x , ) u j ( x r , t ) ) (3 . 9 )and to tak e Fo urier transforms with respect to space variables only, e.g.

@ij (k , )= ( 2 ~ ) - ~ij(r, ) exp (-ik a r ) dr . (3 .10)In a field of stationary (m ain tain ed ) turbulence these functions may, of cou rse, stillbe defined, and they are then independent of t ; when the nota t ion Ri j ( r ) ,O j j ( k )sused, it is this type of ins tan tan eo us correlation tha t is implied; the relation betwee nQ i j ( k ) nd Q i j ( k ,U ) s then simply

I

Q i i ( k ) Q i j ( k , ) d o . (3 . 11 )T he kinetic energy density of th e turbulence (pe r unit mass) is

$(U) = :Rii(O, 0)= E ( k ,w ) dk dw (3 .12)where

E ( k , w )= $@ii(k ,w ) . (3 .13)W e may define also the energy spectrum function E ( k )by

E ( & ,U ) d S( k )= I wS ( k )

(3 .14)where S ( k ) s the surface of a sphere of radius k in k space. Then (3 .12) becomes

$(U) = J E ( k ) d k0 (3 .15)a nd E ( k ) d k m ay b e th ou gh t of as the contribution to $(U) f rom wavenumbermagnitudes in the interval (k, k + d k ) .Clearly, the function E ( k ,U ) nvolves only the symmetric part of a i j ( k , ) . W eshall find tha t, in certain con texts, and particularly in the magnetic-field context ( 8 8


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634 H K Mofattbelow), the antisymmetric part

a$) (k , w ) = i [ @ i j ( k ,w ) - @ J k , w ) ] (3.16)can play an imp ortan t, and indeed a dom inant, role. Like the antisymmetric part ofR~;(T)iscussed in 2, a non-zero antisymmetric part of a i j ( k , w ) s generallyassociated with th e action of C oriolis force s in a rotating system . Th is antisy mm etricpart can be expressed in terms of a single real scalar function H ( k , w ) , he helicityspec trum function, defined by

H ( k ,w )= -ik,Eij,cPij(k, w ) (3.17)(see Moffatt and Pro ctor 19 82 ); we then have

@!;)(k,W ) =iiEjjkkkH(k, w ) / k 2 . (3.18)The major qualitative difference between H ( k ,w ) and E ( k ,w ) is that H ( k ,w ) is apseudo-scalar, i.e. it changes sign under any change from a right-handed to a left-handed frame of reference (parity transformations). A field of turbulence which isstatistically invariant under such transformations is said to have the property ofreflexional symm etry. Clea rly H ( k ,0) can be non-zero only if the turbulence lacksreflexional symm etry, i.e. only if, in so me sen se, right-handedness and left-handed-ness ar e distinguishable in the statistics of th e velocity field.This stateme nt acquires precision from th e fact tha t

H ( k , w ) = ( 2 ~ ) - ~U(& t ) * w ( x + r ,+ T ) ) e x p [ - i ( k * r - w ~ ) ] d rT (3.19)(where w = V A U, he vorticity field) as may easily be deduced from (3.5) and (3.17).Equivalently,

I(U x, t ) w (x + r , t +T ) )= H ( k ,w ) exp [i(k r -W T ) ] d k d r (3.20)Ian d, in pa rticular,

( U * U )= H ( k ,w ) d k dw .I (3.21)Clearly U * w >0 locally (in a right-h and ed fra m e of refe renc e) if the streamlines have(locally) a right-handed helical structure. T he function H ( k ,w ) defined by (3.1 9) isevidently a sensitive measure of the correlation between velocity and vorticity field.H may, of course, be positive in o n e pa rt of th e (k, ) space and negative in anotherpa rt. It is, how ever, boun de d in magnitude by the inequality

IH(k,w ) l s 2 k E ( k , w ) (3.22)a consequence of Cra me rs theo rem (see Moffatt and Proctor 19 82 ). No te that theconditions (3.7) imply th at E and H both satisfy th e s am e symmetry conditions:

E ( - k , - w ) = E ( k ,w ) H ( - k , - w ) = H ( k ,w ) . (3.23)The m anner in which Coriolis forces in a rotating system are responsible for thegeneration of a n on -ze ro helicity sp ect rum in a field of turbulence has bee n discussedat length in the lite ratu re of dy na m o theory (see particularly Moffatt 1 97 8 ch ap 10-12,Parker 197 9 chap 18, Krause a nd R adler 198 0 chap 9) and this discussion need notbe repeated here . As regards the dynamics of helicity, we simply draw attention tothe fact that th e mean helicity (U - w ) s an inviscid invariant of the dynam ical eq ua tion s


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Transport effec tsassociated with turbulence 635of motion (Moreau 1961), this invariance reflecting the invariance of the topologicalstructure of the vorticity field when viscous diffusion is neglected (Moffatt 1969).

4. Spectral theory for convected scalar fieldSuppose now that there is some source of temperature fluctuations S(x, t ) on a scalelarger than the scale ZO of the energy-containing eddies of the turbulence, and that( S )= 0, so that the turbulence generates a statistically homogeneous and stationarytemperature fluctuation field e(r, ) satisfying ( 8 )= 0. Equation (1.1) s then replacedby

(4.1)and the problem that presents itself is to determine the statistical properties of 0 interms of those of the velocity field U. This problem was initially considered by Obukhov(1949), Corrsin (1951) and Batchelor (1952) and was further developed in thefar-reaching studies of Batchelor (1959) and Batchelor et a1 (1959), whose results wesummarise in this section.

The argument is essentially an elaboration of the Kolmogorov picture of thedynamics of turbulence (see figure 5). In this, the all-important parameter is E , the

aelat + U ve = Tv2e+s(x, t )

Figure 5. The Kolmogorov picture of the dynamics of turbulence; energy cascades through the spe ctrumat a rate e, and is dissipated at wavenumbers d ) ( / c , , ) ; (logarithmic scale) .rate of dissipation of energy per unit mass by viscosity. In a statistically steady state,this is also the rate of supply of energy to the turbulence on scales of the order of loand greater. This energy cascades through non-linear interactions towards smallerand smaller length scales and ultimately to the viscous length scale I, at whichmolecular dissipation becomes effective; I , is determined in order of magnitude by Eand v only, and hence, on dimensional grounds,

1, - v 3 / E y 4 . (4.2)E - 30/10 (4.3)Moreover, E satisfies the important semi-empirical relationship


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636 H K Moffattwhere u o= (U), and so (4.2) becomes

1, - R e -3/410where R e = u o l o / v is the R eynolds num ber of the turbulence, assumed large.

In spectral terminology, we intro duc e wavenum bers

(4 .4)

k , = 1; - e 3 l 4 k o (4 .5)- -0 - 1 0and define the inertial range

within which th e statistical pro pert ies of the turbulence are de term ined solely by E ,the ra te of cascade of e nergy, an d the local wav enum ber k . In particular, the energyspectrum function E ( k ) s given byE ( k )= C E ~ / ~ , ! - ~ / ~ko> 1. If 7 < v, on the other hand,then as shown by Batchelor (1959) it is the root mean square strain rate C - ( e / v ) l l 2which is the relevant pro pe rty of the velocity field for determ ining t he small-scalefea tur es of th e 8 field; hence, dimensionally,

1,- (7/C) - ~ v / E ) / ~e-/2Ke-/410. (4 .13)Th e corresponding wavenumber k , = 1, is described as the conduction cut-off.


