## Abstract

Nambu field theory, originated by Névir and Blender for incompressible flows, is generalized to establish a unified energy–vorticity theory of ideal fluid mechanics. Using this approach, the degeneracy of the corresponding noncanonical Poisson bracket—a characteristic property of Hamiltonian fluid mechanics—can be replaced by a nondegenerate bracket. An energy–vorticity representation of the quasigeostrophic theory and of multilayer shallow-water models is given, highlighting the fact that potential enstrophy is just as important as energy. The energy–vorticity representation of the hydrostatic adiabatic system on isentropic surfaces can be written in complete analogy to the shallow-water equations using vorticity, divergence, and pseudodensity as prognostic variables. Furthermore, it is shown that the Eulerian equation of motion, the continuity equation, and the first law of thermodynamics, which describe the nonlinear evolution of a 3D compressible, adiabatic, and nonhydrostatic fluid, can be written in Nambu representation. Here, trilinear energy–helicity, energy–mass, and energy–entropy brackets are introduced. In this model the global conservation of Ertel’s potential enstrophy can be interpreted as a super-Casimir functional in phase space. In conclusion, it is argued that on the basis of the energy–vorticity theory of ideal fluid mechanics, new numerical schemes can be constructed, which might be of importance for modeling coherent structures in long-term integrations and climate simulations.

## 1. Introduction

Nambu (1973) proposed a generalization of Hamiltonian mechanics for ordinary differential equations obeying Liouville’s theorem of nondivergent motion in phase space. This so-called Nambu mechanics is characterized by the existence of several conserved quantities and by the possibility of describing dynamical systems with even and odd numbers of degrees of freedom *N*. In the most simple case, *N* = 2, this reduces to the well-known canonical form of Hamiltonian dynamics. Twenty years later this concept was picked up in ideal fluid mechanics by Névir and Blender (1993) to establish a Nambu field approach for partial differential equations. Here, the nondivergent vorticity equation is written using the conservation laws of kinetic energy and enstrophy. Moreover, the Helmholtz equation of a 3D nondivergent barotropic fluid is written in a trilinear bracket representation applying the conservation of kinetic energy and helicity. Whereas the noncanonical form of Hamiltonian fluid mechanics deals with the vorticity-conserved quantities enstrophy and helicity as Casimir invariants of a singular Poisson bracket, the Nambu approach highlights the vorticity-conserved quantities on the same fundamental level as energy. The Nambu representation can include forces like the Coriolis force in terms of vorticity-conserved quantities. Of course, a noncanonical form of the 3D compressible, adiabatic Euler equation in momentum form can also incorporate the Coriolis terms (Morrison and Greene 1980). However, this approach is not related to any vorticity conservation of the system. Moreover, a related Lie–Poisson representation of these equations using the density of momentum is incompatible with the Coriolis force (Morrison 1998). Salmon (2005, 2007), inspired by the Nambu bracket approach, made a first application of this theory to establish numerical schemes that conserve energy and vorticity-related quantities in a numerical model. He showed that the Arakawa Jacobian (Arakawa 1966) can be derived systematically using the twofold antisymmetry of the Nambu representation of the 2D vorticity equation. He also developed the Nambu field brackets of the shallow-water model. The antisymmetry properties of the Nambu bracket can also be retained throughout the spatial and probably even the temporal discretization process to conserve energy and enstrophy in the discrete analog. This approach can be generalized to other fluid dynamical systems. Recently, Bihlo (2008) incorporated dissipative terms in the theory and showed that the Rayleigh–Bénard convection can be represented as a Nambu-metriplectic problem.

Before applying the Nambu representation to a compressible, adiabatic fluid in order to develop new numerical schemes (Gassmann and Herzog 2008), a physical basis of this theory has to be established. Many results of the present paper can be found in the habilitation thesis of one of the authors (Névir 1998). In particular, the explicit inclusion of the continuity equation and the first law of thermodynamics is investigated there. In this paper we investigate the role of the conservation of mass, entropy, and Ertel’s potential enstrophy. Also, the importance of helicity as a conserved quantity for barotropic fluids and a constitutive (albeit not conserved) quantity of baroclinic fluids is analyzed. In section 2 the energy–vorticity representation of the quasigeostrophic model is presented. In section 3 the energy–vorticity structure of multilayer shallow-water models and the primitive hydrostatic model on isentropic surfaces are discussed. In section 4 the energy–vorticity representation of a 3D, compressible, adiabatic, nonhydrostatic fluid is given. In section 5 we show the outstanding role of Ertel’s potential enstrophy. Moreover, this section summarizes the major results of the paper and introduces ideas toward a new classification of fluid dynamical models. Section 6 looks toward the future, discussing possibilities for numerical applications of this theory that might be of importance for structure formation in long-term integrations and climate simulations.

