A.Y.K. Chui and H.K. Moffatt- The energy and helicity of knotted magnetic flux tubes

8/3/2019 A.Y.K. Chui and H.K. Moffatt- The energy and helicity of knotted magnetic flux tubes

1/21


2/21

610 A . Y. K . Chuz and H. K . Moffat tA particularly interesting situation arises if the field B is confined to a single

closed flux tube which may be knotted in the form of a knot K (for examples, seefigures 4, 8, below). We shall suppose th at the field lines lie on a family of surfacesx = const. (0 < x < 1) nested around the axis C of the tube (which is itself a closedfield line in the form of the knot K ) . Let V be the volume of the tube and let @ bethe flux of B across any cross-section. It has been shown (Moffatt 1 9 9 0 ~ )hat if sucha tube is constructed in a standard way with uniform axial field in the tube and netangle of twist 27rh of the field around C (a construction that will be made precisein 54 below), then the helicity of the field is 3-1 = h@ and the minimum energythat can be attained under frozen-field volume-preserving distortions (isotopies) (asrealizable in an incompressible fluid) has the form

M,,, m(h)g2v-l3, (1.1)where m(h) s a function determined (in principle) solely by the topology of K . Non-trivial topology of K and/or non-zero twist h guarantee tha t m(h)> 0 (Freedman1988). An explicit lower bound on m(h) n terms of crossing number is given byFreedman & He (1991). Equation (1.1) s in effect a dimensional necessity, the quan-tities @, V and h being invariant during the distortion process. The dimensionlessfunction m(h)may be described as the ground-state energy function for K ; moregenerally, there may exist a set of functions

{mo(h) ml ( h ) m2 ( h ) ( 1 . 2 )corresponding to different local minima, which may be described as the energyspectrum of K .Determination of the function m(h) or given K is a fundamental objective which isbeyond the power of current super-computers. We address the problem in the presentpaper by placing additional mild constraints on the field structure, by means of whichconsiderable analytical progress may be made. Specifically, we shall suppose (in $ 5 etseq) that the magnetic surfaces x = const. are of circular cross-section and invariantalong the tube axis. These assumptions permit reduction of the expression for Mto a form for which computational minimization is straightforward. We illustratethe procedure in $ 8 for the family of torus knots. Due to the additional constraintsintroduced, the function m(h) btained by this technique can be regarded only as anupper bound for m ( h ) ; ut we believe that this provides a reasonable approximationto m ( h )provided lhl is not too large, and that the behaviour revealed by m(h) sthen qualitatively correct .

The key technique in our approach is the use of a non-orthogonal coordinate system(s,x, ) specially adapted to the problem; s is arclength on C, x is the flux functionintroduced above, and 4 is a polar angle in the cross-section of the tube with thespecial zero-framing property that the ribbon 4 = 0 is untwisted, i.e. the curveC ( x = 0) and the curve {4 = 0 , x = 1) have zero linking number. This essentialproperty allows for an unambiguous decomposition of B into toroidal (or axial) andpoloidal (or meridional) ingredients, and the helicity FI of the field is then expressibleas a function of toroidal and poloidal fluxes (equation (3.14) below). A result of thistype was obtained by Kruskal & Kulsrud (1958) and by Berger &. Field (1984); butthe present paper makes it clear that the result is valid for knotted tubes only if thetoroidal/poloidal decomposition is defined via the zero-framed coordinate system asindicated above.This paper provides a natural development of an earlier preliminary study (Chui &Proc R Soc. Lond A (1995)


3/21

Knotted magnetic f lux tubes 611(a) (b) Section s = cs t

\ . ...... .,Figure 1. (a ) Portion of a flux tube 7 with magnetic axis C and magnetic surfaces x = const.( b ) Section s = const. of flux tube.Moffatt 1992) in which the ground-state energy function m(h)of an unknotted closedflux tube was determined, on the assumption t hat the minimizing configuration is axi-symmetric. Kink mode instabilities may invalidate this assumption. The techniquesdeveloped below are well-adapted to the identification of new helical equilibrium (ornear-equilibrium) states consequent upon such instabilities. This problem is brieflytreated in 5 6 for two cylindrical configurations (free and line-tied),and analyticalresults are obtained which are in broad agreement with earlier studies of Anzer (1968)and Hood & Priest (1979). This theme will be more fully developed in a separatepaper.

2. Flux coordinates and zero-framingOur first objective is to introduce an appropriate coordinate system to describe a

magnetic field B which is confined to a closed, possibly knotted, flux tube 7 of smallcross-section. Obviously we must suppose tha t this tube is nowhere self-intersecting.We shall suppose further that the lines of force of B (or B-lines) ie on a family ofnested surfaces x(z) const. within 7 , here x akes values in the range 0 < x < 1.These surfaces will be denoted S,. The innermost (degenerate) surface x = 0 is aclosed curve C , the magnetic axis (figure la). The outermost surface x = 1 is 8 7 ,the boundary of 7 .A possible choice for x is

x = V / V 7 ,where V is the volume inside the surface labelled x and V7 is the volume of thewhole tube 7 ; ut clearly this is not the only possibility.