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Transport effectsassociated with turbulence 6 3 7Now let T ( k ) be the spectrum funct ion of 8, defined in a manner analogous toE ( k ) , nd with th e property

m(8 ) = r ( k ) k .

0(4.14)

In the spectral rangeko


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Transport effects associated with turbulence 639assum ption tha t the convec ted field-here @-is weakly inh om og en eo us o n a scaleL much gre ater than th e scale lo of the background turbulence. Att entio n is the ngenerally focused on the evolution of the ensemble-average field which evolves onthe length scale L, and on a time scale large compared with the time scale to - l o / u ocharacteristic of the energy-containing edd ies of the turb ulen ce. Th e situation isdepicted in figure 7 which shows the mean isoscalar surfaces (0) constant, which

Figure 7. Double -length scale basis for mean-field theory; the scale 10 of the turbulenc e is small compa redwith the scale L of the m ean field. Averaging is carried out ove r a cube of side a where lo


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640 H K MoffuttSubtracting (5 .3) from ( 1 . 3 ( u ) ) gives an eq ua tion for th e fluctuation field:

a6-= -U * G -V (U$-F) TV$ ( 5 . 5 )where G = VOo, th e local mean-field g radient. This equ atio n establishes a linearrelationship between G and 6 (assuming that any transient effects associated withinitial conditions have had time to disap pea r), and so likewise between G and F. fG were strictly uniform the n this linear relationship could only be of the form

at

where Dij is a tensor de term ine d (in principle) by the statistical properties of th eturbulence and the parameter 7, which intervenes in the solution of ( 5 . 5 ) . Th e heatflux F is then also strictly uniform, and note that, from ( 5 . 3 ) , Oo(x)= G * x thenremains constant in time.If G is not strictly uniform, then ( 5 . 6 ) must be regarded as th e leading term of aseries of th e form

( 5 . 7 )an expansion which may be expected to converge rapidly when lo /L s sufficientbsmall. T he tensors Dij,Eijk,Dit, . . . ,are all deter min ed by the statistical prop ertiesof the turbulence and T ; indeed, they may themselves be regarded as statisticalprop erties of the turbulence de pen den t on the auxiliary para me ter 71, and they areuniform (in x ) and constant (in t ) if the turbulence is homogeneous and stationary.T h e time derivatives in (5 .7) may b e replaced by space derivatives by using the spac ederiv ative of (5.3), iz

Substitution in ( 5 . 7 ) gives( 5 . 8 )

( 5 . 9 )For homogeneous turbulence, the two leading terms are not affected by this substitu-tion. (N ote, however, that if the turbulence is inhomogeneous, so that Dkl is a func tionof x, then (5 .9) becomes

Fi = -D!yGj +E$EaGj/axk , . . ( 5 . 10 )where

and( 5 . 11 )

( 5 . 12 )and th e leading term s are evidently both m odified by virtue of th e inhomo geneity.)


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Transport effectsassoc iated with turbulence 64 1Since the coefficient Dii does not depend on G, it may be calculated on thesimplifying assum ption tha t G is strictly unifo rm , so tha t the relationship (5.7) reducesto (5.6). Similarly, in order to calculate &k, it would be legitimate to assume thatdGj/axk s uniform (and therefore constant in the case of homogeneous turbulence)

so tha t only the first two term s of (5.7) survive; and so on.Attention naturally focuses on th e leading term given by (5.6). If t he turbulenceis isotropic, i.e. its statistical properties ar e invariant under rotations of th e fram e ofreference, then Dij s likewise isotropic, i.e. Dii=D&, and (5.6) becomesF = -DG = -DVOo (5.13)

a linear diffusive relationship, in which D must be interpreted as the turbulent, or'eddy', diffusivity.If the turbulence is not isotropic, then it is tempting to regard Dii as simply ananisotropic diffusion tensor and indeed its symm etric part(5.14)p' =3(Dij+Dji)

does fulfil this role. Ho we ver, Dii may also have an antisymmetric partD!?)~ @ . - D . . ) = E . .1 11 v kD(ka) (5 .15)

say, which makes a corresponding contributionF(')=D'"'A T O o (5 .16)

to the heat flux in (5.6). Note that D(') is a pseudo-vector, so that D(')*e s apseudo-scalar, where e is any fixed unit vector. H en ce (cf the discussion con cernin ghelicity in D 3), D") can be non -zero only if t he turbule nce lacks reflexional sym metry.It will be shown in the following section that the antisymmetric part of Dij isintimately related to the antisymmetric part of the Lagrangian correlation functionR f ' ( 7 ) ntroduced in 2, which is non -zero only if On sager's (1 93 1a , b) principle ofmicroscopic reversibility is violated. A s m ention ed p reviously, when Coriolis forcesare dynamically significant, this principle m ust b e ab and on ed, and it is the refo re notsurprising th at a contribution of the form (5.16) can then app ear.In homogeneous turbulence, Dij s uniform, and therefore so is D"). From (5 .16)we then have

V .F(')= 0 (5 .17)so that F") makes no contribution t o t he m ean-field e qua tion ( 5.3 ); this is because(5.16) describes h eat tran sfe r parallel to surfaces of constant Oo. However, if theturbulence is inhomogeneous, then D") (which should now be derived from themodified tensor DjF) of (5.1 1)) will be a function of x, and so

V * F ' " ) = V . ( V O o ) (5.18)where

v = v AD(') (5.19)and this, substituted in (5.3), mplies convection of Oo by an 'effective' velocity V , aphenomenon which has an important cou nte rpa rt in th e magnetic-field context to beconsidered later (see 8). The velocity V is evidently solenoidal; t he correspondingeffective convection velocity for magnetic field is noteworthy in being, in general,non-solenoidal.


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642 H K MofattThe above discussion is quite general and is based on the assertion that a linearrelations hip is established be twe en t he fluctuating field 8 and th e mean field gradientG (see equation ( 5 . 5 ) ) . I t does not depend on any assumption concerning the'smallness' of 8; and ind eed the fluctuations in 8 may be expected t o be large when

the P6clet number is large, because then fluid elements carry their O value over agreat distance, and therefore over a large change in O0 G - x, before moleculardiffusion begins to be effective.Th e sam e linear relationship may, of course, be exploited in constructing otherstatistical prop erties of 8; for example, (6') must be a quadratic functional of G, i.e.at leading o rde r in lo/L,(6') = qijGiGj (5.20)

where Qii is again a tensor d eterm ined exclusively by the statistical prop erties of th eturbulence and by 77. This sort of state m ent is entirely com patible with th e results of 4 (e.g. (4.13) and (4.1 8)) ; but it may be applied som ewhat mo re generally, in a waythat incorporates properties of the energy-containing eddies of the turbulence (onscales of the order of I o ) as well as prop erties o n much smaller scales.

