H.K. Moffatt- Structure and stability of solutions of the Euler equations : a lagrangian approach

8/3/2019 H.K. Moffatt- Structure and stability of solutions of the Euler equations : a lagrangian approach

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Structure and stability of solutions of the Eulerequations : a lagrangian approachB Y H. K. M O F F A T T

Department of Ap plied Mathem atics and Theoretical Ph ysi cs, Xilver Street,Cambridge CB 3 9 E W , U.K.

This paper reviews methods that are essentially lagrangian in character fordetermination of solutions of the Euler equations having prescribed topologicalcharacteristics. These methods depend in the first instance on the existence oflagrangian invariants for convected scalar and vector fields. Among these, thehelicity invariant for a convected or frozen-in vector field has particularsignificance. These invariants, and the associated topological interpretation arediscussed in $91 and 2 . In $ 3 the method of magnetic relaxation to magnetostaticequilibria of prescribed topology is briefly described. This provides a powerfulmethod for determining steady Euler flows through the well-known exact analogybetween Euler flows and magnetostatic equilibria. Stability considerations relatingto magnetostat ic equilibria obtained in this way and to the analogous Euler flows arereviewed in $4. In $ 5 the related relaxation procedure is discussed; for two-dimensional and axisymmetric situations this technique provides stable solutions ofthe Euler equations for which the vorticity field has prescribed topology. Theconcept of flow signature is described in $6: this is the relevant topologicalcharacteristic for two-dimensional or axisymmetric situations, which is conservedduring frozen-field relaxation processes. In $47 and 8, the formation of tangentialdiscontinuities as a normal par t of the relaxation process when saddle points of thefrozen-field are present is discussed. Section 9 considers briefly the application ofthese ideas to the theory of vortons, i.e. rotational disturbances that propagatewithout change of structure in an unbounded fluid. The paper concludes with a briefdiscussion, with comment on the possible development of the results in the contextof turbulence.

1. Lagrangian invariants for convected scalar and vector fieldsA lagrangian approach to problems of fluid mechanics naturally forces us to focus onmaterial domains , i.e. on curves, surfaces or volumes (or more generally any set ofmarked fluid particles) tha t move with the fluid. Of particular interest arequantities which, when integrated over a material domain, are constants of themotion. Such integrals may be described as lagrangian invariants of the flow.We consider a velocity field u ( x , t ) n some fluid domain D and i ts associated densityfield p(x , ) ( 0) satisfying the equation of mass conservation

D ~ / D ~a p / a t + u . v p = - p v . u . (1.1)Let 8(x, ) be a scalar field (e.g. dye concentration),which we assume to be passively

32 1Phil Trans. R. Soc. Lond. A (1990) 3 3 3 , 321-342Pranted an Great Bratain

www.moffatt.tc


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322 H . K . J l o f f d tconvected by the flow, without any molecular diffusion. The flux of 8 is u8, o that8 satisfies the same conservation equation as p , namely

D e / m = a e / a t + u w = - m u .D/Dt(O/p) = - ( 8 / p ) V . ~ + ( b / p ~ ) p V ~ ~0.

( 1 . 2 )It follows from (1.1)and ( 1 . 2 ) that

(1.3)Kow let S be any closed material surface imbedded within D, containing a (material)volume V , and consider, for any function F , the lagrangian integral

(1.4)Obviously.

B (O / p )D/Dt(O/p) p dV = 0.x=s,Hence, for arbitrary F, I , is a lagrangian invariant of the flow.The topological structure of the field 8 / p is associated with the structure of thefamily of surfaces 8 / p = const., each of which moves with the fluid. The volumeintegral (1.4)yields a family of surface integrals through the choice

F ( O / P ) = CY8/P-- ( O / P ) C ) > (1.6)where ( B / p ) , labels one of these material surfaces, 8 , say. Let n be the unit normalon 8 , then writing dV = dndX = d(8/p)dS/(n.V) O/p ) , 1 . 7 )the integral (1.4) converts to the surface integral

1.8)and this also (for each 8,) is a lagrangian invariant.invariants (1.4)and (1.8)take the simpler formWhen the flow is incompressible, and of uniform density, then (with p = 1) the

In this case, the gradient G = V 8 satisfies the equation (Batchelor 1952)D G , / D ~= - ~ ~ a u , / a x , , (1.10)

and i t is interesting to note (from 1 . 9 ) hat this equation has the lagrangian invariantIc = l s c ( n * ) - l dS. (1.11)

Suppose now that B ( x . ) is a solenoidal vector field ( V .B = 0) which is convectedby the flow u ( x , ) with conservation of the flux of B through every closed materialcurve C. This is the fundamental property of a magnetic field in a perfectlyPhi l . Trans R. oc . Lond. A (1990)


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The Euler equations : a lagrangian approach 323conducting fluid medium, and we shall use the language of magnetohydrodynamics,although the results are more generally applicable. We shall suppose that B has a(single-valued) vector potential A satisfying

B = V A A , V . A = 0. (1.12)The equations satisfied by A and B are

(1.13)where y is a scalar potential field, which we assume also to be single-valued. Theequivalent lagrangian form for these equations is

and

(1.14)

(1.15)For any unknotted closed material curve C spanned by an orientable surface C wehave

I , = A - d x = B . n d S = const.k J, (1.16)This is the frozen flux theorem of Alfven (1942); for an arbitrary material closedcurve C, I , is a lagrangian invariant.

The solution of (1.15)may be expressed in terms of the lagrangian particle pathx ( a , ) which s tarts from position a a t time t = 0. Writing B = B/p, this solution (ina form anticipated by Cauchy) is

B , ( x , ) = i j l j(a, ) c ? ~ , / c ? ~ ~ , (1.17)a form tha t makes evident the rota tion and stretching of the field by the deformationtensor ax,/aaj , during convection from (a , 0) to (x, t ) . The equivalent solution of(1.14) is

for some scalar field x, constrained by our adoption of the gauge conditionV ' A = 0. Note the appearance of the inverse deformation tensor c?al/axiin (1.18),with the consequence that

A , ( x , ) = A , ( a , O)~a4/ax,-~x/ax, (1.18)

A ( x , ) * B ( x , ) = A ( a , O ) . B ( a ,O ) - B ( x , t ) . V x (1.19)a result obtained by Elsasser (1946), although with a different, and rather special,choice of gauge for which x = 0.If we now multiply (1.19) by p , and integrate over a material domain V using

J B . V x p d V = J BSVxdV =V V

we obtain

Phil. Trans. R. Soc. Lond. A (1990)

(1.20)

(1.21)


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324 H . K . Xof fa t tI n particular, if S is a magnetic surface on which n *B = 0 (a condition t ha t persistssince B is a convected field) then

H - A * B d V= const.- 1 (1.22)

This type of invariant was first obtained by Woltjer (1958).H , is the helicity of themagnetic field within the material volume V (Moffatt 1969) and it is clearly alagrangian invariant.