Let s represent arclength on C measured from some arbitrary point of C , and letX(s) be the parametric representation of points of C. Since C is a closed curve,X(s+ L ) = X(S), (2.2)

where L is the length of C . Let c(s), ~ ( s )e the curvature and torsion of C . Weshall suppose, unless otherwise s tated. tha t

c ( s )> 0 (for all s ) , (2.3)i.e. that C has no points of inflexion. Let t = dX/ds be the unit tangent vector,Proc. R. oc . L o n d A (1995)


4/21

612 A . Y. K . Chui and H. K . Moffat t(a) n

a ribbon ofconstant 0 /constant 4 (4 = 0)

Figure 2. (a ) Ribbon 8 = const. with boundaries C an d re which have linking number N . ( b )Ribbon 4 = const. with boundaries C an d I74 which have zero linking number.n the unit principal normal, and b = t A n the unit binormal, satisfying the Frenetequations

-rnc n , - - c t + r b , - -n dbtds ds ds-Clearly an arbitrary point z in 7 may be expressed in the formz = X ( s ) + r c o s Q n + r s i n Q b , (2.5)

where ( T , Q ) are plane polar coordinates in the plane through X(s) defined by thevectors (n ,b) with Q measured from the direction of n (figure l b ) .

Consider a ribbon surface Q = const., 0 < x < 1 (figure 2a). The boundaries ofthis ribbon are the curve C and its curve of intersection I0 with 87 . If the tube isknotted. then in general C and I0 are linked. Let N be the Gauss linking numberof (C,Fe) which may be positive or negative (or exceptionally zero); in constructingN, we adopt the same orientation of C and I70 going the long way round the tube7 .

We now define a new angle variable@ = Q + a n N s / L . ( 2 . 6 )

This has the property that the ribbon q = const. (i.e. Q = const.-anNs/L) isuntwisted in the sense that its boundaries C at x = 0 and F# at x = 1 have zerolinking number (figure 2b) . Choice of 4 (rather than Q) as the appropriate anglevariable is what topologists describe as zero-framing.

Through each point ( s , , q5), there is a unique magnetic surface, i.e.x = x ( s , ,4 )

T = R(s ,x, )

(2.7)

(2.8)We shall suppose that we may invert uniquely for T in the form

(This requires merely that the surfaces x = const. are not folded upon themselveswithin 7 . )t is convenient to use x as a coordinate in place of T , because by definitionB .Ox = 0, so that B has no component in the direction of V x . R(s ,x, ) may bedescribed as the shape-function of the flux tube.We now adopt (s,x, ) as our system of coordinates, so that (2.5) becomesz = X ( s ) + R ( s , x , $ )o s ( 4 - 2 n N s / L ) n ( s ) + R ( s , ~ , @ )in(4-2nNs/L)b(s). (2.9)

Proc. R Soc. Lond. A (1995)


5/21

Knotted magnetzc flux tubes 613Let us consider the fundamental properties of this system (for relevant backgroundconcerning non-orthogonal coordinate systems, see Bradbury 1984). Note first that

d z = e l d s + e z d x + e 3 d $ , ( 2 . 1 0 )where the basis vectors el , e2, e3 are given by

(2.11)

(2.12)1l = (1 - Rc cos 8 ) t + (R, cos 8 - R r *sin 8)n+ (R, sin 8 + R r *cos 8 ) be2 = RX(cos8n+ sin8b) ,e 3 = ( - R s i n 8 + R q c o s 8 ) n + ( R c o s ~ + R ~ s i n 8 ) b ,

with8 = $ - 27rNs/L, T * ( s ) = r ( s )- 27rN/L.

Note that el is dimensionless. while e2 and e3 have the dimensions of length. Themetric tensor is

R,R, R; RxR6 ) (2.13)so that the coordinate system ( s , x . + ) s clearly in general non-orthogonal. Thedeterminant of ( g z J ) s

g = R2R i (1- R c c 0 ~ 8 ) ~ . (2.14)and the Jacobian of the transformation to coordinates (s,x. $) is

(1- R cc0 s 8 )~ R 2 r*2+ RZ R,R, R2r*+ R,R$( R 2 r*+ R,R$ R,R$ R2+ R$( g J = (et . e3)=J = ( e l ~ e 2 )e 3 = fi = RR,(1 - Rccos8). (2.15)

Thus J > 0 and the transformation is well defined provided~ ( s )1 all s , all z E 7 , (2.16)

i.e. provided 7- has sufficiently small cross-section. We shall assume (2.16) to besatisfied.Since B . V x = 0 , the field B has only two components relative to the system(2.17)

( s .x . $1: B = Blel + B3e3.The condition V B = 0 implies that

Hence there exists a flux function $(s. x, 4) such that(2.18)

(2.19)By analogy with the terminology for an unknotted flux tube in the form of a torus,we define

BT = B1el , B p = B3e3. (2.20)as, respectively. the toroidal and poloidal ingredients of B . Let T(x) be the toroidalflux across any section s = cst. inside S,, and let P ( x ) be the poloidal flux acrossProc. R. Soc. L o n d . A (1995)


6/21

614 A . Y. . Chui and H. K . Moffatt

F ig u re 3. Siefert surface C for the trefoil knot. in tersecting the tube surface in the curve r:C"i s that par t of C outside 7 .any ribbon d = cst. bounded by the magnetic axis C and S,. Then (cf. Bateman1978, pp. 127-128) - 0 dT s d P

27r d x L d xq ( s ,x,d) = $ ( s , x,d) -- - - - (2.21)where $ ( s , x, 4) is a single-valued function, periodic in s (with period L ) and in 0(with period 27r). Hence from (2.17) and (2.19).

