H.K. Moffatt- The Topology of Turbulence

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  • 8/3/2019 H.K. Moffatt- The Topology of Turbulence


    COURSE 6


    H.K. OFFATTIsaac Newton Inst i tute forMathematical Sciences, 20 ClarksonRoad, Cambridge, CB3 OEH,U.K.

    P H O T O : height 7.5cm, width l lcm


  • 8/3/2019 H.K. Moffatt- The Topology of Turbulence


    Contents1 Introduction 32 The family of helicity invariants 42 .1 Chaotic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2.2 Simply degenerate fields . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Doubly degenerate fields . . . . . . . . . . . . . . . . . . . . . . . . 6

    3 The special case of Euler dynamics 74 Scalar field structure in 2D flows 85 Scalar field structure in 3D flows 96 Vector field structure in 3D flows 107 Helicity and the turbulent dynamo 11

    7.1 The kinematic phase . . . . . . . . . . . . . . . . . . . . . . . . . . 127.2 The dynamic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 158.1 The analogy with Euler flows . . . . . . . . . . . . . . . . . . . . . 179.1 Interaction of skewed vortices . . . . . . . . . . . . . . . . . . . . . 19

    8 Mag netic relaxation 16

    9 T h e blow-up problem 1 8

  • 8/3/2019 H.K. Moffatt- The Topology of Turbulence



    H.K. offatt

    1 IntroductionTopological considerations enter t he stu dy of turbulence thr ou gh quantities,often expressed as integrals over the fluid domain, which in idealized (non-dissipative) circumstances, are c ons tant in time , i .e . invariants of the flow.The simplest example occurs for two-dimensional incompressible inviscidflow for which the vorticity field w ( z ,y, t) satisfies the equation

    Dw a w- + u * v w = o .Dt - atHere U = (u(z,y,t),( z , y , t ) , O ) is the velocity field with w = a v / d y -au/az. Equat ion (1.1) tells us that the vorticity associated with any fluidelement is constant. The isovorticity lines w = cst. are frozen in thefluid, so t h a t t h e a r ea A(w) inside any closed isovorticity line w = cst .is conserved. T hi s function, described (M offatt 1986) as the signatureof the vorticity field, may be thought of as a topological invariant underth e continuous deform ation of th e vorticity field by its self-induced velocityfield U .If F ( w ) s an arbitrary function of U , then

    d Dw

    where SL is a patch of fluid bounded by a closed curve CL hat moveswith the fluid. We use the suffix L here for Lagrangian. This gives thefamily of invariantsI { F } = F ( w ) d z d yLL (1 .3)

    (of which th e best know n, with F ( w ) = w 2 and SL th e w hole fluid do main,is the e ns tro ph y of th e flow).@ EDP Sciences, Springer-Verlag 2001

  • 8/3/2019 H.K. Moffatt- The Topology of Turbulence


    4 New T rends in TurbulenceTwo imp orta nt question s arise from these elemen tary considerations: (i)how are these families of invariants to be generalized to three-dimensionalflows? (ii) How ar e these invariants modified, a nd w ha t role do they play,when weak dissipative (e .g . viscous) effects a re tak en int o consideration?In the se lectures, we first provide some tenta tive approa ches to answerthese questions. Then we discuss some particular situations in which topo-logical considerations play a central role.

    2 The family of helicity invariantsLet v(x, ) be an arbitrary three-dimensional incompressible flow and letB(x, ) be an ar bit rar y solenoidal field satisfying th e evolution equa tion- V A ( V A B ) ,* B= 0.dBatIt is well-known that this equation describes frozen-field transport of B,the flux of B across any material surface element being conserved. If B isinterpreted as the magnetic field in a perfectly co ndu cting fluid, then thisis none oth er th a n Alfvens theorem in m agnetohydrodynam ics.Now let A be a vector potential for B satisfying V . A = 0; then un-curling (2.1) gives

    for some scalar field cp. Equat ions (2.1) and ( 2 . 2 ) can be wri t ten in equiva-lent Lagrangian formD B j - B.-Vi- -D t d ~ j

    where D I D t = d / d t +v .V. We define the helicity Ft of the field B withinany (Lagrangian) mag netic surface SL on which n - B = 0 byFt = lL . BdV.

    It is easily shown that this integral is gauge invariant ( i . e . invariant underreplacement of A by A + V$). It is further easily shown th a tD7-idt = lL t(A. B ) d V = (n B)(-cp + v . A ) d S ( 2 . 5 )

  • 8/3/2019 H.K. Moffatt- The Topology of Turbulence


    H.K. offatt: T h e Topology of Turbulence 5and this vanishes since n B = 0 on SL. Hence 3-1 is invariant (Woltjer1958) .The invariant has topological interpretation (Moffatt 1969; Arnold1974); thus for example if the field B is identically zero except in two fluxtubes carrying fluxes @ I , @ 2 , an d linked with (G au ss) linking number n,then

    3-1 = 1 2 n @1 @2 (2 . 6 )(where VL s taken to b e a volume containing both tubes) . This s tatementrequires immediate qualification: it is tr ue provided the B-lines with in eithertu be are closed curves w ith no linkage between any pair; in th is case, eachtu be on its own makes no c ontribu tion to t he helicity. It is ju st th e linkageof the tubes tha t contr ibutes , via h e formula (2.6).Suppose instead that we have a single flux tu b e w ith flux @ an d axis inth e form of a circle, th e B-lin es in th e tu b e being all parallel circles. Th enth ere is no linkage an d t h e helicity is zero. Imagine now th a t we cu t thetub e, twist one cut end thro ugh a n angle 27r an d rejoin th e ends. Now eachpair of B-lines is linked, with linking number 1; if we imagine this twistedtube as buil t up through the addit ion of incremental f luxes dp, the totalhelicity is given by