The lines of force of B(x. ) (or B-lines) t any instant t are given by solution ofthe differential system

dx/B, = dy/B, = dz/B,. (1.23)I n general, this system is non-integrable, and the B-lines then wander chaoticallythroughout the fluid domain ; exceptionally, however, the B-lines may be closedcurves or may lie on a family of closed surfaces. The following examples illustratethese possibilities : in each case the domain of definition of B is the sphere r < 1, and( r , 0 , v ) are spherical polar coordinates.

(1.24)For this simple field, the B-lines are circles about th e polar axis ; the topology of thefield is tr ivial n t he sense tha t every B-line is an unknotted closed curve, and anytwo B-lines are unlinked ; the helicity is obviously zero.

Example 2(1.25)

where the function X ( r , 0 ) is described as the flux function of the meridional (orpoloidal)part of the field. In this case, B-lines lie on tori X ( r , 0) = const. and ingeneral cover these tori, although exceptionally they may be closed curves (torusknots) . The field is contained in the sphere provided ~ ( 1 ,) = 0, and the helicity isgiven by

(1 .26 )Example 3

I n cartesian coordinates.B = ( E X - ~ X Z J , ~ ~ x ~ + ~ z J ~ + x ~ + x z J - ~ ,E X + ~ Z J Z - X Z J ) . ( 1 . 2 7 )

where a(# 0) is constant. This is an example of a non-integrable field, whoseproperties have been studied by Bajer & Moffatt (1990) and Bajer et al. (1990).Figure1a hows a portion of a single B-line (which is confined to the sphere r < 1)and figure1b shows a section of this streamline by the diametral plane x = z , i.e. a PoincarBsection ) . About 8000 successive points of section for this single B-line are shown, andthe chaotic wandering is quite evident, although some structure within the chaos isPhd Trans R Soc Lond A (1990)


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The Euler equations : U. agrangian approach 325

Figu re 1 . (a) single c haotic B -line of the field (1 .26 ) . b ) Poincark section of this B-line(see Baje r et al. 1990).

Figure 2 . Possible configurations of linked flux t ubes . (a) imple l inkage that gives non-zerohelici ty. ( b )Son-tr ivial l inkage with zero l inking number and therefore zero hel ici ty.evident also. The topology is clearly nontr ivial; and the helicity in r < 1 , may becalculated to be H =- 6 ~ ~ / 3 5 . ( i 28)

For a chaotic field of this kind. there is but a single helicity invariant of the form(1 .22) or each subdomain V within which a B-line is space-filling. If, however, the B-lines lie on a family of surfaces 8 = const., then, as for the invariant ( 1 , 4 ) ,we mayconstruct associated surface invariants

(1 .29)There is a further generalization of the helicity invariant ( 1 . 2 2 ) ,which may be

constructed as follows (Moffatt 1 9 8 1 ) : let B, = V A A , and B, = V A A , be twoindependent solenoidal vector fields convected by a flow u ( x . ) , and let X be a closedmaterial surface, on which n .B, = 0, n.B, = 0, (1 .30)and containing the material volume V . Then

Bl* A2d V= B2* Ald V= onst.JThis is the cross-helicity between the fields B, and B,.Phil Trans R Soe Lond. ,4 (1990)

17

(1 .31)

VOI 333 A


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326 H. K . Xof fu t t2. Welicity invariants and topological structure

The relation between the helicity invariant (1.22)and the topological structure ofthe field B is made transparently clear (Moreau 1961; Moffatt 1969) throughconsideration of the particular situation in which B is identically zero except in twoflux tubes of small cross-section whose axes are the unknotted closed curves C,. C,which may be linked (figure 2a) .Obviously, the linked configuration is topologicallydistinct from the unlinked configuration. and the degree of linkage is conservedunder frozen-field distortion. We suppose that the field lines have no net twist withineither flux tube; this means th at either tube may be continuously deformed to acircular tube (like a bicycle tube) within which the B-lines are all circular andunlinked; i.e. the topology of the field within either flux tube in isolation is trivial.I n the limit as the tube cross-sections tend to zero, the volume integral (1.22) withVthe whole space) degenerates to th e sum of two line integrals round C, and C, viathe substitutions Bd V- t @, dx on C,, @, dx on C,, where @, and @, are the magneticfluxes in the tubes ; and since for example

A * d x= B a d ~ ,kl s,,where 8 , spans C,, and t he l atter integral equals &@, it follows that

H , = *2n@,@,. (2.1)where n is the relative winding number of C, and C,, and the + or - depends on thefield directions in the two tubes. Helicity therefore clearly has a topologicalcharacter, and the invariance of helicity is a consequence of the topologicalinvariance intrinsic to frozen-field distortion.The argument in this simple form is of course restr icted t o fields having closed B-lines. The argument has however been adapted to more complex topologies byArnold (1974).who has shown how t o give meaning to the double limit

(2.2)

when the curve C, winds infinitely around the (still closed) curve C,. Arnolddescribes this limit as the asymptotic Hopf invariant , the relation betweenintegrals of the form (1.22)and the Hopf (1931) invariant of topological mappingshaving been established earlier by Whitehead (1948).Relicity thus plays a fundamental role in the topological classification ofsolenoidal vector fields. However. the complete set of helicity invar iants is not nearlysufficient t o provide a complete classification. This may be easily seen by consideringthe case when C, and C, have the more complex linkage shown in figure 2 b. Here thetoroidal flux across the surface 8, spanning C , is zero, and so the helicity integral iszero as for the case when C , and C , are unlinked. And yet the two cases are clearlytopologically distinct. The topology of figure 1 b is invariant under evolutiongoverned by (1.13 ) , and so any topological invariant that distinguishes thisconfiguration from the unlinked configuration must be somehow contained in(1.13 ) . How can we construct such an invar iant !We develop one possible approachin the following sections.Phal Trans R Soe Lond A (1990)