( 2 . 2 2 )1L. -P'(x)Here, the first term is the 'mean' field with prescribed fluxes. and the second termis the 'fluctuation' field with zero net fluxes.

3. Th e helicity of the flux tubeThe helicity of the field B s

3-1= A . B d V ,J.where A is a vector potential of B. i.e. B = curl A . We must impose upon A acondition that guarantees that the magnetic flux threading the space I* utside I szero. Let C be a Siefert surface (two-sided and non-self-intersecting) of C (figure 3 )(see, for example. Scharlemann 1992). Let T be the curve of intersection of C and8 7 . The linking number of any closed curve C* with C may be computed as thealgebraic sum of the number of crossings of C by C* (taking account of the senseof crossing). If C* lies wholly on C . then the linking number of C and Cx is zero.Hence in particular the linking number of r and C is zero. and so r may be deformedcontinuously on d l o coincide with I'9, the intersection of o = 0 and dl.Now letC" be the part of C bounded by To nd lying entirely in I*.lainly, the flux of Bacross C" is zero, i.e.

This is the condition that A must satisfy: subject to this constraint and to prescribedtoroidal flux @. the helicity integral (3.1) is gauge-invariant. (Rote that ( 3 . 2 ) issatisfied for knotted tubes precisely because the linking number of To nd C is zero;it is no t satisfied if coordinates (T , x. 0) are used instead of ( T , x.o).)The form ( 2 . 2 2 ) of B allows us to seek a vector potential of the form A =Proc. R. Soc L ond . A (1995)


7/21

Knotted magnetic flux tubes 615( A l : 0: A3) where: from B = curl A,

Let(3.3)

(3.4)like 4. this function is single-valued and periodic in s and 6. rom (3.3) and ( 2 . 2 2 ) ,we then find

(3.5)11 = - q s ( s . X. 6) L-lP(x) + Cl(s.6).A3 = - ~ Q ( s . . d) + ( 2 ~ ) - l T ( x ) C ~ ( S .).where C1 and C3 are 'constants of integration'. Choosing C3 = 0 ensures that A iscontinuous at x = 0: and choosing C1= P ( l ) / L ensures that the condition (3.2) issatisfied. This gives then -A1 = - q s + P I L . A3 = - ~ 4 T / 2 r , (3.6)where P(x) = P(1)- P(x) is the 'complementary' poloidal flux across a ribbon6 = const. outside S,.Now let

7-l(x*)= l


8/21

616 A . Y. K . Chui and H. K . MoffattThus the helicity inside each magnetic surface S, is determined solely by the toroidaland (complementary) poloidal flux functions T ( x ) nd P ( x ) .A result equivalent to (3.12) was obtained by Kruskal & Kulsrud (1958). and byBerger & Field (1984) by an argument based on linkage of field lines for the caseof an unknotted tube. From the discussion of this section, however. it emerges thatthe result is true also of a knotted flux tube, provided the poloidal field is properlydefined through use of a zero-framed coordinate system.

The total helicity is(TP- T P ) X = 2 (3.14)

using integration by parts. For a field with uniform twist h. P ( x ) = hT( x ) ,and(3.14) becomesXtot= 2 h l T ( x ) T ( x ) x = h@. (3.15)

where Qj = T(1) is the total toroidal flux. The result (3.15) is already known forunknotted twisted flux tubes (Moffatt 1990~):ere we have shown it to be equallytrue for knotted flux tubes. provided again that the poloidal flux is defined as theflux through a ribbon (a5 = cst.) which is not twisted with respect to the tube axis.

We can now show explicitly that the helicity of a purely toroidal field B1el is zero.for then

1

so that(3.16)

(3.17)for some f ( x .6) eriodic in 6. ut 4 s periodic in s. hence dP/dx = 0, i.e. P G 0.Hence from (3.13).X ( x ) = 0. Similarly for a purely poloidal field T = 0 and so againX ( x ) 0. Again, we emphasize that these results hold only by virtue of the factthat the poloidal field is defined with respect to the zero-framed coordinate system.

4. Standard flux tu be with prescribed helicityIt is useful in what follows to define a .standard flux tube knotted in the form of

an arbitrary knot K , carrying toroidal (or axial) flux @, and having prescribed totalhelicity h @. A technique for achieving this using Dehn surgery (cutting. twistingand reconnecting) was described by Moffatt (1 9 9 0 ~ ) . ere, we obtain the form of theresulting field B eferred to the system of coordinates ( s ,x .6) ntroduced above.

We first suppose that the shape function R(s ,x. 6) is independent of both s and@. so that (2.8) becomes simply

The tube is then uniform in structure along its length, and all magnetic surfaceshave circular cross-sections ( s = const., x = const.) . The volume of the tube insideS, is

T = R ( x ) . (4.1)

V ( x )= lLs lzTd l x i d ~ r R 2 ( x ) L . ( 4 4Proc. R. Soc. L o n d . A (1995)