    * c 53-1 = 1 2 1 - p d p = &Q2 (2.7)th e + or - being chosen according as the twist is right- or left-handed.linear in h and is therefo re given by

    If the angle of twist is 27rh, instead of 27r, then clearly the helicity is

    3-1 = & h a2 . (2 . 8 )Retu rning now t o th e two linked flux tub es, we may allow for Ltwistsh land h2 in the two tubes, and the total helicity is then

    3-1 = &hi@: f 2@; & [email protected]@2 (2.9)where, for each te rm , th e sign is chosen according as the twist (or linkage)is right- or left-handed.2.1 Chaotic fieldsIn general, a field B in Iw3 may be expected to have chaotic field lines,i .e. field lines which do not lie on surfaces except in some sub-regions ofregularity of the field. Examples of such fields may be found in Dombre

  • 8/3/2019 H.K. Moffatt- The Topology of Turbulence


    6 New Tr end s in Turbulenceet al. (1986) (the ABC-field) and in Bajer and Moffatt (1990) (the STF-field). Let V, be a subdomain in which the B-lines are chaotic; then forsuch a subdo m ain, we have the single helicity invariant (2. 4), an d the reappears to be no analogue of the infinite families of invariants A ( w ) andI{F} ntroduced in Section 1. It may be noted that it is for this situationth a t A rnold (1974) established th e generalized i nte rpr eta tion of (2.5) as th eLasymptoticHopf invariant ( i .e . asymptotic linking number) of the fieldB in the chaotic region.2.2 Simply degenerate f ieldsIn m any situ ati on s of interest however, the B-lines do lie on a family ofsurfaces. Suppose this family is th e family $(x, )= 0; th en evidently, sinceth e B -lines move w ith th e fluid, we mu st have

    D $ / D t = O and B . V $ = O . (2.10)We ma y clearly define an invariant helicity function

    f i ( $ )=/ A . B d VVL %!J) (2.11)where VL($) s the volume inside the Lagrangian surface $ = cst. Herewe have an obvious analogue of the signature function A ( w ) of Section 1.Equally however if we defineh($) = dfi /d$ (2 .12)

    so t h a t h($)d$ represents th e (invariant) helicity tra pp ed between surfaceslabeled $ an d $+d$, th en we m ay construct a family of invariants analogoust o (1 .3) , namely

    where F is a n a rb itra ry function of h; for

    by virtue of (2.11)



    2.3 Doubly degenerate f ieldsIn special circumstances, it m ay ha ppe n t ha t every B-line is a closed curve;for example, the twisted flux tube described above, with angle of twist

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    H.K. M offatt: T h e Topology of Turbulence 727rn ( n an integer) is a field for which each B-line is an unknotted closedcurve , each pair of B-lines having linking num ber n. More generally, we m ayconsider a situation in which the B-lines are the intersections of surfaces$ = cst. and x = cst. (a doubly-infinite family), with

    D $ / D t = 0, D x / D t = 0, B = V $A VX . (2.15)Th en (cf. (2 .11)) , we may define

    f i ( A + , A x ) = / A * B d V (2.16)where A$ = $1 -$z, A x = x1 - X Z , an d VL is the tube-like volume boun dedby t h e surfaces labeled $1, $2 , X I , XZ. Defining h($,x) by

    VL (A$ ,AX )

    (2.17)for incremen ts d$, d x , we now have a family of invariants (cf (2 .13 ) )of theform

    I { F ) = / F ( h ( $ ,X))d$dX. (2.18)Note that the greater the degree of degeneracy of the field, the richer isth e family of topological invariants th a t it exhibits. These invariants are allcon struc ted from ( 2 . 4 ) , in which the Lagrangian volume VL is constrainedonly by th e requirement th a t i ts surface SL be a surface on which n . B = 0;the greater the degeneracy of the field, the greater is the freedom in thechoice of such volumes.

    VL ( A $ , AX )

    3If, in the above theory, we choose v = u(x,t) , a velocity field evolvingunder t h e incompressible Euler eq uatio ns, and w = V A U, he correspondingvorticity field, then (2.1) is clearly satisfied since

    T h e special case of Euler dynamics

    T h e special feature here is th a t th e field w is transpo rted by its own self-induced velocity field U. This coupling of U t o w makes (3.1) nonlinear;bu t th is does not invalidate the m anip ulatio n leading to ( 2 . 5 ) which, withFI = iL. dV,

