H.K. Moffatt Vortex and MagnetoDynamics: A Topological Perspective

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VORTEX AND MAGNETODYNAMICS  A TOPOLOGICAL
PERSPECTIVE
H.K. MOFFATTIsaac Newton Institute for Mathematical Sciences
20 Clarkson Road, Cambridge CB3 OEH
1 Introduction
The subject of vortex dynamics] within the broader field of fluid dynamics, was
initiated by the pioneering studies of Helmholtz (1858) and Thomson (Lord Kelvin)(1869) on the laws of vortex motion. These laws are encapsulated in Kelvin’s
circulation theorem which applies to the motion of an ideal (i.e. inviscid) fluid in
which pressure p and density p are functionally related, i.e. p = p ( p ) , and any body
forces acting on the fluid are irrotational. Under these conditions, Kelvin showed
that the circulation round any closed circuit C moving with the fluid is conserved:
IE = k u . dx = cst.
Here, u(x,t) represents the velocity field. The circulation IE may equally be ex
pressed as the flux of vorticity w = curl U across any orientable surface S spanning
c:
IC =1 .ndS,
and, since this applies to every material circuit C, including the boundaries of
infinitesimal surface elements, it is readily deduced that “vortex lines are frozen inthe fluid”, i.e. the velocity field w ( x , ~ ) s transported with the flow.
Kelvin immediately recognized that this result implied also the conservation
of any linkage or any knottedness that might exist in the vorticity field at some
reference instant t = 0. For example, if the vorticity field in a fluid is zero every
where except in a closed tube which is knotted in the form of a knot K , then this
topology of the vorticity field is conserved for all time; (it should be noted here
and subsequently that this sort of result holds only for so long as the fluid can be
regarded as truly inviscid; this is an idealisation that is never realised exactly in
practice] except perhaps in liquid helium I1 in which quantum effects provide al
ternative complications). It was this insight that led Kelvin to propose his ‘vortex
theory of atoms’ in which a correspondence is conjectured between atoms of differ
ent elements and knots of different knot types. This theory, although subsequently
abandoned] provided a powerful stimulus for the major study of the classification
of knots undertaken by Tait (1898, 1900) and in the subsequent development of
topology as a distinct branch of mathematics.
www.moffatt.tc
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2 Helicity and its topological interpretation
Remarkably, almost 100 years elapsed following Kelvin’s great paper before the
discovery of a n invariant of the Euler equations of fluid motion which is trulytopological in character and which indeed provides a natural bridge between fluid
dynamics and topology. This invariant (J.J. Moreau 1961, Moffatt 1969) is the
hel ic i ty of a flow, defined as follows. Let S be any closed orientable surface moving
with the fluid on which w . n = 0 , i.e. the vorticity field is tangential to S; S may
be described as a ‘vorticity surface’, a condition that clearly persists if it holds for
t = 0. The helicity of the flow in the volume V inside S is then defined by
3c = l u  w d V , (3)
and this quantity is conserved under evolution governed by the Euler equations. To
show this, it is best to express the Euler equations in Lagrangian form, viz.
where h = S p  l d p (the need for the ‘barotropic’ condition p = p ( p ) may be seen
here). The curl of (4 ) coupled with the equation of mass conservation in the form
leads to the vorticity equation in the form
Now from ( 3 ) , it is readily shown that
and, on using (4) ,and (6)., this reduces to
on using the essential condition n . w = 0 on S. Hence, 3c is indeed constant.
Note that this result does not require that the flow be incompressible, although
it does hold also in this special case (with p = cst.). In general, it holds under
precisely the same conditions that govern Kelvin’s circulation theorem: inviscid
fluid, barotropic flow, and irrotational body forces. It should be evident therefore
that 31 must admit a topological interpretation.
