H.K. Moffatt Topological Dynamics of Fluids

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8/3/2019 H.K. Moffatt Topological Dynamics of Fluids
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Topological Dynamics of FluidsH.K. MoffattDepartment of Applied Mathematics and Theoretical PhysicsUniversity of Cambridge, Silver Stree tCambridge CB3 9EW, U.K.
AbstractCertain fluid dynamical problems having a pecularily topological flavour are reviewed. Firs t, some properties of th e helicity invariant of th e Euler equa tions aredescribed. T he n the technique of magn etic relaxation for th e indirect determina tion of stea dy Euler flows is summarised. Applications t o th e relaxation of twodimensional fields characterised by an invariant signature function, of knottedmagn etic flux tub es t o minimum energy state s, an d of chaotic threedimensionalfields a re briefly discussed.
1 IntroductionLet us consider an incompressible fluid moving with velocity field u(x, ) whereV  U= 0. Let p be th e density of th e fluid (assumed uniform a nd con stant), p(x, t )the pressure field, V(x, t ) the potential of any conservative forces (includinggravi ty) , and f (x , t ) any othe r nonconservative forces applied t o t he fluid. T henthe evolution of the flow is'described by the NavierStokes equation
(1.1)dU U AW  V h + v V 2 u + fdtwhere w = V A U is the vorticity field, h = p / p + $u2+V, and U is the kinematicviscosity of the fluid. Th is famous equation describes both lam inar and tu rbulen tflows. Only a handful of exact solutions are known, an d these usually correspondto highly idealised boundary and/or initial conditions. In practical contexts,and particularly for turbulent flow fields, recourse must generally be had tonumerical techniques, which are highly developed. These are however viableonly for &her modest values of th e Reynolds number Re, defined as Re = U L / uwhere U is a typical velocity an d L a typical leng thscale of th e flow field. In m anypractical contexts, Re 2 106, r even (in meteorological contexts) Re 2 108,and numerical techniques have to be coupled with semiempirical assumptionsof dubious validity. Asymptotic theories (for Re >> 1) are still a t a rudimentarystage of development, but the hope for the fu ture must lie in a development ofthese asym ptotic theories, and incorporation of th e results in improved numericaltechniques.
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466 H.K. MoffattIn the limit Re t 3 (or equivalently U t ) and assuming f = 0, (1.1)reduces to the Euler equation of classicalhydrodynamics
all = u A w  V ~atAs is wellknown, this is a singular limit which fails to describe boundary layersat the fluid boundaries, or the structure of vortex sheets, or other singularities,in the fluid interior. Excluding such singular regions, however, eqn. (1 .2) may beexpected to provide a reasonable description of the evolution of the flow.
The curl of (1.2) provides the vorticity equation V A ( U A W )uat
which embodies the results of Helmholtz (1858) and Kelvin (1869), pecificallythat vortex lines move as if they were composed of fluid particles  n modernparlance, the vortex lines are frozen in the fluid. As Kelvin recognized, thismeans that all topological structures associated with the vorticity field are invariant: two closed vortex tubes remain linked for all time if they are linked attime t = 0; a single vortex tube remains knotted in the form of a knot K ifit is so knotted at time t = 0; and so on. Equation (1.3) may be regarded asthe master equation that guarantees invariance of the complete topology of thevorticity field u(x, ). This is a powerful statement, because, for example, allknot invariants may be regarded (although some may consider it eccentric to doso) as invariants of the nonlinear evolution equation (1.3).
When we adopt a topological (asopposed to an analytical or differential) approach to fluid dynamics, we seek to exploit the topological invariance embodiedin (1.3). This forces us to focus on the global (as opposed to local) propertiesof the vorticity field and to identify specific invariants of (1.3) which have atopological character.It proves fruitful to widen the investigation by considering an arbitraryfrozenin solenoidal field B(x, ) (V * B = 0) satisfying the evolution equation(cf. (1.3))
This is the equation satisfied by the magnetic field B(x,t) in a perfectly conducting fluid moving with velocity u(x,t); t also serves as a model for anyother situation in which a vector field B is transported by the fluid, the flux ofB through any material surface element being conserved. The field U may beindependent of B (in which case B is a passive vector field and eqn. (1 .4) islinear); or U may be influenced by B through dynamic effects (e.g. through theLorentz force in the magnetohydrodynamic context), and then (1.4) s in effecta nonlinear evolution equation, like (1.3) .