  • 8/3/2019 H.K. Moffatt- The Topology of Turbulence


    8 New Trends in Turbulencenow becomes

    = l L ( n . ) ( - p + 2u2) dV (3 . 3 )Here p is the fluid pressure, and for simplicity of notation we have takenthe fluid density to be p = 1. Thu s again, H = cst. provided n - w = 0 onSL (a condition that persists under Euler evolution).Th us th e helicity FI is an invariant of the nonlinear system (3.1). How-ever, it should be int erp rete d within t he b roader canvass of Section 2 , in -volving the system (2.1), which (for any given v) is evidently linear in B.From thi s point of view, it is evident th a t , und er artificial Euler dynam icsfor which (3.1) is replaced by

    with w = V A U and v an ar bi tra ry field satisfying D .v = 0, th e helicity (3.2)is still invariant; this is because w is now tra nsp orte d by th e v-field an d it stopology is still conserved. In p arti cu lar, if v is some functional of U (asfor example in the artificial Euler dynamics of Vallis e t al. 1989), helicity(and w-topology) are still conserved. The more recent averaged Eulerequations of Marsden e t al. (2000) appear to have similar conservationproperties.4 Scalar field structure in 2D flowsT h e str uc tu re of a scalar field s(x) m ay b e described in term s of th e criticalpoin ts of th e field where V s = 0 and t h e s tru ct ur e of th e iso-surfaces s = cst .near th ese p oints (for details, see M offatt 2001). Consider first th e situ atio nin a 2 D periodic domain (as frequently adopted in numerical experimentson 2 D turbulence). Here, the field s may be the streamfunction I) of theflow, or the vorticity field w = -V21), or it m ay be some other scalar fieldof physical significance, such as the pressure field p . The critical pointsare either elliptic (ex trem a) or hyperbolic (saddle points). In a periodicdomain, the number of extrema is equal to the number of saddle points(a consequence of Eulers index theorem). In an evolving situation, pairsof critical points (always one saddle and one extremum) can appear ordisappear (through saddle-node bifurcation). If s is the vorticity field w ,th en it is obvious th a t such bifurcation (with consequent change of topologyof th e w-field) can occur only throu gh t h e agency of viscosity.In 2D turbu lenc e, one may distinguish between t he eddies th at maybe observed in ins tan tan eo us streamline plots I)= cst., and vortices t h a tmay be observed in instantaneous isovorticity plots w = cst. Indeed, if

  • 8/3/2019 H.K. Moffatt- The Topology of Turbulence


    H .K . M offatt: T h e Topology of Turbulence 9we define an eddy as the region of closed streamlines circulating round anextremum of $, then the number of eddies N+ in the domain is preciselythe number of extrema of $. We may define a vortex similarly in terms ofth e w-field, so th a t th e num ber of vortices N , is th e num ber of ex trem a ofw (some positive, som e neg ative). In a typica l field of 2D turbulence a t highReynolds nu mbers, th e vorticity field shows much more stru cture t h an th evelocity field; this visual property suggests that N , is much greater thanN$. n fact , these numbers depend on the energy spectrum E ( k ) of theturbulence. If there is an inertial range in which

    E ( k )N k- ( k i N $ is evident. The condition 1 < X < 5 issatisfied for all reasonable models of 2D turbulence (e.g. the enstrophy-cascade model of Kraichnan 1967 an d B atchelor 1969 with X = 3; or thespiral wind-up model of Gilbert 1988 for which 3 < X < 4).5 Scalar field structure in 3D lowsThe critical points of a scalar field s(x) in 3D are again elliptic or hyper-bolic, the elliptic points being either maxima or minima of the field. Thehyperbolic (sadd1e)points are of two distinct types depending on whethers decreases in one principal direction away from the saddle (and increasesin the other two directions), or vice versa. Bifurcations of such a field cancreate (or destroy) critical points in pairs: a minim um an d a saddle point ofthe first kind; a maximum and a saddle point of the second kind; or a pairof saddle points, one of each kind. Each such topological transition is com-patible w ith E ulers index th eo rem , which for th e case of a space periodicfield, takes th e form

    720 - n1+ 722 - 23 = 0 ; (5.1)here 720 is the number of minima, 723 the number of maxima and 721,722 th enumber of saddle points of first and second kinds respectively. (na is thenumber of critical points, always assumed non-degenerate, with index a;Q! is the number of negative eigenvalues of t h e m a t rix d2s/dxidxj t t h ecritical point.) I t is worth noting th a t, since saddle points m ay be created inpairs, whereas extre m a cannot be created w ithou t creatin g an equal numberof saddle points, the number of saddle points will in general be greater (andpossibly much grea ter) th a n the num ber of extrem a.

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    10 New Trends in TurbulenceA simple measure of th e topological complexity of a scalar field s(x) in

    3D is given by the number N of critical points of the field per unit volume(irrespective of ty pe ). If s(x) s a passive sca lar field sub jec ted t o convectionwith negligible diffusion, then D s l D t = 0, and the number N is conserved(since topological transitions involving either increase or decrease of N)cannot occur.In the presence of weak molecular diffusivity K , the field s is governedby th e advection-diffusion e qu ationD s l D t E a s / a t + U * V S = KVS. ( 5 4

    Following Batchelor (1959) and Batchelor et al. (1959), we may supposethat gradients of s are mainta ined at a large length-scale, and that fluctu-ations of s, relative t o its m ean , cascade to very small diffusive scales, atwhich they are el iminated. In these circumstances, th e numb er N may beexpected to at tain a large mean value determined by the detailed processof molecular diffusion (which both creates and destroys critical point pairsin equal measure). When K 2 v (kinematic viscosity), the wave-numberbeyond which molecular diffusion becomes important is k, N ( E / K ~ ) ~ / ~th econduction cut-off) and th e number N of critical points in a periodicityvolume V may be es t imated as

    N N k2V = ( E / K ) ~ / ~ V . (5.3)Th e si tuat ion when K

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    H.K. Moffatt: The Topology of Turbulence 11B-lines cannot be expected to remain on surfaces (it would be interesting toinvestigate for what special class (if any) of flows U, he B-lines do remainon surfaces!). Generically, the B-lines must be expected to have a chaoticcharacter. Two such fields (already mentioned in Sect. 2) have been studiedin some detail. The first is the ABC-field.