## 2. Energy–vorticity theory of quasigeostrophic models

A central quantity in quasigeostrophic (QG) dynamics is Ertel’s potential vorticity (EPV) in the quasigeostrophic approximation. In a system with pressure *p* as a vertical coordinate, this quantity is given by

Here, *ϕ* is the geopotential height field, *σ _{o}* is the constant stability parameter,

*f*= 2

*ω*sin

*ϕ*is the varying Coriolis parameter, and

*f*is a constant value of

_{o}*f*at a central latitude

*ϕ*. In this approximation the quasigeostrophic PV is a sum of the relative geostrophic vorticity , a potential term describing the influence of the vertical structure, and the varying Coriolis parameter. This quasigeostrophic potential vorticity Π

_{o}_{QG}can be written in a system with a stretched vertical coordinate

*z**, whose total differential is

Here, *N*_{z,o} is the related Brunt–Väisälä frequency and *z* is the geometrical height. With (2) the stretched vertical coordinate can be written as *z** = *zL _{D}*/

*H*, where

*L*is the internal Rossby radius of deformation. With the values of

_{D}*f*≈ 10

_{o}^{−4}s

^{−1},

*N*

_{z,o}≈ 10

^{−2}s

^{−1}, and

*H*≈ 10 km, the scale of the coordinate is approximately 1000 km and thus comparable to the magnitude of the horizontal length of large-scale motions. Taking into account the fact that the properties of a space are properly defined in terms of a metric and not in terms of the used coordinates, and in order to show an exact mathematical analogy to the barotropic nondivergent vorticity equation, the QG theory can be rewritten. To that end, an adapted scalar product that appropriately scales the horizontal and the vertical dimensions is introduced:

Using this metric, the quasigeostrophic potential vorticity can be written as

The central evolution equation in QG theory is the Lagrangian conservation of quasigeostrophic potential vorticity Π_{QG}, which is given by

As in the nondivergent 2D case, the nonlinear advection can be expressed as Jacobian *J* of the geopotential height field and QG potential vorticity. The constitutive conserved global quantities of this model are the total energy ℋ and the QG potential enstrophy ɛ* _{p}*:

Here, *dτ* = *dx dy dz* is the infinitesimal volume element. According to (6), the total energy of the quasigeostrophic model is given by the sum of the geostrophic kinetic part and a potential part, describing the vertical structure of the geopotential height field. Throughout the paper potential enstrophy is denoted ɛ* _{p}*; however, this globally conserved quantity depends on the considered model. The functional derivatives of the two quantities energy and potential enstrophy are given by

Inserting these functional derivatives into (5) yields the energy–vorticity representation of quasigeostrophic theory:

With this representation, the temporal evolution of an arbitrary functional ℱ[Π_{QG}] can be written in terms of the following trilinear bracket which is completely antisymmetric:

An important and interesting result is that according to (10) the energy–vorticity representation of 3D quasigeostrophic dynamics is structurally similar to the 2D nondivergent vorticity equation derived by Névir and Blender (1993). In both cases the constitutive quantities are total energy and a positive definite (potential) enstrophy. This similarity can be used to establish classes of fluid mechanical models with similar representations. For example, a classification can be applied with regard to the important cascades of turbulence. It is well known that a 2D incompressible flow is characterized by a direct cascade of enstrophy to small scales and an indirect cascade of energy to large scales. According to the similarity in the energy–vorticity representation, the same behavior of turbulence could be conjectured for large-scale atmospheric motions described by the quasigeostrophic model. At least, there are hints that confirm this hypothesis. McWilliams et al. (1994) discovered the emergence of coherent vortices in a high-resolution numerical simulation based on the quasigeostrophic equations for a Boussinesq fluid in a uniformly rotating and stably stratified environment. The evolution of this turbulent 3D flow toward a final configuration that is structured shows a behavior of self-organization similar to that known from the 2D nondivergent case.

