Andrew D. Gilbert and Mitchell Berger- Topological fluid mechanics

8/3/2019 Andrew D. Gilbert and Mitchell Berger- Topological fluid mechanics

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MAGIC course, spring 2010

Topological fluid mechanics

Andrew D. Gilbert and Mitchell Berger

Mathematics Research Institute,

School of Engineering, Mathematics and Physical Sciences,

University of Exeter, U.K.

1 Why topological fluid mechanics?

To the authors knowledge, the term was first coined for an IUTAM conferencein Cambridge, U.K. in 1989, by Keith Moffatt and Arkady Tsinober [13].The aim of the conference was to bring together scientists interested in fluidmechanics and the sister subject of magnetohydrodynamics (MHD) who werewrestling with problems of understanding the complexity of fluid flows andmagnetic fields.

Overall the aim is to approach such problems from a general viewpoint thatcomplements the more classical approaches. So for example a classical questionwould be to understand the instability of plane parallel flows, introducingproblems involving eigenmodes and eigenfunctions for specific profiles, andsome general results such as Rayleighs inflection point theorem.

On the other hand the type of question that topological fluid mechanics wouldaim to address is: what is the general structure of a steady inviscid fluid flow?Can stability be proven for whole classes of two- or three-dimensional flowsin arbitrary shaped regions? Can we construct flows with general streamlinetopology? If a magnetic field is tangled and knotted, how much energy may

be realised if it is allowed to reconnect?

Such questions have relevance in understanding a whole range of phenomena,from the structure of turbulent flow, to superfluid flow (in which vortices forma tangle of lines), to the magnetic fields of the solar corona and generation offield deep in the Earth or Sun. In each case a range of methods of looking atgeneral fields and their structure have proved useful and the aim of this courseis to set out some of the theory and applications.

In a sense the term topological fluid mechanics is something of an umbrella

Preprint submitted to Elsevier 8 February 2010


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term, and some of the methods and ideas could perhaps be better describedas geometrical. Nonetheless it is a useful title to bring together a whole rangeof topics that take a general view of whole classes of fluid flows and magneticfields and their properties. This includes many works with a distinctly puremathematical flavour, and some areas of study rapidly move into studying

general inviscid flows on manifolds of arbitrary dimension, or to quantifyinghow big groups of diffeomorphisms are. However our approach will be fromthe angle of applied mathematics, with a close eye on applications and notbeing concerned about technicalities.

We have benefitted from a range of sources of information, and in particularhave made use of material in the unpublished lecture notes of Steve Childress[3] for the beginning of the course, with his permission.

2 Topics

The course will be given by Andrew Gilbert and Mitchell Berger and coverthe following topics:

Part I (AG): basics, helicity and relaxation

Background and motivation, hydrodynamics and magnetohydrodynamics.Revision of Kelvins theorem and magnetic analogies.

Fluid, magnetic and cross helicity, geometrical interpretation. Magnetic relaxation.

Part II (MB): knots, tangles, braids and applications

Link, twist and writhe of flux and vortex tubes. Braiding of flux and vortex tubes. Vortex tangles in quantum fluids and vortex tubes in turbulence, crossing

numbers. Chaotic mixing, stirrer protocols, pA maps and topological entropy.

Part III (AG): dynamics of point and line vortices

Point vortex dynamics, invariants, integrability.

Vortex tube dynamics, local induction approximation, invariants, solitons. If time: introduction to Lagrangian fluid mechanics.

These Latex notes cover the part I, which is the first three lectures, and thereare exercises and bibliography at the end. The notes are pretty informal and Ihave not cross referenced equations, preferring to repeat them where necessary.This is because these notes are a basis for the lectures I will give: the lectureswill also have an informal style but be more bullet point orientated and includepictures and sketches. Finally there may be (will be!) errors and slips in thenotes and lectures: please do not hesitate to bring these to my attention.

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3 A few general notes

The summation convention is in force in these notes unless otherwise stated.We sum over repeated (dummy) indices in each product. Single (free) indicesare not summed and label different equations.

We will refer to position with vector r or x and use either (x,y,z) or(x1, x2, x3) interchangeably. Note that r = |r|.

We may use components of a vector u as (u1, u2, u3), especially in generaltheory, but also (u,v,w), particularly in examples.

We use the convenient abbreviation t for /t and i for /xi. For a vector b we often write b2 for b b = |b|2. We will often write something like uA. This can be thought of as (u)A

where u = uii is an operator acting on the vector field A. Or, we canthink ofA as a matrix whose ijth component is iAj. In this case we aresimply dotting u into the first index by taking uiiAj. In the latter case

we consider A as defining a matrix and we define |A|2

= (iAj)(iAj),that is the sum of the squares of the entries of the matrix.

We assume a good knowledge of vector calculus identities: many are givenbelow.

4 Vector calculus identities

Here is an incomplete list of useful identities:

() = +

(u) = u+ u

(u) = u + u

(u v) = u v + v u + u ( v) + v ( u)

u ( u) = 12u2 u u

(u v) = v u u v

(u v) = u v v u + v u u v

( u) = ( u) 2u

= 0 u = 0

We also make much use of Gauss theorem (the divergence theorem)

V

u dV =S

n u dS

where V is a volume enclosed by a surface S with outward normal n. Or we

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have it in a more general form

V

i dV =S

ni dS

where i = /xi and ni is the ith component of the normal vector n. Here can be any quantity, so could have a set of indices itself.

We also occasionally use Stokes theorem

S

n u dS =C

u dr

where S is a surface with boundary C and the direction of the normal n toS is related to the direction of integration around C by the usual right-handrule.

