magnetic hydrodynamics with asymmetric stress tensor

Magnetic hydrodynamics with asymmetric stress tensorYuly Billig Citation: Journal of Mathematical Physics 46, 043101 (2005); doi: 10.1063/1.1857065 View online: http://dx.doi.org/10.1063/1.1857065 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/46/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Classification of integrable Hamiltonian hydrodynamic chains associated with Kupershmidt’s brackets J. Math. Phys. 47, 103507 (2006); 10.1063/1.2354590 Old and new results for superenergy tensors from dimensionally dependent tensor identities J. Math. Phys. 44, 6140 (2003); 10.1063/1.1624094 Super-energy tensor for space–times with vanishing scalar curvature J. Math. Phys. 40, 3053 (1999); 10.1063/1.532743 A propulsion–mass tensor coupling in relativistic rocket motion AIP Conf. Proc. 420, 1509 (1998); 10.1063/1.54960 Stress-energy-momentum tensors in constraint field theories J. Math. Phys. 38, 847 (1997); 10.1063/1.531873

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1218966096/x01/AIP-PT/CiSE_JMPArticleDL_120314/Awareness_LibraryF.jpg/47344656396c504a5a37344142416b75?x

http://scitation.aip.org/search?value1=Yuly+Billig&option1=author

http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov

http://dx.doi.org/10.1063/1.1857065

http://scitation.aip.org/content/aip/journal/jmp/46/4?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jmp/47/10/10.1063/1.2354590?ver=pdfcov



http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.54960?ver=pdfcov


Magnetic hydrodynamics with asymmetric stress tensorYuly Billiga!

School of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6

sReceived 12 February 2004; accepted 3 December 2004; published online 25 March 2005d

In this paper we study equations of magnetic hydrodynamics with a stress tensor.We interpret this system as the generalized Euler equation associated with an Abe-lian extension of the Lie algebra of vector fields with a nontrivial 2-cocycle. We usethe Lie algebra approach to prove the energy conservation law and the conservationof cross-helicity. ©2005 American Institute of Physics.fDOI: 10.1063/1.1857065g

I. INTRODUCTION

Magnetic hydrodynamicssMHDd describes evolution of a fluid or plasma, carrying a mag-netic field. This theory is used to model the processes in the Solar corona,22 as well as to designtokamaks.14 There are numerous books treating various aspects of the subject, see e.g., Refs. 9 and20.

The MHD equations are derived from the Euler equation of motion of an incompressible fluidand the Maxwell’s electrodynamics equations, and describe evolution of a fluid with the velocityvector fieldv and the magnetic fieldB,

]v

]t= − sv · ¹ dv + sB · ¹ dB − ¹ p,

]B

]t= − hv,Bj,

divsvd = 0, divsBd = 0. s1.1d

In this paper we study another system of PDEs, where we add extra terms into the evolutionequation for the velocity,

]v

]t= − sv · ¹ dv + sB · ¹ dB + o

i,j

]Bi

]xj¹

]v j

]xi− ¹ p. s1.2d

The additional terms here are of the third degree in derivatives and one can draw certain parallelsbetweens1.2d and the Korteweg–de Vries equation,

ut = uux + uxxx. s1.3d

We interpret the extra terms ins1.2d as a contribution of a stress tensor,

Tki = oj

]Bi

]xj

]v j

]xk. s1.4d

This stress tensor is not symmetric, which indicates that the particles of the fluid should possesselectric or magnetic momentum.

adElectronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS46, 043101s2005d

46, 043101-10022-2488/2005/46~4!/043101/13/$22.50 © 2005 American Institute of Physics


216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

http://dx.doi.org/10.1063/1.1857065

In spite of the additional higher-order terms, the new system of PDEs retains many features ofthe original system. In particular, we show that it still admits the Alfvén wave solutions.

We follow an approach developed by Arnold to give an interpretation of the MHD equationswith the stress tensor as a generalized Euler equation. A generalized Euler equation is an equationfor the geodesics on aspossibly infinite-dimensionald Lie group supplied with a Riemannianstructure. This equation describes the evolution of a tangent vector of the geodesic in the Liealgebra of the Lie groupssee Sec. III for detailsd.

The Lie algebra that corresponds to the MHD equations with the stress tensor is an Abelianextensiongstd of the Lie algebra of the divergence zero vector fields twisted with a nontrivial2-cocyclet. This Lie algebra was studied in the framework of the representation theory of thetoroidal Lie algebras and the cocycle plays a prominent role there.

