H.K. Moffatt- Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology....

8/3/2019 H.K. Moffatt- Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations

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J . Fluid Mech. (1986),vol. 166, p p . 3 5 S 3 7 8Printed in &eat Britain

359

Magnetostatic equilibria and analogousEuler flows of arbitrarily complex topology.Part 2. Stability considerationsBy H. K. MOFFATT

Department of Applied Mathematics and Theoretical Physics, Silver Street,Cambridge CB3 9EW(Received 24 August 1985)

The stability (i ) of fully three-dimensional magnetostatic equilibria of arbitrarilycomplex topology, and (ii) of the analogous steady solutions of the Euler equationsof incompressible inviscid flow, are investigated through construction of the secondvariations S2M and S2K of the magnetic energy and kinetic energy with respect toa virtual displacement field q ( x )about the equilibrium configuration. The expressionsfor S2M and S2K differ because in case ( i) the magnetic lines of force are frozen in thefluid as it undergoes displacement, whereas in case (ii) the vortex lines are frozen,so that the analogy between magnetic field and velocity field on which the existenceof steady flows is based does not extend to the perturbed states. It is shown th at thestability condition S2M > 0 for all q ( x ) for the magnetostatic case can be convertedto a form tha t does not involve the arbitrary displacement q ( x ) ,whereas the conditionS2K > 0 for all q for the stability of the analogous Euler flow cannot in general beso transformed. Nevertheless it is shown that, if S2M and S2K are evaluated for thesame basic equilibrium field, then quite generally

S2M+S2K> 0 (all non-trivial q ) .A number of special cases are treated in detail. In particular, it is shown that the

space-periodic Beltrami fieldBE= (B, cosaz+B, sinuy, B, cosax+ B, sinuz, B, cosay+ B, sinux)

is stable (i.e. S2M > 0 for all q) and tha t the medium responds in an elastic mannerto perturbations on a scale large compared with a-l . By contrast , it is shown thatS2K is indefinite in sign for the analogous Euler flow, and i t is argued that the flowis unstable to certain large-scale helical perturbations having the same sign of helicityas the unperturbed flow. It is conjectured that all topologically non-trivial Euler flowsare similarly unstable.

1. IntroductionIn a previous paper (Moffatt 1985, hereafter referred to as M85),he existence of

magnetostatic equilibria, and hence of analogous steady Euler flows has beenestablished, through consideration of the relaxation of a magnetic field B ( x , ) in aperfectly conducting, but viscous, fluid contained within a bounded domain 23 withfixed boundary B. he magnetic energy stored in an arb itrary initial field B , ( x ) isconverted to kinetic energy of motion and hence, via viscous dissipation, to heat.However, if the topology of the magnetic field is non-trivial, topological constraints

www.moffatt.tc


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360 H . K . Moffattplace a lower bound on the magnetic energy, and the system asymptotes to amagnetostatic equilibrium B E ( x ) atisfying

jE B E = V p E , jE = V x BE, V . B E= 0 in 9,\ (1.1)for some scalar (pressure) field p E ( x ) .An essential property of the magnetostaticequilibria satisfying (1 l ) s that they may contain tangential discontinuities of B E(i.e. current sheets) imbedded within 9 as well as on the boundary a9),venalthough the initial field B , ( x ) is everywhere differentiable (Cl) .?

The physical nature of this relaxation process implies that, in general, theequilibrium described by (1.1)should be stable with respect to small displacementsof the fluid which conserve the frozen-in character of the magnetic field, since.otherwise the decrease in magnetic energy by the above mechanism may (and ingeneral will) continue. Thus we expect that the magnetic energy should be a minimumwith respect to small frozen-field perturbations, and this should be true even if theequilibrium field involves current sheets. We are of course ignoring here thepossibility of resistive instabilities which may occur as a result of the finiteconductivity of real fluids.

To each magnetostatic equilibrium B E ( x ) , here corresponds, via the analogyB E ( x ) ) E ( x ) , solution of the steady Euler equationsu E x m E = V h E , m = V x u E , V * u E = O i n 9 , n * u E = O o n a 9 , (1.2)

describing the steady flow of a n inviscid incompressible fluid within 9. ote here th at9 ay be simply or multiply connected, and th at C9 may consist of several disjointparts.

As emphasised in M85, he analogy between magnetostatic equilibria and Eulerflows exists only for the steady states, and does not extend to questions of stabilityabout these steady states . Thus , the fact th at a magnetostatic equilibrium is stableis no guarantee that the corresponding Euler flow is stable. Indeed, if current sheetsare present in the former, then vortex sheets are present in the lat ter, and these arelikely to be unstable by the Kelvin-Helmholtz ideal-fluid instability mechanism.