5 Governing equations for fluid flow

We consider fluid flow u(r, t) with constant density, governed by the NavierStokes equation in the form

tu + u u = p + 2u

u = 0

The latter condition, of a solenoidal or divergenceless vector field, is used agreat deal below. The viscosity will always be a constant. We will often makeuse of the material derivative

Dt =D

Dt= t + u

for which the equation becomes

Dtu = p + 2u

The flows may be in infinite space, or in a bounded domain D with boundaryS in which case we apply a no-slip condition

u = 0 (r S)

We will usually focus on inviscid or ideal fluid flow, governed by the Eulerequation, with zero viscosity (this is a highly singular limit!), namely,

tu + u u = p

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This can also be written in the attractive form

tu = u P

where the Bernoulli function is

P = p + 12u2

and the vorticity is given by = u

which is another solenoidal vector field

= 0

Note that by taking the divergence, the pressure is given by

2P = (u )

Actually we have lapsed here: P is the Bernoulli function, but we often slipinto calling the pressure. Whatever it is called, obtaining P is thus a non-local procedure: the flow and vorticity at all points in the domain affect allother points. In a numerical code, this Poisson equation has to be solved eachtime-step.

Taking the curl eliminates the pressure and gives the vorticity equation

t = (u )

orDt = t + u = u

This latter equation shows the is transported with the fluid flow (left-handside) but also undergoes stretching and rotation (right-hand side). Technicallyit is Lie-dragged in the flow and for this reason vorticity (rather than velocity)plays a central role in discussion of topological fluid mechanics. In other wordsgiven a flow u, the above equation carries vectors as though they each jointwo infinitesimally close fluid particles. Here the Lie bracket of two vectorfields is defined as

[u,v] = u v v u

It is antisymmetric [u,v] = [u,v] and satisfies the Jacobi identity

[[u,v],w] + [[v,w],u] + [[w,u],v] = 0

For more information, see books on differential geometry, for example that ofSchutz [16].

In the ideal case we also have boundary conditions: in a finite domain D weimpose a no normal flow condition

n u = 0 (r S)

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where n is a normal vector to S pointing outwards from D. As setting = 0reduces the order of the equation, we are forced to reduce the number ofboundary conditions and can no longer require that the flow be no-slip at theboundary. Note that we will use D for the domain, the entire volume, in whichwe are working, and V if we are using a subvolume of this. We will use S for

the surface of either D or V: which will be clear from the context.

Sometimes it is convenient to work in unbounded space D = R3, for whichwe typically assume that the vorticity field is localised, so falling off with rfaster than any negative power of r so that for any n,

|(r, t)| = o(rn) (r )

For working in infinite space we start with a finite sphere SR of radius Renclosing a volume VR. We then derive an exact result valid in VR and thenallow R to tend to infinity. A condition such as that the vorticity field be

localised will typically mean that the surface integral over SR vanishes asR , leaving us with a result valid over all space D = R3.

Sometimes it is instead convenient to work in periodic space, for example weset D to be the periodic cube T3 defined as [0, 2]3 with periodic boundaryconditions so that surface integrals sum to zero on opposite faces by periodicity(as the normal vector n changes sign, all other quantites remaining the same).One may have to take care defining potentials in this geometry, for examplemagnetic vector potential A below.

Note that in a finite domain D with the boundary condition u n = 0, the

divergence-free field u is orthogonal to any gradient in the sense thatD

u dV =D

(u) dV =S

u n dS = 0

using Gauss theorem (the divergence theorem).

6 Cauchy solution

A particle or fluid element will be carried by a fluid flow. It will start at say

a at time t = 0 and at each moment it will have position x(t) and velocityu(x, t). Its position x(t) is then given at later times by the differential equation

dx

dt= u(x(t), t)

and initial conditionx(0) = a

It is convenient often to use a label a to label the initial condition, so weessentially follow the whole family of trajectories x(a, t) from all the initial

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positions a and writex

t

a

= u(x(a, t), t)

and initial condition

x(a, 0) = a

Thinking about fluid dynamics in terms of these Lagrangian coordinates agives the Lagrangian picture of the flow, which has some advantages, thoughmany disadvantages (in that the pressure transforms in a complicated way inLagrangian coordinates). Within the Lagrangian picture one can solve for thevorticity field if the velocity field is known. Of course the two are linked ina complex way for ideal fluid flow! But life is a bit easier in the kinematicdynamo problem: see later.

What about an infinitesimal vector that joins x and x + x? We get the

equation for this by linearising the above ODE to yield

x

t

a

= x u(x(a, t), t)

We have Taylor expanded in x and dropped quadratic and higher order terms.This essentially defines an infinitesimal vector: it is always so short that wecan neglect these higher order terms (one can imagine rescaling it). This is thevector used for example for measuring Liapunov exponents (and technicallyit is in the tangent bundle of the domain in question).

This is essentially the same equation as that for vorticity in the form

Dt = u

confirming that vorticity vectors are carried like infinitesimal vectors in theflow.

We can also obtain a similar equation for the Jacobian: given x(a, t) we mayset

J(a, t) = x/a or Jij =xiaj

Differentiating the equation for particle motion with respect to a gives

J

t

a

= (u)TJ orJij

t

a

= kui(x(a, t), t) Jkj , J(a, 0) = I

(with I as the identity matrix). Here u is the matrix of derivatives of u,with

(u)ij = iuj

and T gives its matrix transpose.

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This gives us solutions to transport of vectors and scalars in the flow. Firstconsider a scalar field that is carried in the flow passively according to

Dt = t + u = 0

with initial conditon (a, 0) = 0(a). The solution is given by

(x(a, t), t) = 0(a)

as can be checked by differentiating with respect to time at constant a.

Similarly suppose the initial vorticity is given by (a, 0) = 0(a) and thevorticity (x, t) obeys

Dt = t + u = u

Then the solution to this, the Cauchy solution, is given by

(x(a, t), t) = J(a, t)0(a) or i(x(a, t), t) = Jij(a, t)0j(a)

or, taking x = x(a, t) as read, we have

(x, t) = J(a, t)0(a) or i(x, t) = Jij(a, t)0j(a)

This natural link between the motion of particles, infinitesimal vectors andJacobians, and the equations governing vorticity and (later) magnetic fields is

at the heart of the notion of Lie-dragging. It is also natural to think of vortexlines, that is integral curves of the vorticity field given by

dx11

=dx22

=dx23

at a fixed time t, being carried materially by the fluid flow. If we mark vectorson these lines, joining material points, then as the points move apart or rotatethe vectors stretch or rotate accordingly.