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Topological Dynamics of Fluids2 The Helicity Invariant
467
Consider a closed material surface S within the fluid on which n w = 0, acondition that persists for all t when w satisfies (1 .3) . Let V be the volumeinside Sw;we may t he n define th e helicity H of the flow within V as
r
and there is a family of such integrals corresponding to any family of vorticitysurfaces S,. It is known (Moreau 1961, Moffatt 1969) that, under the Eulerevolution (1.3), H is invariant:
an d th at t his invariance is a consequence of th e conserved topology of th e vorticity field. Thu s, for examp le, if t h e vorticity field consists of two (possibly linked)vortex tu be s in each of which th e vortex lines ar e untwisted, then
H= CSt., (2.2)
H= f2nK1K2 , (2.3)where ~1 , ~2 are th e (conserved) circulations of t he tube s, n is the Gauss linkingnumber of their axes, and th e + or  s chosen according as th e linkage is positiveor negative. Eqn. (2.3) provides a clear bridge between an invariant of classicalfluid dynamics and t h e fundamen tal topological invariant n of two linked curves.
A further bridge is provided throug h consideration of a single knotted vortextu be of circulationK, th e ax is C of th e tub e being k notte d in th e form of a knotK (Moffatt & Ricca 1991). Here we cannot avoid consideration of the twist ofth e ulines within th e tube. Suppose th at this twist is uniform in the sense th a teach pair of wlines in 7 as th e same linking num ber n. Then the helicity ofth e tu be may be calculated explicitly in th e form H = f n K 2 , (2.4)
where again th e + or  s chosen according as th e linkage is righthanded or lefthanded. In this situatio n, th e number n may be related to geometric properties ofth e ribbon whose boundaries are th e axis C (itself a vortex line) and any o thervortex line in the tube. Let N(s) be a unit transverse vector on this ribbon,where s is arclength on C. The n, following Fuller (1971) we may define the twistof the ribbon by . .where N = dN/ds. We m ay also define th e writhe W T of C by
d x Ad x ) . ( x  x )wr=ff n c c lx  13
Then, as shown by CQ ugk ean u (1961) and White (1969),n = W r + T w , (2.7)

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468 H.K. offattso th at th e sum of writhe and twist is a topological invariant of the ribbon. Thetwist may be fu rthe r decomposed in the form
T w = 7 + N o , (2.8)where7 = / r ( s ) d s , (2.9)
~ ( s )s the torsion on C, and NO s the number of rotations of N(s) relative t o th eFrenet tr iad o n C in one passage round C. It is interesting to note here th at if Cis deformed through an inflexion point, then both 7 and NO ar e discontinuous(by f l and r f l respectively) but 7 NO s continuous. The selflinking numberSL ntroduced by Pohl (1968) (half the algebraic sum of the crosstangents onC) may be identified with Wr+7 , an d is likewise discontinuous in d eform ationof C thro ugh an y inflexional configuration.3 Relaxation With Conservation of TopologyRegarding (1.2) as a dynamical system in the phasespace of solenoidal velocityfields, attention first focuses on the fixed points of this dynamical system, i.e.on the steady Euler flows satisfying
U A W = V ~ (3.1)These flows ar e characterised by th e property th at their energy
E = f / u 2 d Vis sta tio na ry under isovortical displacements, i.e. perturbation unde r whichthe vorticity field is frozen. Moreover, if E is extremal (either maximum orminimum) under such perturbations, then the flow is stable (Arnold 1966). Itmakes sense therefore to att em pt t o construct relaxation processes which drivethe system towards a stat e of statio nary energy, and then to consider the na tur eof the s tatio nar y point, whether saddle or extremum. One such process has beenproposed by Moffatt (1985), following an earlier suggestion of Arnold (1974).This first involves consideration of the relaxation of a magnetic field B in aperfectly conducting , but viscous, fluid, a process governed by the equationsB = V A ( V A B )
at6% 1 1 v .Vv =   V p + j A B + v V 2 vat P P (3.3)(3.4)where v is the velocity field (the change of notation here is deliberate), andj = V AB is th e current distribution (t he usual factor p i 1 is here irrelevantand is om itted ). Viscosity is included in order to provide a means of dissipatingenergy which now has both kinetic ( K ) nd magnetic (M ) contributions. Eqn .

