Action Principles for Extended MHD Models
I.Keramidas Charidakos, M.Lingam, P.J.Morrison, R.L.White, A. Wurm
aa r X i v : . [ phy s i c s . p l a s m - ph ] J u l Action Principles for Extended MHD Models
I.Keramidas Charidakos, M.Lingam, P.J.Morrison, R.L.White and A. Wurm Institute for Fusion Studies and Department of Physics, The University of Texas at Austin, Austin, TX 78712,USA Department of Physical and Biological Sciences, Western New England University, Springfield, MA 01119,USA
The general, non-dissipative, two-fluid model in plasma physics is Hamiltonian, but this property is sometimeslost or obscured in the process of deriving simplified (or reduced) two-fluid or one-fluid models from the two-fluid equations of motion. To ensure that the reduced models are Hamiltonian, we start with the generaltwo-fluid action functional, and make all the approximations, changes of variables, and expansions directlywithin the action context. The resulting equations are then mapped to the Eulerian fluid variables using anovel nonlocal Lagrange-Euler map. Using this method, we recover L¨ust’s general two-fluid model, extendedMHD, Hall MHD, and electron MHD from a unified framework. The variational formulation allows us to useNoether’s theorem to derive conserved quantities for each symmetry of the action.
I. INTRODUCTION
Fluid models are ubiquitous in the study of plasmas. Itis desirable that the non-dissipative limits of fluid mod-els be Hamiltonian, but this property is often lost in theprocess of deriving them (see e.g. Ref. 1 and 2). Oneway to ensure the Hamiltonian property of such mod-els is to derive them from action principles, i.e, to startfrom a Hamiltonian parent-model action and to make allthe approximations and manipulations directly in the ac-tion (see e.g., Refs. 3–6). The equations of motion arethen obtained as the stationary points of the action undervariation with respect to the dynamical variables. Thisis known as
Hamilton’s principle in particle mechanics.Deriving the equations of motion of fluids andplasmas from an action principle has a rich his-tory. The reasons that such a formulation is pur-sued, even after the equations of motion are alreadyknown, are numerous. Finding conservation laws usingNoether’s theorem, , obtaining variational principlesfor equilibria , performing stability analyses ,or imposing constraints on a theory is straight-forward inthe action functional, but often not easily done directlyin the equations of motion.Fluids can be described within the Eulerian or the La-grangian viewpoints. The former describes the fluid interms of, e.g., the evolution of the fluid velocity field ata position x , whereas the latter tracks the motion of in-dividual fluid elements. The map connecting these twoviewpoints is known as the Lagrange-Euler map. Ac-tion functionals are naturally expressed in terms of La-grangian variables, whereas the equations of motion offluid models are Eulerian.Here we are interested in fluid models describing twocharged fluids, e.g., an ion and an electron fluid, inter-acting with an electromagnetic field. The general, non-dissipative two-fluid system is Hamiltonian, and thereforeit is desirable that any model attempting to give a re-duced description of it, should also be Hamiltonian and,consequently, not possess spurious forms of dissipation.We will use the Lagrangian viewpoint and construct ageneral two-fluid action functional. Any subsequent or- dering, approximation, and change of variables will bedone directly in the action before deriving the equationsof motion using Hamilton’s principle. To ensure that thefinal equations of motion are Eulerian, we only constructactions that can be completely expressed in terms of Eu-lerian variables. This general requirement was elucidatedin Refs. 5 and 6, where it was termed the Eulerian Clo-sure Principle .This paper is organized as follows: In Sec. II we reviewthe Lagrangian and Eulerian picture of fluid mechan-ics, including the derivation of the two-fluid equationsof motion through Lagrangian variations of a two-fluidaction functional. Starting from this action, we derivea new one-fluid action functional using careful approx-imations, e.g., imposing quasineutrality, and a changeof variables in Sec. III. Here we also introduce a newLagrange-Euler map and impose locality in order to de-rive Eulerian equations of motion in the new variables.In Sec. IV we show in detail how to derive various fluidmodels, e.g., extended MHD and Hall MHD, from thisnew one-fluid action functional. Sec. V contains a dis-cussion of Noether’s theorem applied to the new actionfunctional, and finally, Sec. VI our conclusions and somediscussion of future work.