Infinite-dimensional groups associated with Abelian extensions of the Lie algebra of vectorfields are discussed in Ref. 8.

The Lie algebragstd has nice properties and this translates into nice properties of the PDEs.In particular we establish the energy conservation law and the cross-helicity conservation forMHD with the stress tensor.

Ovsienko and Khesin showed in Ref. 21 that the generalized Euler equation for the Liealgebra of vector fields on a circle yields the nonlinear wave equationut=uux, while incorporationof the Virasoro cocycle into the Lie algebra leads to the Korteweg–de Vries equations1.3d. In away, the present paper may be viewed as a higher-dimensional generalization of Ref. 21.

The paper is organized as follows: in Sec. II we discuss the properties of the systems1.2d,derive the expression for the stress tensor, list the conservation laws and describe the Alfvén wavesolutions. In Sec. III we review the generalized Euler equation for an arbitrary Lie algebra and weapply this method in Sec. IV to an Abelian extension of the Lie algebra of the divergence zerovector fields, derivings1.2d, and establishing the conservation laws in a purely algebraic way.

II. MAGNETIC HYDRODYNAMICS WITH A STRESS TENSOR AND ITSPROPERTIES

Evolution of an incompressible fluid carrying a magnetic field is given by the equations ofmagnetic hydrodynamicssMHDd,

]v

]t= − sv · ¹ dv + sB · ¹ dB − ¹ p,

]B

]t= − hv,Bj,


Here B is the magnetic field,v is the velocity vector field of the fluid, andp spressured is anauxiliary function which is chosen in such a way that the equation divsvd=0 is satisfied. Theformal dot productv ·¹ represents the differential operator

v · ¹ = oj

v j]

]xj.

The PoissonsLied brackethv ,Bj of two vector fieldsv=o jv jsxds] /]xjd andB=o jBjsxds] /]xjd isgiven by

hv,Bj = adsvdB = sv · ¹ dB − sB · ¹ dv = ojSv j

]B

]xj− Bj

]v

]xjD .

In the three-dimensional space, the first equation ins2.1d may be also written as

043101-2 Yuly Billig J. Math. Phys. 46, 043101 ~2005!


216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

]v

]t= v 3 curlsvd + curlsBd 3 B − ¹ p.

For a conducting medium, the curl of the magnetic field is equal to the electric currentsAmpère’slawd, and so the expression curlsBd3B represents the Lorentz force, with which the magneticfield acts on the current.

The configuration space for a flow of an incompressible fluid is the group of volume-preserving diffeomorphisms. In his remarkable paper,2 Arnold interpreted the Euler equation foran ideal fluid as the geodesic equation on this Lie group. The geodesic equation describes theevolution of the tangent vector of the geodesic curve, and this tangent vector belongs to the Liealgebra, which is the Lie algebra SVect of the divergence zero vector fields, in case of the groupof volume preserving diffeomorphisms.

Using Arnold’s method, Vishik and Dolzhanskii24 ssee also Ref. 17d showed that the MHDequationss2.1d also may be interpreted as a geodesic equation for a certain infinite-dimensionalLie group. The Lie algebra that is used to write this equation is the semidirect product

SVect% V1/dV0

of the Lie algebra of divergence zero vector fields SVect with its dual space—the factorV1/dV0

of the differential 1-forms modulo the exact 1-forms. We review this construction in detail in Sec.III.

The Lie algebra that is associated with the MHD equations,

g = SVect% V1/dV0,

has recently attracted much interest in representation theory. It turns out that representations ofgmay be used for constructing modules for toroidal Lie algebrasssee, e.g., Refs. 5, 6, and 11d. Itwas also discovered that this Lie algebra has an important deformation—the Lie bracket ing maybe twisted with a Virasoro-like 2-cocyclet. The twisted Lie algebragstd still has nice properties,and its representation theory is even better than forg itself.