Given that we can solve the vorticity equation by means of the Cauchy

solution it may seem attractive now to use this to solve the Euler equation.The problem though is that it is very hard to invert the vorticity solutionto give the flow field (this being a non-local procedure) and it is only invery special cases that these Lagrangian methods can be used to solve fluidproblems.

Note that for similar reasons there is no obvious means to transport flowvectors u in the fluid. Whereas the equations for vorticity permit a local cal-culation of the evolution of vectors, those for u involve the non-local pressureterm, which requires a global calculation.

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7 Transport of gradients or one-forms

There is another natural means of transporting vectors in a flow. Supposewe differentiate the equation

Dt = t + u = 0

for a passive scalar to obtain the equation for its gradient, say g = . Thissatisfies

Dtg = tg + u g = (u) g

Here u is the matrix of derivatives ofu, with

(u)ij = iuj

Thus gradients are transported differently from vorticity or magnetic field

vectors. This is natural when you think how they are defined: they point inthe direction of greatest increase of at a point: if there is stretching in thatdirection then the gradient is reduced! Technically we ought to distinguishvectors that are stretched like vorticity, obeying the Lie dragging equation

Dtb = tb + u b = b u

and one-forms or covectors which obey a different Lie-dragging equation

Dtg = tg + u g = (u) g

Books on differential geometry (also modern books on relativity) make thisimportant distinction. We do not want to get too sidelined by the manifoldsand differential geometry, but it is useful to know that there are these twonatural kinds of vectors, with different stretching properties in the flow. Whenthe properties are important we will refer to one-forms as distinct from vectors,but we may be a little lax sometimes.

Furthermore if we take a scalar product we can check that

Dt(b g) = b (u) g b (u) g = 0

so the scalar product of a vector field and one-form field is a scalar field with

Dt(b g) = 0

If we consider 3 infinitesimal vectors transported in a fluid flow, b, c and d,then these demarcate an infinitesimal area element, whose volume is preservedas u = 0. Thus we have

Dt(b c d) = 0

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Now, this is true for all choices of vector fields b, c and d and so it can beshown that bc gives a one-form which may be thought of as an area element,say dS= n dS. We obtain the result that area elements are transported likegradients, according to

Dt dS= (u) dS

Again, thinking of how an area element is transported under stretching orshearing gives some intuition of what is going on, and why the transportequation is different from that for vectors. Wavevectors are also transported inthe same way, in other words as one-forms. We can also write for infinitesimalvectors again and for infinitesimal volumes,

Dt dr = dr u

Dt dV = 0

Finally we have the Cauchy solution for vector fields, and something tells usthat there should be something analogous for one-form fields. Suppose g is aone-form field and b is an arbitrary vector field, then from the Cauchy solutionfor a vector field, say

b(x, t) = J(a, t)b0(a) or bi(x, t) = Jij(a, t)b0j(a)

and the fact that Dt(b g) = 0 gives

(b g)(x, t) = (b0 g0)(a) or (bigi)(x, t) = (b0ig0i)(a)

one can check that

g(x, t)J(a, t) = g0(a) or gi(x, t)Jij(a, t) = g0j(a)

Or, if we define the matrix K(a, t) to be the inverse ofJ(a, t),

JK= KJ= I or JijKjk = KijJjk = ik

then one can check that

g(x, t) = g0(a, t)K(a, t) or gi(x, t) = g0j(a)Kji(a, t)

An alternative route is to obtain the equation for K as

K

t

a

= K(u)T orKij

t

a

= Kikjuk(x(a, t), t), K(a, 0) = I

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8 Kelvins theorem

Kelvins theorem states that in ideal flow, the circulation of the fluid flowaround any material curve C(t) is conserved in time.

C =C(t)

u dr

By virtue of Stokes theorem this may be written as an integral over a surfaceS(t) that spans C(t), with an appropriate choice of outward normal

C =S(t)

dS

The invariance of these follows easily from the discussion above. We know thatsince is a vector and dS is a one-form

Dt( dS) = 0

and thusdCdt

=S(t)

Dt( dS) = 0

Note that we had to take a material derivative inside the integral. To appre-ciate this or visualise it, it is perhaps most natural to break up the surfaceS(t) into many infinitesimal surface elements dS and apply the infinitesimalversion to each.

9 Energy and helicity invariants for fluid flow

We have conservation of (kinetic) energy

EK =D

12u2 dV

with the no normal flow boundary condition, and (kinetic) helicity

HK =D

u dV

with the boundary condition that

n = 0 (r S)

This boundary condition states that vortex lines must be parallel to the bound-ary. In a sense this is an unnatural boundary condition, and it would not bepreserved if viscosity were present. However in ideal fluid flow it is is preserved

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if it is the case initially: to understand it we need to know more about theinterpretation of the helicity integral.

To check the conservation of helicity, we take

dHKdt =

D[(

tu) + u (t)] dV

=D

[(u P) + u (u )] dV

=D

u (u ) dV

with the n = 0 condition used to remove the P integral (recall theorthogonality property discussed earlier). We now use = u, the identity

(a b) = b a a b

and apply Gauss theorem to give

dHKdt

=S

(u ) u n dS

=S

[u2 (u )u] n dS = 0

This vanishes provided the natural boundary condition un = 0 holds and thecondition that vortex lines do not penetrate the boundary, n = 0. Underthese conditions HK is constant, independent of time.

It is interesting to look at the evolution ofHK(V) where now V(t) is a materialvolume, carried in the fluid flow. In this case we calculate as follows

dHK(V)

dt=V

[(Dtu) + u (Dt)] dV

=V

[p + u ( )u] dV

=V

(p + 12u2) dV

=S

(p + 12u2) n dS = 0

provided that n = 0 on the boundary S(t) of the volume V(t). Suppose thatinitially the volume V(0) is bounded by vortex lines, lying on the surface S(0).Then, as vortex lines are carried by the fluid flow the material surface S(t) willcontinue to be composed of vortex lines and so the boundary condition n =0 will continue to hold. Thus for a material volume V(t) bounded by vortexlines HK(V) is conserved. When we discuss the topological interpretation ofHK this will become clear. Here we just note that there could be whole familiesof surfaces, for example nested tori, giving a continuous family of invariants,labelled by the surface.

