Nambu brackets in fluid mechanics and magnetohydrodynamics

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J. Phys. A: Math. Theor. 43 (2010) 305501 (8pp) doi:10.1088/1751-8113/43/30/305501

Nambu brackets in fluid mechanics andmagnetohydrodynamics

Roberto Salazar1 and Michael V Kurgansky2,3

1 Center for Quantum Optics and Quantum Information, Department of Physics,University of Concepcion, Casilla 160-C, Concepcion, Chile2 Departament of Geophysics, University of Concepcion, Casilla 160-C, Concepcion,Chile3 AM Obukhov Institute of Atmospheric Physics, Moscow, Russia

E-mail: kurgansk@udec.cl

Received 17 February 2010, in final form 21 May 2010Published 24 June 2010Online at stacks.iop.org/JPhysA/43/305501

AbstractConcrete examples of the construction of Nambu brackets for equationsof motion (both 3D and 2D) of Boussinesq stratified fluids and also formagnetohydrodynamical equations are given. It serves a generalizationof Hamiltonian formulation for the considered equations of motion. Twoalternative Nambu formulations are proposed, first by using fluid dynamical(kinetic) helicity and/or enstrophy as constitutive elements and second, by usingthe existing conservation laws of the governing equation.

PACS numbers: 47.10.−g, 47.10.ab, 47.10.Df, 47.65.−d, 52.30.Cv, 52.65.Kj

1. Introduction to the Nambu formalism

Nambu [1] proposed a generalization of the Hamiltonian mechanics taking as basis the factthat the said mechanics is not the only formalism which leads to the Liouville theorem, whichplays a fundamental role in statistical mechanics.

Nevir and Blender [2] established a Nambu field approach to ideal fluid mechanics byusing conservation laws of kinetic energy and enstrophy/helicity for 2D/3D non-divergentbarotropic fluid flows, correspondingly.

An important feature of the Nambu formalism is the non-degeneracy of its brackets.In Hamiltonian mechanics, the vorticity conserved quantities, enstrophy and helicity, are theCasimir invariants of singular Poisson bracket (PB), whereas in Nambu mechanics it is possibleto put these conserved quantities on the same fundamental level as energy.

This non-degeneracy allowed Salmon [3, 4] to make a first application of this theory. Heestablished that the discretization of the Nambu brackets (NBs) leads to numerical schemesthat conserve energy and vorticity-related quantities. He showed that the Arakawa Jacobiancan be derived systematically from the antisymmetry of the Nambu representation. Also the

1751-8113/10/305501+08$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. A: Math. Theor. 43 (2010) 305501 R Salazar and M V Kurgansky

Nambu representation includes forces like the Coriolis force in terms of vorticity-conservedquantities. Recently, Nevir and Sommer [5] extended the NB approach to the full set of theideal hydrothermodynamic equations, describing a 3D compressible nonhydrostatic fluid.

To define the Nambu formalism consider, following [1], a triplet of the dynamical variablesr = (x, y, z) that spans a 3D phase space. Then introduce the two functions H and G of(x, y, z), where H is the Hamiltonian of the system and G a Casimir. In general, the Nambuequations are

drdt

= ∇H × ∇G. (1)

Thereafter, we have for an arbitrary function F = F(x, y, z)

dF

dt= ∂(F,H,G)

∂(x, y, z)≡ ∇F · (∇H × ∇G). (2)

The term on the right-hand side of equation (2) is usually written as [F,H,G] a trilinearcompletely antisymmetric form and is called the Nambu bracket (NB). From equation (1) it isclear that ∇ · r = 0 and therefore the Liouville theorem is satisfied.

For this scheme to be successful, it is necessary to choose properly the Casimir andthe construction of the trilinear completely antisymmetric bracket [· · ·] so as to obtain theequations of motion for the system in question.

We call NBs of first kind (NB I) the NBs (2) or the sums of NBs which are related todifferent triplets (xk, yk, zk), k = 1, . . . , N , that involve the same conserved quantities H andG [1]:

dF

dt=

∑k

∂(F,H,G)

∂(xk, yk, zk)=

∑k

[F,H,G]k. (3)

The sums of NBs that involve different conserved or constitutive quantities (Hi,Gi),i, . . . ,M, are called the NBs of second kind (NB II) [1]:

dF

dt=

∑i

∂(F,Hi,Gi)

∂(x, y, z)=

∑i

[F,Hi,Gi]. (4)

Like in [5] we call the constitutive quantities the quantities which form the basis of theconstruction of NBs but are not necessarily the constants of motion.