9/21

Knotted magnetic flux tubes 617

Figu re 4. Trefoi l knot in the pulled-t ight configurat ion for which the radius of cu rva tu r e p ( s )of C is nowhere less than 2eL.We choose V = xV7 (see (2 .1) ) , so that

where E = (I+/7rL3)li2. In practice, for any knotted flux tube, the parameter E issmall. Consider, for example, the trefoil knot in the pulled-tight configuration shownin figure 4. The cross-section is assumed circular with radius EL. The constraint ofnon-self-intersection implies that the radius of curvature p ( s ) of the magnetic axismust be nowhere less than EL. The tube can be cut into three equal portions eachof length L/3 M pA0 where A0 2 3 ~ / 2nd p > EL. Hence, for this case,

E < (97r)- = 0.035. (4.4)For any more complex knot, E will be even smaller.The standard flux tube is now defined by the choices of T ( x ) , ( x ) :

(4.5)To construct, a particular field Bo with this signature, we substi tute (4.1). (4.3)and (4.5) in ( 2 . 2 2 ) . aking 4 0. This gives(L :0,7rh)./hBo = 1- R ( x ) c ( s ) os(# - 27rNs /L)

Here the components BA and B: have different dimensions, but B;el and Bie3 havethe same dimensions (those of flux divided by area) as required. The total helicityof the field Bo is. from (3.15). given by3-1 = h@. (4.7)

The parameter h may be interpreted as ( 2 ~ ) - imes the angle of twist requiredto generate the poloidal field (4.5b) from the toroidal field (4 .5a) , starting from asituation with P = 0. We may therefore refer to h as the twist parameter of thestandard flux iube.5 . Minimization of magnetic energy

Consider now the magnetic energy of the fieldM = - l B d V ,2

Proc . R. Soc. L o n d . A (1995)


10/21

618 A . Y. . Chuz and H. K . MoffattWe suppose that the flux tube 7 is immersed in a perfectly conducting incompressiblefluid: its signature

is then invariant under volume-preserving frozen-field distortions (Moffatt 1990b) .Moreover the knot type IC of the magnetic axis is invariant. and the tube 7 must notintersect itself. We seek to minimize M subject to these constraints, the minimumthen corresponding to a state of stable magnetostatic equilibrium. Kote that thevolume V ( X )s given by

s = { V ( X ) .T(X) P(XH ( 5 4

or equivalently, using ( 2 . 1 4 )dvX = l Z T l i R R X ( lR c ( s ) ) c o s ( @ - 2 ~ N s / L ) d s d 4 . (5.4)

Also, definingbl = & B1= -Gp + T / ~ T b3 = & B3= G s + P /L ( 5 . 5 )

we have

M is a function of 6 (through bl and b3). of R (through the constraint ( 5 . 4 ) )and ofX ( s ) (through g z J ) ; he problem is thus to determine 6, and X ( s ) uch that Mis minimized subject to the above constraints.

This variational problem is extremely difficult. and in order to make progress itis necessary to impose additional mild constraints on the class of field structuresto be considered. The minimum energy thus obtained will be greater than the trueminimum. but will at least provide an upper bound on this. The field structuresdetermined by this procedure are constrained magnetostatic equilibria: if the con-straints are removed. then in general the field will evolve to a lower energy state oftrue magnetostatic equilibrium. This evolution may involve simply a small adjust-ment. or it may involve an instability with relatively large associated reduction ofmagnetic energy. We shall comment further on these possibilities below.

( a ) Axzally uniform f lux tubesThe first additional constraint that we impose is that the flux tube be uniformalong its length, i.e. that 4 nd R be independent of s.The magnetic surfacesx = x ( T , ~ ) re then the same at every section s, nd B depends on s only through

the dependence of g t J on c ( s ) and T ( s ) .Under this assumption. ( 5 . 5 ) givesbi A ( x , @)+ T ( x ) / ~ T ,3 1 ( x ) / L . (5 .7 )

where A ( x ,4 ) = -4p; ince G s periodic in 4,i2=X , 4 )d 4 = 0 .Since bl and b3 are now independent of s3 5 . 6 ) now gives

IL L = [ 6 1 l b : + 2613 bl b3 + ~ 3 3g)2Proc. R. Soc . L ond . A (1995)


11/21

Knotted magnetic flux tubeswhere

Substi tut ing (5.7) in (5 .9))we now obtain M in the formM = 21AA2+ 2B A + C) dX d 4 ,

619

(5.10)

(5.11)where

We may now partially minimize M with respect t o A ; subject to the const ra int (5 .8) .Th is is achieved in straightforward m an ne r by introducing a Lagran ge multiplier X(x)an d m inimizing

;(AA2+ 2 B A + C )- X(x)A(x,4 ) . (5.13)Th is gives

X - BA = -Hence the p artially minimized energy is obta ined in th e form

where

and

(5.14)

(5.15)

(5.16)

(5.17)Th e uniform flux tu be constraint seems reasonable for an y flux tu be whose cross-sectional radius 6 s everywhere small compared with l /c(s) ; however, if the con-straint is removed, the tu be may be su bject t o ballooning instabilities or t o kink-mod e instabilities which increase c (s ) : in either case, evolution t o a significantly lowerenergy state may occur.