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It would be nice to illustrate this with some exact solutions for fluid flowssatisfying the unsteady Euler equations, but such solutions are few and farbetween! Instead we will content ourselves with the steady Euler equationsand a family of flows (a special case of the family of ABC flows we will meetsoon),

u = A(0, sin x, cos x) + B(cos y, 0, sin y)There are two parameters A and B (though only the ratio is important as wecan rescale u and time t) and the flows may be thought of as defined in a2 periodic cube, i.e. a 3-torus T3. This flow has the Beltrami property that = u u, vorticity and velocity are everywhere parallel, and so it is asolution of the steady Euler equation. The flow may also be written as

u = (y, x, ), = A cos x + B sin y

and so we see that in the (x, y)-plane the motion is along curves of constant ,while the vertical motion, that is in the z-direction, is constant on each curveof constant . In full 3-dimensional space, helical streamlines lie on cylindersof constant and these are also vorticity surfaces. The helicity inside eachsurface is constant, as it must be trivially for a steady flow! But in a moregeneral context we have to imagine such a flow evolving with time and thehelicity inside each surface of constant would be conserved.

10 Ideal MHD and gauges

Before going on to study the topological interpretation of fluid helicity, we setup the analogous quantity of magnetic helicity. We consider the equations ofmagnetohydrodynamics (MHD), which in suitable units take the form of theNavierStokes equation

tu = u +j b P + 2u

with the addition of the Lorentz force j b and the magnetic induction equa-tion

tb = (u b) ( b)

Here we take the viscosity and the magnetic diffusivity to be constants

(otherwise the above forms need modifying). The limit 0 is called theperfectly conducting limit as 1 is proportional to the conductivity of thefluid, and so is large in plasmas and liquid metals.

The two fields are divergenceless

u = b = 0

The first of these is an approximation (no fluid is entirely incompressible)while the second is a fundamental law of physics (no magnetic monopoles).

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The electric current is given by

j = b

Note that for divergenceless fields the two forms of the final term are equivalent

as a result of the vector identity

( u) = ( u) 2u

One can swap between these as needed.

There are a number of ways of rewriting the equations, for example as

Dtu = tu + u u = b b PM + 2u

Dtb = tb + u b = b u + 2b

wherePM = p +

12b2

now includes the magnetic pressure.

In the ideal case we set = = 0 and consider the perfectly conducting andno normal flow boundary conditions, namely

n u = n b = 0 (r S)

for a finite domain D with boundary S. This requirement gives no flow through

the boundary and also no magnetic field lines penetrating the boundary. Thelatter is natural for theoretical discussion but not for many applications, forexample the solar magnetic field which penetrates the photosphere, into themagnetosphere; see later.

Alternatively we may impose that the vorticity and magnetic fields fall offrapidly (faster than any power of radius) in an infinite domain, so that (atany time t) for any n,

|(r, t)|, |b(r, t)| = o(rn) (r )

We now focus on the ideal case = = 0.

tu = u +j b P

tb = (u b)

or

Dtu = tu+ u u = b b PMDtb = tb + u b = b u

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We note first that given b = 0 initially, by taking the divergence of theinduction equation it is preserved in time,

t b = 0

This is why there is no pressure needed in the magnetic induction equation.For u pressure is defined to constrain u to remain divergenceless.

Also, the induction equation is completely analogous to the vorticity equation,except that the magnetic field is not directly linked to the flow field u, whereasfor vorticity = u. Of course we did not use that link earlier in ourdiscussion of Lie-dragging and so we still have the Cauchy solution for themagnetic induction equation

b(x, t) = J(a, t)b0(a) or bi(x, t) = Jij(a, t)b0j(a)

In fact in kinematic dynamo theory we assume the field is so weak that the

Lorentz force term may be neglected in the NavierStokes equation, and theflow field u may be specified independently of the magnetic field. In this casethe Cauchy solution can be really useful if the flow is not too complicated.

We also have the analogue of Kelvins theorem, namely Alfvens theorem thatthe flux S of magnetic field through any material surface S(t) is conserved,

S =S(t)

b dS

We have the usual conservation of energy with

EK + EM =D

12u2 dV +

D

12b2 dV = constant

What of helicity? Just as = u, plainly we need to obtain a field A with

b = A

The existence of such a vector potential follows from b = 0, at least locally.This definition ofA can be done globally in infinite space or in a boundeddomain provided that it is simply connected. However there can be problems indefining A (as a single-valued function) if the domain D is multiply connected.

For example if we work in three-dimensional periodic space D = T3 we cannotuncurl a periodic b to obtain a periodic A unless it b has a zero average overD: no mean field. We will not get side-tracked by these considerations (anotherway in which topology intrudes into MHD) here, but they may reemerge lateron.

Instead we focus on a different issue, that of gauge freedom: we can alwaysadd a gradient to A without affecting b:

A A + (r, t)

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We uncurl the ideal induction equation in the form

tA = u b + = u ( A) +

where the presence of corresponds to the freedom to choose the gauge:

whatever we put here will disappear when we take the curl to obtain theinduction equation for b. There are a number of choices for : one is to make A = 0, in which case we require this to hold for the initial condition forA and then evolve A with determined by

2 = (u b)

This is a non-local equation, rather like that for the pressure in the Eulerequation.

11 Magnetic helicity

Given A with some choice of gauge we can define the magnetic helicity overthe domain D by

HM =D

A b dV

We then have

tHM =D

[(u b + ) b + A (u b)] dV

Now the integral of () b vanishes as usual with n b = 0 on the boundary

and so the integral is reassuringly independent of the choice of gauge. Alsou b b = 0, removing one term above, and with this we have

((u b) A) = A (u b)

from which after applying Gauss theorem

tHM =S

n (u b) A dS =S

n [(u A)b (b A)u] dS = 0

by virtue of the boundary conditions: n b = 0 so no field lines leave throughthe boundary, and n u = 0 so there is no flow to transport helicity density

b A out of the domain.