The Nambu field bracket features two algebraic properties, which are generalizations of the antisymmetry and the Jacobi identity for Poisson brackets. The antisymmetry condition for Nambu brackets is

Thus, cyclic permutations of the three functionals leave the sign of the bracket unchanged. The Nambu bracket for the quasigeostrophic model is totally antisymmetric for the same reasons as for the 2D nondivergent vorticity equation (Névir and Blender 1993). Conservation of energy and potential enstrophy is manifest in this representation:

Retaining the twofold antisymmetry through discretization, this feature can be exploited for construction of conservative numerical schemes (Sommer and Névir 2009).

The second property is a generalized Jacobi identity or fundamental identity proposed by Takhtajan (1994) for finite-dimensional Nambu systems. For the energy–vorticity representation of QG dynamics, this identity is given by

Note that in general the twofold antisymmetry (11) and the fundamental identity (14) hold under periodic or other suitable boundary conditions. A physical consequence of the fundamental identity (14) can be given: Provided that the three functionals ℱ_{1}, ℱ_{2}, and ℱ_{3} are conserved quantities with vanishing Nambu brackets, the functional {ℱ_{1}, ℱ_{2}, ℱ_{3}} generated by the trilinear bracket is also a conserved global quantity. This is a technique for exploiting nontrivial conservation laws and can be applied in the framework of group-theoretical methods with applications to fluid mechanics. Under the precondition that (14) holds, Névir (1998) could derive the angular momentum algebra in case of the nondivergent 3D Helmholtz vorticity equation using helicity as a vorticity-conserved quantity.

As far as we know, no proof of the Nambu fundamental identity in the case of fluid dynamical systems has been published. In the case of the continuous Nambu bracket of the barotropic vorticity equation, a proof of this fundamental identity has been given by A. Bihlo (2008, personal communication). It also applies to the bracket of the quasigeostrophic model. To prove (14), the following identity of a Jacobian *J* acting on arbitrary fields *a*, *b*, *c*, *d* is needed:

The fundamental identities for the shallow-water and the nonhydrostatic system have not been checked so far and are left for future work. Whether they are satisfied is an interesting question, since this identity is a precondition for the phase space flow to be nondivergent with respect to the Poisson or Nambu form.

## 3. Energy–vorticity theory of multilayer and hydrostatic fluids

Whereas the wind in the previous section on the quasigeostrophic system was purely solenoidal, we will now consider systems including a divergent component. A Nambu representation for the shallow-water equations has been proposed by Salmon (2005, 2007). His result can be generalized to a multilayer shallow-water and a hydrostatic model.

### a. Multilayer shallow-water system

Consider the dynamics of *n* shallow-water layers^{1} numbered from bottom to top, given by

and

where *f* is the Coriolis parameter, **v*** _{i}* is the horizontal velocity in the

*i*th layer,

*h*its thickness,

_{i}*ρ*its density,

_{i}*p*the pressure at a reference height, and its height above bottom.

_{i}With vorticity *ζ _{i}* =

**k**·

**∇**×

**v**

*, the Weber transform*

_{i}and the Bernoulli function

Eq. (16) can be written as

This vector equation can be brought into scalar form using vorticity and divergence *μ _{i}* :=

**∇**·

**v**

*as prognostic quantities:*

_{i}Total kinetic energy and potential enstrophy are simply summed up over the layers:

with potential vorticity *q _{i}* := (

*ζ*+

_{i}*f*)/

*h*and piecewise constant density

_{i}*ρ*(

*z*) =

*ρ*for

_{i}*z*in layer

*i*.

Potential energy is integrated layer-wise:

Calculating the corresponding functional derivatives of the total energy ℋ = ℋ_{kin} + ℋ_{pot} and potential enstrophy ɛ* _{p}* results (aside from density factors) in the same structure as the single-layer shallow-water equations (Salmon 2005):

Here *χ _{i}* and

*γ*are the streamfunction and velocity potential of the layer momentum, respectively, defined by

_{i}Relying on the abovementioned representation by Salmon (2005), the Nambu representation of models with layer-specific vorticity, divergence, and thickness as prognostic variables can be given as follows:

with the bracket definitions

Here, ℱ is an arbitrary functional depending functionally on vorticity, divergence, and height of the layers.