In this paper we study the system of PDEs that comes from the geodesic equation forgstd. InSec. IV we show that this Lie algebra yields the following system of PDEs:

]v

]t= − sv · ¹ dv + sB · ¹ dB + o

i,j

]Bi

]xj¹

]v j

]xi− ¹ p,

]B

]t= − hv,Bj,


Let us discuss the properties of this system of PDEs.First of all, we note that the new term

oi,j

]Bi

]xj¹

]v j

]xi= o

i,j

]2

]xi ] xjsBi ¹ v jd s2.3d

can also be written as

− oi,j

]v j

]xi¹

]Bi

]xj, s2.38d

since the difference of the two expressions is the gradient ofoi,js]v j /]xids]Bi /]xjd and may beabsorbed into¹p.

043101-3 Magnetic hydrodynamics J. Math. Phys. 46, 043101 ~2005!


216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

It is curious to note here that in the one-dimensional case the passage from the Lie algebra ofvector fields to the Virasoro algebra leads to the transition from the nonlinear wave equation

ut = uux

to the Korteweg–de Vries equation

ut = uux + uxxx,

as shown by Ovsienko and Khesin in Ref. 21.Just like the dispersion termuxxx in the KdV, the new termoi,js]Bi /]xjd¹ s]v j /dxid that we get

in s2.2d has a triple derivative inx.Next we are going to show thats2.2d describes magnetic hydrodynamics with astress tensor.

Indeed, for a stress tensorTki, the equations on the velocity field ins2.1d will becomessee, e.g.,Sec. 1.7 in Ref. 23d

]vk

]t= − sv · ¹ dvk + sB · ¹ dBk + o

i

]Tki

]xi−

]p

]xk. s2.4d

Proposition 1:The systems2.2d describes magnetic hydrodynamics with a stress tensor. Thestress tensorTki may be written as

Tki = oj

]Bi

]xj

]v j

]xks2.5d

or as

Tki8 = − oj

]vi

]xj

]Bj

]xks2.58d

or as a linear combinationaTki+bTki8 with a+b=1.Proof: We will prove the statement of the Proposition for tensorTki given bys2.5d. We write

the contribution ins2.4d from the stress tensors2.5d,

oi

]Tki

]xi= o

i,j

]2Bi

]xi ] xj

]v j

]xk+ o

i,j

]Bi

]xj

]2v j

]xi ] xk.

Since divsBd=0, the first term on the right-hand side vanishes, and we get precisely the firstequation froms2.2d.

The stress tensors2.58d will yield the additional term written in the forms2.38d. The proof inthis case is completely analogous.

We point out that the stress tensors we obtain here are not symmetric,TkiÞTik. In a typicalsituation in hydrodynamics, stress tensor is symmetric. The derivation of the symmetry of a stresstensor is based on the consideration of the angular momentum for an infinitesimal region of thefluid. Provided that the external force does not cause angular acceleration of individual particles ofthe fluid, the stress tensor must be symmetric. However, this need not to be the case if there is aforce acting on polarized fluid particlesssee discussion in Sec. 1.3 in Ref. 4 or Chap. 8 in Ref. 23d.The asymmetry of stress tensors2.5d indicates that the particles of the fluid are polar, i.e., possesselectric or magnetic momentum. The derivation of the stress tensor from the first principles israther delicatessee, e.g., Chap. 7 in Ref. 23d, and this paper does not describe precisely thephysical situations when the stress tensorss2.5d and s2.58d would occur.

Introduction of the terms2.3d which has the third order in derivatives, into the equations willclearly change the behavior of the solutions in a substantial way. It is quite surprising that theconservation laws of magnetic hydrodynamics still hold for the systems2.2d.

Our next goal is to study the conservation laws for magnetic hydrodynamics with the stresstensors2.2d. However before we do that, let us discuss the class of solutions that we consider here.



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

We require that the functionsvisxd ,Bisxd are defined in a domainD,Rn and belong to theintersection of the Sobolev spacesùk=1

` H0ksDd. We recall that the spaceH0

ksDd is the closure in theSobolev spaceHksDd of the functions of classCk with compact supportssee, e.g., Ref. 1d. Thiswill ensure that all functions and their partial derivatives of all orders are square-integrablefbe-long toL2sDdg and satisfy the vanishing conditions on the boundary ofD sif D is unbounded thismeans that at infinity the functions go to zero faster than the inverse of any polynomiald. Choosingthis class of functions will allow us to carry out integration by parts with the boundary termvanishing.

Alternatively, we may consider periodic boundary conditions, as it is often done in turbulencetheory.