For ideal MHD, the kinetic helicity HK is no longer conserved but a quantitycalled the cross-helicity is:

HX =D

u b dV

What of our magnetic vector potential A: is it a vector or a one-form? Asit stands it is neither, in the sense that it is generally not transported locally:

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for example if one imposes the gauge A = 0 then one has to solve a global,Poisson equation for the gauge . However we can write

tA = u ( A) + = u A+ (A) u +

With the choice = u

A

we have = (u) A (A) u

and soDtA = tA + u A = (u) A

which is the equation for the Lie-dragging of a one-form. A is a one-form fieldwith this particular choice of gauge and then we not only have conservationof the integral of helicity but also of its value, the helicity density hM = A badvected in the fluid flow

DthM = thM + u hM = 0

Although this is attractive, having a materially conserved quantity, it doesdepend on following the field A in this gauge, and as it evolves it may becomevery fine-scaled. Typically people would rather follow the magnetic field b andthen compute A in a suitable gauge such as A = 0: in this case we canonly use the integrated form HM of the helicity as a conserved quantity.

As in the case of kinetic helicity we can establish that HM is conserved if takenover a material volume V(t) contained in the domain D,

HM(V) =V(t)A b dV

We can use the form

Dtb = b u, DtA = (u) A +

for an arbitrary gauge and obtain (with Dt dV = 0)

dHM(V)

dt=V(t)

[b (DtA) + (Dtb) A] dV =V(t)

b dV = 0

using Gauss theorem with n b = 0 on the boundary of V. As for vorticity,if the surface S volume is composed of magnetic field lines, then the helicitywithin is conserved.

Historical note: the conservation of magnetic helicity was noted by Woltjer in1958: the attractive name of helicity was chosen because HM or HK changesign under reflection, being pseudo-scalars, by Moffatt in 1969.

For example consider the flow

u = (y,x,x2 + y2) = (0, 0, 2) u = 2(x2 + y2)

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and its mirror image under inversion r r

u = (y,x, x2 y2) = (0, 0, 2) u = 2(x2 + y2)

12 Topological interpretation of helicity as linking number

We now consider the interpretation of helicity. We can work with either thefluid helicity

HK =D

u dV

or the magnetic helicity

HM =D

A b dV

in the whole domain D where the flow or field is defined. We shall choose to

work with the magnetic helicity.Consider two thin linked tubes Tj in D, of magnetic field as depicted, withcentre lines Cj for j = 1, 2. Here b = 0 outside each one, and each carries aflux j > 0: the direction of traversal along Cj is in the direction of the field.The flux is the integral

j =Aj

b dS

where Aj is a surface spanning Tj with normal in the direction of the field,i.e. a cross sectional area of the tube. We also assume that, individually, eachtube is unknotted.

Focussing on the fact that b is non-zero only in the tubes, we have two con-tributions to the helicity integral, HM = H1 + H2, where

Hj =Tj

A b dV

We can approximate these as integrals along the centrelines Cj of the tubes,replacing b dV with j dr, as

Hj = j

Cj

A dr

We may now use Stokes theorem: the integral ofA around Cj is the integralof the field b = A through a surface Sj spanning Cj

Hj = j

Sj

b dS

and so is plus-or-minus the flux of the other tube, depending on orientation.For the picture shown and j = 1 we have a plus sign and so H1 = 12.Likewise H2 = 12 and the total helicity is HM = 212. If the tubes are

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not linked the value is zero, and if one set of arrows is reversed (or the wholesystem is reflected) then HM changes sign.

In general we have the result for a whole collection of tubes (each individuallyunknotted) labelled by j that

HM =i

j

ijij

where ij are integers that label the signed number of times that the centreline Cj of the jth flux tube crosses the surface Si spanning the centre line Ciof the ith flux tube. Here the sign is +1 when the direction of Cj agrees withthe outward normal of Si, obtained from a right-hand rule from the directionof the centre line Ci

Note that it is important that the flux tubes individually are not knotted(in knot theory these are called unknots) and we set ij = 0 for i = j. If a

tube is knotted then we may break it up into a collection of unknotted tubes(see pictures) by inserting links and then compute the helicity this way. Thisdefines the value of ij for i = j in this more complicated case.

A number of notes are in order. First, the topological nature of the invariant,in terms of the linkage of magnetic field lines or of vortex lines, explains whyit is invariant under ideal, frozen field evolution: linkages cannot be broken ifmagnetic field or vortex lines move materially with the fluid. It also explainswhy the condition that the field, or vortex, lines must satisfy n = b n = 0,either on the surface of the domain D for the total helicity, or on the surface ofa moving material volume V(t). We cannot define linkage (easily) if the field or

vortex lines do not close up. Note, though, that some important applicationsinvolve field lines that do intersect surfaces, for example in the solar corona,and ways of measuring linkage have been adapted to this: see later.

There is a connection between helicity and an invariant known as the Gausslinking number. Consider a magnetic field b in unbounded space that occupiestubes Tj with centrelines Cj and has fluxes j . The corresponding A (in thegauge A = 0) may be obtained at any time by a BiotSavart integral (ascan the velocity from the vorticity) namely,

A(r) =

1

4D

(r r) b(r)

|r r|3 dV

If we replace b(r) dV in the jth tube by j dr then we obtain

A(r) =j

j4

Cj

(r r) dr

|r r|3

We now plug this into

H =i

i

Ci

A dr

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from above to obtain

H =i

j

ijij

with

ij =1

4Ci

Cj

(r r) dr dr

|r r|3

This integral gives an integer, which is the Gauss linking number of curve iwith curve j for i = j.

Note that if the linking number ij between any two curves is non-zero thenthey are linked, and cannot be smoothly separated. However one can obtainconfigurations such as the Borromean rings, which are 3 linked rings: theycannot be pulled apart but have the property 12 = 23 = 31 = 0.

Furthermore we have assumed there is no linkage associated with the field

lines inside a given flux tube: this is not always correct and leads on to howone measures the twist of field in a tube (which may itself be in a complicatedknot) and the related topic ofwrithe. This will be taken up later in the course.