The twofold antisymmetry of the last two brackets is obtained by addition of cyclic permutations of the three functionals ℱ, ℋ, and ɛ* _{p}*. Layer density is a fixed parameter rather than a prognostic quantity and therefore the first two brackets are still canonical. The first term (26) is structurally identical with the Nambu bracket of the barotropic vorticity equation (Névir and Blender 1993) and of the quasigeostrophic model. Note that this representation does not rely on any rigid-lid approximation.

### b. Hydrostatic model

Salmon (2005, 2007) mentioned that because of their close analogy, the hydrostatic system can be written in Nambu form once the form of the shallow-water system is known. Here, we show how the potential energy expression, its functional derivatives, and the vertical coupling follow from the layer interaction in the multilayer shallow-water system (see previous subsection) and give an explicit formulation.

In contrast to the multilayer models, the vertical structure in the hydrostatic system is no longer measured by the layer index *i* but by a continuously varying coordinate *z*. Furthermore, the assumption of constant density in the layers is abandoned, and, perhaps most important, the twisting term in the corresponding 3D vorticity equation has to be taken into account. Searching for a Nambu representation of this hydrostatic model, it turns out that the vertical velocity in the twisting term becomes a major obstacle. This difficulty can, however, be avoided by changing to isentropic Θ coordinates, where the dynamics is effectively two-dimensional. The multilayer representation can be used straightforwardly to construct a Nambu representation for this system. We start with the prognostic equations for the horizontal velocities and the pseudodensity:

The pseudodensity *ρ̂* and the potential temperature Θ are defined as follows:

The Montgomery potential *M* in (29) can be written in the form

which shows the analogy to the pressure term in the shallow-water equation of motion (16)

The essential difference is that in the multilayer shallow-water case, the densities *ρ _{i}* in (31) are fixed parameters whereas in (30) they are diagnosed [together with

*z*(Θ)] using the hydrostatic relation ∂

_{Θ}

*p*= −

*gρ̂*and the ideal gas equation.

The entire system can only be written in isentropic coordinates when neither the bottom nor the top layer is intersected. Although for the top layer the assumption of a free isentropic surface is not too limiting, the restriction that orography may not intersect the bottom layer is more severe. The practical methods to cope with this difficulty are, however, beyond the scope of this text. For details of the massless-layer approach to this problem, refer to Hsu and Arakawa (1990). Using again the Weber transform and the Bernoulli function

the equation of motion (29) takes the form

The streamfunction and velocity potential on isentropic surfaces can be defined analogously to the shallow-water case:

This is obviously the same structure as that of the shallow-water equations, and indeed the functional derivatives of the energy expressions are commensurate:

Ertel’s potential vorticity can be written on Θ surfaces as

Ertel’s potential enstrophy and its functional derivative are therefore given by

Hence, the functional derivatives of the hydrostatic system on isentropic surfaces are structurally completely analogous to those of the shallow-water equations, and it is not difficult to write the system in Nambu representation using vorticity, divergence, and pseudodensity as prognostic variables:

The bracket operations are given by

This shows that the close similarity between the single-layer shallow-water system and the hydrostatic system in isentropic coordinates described by Salmon (1998) is also reflected in Nambu representation, as noted in Salmon (2005, 2007). The multilayer system perfectly fits into this scheme and the vertical Nambu structure of the three-dimensional hydrostatic system can be naturally derived from it. Compared to the nondivergent or quasigeostrophic vorticity equation, this representation consists of a bracket sum to represent the divergent part. Both models are constituted by specific forms of total energy and (potential) enstrophy.

## 4. Energy–vorticity theory of compressible, adiabatic fluids

In a next step we will extend the Nambu bracket approach to the full set of the ideal hydrothermodynamic equations, describing the motion of a 3D compressible, adiabatic, and nonhydrostatic fluid. In this case, a thermodynamic equation like the conservation of entropy has to be included explicitly in the system of prognostic equations, which poses an additional challenge for the derivation of a Nambu representation. The first noncanonical Hamiltonian representation of this model was found by Morrison and Greene (1980). In a review article, the noncanonical Hamiltonian representations of various fluid dynamical models are summarized by Shepherd (1990). The conventional form of this set of equations in a rotating coordinate system reads as