There are several conserved quantities for the MHD system—mass, momentum, magnetichelicity, as well as energy and cross-helicity. It turns out that all of these are also conserved for theMHD with the stress tensor. Since we consider the case of an incompressible fluid, the conserva-tion of masssvolumed holds trivially. The derivation of the conservation of magnetic helicityinvolves only the evolution equation on magnetic fieldB, and is exactly the same for bothsystems. Let us prove the conservation of momentum for the new system.

Proposition 2:The total momentum is a conserved quantity for the MHD system with thestress tensors2.2d,

ED

vsxddV ; const.

Proof: With the help of Proposition 1, we can write the first equation using the divergenceoperator,

]vk

]t= o

i

]

]xis− vivk + BiBk + Tki − pdkid.

By the divergence theorem, we get

]

]tE

DvksxddV =R

]D

Rkiei · dn,

whereR is a 2-tensor

Rki = − vivk + BiBk + Tki − pdki,

n denotes the unit outward normal vector, andei’s are the standard basis vectors. Since the vectorfields we consider vanish on the boundary ofD, the last integral is zero.

Next we state the corresponding theorem for the conservation of energy and the cross-helicityconservation.

Theorem 3:The systems2.2d of magnetic hydrodynamics with stress tensors2.5d in a domainD,Rn with appropriate boundary conditionsssee discussion aboved has the following two firstintegrals:

EDo

ivisxd2 + o

i

Bisxd2 dV ; const senergy conservationd s2.6d

and

EDo

ivisxdBisxddV ; constscross-helicity conservationd. s2.7d

We will give the proof of this theorem in Sec. IV. These conservation laws will be derivedfrom the properties of the Lie algebragstd.



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

It is interesting to note that unlike the case of the Navier–Stokes equation, introduction of thestress tensor ins2.2d does not lead to the dissipation of energy, and the energy conservation lawstill holds.

For the topological interpretation of helicity and cross-helicity, see Refs. 18 and 19.In the conclusion of this section we show that systems2.2d admits Alfvén wave solutions.Alfvén waves solutions are obtained as a perturbation of a steady-state constant solution

vsxd=0,Bsxd=B0. If we take an expansionv= vsxd ,B=B0+Bsxd near this equilibrium state, wewill get the following system:

] v

]t= − sv · ¹ dv + sB0 · ¹ dB + sB · ¹ dB + o

i,j

]Bi

]xj¹

] v j

]xi− ¹ p,

]B

]t= − sv · ¹ dB + sB0 · ¹ dv + sB · ¹ dv,


Next we setB= v. In this case the termoi,js]Bi /]xjd¹ s]v j /]xid is a gradient ofoi,js]vi /]xjd3s]v j /]xid, and we can eliminate it by setting

p = oi,j

] vi

]xj

] v j

]xi. s2.9d

Now the first two equations ins2.8d reduce to a single equation

] v

]t= sB0 · ¹ dv. s2.10d

Finally, by taking an arbitrary divergence zero vector fieldwsxd, we can construct a solution ofs2.8d,

vsx,td = wsx + B0td.

This traveling wave solution is called the Alfvén wave. The only difference with the classicalMHD systems2.1d is the change of the pressure terms2.9d.

III. GENERALIZED EULER EQUATION

In this section we are going to review the geodesic equation approach to hydrodynamicsdeveloped by Arnold. In the key paper,2 Arnold gave an interpretation of the Euler equation for anincompressible ideal fluid

]v

]t= − sv · ¹ dv − ¹ p,

divsvd = 0, s3.1d

from the perspective of infinite-dimensional Lie groups. He showed that the Euler equation maybe interpreted as the geodesic equation on the group of diffeomorphisms, where the Riemannianstructure on the group is given by the energy functional.

We will describe this approach here following Ref. 3.Let G be a Lie groupspossibly infinite-dimensionald, and letg be its Lie algebra. Consider a

map fromg to its dual



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

A: g → g* sinertia operatord,

such that it defines a positive-definite symmetric bilinear form ong,

kXuYl = 12AsXdY + 1

2AsYdX, X,Y P g.

The corresponding quadratic formkXuXl=AsXdX is called the energy functional.The Lie algebrag acts on its dual space via the coadjoint action,

sad*sXdudsYd = − usfX,Ygd for X,Y P g, u P g* .