6. Lagrangian analysis of the fluctuation fieldIn this section, we exam ine the form of the tenso r Dii,on th e assumption t hat diffusiveeffects are entirely negligible. T he tr eatm ent here is closely analogous t o tha t presentedby M offatt (1974) for the corresponding magnetic problem (see 0 9 below).W e suppose that a t time t = 0, 0 s non-ra ndom , i.e.

where a ( x , t ) is the initial position of th e particle w hich passes through x at time t.For given (x, t ) ,a varies from one realisation to another, and should be regarded asa random variable in (6.3).On th e assumption tha t th e scale of Oo is large, we now expand @(a, ) in Taylorseries about (x, 0):

(6.4)o(a,0) = (1-x v +i(X V)'- . .)OO(X, 0)where X = x -a , and substitute in (6.3), giving

F j (X , t )=-Dij(t)Gj(X, O)+Ejjk(t)Gj,k(X,)-. . . (6 .5)where

r f


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Transport effects associated with turbulence 643and

We have already enc ou nte red symmetrised forms of th e tensors Dij and Eiik (seeequations (2 .7) and (2.12)), and it is reassuring to find the same tensors appearingnaturally in the two-scale treatment. As noted earlier, the integrals (6 .6) and (6.7)approach their asymptotic ( t = a) alues rapidly for t a ,, th e Lagrangian correlationtime, and it is these asymptotic values which may now be used in (6 .5 ) ; hese will beden oted simply by Dijand Eiik.Th e time scale on which the m ean field Oo changes is then given bytM - 2 /Dwhere D -U& i.e.

tM - L/uo tJ2 tc . (6 .8)Provided tcc tM ,G(x,0 ) n (6.5) may be replaced by G(x , t ) (th e correction affectingonly higher terms in the expansion, as commented earlier) and (6.5) then takes theexpected form

This derivation of the expansion (6.9) and of the tensors Dii and Eijk, althoughmaking no appeal to the effects of molecular diffusivity, in fact depends o n theseeffects in quit e a subtle way. T he reason is tha t we have assumed tha t ther e is aninitial referen ce instan t t = 0 at which th e 0 ield is non-random , and we have supposedthat t he elapsed time t from this instant is small compared with t ~ ,s given by (6.8).For t = O( tM) nd g reat er, the validity of th e proc edu re leading to (6 .9) must be insome doubt when 77 = 0. However, when 7 is non-zero but still extremely small(compared w ith uolo),molecular diffusion always acts in such a way as to elim inate 8fluctuations with a consequent loss of memory of initial conditions. As shown byBatchelor (1952),(6.10)(vo))(~(0))((ve))- xp (it[)

where 5 is the mean logarithmic rat e of extension of line elements, i.e.a -5 = lim - n SL(t)r+m a t (6.11)

and this quantity, being determ ined by th e smallest scales of the motion, is determ inedin order of magnitude by5- E / v ) 1 2 . (6.12)

T he length scale of t he 8 field, I(?), is th er efo re given in orde r of mag nitude byI-L xp ( - $ t o (6.13)


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644 H K M o f f a t tand mo lecula r diffusivity effects beco me significant whe n

2rl7 lot-' d t -- exp (4 l )- 11L 2 5becomes of the order of unity, i.e. when

(6 .14)

(6 .15)--In ( ~ ' l / 2 7 7 ) .Provided this time is small co m pa red with fM, the initial conditions should b e irrelevantin the derivation of (6 .9 ) . Using (6 .12) and (6 .8 ) , and putting t, - l o / u o (probably alow estim ate ), this criterion reduces to

In Pe > 1and L/ lo > 1 , as has bee n assumed. This mea ns that an extremely small value of 7will in gen eral b e sufficient to ach ieve th e require d loss of m em ory of initial con ditions ,to m ake (6 .9 )valid for all t.It is natural now to focus attention on the leading term of the expansion (6 .9 ) .For t >> t,, we have

00Dij-Ui R;"(T) T (6 .17)

and in the case of isotropic turbulence, Dij=DOSij,here(6 .18 )

in agre em en t with the value of DO iven by (2 .4 ) and (2 .9 ) .W h e n t . (6 .22 )Th e complete series for D i j ( t ) bviously involves Eulerian m ea n produc ts of th e velocityfield of all orders .


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Transport e ffec tsassociated with turbulence 6 456.1. Th e antisymmetric part of DijThe pseudo-vector D) efined by (5.15) can be evaluated simply from (6.6) and isgiven by

Dca ) ( f )$ (0 (a, ) A 0 (a,7)) T (6.23)and it is easy to see that this will be non-zero if there is any systematic tendency forparticles t o veer to o ne side rather than the o the r in their otherwise random path(as in figure 2); no te tha t a particle path tha t is clockwise in a right-handed fra m eof reference is anticlockwise in a left-handed frame. A preference for one senserath er than the o th er is the refo re a symptom of a lack of reflexional symm etry in theturbulence. This lack of reflexional symm etry as it app ear s in (6 .23 ) may also beinte rpre ted as a lack of symmetry with respect to time reversal; for, from (6.6 ) an d (2.5),

0

(6.24)

Hence , if R f f (7)s invariant und er th e time reversal T -* -7, then Dij=Dji (cf Onsager1931a b). A motion like that sketched in figure 2 is clearly not statistically invariantun der time reversal, since reversing th e arrows on t he streamlines converts a systematicclockwise motion into a systematic anticlockwise motion.For t


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6 4 6 H K Moffattwhere $ ( x , y, t ) s a stationary ra nd om function of its th re e arguments, an d

w = (0, , V2$). ( 6 . 3 1 )It is evident from ( 3 . 1 9 ) hat H ( k ,U ) s identically zer o for such a flow (an d of cou rseU - w =0 also). In this situation, only the second te rm o n the right-hand side of ( 6 . 2 7 )survives, and we h ave

D(=)( t )- : t 2 ( U 2 w ) + ~ ( t 3 ) . ( 6 . 3 2 )6.2. Limitations of he Lagrangian analysisAlthough the above analysis provides an expression for the eddy diffusion tensor Dijwhich a ppe ars altoge ther plausible, and consistent w ith the T aylor app roac h, i t doe snot s tand up to at tempts to construct ot he r statistical qua ntities; even t he simplest ofthese , the m ean squ are of the fluctuation f3 =-X G, is given by( 0 2 )= GiGj(XiXj)- tDijGiGj ( 6 . 3 3 )which apparently increases without limit. This divergence arises partly becausemolecular diffusion ha s been neglected, a nd partly because t he gradient G has beensupposed uniform over the full extent of a particle trajectory throughout a t ime t(even though this increases in root mean square like t l") . In fact, we have seen inQ 4 that the equilibrium level of ( 8 ' ) is given by ( 4 . 2 1 ) (with possibly th e log arithmiccorrection ( 4 . 2 2 ) ) ; he process by which the transition occurs from th e linear growth( 6 . 3 3 ) o th e asym ptotic steady level ( 4 . 2 1 ) s clearly contro lled by molecu lar d iffusivity.Certain aspects of this process will be examined further by means of a simplifiedmodel problem, in 12.