In general the choice of constitutive quantities is different for the NB of first andsecond kind. In this paper we explore this remark in the context of fluid dynamic andmagnetohydrodynamic (MHD) equations.

Our main result is an extension of the Nambu formalism onto 3D and 2D Boussinesqfluids and also onto the MHD case. For the 2D Boussinesq fluids and MHD we find that bothNB I and NB II are possible to construct and we compare both options.

2. Boussinesq approximation in 3D

The Boussinesq approximation is used in the study of buoyancy-driven flows. It states thatsufficiently small density ρ deviations from a reference density ρo can be neglected, exceptfor the case where they appear in terms which include the acceleration due to gravity. Thegoverning equations under this approximation are

∂v∂t

= −∇� + b − wa × v (5)

∇ · v = 0 (6)

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J. Phys. A: Math. Theor. 43 (2010) 305501 R Salazar and M V Kurgansky

∂b

∂t= −v · ∇b. (7)

Here, v and b = −g(ρ − ρo)/ρo are the velocity and buoyancy, respectively; b = bk wherek is the unit vector in the positive z direction, opposite to the direction of gravity accelerationof magnitude g; −∇� includes the conservative forces and the kinematic units (ρ0 = 1) areassumed; wa = ∇ × v + 2� is the total (absolute) vorticity in a reference frame rotating witha constant angular velocity �; 2� stands for the so-called planetary vorticity.

We propose the following NB II formalism (see the explicit formulas in the appendix):

dF{v, b}dt

= [F,H,G]v,v,v + [F,H,L]b,v,b , (8)

where we use the 3D energy H, 3D kinetic helicity in a rotating frame, G [6, 7], and the globalbuoyancy L :

H =∫

d3x

{v2

2− bz

}

G = 1

2

∫d3x {(∇ × v + 4�) · v}

L =∫

d3xb.

This is valid for all functionals F{v, b} = ∫d3xF(v, b), with F being an arbitrary function,

that satisfy the condition

∇ · δFδv

= 0.

If we want to recover the momentum equation (5), we have to substitute F → ∫d3xv

into (8).At the end of the first section we made a remark about the difference between NB I and

NB II. In fact not all the constitutive elements of NB II in equation (8) are the conservedquantities. In (8) and in all our further constructions of NB II the Hamiltonian H has a highestranking hierarchy, since it appears in all the NBs while the other constitutive elements are inonly one of them.

A feature of NB II is the explicit separation of the terms in the equations onto a kineticpart [· · ·]v,v,v and a buoyant one [· · ·]b,v,b. It is clear that after taking the limit b → 0 ourgoverning equations (5)–(7) describe the completely incompressible fluids and also in thislimit L → 0, whereas H reduces to the kinetic energy. This observation and also the factthat the NB [· · ·]b,v,b has two functional derivatives with respect to b show that this buoyancybracket vanishes identically in this limit case. This shows directly that the kinetic helicity isconserved in completely incompressible (homogeneous by density) fluid flows, as should be.So even when the kinetic helicity is not conserved in heterogeneous (by density) Boussinesqfluid flows, the formalism of NB II provides it a non-trivial physical role. Also one can see abreak in the conservation of kinetic helicity as buoyancy is not neglected.

The same happens with the NBs of Nevir and Sommer [5] for general ideal fluids. Herewe need to comment that the energy–entropy NB in [5] contains a minor flaw. We proposeto fix this problem by introducing a new energy–entropy NB of the form [F,H, S]σ,v,σ . Thisnew bracket is shown in the appendix. It is equivalent to our NB, [F,H,L]b,v,b, if instead ofb we substitute the specific entropy s = σ/ρ and also replace the global buoyancy L with thetotal entropy S.

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J. Phys. A: Math. Theor. 43 (2010) 305501 R Salazar and M V Kurgansky

3. Boussinesq approximation in 2D

One way to write the equations of motion for 2D ideal fluids in the Boussinesq approximationis

∂ω

∂t= −J (ω,ψ) − J (b, y) (9)

∂b

∂t= −J (b, ψ), (10)

where b is the buoyancy, ψ is the stream function and ω = ∇2ψ is the 2D vorticity. Here,J (A,B) = ∂(A,B)/∂(x, y) stands for the Jacobian of the variables A and B in the (x, y)

coordinates, where y is the vertical coordinate.For this system, we find the following NB I formalism:

dF{ω, b}dt

= [F,H,G]ω,ω,b , (11)

where we use the 2D energy H and 2D helicity analogue (cross-helicity) G

H =∫

d2x

{(∇ψ)2

2− by

}

G =∫

d2xb∇2ψ.