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The Euler equations : lagrangian approach 3273. Magnetic relaxation

The style of argument presented in this section was suggested by Arnol'd (1974)and developed by Moffatt (1985),and is an essential preliminary to the considerationof steady Euler flows.We consider a fluid that is incompressible ( V . u= 0 ) and contained in a domain Dwith boundary i?D on which u . n = 0 . We consider again a magnetic field B ( x , )satisfying

a condition that persists under evolution governed by the frozen-field equation(1.13b).We now focus attention on the energy of this field given byB * n = O on aD, (3.1)

M ( t )= - B 2dli,YDwhich by elementary manipulations satisfies the equation

B .V A ( u A B )dli =- v ( j B )dli,s(3.2)

(3.3)where j = V A B , the current associated with B . (Note that a current sheetj,= - n A BI,, may flow on the boundary.)Let us adopt the simplest possible equation of motion for the fluid which (i)incorporates the Lorentz force j A B per unit volume appearing naturally in (3.3) (ii)allows u ( x , ) to remain solenoidal for all t : and (iii) includes a term tha t dissipatesenergy. This is (3.4)/at = - p+j B - kvwhere k > 0 and p ( x . ) is a (pressure) field satisfying

V 2 p = V * ( j A B ) in D ,a p / a n = n . ( j A B ) on aD .Let K(t ) be the kinetic energy of the flow:

K ( t )= - u'd?'.:IDFrom (3.4), his satisfies %=t su.(jAB)d?'--2kK,so that , from (3.3)and (3.7),d/d t ( i l / l ( t )+K(t))= -2kK.

(3.5)

Hence, for so long as K(t)> 0, the total energy M + K is monotonic decreasing, andbeing positive must tend to a limit. Hence. no matter what the init ial conditions maybe.(3.9)

Suppose now that these initial conditions areB ( x ,0 ) = B , ( x ) , U ( X , 0 ) = 0 , (3.10)

where B , ( x ) is an arbitrary solenoidal field of finite energy. In general, the initialLorentz force (V A B , ) A B , is not irrotational and so cannot be compensated by thePhil. Trans . R. Soc. Lond. 4 (1990)


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328 H . K.Noj?fattpressure term in (3.4).The fluid must therefore move, and as it does so. it Ponvectsthe field B as- a frozen-in field. Thus the kinetic energy K( t ) nitially increases fromzero (at the expense of magnetic energy) but ul timately tends t o zero again accordingto (3.9).For all finite t . the flow v ( x . ) remains smooth. and induces a continuousvolume-preserving mapping x --f X ( x . ) associated with the particle paths. For allfinite t . the field B evolves through topologically equivalent states, but as t --f CO , themapping may develop discontinuities and equally the convected field may developsingularities, although the field energy is strictly under control and bounded fromabove by its initial value.For any non-trivial field topology. the magnetic energy is also bounded below andaway from zero (Arnold 1974; Freedman 1988). In order to understand this.consider a portion of a flux tube of length L and cross-section A carrying uniformfield B : the flux 0 =BA and the volume I=LA are conserved under frozen-fielddistortion of the kind considered. The contribution to magnetic energy from thisportion of tube is 3 J = +V1Qj2L2.ence the energy decreases th rough reduction ofL , i.e. contraction of the tube. together with corresponding increase of its cross-section. The energy can tend to zero only if this process is carried to the limit in whichevery closed B-line contracts to a point. as is possible (figure 3a) if the topology istrivial. It cannot happen however if the topology is nontrivial (figure 3b); for thenthe contraction of a flux tube is inevitably impeded by the growth of cross-sectionof any other flux tube with which it is linked. This argument is given expression inthe formal language of topology by Freedman (1988).It is clear from this description of the process of magnetic relaxation that. ast --t CO. tangential discontinuities of B must form wherever linked flux tubes areultimately brought into contact. and t ha t the asymptotic equilibrium state B()(x)characterized by the magnetic energy LW(E)(0 ) will generally contain tangentialdiscontinuities (cur rent sheets) imbedded within the domain 1).This appears to bethe case even if the initial field B o ( x ) s C . because the rearrangement of B-linesduring relaxation will still generally tend to produce tangential discontinuities. Anexample may make this clear: let D be the cylinder s < a (in cylindrical polarcoordinates (s,97, ) ) and let B o ( x )= (0.B,,(s). Boz(s)) .here Bo,(s)an d Bo,(s) re Cfunctions of non-overlapping bounded suppor ts, as indicated in figure 4. Relaxationcan proceed through rearrangement of the circular B-lines in the region so< s < a.the stronger field lines from the outer region near 5 = a displacing the weaker fieldlines in the inner region near s = so . Minimum energy is achieved by a field of theform

B E @ ) = (0. B E ( s ) .Bo,(s)) (3.11)where s - l B E ( s ) s the rearrangement of s ~ ~ R , , ( s )ha t makes(3.12)

In this state, there is clearly a tangential discontinuity of BE of magnitudesomax IB,,(s)/sl across s = so .Xote tha t the rearrangement is achieved by a flow ofthe form u ( x . ) = (v,(s, . t ) , 0. U,(& 2 . t ) ) ( t > 0). (3.13)but the asymptotic state is z-independent.

In general, therefore, we are driven to the conclusion that B relaxes to amagnetostatic equilibrium sta te BE(x ) hich is topologically accessible from B o ( x )nPhd TTCk72S R SOCLand A (1990)


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The Euler equations : lagrangian appyoach 329

Figure 3 . The process of magnetic relaxation. (a )Trivial topology, for which magnetic energycan decrease to zero. ( b ) Xon-trivial topology providing an impediment to relaxation.