That it does so is best seen through consideration of the simplest possible ‘pro
totypical’ linkage of vortex lines: consider the situation in which w is zero except
inside two unknotted but linked vortex tubes of circulations “1 and 1c2 and of small
crosssections; and suppose that the vortex lines within each such tube are them
selves unlinked closed curves. Then 31 may be evaluated by first integrating across
the crosssection of each tube, then along their axes Cl and C2. The result is
31= f2nK1K2, (9)
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where n is the number of times that C1 winds round Cz before closing on itself (the
Gauss linking number of C1 and Cz), and the + or  is chosen according as this
linkage (which is oriented by the direction of vorticity within each tube) is right
handed or lefthanded. The velocity field U can be expressed in terms of vorticity
w by the BiotSavart law, and this leads to the wellknown expression for n as an
integral:
(dx1 dxz) . (XI  x2)
n = L f f Ixl  x2I3 (10)4 m c1 Ca
This is the fundamental topological invariant of the two closed curves C1 and C2,
and the bridge between topology and fluid dynamics is therefore established by the
simple result ( 9 ) .
The situation is not so simple when knotted, as opposed t o linked, vortex tubesare considered (Moffatt lk Ricca 1992). Suppose now that w is zero except in a
single closed vortex tube whose axis C is knotted in the form of a knot K . Suppose
that each vortex line in the tube is a closed curve ‘nearly parallel’ to C, by which
we mean simply that if C’ is one such curve, then C and C’ form the edges of a
closed ribbon of small width. Let N(s) be a unit spanwise normal directed from Cto C’ on this ribbon,where s is arclength on C; then the twist of the ribbon (see,
for example, Fuller 1971) is defined by
1T w =  N(s) A N’(s)) . d x .
2r c
This twist can be decomposed in two parts:
T w =& T ( S ) ~ S N
where ~ ( s ) s the torsion on C and N , n integer, is the number of rotations of
N(s) relative to the Frenet triad ( t ,n ,b) of unit vectors on C. If such a ribbon is
cut and one of the cut ends twisted through 2 x and then rejoined, then N changes
by kl, epending on the sense of twist. N may be described as the ‘intrinsic twist’
of the ribbon.
Now suppose that the vortex tube is ‘uniformly twisted’ in the sense that every
pair of vortex lines C’,C” has the same value of intrinsic twist N . Then the result
analogous to (8) is the following:
7l = hlc2 , (13)
h = W r + T w , (14)
where
Tw is given by ( l l ) , nd the writhe Wr is given by
( d X A d X ’ ) . (X X’)
7 (15)W r =  j f
4r 1 x113
i.e. by the Gauss formula but with the integral taken twice round the same curve.
Under continuous deformation of C, both W r and T w vary continuously, but their
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Figure 1. Conversion of writhe to twist through continuous distortion of a ribbon: (a) W r =1,Tw = 0; b) W r +Tw = 1; (c) Wr = 0, w = 1 [From Moffatt and Ricca (1992)].
sum is invariant (C dug Br ean u 1961, W hite 1969). Again therefore it is conservation
of helicity that actually underlies (via (14) and (15)) the essentially topological
invariance of writhe plus twist.
There is a further subtlety in relation to the decomposition of twist (12) in
which the first term depends only on C, while the second depends on the mutualconfiguration of C and C’. Under continuous deformation of the r ibbon, i t maya t discrete ins tan ts pass thr ou gh ‘inflexional configurations’, i.e. configurations
for which C contains an inflexion point at which the curvature C ( S ) is zero, and
th e torsion is undefined. As the ribbon passes through such a configuration, the
integral of the torsion ~ ( s )um ps by f 2 n , but there is a corresponding ju m p 71 in
the integer N , so t ha t t he s um in (12) varies continuously th rou gh th e tra nsition .T h e delicate interplay of writhe, torsion and intrinsic twist can be visualised in the
process of stretching a twisted ribbon (see figure 1).