Our aim in this paper is to investigate the relationship, if any, between the twostability problems. We shall show in fact that there is an interesting complementaritybetween the two problems, with some points of comparison, and also, as expected,some vital differences. The natural tools for this investigation are certain variationaltechniques, developed in the magnetostatic context by Bernstein e t al. (1958),andin the Euler flow context by Arnold (1966~) .hese techniques differ in the twocontexts in a rather subtle manner. We shall therefore develop them ab i n i t i o here,in order to highlight points of comparison and points of contrast. We shall thenillustrate the techniques for three distinct field (or flow) geometries: ( i ) the two-dimensional situation; ( i i ) the stability of a cylindrically symmetric field (and of itsEuler-flow analogue) to axisymmetric disturbances ; and (iii) the stability of the

t It is conceivable, as pointed out by a referee, that the general magnetostatic equilibrium (andtherefore the general analogous Euler flow) may have a more complex structure th an indicated here;e.g. tangential discontinuities may conceivably be stacked on top of each other in such a way tha tthe field is almost nowhere differentiable. While accepting this possibility, we restrict at tention inthis paper to fields B E that have at most a finite number of discontinuities per unit length on anystraight line transversal. The current dens ityjEhas a S-function structure a t each such discontinuity,and volume integrals such as (2 .21) below must be interpreted in the obvious way when suchdiscontinuities are present.

n * B E = O o n a 9 , J


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Magnetostatic equilibria and Euler flows of complex topology. Part 2 361

FIQURE. Relationship between C(x,7 ) and q ( x ) (see (2.5))

space-periodic Beltrami flows, whose structure has recently been investigated byDombrk et al. (1986).

I n general, the results are consistent with the conjecture, hinted a t in M85, ndnow made explicit here, that a three-dimensional Euler flow of any significantcomplexity is in general unstable ; in the language of dynamical systems these Eulerflows may be regarded as the unstable fixed points in the function space in whichsolutions of the unsteady Euler equations evolve. Intrinsic instability of complexthree-dimensional flow in the inviscid limit is of course one of the hallmarks ofturbulence; and it is very much with an eye to application in the turbulence contextthat this investigation, which might otherwise seem somewhat artificial, is pursued.

2. Stability of magnetostatic equilibriaThe magnetic energy associated with a field distribution B ( x ) s

and the equilibrium B E ( x ) atisfying ( 1 . 1 ) will be stable if the corresponding energyME is minimal with respect to small displacements of the fluid in 9 which perturbthe field in a ' frozen-field 'manner. We restrict attention to incompressible fluids, andsuppose that the displacement occurs through the action of a steady solenoidalvelocity field ~ ( x )V * u= 0 ,n * u= 0 on 3 9 )which acts during a small time interval 7 .

Let t (x, ) be the displacement of the fluid particle initially a t x ; thena t- u ( x + { ) = U ( X ) + t ' V U + ...,at

so that i ( x ,T ) = ~ U ( X ) + $ - ~ U * V U + O ( ~ ~ ) . (2 .3 )Equivalently t ( x , ) = tt(x)+2tt'vs+0(r3), (2.4)

where ~ ( x ) T U ( X ) .Equation (2 .4 )may of course be inverted to givet t ( x )= t - - i C * V t + 0 ( i 3 ) , ( 2 . 5 )


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Magnetostatic equilibria and Euler flows f complex topology. Part 2 363magnetostatic equilibrium, i.e. th at S'M = 0. This is easily verified: note first that,using n.BE = n'q = 0 on i323,

andHence

c P

(2.17)

(2.18)

Consider now the expression for S 2 M :S 2 M = AJ[(S1B)2+BE*V~(qxSIB)]dV, (2.20)

so th at , integrating the second term by partsS2M = AJ[(S1B)2-(qjE)-SIB]dV. (2.21)2

The equilibrium BE(x)s stable t o small disturbances if S2M > 0 for all q ( x )satisfying(2.6).Clearly therefore, in order to find a useful criterion for stability, we need to placea bound on the magnitude of the second term of the integral in (2.21).There are twoprocedures th at we may follow, the first of which is the more straightforward, andthe second of which is more natural ly related to the corresponding Euler-flow stabilityproblem considered in the next section.Procedure A

Firs t note th at, from the Schwarz inequality,9 1 [ ( qxjE)-SIBdVI< { ] ( q xjE)zdVs(SIB)zdVj!. (2.22)Secondly, we seek the (non-trivial)displacement q , ( x )satisfying (2.6)which minimizesthe ratio

s(SIB)ldV J[Vx (qxBE )I2dVJ ( qxjE)2 V s [ qx (V x BE)Iz Vh{tl(x)l= - (2.23)

for fixed B E ( x ) . o this end, let q = q l ( x ) + S q ( x )where Sq is a (virtual)displacementwhich also satisfies (2.6); hen q l ( x ) s determined by the variational statementS {[V x ( q x BE)Iz-Al[q x (V x BE)]'} dV = 0, (2.24)

where A, is a Lagrange multiplier. Hencel { V x (qx B E ) - V x (Sq x BE)Al q x (V x BE)*Sq (V x BE)}V = 0. (2.25)