Here, **v** denotes the 3D velocity, *ρ* the density, *s* the specific entropy, *p* the hydrodynamic pressure, *ϕ*^{(s)} the sum of the potential of gravitational and centrifugal force, and ** ω** the angular velocity of the earth’s rotation. In this set of equations, (37) is the Eulerian equation of motion in a rotating coordinate system, (38) is the continuity equation, and (39) is the Lagrangian conservation of specific entropy. Introducing the vector of field variables

**u**= (

**v**,

*ρ*,

*s*), the noncanonical representation of this set of equations and the related Poisson bracket can be written as

The Hamiltonian ℋ is the total energy of the fluid mechanical system given as a sum of a kinetic, internal, and potential energy:

Here, *e* is the specific internal energy. In this representation, the Hamiltonian functional is the only constitutive conserved quantity that, according to the first theorem of Noether [1918; see Travel (1971) for an English translation], is related to the invariance of the fluid dynamical equations to time translations. The matrix operator 𝗗(**u**) in (40) is an antisymmetric differential operator that takes the role of the Poisson tensor in the case of partial differential equations. The functional derivatives of total energy with respect to velocity, density, and specific entropy are given by

Here, *T* is the temperature and *h* = *e* + *pυ* is the specific enthalpy. Because of the importance of these equations as dynamical core of modern nonhydrostatic weather and climate models, we will give an explicit form of (40) together with the structure of the antisymmetric differential operator 𝗗(**u**) following Shepherd (1990):

Here, **v** = (*υ _{x}*,

*υ*,

_{y}*υ*) are the three velocity components. The components of the absolute vorticity vector can be written as

_{z}where *f* = 2*ω* sin*ϕ* and *l* = 2*ω* cos*ϕ* are two Coriolis parameters, depending on the latitude *ϕ*. The matrix operator 𝗗(**u**) in the noncanonical representation (40) defines a bilinear Poisson bracket:

For two arbitrary functionals ℱ[**u**] and 𝒢[**u**] depending on variables **u** in phase space, this bracket reads as

A characteristic feature of the operator 𝗗(**u**) in (40) and the corresponding Poisson bracket (45) is its degeneracy. Physically, this degeneracy is linked to the specific particle relabeling symmetry (Névir 2004). This is a fundamental fluid dynamical symmetry in the Lagrangian or material representation of fluid mechanics (Salmon 1988) and is responsible for the vorticity-conserved quantities whose accordant symmetry is hidden in the Eulerian representation. The compliance of this symmetry can be expressed with the Lin constraints of conservation of particle labels (Lin 1963) in an Eulerian variational principle of fluid mechanics (Seliger and Whitham 1968), which otherwise would describe only the subclass of irrotational motion. Mathematically, this property is related to distinguished functionals or Casimir functionals that commute with arbitrary functionals ℱ. Thus, these Casimirs also commute with the Hamiltonian and therefore are conserved quantities in noncanonical Hamiltonian mechanics. The degeneracy and the related Casimir functionals are defined by the following equation:

In the case of a compressible, adiabatic fluid, the Casimir functionals are mass-weighted integrals of arbitrary functions Ψ(Π, *s*) of Ertel’s potential vorticity Π and specific entropy *s*:

Special Casimir functionals are mass ℳ, entropy 𝒮, and Ertel’s potential enstrophy ɛ* _{p}*:

A challenge in the application of noncanonical Hamiltonian mechanics for the construction of numerical schemes that also conserve all quantities of the dynamical system in the discrete analog is the degeneracy of the Poisson operator 𝗗. For example, the conservation laws of mass, entropy, and Ertel’s potential enstrophy are hidden in the noncanonical representation. The main goal of the energy–vorticity theory of fluid mechanics is to expose all global constitutive quantities explicitly. A new aspect of this model is that the degeneracy of the Poisson bracket is related not only to the vorticity-conserved quantities but also to the conservation of mass and entropy. Below, we will give a clear mathematical derivation of this approach. At first, we carry out a coordinate transformation in phase space and use the fields of velocity **v**, density *ρ*, and entropy density *σ* = *ρs* as new variables:

Applying the chain rule for functional derivatives, we transform the derivative of an arbitrary functional with respect to the old variables into a derivative according to the new set of variables:

With *σ* = *ρs* we get the following transformation rules of the functional derivatives:

Applying these transformations, the singularity of the Poisson bracket (46) can be eliminated by decomposing it instead into a sum of three nonsingular Nambu brackets. The conservation of mass ℳ, entropy 𝒮, and the constitutive global absolute helicity *h _{a}* given by

enter the representation on the same fundamental level as the Hamiltonian. Indeed, the functional derivatives of the Hamiltonian (41) and these three global quantities with respect to velocity, density, and entropy density,

can be used to establish three trilinear brackets. According to the meaning of the characteristic functionals, these brackets will be named the energy–helicity bracket, the energy–mass bracket, and the energy–entropy bracket and are given by

To satisfy the requirements of a Nambu representation, the brackets have to be completely antisymmetric. The energy–helicity bracket is obviously a Nambu bracket because of the related property of the mixed product. The twofold antisymmetry of the energy–mass bracket and energy–entropy bracket can also be obtained by addition of cyclic permutations of the three arguments. Using these three brackets, the temporal evolution of an arbitrary functional ℱ[**v**, *ρ*, *σ*] can be obtained in the following way:

Replacing the functional ℱ[**v**, *ρ*, *σ*] by velocity, density, and entropy density yields the Eulerian equation of motion, the continuity equation, and the first law of thermodynamics in Nambu bracket representation:

The Eulerian equation is decomposed in a sum of all three brackets, the continuity equation is established only by the energy–mass bracket, and the first law of thermodynamics only by the energy–entropy bracket. Of course, the Nambu representation of the equations for a baroclinic compressible fluid, including the first law of thermodynamics, are generalizations of the original idea of Nambu. But in his pioneering work, Nambu (1973) himself points out two possibilities of more general forms, which may have many triplets and many Hamiltonian pairs.

The representation (56) is a first attempt to apply the ideas of Nambu to the basic equations of atmospheric fluid mechanics. A benefit of this new representation is the disclosure of the internal dependence of the primitive equations among each other: For a barotropic flow, the first law of thermodynamics and therefore the energy–entropy bracket is omitted. Considering Eq. (56), the energy–entropy bracket also has to be removed in the Eulerian equation of motion, giving the energy–vorticity representation of a compressible barotropic fluid:

In this case the absolute helicity is a globally conserved quantity, and the evolution of the fluid is determined solely by the three conservation laws of energy, helicity, and mass. A further approximation is an incompressible 3D fluid, which requires the cancellation of the continuity equation and the corresponding energy–mass bracket. Therefore, in the Eulerian equation only the energy–helicity bracket remains. Note that these two conservation laws constitute the energy–vorticity representation of the incompressible Helmholtz equation. Hence, a hierarchy of Nambu field representations of ideal fluid mechanics is established, which relates incompressible to barotropic and also compressible, adiabatic flows. Of course, the Poisson bracket (46) can also be seen as the sum of a compressible, rotational, and baroclinic term. However, the explicit appearance of the energy–mass bracket and the energy–entropy bracket in the system of equations emphasizes the specific importance of the two conserved quantities—mass and entropy—for each term.

## 5. Ertel’s enstrophy as a super-Casimir functional

In addition to energy, we have used helicity, mass, and entropy as constitutive global quantities to establish a trilinear bracket representation of a 3D compressible adiabatic fluid. In this context, the role played by Ertel’s potential vorticity is a special one. The conservation of Ertel’s potential vorticity or the related potential enstrophy is interpreted as a geometrical conservation law in the phase space of the primitive equations, generated by the particle relabeling symmetry of fluid mechanics (Salmon 1988; Névir 2004). Moreover, this quantity is related to the combined conservation of circulation on isentropic surfaces of mass and of entropy. Thus, conservation of Ertel’s potential vorticity in the Lagrangian sense or the related global conservation of Ertel’s potential enstrophy ɛ* _{p}* is a fundamental higher constraint of the complete set of equations. We can derive the following constraint using an arbitrary functional ℱ[

**v**,

*ρ*,

*σ*]:

Substitution of the arbitrary functional ℱ introduced in (56) by the Hamiltonian ℋ gives the global conservation of Ertel’s potential enstrophy ɛ* _{p}*. The condition (62) and the resulting conservation law can be interpreted as a constraint in phase space, pointing to the mutual interconnection of the three underlying prognostic equations. The temporal evolution of the rotational part of the 3D velocity and the density of mass and entropy are not independent, meaning that the evolution of the whole hydrothermodynamic system has to take place in such a way that Ertel’s potential vorticity is conserved in a Lagrangian sense. Finally, the particle relabeling symmetry, and with it the conservation of Ertel’s PV, indicates that particles constitute the fluid mechanical field, in contrast to fields of electrodynamics. This constraint is clearly revealed in the frame of the energy–vorticity representation and could be used as diagnostic tool to evaluate several conservation laws simultaneously. A practical application of this conservation law could be the estimation of the skill of a numerical simulation concerning the interconnection of all primitive equations.