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Topological Dynamics of Fluids 469(3.3) however guarantees that the topology of the field B s conserved. If thistopology is nontrivial, the n i ~ 9 hown by Freedman (1988), the m agnetic energy
M = A B 2 d V (3.5)is bounded away from zero. Here, V now represents the whole fluid domain,and we may assume tha t M is finite. Since K +M is monotonic decreasing andbounded below, it mu st tend to a constant. Th us in th e absence of any bound aryforcing and assuming nonpathological behaviour, the velocity v must tend tozero everywhere, since otherwise viscous dissipation would continue to reduceK +M. In this limiting situation, (3.4) reduces to
j A B = V p , (3.5)i.e. we reach a state of ma gne tosta tic equilibrium in which th e field B(x) has,with one qualification, the same topology as t ha t of the initial field for therelaxation problem. T h e qualification is tha t tang enti al discontinuities of B mayappear during the process of relaxation (and consideration of the relaxationof fields with linked flux tubes indicates that such discontinuities do appeargenerically  Moffatt 1985).
Note now the structural similarity of (3.1) and (3.5), which are indeed isomorphic under the identifications
u w B , w t j , h c t p o  p (3.6)for arbitrary constant PO.Th is means th at t o every magnetostatic equilibriumobtained by th e process of magn etic relaxation, th ere corresponds a steady Eulerflow obtained by replacing th e m agnetostatic B(x) by u(x).Tangential discontinuities of B become vor tex sheets in this analogue Eul er flow; and the topologyof U is the same as the topology of B, which may be arbitrarily prescribed viathe initial condition for th e relaxation problem.
This procedure for obtaining solutions of the steady Euler equations is oneof considerable subtlety, involving as it does an interplay between two distinctanalogies  irst that between frozen fields B(x, ) and w ( x , ) n unsteady evolution, and second, th at between B(x) and u(x) in equilibrium states. The latteranalogy does not extend to unsteady evolution, an d so th e stability criteria forma gne tost atic equilibria and for steady Euler flows ar e quite distinct; the firstinvolves consideration of isomagnetic disturbances, while the second involvesconsideration of isovortical disturbances (Moffatt 1986,1990a). A magnetostaticequilibrium obtained by the process of magnetic relaxation is a minimum energy st ate with respect t o isomagnetic p erturbation s, and is therefore stable inan ideal fluid. The analogous Euler flow may however be unstable, and indeedis presumably unstable (to KelvinHelmholtz instabilities) if vortex sheets arepresent.

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470 H.K. Moffatt4 TwoDimensional RelaxationThe process described above has been implemented numerically (with a simplified dissipation mechanism) by Bajer (1989) and by Linardatos (1993)for twodimensional configurations. If th e initial field B(x,0) has any saddle points, the nit is found tha t in general th e Xty pe struc ture near a saddle point collapses withformation of a curren t sheet of finite extent as t t 00. Th e technique thu s leadsto numerical determination of steady Euler flows with imbedded vortex sheetsof finite extent  which may be expected to be unstable for th e reason indicatedabove.
If th e initial field B(x,0) has only one neutral point of Otype, an d no saddlepoints, t he n t he field lines have a natura l tendency to relax t o circular form, butthis tendency is impeded by th e boundary constraint Bn= 0 if th e bou nda ry isnoncircular. The asymptotic equilibrium represents a compromise between thesetwo conflicting demands. T he field lines may be represented by a family of curvesx = cst. which move with t he fluid during th e relaxation process. T h e ar ea A($inside each such cu rve is constan t (t he flow being incompressible). For this simplefield structure, the field topology is totally described by this invariant functionA ( x ) , he signature of th e field (Moffatt 1990a); under t he assumed conditions,A ( x ) s monotonic.