    B = (Csin kz + B cosk y , A sin kz +C cos k z , B sin ky +A coskz) (6.2)which, for arbitrary constants A,B,C, atisfies the Beltrami conditionV A B = kB. The chaotic character of the B-lines of this field (whenABC # 0) was conjectured by Arnold (1965), explored numerically byHQnon (1966), and analyzed further in detail by Dombre e t al. (1986). Thesecond is the STF-field

    B = (az- 8 z y , 11z2+ 3y2 + z2 + z y - 3, -cm+ 2yz - y ) (6.3)(Bajer and Moffatt 1990), which, as may be easily verified, satisfies V +B = 0 and n B = 0 on 1x1 = 1. (The field also satisfies the Stokescondition V 2 ( V ~ B ) 0.) When the parameter cy equals zero, the fieldB is doubly degenerate, each B-line being a closed curve; this imposes acertain character on the chaotic wandering of B-lines for cy # 0. Note thatthe STF-field was originally constructed as a velocity field in a sphericalball 1x1 < 1 having a combination of stretch, twist and fold ingredients -hence the STF-label.Apart from these two fields, whose properties are now reasonably welldocumented, very little is known concerning the generic structure ofsolenoidal vector fields in 3D. The beginnings of a general structural theorymay perhaps be seen in the work of Ghrist (1997) and Kuperberg (1999);but these are only the beginnings, and the general classification problem forsuch 3D fields is still wide open.7 Helicity and the turbulent dynamoThe evolution of a magnetic field in a conducting fluid moving with velocityu(x, ) s governed by equation (6.1). It has long been recognized that fieldintensification will result from the stretching of B-lines associated with theexponential separation of initially neighboring particles in turbulent flow.However, this intensification is achieved at the expense of a reduction of scalein the B-field, and a consequential enhancement of the effect of diffusionrepresented by the term 7V2B in (6.1); this ultimately becomes importantno matter how small the diffusivity parameter 7 may be. The questionthen arises whether sustained intensification of magnetic field (or dynamoaction) can arise, and if so through what mechanism that somehow bypassesthe enhanced diffusivity effect.

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    1 2 New Trends in TurbulenceAs pointed out earlier, if the velocity field U can be specified indepen-den tly of B, then (6.1) is linear in B, which behaves as a passive vec torfield. Th e only constraint t h a t need be imposed on U is the kinematicco nst rai nt of incompressibility V . U = 0; and the corresponding phase offield evolution is covered by th e te rm kinem atic dyna mo p roblem . If th efield grows exponentially, then this phase cannot last forever, because theLorentz force j A B (with j = pOIV A B) obviously reacts back u po n th e d y-namics of the flow, i .e . U can no longer be specified independently of B;we the n enter upon th e phase of the fully magn etohydrodynamic dy na m opro ble m , when it becomes necessary t o specify also the na tu re of th e forcesag itat ing th e fluid (or equivalently th e n at ur e of the source of kinetic energy

    of the turbulen ce) .7.1 T he kinemat ic phaseA double-length-scale approa ch is usually ad opted a s a sta rti ng point. I t issupposed that the dominant scale of the turbulence ( i . e . the scale of th eenergy-containing ed dies) is l o , and one focuses on the evolution of thefield B(x,t) o n a scale L much greater than lo. Th is is th e mean-fieldapproach pioneered by Steenbeck et al. (1966) (see Moffatt 1978; K rau seand Radler 1980). If the turbulence is homogeneous with zero m ean , thena mean electromotive force &(x, ) s generated on the scale L , given by anexpa nsion of th e form

    &i = (U A b)i = QijBj+ PijkaBj/axk + . . . , (7.1)where a i j , P i j k , . . are pseudo-tensors determined (in principle) by thestatistica l properties of the turbu lenc e and the param eter 7 . T he field b isth e small-scale magnetic pertu rba tion induced by th e flow U across B. T h eim po rtan t feature of (7.1) is th e linearity between th e fields & and B. Thislinearity results from the linearity of (G.1) ; no assumption need be madeconcerning the relative magnitudes of Ibl and IBl in order to deduce thes t ructure (7.1) . Conventional estim ates suggest t ha t

    b = O(R,)B, ( 7 4where

    th e m agnetic Reynolds num ber of the turbulence. In situatio ns of interestin planetary physics and astrophysics, R, is a least of order unity, andgenerally much greater than unity; in these circumstances, the fluctuatingingredient of the field b may b e much greater than the mean B.

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    H.K. M offatt: T h e Topology of Turbulence 13If t he turbulence is isotropic as well as homogeneous, th en th e pseudo-tensors a i j , P i j k , . . . must also be isotropic, i .e .

    f f t j = a d i j , p i j k = pfi jk, . . . (7.4)where a is a pseudo-scalar and ,49 a pure scalar (the pseudo property of&k being taken up by the pseudo-tensor E z j k ) . T he expansion (7.1) thentakes th e simpler form

    & = a B - p V A B f . . . , (7.5)V A I = ~ v A +pv2B.

    so tha t (wi th a,L3 constants by virtue of homogeneity),(7.6)

    T h e mean-field equation for B then becomesaB/at = a V B f VTV2B, (7.7)

    where TIT= q+p, a diffusivity augm ented by th e turb ule nt con tribution of p;one would clearly expect p t o be positive, although there is no gu aran tee ofthis, a t this level of ar gum ent.Equation (7.5) clearly admits exponentially growing modes of force-free s tru ct ur e satisfying

    V A B =KB , (7.8)

    (7.9)where K is a co nst an t; for such modes satisfy

    aB/at = aKB - V T K ~ B ,and hence B N ept where

    p = aK - V T K ~ , (7.10)showing exponential growth provided

    la/KI > TIT* (7.11)This condition is satisfied if K has the same sign as a and the scale L NIK - l I of B is sufficiently large (consistent with the two-scale assumption

    This dy nam o mechanism is due t o the generation term a V B in (7.5),or equivalently to the term aB n (7.3). This is the famous a-effect; agrowing understanding of this generic phenomenon has been one of themost dr am at ic developments of turbulence the ory of th e last half-century.