7. First-order smoothing theoryT h e Lagrangian ap pro ach described in th e preceding section involves (in effect) neglectof the diffusion term q V 2 8 in the fluctuation equation ( 5 . 5 ) . An al ternat ive ap proach,which can provide some useful insights, is based (at lowest order) on neglect of theterm V * ( U 8 - F ) which involves the pr od uc t of fluctuating fields; m or e formally, weconstruct a solution of the form

e = x t P ) ( X , t ) F =W( " ) ( x , ) ( 7 . 1 )ae(')/at-qv2e(o)- . G (7.2)

( 7 . 3 )

F ( " ) ( U e ( n ) ) , ( 7 . 4 )

wherea e ( n + l ) / a t - q V 2 e ( n + l ) = - V . ( U e ( n ) - F ( n ) )

with

A t lowest orde r, this is known as the first-order smoothing approximation, or th equasi-linear approximation. Th e Fourier transform of ( 7 . 2 ) ( t reat ing G as uniform)is(- iw + qk2 ) , $ (0 )= - .G (7.5)


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Transport effects associated with turbulence 647and so, using (3 .6) ,

Ff o = (ui8())= - (-iw + qk2) -1@ i i (k ,w ) G i d k dw . (7 .6)IH e n c e

f7:) = -D!?)G.1 I (7 .7)where

iw + T k 2I I w 2 + q kD!?= 2 4 @ij(k,0)dk dw.

T he sym metric part of this is given byD!?= Qii(k,w ) dk dw

(7 . 8 )

(7 .9)and is a positive-definite tensor (or at any rate non-negative-definite) of a genuinelydiffusive characte r. No te that, as q +0, the integrand becomes singular at w = 0; butintegration through the singularity gives

D!? - ~ @!;(k, 0) d k (7 .10)which is in fact identical with the linearised form of (6 .17) (when the Lagrangiancorrelation deg enera tes to an E ulerian correlation). T he result (7 .10) may thereforebe se en as the meeting point of the Lagrangian theory a nd th e first-order smoothingtheory, which of course must agree when both diffusive effects and fluctuation-interactio n effects are negligible in (5 .5 ) .Th e antisymmetric part of (7 .8) is given by

D j O a ) = E , . kalp' (7 . 11 )where

w k H ( k , w )k 2 ( w 2 + q k ) d k d w (7 . 12 ) t

which admits comparison with (6 .27) and (6 .29) . Again, we se e tha t it is helicity th atgives rise to an antisymmetric ingredient of D,.Th e general treatm ent of eq uation s (7.1)-(7.4)at higher order s (n= 1 , 2 , . . . ) leadsto heavy no tation, and w e shall therefore l imit attention h ere to two simple exampleswhich shed light on so m e of th e physical effects involved. It is well kno wn in themagnetic context (Roberts 1970, 1972, Childress 1970, 1979) that there is a closeparallel between the action of turbulence on a convected field and the action of aspace-periodic velocity field. T he the rm al problem lends itself m ore readily to similarinvestigation, since even with a two-dimensional velocity field of the form (6 .30) ,non-trivial results may be obta ine d. Suppo se first tha t

4 = 4,,sin kx sin k y (7 .13)so tha t the velocity field consists of a squ are ar ray of ed dies (figure 8 ) . A s before , in+ I am indebted to Pe te r W at t e rson , who drew my a t t en tion to t he fac t t ha t t he an t i symmet r ic par t of D,,can be non-zero even unde r the f i rst -order smoothing approximat ion.


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6 48 H K Mofatt

G

X

Figure 8. Sketch of stream line configuration for the st reamfunct ion (7 .13),for which adjacent eddies rotatein opposi te senses. Th e heat flux i s parallel to the m ean tem perature gradient .calculating Dij,we may suppose that G is uniform an d steady, and th e steady solutionof (6.2) is then

(7 .14)Hence

the averag e being take n over a squ are cell. This, of co urse, is just the cou nte rpa rt ofthe mo re general result (7.7) and (7.8) . Now, however, we may continue the proc edu reto higher ord er in the relevant PCclet num ber Pe = (lr0/7. W e find that F 0 and(7.16)

All the odd terms vanish, because changing the sign of (lro in (7.13) is equivalent toa change of origin which clearly can have no effect on the mean diffusive propertiesof the flow, Evidently the diffusion tensor Dii or which Fi = -DiiGj has the form

where, from (7.15) and (7 .16) ,(7.17)

F ( P ~ )kpe - A p e 4 0(pe6) (7.18)for small Pe. T h e radius of conv ergence of this series is probably close to Pe = 4 .Incidentally the high PCclet nu m be r limit for this problem can be trea ted by bou nd arylayer meth ods similar to those developed by Proctor (197 5) in a related prob lem ; thissuggests tha t? F(Pe)-Pe12 as Pe +CO (7.19)iI am indebted to Dr M R E Proctor for this observation.


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Tran sport effects associated with turbulence 649so that DijK 7 as 7)+ 0 (all oth er param eters being fixed). It should be emph asisedhowever that th e behaviour (7 .18) and (7 .19) is critically de pe nd en t o n th e assumedsteadiness of the velocity field (7 .13); unsteadiness is an essential feature of realturbulent flows, so that th e results (7 .18) and (7 .19) are n o m ore than suggestive inthe turbulenc e context.We have seen in 0 6 that a motion which exhibits a preferred 'sense of turning'gives a non-symmetric Dip Le t us now see wh ethe r this behaviour arises also for smallPe. To get an array of eddies all with th e sam e sense, we need a 9 which is a periodicfunction of x and y and such that U =&,h/ay does not change sign as x varies (forfixed y ) and ZI = -a*/& does not change sign as y varies (for fixed x ) . A suitablesimple choice is

(7 .20 )the stream lines of w hich (9= con stant ) are sketched in figure 9. All the eddies rota tein the sam e sense, and a change of sign in $o reverses this sense and is not equivalent

= *o sin' k x sin2 k y

'I

Figure 9. Sketch of streamline configuration for the streamfunction ( 7 .20 ) , or which all eddies rotate inthe same sense . T he hea t flux has a component perpendicular to the mean temp erature gradient .

to a me re change of origin; put ano ther way, the motion is not invariant un der timereversal (cf the discussion in 6.1 above).Fo r this motion, following the abo ve pro ced ure we findF'O'= - 3 P e 2 ( G l , G z, ) (7 .21)6 4

where, as before, Pe = $ o / q . Now, however, the contribution F"' does not vanish;we find

whereD"'= (0 ,0 , - q P e 3 / 2 5 6 ) . (7 .23)


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650 H K MoffattTh e contribution F ( l ) o the total heat flux is perpendicular to the local mean tem-pe ratu re gradient. N ote tha t, for the motion described by the strea m function (7.20)

( U ZW )= (0, 0, & k4/4) (7.24)so that

(7.25)which admits suggestive comparison with (6.32).T he physical reason for the app earance of this transverse co mpon ent of he at fluxis indicated by the sketch of figure 10: th e isotherms are d istorted antisymmetrically

Figure 10. Sketch of the distorted isotherms when $ is given by (7 .20)and G = (0,Gz,). The temperatureexcess at P is convected to the left and the deficit at Q is convected to the right, implying a net componentof heat flux in the negative x direction.in each cell; hot fluid is convected to the left in region P and cold fluid to the rightin region Q; this provides a net h ea t flux to the left which is not co mp ensate d by heatflux arising in any o the r parts of t he cell.