A result of great interest has recently been proved by Davidson (1994): this isth at Euler flows obtained v ia twodimensional magnetic relaxation ar e stabl e (t otwodimensional d isturban ces) provided t he initial field has t he simplest topology as described in th e preceding paragraph. Thus for arb itrar y twodimensionalboundary and arbitrary (monotonic) signature function A ( x ) ,we have a meansof obtaining a corresponding sta bl e Euler flow. Th e inference is th at the instabilities to w hich more general Euler flows may be sub ject a re necessarily localisedin the neighbourhood of vortex sheets  but this requires further investigation.Davidsons result applies equally to axisymmetric configurations in which
th e fields (magnetic or velocity) have only meridional components  such fieldsyielding st ab le Euler flows. Th e argument fails however if zonal field comp onen tsare included: axisymmetric Euler flows with swirl obtained via the magneticrelaxation technique may be unstable.
5 The Minimum Energy of Knotted Flux TubesThe magnetic relaxation technique may be applied to a magnetic field confinedto a single flux tube T knotted in the form of a knot K . Let 9 e the flux ofB in t he tube , conserved under frozenfield evolution. If t he twist h of th e fieldlines in 7 is uniform, then the magnetic helicity (also conserved) is given by

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Topological Dynam ics of Fluids 471If h is an integer n, then this is the linking number of any pair of Blines in 7as discussed in $2 above. However h need not be an integer (the Blines beingthen ergodic on nested toro idal surfaces within I) .
If the f lux tub e is allowed to relax to a minimum energy state in an incompressible med ium, the n i ts volume V is conserved (like @ and h).On dimensionalgrounds, the minimum energy Mmin s proportional to G 2 V  $ ; hus (Moffatt1990b)where m(h) s a dimensionless function of the dimensionless parameter h, thisfunction being d eterm ined solely by the topology of th e knot K.
Mmin =m ( h ) ~ ~ ~  + (5.2)
Th e function m(h) (described by Moffatt 1990b as th e groundstate energyof the framed kn ot) m ay b e determined, in principle, by numerical implemen tation of the m agnetic relaxation procedure. Th e need t o ensure strict conservationof field topology however makes for practical difficulties that have not yet beenovercome. A variational technique has recently been developed (Chui & Moffatt1995) using a local nonorthogonal coordinate system attac hed to the f lux tub e,to ob tain the minimum energy function R ( h ) nder certa in additional mild constraints , this providing a n u pper bound o n, and a reasonable approximation to,m(h)when h is not too large. Th e function B ( h ) has been computed for various toru s kno ts; for each kn ot considered it is a convex curve with minimum ath = h, # 0; and the evidence available so far indicates that when h = 0(1),B ( h ) ncreases with increasing knot complexity, as is t o be expected.
T he concept of a relaxed minimum energy sta te for a knotted tube or r ibbonis appealing in many physical and biological contexts; the procedure outlinedabove provides a viable approach for the actual determination of the minimumenergy and the corresponding geometrical configuration. The concept may alsobe extended in a natu ral way to l inks.
6 Relaxation of Chaotic FieldsAn arbitrary solenoidal vector field B defined within a finite domain 2) a n dwith B .n = 0 on dD in general exhibits the phenomenon of chaotic field linewandering, i.e. the Blines do not lie on surfaces nested within D, ut a re morenearly spacefilling in c ha rac ter. Poinc arQ sections of such a field exhibit th efamiliar sca tter of points (for a single Bline) ch aracteristic of chaotic behav iour.The components of B need not be particularly complicated functions of thecoordinates (z,y,z) to give this sort of behaviour; thus, for example, if eachcomponent is a quadratic function of (z,, z ) , then the field lines in generalexhibit chaos (Ba jer & Mo ffatt 199 0), althoug h frequently islands of reg ularityare ap pare nt within t he chaotic sea.