Figure 4. Relaxation of a cylindrically symmetric field t o a state of minimum magnetic energy.the sense th at it is obtained from B,,(x) through convection and distortion by a flowu ( x , ) (0 d < a) hich dissipates a finite fraction of the initial field energy. B E ( x )may have tangential discontinuities, or possibly more awkward singularities (e.g.accumulation points of tangential discontinuities) ; it would be nice to characterisethe properties of the function space in which the relaxed fields B E ( x ) eside; this isas yet an unsolved problem.Being a magnetostatic equilibrium (with U = 0 ) . B E ( x ) atisfies the equation

j E ABE= V / \ B E ) ~ B E = V p E (3.14)for some scalar fieldp E ( x ) . he field is characterised by it s magnetic energylWE whichis clearly minimal with respect t o small frozen-field per turbat ions of the medium (fora full discussion see Moffatt 1986a).There may be more than one such equilibriumtopologically accessible from an initial field B , ( x ) by different routes in functionspace (by adopting a non-zero initial condition for u ( x , ) . or by using a differentdissipative mechanism in (3.4)or by varying the relaxation process in some otherway). Among these equilibria, however. there is always one whose energy H E (> 0)is least (or more than one with equal least energy): such a s tate (or states) are themost stable (or 'ground') states available to the field with it s prescribed topology.These considerations have interesting implications for the theory of topologicalinvar iants of knots and links in R3.Suppose for example tha t we have a n arbi traryknot K , and let Tc(K)be a tubular neighbourhood of the knot of cross-sectionA =m2. et B,,(x) be a field of magnitude B, within Fe(K) ligned along the axis ofthe tube. so th at the magnetic flux is d j =BO A. he volume of Tc(K)s LA where L isthe length of the knot. We now let this field relax a s already described. During thisprocess d j and V are invariant. and so also is the helicity of the field, which. fordimensional reasons, must be of the formH = hdj' (3.15)Phzl T r a m R Soc Lond 4 1990)


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330 H . K . Xof fat twhere h is a dimensionless number (which may be positive or negative or zero). Asshown by Berger & Field (1984), with a particular convention for the field twistwithin the tube, h = N+-A- where A T k are the numbers of positive and negativecrossovers in a plane projection of the knot. Even more interestingly, the field relaxesto a minimum energy state compatible with the knot topology, and the energy M Ein this s tate, again for dimensional reasons, must have the formJp= mcpv- i , (3.16)where m(> 0 ) is a real number determined solely by the original knot topology. Thenumber m is a topological invariant, and generally (excluding mirror symmetries)two topologically distinct knots will yield distinct values of m. Of course if there ismore th an one minimum energy state for a given knot, then we obtain a sequence ofnumbers (m,, m,, m2, ..) with 0 < m, < m, < m2 ... characterizing these states.Similar considerations apply to links. We may then talk unambiguously of th eenergy spectrum of knots and links, a point of view developed in more detail byMoffatt (1990b).Parallel developments on the purely topological side are given inFreedman & He (1990a)b).

4. Stability of magnetostatic equilibria and of analogous Euler flowsThe techniques described in $ 3 provide a means by which magnetostatic

equilibrium states of prescribed magnetic field topology may (at least in principle) beconstructed. It is well-known that such states are characterized by a magnetic energythat is stationary with respect to small frozen-field displacements. and t ha t stabilityis assured if the magnet ic energy is in fact minimal with respect to such displacements(Bernstein et al. 1968). The magnetic relaxation technique will generally yield suchstable states, since the magnetic energy may be expected in general to decrease to aminimum rather than a saddle point within the subspace of fields that aretopologically accessible from the init ial field.

Let c(x)be a small volume-preserving vi rtual displacement of the medium. Then,as described in Moffatt ( 1 9 8 6 ~ ~ ) )he first and second order varia tions of B about theequilibirum state BEare given by

dlB = V A ( 6 A BE). d 2 B= 4V A (5 dB1). (4.1)and the corresponding variat ions of magnetic energy are

(4.2)

(4.3)It is easy to show that S1iM = 0 when BE satisfies the magnetostatic equilibriumconditions. and stability is then guaranteed provided

62144> 0 (4.4)for all admissibledisplacement fields


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The Euler equmtions : lagrangian approach 331The magnetostatic equilibria considered above have a double interest, since toeach such equilibrium. there corresponds (by exact analogy) a solution of the s teadyEuler equations of classical (inviscid) fluid mechanics. Thus, by the subst itut ions

B E + u E . j E + w E , p E + - - h E , (4.6)

j E / \ B E= V p E J -E - V A B E V * B E = O (4.6)

U E A w E = V h E , w E = V A u E , V * u E = O (4 .7)

the magnetostatic equations

are replaced by the steady Euler equations

for an incompressible inviscid fluid. Here, hE is the (Bernoulli) total 'head' in theanalogue fluid. The minus sign in the analogue relation p E+- Eis to be particularlynoted ; physically, this is related to the fact that the Lorentz force associated with acurved magnetic flux tube acts towards the centre of curvature. whereas thecentrifugal force in the analogous curved stream tube acts away from the centre ofcurvature. This change of sign is immaterial as far as the structure of equilibriumstates is concerned ; but i t is of critical importance in relation to the stability of thesestates.The stability problem for the analogous Euler flow u E ( x )s different from th at forthe magnetostat ic field B E ( x ) ecause, under the unsteady Euler equations

h / a t = U A m - V h , V * U 0. (4.8)it is the vorticity field, rather than the velocity field, that has a (frozen-in' character.Thus. under virtual displacements


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332 H . K. Jfof fat tfor all volume-preserving displacements 5 . Thus the kinetic energy of an Euler flowthat is the analogue of a stable magnetostatic equilibrium may be maximal orminimal or neither (i.e. a saddle) with respect t o virtual displacements, the vorticityfield being frozen in the fluid.

According to an argument of Arnol'd (1966) the Euler flow is stable if the kineticenergy is maximal or minimal, i.e. ifS2K >, 0 for all admissible 5 , (4 .16 )or P K < 0 for all admissible 5 . (4 .17 )The reason is that the perturbed flow u ( x . t ) evolves under the kinetic energyconstraint K = const. If K is maximal when U = u"(x), hen the perturbed flowremains permanently in a neighbourhood of u " ( x ) (a t least with respect to an energynorm) : similarly if K is minimal. However, if K is neither maximal or minimal, the'surfaces' K = const. have hyperbolic structure near the equilibrium point u " ( x ) inthe relevant function space, and the condition K = const. therefore places noconstraint on the magnitude of the perturbation u ( x , t )- " ( x ) ; hence if S2K isindefinite in sign, the flow u " ( x ) may (and presumably will) be unstable.