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364 H . K . MoffattIntegrating he first term by parts and using the boundary conditions n .BE= n ' q = 0on a9 (a procedure that will be used repeatedly in what follows), an d rearranging,gives s6q*{BEx V x V x ( q xBE)]-h,(Vx BE)x [qx (V x BE)]}dV= 0. (2.26)Since this holds for arbitrary solenoidal 611, i t follows that the minimizing q = q , ( x )

BEx [V x V x (BEx ql)]+Al(Vx BE)x [q l x (V x BE)]= VY, (2.27)for some scalar field Yl(x). Moreover, when this equation is satisfied, together withthe boundary condition q l * n= 0 on a9,calar multiplication of (2.27) by q1 andintegration over 9 hows tha t h{tll(X)} = 4, (2.28)so th at A, is real and positive, and is determined uniquely by ql(x).It now follows from (2.21)-(2.23) nd (2.28) hat

a2M2 ( l -A ; i ) (S1B)2dV. (2.29)Hence if A , is the smallest strictly positive eigenvalue of (2.27), hen A , > 1 impliesthat 6'M > 0 for all non-trivial displacement fields q ( x ) , .e. that the magnetostaticequilibrium is stable.

sProcedure B

Instead of q x j E in (2.21),we may use the solenoidal part of q xjE efined by(11 x j E ) s = 7 X j E +vq5,where q5 is chosen so that (2.30)

V.(q xjE) s= 0, n - ( q jE)s= 0 on 3 9 . (2.31)These conditions imply that $(x) ( = q5{q(x)}s the linear functional of q (unique upto an additive constant) determined by the Neumann problem

V'$ = -V*(q xjE ), aq5 = - n * ( q x j E ) on a 9 .sVq5.S1BdV = V*($SIB)dV= q5q06lBdS= 0,s sinceit follows from (2.21) hat

6'M = -j {[Vx (qx BE)]'- ( q x ~ ~ ) ~ * V( q x BE)}dV.'sNow we may put a bound on the second term as in Procedure A . First

(2.32) 0(2.33)

(2.34)

(2.35)and secondly, we seek the non-trivial displacement q , ( x ) which minimizes the (new)ratio

s[V x ( q x BE)]' dVA = (2.36)

J (q X j E ) : d v '


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Magnetostatic equilibria and Euler j lows of compl ex topology. Part 2 365The corresponding variational problem is now

SJ{[V x ( q x BE)I2-A,(q x j E ) i } d 8= 0, (2.37)or, using (2 .30) ,

J{[V x ( qx BE)]-V (Sq x BE) ,(q x jE)s*Sq x j E+ V S $ ) }d8 = 0, (2.38)where S$ = $ {Sq(x ) } .Note now (cf. 2.33) that

J[(T X j E ) , ' V W 1d V = 0, (2.39)so that (2.38)may be manipulated t o the form

SSq*{BE [V x V x (qx B E ) ]AdEx (qx j E+ V$)} dV = 0. (2.40)Hence, in this case, the minimizing q = q , ( x ) satisfies

BEx [V x V x (BE q , ) ] + A 2 j E x ( q xjE+V$,) = VY, (2.41)for some scalar field Y,. Again it is easily verified that

A{qz(x)>= A,, (2.42)so that A, is real and positive; moreover, now

S 2 M 2 ( - A ; + ) (S1B)28, (2.43)so th at if A, is the smallest strictly positive eigenvalue of (2.41) hen A, > 1 is sufficientfor stability of the magnetostatic equilibrium to small disturbances.

JNote tha t, by virtue of (2.39)with S$ replaced by $,

Jot xjE)idV = J M x ~ 2 - V W I d ~ .~ [ V x ( q , ~ B ~ ) ] ~ d 8[ V Xq 2 x B E ) I 2 d 8

Hence A, = 2 3 A,, (2.45)

(2.44)9

J"e2X A Z - (v$2)218 J (q 2X j E Y cl8with equality only when q2 = ql and V $ 2 = 0. Hence the condition A, > 1 forstability is stronger than the condition A, > 1 . However, it is the form (2.34) thatwill be more useful for comparison with the Euler-flow stability problem, to whichwe now turn.

2.1. The case of force-freeJiel dsThe particular case in which the magnetic field BEsatisfies the 'force-free' condition

V x BE= &BE (2.46)has been studied by Molodensky (1974), who showed that, if the disturbance isconfined to a region in the interior of 9 f maximum diameter L then, in the notationof the present paper,

P M 2 ( l - l~ l (L) (SIB)*dV (2.47)s,


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366 H. K. Moffutt(a result tha t may be deduced from (2.21)), o tha t the field is stable provided

L < lal-1. (2 .48)The case when the disturbance extends to the boundary i3B is not covered byMolodenskys analysis. In two further papers, however, Molodensky (1975, 1976)examined the particular case of a spherical domain, and generalized the inequality(2.47) to cover the case when a is a function of position satisfying BE*Va 0.