To present our major results, Table 1 summarizes the different models written in an energy–vorticity representation, together with the prognostic fields, the specification of the total energy, the different vorticity- and mass-related quantities, and the thermodynamic characteristic. Depending on the dimension in Euclidian space and the thermodynamic properties of the model, a total energy can always be assembled in terms of a kinetic, potential, and internal part. A mass-related conservation can be described by the conservation of area *A*, volume *V*, and mass ℳ. In general, the vorticity conservation can be classified by an enstrophy *ε* or a helicity *h*. The declaration of a mass- and vorticity-related quantity in the framework of the energy–vorticity theory is noteworthy. In classical as well as relativistic mechanics, Casimir functions of the Galilei and Poincaré groups are given by the mass and spin of a particle. Moreover, these Casimir functions are not only constant in time but are invariant under all space and time transformations given by the two groups. Thus, elementary particles are irreducible representations of these groups. Therefore, it is tempting to use the different kinds of mass and vorticity conservation laws also for an irreducible classification of fluid mechanics. Table 2 gives an overview of this classification for the discussed models. According to the different topological nature of the vorticity conservation, an enstrophy classification is apparently related to layered models. The function *ψ* in Table 2 determines the layered structure *ψ* = const of fluid mechanical systems that are characterized by an enstrophy conservation. Here, *ψ* is identical with the arbitrary function introduced by Ertel (1965) in his vorticity theorems. In contrast to this, models with a helicity classification are lacking this layered structure.

## 6. Conclusions and outlook

In this study the energy–vorticity theory of fluid mechanics is extended to a broader class of models, including the quasigeostrophic equations, multilayer shallow-water models, and the hydrostatic, adiabatic system on isentropic surfaces. Furthermore, a Nambu representation of the primitive equations describing the motion of a 3D nonhydrostatic, compressible, adiabatic fluid is proposed. In particular, the last model serves as a dynamical core of present regional and future global weather and climate models. The Nambu approach is in principle not restricted to the use of energy, although by using energy in the brackets the temporal evolution of the system in terms of the prognostic equations is obtained. Replacing total energy in the Nambu brackets by linear or angular momentum yields the evolution of the system under local translations or rotations in 3D Euclidian space. Generally, this is a consequence of a very fundamental theorem found by Noether (1918), which states that symmetries in nature are related to conservation laws.

The local representation of nonlinear terms by globally conserved quantities as described above also has a very practical application. Following a proposal of Salmon (2005, 2007), it is possible to construct discrete schemes for a given dynamical system that reproduce the corresponding conservation properties of energy and vorticity quantities. This method generalizes the ideas of Arakawa and is currently being investigated in several research projects. In a recent study, Sommer and Névir (2009) developed a conservative scheme for the shallow-water system on a staggered geodesic grid based on a Nambu representation. They demonstrate that in comparison to traditional discretizations, such a scheme can improve stability and the skill to represent conservation and spectral properties of the underlying partial differential equations. The algebraically exact conservations of energy and potential enstrophy of the discretized ordinary differential equations have the skill to simulate a reasonable flux of energy and enstrophy throughout the scales. These results demonstrate that this method has an influence not only on the very small but also on the global scale, which might be of importance for structure formation in long-term integrations and climate simulations.

## Acknowledgments

The authors are grateful to three anonymous reviewers for their helpful comments, which improved the paper substantially. We are also grateful to Anne Kubin for proofreading the manuscript.

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## Footnotes

*Corresponding author address:* Peter Névir, Freie Universität Berlin, Institut für Meteorologie, Carl-Heinrich-Becker-Weg 6-10, 12165 Berlin, Germany. Email: peter.nevir@met.fu-berlin.de

^{1}

For simplicity without bottom topography.