( b Flux tubes wzth czrcular cross-sectaonThe second additional constraint that we may impose is that the shape functionR be independent of 4: then (cf. (4.1)-(4.3)). with th e choice V (x ) = XVT, th e

constraint (5.4 ) gives R(X) = ELXl2, (5.18)Proc. R. oc . L o n d . A (1995)


12/21

620 A . Y. K . Chua and H. K . M o f a t t4 L/N(1+0)

N s )Figure 5 . Helical flux tube with constant curvature and torsion; periodic end conditions areimposed. For the free situation, L is constant: for the line-tied situation, zo = L(l+ o2) - I 2is constant.where E = ( v ~ / . i r L ~ ) ~ / .nd we have seen in 54 that. for any knotted tube. E isnecessarily quite small. The metric tensor (2.13) then simplifies to

(1- E L X ~ / ~ C C O S Q ) ~E2L2Xrr2 o E ~ L ~ X T *c2L2Xr* 0 E2L2X

(5.19)e 2 / 4 ~ 0g23) =

andfi=E2L2(1 -ELX1/2CCOSQ). (5.20)

The circular cross-section constraint also appears to be reasonable providedSc(s) 0 ; c > 0 ; so that the helix is right-handed.This curve is not closed, but i t satisfies the periodicity condition X ( s+ L ) = X ( s ) ,and this is sufficient for present purposes. The unit principal normal isn ( s )= ( -cosks , -s inks , 0) ( 6 . 3 )Proc. R. oc. L o n d . A (1995)


13/21

Knotted magnetic flux ubes 621and this makes N complete rotations as s increases from zero to L . The curvatureand torsion of C are easily found to be

( 6 . 4 )ka k, r = JiTT= dT-T-3and both are constant.We consider a flux-tube that is uniform in s and of circular cross-section, so thatas in $ 5 ,The only restriction that we place on E is ( 2 . 1 6 ) ) .e.

3 1 1 2R(X)= ELX1 / 2 , E = (VT/7rL ) .p = E LC < 1 .

We wish to evaluate K~~ exactly; thus

Similarly,

Since the ~ i jre independent of 4, we now obtain from ( 5 . 14 ) - ( 5 . 16 )

which can be further reduced to

using (6.7) and ( 6 . 8 ) ,where p is defined by ( 6 . 6 ) .The structure of this expression now indicates the manner in which a straight tube(for which a = 0) may be unstable to helical deformation. As a increases from zero,the quantityr* = r - k = k [ ( l + a2)-'I2- 11 ( 6 . 11 )

varies between zero and - k , and the consequent variation of the factor2( r * 271/ + ;

in ( 6 . 10 )may be such as to decrease ILL* .hT' = h @ ) or which ( 6 . 10 )reduces further toThis may be seen explicitly for the case of the standard flux tube (where PI =

(1 + 2L2(T*+ 2 r h L ) 2 F ( p ) }227r2LM * =-where

( 6 . 12 )

( 6 . 13 )We now consider two different cases: ( a ) instability of a flux tube with constantlength and ( b ) instability of a line-tied flux tube.Proc . R. Soc. Lond. A (1995)


14/21

622 A . Y. K . Chui and H . K . M o f a t t

0 1240 122 I- _ _ _ _ -_ - -j

0 01 0 2 0 3 0 4 0 5P P

Figure 6. The energy function m*(p) for the free state with n/ = 1 , E = (47r- ' , for variousvalues of h. (a)Note the loss of stability at p = 0 (i.e. U = 0) as h increases through unity.( b ) Note the appearance of a new constrained equilibrium stat e when h increases through thecritical value h, = 1 2 (from (6.18)).( a ) Flux tube with constant lengthIn this case L is a constant; the function F ( p ) is monotonic increasing in the range

F ( 0 ) = 1 1 2 , F(1)= 4 / 3 . (6.14)Hence, if T - +27rh/L vanishes for any value of T in the accessible range (0 , k ) ) henM * is minimal for that value (and for the corresponding value of 0).This conditionis satisfied if

0 < h < L k / 2 ~ N ; (6.15)

0 < p < 1 between its limiting values

and the corresponding helical configuration is then characterized by the values(h2- 2Nh)1/2

A f - h27r 2TL (h2- 2 N h ) l l 2 0 = (6.16)= L ( N - h ) ) c = -

The cylindrical sta te with 0 = 0 is an equilibrium, but this is unstable if(dM*/dc2),=o< 0 . (6.17)0 < hNE2< 3/47r2. (6.18)

This type of instability has been found previously (Anzer 1968). However the alter-native lower energy states (6.16) (for N = 1 , 2 , 3 , . .) are believed to be new.

By way of illustration. consider the particular case N = 1,E = (47r - l . The criticalvalue of h as given by (6.18) is h, = 12. The corresponding range of p is 0 < p < i ,and (6.12) gives

From (6.1 1), (6.12) and (6.13), this criterion yields as a condition for instability

= 1+ i ( d m h - ) F ( p )* =-M *87r Q2 (6.19)Figure 6 shows m* as a function of p for various values of h, indicating the mannerin which the minimum energy state changes as h increases. Note, however, that thisis just for N = 1. Other modes with N 3 2 may be unstable for values of h for whichthe N = 1 mode is stable.The expression (6.10) may be used to analyse the stability of flux tubes of ar-bitrary signature { T ( x ) , ( x ) } .This topic is important in the context of plasmacontainment: and will be treated in a separate paper.Proc. R. Soc. Lo n d . A (1995)


15/21

Knotted magnetic flux tubes 623~ 1 0 . ~

8 . / h1 / h

= 5.00= 5.03= 5.06= 5.09

Figure 7 . The energy function m*(L)- m*(l ) for the line-tied state with Af = 3 , V = n/400.Note the change of stability of the state L = 1 (Le. ~7= 0) when h increases through the criticalvalue h, = 5.03.( b Line-tied equilibria