8. Doub le-scale analysis for the convected vector fieldW e turn now to th e case of t he convected vector field B(x, ) satisfying equation (1.2).This has been treated very thoroughly in many previous studies (e.g. Moffatt 1978,Krause and Radler 198 0) and so only the'main points will be recapitulated here Th efield B is decomp osed in to its mean part ( B )= B o ( x , ) and its fluctuating part b, andthe m ean and fluctuating parts of (1.2) are then

(8.1)db/at=VA(U ABo)+VA(UA b - ( U A b ) ) + q V 2 b (8.2)

where 8' = (U A b ) . Just as for equation (5.5 ), equation (8.2 ) establishes a linear

aBo/at= v A 8+ 7 7 ~ ' ~ ~


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Transport effec tsassociated with turbulence 65 1relationship between b and B o , and thence between 8 and B o ; and the expansionanalogous to (5 .7) is

g i = a i j ~ o jPi jkaBOj/aXka!,?) Boj /at+ . . . (8.3)Elim ination of the tim e derivative using (8 .1) gives the modified expansion, analogousto (5 .9) ,

For strictly homogeneous turbuience, this takes the formtzi= ~ u ~ ~ B ~ ~pi;) aBoj /axk+ . . .

where( M ) (1)P ij k - i j k + ip e p k r n a r n i

( 8 . 5 )

so that here there is a non-trivial modification of the second term of the expansion.Fo r inhomogeneous turbulence, the leading term is modified (cf (5 .11) ) hus:

and th ere is a similar furth er modification t o P i j k .The coefficients al j , i j k , . . , in (8.3) are pseudo -tensors (pseudo because Bo isan axial vector whereas d s a polar vector) which again are determined by thestatistical properties of the turbulence and by the parameter g . Consider first theleading coefficient aij which may be determined on the simplifying assumption thatBo is uniform and steady. Again, this may b e split into symm etric and antisymmetricparts:f f . . =a ! ? )1 11 - -E, . i i k Y k (8.8)

so that, at this orde r,g i = a ;?Boi+ y A B d i . (8.9)

H e r e a$ is a pseudo-tensor so that(s) ( 1 ) ( 2 )a i j i e ,

is a pseudo-scalar, for any unit vectors e and e ( 2 ) ,This obviously means that a$)can be no n-z ero only if t he turbulenc e lacks reflexional symm etry, and a relationship,direct or indirect, between a!;) and the helicity spectrum function H ( k ,o ) ay beanticipated. This domi nan t im por tanc e of helicity in determining the leading term of(8.3) is noteworthy; the appearance of the mean electromotive force a $ ) B o j s whatis known as the a effect in dynam o theory (Steenbeck an d Kr ause 1966) , and is thekey to th e understanding of how m agnetic fields can be ge nera ted by turb ulen t fluidmotion (see, for example, M offatt 1978, Parker 1979, Krause and R adler 1980) .Th e vector y appearin g in (8 . 9 ) s a polar vector, and as such should be det erm inedby the reflectionally symm etric ingredients of th e turbulence. It leads to a ter mV A ( y A B ~ )n the mean-field equation (8 , 1 ) , mplying an ap pa ren t convection of themean field Bo relative to th e fluid with effective velocity y. Obviously this effect can


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652 H K Moffattarise only if th e turbule nce is non -isotrop ic, th e direction of y appearing as a 'preferre ddirection'.The second term of (8.3) is also of great importance in dynamo theory, since itincorporates the effectiveeddy diffusivity acting o n th e m ean field, as well as certainsubsidiary e ffec ts. T o see this, it is sufficient o consider th e case of isotropic turbulencefor which p i j k must ta ke the isotropic form

P i j k = P E i j k (8.10)where p is a pure scalar. The corresponding contribution to d" in (8.3) is then- p ( V r \ B , ) and the corresponding contribution to V A & in (8.1) (assuming p isuniform) is p V 2 B o , o th at th e total effective diffusivity acting on B O s 77 +p.

9. Lagrangian analysis for the convected vector field (q=0)Following the proc edu re of 6, let us now suppose that at t ime t = 0, B is non-random ,i.e.

B(x,0)= Bo(x,0 ) . (9 .1)Th e Lagrangian solution (1.11) hen gives

B i ( x , )= Boj(a, ) axi/aaj (9.2)from which we may construct

gi(x, t )= E i j k ( V j ( a , t)BOl(a,0) a x k / a a l ) . (9.3)In order to calculate a i i ( t ) a function of t bec ause of th e special initial con dition(9.1)) ,Bo n (9.3 ) may be re gard ed as uniform, and we then have

Again it is illuminating to consider the initial development of this function. For tsmall, we may use th e Taylor series (6.1 9) for u(a, ) ;at leading or de r this gives

T h e symme tric pa rt of this is characterised by(9 .6)2(Y E ~ a ' i i -$(U * v A U ) t = o t +O(C )

which shows up very clearly the relevance of helicity in this context. T he a ntisymm etricpart is given by

yp =-;jlpajl= -;v * ( U U P ) t+0(t2). (9.7)Th e leading term here is non -zero only if the turbulence is inhomogeneous; suppose,for example, that, with respect to Cartesian coordinates O x y z , U = ( U , 0, w), and thatthe turbulence is homogeneous in the ( x , y ) plane but inhomogeneous with respect


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Transpo rt effects assoc iated w ith turbulence 6 53to the z direction. The n, from (9.7) ,

I dwhere ~ ( z ,) = --- W 2 ) t+ O(t2) (9 .8 )2 dzi.e. a flux convection velocity directed down the gradient o f turbulent intensity (seefigure 11). This is the diamagnetic effect first identified by Z el'dovich (1 95 7) ; it is

Y = ( O , O , Y ( Z , t ) )

t 'Gradiento f t u r b u l e n ti n t e n s i t y

Figure 11. Schemat ic representat ion of th e flux convection effect associated with inhomog eneo us turbulenc e.

closely associated with the well-known phenomenon of f lux expulsion, studied in th econtext of deterministic flows by Par ker (19 63) , Weiss (1 966 ) and o thers . T he effectwas recognised by Ra dler (1968 ) and i t has recently been incorpo rated in d ynam omodels by Nightingale (1983). Localised (inhomogeneous) turbulence tends to expelmagnetic flux with an effective velocity y . It is imp ortant to n ote that , in gen eral inthis situation,v - y z o (9 .9 )

(even although the background turb ulen t field is strictly solenoidal).Th e expression (9.7 ) is, of course, valid only for small times t ; bu t, assuming thatstatistical equilibrium is established for t >> t , where t , is a Lagrangian correlation time,a reasonable approximation to the asym ptotic form of y should b e given by

(uu t c ( t >> rc). (9.10)It is interesting th at it is th e divergenc e of the Rey nolds stress ( u u ) that appears here ,a quantity w hich also governs the evolution of the mean velocity field U(x, ) n thedynamics of inhomogeneo us turbulence (the mean pressure field being responsiblefor the maintenance of a solenoidal U ) .If th e turb ulen ce is strictly homogeneous, the n the leading term of (9.7) vanishes;ther e is, however, an effect at ord er t 2 ,viz

= a(UU :V v u ) t 2 +o ( t 3 ) (9 .11)involving a cubic velocity correlation. This m ea n quantity vanishes if the turb ule nc eis isotropic; b ut fo r non-isotropic tu rbu lenc e it simply provides a uniform convectionof mean magnetic field.