We assume that the spaceAsgd is invariant under the coadjoint action, and we make a conventionthat in what followsg* stands forAsgd fthis is a slight abuse of notations since in the infinite-dimensional caseAsgd is typically smaller than the formal dual ofgg. Since the kernel ofA istrivial fotherwise, the quadratic formkXuXl=AsXdX is not positive-definiteg, then with the aboveconvention the operatorA is invertible.

The bilinear formk·u ·l may be left-translated fromg=TesGd to the tangent spaces at all pointsof G. This gives a Riemannian structure onG, and allows us to consider the geodesics on thisgroup.

Next we are going to write the equation for the geodesics onG, which describes the evolutionof the tangent vectorXPg to the geodesic curve. It turns out however, that it is easier to write theevolution equation for the covectoru=AsXdPg* rather than forX itself. The generalized Eulerequation is the evolution equation foru=AsXd which is written using the coadjoint actionfsees6.4d in Ref. 3g

ut = − ad*sA−1udu. s3.2d

WhenG=SOs3d this equation turns into the equations of motion of a rigid body with a fixedpoint.

Let us discuss equations3.2d in the context of fluid dynamics. Evolution of an incompressiblefluid in domainD,Rn from time 0 to timet is given by a volume-preserving diffeomorphism ofD. Thus the group of the volume preserving diffeomorphismsG=SDiffsDd is the configurationspace for this motion. The Lie algebra of the group SDiffsDd is the Lie algebra of the divergencezero vector fields SVectsDd. As the energy functional we take the kinetic energy,

Koi

visxdU ]

]xiUo

ivisxd

]

]xiL =E

Do

ivisxd2 dV. s3.3d

The dual space for the divergence zero vector fields is the factor space of differential 1-formsof D by exact 1-forms,

SVectsDd* = V1sDd/dV0sDd.

The pairing between SVectsDd andV1sDd /dV0sDd is given by the integral

oj

wjsxddxjSoi

visxd]

]xiD =E

Do

j

wjsxdv jsxddV. s3.4d

It is easy to check that exact 1-forms vanish on the divergence zero vector fields, and so the valueof the integral on the right-hand side is independent of the choice of a representative in a class of1-forms modulo dV0sDd.

The inertia operator

A: SVectsDd → V1sDd/dV0sDd

that corresponds to the energy functionals3.3d is written as follows:



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

ASoi

visxd]

]xiD = o

ivisxddxi .

It is possible to check that in these settings, Eq.s3.2d turns into the Euler equations of motion ofan ideal fluid.

Ebin and Marsden10 used this interpretation of the Euler equations to prove existence ofsolutions for the motion of an ideal fluid for small times. A variational approach to the generalizedEuler equation is presented in Ref. 16.

The equations of the magnetic hydrodynamicss2.1d can also be obtained as a special case ofthe generalized Euler equations3.2d. To construct the corresponding Lie algebra we take the Liealgebra SVectsDd together with its dual space,

g = SVectsDd % V1sDd/dV0sDd.

The Lie bracket of two 1-forms is set to be zero. The Lie bracket of two vector fields is defined inthe usual way,

Foi

visxd]

]xi,o

j

wjsxd]

]xjG = o

i,jvi

]wj

]xi

]

]xj− wj

]vi

]xj

]

]xi, s3.5d

and the Lie bracket of a vector field with a 1-form is given by the Lie derivative,

Foi

visxd]

]xi,o

j

wjsxddxjG = oi,j

vi]wj

]xidxj + o

j

wjdsv jd. s3.6d

It is easy to see that the space dV0sDd is invariant under the Lie derivative action, so the aboveformula may be taken modulo dV0sDd.

The Lie algebrag is a semidirect product of SVectsDd with its moduleV1sDd /dV0sDd. ThespaceV1sDd /dV0sDd forms an Abelian ideal ing.

A really important feature of the Lie algebrag is the existence of an invariant symmetricnondegenerate bilinear form. In contrast, the Lie algebra SVect does not possess such a form. Theinvariant form ong is defined as followsfcf. s3.4dg:

Soi

visxdU ]

]xiUo

j

wjsxddxjD =EDo

ivisxdwisxddV,

Soi

visxdU ]

]xiUo

j

wjsxd]

]xjD = 0, SUo

ivisxddxiUo

j

wjsxddxjD = 0. s3.7d

One can verify that the bilinear forms3.7d satisfies the invariance property,

sfX,YguZd = sXufY,Zgd for all X,Y,Z P g.