A final issue here is that we have focussed on field that is concentrated indiscrete flux tubes (or vorticity in discrete vortex tubes). But what of fieldin a continuous distribution, for example take the vorticity field = u formembers of the ABC family above. In this case very few field lines actuallyclose. The work above has been generalised by Arnold to cover this case, andhe refers to this as the asymptotic Hopf invariant. We will not reproducethe proof here, but the idea is to take magnetic field lines starting at points

a and integrate trajectories x(a, t) along them for a time T, say. One thenuses a set of paths to link the end points x(a, T) to the initial points a:these paths are to be short and straight. This then gives a collection of closedcurves whose Gauss linking numbers may be computed and integrated. Anappropriate limiting procedure taking T yields the helicity or Arnoldsasymptotic Hopf invariant as the limit of the Gauss linking number, for acontinuous magnetic field.

13 Beltrami flows and the ABC flows

We now return to fluid flows: our aim is to look at the structure of some steadyEuler flows and how this is constrained.

Consider an ideal fluid in a finite region D with helicity HK = 0. What is theminimum value of the enstrophy

K =D

2 dV

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for this given value of helicity? By the CauchySchwartz inequality we have

H2K =

D

u dV2

D

u2 dV

D

2 dV

= 2EKK

We also have a Poincare inequality that for a given finite region D there is aconstant L such that

K =D

2 dV L2D

u2 dV = 2L2EK

for any u with n u = 0 on the boundary. The number L1 is the smallestvalue such that

= u = L1u

has a non-trivial solution in D subject to the no normal flow boundary con-dition u n = 0.

From these two inequalities we have

H2K 2EKK L22K

and so can write a lower bound for the enstrophy

K L1|HK|

in terms of the helicity. Plainly equality occurs when

= u

everywhere in D, where = L1 is a constant.

Such flows with the vorticity and velocity everywhere parallel are called Bel-trami flows and satisfy the steady Euler equation since u = 0 identically.Here the flow and vorticity are proportional by a constant, namely .

The condition u = u gives an eigenvalue problem, and the least eigen-value gives L1. We have already met one such solution which is a special caseof the ABC (or ArnoldBeltramiChildress) family of flows, taking the form

u = A(0, sin x, cos x) + B(cos y, 0, sin y) + C(sin z, cos z, 0)

(in periodic space T3 with no mean flow). If one ofA, B or C is zero, we have aflow with regular streamsurfaces, seen earlier. In a certain plane there will be arepeated pattern of cats eyes. However if all three are non-zero, then typicallywe see a complex mixture of regular vortices, with stream surfaces that takethe form of tori, interlaced with regions of chaos. To make life difficult theflow is highly three-dimensional. It has been studied by means of Poincaresections, in which a trajectory x(a, t) is follows and points plotted wheneverthis intersects one of a family of planes, for example a plane of constant x.

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Amazingly, this is a steady Euler flow, and raises the question: just how compli-cated can steady Euler flows be? Are there any constraints on their structure?An answer to this question (and pretty much all that is known) is provided ina rigorous result of Arnold, which we set out informally as follows. We wormin a finite domain D which could be a sphere, or could be periodic space T3.

A steady Euler flow obeysu = P

and sou P = P = 0

Now, provided P does not vanish identically in a 3-dimensional region ofspace, there will exist a family of 2-dimensional surfaces S(P0) on which P =P0 is constant. The above result says that streamlines and vortex lines willbe tangent to such surfaces. To understand what form the surfaces can takewe need to be careful about points where P = 0. We assume there areonly finitely many such points in the domain D under consideration. We then

choose a value P = P0 = constant, to give a surface S(P0) which does notintersect any points with P = 0 (in other words we avoid the finite numberof exceptional values of P0 where this occurs). We also assume the surfacedoes not intersect the boundary of D. The vorticity and velocity vectors aretangent to this surface S(P0) and can vanish nowhere on it. It can be shownthat the only such surface is topologically a torus, and that the velocity andvorticity fields wind around it. As we vary the value of P0 we obtain nestedtori S(P0), provided we do not encounter a point where P = 0: at such apoint surfaces may intersect.

Note that the reason the surfaces of constant P must be tori is linked tothe non-vanishing of the flow field u (or ) on the surfaces. For example thesurfaces cannot be spheres as it is impossible to define a non-vanishing vectorfield: this is the famous result that one cannot comb a hairy ball, or that atany time the horizontal wind velocity must vanish at at least one location onthe earths spherical surface. (For more information, in particular the link tothe Euler characteristic of a surface, see the book of Arnold and Khesin [1].)

This analysis assumes that P varies in space and P = 0 at only finitely manypoints, so that nested two-dimensional surfaces P = P0 = constant may bedefined. But what ifP is constant and so P = 0 over a whole 3-dimensional

volume V, a subdomain of D? In such a case we must have that u = 0everywhere in V, so that = u

for some function (r): the flow must be Beltrami. But given that u = = 0, we can take the divergence of or u and rapidly show that

u = = 0

Now the argument repeats itself: if(r) defines two-dimensional surfaces S()and with = 0 assumed only at a finite number of points (which we avoid),

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then the velocity and vorticity vectors must be tangent to the surfaces andnon-vanishing. The surfaces are again tori.

The final possibility then is that (r) is in fact a constant, say (r) = , overa whole 3-dimensional region V and in this case we have

= u

there, which is the eigenvalue problem giving us the ABC family of flows wehad earlier in the particular case when V is all of periodic space. In other

domains there will be other families of flows: for example there are analoguesof ABC flows in spherical geometry.

In conclusion, flow lines and vorticity lines lie on nested tori except in thespecial case of constant Bernoulli function P = p + 1

2u2 and = u with

= constant. The ABC flows provide an example of this: for example thevalues A = B = C = 1 show a complex topology of fluid trajectories. Theseinclude invariant surfaces on which some stream lines sit, but interleaved withthese are thin bands of chaotic trajectories. There is no mathematical reasonwhy there should be any surfaces at all, and in fact the amount of chaos inthe flow varies with the parameters. At another extreme, if C = 0 the flowshows a cats eye structure as mentioned before. Here there are two distinctfamilies of tori on which streamlines (or vortex lines sit).