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6 5 4 H K Moffatt9.1. Ef ec ts associated with mean-f ield gradientWe now turn to the development of the tensor P i j k appearing in the fundamentalexpansion (8.3). T o analyse this, we suppose that, at t = 0,

B ~ ~ ( x ,)= B~~ xj(aBoi/axi) (9.12)in which Boiand aBoi/axjon the right-hand side will be take n as uniform. Th en from(9.3) ,

=aiBoi(x, 0 ) P i lm a B d a x mwhere ail(t) s as previously determ ined , and

P i rm ( t ) = -E i j l ( v Jm) -Ei jk (v Jm axk/aal). (9.13)(Here X = x - a is th e Lag rang ian particle displacement-see figure 1 )T h e effective eddy diffusivity is (from (8.10 ))P E i lmPilm (9.14)= :(U * X )- U * Va)(4X2) U X)V, X).

T h e first contribution he re is the eddy diffusivityD = f ( u X ) (9.15)

that acts upon a scalar field (see 0 6). But we now have an additional contribution,cubic in the displacement X, which acts only o n the vector field. This distinctionbetw een tu rb ul en t diff usivities for scalar and vecto r fields was first identified by Moff att(1974).Again, we may examine the Taylor expansion of (9.14 ) for small t. Using (6.19),we findD ( t ) = $ ( ~ ~ ) o t$ (U * Du/Dt)ot2+O(t3) (9.16)

and(9.17)

The distinction between D ( t )and P ( t ) therefore appears in general at order t 2 , thedifference being cubic in U in inhom ogeneo us turbulence, and quartic in hom ogene ousturbulence.T he tensor (9.1 3) also contains a skew-diffusion effect, some wha t an alogous toth e effect described by the a ntisymm etric pa rt of Dij see 6). This is here rep resen tedby the pseudo-vector - Pimm=D a - ((U A Va)X2> (9.18)where D(a)s given by (6.23). The corresponding contribution to d has the formV A (s A B) where si =$Pi,,. This effect is close to, although not identical with, theeffect discovered by Ra dler (196 9) (and known as the R adle r effect), viz the ap pea ranc eof a contributio n to d erpendicular to V A B in turbu lence tha t is statistically axisym-metric. Skew effects of this kind can ap pe ar only in turbulence tha t lacks reflexionalsymmetry.


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Transport effectsassociated with turbulence 65510. First-order smoothing theory for a convected vector fieldFollowing the proced ure of 0 7, we may expan d b and 8' in the form

(10.1)(10.2)

ab'"+"/at - qV2b("+')= v (U 6'"'- &")) (10.3)with 8 '" )(U A b'"'),and may t hen (in principle) solve iteratively fo r b"), b") , b"), . . . .If the turbulence is homogeneous, the procedure is straightforward and, at lowestord er, we obtain

@ = Y $))Boj+p i2 a B o j / a X k + . . . (10.4)where

(10.5)and

W 4~'kikjkk ki8jk - 16ik)H dk do . (10 .6 )+ 0 2 + q 2 k 4 w 2 + q 2 k 4 + 2 k2T he detai led derivation of these results may be fo und in Moffatt and Proc tor (19 82).Th ey should be c om pared with the results (7.8)-(7.12) for the scalar field case. No tetha t t he coefficient a,,@)s symm etric in th e suffixes ( i , ) i.e. there is no 'flux convection'effect at this level of approximation) a nd that i t depend s only on th e helicity spe ctrumfunction H ( k , w ) . T he coefficient @$?, on th e o the r han d, involves two parts, the first(antisymmetric in ( i , ) ) involving the sym metric part of (P,,, an d th e second involvingagain the helicity sp ectr um function. T he form er part provides a (generally anisotropic)ed dy diffusion effect closely analogous to th e effect repre sen ted by (7 .9) ; and thelatte r provides the 'skew-diffusive' effects such as the R adl er effect referre d to in theprevious section.If this approach is pursu ed to th e next level then we ob tain 8")n the form

= Y f;"Bo,+p f,li Boj/axk + . . . (10.7)where a!:) and @$! are cubic mean functionals of the velocity field. T he main newqualitative effect that appears at this level is that, if the turbulence is non-isotropic,af;) s in general non-symmetric, and there is therefore a flux convection effectrepresented by the velocity yi ' ) =-&,]k(Yi]''), which is again a cubic m ean fun ctionalof U (cf (9.11)).It may be anticipated, however, from 9 that if the turbulence is inhomogeneous,then a stronger flux convection effect may be expected, directed down the gradientof tu rbu len t intensity. This effect do es indee d app ear at the first-order level (repre sen -ted by th e coefficient (Y~,")) and was so identified by Ra dler (1 96 8) . It will be sufficient


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656 H K Mofa t the re to give a simple examp le, using again a two-dimension al mod el (cf the examp lesof 7 ) . L et U = (d$/az, 0, - a $ / a x ) where (figure 12)

CC,= Cclo sin kx sin kz f ( 2) (10.8)where f ( z ) is a slowly varying function of z (i .e. lf '(z)/f(z)l


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Transport effectsassociated with turbulence 657If mo reov er the energy an d helicity of th e turbu lenc e are con cen trated in the regionof ( k ,U ) pace for which u2t,du e to the development of unbound edfluctuations in the fluctuation fields or the ir spatial derivatives. T h e second is aperturbation approach whose convergence is presumably limited to values of Pe orR, of th e ord er of unity or less. W hat we need now is a tech niq ue whereby resultsvalid fo r Pe -s1may som ehow be pushed into the interesting and practically im portantregime Pe >> 1. One possible technique, which has been described in the scalar fieldcontex t by Moffatt (1 98 1) , is tha t of successive averaging over a n um be r of widelyseparated scales. This technique is akin to the renormalisation group procedureswhich have ach ieved considerable success in the theory of phase transitions. T he widespectrum of scales present in any turbulent field suggest that successive averagingover regions of gradually increasing scale may, at the least, provide some valuableinsights. W e describe this proc edu re here for the magnetic context, the primary aimbeing to determ ine expressions fo r the transpo rt coefficients a = $aii and P = & i j k P i j kvalid when R, >> 1.W e shall restrict atten tion to the case of isotropic (but no t necessarily reflexionallysymmetric) turbulence with energy and helicity spectra satisfying the realisabilityconditions IH(k)l G 2 k E ( k ) E ( k )2 0 (all k) . (11.1)We replace E ( k ) and H ( k ) by discrete spe ctra (see figure 13 ) conc entrated a roundwavenumbers k l = lT1,k 2= 1; , . . ,k , = 1, where