    L >> 1 0 ) .

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    14 New Trends in TurbulenceThe fac t tha t a is a pseudo-scalar implies that the effect can occur onlyin turbulence that lacks reflexional symmetry. The mean helicity H =( u . w ) of th e turbulence (also a pseudo-scalar) is in general non-zero in suchcircumstances, and a link between a and FI is to b e expected.To illustrate the essential mechanism by which an a-effect is produced,let us calculate a for th e case of an A B C -t yp e velocity field (w ith A =B = c u g )U = uo(s in(kz- wt) + cos(ky -wt), sin(kx - wt) + cos(kz - wt),

    sin(ky -wt) + cos( kx - wt)), (7.12)for which w = k u , a nd so

    H = (U - W)ki12 = 3 k 4 . (7.13)The helical character of (7.9) is evident in that the motion consists of asuperposition of th re e circularly polarized waves propagating parallel to th eaxes O x, Oy ,Oz.To calculate a , it is legitimate to assume a uniform mean field B, andth e eq uatio n for the fluctuating field b (from 6.1) is then

    db/dt = (B V)U+ V A G + qV2b, (7.14)where G = U A b- ( U A b). If the wave amplitude uo s sufficiently weak, th enthe nonlinear term V A G may be neglected (the first-order smoothingappro xim ation); th e field b may th en be found by elementary methods, andE = ( U A b) con struc ted. T he result is indeed E = aB , where

    .=-I(k 2 ).3 w 2+ q 2 k 4 (7.15)showing the expected dependence on mean helicity. Note also (i) the neg-ative sign in (7.15), i .e . positive helicity generates a negative a-effect, and(ii) the fact that (for w # 0 ) , a --+ 0 as 77 --t 0 , i .e . an a-effect (at leastin this first-order smoothing approximation) requires a non-zero moleculardiffusivity q.T he helicity (b . V A ) of the fluctuation field may be calculated underfirst-order smo othing (Moffatt 1978, Sect. 1 1.2). Not surprisingly, it ha s th esame sign as the driving kinetic helicity (U . w). Now look again at themean-field equation (7.5) (with now slowly varying B); if (say) ( U . w) ispositive, th en a is negative an d so negative helicity B .V A B is generated inth e large-scale field. It seems th a t th e positive ma gnetic helicity th a t is gen-era ted (a nd dissipated) on scales of order l o is a t least partly compensated

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    H.K. M offatt: T h e Topology of Turbu lence 15by negative m agne tic helicity genera ted on th e large scales of orde r L >> 10.We know that if 7 = 0, then total magnetic helicity is conserved. It wouldappear that dissipation of small-scale helicity is essential for dynamo actionprecisely because this allows the sustained increase of magnetic helicity (ofopposite sign) in the large-scale field.This manifestation of the effect of helicity in the turbulence was rec-ognized in the seminal paper of Pouquet e t al. (1976), who used theeddy-damped quasi-normal Markovian (EDQ NM ) closure scheme t o an-alyze spectral evolution of fully magnetohyd rodynam ic turbulence. M orerecent direct numerical simulation of MHD turbulence (e.g. Brandenberg1992), although limited to rath er modest R eynolds and magnetic Reynoldsnumbers, shows similar trends.7.2 The dynamic phaseLet us now consider briefly what happens when dynamo action occurs as aresult of the a-effect, and a large-scale magnetic field grows exponentiallyuntil the stage at which the back-reaction of the Lorentz force on the flowbecomes im po rta nt . As previously indicated, it is necessary at th is s tage tospecify th e na tu re of th e source of energy for th e m otion. L et us supposefor th e sake of a rgum ent th a t there is a random body force f( x , ) o n th escale 1 0 , and th a t t his force is independent of bo th U an d B.T he large-scale field B evolves relatively slowly, an d m ay, for th e purp oseof this analysis, be treated as constant. When B is sufficiently strong, itseffect is t o severely con trol th e am plitude of motion on scales of orde r 1 0 . I t isreasonable the n t o linearize the e quations for th e velocity U and fluctuatingfield b. These equ ation s (in units such th at pop = 1) become

    (7.16)where P is th e sum of fluid an d magnetic pressure. T he se are ju st th eequations for forced Alfven waves traveling on the field B; indeed if weconsider a single Fourier com ponent f xp i (k .x-wt) o f f , the correspondingsolution of (716) is

    Iu/dt = - V P +B * Vb + vV2u+ fdb/dt = B * V U + vV2b

    ii = D- l ( - iw +#)f, b = D - q B k ) f , (7.17)where

    D = w 2 + ~ W ( V q ) k 2 - v7k4 - ( B .k ) 2 . (7.18)Where v and v are b oth small, there is a sh arp resonance near w = fB. ,th e frequency of freely propag ating non-dissipative Alfven waves. W hen we

