H.K. Moffatt- Some Remarks on Topological Fluid Mechanics

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    SOME REMARKS ON T O P O L O G I C A L F L U I D MECHANICS

    H. KE IT H MOFFA T TIsaac Newton Inst i tute f o r Mathematical Sciences20 Clarkson Road, Cambridge, CB3 OEH, U KE-mail: [email protected]

    Abstract. Some fluid dynamical problems having a topological flavour arebriefly reviewed, and some further problems having at least a topologicalstarting point are posed.

    1. I n t r o d u c t i o nTopological Fluid Mechanics is primarily concerned with structures withina flow field which retain some coherence over a significant period of time.Under circumstances that may be described as ideal relative to the type ofstructure considered, this significant period of time is infinite; but insofaras circumstances are never ideal in reality, we must be equally concernedwith the manner in which structural (or topological) properties of a flowmay change with time (generally under the influence of some diffusive pro-cess).

    These statements suffer from a degree of imprecision that can be re-moved only through consideration of particular problems. The purpose ofthis brief paper is to set out a number of such problems, all of which haveat least a starting point that can be described as topological, and most ofwhich are unsolved. There is no shortage of challenging problems of thistype for which a combination of analytical, computational and experimental(ACE) techniques will be required if real progress is to be made.2. Particle PathsFor any given flow field u ( x , t ) , whether in a finite or infinite domain, thepath X = X(x, ) of the particle initially at position x is determined by the

    3R.L. Ricca (ed,),A n Introduction to the G eometry and Topology of Fluid Flows, 3-10.0 2001 Kluwer Aca dem ic Publishers. Printed in the Netherlands.

    www.moffatt.tc

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    XXI Century Problem 1

    1t-03 t= l im -In

    For a given bounded do ma in V and a give n

    (4)is positive; here < is an infinitesimal material line element originally at x;X repre sents th e as ym pto tic ra te of s tretching of this m aterial line element.No te th a t i f XI,x2 l ie on the same streamline, then X(x1) = X(x2). Inregions of 'regularity' of t h e flow, 1" grows merely linearly with time, and

    A quan t i ty of key im po rta nc e in relation t o th e stirring efficiency of aflow is t h e volume fractio n p of the fluid domain in which X(x) > 0. Thisvolume fra ction is a fund am enta l s t ruc tura l param eter ; i t i s topological inthe sense that it is invariant under continuous volume-preserving deforma-tions of the flow field (which deform the regions of chaos without changeof volume) . We m ay th us pose a first problem of topological character:

    X(x) = 0.

    f y i n g V .U = 0 in 27 and n .U = 0 on 827, to determ ine the volume fractionp of V o r which X(x) > 0 , i.e. for which the flow has chao tic particle pa ths.This problem has obvious generalisations for space-periodic flows andfor time-periodic two-dimensional flows.

    3. Scalar Field ProblemsLet O(x,t) e a scalar field that is convected by a continuous velocity fieldU, nd sup pose for th e mom ent t ha t m olecular diffusion is negligible. Then0 i s constant for each fluid particle, i.e.

    DO 80- -4- u . o o = oD t d t ( 5 )To be specific, suppose t h a t t h e flow domain 2) s in Ill3, a n d t h a t u ' n = 0,0 = cs t . on dV. Eq u a t io n (5) of course implies t h a t th e surfaces 0 = cs t .ar e tran sp ort ed with th e flow. Th eir topology is therefore conserved. Howis thi s topology to be described?

    A s ta r t has been made [14] through consideration of the saddle pointsSi of th e f ield B and the homoclinic iso-scalar surfaces C ; through theseS;. Toge ther wi th 827, t h e C i divide V in to a number of subdomains Da(a 0 , 1 , 2 , . , .) where Do i s the subdomain that is bounded externally by8D, nd each Da (a= 1 , 2 , . . ) is bounded by parts of one or ( a t mo s t )tw o of th e homoclinic surfaces. T h e volume of each D, is conserved under( 5 ) as is t h e topology of th e surfaces E;.

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    X X I Century Problem 2

    Knowledge of the relative configurations of the C; and of the volumesof the V, is an important first step in classifying the possible topologiesof the 0-field. This is not all however. Within each V a re a family ofsurfaces 0 = cst . an d we m ay define a signature function V,(O) in V awith the p roper ty th a t (dV,/d0)60 is th e volume between surfaces labelled0,O + SO; the function V,(O) is then defined up to a constant C, whichm ay be chosen so th a t t h e sign ature function varies continuously in movingfrom one subdom ain t o another . T he set of s ignature functions {Va(0)}thus defined is clearly invariant under the evolution (5) nd is therefore atopological property of the field 0 .There are now two interesting directions that merit investigation. Weind icate th ese in th e form of problems:

    Suppose now that molecular diflusivity K.

    What transitions in the topology of the set of homoclinic surfaces {E;}are possible as a result of this diffusion; a n d how does the set of signaturefunct ions {Va(0)} volve, particularly during such a change of topology?