11


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6 5 8 H K M of a t t

Figure 13. Replacement of the cont inuous spectrum E ( k )by a discre te spectrum centred o n wavenum bersk , , k 2 , . . , k , , with k l >> k 2 >> . . . (the scale is logarithmic).first-order smoothing technique of 10 is applicable, and the resulting mean-fieldequat ion for B1= (B ) I s

dBl/at = V A (U , B l ) + a l V A B l+ ( r , +P1)V2B1 (11 .5 )where, f rom (10.13) ,

2PI =- k-E(k) dk .377 A k i(11.6)(11.7)

No te tha t by virtue of ( l l . l ) ,

We now repeat the first-order smoothing procedure by averaging over the scale= 0, and we assume that u 2 is statistically2 , i.e. we let U1= U2+ 2 whereisotropic in a fram e of reference moving with th e local velocity U2 nd that

(11.9)1/2Rrn2=(u2)2 M77 + P J l < 1.N ote th at we now get a little help in satisfying this inequality through the ap pe ara nc eof the renormalised diffusivity 77 + P I . Applying first-order smoo thing to (1 .5) leadsto the mean-field equ ation

aBz/a t = V A (U2 B z )+ a 2 V A B Z+(7 P 2 ) V 2 B 2 (11 - 1 0 )where now (with r = 1)

(11.11)


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Transport effects associated with turbulence 6 5 9Th e impor tant point here is that the structure of ( 1 1 . 1 0 ) s identical with th at of ( 1 1 . 5 ) ;only t he suffix 1 has been replaced by th e suffix 2 . It is evident ther efo re that we may(in principle) repea t the procedure any num ber of times, always obtainin g a m ean-fieldequation with the sam e structure. Mo reover equations (1 . 1 1 ) and ( 1 1 . 1 2 ) remainvalid for r = 2 , 3 , 4 , . . . .W e now ta ke t he bold st ep (which cannot be justified with any degre e of rigour:the reasonableness of the end product must justify the means!) of returning to acontinuous spectrum. Th en it is clear that th e difference equa tion s ( 1 1 . 1 1 ) and ( 1 1 . 1 2 )must be replaced by differential equations?

d a 1 2 a E ( k ) + ( q + P ) H ( k )-= -d k 3 k 2 ( q+ P ) 2 - a 2 ( 1 1.13)d p 1 2 ( q + p ) E ( k ) + a k - H ( k ) ( 11 . 1 4 )

and a ( k ) and P ( k ) must now be int erp rete d as the values of a and p associated withall scales s k - . Fo r given E ( k ) and H ( k ) ,we may integrate these eq uatio ns inwardsfrom k = CO, with initial conditio ns a CO)= 0, p ( 00 ) = 0. W e may easily see what musthapp en. No te first that, from ( 1 1 . 1 3 )and ( 1 1 . 1 4 ) ,

- = - -d k 3 k2 ( 7+ /3 ) - a2

so that

Hence we may eliminate ( q+ p ) from ( 1 1 . 1 3 ) ,obtaining- 2 2 a 2 ( 2 a 2 E - h H )dk = 3 ( k h - a 2 ) ( k h +a2) *

( 1 1 . 1 5 )

( 1 1 . 1 6 )Figure 14 shows what ha pp en s if we integrate this inw ards from k = CO: a 2 ncreasesuntil the value of k (=k,, say) at which a2 k H / 2 E ; or k < k,,

Ikh >a h H / 2 E (1 . 1 7 )and a must decrease to ze ro as k + 0. This decrea se is in fact ultimately line ar in k ,so tha t a - , p + 7 - - l l 2 (assuming h (0)# 0) as k + 0. This unbound ed behaviourof p as k + 0 is associated with a sym ptotic vanishing of the den ominator in ( 1 1 . 1 4 ) ;the condition

k 2 ( q+p) O according to ( 1 1 . 1 3 )and ( 1 1 . 1 4 ) ,but

( 11 - 1 9 )and the associated magnetic fluctuation increase (without limit) as the length scaleincreases.t Similar equations have been obtained by S Vainshtein (private comm unication).


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660 H K MoffattI

kFigure 14. Form of the function ru2(k) s determined by integration of equation (11.16).

In any finite geometry (e.g. a stellar convection zone) the turbulence ishomogeneous only on scales small compared with the external geometric scale (e.g.the stellar radius or the dep th of th e convection zone). Eq uation s (11,13)and (1 .14)can then b e used to represent (an d param etrise) th e effects of turbulen ce o n scalessmaller than some scale lo= kO1 satisfying this criterion . Ingre dien ts of th e mo tionon scales larger than fo can be described explicitly only in large-scale numericalcomp utations such as those of G ilman an d Miller (1 98 1).T he importance of equ ation s such as (1 . 13) and (11.14 ) is that, n o m atter howsmall may be , they lead to values of a and p of th e or de r of (uZ) l / and 1 0 ( u ~ ) ~ / ~ ,respectively, both ind epe nde nt of 7, lthough ultimately the process by which a andp are generated is diffusive in character, the first step in the averaging processrequiring th e stro ng diffusion co ndition (1 .4) .12. Development of intermittency in convected fieldsAs we have remarked in 8 6, expansions of the typ e (6.9) and (8.3) would ap pear tobe reliable only if m olecu lar diffusion effects acting on th e conv ected field ar e operativ ebefore a fluid particle has moved ou t of a region in which th e convected-field grad ientis app reciab ly uniform . If this conditio n is not satisfied, then t he series (6.9) and (8 .3 )will conve rge very slowly, or m ay even diverge, and t he m ean-field a pp roa ch will notthen be viable. In this final section, we shall analyse som e of the effects that m ay arise.W e assume an initial 8 field of th e for m

e(x, 0)= eoCOS K ~ X (12 .1)having a single length scale L o - K i l , and we suppose that this field is subject toconvection by the steady velocity field

= (U(Y), 90) (12.2)where U( ) is a stationary random function of y with zero mean, ( U )= 0 (here ( . . . )will indicate an average over y). This is clearly a highly idealised model problem;but in choosing a steady velocity field, we focus attention on the effects of persistenceof a tu rb ul en t velocity field, in which case the es cap e of fluid particles fro m regionsof un iform field gradient will occur most rapidly. N ote th at, in the two -dimensiona lsituation considered he re, the vector potential (0, 0, A )of a m agnetic field B satisfies


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Transport effects associated with turbulence 66 1the same equat ion as 8, so that results ob tained for th e scalar f ield problem may beimmediately carr ied over to th e vector f ield pro blem .The root m ean squa re vort ici ty

00 = ( (du/dy)*) (12.3)plays an important part in this problem, and also the associated shear t ime scale,t ,= w o . Th e relevant PCclet num ber is