3 Magnetic relaxation
For simplicity, let us focu s now on th e case of incomp ressible (or volumepreserving)
flow with p = cst . , for which V + U = 0 , and equation (6) may be writ ten in the
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equivalent form
dW/dt = v A ( U A W ) . (16)
This equation is of course nonlinear (through the dependence of w on U ) . It provesfruitful to consider an associated linear equation having a very similar structure,
namely
bB/dt = V A (V A B) (17)
where V .B = 0, V .v = 0 and B and v are otherwise independent fields. Equation(17) means that the field B(x,t) is transported by the ‘velocity’ field v(x,t ) , heflux of B through every material circuit being conserved. Equation (17) is in factthe equation satisfied by a magnetic field B in a perfectly conducting fluid moving
with velocity v. This interpretation may be helpful, but is by no means essentialto the argument; we shall however use the terminology of magnetohydrodynamics(M HD) in what follows. The important property of (17) is tha t, no matter whatthe field v may be, it conserves the topological structure of B , at least for all finitetime; the question of what may happen as t + 03 is of particular interest, and willbe discussed below.
Let us define the ‘energy’ M of the field B in the obvious way, i.e.
M = iJB2dV (18)
n . B = O , n . v = O on 82). (19)
where the integral is taken over the domain 2)of fluid, and it is supposed that
We pose the question: can we choose v(x,t) in such a way that the energy Mdecreases to a minimum compatible with the conserved topology of B? In fact,there are various possible ways of choosing v to achieve this end. The simplestchoice (irrelevant constant factors being set equal to unity) is
v =jAB  v p (20)
where j = V A B the current distribution, if B is indeed a magnetic field), and p
(the ‘pressure field’) is chosen so that 0 . v= 0 and n . v= 0 on d D . We may write(20) in the more compact form
v = (jAB)s (21)
where the notation (. . .)s is used to denote the ‘solenoidal projection’ of a vectorfield. With this choice of v, equation (17) becomes
dB/at = VA((jAB)sAB), (22)
an evolution equation with cubic nonlinearity. It follows moreover that
dM
dt J B . V A ( ~ A B ) ~ V
=  1v2dV (23)
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Figure 2 . Relaxation of two linked flux tubes which lose energy through contraction.
on using th e boun dary condition (19) to eliminate th e pressure term . T hu s th e
energy of the field B does decrease monotonically for so long as v $ 0.
However the conserved topology of B implies th at (if this topology is nontriv ial)
there is a positive lower bound for M (Freedman 1988). Again, the prototypical
configuration of two linked flux tubes makes this clear (figure 2): the field ‘relaxes’
as a result of contraction of the Blines (due to the Maxwell tension associatedwith the Lorentz force); both the fluxes @I and @ 2 and the volumes Vl, Vz are
conserved du ring thi s process, which must evidently be arrested when t he two tu be smak e contact with each oth er. ‘Nontriviality’ of the topology means simply th a t
there exist field lines which cannot be continuously contracted to a point without
‘trapping’ other field lines in the process.Thus , as t + CO, we m ust conclude th a t, for any nontrivial field topology ,
/ v 2 d V + 0 and M + M E , (24)
where M E (> 0) is the asy m pt ot ic (relaxed) energy. Unless singularities of v appear
dur ing th is relaxation process (a possibility that appears extremely unlikely, buthas not as yet been rigorously e limin ated ), i t follows furth er th at
v + 0 and B ( x , t )+ BE(.)
as t + 03. From (20), th e field BE(x) atisfies
i.e. it is a m ag ne tos tatic equilibrium with pressure p E ( x ) .T he linked tub e exam ple
suggests rather strongly that in general, the relaxed field B E ( x )may contain tan gential discontinuities (as where the two tubes ul t imately make contact); these
tangential discontinuities of B E are current sheets, and we should here emphasise
t ha t i t is the assum ption of perfect conductivity th at perm its the appearan ce of
such current sheets. If the least resistivity is pe rm itted in th e fluid, then th e curr en t
sheets will diffuse to finite thickness, an d m ay be sub jec t t o ‘resistive instabilities’;here we deliberately exclude such resistive effects.