T h e situ atio n is rath er different if th e field lines a re tie d a t fixed planes x = 0L = x o m , (6 .20)

and we must take account of this variation of L with LT in considering the energyfunctional M * . We may take as unit of length xo = 1, so t h a t L 2 1. Wi t h E =( V / T ) / ~-3/2, 6 .12 ) gives

and x = xo (as considered by Hood & Priest 1979). In th is case,

2m* = -_v M * ~2 +___TN2v(/= + $ - 1) F ( p ) (6 .21)sp2

and p is related to L throughp = ELC ~ ( T V ) ~ / L T ( ~ +T ) - ~ / *= ~ ( T V ) ~ / N ( L ~ L-5 /2 . (6 .22)

The quan t i ty m * ( L )- m*(1) s plotted in figure 7 as a function of L for N = 3 ,V = 7r/400. This indicates a change of stability of the cylindrical state ( L= 1) whenh increases throug h t he critical value h, x 5.03. For h > h, a new constrained helicalequilibrium is established as indicated by th e corresponding minimum of m* ( L ) .Th ere ar e essentially thr ee con straints in the above tre atm en t of helical flux tube s:(i) the uniform flux co nstrain t; (ii) the circular cross-section co nstrain t; and (iii) thecon strain t th at th e magnetic axis should b e one of th e two-parameter family of he-lices (6 .1 ) . If con stra int (i) alone is removed, the equ ilibrium should survive because auniform flux tu be is compatible with uniform curv ature an d torsion. If co nstrain t (ii)alone is removed, then a small but uniform adju stm ent of cross-section is to b e ex-pec ted; and if t he con strain t (iii) alone is removed, then superkink typ e instabilitiesmay conceivably deform C out of the two-parameter family ( 6 . 1 ) , with a possiblylarge reduction of magnetic energy.7. Expansion of M * in powers of e2

As observed in 4, the parameter E = ( V . / T L ~ ) ~ / ~s necessarily small for a kn otte dflux tub e. It makes sense therefore t o expand th e expression (5 .14 ) for M * as a powerProc. R. Soc . L ond . A (1995)


16/21

624series in E. We find that

A . Y. K . Chui and H . K . Moflat t1M * = - (2 mo+ m 2 ~ '+ . . .) ( 7 4

where th e coefficients mo and m2 are determined below (equations (7.12) an d (7 .13 ));here and subsequently, + . . indicates term s tha t may b e neglected.Firs t, we need t o calculate r ; l l (x ,4 ) . From (5.17) an d (5.1 8), we have( 7 4

Here,cos 0 = cos(+ + 27rNs/L) = co s + cos(27rNs/L) - in + sin(2-/rNs/L) , (7.3)

so t h a t

= -(1- $EX112L(aNcos+- b N s i n + ) + e 2 L 2 X f r * ' d s + . . . ) ) (7.4)2L

where aN and bN are the N t h Fourier coeff icients of c(s), i.e.C(S) C O S ( ~ T N S / L )s ) bN = - iL(s) sin(2-/rNs/L)d s . (7.5)

whereCN = a,%( ibN = - c(s) e x p ( 2 7 r N s l L ) d s .i L L (7.7)Secondly, we need to calculate ~ 1 3nd ~ ~ 3 ;owever, since these ar e bo th 0(1), eneed only calculate the leading terms of the integrals

dq5 = 27rc2Lx r*d s + . . ,f12nr ; 33 - %) + = 27rxL + . (7.9)For simplicity, we now restrict attention to the standard flux tube for whichP' = hT' = h@ . Subst itu t ion of (7.6 )) (7.8) and (7.9) in (5.15) and (5.16) thengivesH(x)= 47rh2 G 2 x /L + . . . . (7.11)

Hence finally from (5.14))we ob tain th e result (7.1) withmo = @ ' / 2 7 r ~ , (7.12)

Proc. R . Soc. Lond . A (1995)


17/21

Knotted magnetic f lux tubes 625

Figure 8. Flux tube in the form of the torus knot T4,s.

(7.13)Once again, we note the appearance of the factor (7 + 2 ~ h / L ) ~cf. equation (6 .11 )for the case of the helix). Thus, again, M * is decreased by deformations that reducethe mean-square value of r*+27rh /L on C. Now, however, we note an additional new,and somewhat surprising, phenomenon: an increase of curvature on the scale L / 2 7 r Nincreases ICN / and therefore tends to decrease m2. Under general deformations of C ,however, there may be a trade-off between the positive and negative contributionsto (7.13), and it is the balance between these effects that determines the minimizingconfiguration.We have supposed throughout that C has no points of inflexion. We may now,however, consider what happens if C is continuously deformed in such a way thatan inflexion point s = s* appears at one instant t = t*, say. As shown by Moffatt& Ricca ( 1 9 9 2 ) , both N and ( 2 ~ ) ~ ~r d s are discontinuous in the passage throughthis inflexion, but in just such a way that the quantity

(7.14)varies continuously. However, the integral f * ds diverges as C approaches theinflexional configuration, so that 6 1 1(x, ) is then undefined. The coordinate system(s,x, ) is simply not appropriate to deal with this situation.