If such a field is allowed t o relax, its chao tic topology being conserved, th enas we have seen the asy m ptotic field satisfies the magn etostatic equation (3.5)

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472 H.K. Moffattwith the consequence B .V p = 0 . This means that the Blines lie on surfacesp = cst.; but by definition within any chaotic region in V , he Blines do notlie on surfaces. The only escape from this apparent contradiction is to concludethat V p = 0 within any such region, i.e. from (3.5), = a(x)B for some scalarfield a ( x ) . Now since V . j= 0 and V * B = 0 we have immediately B  Vcu = 0 ,so that by repetition of the argument, V a = 0 also in the chaotic region. Thusin this region we have
V A B = C Y B (6.1)i.e. B is a Beltrami field with constant a. And yet in this same region, B ischaotic, It is a completely remarkable, and as yet illunderstood, phenomenon,that such a simple equation as (6.1) can apparently describe fields having anarbitrary degree of chaos. The islands of regularity may hold the key to thispuzzling conclusion, a conjectural note on which it may be appropriate to terminate this lecture.
References1.
2.3.4.5.6 .7.
8 .9.
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Arnoldd, V.I. 1966 Sur un principe variationel pour les Bcoulements stationnairesdes liquides parfaits et ses applications aux problkmes de stabilitk nonlinCaires.Arnold, V.I. 1974 The asymptotic Hopf invariant and its applications [Englishtranslation: Sel. Math. Sov. 5 , 327345 (1986)l.Bajer, K. 1989 Flow kinematics and magnetic equilibria. PhD thesis. CambridgeUniversity.Bajer, K. & Moffatt, H.K. 1990 On a class of steady confined Stokes flows withchaotic streamlines. J. Fluid Mech. 212, 37363.Cdugtireanu, G. 961 Sur les classes disotopie des noeuds tridimensionnels etleurs invariants. Czechoslovak Math. J. 11, 88625.Chui, A. & Moffatt, H.K. 1995 Energy and helicity of knotted magnetic fluxtubes. Preprint.Davidson, P. 1994 Global stability of two dimensional and axisymmetric Eulerflows. J . Fluid Mech. 246, 73305.Freedman, M.H. 1988 A note on topology and magnetic energy in incompressibleperfectly conducting fluids. J. Fluid Mech. 194, 49551.Fuller, F.B. 97 1 The writhing number of a space curve. Proc. Natl . Acad. Sci.Helmholtz, H. von 1858 Crelles Journal 55 . [English translation: Phil. Mag.Kelvin, Lord (then W. Thomson) 1869 On vortex motion. R a m . Roy. Soc. Edin,
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12. Linardatos, D. 1993 Determination of twodimensional magnetostatic equilibriaand analogous Euler flows. J. Fluid Mech. 246, 569591.
13. Moffatt, H.K. 1969 The degree of knottedness of tangled vortex tines. J . FluidMech. 35, 117129.14. Moffatt, H.K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology Part 1. Fundamentals. J. Fluid Mech. 159, 59378.

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Topological Dynamics of Fluids 47315. Moffatt, H.K. 1986 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations. J. Fluid Mech. 166,16. Moffatt, H.K. 1990a Structur e and s tabil i ty of solutions of the Euler equations:17. Moffatt, H.K. 1990b Th e energy spectrum of knots an d links. Nature Lond. 347,18. Moffatt, H.K. & Ricca, R. 1991 Helicity and th e CdugB reanu invariant. PTOC.Roy. Soc. Lond. A 439, 11429.19. Moreau, J.J. 1961 Co nsta nts dun ilot tourbillonn aire en fluide parfait baro trope.C . T .hebd. Seanc. Acad. Sci. Paris 2 5 2 , 28102812.20. Pohl,W.F. 1968 T h e selflinking nu mb er of a closed space curve. J. Math. Mech.21. White , J .H. 1969 Selflinking an d t h e G auss integral i n higher dimensions. A m .
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