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4 Magnetic knots
The above relaxation process is particularly intriguing and illuminating when we
consider the case of an initial field Bo(x) with ‘knotted tube’ topology. If the field
lines within such a tube are ‘uniformly twisted’ so that the helicity (cf (13) ) is
given by 31 = h Q 2 ,where Q is the axial flux of Bo, then the key parameters that
rem ain constant d urin g th e relaxation process are h , Qp and the volume V of the
tube (the fluid being still supposed incompressible). Hence the asymptotic energy
M E is determined by these three parameters, no others being available. There is
only one dim ensional possibility:
M E = m(h)aW1/3 , (26)
a result first obtained by Moffatt (1990) . Here m(h) s a function of the dimensionless parameter h , and this function is determined (in principle) solely by the
topology of the tube knot I ( . I t may of course happen that there are mult iple
equilibrium state s, which m ay be ordered so t ha t
0 < mo(h)5 m1(h)5 mz(h)5 . . . .
m(h)= (mo(h),( h ) ,2(h), . .) .
(27)
(28)
We m ay then talk of the ‘energy spec trum ’ of the kn ot,
This type of argument may now be carried somewhat further. Suppose weconsider the relaxed state of lowest energy mo(h)@zV1/3. h e corresponding
magnetostatic equilibrium BE(x)can exist in an incompressible fluid a t rest. Let
us suppose that that fluid is ideal (i.e. inviscid as well as perfectly conducting).
T h e equil ibrium is s table (being one of m inimum m agnetic energy). We may ask,
in the spirit of Kelvin, what are the normal modes of vibration about such an
equilibrium? If we linearise the equations of ideal MHD,
aB/at = V A ( U A B ) (29)
abo ut the equil ibrium state U = 0, B = BE(x), hen, wri t ing
U = ul(x)eiwt,B = BE(x)+ B1(x)eiut (31)
we obtain
iwBl = Vh(ulhBE) , (32)
i w u l =Vpl+j lABE+jEAB1, (33)
an eigenvalue problem (when coupled with appropriate boundary conditions) for
the pa i r of fields {u1(x) ,B l( x) }. Note that in the notation used here, the field
B has been scaled so that its units are those of velocity (actually, B is the local
Alfven velocity).
Here again there is presumably a spectrum of frequencies 0 < W O 5 w1 5w2 5 . . ., the fundamental frequency W O being of gre ate st intere st. Since th is is
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determined in principle by the field BE(x)which is in turn determined uniquely
by h , 4 and V , we may conclude on dimensional grounds that
WO
= n o ( h ) a ? V  l (34)where again no( h ) s a dimensionless function of h determined solely by the topologyof the knot I<’.Of course there will presumably be a matrix f i j j (h ) of such functions,where j labels the member of the spectrum of relaxed fields, and i labels the normal
mode of vibration of this member. It is of course one thing to assert the existenceof such functions; it is an altogether more difficult matter , as yet beyond analytical
or computational capabilities, to determine and evaluate them.
5 The relaxation of chaotic fields
The situation considered in $4 n which all magnetic field lines are closed curves isvery exceptional. The field lines of an arbitrary field B(x) are the trajectories of
the system
and if the components B,, By , , are nonlinear functions of ( x , y , ) , then ingeneral these trajectories are chaotic within the domain 2) of definition of the field.
An example of such a chaotic field within a sphere 1x1 < 1 has been studied byBajer & Moffatt (1990); the field is quadratic in the space variables, i.e.
Bj = c i j k x j x k , (36)
the tensor C i j k being such that V . B = 0 and n . B = 0 on 1x1 = 1. Even forthis simple form of nonlinearity, the lines of the force of B are chaotic: they are
not closed curves, neither d o they lie on a family of surfaces. The system (35)is technically ‘nonintegrable’. Figure 3 shows a PoincarC section of the field fora particular choice of C i j k .  it shows the points in which a single field line of B
intersects the plane of section; the widespread scatter of these points is a familiarsymptom of chaotic behaviour; at the same time, one should note the existence of
a certain order within th is chaos, an order that can be analysed and understood bymeans of ‘adiabatic’ techniques.