A particular force-free field is treated in $ 6 of the present paper, and is shown tobe stable to all disturbances, irrespective of lengthscale.

3. Stability of analogous Euler flowConsider now the analogous Euler flow uE(x) atisfying (1.2).As shown by Arnold( 1 9 6 6 ~ )he stability of this flow to perturbations governed by the Euler equations

of inviscid flow may be investigated through consideration of the kinetic-energyinvariant K = - S U 2 d V . (3.1)Under perturbations of the flow uE (x ), he vortex lines are frozen in the fluid (notthe streamlines as would be required if the U*B analogy were to persist for thestability problem), and the energy K is stationary for U = uE with respect to suchperturbations (see below). The question of stability leads naturally to considerationof the second variation S2Kof the energy with respect to the initial displacement fieldq ( x ) .If K is a minimumt with respect to all admissible perturbations (i.e. if a2K isa positive-definite functional of q ) then U remains in a neighbourhoodof uE in thesense th at the norm I I u - u ~ ~ / = IK-KEli x 18*Kli (3.2)remains constant, and the flow is then (in this sense) stable.

We consider then a displacement field q ( x )as in $ 2 , satisfying (2.6). Regarding thisas a virtual instantaneous displacement which carries the vortex lines of theequilibrium field w E in a frozen-in manner, the perturbed vorticity field, by analogywith (2.9), s given by

where= W ~ ( X )810+ 620 + ( ~ 3 ) ,

810 = v x (qx WE) ,S2W = ;v x ( qx 810).(3.3)(3.4)(3.5)

Again we suppose that the displacement is non-trivial in the sense that 8lw + 0. Byuncurling (3.3), the corresponding perturbed velocity field is

(3.6)= UE(X)+ S ~ U PU + 0(73),where

where the suffix s denotes solenoidal part of defined as in (2.30)-(2.32).t Arnold (1 966 a) argued t h at s tabi l i ty is ensured if K i s e i the r a min imu m or a m a xi m um wi t hrespec t to smal l pe r turba t ions ; bu t in a subsequen t paper (Amold 19663, see also M c I n t y r e &Shepherd 1985)he showed th a t in th e two-d imensiona l case, maxim al i ty of K is not in fa ct sufficientfor s tab i l i ty . We d i scuss th i s po in t in the contex t of cylindrically sy mm etri c f low in $ 5 of th i s paper .


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Magnetostatic equilibria and Euler j lows of complex topology. Part 2 367The corresponding expansion for K is then

whereK = K E + s ~ K + P K + o ( ~ ) ,

S IK = uE*S1udV,S2K = ~ ~ [ ( S 1 u ) 2 + 2 u E * S 2 u ]dV.

s (3 .9 )(3.10)(3.11)

These expressions were obtained by Arnol'd ( 1 9 6 6 ~ ) .t is easily shown that, by virtueS IK = 0, (3.12)f (1 .2 ) ,

i.e. the kinetic energy is indeed stationary with respect to admissible variations aboutthe equilibrium state U = u E .This is theorem 1 of Arnol'd ( 1 9 6 6 ~ ) .Consider now the second variation S2K.This may be readily manipulated to the

(3.13)form

S2K = - s [ ( qx o E ) E - ( q x oE);V x ( qx uE)]dVusing q * n= 0 on 3 9 . Remarkably, the second term in the integral is the exactanalogue of the second term in (2 .34) ,under the analogy uEe,BE,oE jE.The firstterm is, however, different from the first term in (2 .34) ,and i t is this difference whichpermits instability of the Euler flow, despite the fact that the analogous magnetostaticequilibrium is stable.

12

If we make the substitutionsBE+uE, j E + o E

in (2.34),we obtain(3.14)

(3.15)apd of course, if the analogous magnetostatic equilibrium is stable, then S2Mall q . From (3.13) and (3.15),we have 0 for

S 2 K + S2M = - [V (qx u E )- qx w ~ ) , ] ' V > 0,2S (3.16)a result that is of some interest, although unfortunately it does not give sufficientinformation about S2K to discriminate between stable and unstable states .If we apply the argument of Procedure B in $2 to place an upper bound on thesecond term of (3.13),we find (contrast (2 .43) ) that

(3.17)where A, is still the smallest eigenvalue of (2.41).Again the statement (3.17) is notparticularly helpful because i t appears t o be impossible to convert the integrand toa form that is positive definite except for certain very simple choices of u E ( x ) . hisdifficulty (which was recognized by Arnol'd 1966a) is perhaps no more than a firstindication that, in general, three-dimensional Euler flows of any complexity have anassociated S2K that is indefinite as regards sign, so that the invariance of K placesno constraint on the growth of perturbations to the flow. We shall evaluate S2Kexplicitly for certain basic flows in the following sections in order to test this assertion.