5. The relaxation procedure of Vallis e t al. (1989)The magnetic relaxation procedure described in $ 3 is 'natural' to the problem of

locating stable magnetostatic equilibria because it respects the frozen-in character ofthe magnetic field. which is the essential feature of the stability problem. It is notnatural to the problem of locating stable Euler flows. because magnetic relaxationoccurs in a subspace tha t does not span the space of perturbations governed by theunsteady Euler equations in which the vorticity field is frozen-in.It is important therefore to enquire whether there are alternative relaxationprocedures that are natural to the Euler equations in the sense that relaxation toequilibrium occurs in the subspace in which perturbations most naturally evolve, i.e.the subspace of flows u ) ( x , ) for which the vorticity field u)(x, t ) is topologicallyaccessible from some initial reference field u),,(x).One such procedure, which we shalldescribe as 'VCY relaxation'. has been devised by Vallis e t a l . ( 1 9 8 9 ) .Suppose th atthe vorticity U) = V A U is artificially constrained to evolve under the frozen-fieldequationwhere

h O / a t = v A ( U A U)),U = U+ a a u p t

( 5 . 1 )(5 .2 )and a is a constant . Under evolution determined by ( 6 . 1 ) , the topology of thevorticity field is certainly conserved ; in particular, the helicity

A?= u . u ) ~ Vs,, ( 6 . 3 )is invariant , being still a measure of the 'degree of knottedness ' of the vorticity field.The kinetic energy K of the flow is not, however, conserved when -a # 0 ; in factelementary manipulations yield

u2dV = --a (c?u/c?t)zlr.sPhil. Trans . R. Soc. Lond. A (1990) ( 5 . 4 )


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The Euler equations : a lagrangian approach 333Hence, if a > 0 ( < 0 ) ) then K is monotonic decreasing (increasing) for so long asau/i?t # 0. Unfor tunately, th is is not sufficient in general to guarantee the existenceof non-trivial steady states. Cnless either a non-zero lower bound or an upper boundcan be placed on K . there is no guarantee th at K does not simply tend to zero whena > 0, or that K does not increase without limit when a < 0. I n either case, no usefulconclusion may be drawn.There are, however, two special circumstances in which an upper bound can beplaced on K , and useful conclusions may be drawn.

( a ) Two-dim ensional f lowsSuppose th at

U = (&f?/ay, -a$/ax, 0)andU ) = (0 , 0,

(6.6)(6.6)

where w, = -V2$ and $ = $(x, y , t ) . Then (5.1)describes convection of the vortexlines by the flow U , and the enstrophy of the flow;

(5 .7)is conserved. Moreover the kinetic energy is bounded by a Poincard inequality of theform K 6 kAQ = const., (6.8)where A is the cross-sectional area of the (two-dimensional) domain D, and k is adimensionless number of order unity determined solely by the shape of D. Hence,choosing a < 0 in ( 5 . 2 )and (5.4).K is monotonic increasing an d bounded above a ndtherefore tends to a constant. Hence from (5.4), u/i?t-0 (a t least almost everywhere)and so as described by Vallis et al. (1989)) tends to an equilibrium sta te u " ( x )whosevorticity field d ( x ) s topologically accessible from the initial field oo(x) .n this two-dimensional context, the vorticity field U)" = (0, 0, @) is obtained simply byrearrangement of the vortex lines of U),, = (0, 0, w o ) : the word 'isovortical' isfrequently used in this context to describe two vorticity fields wl(x), w z ( x ) uch tha t

W 2 i W = W l i 4where x - X is an area-preserving orientable continuous mapping in t he plane (i.e. arearrangement in the above sense).

Flows obtained b y th is procedure will in general have maximal energy with respectto isovortical perturbations. and will therefore be stable, by Arnol'd's (1966)criterion.M7e note tha t the procedure described above has close points of contact with theprocedure used by Campbell & Kadtke (1987) (see Aref e t al . 1988) to determineabsolutely sta tionary configurations of systems of point vortices, i.e. steady solutionsof the Euler equations of prescribed (and very par ticular) vorticity topology.( b ) A x i symme t r i c f lows

Suppose now tha t, in spherical polar coordinates ( r , 0, $),(6.9)

Phi l . Trans.R. Soc. L ond . X (1990)


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334 H . K . Moffattandwhere $ = $(.I.,0 ; t ) and

U = (0, 0, wp,(r,0; t ) ) ,

I n this case (5.1) mplies that

whereD a- -+ v . v ,Dt a t

so that, in particular,

Hence the enstrophy is bounded above, sinceconst.

( r dV

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

(5.15)Hence K is also bounded above, since an inequality of the form (6.8) still applies. Abeing now the area of a meridional section of the axisymmetric domain D.

Hence in this case also VCY relaxation with U < 0 will in principle yield stableEuler flows for which again the vorticity field u E ( x )s topologically accessible fromthe initial field o o ( x ) .The procedure of Vallis et a l . (1989) has been placed in a more general context byShepherd (1990) who shows how any hamiltonian dynamical system may be modified

in such a way as to drive the system towards an energy maximum or minimum (ifsuch a state exists) while conserving those invariants (known in two-dimensionalcontexts as Casimirs) that are essentially topological in character. The technique cantherefore be applied not only to the Eider flow problem, but also to more complexsystems of equations involving effects of stratification and/or compressibility. Amore elaborate relaxation procedure has also been advocated (Moffatt 1989) toestablish the existence of steady solutions { u ( x ) . B ( x ) } of the magnetohydro-dynamic equations of an ideal fluid. the topology of both fields U and B beingprescribed in a compatible manner.