We can use this form to identify each elementXPg with a linear functionalsXu ·d in g* . It iswell known that when the bilinear form that is used to identifyg* with g, is invariant andnondegenerate, the coadjoint action becomes isomorphic to the adjoint action. In this case thegeneralized Euler equation takes the following form:

Xt = − adsA−1XdX, s3.8d

where nowXPg and the inertia operatorA now mapsg to g.The generalized Euler equations3.8d yields the equations of magnetic hydrodynamicss2.1d if

we chooseA to be the following involution ong:



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

ASoi

visxd]

]xiD = o

ivisxddxi ,

ASoi

wisxddxiD = oi

wisxd]

]xi. s3.9d

Note that in the second equality we choose asuniqued representative in a class modulo dV0

satisfyingois]wi /]xid=0, so that the right-hand side is a divergence zero vector field.We see that the inertia operators3.9d satisfiesA−1=A, and the energy functionalsAXuXd for

X=oivisxds] /]xid+oiwisxddxi Pg is given by the integral

EDo

ivisxd2 + o

i

wisxd2 dV. s3.10d

The Lie algebra

g = Svect% V1/dV0

appears in the study of toroidal Lie algebrassRefs. 5–7, 11, and 15d. The representations ofg arean essential ingredient for constructing the representation theory of toroidal Lie algebras. It wasdiscovered however thatg admits a nontrivial deformation with aV1/dV0-valued 2-cocycle onSVect, and one gets a better representation theory for the deformed algebra than forg itself.

In the next section we will describe this deformation ofg and study the associated generalizedEuler equation.

IV. ABELIAN EXTENSIONS OF THE LIE ALGEBRA OF VECTOR FIELDS

In the preceding section we have constructed a semidirect productg of the Lie algebra ofdivergence zero vector fields withV1/dV0. It turns out that on the same vector space

g = Svect% V1/dV0

we may deform the Lie bracket in a nontrivial way. When we define the Lie bracket of two vectorfields, we are going to add to the right-hand side ofs3.5d a correction term which has value inV1sDd /dV0sDd,

Foi

visxd]

]xi,o

j

wjsxd]

]xjG = o

i,jSvi

]wj

]xi

]

]xj− wj

]vi

]xj

]

]xiD + tSo

ivisxd

]

]xi,o

j

wjsxd]

]xjD .

s4.1d

In order to get a Lie bracket,t must be a 2-cocycle on SVectsDd with values inV1sDd /dV0sDd.The following cocycle plays an important role in the representation theory:

tSoi

visxd]

]xi,o

j

wjsxd]

]xjD = o

i,j

]vi

]xjdS ]wj

]xiD . s4.2d

This cocycle may be viewed as a higher-dimensional generalization of the Virasoro cocycle.Just as the Virasoro cocycle, it has a triple derivative inx, and in facts4.2d reducès to the Virasorococycle for the Lie algebra of vector fields on a circle.

The cocyclet can be constructed using Gelfand–Fuks cohomology.12,13 In its explicit form itfirst appeared in Ref. 11 in the context of the representation theory of toroidal Lie algebras.

We will denote the Lie algebra with the Lie bracket deformed by the cocyclet by gstd. Notethat



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

gstd = SvectsDd % V1sDd/dV0sDd

is no longer a semidirect product, but the subspaceV1sDd /dV0sDd still forms an Abelian ideal. Asbefore, the action of SvectsDd on V1sDd /dV0sDd is given by the Lie derivative formulas3.6d.

Proposition 4:The bilinear form ongstd given by s3.7d is invariant.Proof: We need to establish the invariance property,

sfX,YguZd = sXufY,Zgd.

There are three nontrivial cases to be considered,

sid X,YPSvectsDd ,ZPV1sDd /dV0sDd,sii d X,ZPSVectsDd ,YPV1sDd /dV0sDd,siii d X,Y,ZPSVectsDd.

We will verify the invariance only for the last case, since only this case will involve thecocyclet, and leave the first two cases as an exercise to the reader.

Suppose

X = oi

ui]

]xi, Y = o

jv j

]

]xj, Z = o

k

wk]

]xk.

SincesSVectuSVectd=0, then we get thatsfX,Yg uZd=stsX,Yd uZd and sXu fY,Zgd=sXutsY,Zdd.We have

sfX,YguZd = Soi,j ,s

]ui

]xjU ]2v j

]xi ] xsdxsUo

k

wk]

]xkD =E

Doi,j ,k

wk]ui

]xj

]2v j

]xi ] xkdV.