    Suppose that the velocity f ield U is itselfdriven b y inhomogeneity of the 0-field,X I Century Problem 3according to some well-defined dynamical prescription (e.g. 0 could repre-sent temperature variation in a gravity field, the flow being driven b y thebuoyancy force i n the Boussinesq approximation (see, fo r example, [ . I ) .Th e problem is to exam ine the evolution of the 0-field in the neighbourhoodof its saddle-points, to determine whether singularities of VO can develop,and to examine the influence of weak molecular diffusivity K . in controllingthe approach to such singularities.

    4. Vector Field ProblemsEa ch of th e above scalar field problems has a counte rpart in t he c ontext of atra ns po rte d vector field, such as th e m agnetic field B(x, ) n a conductingfluid. T h is field is d ivergence-free, i.e. V .B = 0 , and satisfies th e induc tionequat ion

    (7)dB- = V X ( u x B ) + q V 2 B ,d tthe coun te rpar t o f ( 6 ) . In the diffusionless (perfectly conducting) limitq = 0, (7 ) implies t h a t B -lines a re frozen in t h e fluid, th e flux of B

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    rxxI Century Problem 4 1

    through any closed material curve C being conserved. Th is has th e imp or-tant consequence that any linkage between B-lines is conserved - clearly aresult w ith topological co nte nt. T h e simplest measure of net linkage of th efield inside any Lagrangian (i.e. material) surface S on which n a B = 0 isth e m agne tic helici ty

    r

    Consider two linked unknotted flux tubes,

    where V is the volume inside S, and A is a vector potential for B, i.e.B = V x A. T h e in t eg ra l (8) is gauge-independen t, but it is usual t o chooseth e gauge of A so t h a t V A = 0 .

    I xxICenturyProblem 5 1 Con sider tw o flu x tubes oblique to each

    tube being untwisted, so that the helicity X ,M is 2ip2 [ll];we assume herethat the linkage is right-handed. Suppose that the fluid mo tio n brings thetubes into close proximity and that weak difjusion (11 > 0 ) causes reconnex-ion o f B - l i n e s in suc h a way that the two tubes become a single tube carryingf l ux a . In this process, the helicity (or at least some proportion of i t) maysurviv e through the appearance of inter nal twist i n the resultant tube 1151.T h e prob lem i s to dete rmin e precisely what i s the total f ield helicity afterreconnexion, the whole process being governed by equation (7).

    In th e abo ve prob lem , if th e field B is sufficiently weak, the n p resum ablyit may be treated as dynamically passive, the velocity U being then inde-pendentIy prescribable. More realistically, however, the Lorentz force j x Bwhere j = V x B, plays a n imp ort an t par t in the reconnexion process. Th isis part icu larly t h e case when 7 is very weak since th en very str on g stre tch -ing of field lines occurs in conjunction with the flow that brings sections ofth e tw o init ial tu be s in to close proximity. This leads t o

    Just as for the scalar f ield problem, there are circumstances in whichthe velocity field U is entirely driven by the Lorentz force distribution. Ifth e f luid is viscous bu t perfectly conducting, the n t he field energy conv ertst o kinetic e nergy which is dissipate d by viscosity, a nd d uring th is processthe field topology is conserved. There is however an outstanding problemin relat io n t o this m agn etic relaxation scenario th at remains open:

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    Determine mmin for knots of minimum

    XX I Century Problem 6a viscous, perfectly conducting,

    Consider a smooth localised magnetic fieldoffinite and nonzero magnetic helicity inincompressible f lu id initially at rest. It isknown [12] that the kinetic energy tends to zero as t --+ 00. It is required toprove that Ju(x,t)l+ 0 at all points x as t --+ CO.This is almost certainly true, since otherwise the appearance of singu-larities (of implausible form in a viscous fluid) is implied; a proof shouldnot be impossibly difficult.If the initial field is confined to a single knotted flux tube of volume

    V and carrying flux @, and with internal twist such that the helicity is7 - l ~ h a 2 , h e n , on dimensional grounds the relaxed state has minimalma gnetic energy M E given by

    M E = ~ n ( h ) @ ~ V - / ~ (9)

    Th is problem presents a considerable computational challenge. A s t a r thas been made for toru s knots by Chui St Moffatt [6].5. The Finite-Time Singularity ProblemI t is bu t a small ste p from t h e above magnetically active problems t o th eEuler a nd Navier-Stokes problems th a t lie a t th e hear t of fluid mechanics.We simply replace B in ( 7 ) by the vorticity field U, and we take U t o b eth e inverse cu rl of U:

    and of course we replace 7 by kinematic viscosity v.T h e simplicity of th e functional relationship (10) between U and w mightsuggest t h a t, for example, th e problem of viscous vortex tu be reconnexionshould b e n o more difficult th an th e problem of magnetic flux tu b e recon-nexion (w ith L orentz forces included). This however is jus t wishful thinking!Vortex t ub e reconnexion has at tra cted much study, bo th analytical [5] andcomputa t iona l [ l O ] , [18] (see also this volume), and yet we are still in the