The reader may wonder why we cannot have regions where there are two-dimensional surfaces P = constant interleaved with three-dimensional regionsin which P is identically constant. Or one could have a sequence of pointswhere P = 0 that accumulate at some point in space. There are a lot ofpossibilities in three dimensions! In fact the theorem of Arnold excludes this byrequiring that the field u be complex analytic: this means that critical pointsP = 0 (or = 0) cannot accumulate, and that if P (or ) is constant

in any three-dimensional volume then it is constant everywhere. Thus eitherthere are P = P0 surfaces with P = 0 at only finitely many points, or P isconstant with P = 0 everywhere in D: there is no messy mixture allowed.In the latter case either = 0 surfaces exist with = 0 at finitely manypoints, or = 0 everywhere in D. The assumption of analyticity gives clearand clean alternatives at each point in the argument.

More general situations, in which the fields are only required to be infinitelydifferentiable, or arise from magnetic relaxation (see below) do not appear tohave been studied, and are likely to be complicated.

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14 Magnetic relaxation

Consider the following problem: we consider evolution of magnetic field withzero magnetic diffusion = 0, under the full MHD equations including non-

zero viscosity > 0. These are

tu = u +j b P + 2u

tb = (u b)

u = b = 0, j = b

or equivalent. We work in a domain D which is finite with u = 0, n b = 0 onthe boundary S. We have kinetic and magnetic energies

EK =D

12u2 dV, E M =

D

12b2 dV

Then it may be shown that

d

dt(EK + EM) =

D

|u|2 dV

and so the total energy E = EK + EM 0 of the system decreases mono-tonically as t and must tend to a limit, positive or zero. Recall that|u|2 = (iuj)(iuj) 0.

Can the limit be zero? Suppose we start with a given magnetic field b0 andzero flow u = 0: then the Lorentz force j b will tend to drive a flow (field

lines are elastic and tend to contract, reducing magnetic energy). But the fieldlines are frozen in the flow and so any linkages and any helicity integrals (overthe whole domain or over subvolumes) must be conserved: this constrains howsmall the magnetic energy may be. In fact we have from our discussion ofBeltrami flows that

K =D

2 dV L1D

u dV

= L1|HK|where L is the constant appearing in the Poincare inequality. In magneticterms this tells us that

2EM =D

B2 dV L1D

A b dV = L1|HM|

and so the helicity, if non-zero initially, gives a lower bound on the magneticenergy. (In fact a little care must be taken with the definition of L here as itinvolves boundary conditions on A: see Moffatt [9] for more information.)

The conclusion is that non-zero helicity must lead to a non-trivial final stateof this relaxation process, and in fact it has been shown that any linkage offield lines (for example Borromean rings) must do the same (by Freedman [5]).

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In the limit t we expect the kinetic energy to go to zero and leave uswith a magnetostatic equilibrium satisfying

j b = P, j = b

But this is analogous to the steady Euler equation

0 = uE E PE, E = uE

under the replacement

j E, b uE, P PE

and so we find a steady Euler flow from the relaxation process and thenapplying this analogy. We use an E subscript to distinguish the analogousflow from the original one in the MHD system

To recap: we choose a magnetic field with whatever magnetic field line, that ischoose b-line topology, we like but with some non-trivial linkage. This couldbe total helicity, or the helicity restricted to the field inside magnetic surfaces,perhaps a whole family or magnetic surfaces. Or any helicities may be zero andthere could be higher order linkages analogous to the Borromean rings. Wethen do the relaxation process and let t , in which case the flow u 0and so we are left with the relaxed field with the same topology as initiallyand non-zero magnetic energy. This corresponds to a steady Euler flow uE inwhich the stream lines (not the vortex lines) have the same topology as thefield lines b specified initially.

Moffatt [9; 11; 12] gives appealing pictures of relaxing magnetic fields andcorresponding Euler flows. For example we can start by a set of nested surfacesS() in which the helicity will be conserved, being a parameter labellingthe surface. If we then relax this magnetic field we obtain a correspondingEuler flow, and the helicities will be preserved: this helicity function gives thesignature of the vortex.

Nonetheless, despite this appealing picture, there are many unknowns (andprobably unknown unknowns, particularly for general three-dimensional fields).Does u tend to zero pointwise? Can singularities occur in the fields as they

relax in the limit t ? Or rather, how bad can singularities be? How cana sensible limit then be defined?

For example we have the situation described by Bajer (see [12]) where aninitial X-point topology leads to current sheet formation in 2-dimensionalrelaxation. Here discontinuities in the b field emerge in the infinite time limit,corresponding to a singular current distribution.

Suppose a sensible limit can be taken. Then how does the resulting Euler flowuE fit into Arnolds classification: presumably the requirement of analyticity

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will no longer hold and will allow all sorts of complex mixtures of surfaces andchaotic regions, with discontinuities and maybe devils staircases for functionssuch as P, as suggested by Bajer [2]. Also these questions are very hard toinvestigate numerically in the difficult limit of zero magnetic diffusion andt .

15 A variational formulation

To investigate a little more, we temporarily forget about magnetic relaxationand consider any velocity field u and vorticity field = u. This haskinetic energy

EK =D

12u2 dV

Suppose some fictitious, divergenceless, infinitesimal flow field acts and

moves the vortex lines, so we have a new vorticity field + andvelocity field u u + u given by, at leading order

= ( )

u = +

Here the scalar field is fixed to make the field u divergenceless. This iscalled a frozen field or isovortical displacement. The corresponding change inenergy is

EK =D

u u dV =D

u ( + ) dV

With u n = 0 in the domain where we are working, we obtain

EK =D

( u) dV

Now if u is the gradient of some scalar field then we obtain zero by thedivergence theorem: = 0 means that is orthogonal to gradients provided n = 0 on the boundary of the domain. Conversely it can be shown that ifEK vanishes for all divergenceless then u must be the gradient of ascalar field, say

u = P

This is Eulers equation for steady flow and we obtain the variation result:steady Euler flows are stationary points of EK with respect to general isovor-tical displacements.