6. Flow signatureConsider again the magnetic relaxation problem in a two dimensional domain D.

with magnetic field expressible in terms of a flux function x ( x , y , t ) byB = (i?x/i?y. -ax/i?x, 0). (6.1)

The B-lines are then the contours x = const., and in particular we may suppose thatx = O on 2 9 . (6.2)

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The Euler equations :a lagrangian approach 335Let us suppose first th at x0(x, ) = ~ ( x ,. 0) has only one stat ionary point PO n theinterior of D and th at this is a maximum: then B = 0 a t PO nd the B-lines are (ingeneral) elliptic in a neighbourhood of PO.Let A ( x c ) e the area enclosed by the curve x0 = xC where x C is a constantsatisfying 0 < x C< xmax.Obviously, A ( x c )s monotonic decreasing in this interva l.withwhere A , is the area of D. oreover, if B is differentiable, then

4 0 ) =A,, A(Xrnax) 0 (6.3)

A = O(~rnax-~c)when X c + X m a x . (6.4)During magnetic relaxation, the B-lines are frozen in the fluid and the frozen-field equation becomes simply Dx/Dt = 0. If we focus attention on one B-line~ ( x ,. t ) = x C , hen, since the flow is incompressible, the area within this B-line isconstant, i.e. the function A ( x c ) s an invariant of the relaxation process. It istherefore appropriate to describe A(xc) s the signature of the field (Moffatt 1986b).Since relaxation proceeds in such a way as to decrease magnetic energy, it isobvious that IBl must remain everywhere bounded and hence ~ ( x ,. t ) remainsdifferentiable for all t . In the asymptotic equilibrium situation ( t -f C O ) , ~ ( x ,, t )+

xE(x, ). whereV 2 X E = F ( f ) (6.6)

for some (current) function F ( f ) this is the well-known Grad-Shafranov equationdescribing two-dimensional magnetostatic equilibrium. The nature of the relaxationprocess allows us to assert th at , for every signature function satisfying (6.3), 6.4)andA'(xc)< 0, there exists a magnetostatic equilibrium in D ; he function F ( x E )characterising this equilibrium is in principle determined by the signature functionwhich may equally be expressed as a function A ( f ) . ndeed. elimination of xEbetween the equations F = E I ( f ) , A =A ( x E ) (6.6)in general implies a relation F =F ( A ) .

A simple example may make this clear. If D is the elliptic domainD : 2 / a 2 + 2 / b 2< 1then (6.5) is satisfied by

providedThe signature function for this field is

X E b ; y) = Xrnax (1-%'/a2- 2/b2)F ( f )= - B ~ ~ ~ ~ ( l / a ~ +/ b 2 )= const.

(6.7)

(6.9)(6.10)

A ( x )= n a b ( l - ~ / ~ m a x ) . ( 6 . 1 1 )Hence only a field Bo(x;y ) with this (linear) signature can relax t o the equilibrium(6.9)for which all field lines are ellipses.Suppose now tha t we start a t t = 0 with a field Bo(x ,y ) with elliptic streamlines sothat

for some monotonic function G ( . ) . The signature function is then(6.12)

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336 H . K . woffutt

Figure 5 . Relaxa t i on of a field initially confined to an el l ipt ic pathway within an el l ipt icdom a in . t he pa th con t r ac t s t o a m in imum l eng th . t he con t a ined a r ea -4, being conservedHence an initial field with arbitrary (monotonic)signat'ure can easily be const'ructede.g. if

A ( x ~ )nab(l-~c/~rnax) '> (6.14)t'hen

(6.16)(Y )= [1- 1- Y)i]xmaxand t'he required initial field is

x o ( x , Y)= (x2/a2+Y /' b2 ) * I ~ m n x ~ (6.16)This field cannot relax t'o t'he field (6.9);but must relax t'o a smoot'h magnetostat'icequilibrium wit'h non-ellipt'ic B-lines, and non-trivial current' funct'ionF ( x E ) .The reason for the appearance of non-ellipt'ic B-lines can be easily understood : int'he absence of any boundary constraint, each B-line would relax t'o a configurat'ionof minimum lengt'h for prescribed contained area, i.e. to a circle. I n the presence ofthe ellipt'ic boundary, t'his process is impeded; but field lines on which t'he field isrelat'ively strong will, as it' were, win in t'he t'endency to become circular, and acompromise bet'ween this tendency and the boundary constraint will be achieved. Anextreme s ituat' ion is illustrat'ed in figure 6 ; which shows relaxat'ion of a field initiallyconfined to an elliptic 'pathway' wit'hin D ; if the area A , inside t'his pathway is lesst'han nb2, 'hen relaxation t o circular B-lines is possible if however nb2


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The Elder equations: a lagrangian approach 337

field discontinuityF igu re 6. Mechanism by which a tange nt ia l d i scontinui ty ma y form a t an in te rsec t ion ofseparatr ices.

F igu re 7. Relax a t ion of th e pro to type f ie ld ( 7 . 1 ) , shouing the development of a tangent ia ld i scontinui ty (Bajer 1989).i 10T1

0 10 20 kt 30Figu re 8. (a)Relaxation of energy for the process depicted in f igure 7 . ( b ) Plot of V2x against2 . showing evidence of a funct ional relat ion, when kt = 36 (Ba j e r 1989).

neighbourhood of the saddle poin t, and the Lorentz force may a ct in such a way asto cause the angle between th e separatrices to collapse to zero. This process has beenanalysed numerically by Bajer (1989) figure 7 shows the nature of the relaxationprocess, computed on the basis of (1 .1 3 ) and (3.4)for initial flux function

X o ( r ,8) r 2 ( l - r 2 ) cos28,-cos28) , ( 7 . 1 )where 28, is the acute angle between th e separatrices at r = 0. The contours A =const. are shown for cos2Bo= 0.8(8 , = 27') and k t = 0, 36, and the formation of thefield discontinuity is clear. There are severe numerical problems in following thisprocess to the asymptotic limit. but the qualitative nature of the process is clear.During this relaxation process, the energy settles down quite rapidly (figure 8 a )to i ts asymptotic level (about 67 % of its initial value) but fine-scale adjustment toPhzl. Tvans R 8o c Lond X (1990)