Integrating by parts and using the fact thatois]ui /]xid=0, we get

sfX,YguZd = −EDoi,j ,k

]wk

]xi

]ui

]xj

]v j

]xkdV.

On the other hand,

sXutsY,Zdd = Soi

ui]

]xiUo

j ,k,s

]v j

]xkU ]2wk

]xjdxsdxsD =E

Doi,j ,k

ui]v j

]xk

]2wk

]xjdxidV = −E

Doi,j ,k

]ui

]xj

]v j

]xk

]wk

]xidV.

This proves the invariance propertysfX,Yg uZd=sXu fY,Zgd in casesiii d.Now we are going to prove the following theorem.Theorem 5: The generalized Euler equation

Xt = − fAX,Xg s4.3d

for the Lie algebragstd with the inertia operatorA given bys3.9d yields the equations of magnetichydrodynamics with asymmetric stress tensors2.2d.

Proof: We write

X = oi

Bi]

]xi+ o

jv j dxj .

We will fix representatives of classes of 1-forms modulo dV0sDd by imposing a conditiono js]v j /]xjd=0. Then we have



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

AX= oj

v j]

]xj+ o

i

Bi dxi

and

fX,AXg = oi,j

Bi]v j

]xi

]

]xj− o

i,jv j

]Bi

]xj

]

]xi+ o

i,j ,k

]Bi

]xj

]2v j

]xi ] xkdxk + o

i,jBi

]Bj

]xidxj + o

i

BidsBid

− oi,j

vi]v j

]xidxj − o

ividsvid.

Note that the termsBidsBid= 12dsBi

2d and vidsvid= 12dsvi

2d are full differentials and thus may bedropped.

Substituting the obtained expression into the generalized Euler equations4.3d and collectingterms at dxj ,] /]xj, and taking into account that equality of 1-forms is taken modulo dV0sDd, weget the following system of PDEs:

]v j

]t= − o

ivi

]v j

]xi+ o

i

Bi]Bj

]xi+ o

i,k

]Bi

]xk

]2vk

]xi ] xj−

]p

]xj,

]Bj

]t= − o

iSvi

]Bj

]xi− Bi

]v j

]xiD ,

oj

]v j

]xj= 0, o

j

]Bj

]xj= 0.

Rewriting this system in a vector form with the vector fields we gets2.2d.Finally let us prove Theorem 3 and establish the energy and the cross-helicity conservation

laws for MHD equations with the stress tensors2.2d. We will in fact obtain Theorem 3 as acorollary of the following.

Theorem 6: Let g be a Lie algebra with a nondegenerate symmetric invariant bilinear forms·u ·d. Let A be an involution ofg preserving the invariant form

A: g → g, A2 = Id, sAXuAYd = sXuYd for all X,Y P g.

Then the generalized Euler equationXt=−fAX,Xg has the following two first integrals:

sAXuXd ; const s4.4d

and

sXuXd ; const. s4.5d

Proof: Let us evaluates] /]tdsAXuXd,

]

]tsAXuXd = sAXtuXd + sAXuXtd.

Taking into account thatsXuYd=sAXuAYd andA2=Id, we get thatsAXt uXd=sA2Xt uAXd=sXt uAXd.Thus

]

]tsAXuXd = 2sAXuXtd.

Substituting the right-hand side of the generalized Euler equation forXt we obtain



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

]

]tsAXuXd = − 2sAXufAX,Xgd.

By invariance of the form we get

sAXufAX,Xgd = sfAX,AXguXd = 0.

Thus s] /]tdsAXuXd=0 ands4.4d is established.The second conservation laws4.5d is obtained in a similar way,

]

]tsXuXd = 2sXtuXd = − 2sfAX,XguXd = − 2sAXufX,Xgd = 0.

This completes the proof of Theorem 6.Note that fors4.5d we may drop the requirements thatA preserves the invariant form and is an

involution.We obtain Theorem 3 as an immediate corollary to the previous theorem, noting that the

bilinear form s3.7d on gstd is invariant by Proposition 4 and the inertia operators3.9d is aninvolution and preserves this form.