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Magnetostatic equilibria and Euler JEows of complex topology. Part 2 369and so a sufficient condition for the equilibrium to be stable to two-dimensionaldisturbances is f ( A )> -q: throughout B2. (4.14)

Consider now the analogous Euler flow, for which

where A satisfies (4.3)nd (4.4).ence from (3.13),

and so a sufficient condition for stability isf (A) > 0 throughout g2.

(4.15)

(4.16)

(4.17)This is theorem V I I of Arnold ( 1 9 6 6 a ) .Clearly the condition (4.17)mplies not onlystability of the Euler flow, but also (since (4.14)s then also satisfied) stability of theanalogous magnetostatic equilibrium.

Note that the second term of the integrand in (4.16) involves only 11 and not thegradient of q ; his is a special feature of the two-dimensional case which permits theextraction of a sufficient condition for stability to all two-dimensional disturbances.I n the three-dimensional case (for which o E * V q+ 0) this simplification is absent.5. Cylindrically symmetric situationsand the analogous Euler flow is instructive is that in whichA second situation in which the comparison between the magnetostatic equilibrium

or analogouslyin cylindrical polar coordinates ( r , 8, ) . We take 9 to be the domain { r < r < r 2 ,0 < z < zo>. Consider the stability of these states to axisymmetric disturbances,represented by the disturbance fieldi)

(5.3)Calculation of S2M and S2K is straightforward, and we simply state the results :

S2M = - jj9:-$:) r2dr dz,and

S2K = 2n 117: [$ r ~ ) ~ ]-2 dr dz.(5.4)( 5 .5 )

Hence the magnetostatic equilibrium is stable to disturbances of the form (5.3)f02r r < 0,a condition that may be obtained in an elementary manner by considering the changein magnetic energy associated with interchange of two flux tubes.


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370 H . K . MoffattAs regards the Euler flow ( 5 . 2 ) ,note tha t if

ddr rw)2> 0 ( 5 . 7 )

then S2K> 0 for all axisymmetric disturbances, and so the flow is stable to suchdisturbances. This is the well-known Rayleigh stability criterion.As mentioned in the footnote in $ 3 . Arnolds ( 1 9 6 6 ~ )riginal assertion was thatstability is assured if either S2K> 0 or S2K < 0 for all admissible perturbations. Theexample considered here shows that the second alternative cannot be generallycorrect. The reason for the failure of the original Arnold argument in the case whenK is ma xima l for U = uE s that the energy released in perturbing the flow to anadjacent state is then available to augment the disturbance and so to perturb theflow still further - and so on. This cannot happen when K is minimal for U = uE, ndthe condition that S2K is positive-definite is therefore a correct sufficient conditionfor stabi1ity.t

Note tha t, if b ( r ) is replaced by w ( r ) in ( 5 . 4 ) , hen from ( 5 . 4 )and (5 .5 ) .

9consistent with (3 .16 ) .

6. Stability of Beltrami fieldsforce-free (or Beltrami) field

A particular, fully three-dimensional, magnetostatic equilibrium is provided by the(6 .1 )E = (B3 osaz+B, sincry, B, coscrx+B, sinaz, B, cosay+B1 sinax),

for which j E = V x BE = -aBE. ( 6 . 2 )The topological structure of this field has recently been explored by Dombr6 et al.(1986) provided B, B, B, =+ 0, the lines of force are space-filling in a subdomain BCof R3 which is connected from one periodicity box of the field ( 6 . 1 ) to itsneighbouring boxes, like a three-dimensional web. In this section we address thequestion of whether the magnetostatic equilibrium (6 .1 ) is stable; in the followingsection, we consider the same question for the analogous Euler flow.It is known (Woltjer 1958) that minimization of magnetic energy subject only tothe single constraint of conservation of global magnetic helicity leads to a force-freefield satisfying (6 .2 ) or some constant a. However, it is not the case th at every suchforce-free field is the result of such a minimization procedure; and in order to testthe stability of a given force-free field such as ( 6 . l ) ,we need to evaluate the integral(2 .21) for S2M, to see whether or not it is positive-definite. The question has beenconsidered by Arnold (1974)who concluded on general grounds that the field ( 6 . i )is indeed one of minimum energy. We shall here provide explicit verification of thisresult.

t Dr David Andrews (private communication)has shown tha t if the procedure of Arnold (19654is followed fo r the axisymmetric flow considered here, then the condition (5 .7) is again obtainedas the condition th at a certain function S2J be positive definite, bu t that if d(rv)2/dr< 0 , then S2Jis not in general negative definite. This conclusion is entirely consistent with the above discussion.


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Magnetostatic equilibria and Euler flows of complex topology. Part 2 37 1First we write (6 . 1 ) n the more compact form

BE = Z B , eian'x,n

where n takes the values f , & 2 , f , andB - , = B,*, U-, = -an, lanl = lal.