8 . Torus knot flux tubesLet us now apply the above results to the situation in which the magnetic axis Cis a torus knot Tm,n ith parametric representation

X ( t ) = R ( ( I + Xcosnt)cosmt, (I + Xcosnt)sinmt , Xsinnt)ds = RdX2n2+ m2(1+ Xcosnt) dt ,

(8.1)for 0 < t < 27r (figure 8 ) . We suppose X > 0 so that the knot is left-handed. Notethat the arc length s is related to the parameter t by

( 8 . 2 )and the length of C is given by

277L = R i JX2n2 + m2(1+ Xcosnt)2dt = Rl(X) , say. ( 8 . 3 )We need first t o compute the separation function4u14 = IX(4- X ( 4Proc. R . Soc. L o n d . A (1995)


18/21

626(4

A . Y. . Chua and H . K. offatt(b )

d(Li,v) - ,n A

\ d = OFigure 9. ( a ) Local minimum d,,, of the separation function d ( u ,U) or the trefoil knot Tz 3.( b ) Surface plot of d ( u .U)= const. for T z , ~R = 1. X = 0.4).

(shown in figure 9 b for the case of the trefoil knot T 2 . 3 ) . This function has localminima for values of ( U ,U) where the curve approaches near to itself. Let d,,, bethe least of these minima (other than zero) (figure 9a). Clearly, d,,, has the form

d,,, = Rd( ) ( 8 . 5 )for some function d (X ) ;for small A , this function has the form d (X )M 2Xsin(.ir/m).

We may now construct a standard flux tube around C of radius EL. nd provided

it is clear that this flux tube is non-self-intersecting. In the pull-tight situationin which the toroidal energy is minimized, the tube makes contact with itself, i.e.EL= d,,, : this contact condition may be written in the equivalent form

and thus, for given VT! etermines R in terms of A .The energy function M * given by (7 .1) . (7.12) and (7.13) may now be readily

computed as a function of R and A, and hence using (8 .7) as a function of X alone:we may then minimize with respect to X obtaining M,i, in the formMlnin= m(h)CP~V;~~. (8.8)

The function m(h) s shown in figure 10 for the two representations T2;3and T3,2fthe trefoil knot. A s anticipated, these curves exhibit minima for non-zero values ofh,(hx 6 , E m i n M 32 for T 2 , 3 ;h M 0.11, mminM 36.3 for 7 3 . 2 ) consistent with theconjecture of Moffatt (1990a).Note however the unexpected result that for h 5 2.3,the T3,2epresentation has lower energy than the T 2 , 3 representation.

Figure 11 shows the function m ( h ) or torus knots T2,nn 3 , 5 : 7 , 9 ) .For mod-erate values of h ( 5 1 0 ) the energy increases with n> .e. with knot complexity. Forlarger values of h the results are unreliable: because the contact condition (8.7) neednot be satisfied: moreover, just as for the case of the helical flux tube, the torus-knottube may be subject to local kink-mode instabilities so that the simple representa-tion (8.1) of the magnetic axis is no longer valid. These more complex effects will beconsidered in a separate paper.P ro c . R. oc . L ond . A (1995)


19/21

Knotted magnetic flux tubes 627

Figu re 10. T he m in imum ene rgy funct ion m(h) or two representat ions of th e t refoi l kn ot , T 2 . 3a n d T 3 , 2 .

160140

806040

hFigu re 11. Th e m in imum ene rgy funct ion E ( h ) or torus knots T z , , (n 3 ,5 ,7 ,9 ) ; or h 5 10,th e knot energy increases w i th increas ing knot complexi ty .

9. Summary and conclusionsIn this paper, we have developed a general technique whereby constrained mini-

mum energy states of knotted magnetic flux tubes may be identified. This techniquefirst involves definition of an appropriate (non-orthogonal) curvilinear coordinatesystem in the neighbourhood of a closed curve C , under the condition that C has nopoints of vanishing curvature. This coordinate system is zero-framed with respectto C. with the consequence that a natural toroidal-poloidal decomposition of thefield B in a flux tube 7 around C can be established. The field is assumed to havemagnetic surfaces x = const. nested around C (the magnetic axis) and is charac-terized by a signature { V( x). T( x) ,P(x)}where V is volume, T s toroidal flux andP poloidal flux inside the surface labelled x. The decomposition is natural in thatthe helicity of either a purely toroidal or a purely poloidal field is zero; the helicityfunction X ( x ) is given by equation (3.12), he appropriate generalization t o knottedflux tubes of a result first discerned by Kruskal & Kulsrud (1938).

In 5 5 , we have used the above coordinate system in order to obtain an expressionfor the magnei ic energy JUof the field B in terms of its signature and the parametricrepresentation of C . Following the variational principle of Bernstein et a1 (1938), buthere applied to volume-preserving deformations, we have then sought to minimizeM while conserving both the signature of the field and the topology of C . To makeProc R. Soc Lond A (1995)


20/21

628 A . Y. K . Chua and H . K . Moffattprogress, we introduced two mild constraints, restricting attention to axially uniformtubes of circular cross-section: and we obtained explicit results for the case of thestandard flux tube, for which V ( x ) xV7, P ( x )= hT (x )= h@x; ere h is the twistparameter, and the total helicity of the field is hQ2.