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    xxI Century 1

    dark as regards the details of the process. In part icular , we do not knowwhether the vort ici ty f ield within a zone of reconnexion remains finite forall t , or conversely whether a singularity of vorticity may develop withina f inite t ime. The computational work cited above provides quite s trongevidence for the appearance of finite-time singularities for Euler evolution(U = 0), whereas a na lyti cal studies based on the Leray similarity tr an s-fo rmat ion 1161, [19] point t o t he non-existence of finite-time singularitieswhen U > 0. I have argued [13] th a t if finite-time singularities ap pe ar un de rEuler evolution, then the same type of s ingulari ty should appear when Uis positive but sufficiently small, i.e. that weak viscosity may not be ablet o prevent th e form ation of finite-time singulari ties. Th is suggests a prob-lem tha t may p rov ide a helpful s tepping-stone towards the central ($lm)finite-time singularity problem as posed by the Clay Insti tute [8].

    Suppose that there exists a smooth velocityf ield uo(x) of fin ite energy in a bounded

    domain V such that, u nder Euler evolution starting fro m this initial condi-t ion , a singularity of w ( x , ) appears at some finite time t * . Prove that, fo r0 < U < U, where U, is small, a n d with the same init ial condition, w ( x , t )still becomes singular at finite time; or conversely, prove that w ( x , t ) re-main s smooth for all t .References1.2.3.4.5.6.7.8.9 .

    10.11.12 .

    Aref , H. (1984) Stirr ing by chaotic advection. J . Fluid Mech. 143, -21.Arnold, V.I . (1965) Sur la topologie des ecoulements stationnaires des f luides par-fa i ts . C.R. Acad. Sc i . Par i s 261, 7-20.B a j e r , K . & Moffatt , H.K. (1990) On a class of steady confined Stokes flows withchao t ic s t reamlines. J . Fluid Mech. 212, 37-363.Ba tche lo r , G .K. , Canu to , V .M. & Chasnov, J .R. (1992) Homogeneous buoyancy-genera ted turbulence . J . Fluid Mech. 235, 349-378.Bea le , J .T . , Ka to , T. & M ajd a , A.J . (1989) Rem arks on the breakdown of sm oo thsolutions for t h e 3-D Euler equations. Commun. Math . Phys . 94, 1-66.C h u i , A . & M offatt , H.K . (1995) Th e energy and helicity of knott ed m agnetic f luxtubes . Proc. Roy. Soc. Lond. A 451, 09-629.D o m b r e , T., risch, U., Greene, J .M ., HCnon, M ., M ehr, A. & Soward , A .M . (1986)Chao t ic s t r eaml ines in the AB C flows. J . Fluid Mech. 167, 53-391.Fefferman, C. (2000) Exis tence and smoothness of the Navier-Stokes equations.ana.claymath.org/prizeproblems/navierstokes.htm.Hknon, M. (1966) Sur la topologie des l ignes de courant dans un cas particulier .C.R. Acad. Sci . Paris A 262, 12-314.K e r r , R.M. (1993) Evidence for a singularity in the three-dimensional Euler equa-tions. Phys. Fluids 6, 725-1746.Moffatt , H.K. (1969) The degree of knottedness of tangled vortex lines. J . FluidMech. 35, 117-129.Moffa t t , H.K. (1985) Mag neto stat ic equilibrium and analogous E uler flows of arbi-t ra r i ly complex topology. Par t 1. Fundamenta ls . J . Fluid Mech. 159, 59-378.

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    1013.14.15.16.17.18.19.

    M offatt H .K . (2000) T h e interaction of skewed vortex pairs: a model for blow-up ofth e Navier-Stokes equat ions. J . Fluid Mech. 409, 1-68.Moffat t , H.K. (2001) The topology of scalar fields in 2D and 3D turbulence. InGe om etry and Stat i s t ics of Turbulence (ed. T. K a m b e e t al.), pp. 13-22. Kluwer.Moffat t , H.K. & Ricca, R.L. (1992) Helicity and the Cillugilreanu invariant. Proc.R . Soc. Lond. A 439, 11-429.N e t a s , J . , R u i i t k a , M. & S v e r i k , V. (1996) On Lerays self-similar solutions of t h eNavier-Stokes equations. Ac t a M at h . 176, 83-294.O t t i n o , J .M . (1989) The K i nem at i c s of Mixing: Stretching, Chaos and Transport .Cambridge Univers i ty Press .Pelz, R .B. (1997) Locally self-similar, f inite-time collapse in a high-sym me try vortexfi lament model . Phys . Rev . E 55, 1617-1626.T s a i , T.-P. (1998) On Lerays self-similar solutions of the Navier-Stokes equat ionssatisfying local energy estimates. Arch . Rat . Mech . Anal . 143, 9-51.