The general disturbance q admits a Fourier representationq = Z qm eikm'x,m

where 'l-m = l m , k - , - k , . (6 . 6 )

P M = ( ( V x A ) . ( Vx A + a A ) ) ,A = q x BE = Z A , eiKhX

Now, using ( 6 . 2 ) ,and replacing the volume integral in (2 .21)by a space-average,i) denoted (. ..) ,we have (6 . 7 )(6 .8)here

where h denotes the ordered pair (n ,m) , andh

~ ~ = a , + k , , h = q m x B , .V x A +a A = Z (iKhx A , +" A , ) eiKAex,

iKhx A , + a A , = iKhx (am B , ) + a q m x B ,Now evidentlyand h

(6 . 9 )(6 .10)

The second term sums to zero by vlrtue-of the reality e d i t i o n s ( 6 . 4 ) ,and we areleft finally with (6 .13)since this expression is positive for all non-trivial q , the stability is proved.?

This means tha t if the equilibrium (6 .1) is disturbed in some way, then the fluidsystem will execute oscillations, which will be damped if viscous dissipation is present,about this equilibrium. Let us now analyse these oscillations on the assumption that

t A referee has pointed out that derivation of (6 .12) from (6 .7) is correct only if all the K~ aredistinct. It is however not difficult to show that if a,+km = a h + k h (i.e. K~ = K ; ) then thecorresponding contribution to P M istl%&(krn B , ) +d n (%n * % ) I 2so that the conclusion following (6.13) s unaltered.The referee has also pointed out tha t (6 .12)in fact follows directly from (2 .21)without the needto invoke the Beltrami property (6 .2) this means tha t all space-periodic magnetostatic equilibriaare stable to all disturbances of the form (6.5).


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372 H . K . Moffattthe scale L of the disturbance q ( x , ) (now regarded as a function o f t as well as x)is large compared with the scale a-l of BE.From (2.9), he perturbed magnetic field

B = BE+6lB+ O ( r 2 ) , (6.14)sand the associated Lorentz force may be written

i aaxj 2 2x,C/X B), = - (B,Bj) - - -B (6.15)The disturbance on the scale L will be driven by this Lorentz force, averaged overthe innerscale a-l, i.e.

YOW t leading order,jEx BE= 0, so we need only concern ourselves with the term 0of order q in (6.16).The first variation of ( B ,B j ) s

# ( B , B j )= (B,E61Bj)+(By61Bi), (6.17)

(6.18)since the average is over the scalea -1on which q ( x ) s slowly varying. Hence, since

if follows from (6.17) hat 6l(Bi Bj ) = C i j k l ~71 )cxk

whereNote that , from (6.1),

C,jkl = (BF B f ) Sjl+ ( B yB f ) 6,,.B; +B; : 1 .B;+Bi

(6.19)(6.20)

(6.21) aThe relationship (6.19) s a stress-strain relationship characteristic of an anisotropicelastic medium; the second term in (6.16) is accommodated through pressurevariations in the incompressible fluid.

The isotropic situation (B ,= B, = B3) s particularly simple. In this case (6.19)reduces to

(6.22)where M E = +((BE). The associated contribution to (6.16) s

(6.23)ax,

a21a t 2

- 6l(B,Bj)) = $MEV2r,,and the equation of motion of the fluid on the outer scale L is(6.24)- {MEV2q viscous term,


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Magnetostatic equilibria and Euler flows of complex topology. Part 2 373where p is the fluid density, assumed constant. The medium evidently supports waveswhich propagate with wave speed C, given by

c, -( y E ) i . (6.25)These waves are of course damped if due account is taken of viscosity.We may note tha t the method described here may be readily adapted to describeoscillations on a large scale L of an arb itrary stable magnetostatic equilibrium B E ( x )with scale 14 L. For example B E ( x )might be a spatially complex field, statisticallyhomogeneous and isotropic with respect to space averages (the end-product of therelaxation process described in M85, tarting from an initial field B o ( x )with thesesame properties). Such oscillations are still evidently described by the wave equation

i) (6*24)*7. Instability of Beltrami (ABC)flows

Consider now the Euler flow analogous to (6.1),viz.u E = ( U , cosaz+ U , sinay, U , cosax+U, sinaz, U , cosay+ 17, sinax). ( 7 . 1 )

The associated vorticity field ismE = -auE. (7.2)

This is the flow described as the ABC-flow by Dombrk e t al. (1986),after Arnol'd(1965b),Beltrami (1902)and Childress (1970).The flow, having maximal helicity, isa natural candidate for dynamo action, and has been studied in this context byGalloway & Frisch (1984) and Moffatt & Proctor (1985).