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    16 New Trends in Turbulenceconstruct (U b) for a spectrum of forcing, the result is dominated by theamplitude and width of these resonances in ( w ,k) space.The details, which are quite subtle, have been worked out with theadd itional com plication of C oriolis effects in a ro ta tin g body of fluid (Mo ffatt197 2). T h e combined effects of Lorentz and Coriolis forces induce anisotropyin the turbulence; the effective a (e.g. a = i a z i ) an still however becalculated; this is now a decreasing function of mean field strength IBI, onaccount of the decreasing width of the resonant layers in ( w ,k) pace as IBIincreases.This effect in which a (B ) s a decreasing function of jB/ (tending tozero as /BI+ CO) has since been described as a-quenching. Clearly, itleads to s atu rat ion in t he growth of th e mean field. For the simple modelof Section 7.1, thi s s at ur at io n will occur for scale L N K-l when

    a(B)= VTK. (7.19)Note however tha t a field satu rat ing a t this level can still app aren tly grow onscales much larger than K-l . T hu s, under su stained forcing on some scale10 which, either through the intrinsic character of the forcing, or througha C oriolis effect, ge ne ra te s turbu lence with non-zero helicity, we may ex-pect the dynamo -generated magnetic field t o s atu rat e a t successively largerlength scales, th e s pe ctra l sha pe of t he resulting field being given, in quali-tative ter m s, by solving (7.19) for B:

    B = aY-l(7pK) (7.20)and by interpreting this as providing th e sp ectrum (in K-sp ace) of B.Of course the parameter , (contributing to V T ) will also be subject toa measure of quenching, which could also be included in the above type ofanalysis.8 Ma gne tic relaxationA s itu at io n of gr ea t interest arises when we consider a different ty pe of initialvalue problem: suppose th at a t t ime t = 0, a random magnetic field Bo(x)exists in a fluid w hich is a t res t. T he associated curr ent is j, = V A Bo andthe Lorentz force Fo = j o A B 0 is in general rotational ( i . e . V A F O 0).T his force ca nn ot therefore be balanced by pressure gradients, and the fluidwill move; energy is the n dissipated by viscosity. At th e sam e time, t hefield is transported by the flow, and the Lorentz force distribution F(x,t)evolves in time.Th e s i tua tion is of pa rtic ula r interest if th e fluid is a perfect co nductor( i . e . 17 = 0), since then the field topology is conserved during this relax-ation process. E ne rgy is however still dissipated by viscosity. T h e magn etic

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    H.K. Moffatt: The Topology of Turbulence 17helicity is conserved, and this acts as a topological barrier that preventsthe magnetic energy from decaying to zero. We are then faced with a vari-ational problem with an unusual twist: to find the minimum energy stateof a field whose topology is prescribed as that of the initial field Bo(x).The equations of magnetohydrodynamics (with7 = 0 , v # 0 ) provide thenatural dynamics that drives the system towards such a minimum energystate (Moffatt 1985). In the minimum energy state, the velocity is again zero(since otherwise it would continue to dissipate energy); the correspondingfield, BE(x)say, is therefore magnetostatic, i .e.

    jEABE= vpE (8.1)for some scalar (pressure) field p E , and jE = VhBE. The field BE istopologically accessible from Bo(x), in the sense that it is obtained bycontinuous distortion by a velocity field v(x, )(O < t < co)which dissipatesa finite total amount of energy (the difference between the energies of thefields Bo and BE).Although the above relaxation process is simple to describe, and seemstransparently clear, it should be noted that point-wise convergence of thefield B(x, ) o an equilibrium field BE(x)has not been proved, and remainsan open problem.8.1There is an exact analogy between (8.1) and the equation

    The analogy with Euler flows

    U A W = V h (8.2)describing steady Euler flows with w = V A U. The analogy is between thefields BE and U, or equivalently between jE and w) . Note that h in (8.2)must be regarded as the analogue of -pE . To any solution of (8.1), therecorresponds via this analogy a corresponding solution of (8.2). It is thusapparent that (subject to pointwise convergence of the magnetic relaxationprocess) there exists a steady solution u(x) of the Euler equations havingarbitrarily prescribed topology ( i . e . that of the initial field Bo(x) in themagnetic relaxation problem).

    Note here that it is the topology of the velocity field (rather than thatof the vorticity field) that can be prescribed. It would be more interestingif a relaxation procedure conserving the topology of w ( i .e . that of j inthe magnetic relaxation problem!) could be devised, because that would bemore natural for Euler dynamics. Only in 2D flows has this been foundto be possible (Vallis et al. 1989). In this context, an upper bound can beplaced on the energy of a flow of prescribed enstrophy, and a relaxationprocedure that increases energy to a limiting value can be constructed.

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    1 8 New Trends in TurbulenceIn 3D, th e s tead y Euler flows obta ined via the magnetic relaxation tech-nique, are all app aren tly unstable (Rouchon 1991). This is probably highlysignificant for turbulence! Nevertheless, it has been hypothesized (Moffatt1990) that long-lived coherent vortices may be associated with maximalhelicity regions where w is parallel to U (so t h a t (8.2) is certainly satis-fied). R ecen t analysis of 3D tur bu len t flows using wavelet transforms (Fargeet al. 2001) lend some su pp or t to th is hypothesis.