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338 H . K . Moffattequilibrium in the neighbourhood of the cuspidal singularities is relatively slow asmay be seen from the plot of V 2x against x a t kt = 36 (figure 8 b ) . In equilibrium afunctional relation V 2 x = F(x)must emerge. and the scatter of points near x = 0provides an indication of persisting disequilibrium near the cusps. There seems littledoubt however that the equilibrium field structure is nearly attained in figure 7 b atkt = 36 . and only numerical problems limit the accuracy wi th which this equilibriumstat e may be determined.O ur interest here is in the analogous Euler flow, for which the asymptotic fluxfunction X E ( r , 8 ) s replaced by an analogous stream function $."(r. 8). igure 7 b thenrepresents the streamlines of a steady Euler flow. with a non-uniform vortex sheeton the segment between the two cusps. This flow is presumably unstable to aKelvin-Helmholtz type of instability ;but i ts existence is nevertheless of considerableinterest, and it is hard to see how this existence could be inferred other than via the

Kote that the separatrices divide the flow domain into four subdomains D,( i =1, 2 . 3 , 4). within each of which a signature function A,(x)may be defined, which isinvariant during the relaxation process. For a general domain D , he general initialflux function xo (r ,8) s characterized topologically (i) by the topology of the networkof separatrices which divide the domain into, say n subdomainsD,( i = 1 . 2 . , .. n ) ,and(ii) by the set of signature functions {A,(x)}or these subdomains. During relaxation,the topology of the network can change only through the collapse of angles andassociated formation of current sheets ( i.e. tangential discontinuities) as describedabove; the set {A,(x)}emains invariant however and survives to characterise theasymptotic relaxed field.

The general Euler flow obtained by this process may thus be expected to containa number of tangential discontinuities (i.e. vortex sheets) of finite extentcorresponding to all the saddle points of the initial reference function x o ( r .8). hepressure field p E in the Euler flow is continuous across each of these sheets, as maybe seen by integrating the Euler equation across the discontinuity.

magnetostatic analogy and the magnetic relaxation argument. _ -

8. Formation of discontinuities under VCY relaxationConsider again the relaxation process advocated by Vallis et al. (1989),applied now

to a two dimensional flow in the domain D with initial stream-function $ o ( r , 8)satisfying

In general, the initial vorticity field wo = -Vzyk0 does not satisfy the condition$ o = O on i3D. (8 . 1 )

wo = const. on i3D. (8.2)This presents an immediate difficulty, because the VCY procedure involvesconvection of the vorticity w by a velocity field U satisfying van = 0 on i3D; hence if(8 . 2 ) is not satisfied, then w can never become constant on aD, i.e. equilibriumdescribed by w =F ( $ ) can never be attained at the boundary.For this reason, we shall restrict attention in the following discussion to initialfields $.,(Y, 8) for which both the conditions (8.1)and (8.2)are satisfied. Relaxationthen proceeds according t o the equation

Dw/Dt = 0, W ( T . 8, )= w o ( r ,B), ( 8 . 3 )where D/Dt = c?/c?t+u.V and U is defined by (5.2)with CI < 0. Here it is the topologyPhil T r a m R. oc Lond A (1990)


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The Euler equations :a laqranqian approach 339

Figure 9. Axisymmetric vorton with swirl. This flow is characterized by a signature { V ( $ ) ,W($)}.hen V ( $ ) is the volume within a torus y? = const., and W($)s the azimuthal flux withinthe torus.of the initial vortic ity field u o ( r ,0) that is relevant. Again this field defines a networkof separatr ices dividing D into subdomains D,( i = 1. 2 . . ,n ) , nd again we have a setof signa ture functions {A,(@))ow involving th e (conserved) area within any closedcontour w = const. Now during VCY relaxation, there is no impediment to thecollapse of angles between separatrices with associated formation of discontinuitiesof w , but the set {A,(@)}emains as the crucial topological invariant of the process.

In the steady st ate predicted by Vallis et al. (1989) the streamlines must of coursecoincide with the iso-vorticity contours w = const., and so the picture is similar tothat described in $ 7 , only now the tangential discontinuities are discontinuities ofshear- rate and not of velocity. Sumer ical simulations of two-dimensional turbulence(e.g. Brachet et al. 1986) frequently indicate th e presence of layers of very highvort icity gradient. suggesting t hat the steady Euler flows. whose existence is impliedby the VCY procedure, may p lay some kind of at tract ing role in the governingunsteady dynamics.

9. VortonsWe use th e word vorton, as in Xoffatt (1986b), o denote a rotational disturbance.

the vorticity field being of bounded suppor t, which propagates without change ofstructure and with constant velocity in an unbounded fluid. Relative to a frame ofreference th at moves with the vorton. the flow is steady, and as shown by Moffatt(1986b, 1988), a very wide family of axisymmetric vortons. both with and withouta swirl component of velocity abou t the axis of symmetry . may be obtained by themagnetic relaxation technique, coupled with the analogy (4.5).These vortons arecharacterized by a signature { V ( $) , W ( $ ) }within the rotational region D,; outsidethis region, the flow is the unique potential flow settling to a uniform stream atinfinity and matching smoothly to th e rotationa l flow across aD,. The configurationis sketched in figure 9 ; note th at, provided t he initial field has no saddle points offthe axis of symmetry. field discontinuities cannot form during the magneticrelaxation process, and so the vortons obtained by this method are characterized bya continuous velocity field : however. the vorticity field is generally discontinuousacross aD, (as for Hills spherical vortex , or for any of the family of spherical vorticeswith swirl discovered by Hicks (1899)).As discussed in $4 , there is no guarantee th at these vortons are stable, even toP hd Tians R SOCLond A (1990)


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340 H . K .Xof fa t taxisymmetric disturbances, within the framework of the unsteady Euler equations.If the VCY relaxation procedure could be adapted to the vorton configuration (i.e.to an infinite domain with U tending to a uniform stream at infinity) then thisparticular difficulty would be overcome. There are difficulties in achieving this,however, because of the need to choose the frame of reference in which theasymptotic flow (as t - t c c ) will be steady. not a straightforward matter since theVCY procedure (involving an artificial modification of the Euler equations) does notconserve momentum. And yet, as argued by Benjamin (1976). vortex rings arecharacterized by maximality of kinetic energy within the class of axisymmetric flowsattainable by rearrangement of the circular vortex lines (cf. (5.12)); o one wouldexpect the YCY procedure. which increases kinetic energy within just such a class offlows, to be well-adapted to the problem. Further developments may be expected.