Also, we construct a NB II formalism

dF{ω, b}dt

= [F,H,K]ω,ω,ω + [F,H,L]b,ω,b . (12)

In equation (12) we use as constitutive quantities the enstrophy K and the global buoyancy L:

K = 1

2

∫d2x(∇2ψ)2

L =∫

d2xb.

We see that in the limit b → 0 the buoyant NB II vanishes as in the 3D case and the enstrophybecomes a conserved quantity. Also we see that in this limit the NB II becomes a NB Iwhereas the NB I in equation (11) becomes a PB; we show this PB explicitly in the appendix.Schematically,

as b → 0[ · · · ]ω,ω,b → [ · · ·]ω,ω

[ · · · ]ω,ω,ω + [· · ·]b,ω,b → [ · · ·]ω,ω,ω.

Also for the constitutive elements we have

as b → 0G → 0L → 0K(non-conserved) → K(conserved)

H(conserved) → H(conserved).

This suggests that there is a relation between the number of the functional derivatives in theNBs and the appearance of constitutive elements in NBs.

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J. Phys. A: Math. Theor. 43 (2010) 305501 R Salazar and M V Kurgansky

4. Magnetohydrodynamics

MHD studies the electrically conducting fluids. Important examples are plasmas and liquidmetals. Here we treat only the ideal case, where the fluid has small resistivity and theeffect of the electric field is neglected. To this approximation the governing equations arethe momentum equation, the continuity equation (incompressibility is assumed) and the pre-Maxwell equations (the displacement current is neglected):

∂v∂t

= −(∇ × v) × v − ∇(

p

ρo

+v2

2

)+

1

ρoμ(∇ × B) × B (13)

∂B∂t

= ∇ × (v × B) (14)

∇ · v = 0 (15)

∇ · B = 0. (16)

Here, p is the pressure, ρo is the mean density, B is the magnetic flux density, and μ is themagnetic permeability (assumed constant). We construct a NB I formalism

dF{v, A}dt

= [F,H,G]v,v,A, (17)

where A is the vector potential (B = ∇ × A). We make use of the energy H and thecross-helicity G [8]:

H =∫

d3x

{ρov2

2+

B2

2μ

}

G =∫

d3xv · B.

Also, we construct a NB II formalism

dF{v, A}dt

= [F,H,K]v,v,v + [F,H,L]A,v,A. (18)

We use as constitutive quantities the kinetic helicity K and the magnetic-field helicity L [8]:

K = 1

2

∫d3xv · w

L =∫

d3xA · B.

Here, w = ∇ × v is the vorticity. Both constructions (17) and (18) are valid for all functionalsF{v, A} = ∫

d3xF(v, A) that satisfy the conditions

∇ · δFδv

= 0 ∇ · δFδA

= 0.

In this case we analyse the limit B → 0. Again the governing equations now correspondto the completely incompressible fluids limit. The NB I becomes a PB and the NB II becomesa NB I that conserves energy and kinetic helicity:

as B → 0[ · · ·]v,v,A → [· · ·]v,v

[ · · · ]v,v,v + [· · ·]A,v,A, → [· · ·]v,v,v.

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J. Phys. A: Math. Theor. 43 (2010) 305501 R Salazar and M V Kurgansky

Also for the constitutive elements we have

as B → 0G → 0L → 0K(non-conserved) → K(conserved)

H(conserved) → H(conserved)

in absolute analogy with the Boussinesq 2D case.

5. Concluding remarks

In this paper we extended the Nambu formalism onto 3D/2D Boussinesq fluids and alsoonto an ideal MHD case. Whereas for Boussinesq fluids a scalar buoyancy b is a materiallyconserved quantity (tracer), which afterwards enters the momentum equation in the buoyantforce, in an ideal MHD case the magnetic field density B is frozen in fluid motion and entersthe momentum equation in the ponderomotive Ampere force.

Two different methods for constructing NBs have been used. The first one uses the energyintegral and also other existing invariants (constants of fluid motion) for such a construction.This construction was named the NB of first kind (NB I). The second method uses the energyintegral along with non-conservative quantities, like kinetic helicity/enstrophy in 3D/2Dcases, respectively, as constitutive elements for the bracket construction. This is the so-calledNBs of second kind (NB II).