    9 The blow-up problem

    T h e question of smoothn ess of solutions of the Navier-Stokes an d/ or Eulereq ua tion s for an incompressible fluid h as remained open since first posed byLeray (1934). Numerical investigations (e.g. Kerr 1993; Pelz 1997) provideevidence for the blow-up of solutions of th e Euler equ ations a t finite tim e;b u t num erical codes have limited validity where singularities are concerne d,an d num erical results can be no m ore th an suggestive in this context. O nthe analytical side, it is known (Beale et al. 1986) that if any breakdownof regularity of solutions of the Euler equations occurs at some finite timet = t*, he n th e maximum value of th e vorticity magnitude lwmax( t ) lmustblow-up as t + t* in such a way as to make the integral of this quantity(from t t o t* ) iverge. T h e simplest possibility is th a tas t t t*,

    an d thi s is indeed t h e so rt of behavior t h a t has been inferred from num ericalexperiments (Pumir and Siggia 1988; Pelz 1997).Why, i t m ay be added , are we so interested in blow-up in th e turbulenceco ntex t? T h e reason is t h a t if blow-up is a generic feature of any fully 3Dtime-dependent flow a t very high Reynolds num ber, then th e spo tty orintermittent character of turbulent dissipation is immediately understand-able in terms of the behavior near points where singularities of vorticity(a nd the related deformation tens or) occur.Even more interesting is the question of how singularities (if they oc-cu r) mu st in p ractice be resolved. Singularities of the E uler equations m ay

    conceivably be resolved through inclusion of the effects of weak viscosity.B u t if it t ur ns out t h a t singular behavior can persist even for the Navier-Stokes equations ( i . e . even when weak viscous effects are indicated), thenwe must look to o the r physical effects t o resolve such behavior. T h e obviouseffect t h a t sh ould th en be considered is compressibility; for a singularity ofvorticity w ill also imply a sing ularity of pressure (th ere being a large re-duction of pressure in the core of an intense vortex). Compressibility in aga s results in t he pro pa ga tion of ac oustic waves by th e Lighthill mechanism

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    H.K. M offatt: T h e Topo logy of Turbu lence 19(Lighthill 1953), a mechanism t h a t m us t surely prevent the form ation ofany pressure singularity in th e fluid interior. In a liquid there is anot he rmechanism: th e liquid will cav itat e wherever th e pressure falls below th evapor pressure, an d small bubbles will form , an effect th a t will be obviouslydependent on the m ean pressure applied t o th e system. Th is mechanismalso mitigates against the formation of pressure singularities. the detailedmechanism of energy dissipation will be influenced by the presence of suchcavitation bubbles; this deserves study!9.1 In terac t ion o f skewed vorticesAll studie s of th e potential blow-up of vortic ity have focussed o n the b ehav-ior of skewed vortex tu be s, when for one reason or another the se are driveninto close proximity with one ano ther . T he simplest scenario(Moffatt 2000) is that in which two skewed vortex pairs propagate on acollision course towards each oth er. Th ese interac t strongly when the sep-aration between the pairs becomes of the same order as the separation ofth e vortices within each pair. We m ay define an inner interaction zonewhose scale a(t) s a decreasing function of time; and an outer zone, wherethe vortex pairs continue t o p rop aga te with negligible interaction.If all characteristic length-scales (e.g. vortex separations, vortex radiiof cu rva tur e, vortex core rad ii, . . . ) decrease in proportion to a(t) and th i sis a big IF), then the behavior in the inner zone is self-similar and can bedescribed by th e L eray (1934) scaling:

    where I s a cons tan t having th e dimensions of a circulation. T h e corre-sponding vorticity is then

    and th e vorticity equation transforms t oo = v A [(U+:.) A n ] +Ev2n, (9.4)

    where E = u / r . If any sm oo th solution of this eq ua tio n, satisfying accept-able bou nd ary con ditions, can be found , then th e corresponding solution ofth e native Navier-Stokes equation clearly has a singularity (w ith ma ximu mvorticity behaving as in (9.1 )) a t x = 0 as t + t*.B ut wh at are the acceptable boun dary condit ions? Suppose th at1O(X)/N /XI- as 1x1t 3.

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    20 New Trends in Turbu lenceThen, for each fixed finite x, rom (9.2) and (9.3),

    (9.5)If a < 2, this vorticity blows up for all finite x as t + t*, behavior that istotally implausible. If cu > 2, then the vorticity goes identically to zero forall x as t --+ t* , a behavior t h a t is equally implausible. T he only realisticpossibility therefore is that cu = 2, so t h a t

    (and correspondingly IUI N /XI-) as would be realized, for example, byconical expansion of vortex tubes as they leave the interaction zone. Cor-respondingly we require that

    as the inner b ou nd ary condition for th e outer region. Equations (9.6)and (9.7) indicate o nly rad ial behavior; an gular dependence is unconstrained .Ag ain, the re is evidence of this behavior in the num erical work of Pelz (1997)who st ud ied , by vo rte x filament techniques, th e implosion of 6 vortex pairstowards the origin, the whole configuration having cubic symmetry.Against th is scenario, two theorems have been proved by functional an-alytic techniques. NeEas e t al. (1996) have shown t h a t, for E > 0, (9.4) hasno nontrivial solution U(X) in L3(R3) ( i . e . for which [U(X)13s integrableover th e whole X -sp ac e). T hi s imm ediately rules ou t solutions for which[U[N 1XI-q (and 1i-l N IXI-(qfl)) for q > 1; it does not rule out thebehavior (9.6), which, perhaps significantly, lies just outside this functionspace. However, m ore seriously, T sai (1998) ha s shown (aga in for E > 0)th at the th eorem of NeEas e t al. (1996) can be extended to cover the non-existence of nontrivial solutions U(x) f (9.4) in Lq(R3) for all q in therange 3 < q < 03; thi s certainly does exclude the behavior (9.6) . Th e fullimplications of T sais theo rem are as yet unclear (to this w riter!), bu t itdoes imp ly th a t singularities of the N avier-Stokes equations with U > 0cannot in fact be described in terms of the Leray scaling (9.2).T he above rem arks do not apply to the Euler limit ( E = 0 in (9.4 )); inthis limit, we m ay inte grate (9.4) t o give

    for some scalar H(X) the scaled Bernoulli function). Since 0 . V H = 0 ,vortex lines lie on surfaces H = a t . , which in effect define the vortex tubes