10. DiscussionThis paper has focussed on what are essentially lagrangian techniques forobtaining solutions of the steady Euler equations of non-trivial topological st ruc ture.and for certain stability considerations. These techniques rely on identification ofappropriate lagrangian invariants, i.e. integrals or functions defined over materialdomains which remain invariant by virtue of some kind of frozen-field evolution.The three-dimensional magnetic relaxation problem is important because it yieldsstable magnetostatic equilibria and hence (by exact analogy) steady Euler flows.which are probably in general unstable. From the standpoint of the Euler equations.the technique is perhaps a little academic for this reason ; however it has been arguedelsewhere (Moffatt 1990a) th at these flows are important when regarded as unstablefixed points of the (infinite-dimensional)Euler dynamical system. it being a common

feature of dynamical systems th at solutions may remain for a high proportion of thetime in neighbourhoods of such unstable fixed points. The technique is also ofinterest in establishing a bridge between mathematical topology and classical fluidmechanics, a bridge dimly perceived by Kelvin (1869), but now seen to be onreasonably firm foundations. The techniques of fluid mechanics are in fact highlyrelevant t o certain problems (e .g. determination of new knot and link invarian ts)that are purely topological in character (Freedman & He 1990a. b : Moffatt 1990b).Magnetic relaxation yields magnetostatic equilibria with prescribed magnetic fieldtopology, and hence. by analogy, Euler flows with prescribed streamline topology.An alternative procedure. as devised by Vallis et a l. (1989)may yield Euler flows withprescribed vorticity topology. However. the procedure works only when the kineticenergy either has a non-zero lower bound or a finite upper bound; a finite upperbound has been established for two-dimensional flows and (in the present paper) foraxisymmetric flows without swirl.

For two-dimensional magnetic relaxation with flux function x. he appropriatelagrangian invariant is a signature (or area) function A ( x ) , r a set of such funct ions{A, (x)} .or VCY relaxation. this is replaced by A ( w ) (or {At (w)} ) ,he area withincontours w = const. I n either case. it is the signature th at remains invariant duringrelaxation, and that characterizes the asymptotic steady state. Similarly. foraxisymmetric problems the signature is a volume function V ( x ) or (for VCYrelaxation) V ( q )where q = w F / r sin 0 . Axisymmetric flows with swirl obtained viamagnetic relaxation are characterised by a double signature ( V ( @ ) . ( @ ) }where W isthe azimuthal flux within the torus @ = const.Phd Trans R Soc Lond X (1990)


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The E d e r equations : a lagrangian approach 34 1It has been conjectured (Moffatt 1990a) hat turbulence may usefully be regarded

as a sea of interacting vortons. This provides a natural explanation for theapparent suppression of nonlinearity that has been identified in direct numericalsimulations of turbulence (Kraichnan& Panda 1988).It also provides a rationale forthe coherent structures that are such an all-pervasive feature of many turbulentflows (Hussain 1986). In this picture. the region of interaction of vortons is the seatof Kelvin-Helmholtz instabilities, which, via spiral wind-up, lead to inertial transferof energy to small scales. This picture is quite different from the traditionalKolmogorov cascade picture, but has the merit of starting from fully nonlinearsolutions of the Euler equations (the vorton solutions) which may provide a usefulbasis for a more deductive theory of turbulence.

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342 H . K . MoffattKelv in , Lo rd 1869 On vor tex motion. Trans. R. Soc. Edinb. 25, 217-260.K r a i c h n a n , R. H . 8z.P a n d a , R . 1988 Depression of nonlineari ty in decaying isotropic turbulence.J lof fa t t , H. K. 1969 Th e degree of kno ttedne ss of tangle d vortex lines. J . Flu id iWech . 36, 117-129.J Iof fa t t ; H. K . 1981 Some developments in the theory of turbulen ce . J . Fluid Mech. 106. 27-47.Moffat t , H. K. 1985 hlag neto stat ic equil ibr ia and analogous Eule r flows of arbi trar i ly com plexJ lof fa t t . H. K . 198 6a Mag netostat ic equilibr ia an d analogous Eu ler flows of arb i trar i ly complexhloffat t , H. K. 1986 b On the existence of local ized rotat ional dis turbances which propagateJ lof fa t t . H. K . 1988 General i sed vor tex r ings wi th and w i thout swir l. FZuid Dyn. Res. 3 . 22%30.Moffat t ; H. K . 1989 On the ex is tence , s t ruc ture a nd s tab i l i ty of 31HD equi libr ium s ta tes . InTurbulence and nonlinear dynamics ( ed . A l . Meneguzzi, A . P o u q u e t & P. L. Sul em ), pp . 185-195.Elsevier Science Publications.hloffat t ; H. K . 1990a Fixed poin ts of turbulent dynamica l sys tems and suppress ion ofnonlineari tp. In Whither turbulence ? ( ed . J . L. Lum lej-) p p , 250-257. Springer-Gerlag.Mof fa t t , H. K. 1990b The energy spec t ru m of knots an d l inks . Xature, Lond. 347, 367-369.Moreau , J.-J. 1961 Co nsta nts d 'u n ilot tourbi l lonnaire en f luide parf ai t baro trop e. C. r . hebd.S e a m . Acad. Sci., Paris 252, 2810-2812.Shephe rd , T . G. 1990 A general met 'hod for f inding extremal energy states of Hamil tori iandynamical systems with applicat ions to perfect f luids. J . Fluid Mech. 213. 573-587.Gallis. G. K . , Ca rneva le , G. F. & Young, R'. R . 1989 Ex t rem al energy proper ties and cons t ruc t ionof s tab le so lu t ions of the Euler eq uat ions . J . Flu id X e c h . 207. 133--152.Whi t ehead , J. H. C. 1947 An express ion of Ho p fs invar ian t as an in tegral . Proc. nutn A c a d . S c i .C.S.A.3 , 117.Wol t j e r , I,. 1958 d theorem on force-free magnetic fields. Proc. natn Acad. Sci. C.S.A. 44,489-491

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H.K. Moffatt- Structure and stability of solutions of the Euler equations : a lagrangian approach