Suppose now that we adopt such a field as the initial field Bo(x) in the relaxationprocess described in $3. During relaxation, the chaotic character of the field clearlypersists  there is no obvious mechanism by which a field line which is initiallychaotic could, under transport by a continuous velocity field, rearrange itself to lieupon a surface. The inference is that the relaxed field B E ( x )must also therefore
exhibit the above symptom of chaos.But here we are driven to a curious conclusion. The relaxed field B E satisfies
the magnetostatic condition ( 2 5 ) , so that
B E . V p E = O , (37)
i.e. the field lines of B E lie on surfaces pE = cst. If, by the above argument,they do not lie on such surfaces, then VpEmust be identically zero in the region of
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Figure 3. PoincarC section showing the intersections of a single chaotic field line of a quadratic
field of the form (36) with an equatorial plane of section in the sphere 11 ) < 1 (from Bajer &
Moffatt 1990).
chaos, and so jE BE 0. It then follows th at
jE = aBE where BE .V a = 0 (38)
v ~ B ~ = ~ B ~n 2,' (39)
and again, BElines lie on surfaces a = cst. , unless a
that BElines can be chaotic in some subdomain 2,' of 2, only if
0. It therefore appears
with a constant in V'. quation (39) expresses the fac t th at BE is a Beltrami field
in V', very special type of field.
As first pointed out by Arnol'd (1974, 1986), there cannot possibly be enough
generality in th e so lutions of (39) (if applied to V a ther than 2,') to accommodate
th e arb itrary topology th at m ay be assured for the in itial cha otic field Bo(x). Howcan we escape this paradox?
The explanation that has been suggested (Moffatt 1985) is that , within any
chaotic field, there are always 'islands of regularity' (large islands can be clearly
seen in the example of figure 3, but there are many smaller islands also, below
th e level of visual de tect ion ). Under relaxation, t h e bou nd arie s of these islands
may become considerably distorted, and the subdomain 2,' of chaos acquires a
correspondingly complex geometry. On this picture, t h e complexity of the ini t ial
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field Bo translates to complexity of the geometry of the domain D‘ in which therelaxed field is chaotic. Whether this is the correct explanation must await directnumerical simulation of the relaxation process, a computational experiment that
has not as yet been accomplished in three dimensions.
6 Twodimensional relaxation
Numerical relaxation to minimum energy states has however been carried out for
twodimensional fields. These have the advantage tha t the topology can be completely prescribed (Moffatt 1999) in terms of the homoclinic field lines (or ‘separatrices’) through all the hyperbolic neutral points (i.e. ‘saddle points’) of thefield. If placed on the sphere S2 ,each such separatrix is a figureofeight and the
generic separatrix structure consists of two families of nested figureofeights. Under relaxation, the incompressibility condition implies that the area A ( x ) nside anyfield line x = cst. remains constant; the function A ( x ) (or set of such functions fordifferent regions within separatrix loops) is called the signature of the field (Moffatt1986a) and is invariant during relaxation; it is in effect a topological property ofthe field.
It has been pointed out in $3 that tangential discontinuities of B may appearduring relaxation as t 4 00. This can occur also in the twodimensional case: suchbehaviour is located near the saddle points, and results from the collapse of the
separatrices (to zero angle) near the saddle points (Linardatos 1993), a behaviourthat has been subsequently reexamined and confirmed by Vainshtein et a1 (1999).
It may be conjectured that saddle points of a field B will play an equally significant role in threedimensional relaxation; but i t is by no means essential that saddlepoints be present to initiate such discontinuities (see Parker 1994 for an extendeddiscussion of the spontaneous formation of such discontinuities in the importantcontext of the solar corona1 magnetic field).