A hierarchical relationship between NB I, NB II and a corresponding PB is established,with regard to the limit b → 0 and B → 0. Under this limit, constitutive elements in the NBII become conserved quantities and therefore the NB II reduces to the NB I of the simplified(limit) problem, whereas in the NB I all invariants of motion except for the energy disappearand the NB I reduces to the PB.

Getting a deeper insight into the relationship between NB II, NB I and PB, from theperspective of group theory, stands as a future intriguing task.

Finally, we comment on whether the proposed formulation could be extended to non-idealfluids with dissipation (see also [9]). The procedure to do it is most straightforward for theNB I formalism and implies summing up a NB corresponding to the conservative system anda complementary bilinear symmetric bracket 〈F,G〉, where G is a ‘Hamiltonian’ (Casimir) ofthe conservative equations. For example, in the MHD case this complementary bracket whichshould stand on the right-hand side of equation (17) reads

〈F,G〉 = −∫

d3x

{νδFδv

∇ × δGδA

+ νm

δFδA

∇ × δGδv

}.

Here, G is the cross-helicity, ν the kinematic fluid viscosity entering the term −ν∇×∇×vwhich should be added on the right-hand side of equation (13) and νm the magneticviscosity entering the term −νm∇ × ∇ × B which should be added on the right-hand side ofequation (14). Basically the same—but slightly more involved—procedure could be used toextend the NB II formalism onto the case of dissipative systems. Here, the complementarybilinear symmetric brackets would be of the two sorts, each containing one of the twoconstitutive quantities. For the case of 2D/3D Boussinesq fluids the dissipative terms couldbe incorporated into the formalism using the same procedure as in [9].

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J. Phys. A: Math. Theor. 43 (2010) 305501 R Salazar and M V Kurgansky

Appendix. Explicit Nambu brackets

Here, we show the NBs explicitly.The NBs in the Boussinesq 3D case are

[F,H,G]v,v,v =∫

d3x

{δFδv

(δHδv

× δGδv

)}(A.1)

[F,H,L]b,v,b =∫

d3x

{δHδb

∇(

bδFδv

δLδb

)− δF

δb∇

(bδHδv

δLδb

)}+ cyc(F,H,L). (A.2)

We find that an alternative buoyancy bracket can be used

[F,H,L′]b,v,b = −∫

d3x

{δFδb

δHδv

· ∇ δL′

δb

}+ cyc(F,H,L′) (A.3)

if the invariant

L′ = 1

2

∫d3xb2

is used instead of the global buoyancy L.The NBs in the Boussinesq 2D case are

[F,H,G]ω,ω,b = −∫

d2x

{J

(δFδω

,δHδb

)δGδω

}+ cyc(F,H,G) (A.4)

[F,H,K]ω,ω,ω =∫

d2xδFδω

J

(δKδω

,δHδω

)(A.5)

[F,H,L

]b,ω,b

=∫

d2x

{δHδb

J

(δFδω

, bδLδb

)− δF

δbJ

(δHδω

, bδLδb

)}+ cyc(F,H,L) (A.6)

and the corresponding PB is

[F,H]ω,ω = −∫

d2xωJ

(δFδω

,δHδω

). (A.7)

The NBs for magnetohydrodynamics are

[F,H,G]v,v,A =∫

d3x

{1

ρo

δFδv

·(

δHδv

× δGδA

)}+ cyc(F,H,G) (A.8)

[F,H,K]v,v,v =∫

d3x

{1

ρo

δFδv

·(

δHδv

× δKδv

)}(A.9)

[F,H,L]v,A,A =∫

d3x

{1

ρo

δFδv

·(

δHδA

× δLδA

)}+ cyc(F,H,L). (A.10)

The corresponding PB is

[F,H]v,v =∫

d3x

{1

ρo

w ·(

δFδv

× δHδv

)}. (A.11)

Finally the energy–entropy NB that we propose for the evolution equationdF{v, ρ, σ }

dt= [F,H,G]v,v,v + [F,H,M]ρ,v,ρ + [F,H,S]σ,v,σ (A.12)

of a general ideal fluid considered in [5] is

[F,H,S]σ,v,σ =∫

d3x

{δHδσ

∇(

σ

ρ

δFδv

δSδσ

)− δF

δσ∇

(σ

ρ

δHδv

δSδσ

)}+ cyc(F,H,S). (A.13)

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J. Phys. A: Math. Theor. 43 (2010) 305501 R Salazar and M V Kurgansky

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