In order to consider the stability of the flow ( 7 . 1 ) ,we should first evaluate S'K,as given by (3.13). t will be sufficient to consider the particular displacement fieldq = vo (cosk z , sin k z , 0), (7.3)

where k is a constant (positive or negative) ; this satisfiesi) V X ~-kq. (7.4)Using (7.2),and again replacing the integral by a space-average, (3.13)becomes

2d2K= a 2 [ ( A 2 ) - ( ( V q 5 ) 2 ) ] + ~ ( A . VA ) , (7.5)where now A = q x u E with components

( 7 . 6 )A , = v o (U , cos ay+U , sin ax) sin k z ,A , = - 7 ( U, osay+ U , sinax) cos k z ,A , = v o [ (U , cos ax+C, sin 012) cos k z - U , cosaz+U , sin ay) sin k z ] .

and q5 is the space-periodic field satisfyingVzq5 = - V * A = - v o ( a - k ) [ U , cosax sinkz

+ U , s i n a y c o s k z + L ' , c o s ( a - k ) z ] , ( 7 . 7 )i.e. U,T0 cos(a-k)z . (7.8)a-k70'"- { U , cos ax sin k z + U , sin a y cos k z }+a 2 + k 2


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374 H , K . MoffuttFrom these expressions, we may easily calculate

( A 2 )= $[w:+) + t U : l ,andand hence, from (7.5), after simplification,

( A - V x A ) = - + v i ( k + a ) U:+ U: ) ,

(7 . 9 )(7 .10)( 7 . 1 1 )

(7 .12)Since this expression changes sign as k changes sign, it is indejinite as regards sign,and so Arnol'd's ( 1 9 6 6 ~ )ufficient condition for stability is not satisfied. This doesnot necessarily mean that the flow is unstable, although the energy released by theperturbation, when ak > 0, is available to augment the disturbance, and so instabilitymay reasonably be anticipated in this case (following the clue provided by theTaylor-Couette situation discussed in 5 5 above).

To investigate this question further, let us adopt the 'mean-field' approach usedin $ 6 , i.e. suppose that the scale of q ( x ) is large compared with the scale of u E ( x ) ,i.e. th at k < a. (7 .13)The perturbed velocity is given by ( 3 . 6 ) , .e.

U = U E -t- ( q x +0(72), (7 .14)and we aim to calculate the Reynolds stress (u i u j ) o order q , the average now beingover the scale 01-l To this order, we have

(7 .15)(7 .16)

i.e. where, by comparison with ( 7 . 7 ) ,$' = -4 (7 . 17)and (i* j ) denotes repetition of the previous term with i and j nterchanged.

Consider first the term(7 .18)

Evaluating thevarious componentswithuEandqgiven by ( 7 . 1 )and (7 . 3 ) espectively,we find that the only non-zero terms are those for which(&j) ( 1 , 3 ) , ( 3 , I ) , ( 2 , 3 )or ( 3 , 2 ) , (7 .19)and, for these,

'I (7 .20)

E E(u f (q o E ) j > = -"jkl ( u i u1 ) k .

(u,E(q x oE),) ( 1 ++ 3) = $a(U:- U!)y o sin k z ,(u?(qxaE) , )+(2++3) = - t a ( q - U ; ) q 0 c o s k z . )

Consider now the term( u p g ) = -a(.?$). (7 . 21)


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Magnetostatic equilibria and Euler flows of complex topology. Part 2 375Again, using ( 7 .1 ) and (7.8),we find that the only non-zero terms are the four givenby (7.19); nd for these

sin k z ,

cos k z .

( u r g ) + ( l * 3 )( u F g ) + ( 2 * 3 )

Combining these results with (7.20),we find from (7.15)

and a k 2( U 2 u3) = U ;T o cos k z .Now the effective force driving the large-scale perturbation is

i.e. -

where K$ is the kinetic-energy matrix with components

(7.22)

(7.23)

(7.24)

(7.25)

The factor a k 3 ( a 2 + k 2 ) - lappearing in (7.24) is the same as that appearing in theexpression (7.12) or a 2 K ;hus , asexpected, the force (7.24)which drives the large-scaleperturbation is intimately related to the energy (- 2 K ) ha t is available to augmentthis perturbation. The equation of motion that is compatible with this descriptionis

@

(7.26)where q ( x , ) s now regarded as a (slowly varying) function of x and t . Although thisequation is not rigorously established by the above argument (which fails to take fullaccount of the perturbation of the large-scale vorticity field by the small scale-velocityuE),ts structure is indicative of the instability th at may be expected when a2K< 0.

We have carried out the above calculation for the particular displacement field(7.3). However the form of (7.26) now permits us to generalize the result to anarbit rary large-scale perturbation q ( x , ) . For, using (7.4),and expanding (7.26)inpowers of the equation may be written

a27i- 2 1 1p- -K$( 1 +,V2 + + . .)V2(Vx q ) j ,at2 a U U (7.27)a form that is presumably quite general.