We first considered (in 5 6) the case of a helical flux tube with periodic end condi-tions. The fact that both curvature c and torsion T are constant in this case allowsexplicit evaluation of the (constrained) minimum magnetic energy as a function ofthese parameters, or equivalently of the parameters ( k ,G) elated to (c, T ) by (6.4).New lower energy helical equilibrium states have been identified in circumstanceswhere the cylindrical magnetostatic equilibrium (0 = 0 ) is unstable. Criteria forinstability have been obtained for both free and line-tied cylindrical equilibria,in broad agreement with earlier investigations of Anzer (1968) and Hood & Priest(1979). These results inspire confidence in the validity and viability of the technique.

In 5 8. the technique is applied to situations in which the flux tube has the form ofa torus knot T,,,, the curve C having parametric representation (8.1).The magneticenergy of the standard flux tube depends on the parameters R and A , and these areassumed related by the contact condition (8.7). The energy is then minimized withrespect to the single independent parameter A, and the dimensionless energy func-tion m (h ) n the relation M,,, = m(h) 2V113 is thus determined for a variety oftorus knots of low order. These results. displayed in figures 10 and 11, are broadlyas anticipated by Moffatt (1990a). although the fact tha t, for small h , the value ofm(h) s smaller for T 3 , 2 than for T 2 , 3 (the trefoil knot in two different geometricalconfigurations) is unexpected. The function m(h)provides an upper bound on thefunction m ( h ) hat applies when the constraints on tube structure and on the con-figuration (8.1) of the axis are removed. Recall that a lower bound is provided bythe work of Freedman & He (1991).

The technique, as developed in this paper, may be applied to any knotted fluxtube of volume V7 whose axis C may be represented in parametric formx = X ( t ; . p , . . .) , (9.1)where t is the parameter on C (related to arc-length). and A,p, are further geo-metrical parameters on which the magnetic energy M * will depend, and with respectto which M * may be minimized, subject to a contact condition of the form

(9.2)The sole limitation on the technique is that the curve (9.1) must have no inflexionpoint (i.e. no point of vanishing curvature), since otherwise the coordinate systemused is ill-defined. A technique which can cope with deformation through inflexionalconfigurations provides an interesting. albeit elusive, target for future investigation.A.Y.K.C. thanks th e Croucher Foundation (UK) for a three-year scholarship. The work has beenpartly supported under EPSRC Contract GR/J21439.

ReferencesAnzer, U. 1968 The stabil ity of force-free magnetic fields with cylindrical symmetry in thecontext of solar flares. Solar Phys. 3, 298-315.Arnold, V. I. 1974 The asymptotic Hopf invariant and its applications (in Russian). In Proc.Su mm er School i n Differential Equations , Erevan. Armenian SSR Academy of Science.(Transl. (1986) in Sel. M ath. Sow. 5 , 327-345.)Proc. R. Soc. Lo n d . A (1995)


21/21

Knotted magnetic flux tubes 629B a t e m a n , G . 1978 MHD nstabilities. M I T P r e s s .Berger , M. A. & Field , G . B . 1984 T h e topological properties of m agnetic helicity. J . Fluid Mech.Bernste in, I . B., Frieman , E. A . , Kruskal , M. D. K u ls ru d , R. M. 1958 An energy principle forhydromagnetic stabil i ty problems. Proc. R . Soc. Lond. A 244, 17-40.B rad b u ry , T. C. 1984 Mathem atical m ethods with applications to problems in the physical sci-ences. Wiley.C h u i , A . Y . K . & Moffatt , H. K. 1992 Minimum energy magnetic f ields with toroidal topology.In Topological aspects of the dynamics of fluids and plasmas ( ed . H . K . Mo f fa t t et al. ) ,pp , 195-218. Kluwe r.F r e e d m a n , M. E[. 1988 A note on topology and magnetic energy in incompressible perfectlyconducting fluids. J . Fluid Mech. 194, 549-551.F reed man , M. H . & H e, Z . X . 1991 Divergence-free field: energy an d asy mp totic crossing num-bers. A n n . M a t h . 134, 189-229.H o o d , A . W. & P r i e s t , E . R . 1979 Kink instability of solar corona1 loops as the cause of solarflares. Solar Ihys. 64, 303-321.Kruskal , M. D . & K u ls ru d , R. M. 1958 Equilibrium of a magnetically confined plasma in a

to ro id . Phys. Fluids 1, 265-274.Moffatt , H . K. 1985 M agn eto stati c equil ibria and analogous Euler f lows of ar bitra ri ly complextopology. Part 1. Fundamentals . J . Fluid Mech. 159, 359-378.Moffatt , H . K . 19900. T h e energy spe ctru m of kno ts an d l inks. Nature 347, 367-369.Moffatt , H. K. 1990b Structure and stabil i ty of solutions of the Euler equations: a LagrangianMoffat t , H . K . & Ricca, R . L. 1992 Helici ty an d th e C &lug&reanu nvariant . Proc. R . Soc. Lond.Schar lemann , M . 1992 Topology of knots. In Topological aspects of the dynamics of fluids and

147, 133-148.

ap p ro ach . Phil. Trans. R . Soc. Lond. A333, 321-342.A 439, 411-429.plasmas ( ed . H . K . M o ffa tt e t a l . ) , pp. 195-218. Kluwer.

Received 24 D ecember 1994 ; accepted 6 Apri l 1995

A.Y.K. Chui and H.K. Moffatt- The energy and helicity of knotted magnetic flux tubes

Documents

Transcript of A.Y.K. Chui and H.K. Moffatt- The energy and helicity of knotted magnetic flux tubes