We can see that for

X = oi

Bi]

]xi+ o

jv j dxj ,

the first integrals4.4d becomes the energy conservation law

sAXuXd =EDo

ivisxd2 + o

i

Bisxd2 dV ; const,

and s4.5d becomes the cross-helicity conservation,

sXuXd = 2EDo

ivisxdBisxddV ; const.

ACKNOWLEDGMENTS

The author thanks Boris Khesin for getting him interested in this problem, and David Amund-sen for helpful discussions. The author has benefited from the hospitality of l’Institut des Mathé-matiques de JussieusParisd, where part of this work has been done. This research is supported bythe Natural Sciences and Engineering Research Council of Canada.

1Adams, R. A.,Sobolev Spaces, Pure and Applied MathematicssAcademic, New York, 1975d, Vol. 65.2Arnold, V., “Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications àl’hydrodynamique des fluides parfaits,” Ann. Inst. Fourier16, 319–361s1966d.

3Arnold, V. I. and Khesin, B. A.,Topological Methods in Hydrodynamics, Applied Mathematical Sciences, Vol. 125sSpringer-Verlag, New York, 1998d.

4Batchelor, G. K.,An Introduction to Fluid DynamicssCambridge University Press, Cambridge, 1967d.5Berman, S. and Billig, Y., “Irreducible representations for toroidal Lie algebras,” J. Algebra221, 188–231s1999d.6Billig, Y., “Principal vertex operator representations for toroidal Lie algebras,” J. Math. Phys.39, 3844–3864s1998d.7Billig, Y., “Energy-momentum tensor for the toroidal Lie algebras,” math.RT/0201313.8Billig, Y., “Abelian extensions of the group of diffeomorphisms of a torus,” Lett. Math. Phys.64, 155–169s2003d.9Biskamp, D.,Nonlinear MagnetohydrodynamicssCambridge University Press, Cambridge, 1993d.

10Ebin, D. and Marsden, J., “Groups of diffeomorphisms and the motion of an incompressible fluid,” Ann. Math.92,102–163s1970d.

11Eswara Rao, S. and Moody, R. V., “Vertex representations forN-toroidal Lie algebras and a generalization of theVirasoro algebra,” Commun. Math. Phys.159, 239–264s1994d.

12Fuks, D. B.,Cohomology of Infinite-Dimensional Lie AlgebrassConsultants Bureau, New York, 1986d.13Gelfand, I. M. and Fuks, D. B., “Cohomology of Lie algebras of vector fields with nontrivial coefficients,” Funkc. Anal.



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

Priloz. 4, 10–25s1970d.14Kadomtsev, B. B.,Tokamak Plasma: A Complex Physical SystemsInstitute of Physics, Bristol, 1992d.15Larsson, T. A., “Lowest-energy representations of non-centrally extended diffeomorphism algebras,” Commun. Math.

Phys. 201, 461–470, 1999.16Marsden, J. and Ratiu, T.,Introduction to Mechanics and Symmetry, Texts in Appl. Math.sSpringer-Verlag, New York,

1994d, Vol. 17.17Marsden, J., Ratiu, T., and Weinstein, A., “Semidirect products and reduction in mechanics,” Trans. Am. Math. Soc.281,

147–177s1984d.18Moffatt, H. K., “Vortex- and magneto-dynamics—A topological perspective,”Mathematical Physics 2000sImperial

College Press, London, 2000d, pp. 170–182.19Moffatt, H. K., “Some developments in the theory of turbulence,” J. Fluid Mech.106, 27–47s1981d.20Moreau, R.,Magnetohydrodynamics, Fluid Mechanics and its ApplicationssKluwer Academic, Dodrecht, 1990d, Vol. 3.21Ovsienko, V. Yu. and Khesin, B. A., “The super Korteweg–de Vries equation as an Euler equation,” Funkc. Anal. Priloz.

21, 81–82s1987d.22Priest, E. R.,Solar Magnetohydrodynamics, Geophysics and Astrophysics MonographssD. Reidel, Dordrecht, 1982d,

Vol. 21.23Rosensweig, R. E.,FerrohydrodynamicssCambridge University Press, Cambridge, 1985d.24Višik, S. M. and Dolžanski�, F. V., “Analogs of the Euler-Lagrange equations and magnetohydrodynamics connected

with Lie groups,” Dokl. Akad. Nauk SSSR19, 149–153s1978d.



216.165.95.69 On: Mon, 08 Dec 2014 06:09:20

magnetic hydrodynamics with asymmetric stress tensor

Documents