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    H.K. Moffatt: T h e Topology of Turbu lence 21of th e flow in th e inner (Leray) region. T h e self-induced velocity U(X)mustsatisfy

    ( U + t X ) . V H = o , (9.9)i .e . there must be an inward flow across each vortex tube to compensatethe outward transport represented by the term 4X . V H (which can betraced to the space-scaling relating X and x in (9.2)). This inflow canin principle be compensated by outflow along the vortex tubes emanatingfrom the interaction zone. B ut th e $ million question is whether there isany vortex configuration which induces a velocity field which in turn keepthe configuration steady via the condition (9.8). It seems likely that thisquestion will continue to present a profound challenge over the next fewyears, if not decades!References[l] V.I . Arno l d , C.R. Acad. Sc i . Pa r i s 261 (1965) 17-20.[2] V.I. Arnold, Selecta Mathematica Sowetica 5 1986) 327-345 (1974).[3] K. Ba je r and H.K . Moffa t t, J . Fluid Mech. 212 1990) 337-363.[4] G.K. Batchelor, J . Fluid Mech. 5 (1959) 113-133.[5] G.K. Batchelor , Phys . Flu ids . Supp . I I (1969) 233-239.[6] G.K. Batchelor , I . Howell s and A.A. Tow nsend, J . Flu id M ech. 5 (1959) 134-139.[7] J.T. Beale, T. K a t o a n d A . M a j d a , C o m m . Math. Phys . 94 1984) 61-66.[8] A. Brandenberg , Phys . Rev . Let t . 69 1992) 605-608.[9] T. Dombre , U. Fr i sch , J .M. G reene , M. Hknon, A. M ehr an d A.M . Soward , J . Flu idMech. 167 1986) 353-391.

    [10] M. Farge, G. Pel legr ino a nd K . Schneider , Coheren t vor tex ex t rac t ion in 3 Dturbulent f lows u sing orthogonal wavelets (2001) Prep r in t .[ l l ] A.D. Gi lber t , J . Fluid Mech. 195 (1988) 475-497.[12] R.W. Ghrist , Topology 36 (1997) 423-448.[13] M. Hknon, C.R. Acad . Sc i . Par i s 262 1966) 312-314.[14] R.M. Ker r , Phys . F lu id s A 5 (1993) 1725-1746.[15] R. Kra ichnan , Phys . F lu id s 10 (1967) 1417-1423.[16] F. Krause and K.-H. Radler , Mean-f ie ld Magnetohydrodynamics and Dynamo

    [17] K.M. Kuperberg , A Coo counterexample t o the Seifert conjecture in dimen sion three[l8] J . Leray , Acta M ath . 63 1934) 193-248.[19] M.J. Lighthi l l , Proc. Roy. Soc. A 211 (1952) 564-587.[20] J . Marsden , T. R a t i u a n d T. Shkoller, Geom. Funct . Anal . 10 2000) 582-599.[21] H.K. M offa t t, J . Flu id M ech. 35 (1969) 117-129.[22] H.K. M offa t t, J . Fluid Mech. 53 (1972) 385-399.[23] H.K . Mo ffa t t , M agnet ic Fie ld G enera t ion in Elec t rical ly Cond uct ing Flu ids

    Theory (Pe rgam on Press , 1980).(AMS Bul le t in , 1999).

    (Cam bridge Univers i ty Pres s , 1978).

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    nonlinearity, in W hi the r Turbulence? Turbulence a t the Crossroads , edited by J.L.Lumley, Lecture Note s in Phys ics 357 Springer-Verlag, 1990) p p . 250-257.[27] H.K. Mof fa t t, J . Fluid Mech. 409 2000) 51-68.[28] H.K . Mof fat t, T he topo logy of scalar fields in 2D a n d 3D turbulence, in Geometryand Statis t ics of Turb ulenc e, edited by T. Kambe (Kluwer , 2001) ( t o a p p e a r ) .[29] J . NeEas, M. Rpui iEka and V. Sverbk, Acta Math . 176 1996) 283-294.[30] R.B. Pelz , Phys . Rev . E 55 1997) 1617-1626.[31] A . P o u q u e t , U. R i s c h a n d J. Lkorat , J. Fluid Mech. 77 1976) 321-354.I321 A . P u m i r a n d E . Siggia, Phys . F lu ids 2 (1990) 220-241.[33] P. Rouchon, E u r . J . Mech . , B /F lu ids 10 1991) 651-661.[34] M . Steenbeck, F. Krause and K.-H. Radler , 2. Natur forsch 21a (1966) 369-376.[35] T.-P. T s a i , Arch. Rat ional Mech. Anal . 143 1998) 29-51.[36] G.K. Vallis, G.F. Carneva le an d W .R. Young , J . Fluid Mech. 207 1989) 133-152.