Further study looks at the second variation of the energy and if it is a maxi-mum or a minimum then stability can be shown [10]. This has led to powerfulstability theorems in two dimensions [1].

In terms of relaxation, we move the magnetic field lines b with the flow uso as to minimise the magnetic energy EM, so in particular we are making it

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stationary and tending to a stable magnetic equilibrium. When we identify bwith uE we have a stationary point of the corresponding kinetic energy EK ofuE. But this need not be a minimum, and in general there is no reason whythe fluid flow uE should be stable.

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16 Exercises

These vary in difficulty: if you cant do one, then do move on to the next. Mostinvolve vector calculus identities and the use of Gauss theorem. Questions 6,

8, 9 are very easy, and question 5 is potentially quite tricky.

Q 1. Show that in the presence of viscosity , for a flow (no magnetic field) ina bounded domain D with u = 0 on the boundary S,

dEKdt

= D

2 dV = D

|u|2 dV

where |u|2 = (iuj)(iuj).

Q 2. Show that for a flow in infinite space with a localised vorticity distribution

dHKdt

= 2D

dV

Q 3. A divergence free field takes the form u = + where , and are smooth functions such u = 0. Show that if these functions are localisedand so decay sufficiently rapidly at infinity, then the flow field has zero totalhelicity HK.

Q 4. Prove the Jacobi identity of Lie brackets for 3 general vector fields,

[[u,v],w] + [[v,w],u] + [[w,u],v] = 0

Q 5. Show that given arbitrary vector fields b, c and d transported in theusual way, namely

Dtb = b u, Dtc = c u, Dtd = d u

then for incompressible flow u = 0 we have

Dt(b c d) = 0

Hence show that b c is transported as a one-form, namely

Dt(b c) = (u) b c

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[Hint for first part: after a short time t, the vectors b, c and d are moved tob (I+ tu), c (I+ tu) and d (I+ tu) (I being the identity matrix).The change in the volume is given by the determinant of (I+tu) which canbe linked to the trace of u through the identity detM= exp(tr logM).]

Q 6. Show that for two dimensional flows u = (u(x,y,t), v(x,y,t), 0) withvorticity = (0, 0, ) the vorticity equation may be written in the form

Dt = t + u = t J(, ) = 0

where u is written in terms of a stream function u = (y, x, 0) and J(a, b) =(xa)(yb) (ya)(xb) is a Jacobian. From t + u = 0 show that theenstrophy

K =D

2 dV

is conserved for ideal flow and a localised vorticity distribution in the infiniteplane. Show also that the moments of the vorticity distribution

Kn =D

n dV

are also conserved.

Q 7. Show that for ideal MHD the total energy

EK + EM =D

1

2u

2

dV +D

1

2b

2

dV = constant

in a finite domain with no normal flow, perfectly conducting boundary condi-tions

n u = n b = 0

Show that the cross helicity

HX =D

u b dV

is conserved in time with n u = n b = 0 on the boundary, but that the

kinetic helicity obeys dHKdt

= 2D

j b dV

provided n j = 0 on the boundary. Interpret the cross helicity in terms oflinkage by considering magnetic field b confined to tubes Tj.

Q 8. Show that ideal MHD may be written in terms of Elsasser variables

+ = u + b, = u b

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as

t+ + + = PMt + + = PM

+ = = 0

Hence show that any solution of the form u = b or u = b satisfies the idealMHD equations.

Q 9. Consider magnetic relaxation under the following equation for the fluidflow in a porous medium, with constant > 0

tu = u +j b P u

Show that EM + EK decreases in time, for u non-zero.

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References

[1] V.I. Arnold & B. Khesin 1998 Topological methods in hydrodynamics.Springer.

[2] K. Bajer 2005 Abundant singularities. Fluid Dyn. Res. 36, 301317.

[3] S. Childress 2004 Topological fluid dynamics for fluid dynamicists.http://www.math.nyu.edu/faculty/childres/tfd.pdf

[4] T. Dombre et al. 1986 Chaotic streamlines in the ABC flows J. Fluid Mech.167, 353391.

[5] M.H. Freedman 1988 A note on topology and magnetic energy in incom-pressible perfectly conducting fluids. J. Fluid Mech. 194, 549551.

[6] A.D. Gilbert 2003 Dynamo theory. In Handbook of mathematical fluid dy-namics (ed. S.J. Friedlander, J.P. Serre), volume 2, pp. 355441. NorthHolland.

[7] H.K. Moffatt 1969 The degree of knottedness of tangled vortex lines. J.

Fluid Mech. 35, 117129.[8] H.K. Moffatt 1978 Magnetic field generation in electrically conducting flu-

ids. Cambridge University Press.[9] H.K. Moffatt 1985 Magnetostatic equilibria and analogous Euler ows of

arbitrary complex topology. Part 1. Fundamentals. J. Fluid Mech. 159,359378.

[10] H.K. Moffatt 1986a Magnetostatic equilibria and analogous Euler ows ofarbitrary complex topology. Part 2. Stability considerations. J. Fluid Mech.166, 359378.

[11] H.K. Moffatt 1986b On the existence of localized rotational disturbances

which propagate without change of structure in an inviscid fluid.J. Fluid

Mech. 173, 289302.[12] H.K. Moffatt 1990 Structure and stability of solutions of the Euler equa-

tions: a lagrangian approach. Phil. Trans. R. Soc. Lond. A 333, 321342.[13] H.K. Moffatt & A. Tsinober (eds.) 1989 Topological fluid mechanics. Cam-

bridge University Press.[14] H.K. Moffatt et al. (eds.) 1992 Topological aspects of the dynamics of luids

and plasmas. Kluwer.[15] R.L. Ricca (ed.) 2001 An introduction to the geometry and topology of

fluid flows. Kluwer.[16] B. Schutz 1980 Geometrical methods of mathematical physics. Cambridge

University Press.

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Andrew D. Gilbert and Mitchell Berger- Topological fluid mechanics