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376 H . K . MoffattThe isotropic situation, in which

K t = $KESt i , (7.28)is again particularly simple. If we retain only the leading term in (7.27),we then have

P q 2at2 3a- = - K E V 2 ( V x q ) , (7.29)

which may be contrasted to (6.24). n this approximation, any helical mode for whichv x q = - k q , v - q = 0,satisfies (7.30)

(7.31)and is clearly unstable if ak > 0 , consistent with the remarks following (7.12).It is apparent therefore that the flow ( 7 . 1 ) is unstable to large-scale helicaldisturbances having the same sign of helicity as the basic flow. This result is of greatinterest, since it indicates a mechanism for an inverse cascade of helicity from largewavenumbers to small wavenumbers. There is every reason t o believe tha t the samemechanism will be present for an arbitrary Euler flow (the analogue of the arb itrarymagnetostatic equilibrium conceived in the final paragraph of $6).A similar pointof view is adopted by Moiseev e t al. (1984) in a discussion of turbulence with helicity,in which compressibility effects are regarded as important.

8. ConclusionsIn this paper, we have discussed in general terms criteria for the stability of anarb itrary magnetostatic equilibrium and for the stability of the analogous Euler flow.A sufficient condition for stability of the magnetostatic equilibrium is

S2M > 0 (8.1)for all admissible displacement fields q ( x ) ,S2M being defined by (2.14) or (2.21)(Bernstein e t al . 1958).A sufficient condition for stability of the analogous Euler flowis given by Arnolds (1966a) condition

S2K > 0 , ( 8 . 2 )S2K < 0 (8.3)

for all admissible q , where S2K is defined by (3.11)or (3.13).The alternative condition

for all admissible q has been shown by the explicit examples of $5 and $ 7 to beinapplicable, as recognized in the context of two-dimensional flows by Arnold(1966b).If S2M and S2K are evaluated for the same basic equilibrium field, we haveshown (3.16) ha t, quite generally, the inequalityis satisfied.

The two-dimensional situation has been briefly discussed in $4. n this situation,there is a considerable simplification in th at S2K may be expressed explicitly in termsof q alone (and not involving the spatial gradient of q ) .This same simplification occursalso for the cylindrically symmetric situation considered in $ 5 . I n both cases, thispermits the extraction of a useful sufficient condition for the stability of the flow

S 2 M + S 2 K> 0 (8.4)


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Magnetostatic equilibria and Euler j lows of complex topology. Part 2 377((4.17)n the two-dimensional case, and (5.7)n the cylindrically symmetric case).The corresponding conditions for the magnetostatic problem are given by (4.14)nd(5.5)espectively. It is noteworthy in the cylindrically symmetric problem tha t if thevelocity component w(r) (see (5.2))s given by a power law

w( r )= A r A , ( 8 . 5 )-l


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378 H . K . MoffattARNOLD,. I. 1966a Sur un principe variationnel pour les Bcoulements stationnaires des liquidesparfaits et ses applications aux problemes de stabilite non-linkaires. J. Mec. 5 , 24-43 .ARNOLD,. I. 1966b On an aprior i estimate in the theory of hydrodynamical stability. Izv. Vyssh.Uchebn. Zaved. M atema tika 54 , 3-5 [translated in Am. Math. Soc. Tran sl. Ser 2. 79 , 267-269.19691.ARNOLD,. I. 1966c Sur la geometric differentielle des groupes de Lie de dimension infinie et sesapplications a hydrodynamique des fluides parfaits. Ann. In st . Fou rier, Grenoble, 16, 319-361.ARNOLD,. I. 1974 The asymptotic Hopf invariant and its applications (in Russian). In Proc.Summer School in Differential Equations, Erevan. Arm eni an SSR Acad Sci.BELTRAMI,. 1902 Opere Matemutiche. Milano.BERNSTEIN,. B., FRIEMAN,. A., KRUSKAL,. D. & KULSRUD, . M. 1958 An energy principleCHILDRESS,S. 1970 New solutions of the kinematic dynamo problem. J . Math. Phys . 11,3063-3076.DOMBRE, ., FRISCH,., GREENE,. M., HENON,M., MEHR,A. & SOWARD,. M. 1986 Chaoticstreamlines in the ABC flows. J . Fluid Mech. (in press).GALLOWAY,. & FRISCH,. 1984 A numerical investigation of magnetic field generation in a flow

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topology. Part 1 . Fundamentals. J. Fluid Mech. 159 , 359-378.MOFFATT,H. K. & PROCTOR,. R. E. 1985 Topological constraints associated with fast dynamoaction. J. Fluid Mech. 154, 493-507.MOLODENSKY,M. M . 1974 Equilibrium and stability of force-free magnetic field. Solar Phys. 39,3 9 3 4 0 4 .MOLODENSKY, . M. 1975 Equilibrium and stability of the force-free magnetic field 11.Stability.Solar Phys. 43, 311-316.MOLODENSKY, . M. 1976 Equilibrium and stability ofthe force-free magnetic field111.Solar Phys .49 , 279-282.WOLTJER, . 1958 A theorem on force-free magnetic fields. Proc. Natl Acad. Sci. 44, 484-491.

for hydromagnetic stability problems. Proc. R . Soc. Lond. A 24 4, 17-40.

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