Pe = u 0 / K ; q (12.4)an d we shal l suppose that Pe >> 1. T h e isoscalar surfaces 8 = constant are then s tronglydistorted before diffusion becomes effective (see figure 15). We aim to fol low the

- 1

Figure 15. Distort ion of the scalar. field B(x,0)= BO cos K ox by the random velocity field U = ( u ( y ) ,0,0).Not e how high values of M / a y build up, except in the neighbou rhood of the layers y = y r ( r = 0,* l, * 2 , . . . )where U = 0. Diffusion eliminates the B field excep t in these neighbourhoods.detailed develop men t of the field 8 ( x , y, t ) during this initial process of dis tor tio n, an dduring the subseq uen t decay.Th e relevant eq uat ion is

ae ae a a2- + u ( Y ) & = R ( s + 7 )t a y 8an d the solution evidently has th e form

(12.5)


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662 H K Moffattwhere

(12.7)and f ( y , 0)= 1. T he initial phase of develop men t is given by simply putting 77 = 0, sothat f ( , t )= exp (-&U ( y) t ) , nd correspondingly

e(x, Y , t ) =R e{& exp [iKo(x -u(y) t ) l I (12.8)the Lagrangian solution (cf (6.2 )). To im prove on this, let

f ( Y , t )= e x p (- iKou(y)t)F(y,) (12 .9)where we expect that F ( y , ) will be a slowly varying function of t (at least fo r som etime interval). Substitution in (12 .7) leads to the equ ation

a2F aFa y a y- K ~ t 2 u 2 F - i K O t u F + ~2iKotu- -K$F). (12.10)

Th e first term o n th e right-hand side increases most rapidly with t, and if we retainonly this term we may integrate t he equ ation to obtainF ( y , )= exp (-$7$C,342t3) (12.11)

indicating d ecay of the 6 fluctuations o n a time scale(1 .12)l=(TKowo) - P e t,.For t > f l , the function F ( y , ) becomes very sharply peaked in the neighbourhoodof points y r ( r = 0 ,* l, *2, . . . ) where U 0. Equally, the temperature fluctuationsbecome sharply concentrated in these neighbourhoods (figure 16). A more refined

2 2 - 1 1 3 - 1 / 3

Y Iy3Y2Y1YOY- 1

Y-2

Y-3

Figure 16. Development of intermittency in the amplitude I f ( y , ) l of the temperature field, for t,


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Transport effectsassociated with turbulence 66 3analysis of the behaviour in these neighbourhoods shows that ultimately the term77 a2F/ay2n ( 12 .1 0) limits th e decrease of scale in th e y direction; this occurs whent - ( K o L ) / 2 P e 2 / 3 here L is a typical scale of variation of the shear function u ( y ) .Th e solution (12 .11) then requires m odification.

L et us now retu rn t o the L agrangian solution (1 2.8), valid for t > t,, the d om inant contribution to th e integral comes from the neighbourhoodsof the points y r of stationary phase w here u ( y )= 0 ; and evaluation of (12.13) by th emethod of stationary phase yields

(1 .14)where the + or - is chosen according as U: s greater o r less than 0. If N is the m eannumber of extrema of u ( y ) per unit length in the y direction, then ( 12.1 4) may equallybe w ritten

where the overbar represents an average with respect to the suffix r labelling theextrema. (If U ( ) is statistically invariant with respect to change of sign (U + -U),then (12 .15 ) simplifies to(12.16)

but otherwise there may be a phase shift 8 ( t )of 0 in the x , direction.) T h e result(12 .15) provides an explicit repr esen tatio n of 0 n terms of the Eulerian statistics ofth e velocity field U (y ), an d is valid f or t s


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664 H K MoffattReferencesBatchelor G K 1952 Proc. R . Soc. A 213 349-66- 953 The Theory of Homogeneous Turbulence (Cambridge: Cam bridge University Press)- 959 J. FluidMech. 5 113-33Batchelor G K, Howells I D and Townsend A A 1959 J . Fluid Mech. 5 134-9Batchelor G K and Townsend A A 1956 Surveys in Mechanics ed G K Batchelor and R M DaviesChildress S 1970 J. Math . Phys. 11 3063-76- 979 Phy s. Ear th Planet. Interiors 20 172-80Corrsin S 1951 J . Appl. Phys. 22 469-73Gilman P A and M iller J 1981 Astrophys. J. Suppl. 46 211-38Ibbetson A and Tritton D J 1975 J . Fluid Mech. 68 639-72Knobloch E and Rosner R 1981 Astrophys. J . 247 300-11Kraichnan R H and Nagarajan S 1967 Phys. Fluids 10 859-70Krause F and Radler K-H 1980 Mean-field Magnetohydrodynamics and Dynamo Theory (Oxford: Per-Mofiatt H K 1961 J . Fluid Mech. 11 625-35__ 1963 J . Fluid Mech . 17 225-39- 969 J. FluidMech. 35 117-29- 974 J.Fluid Mech . 65 1-10_. 1978 Magnetic Field Generation in Electrically Conducting Fluids (Cambridge: Cam bridge University- 981 J. Fluid Mech . 106 27-47Moffatt H K and Proctor M R E 1982 Geop hys. Astrophys. Fluid Dy n. 21 265-83Moreau J J 1961 C.R. Acad. Sci., Paris 252 2810-1Monin A S and Yaglom A M 1975 Statistical Fluid Mech anics vol 2 (Cambridge, Mass.: MIT Press)Nightingale S 1983 Stellar and Planetary M agneti sm ed A M Soward (New York: Gordon and Breach) toObukhov A M 1949 I z v . Ak a d . Na u k . S S S R Geo gr. Geofit. 13 58Onsager L 1931a Phys. Re v . 37 405- 931b Phys. Rev . 38 2265Parker E N 1963 Astrophys. J . 138 552-75- 979 Cosmical Magnetic Fields (Oxford: Clarendon)Proctor M R E 1975 Non-linear mean-field dynamo models and related topics. PhD Thesis University ofRadler K-H 1968 2. aturf. 23a 1851-60- 969 Monats . Dt. Ak a d. Wiss . , Berlin 11 194-201Roberts G 0 1970 Phil. Trans. R. Soc. A 266 535-58__ 1972 Phil. Trans. R. Soc. A 271 411-54Rob erts P H and Stix M 1972 The Turbulent D yn am o : a translation of a series of papers by FKrause, K -HSaffman P G 1960 J. Fluid Mech. 8 273-83Steenbeck M, Krause F and Radler K-H 19662. aturf. Z l a 369-76Steenbeck M and K rause F 1966 Z. Narurf. 21a 1285-96Taylor G I 1921 Proc. Lon d. M ath . Soc. A 20 196-211Townsend A A 1951 Proc. R . Soc. A 209 418Weiss N 0 1966 Proc. R. Soc. A 293 310-28Zel'dovich Ya B 1957 Sou. Phys.-JETP 4 460-2

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gamon)

p r e s s )

appear

Cambridge

Radler and M Steenbeck, Tech. Note 60 (NC AR , Boulder, Colorado)

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