7 Analogous Euler flows
What, it may be asked, does magnetic relaxation, in a perfectly conducting viscousfluid, have to do with the problem that we started with, namely the flow of aninviscid nonconducting fluid in the absence of any magnetic effects? The answeris it provides a powerful, albeit indirect, method for establishing the existence of
steady Euler flows (i.e. steady solutions of the Euler equations of an incompressiblefluid) having arbitrary streamline (NB not vortex line) topology. For the equationfor such steady flow may be written in the form
U A W = V H (40)
where H = p / p + +U’ is the Bernoulli function, and w = V A U.There is an obvious
analogy between equations (25) and (40) through the identifications
B E + u , j E + w , p o  p E + H (41)
where po is an arbitrary constant. Thus the magnetic relaxation mechanism, whichestablishes the existence of fields BE atisfying (25) (and recall that j E = V A BE),
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simultaneously determines an analogous Euler flow U via the analogy (41). Of
course, care must be taken to ensure th a t th e bou nd ary conditions on the flow a re
compatible with the analogy (see Moffatt 1985).
Th us , for example, since the arguments of $ $3 and 4 establish the existence of‘k no tte d magnetic flux tub e equilibria’ for any kn ot class I(, t follows via th e above
analogy th at steady Euler flows having similarly knotte d st rea m tub es also exist! I t
is no t qu ite as visualised by Kelvin who considered kn ott ed vortex tub es ; there may
exist steady knotted vortex tube configurations, but no technique has as yet been
found to prove the existence of such configurations.Note that the tangential discontinuities of BE (i .e. current sheets) that inay
appe ar during the relaxation process translate via t he analogy (41) to tangen tialdiscontinuities of U , i.e. vortex sheets, imbedded within the Euler flows thus de
termined. Now it is wellknown that vortex sheets are prone to instability (theKelvinHelmholtz instability) and one may infer that the steady Euler flows maybe unstable within the context of the Euler equations despite th e fact t ha t th e a na
logous magn etostatic equilibria are, by their construction, s tab le within th e context
of the magnetohydrodynamic equations in a viscous, perfectly conducting, fluid.
This may appear surprising, but it should be recognised that the analogy (41)
applies only to th e steady state s, but not t o the (different) problems of the s tabil ity
of these steady states. The differences between the two types of stability problem
has been discussed by Moffatt (1986b); and it has in fact been shown by Rouchon
(1991) th at the sufficient condition for stability of an Euler flow obt ain ed by A rnol’d(1966) is never satisfied for flows th a t are fully 3dimensional a nd lack an y obvious
symmetry .T hu s, althoug h the mag netic relaxation technique yields a rich harvest of inform
ation about the existence of steady solutions of the Euler equations, the downside
is that any such solution of nontrivial topology is almost certainly unstable.
8 Relaxation to steady solutions of the MHD equations
Let us consider the full MHD equations for an ideal (i.e. inviscid, perfectly con
du cting ) fluid in th e form
d u / & ! t u ~ k J + j ~ B  V H (42)
dB/dt = v A ( U A B) (43)
and let us now, following Vlad imirov , Moffatt & Ilin (1 999), construc t a relaxation
process th a t yields topologically interesting steady solutions of these equa tions.Note first that there are two classes of topological invariants (or ‘Casimirs’)
associated with (42), (43); these are first the magnetic helicity invariants
X M = A . BdV (44)
where B = V A A cf (3)) , and V is any material volum e on w hose surface B ‘ n= 0 ;
and second the crosshelicity invariants
Xc = u .BdV. (45)
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So we have the curious result that a technique is available for treatment of
the ideal MHD equa tion s, bu t thi s technique fails for w ha t, on th e face of it, is a
simpler system, namely th e Euler equations for ideal fluids. T h e Euler equations, ina sense, emerge victorious, resistant as yet to the above type of general treatment
th at is available for more complex systems. T he gr ea t, an d enduring, difficulty
of the Euler equations lies in their purity, within which the central intractable
nonlinearity continues to defy progress at a fund am en tal level. I t is this p uri ty
and associated intracta bility th at lies at the heart of the still unsolved problem
of turbulence  a problem that will continue to challenge and frustrate for many
decades into the 21st century.
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