II. REVIEW: TWO-FLUID MODEL AND ACTION
In this section, we will briefly review the derivation ofthe non-dissipative two-fluid model equations of motionfrom the general two-fluid action functional. This actionwill be the starting point for deriving reduced modelsfurther below. In this context we establish our notationand, later on, discuss differences with our new procedureand results.Non-dissipative fluids can be described in two equiva-lent ways: The Eulerian (or spatial) point of view, whichuses the physical observables of, e.g., fluid velocity v ( x, t )and mass density ρ ( x, t ) as its fundamental variables anddescribes the fluid at an observation point x in the three-dimensional domain as time passes, or the Lagrangian (ormaterial) point of view, which considers individual fluidelements with position q and tracks their time evolution.As described below, both pictures are related throughthe standard Lagrange-Euler map.From an action functional/variational point of view,the Lagrangian picture is the more natural one, asit represents the infinite-dimensional generalization ofthe finite-dimensional Lagrangians of particle mechan-ics. The equations of motion are then obtained usingHamilton’s principle as the stationary points of the ac-tion, i.e., the first variation of the action with respect tothe variables is equal to zero.We will use the Lagrangian picture as our startingpoint and construct a general two-fluid action functional.To ensure physical relevance of the theory, we only con-struct actions that obey the Eulerian Closure Principle,which states that any action functional of a physical fluidtheory must be completely expressible in Eulerian vari-ables after the application of the Lagrange-Euler map.To simplify our notation (consistent with Ref. 3), wewill avoid explicit vector notation and define the fol-lowing: q s = q s ( a, t ) is the position of a fluid element( s = ( i, e ) is the species label) in a rectangular coordi-nate system where a = ( a , a , a ) is any label identifyingthe fluid element and q s = ( q s , q s , q s ). Here we choose a to be the initial position of the fluid particle at t = 0,although other choices are possible . The Lagrangianvelocity will then be denoted by ˙ q s .The Eulerian velocity field will be denoted by v s ( x, t )with v s = ( v s , v s , v s ) where x = ( x , x , x ) is theposition in the Eulerian picture. Similarly, we definethe electric field vector E ( x, t ), the magnetic field vec-tor B ( x, t ), and the vector potential A ( x, t ). If we needto explicitly refer to components of these vectors, we willuse subscripts (or superscripts) j and k . To simplify theequations, we will also often suppress the dependence on x , a , and t .The action functional described below will include inte-grations over position space R d x and label space R d a .We will not explicitly specify the domains of integration,but assume that our functions are well-defined on theirrespective domains, and that integrating them and tak-ing functional derivatives is allowed. In addition, we willassume that all variations on the boundaries of the do-mains and any surface terms (due to integration by parts)vanish. A. Constructing the two-fluid action
The action functional of a general theory of a chargedfluid interacting with an electromagnetic field should in-clude the following components: The energy of the elec-tromagnetic field, the fluid-field interaction energy, thekinetic energy of the fluid, and the internal energy of thefluid, which describes the fluid’s thermodynamic proper-ties.We will assume two independent fluids correspondingto two different species (ions and electrons with charge e s , mass m s and initial number density of n s ( a )) whichinteract with the electromagnetic field, but not directlywith each other. Therefore the fluid-dependent parts ofthe action will naturally split into two parts, one for eachspecies.The complete action functional is given by S [ q s , A , φ ] = Z T dt L , (1)where T is a finite time interval and the Lagrangian L isgiven by L = 18 π Z d x (cid:20) (cid:12)(cid:12)(cid:12)(cid:12) − c ∂A ( x, t ) ∂t − ∇ φ ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) − |∇ × A ( x, t ) | (cid:21) (2)+ X s Z d a n s ( a ) Z d x δ ( x − q s ( a, t )) × h e s c ˙ q s · A ( x, t ) − e s φ ( x, t ) i (3)+ X s Z d a n s ( a ) h m s | ˙ q s | − m s U s ( m s n s ( a ) / J s , s s ) i . (4)The symbol J s is the Jacobian of the map betweenLagrangian positions and labels, q ( a, t ), which we willdiscuss in more detail below. Here we have expressed theelectric and magnetic fields in terms of the vector andscalar potential, E = − /c ( ∂A/∂t ) − ∇ φ and B = ∇ × A . The first term (2) is the electromagnetic field energy,while the next expression (3) is the coupling of the fluid tothe electromagnetic field, which is achieved here by usingthe delta function. The last line of the Lagrangian L (4)represents the kinetic and internal energies of the fluid.Note, the specific internal energy (energy per unit mass)of species s , U s , depends on the Eulerian density as wellas a function s s , an entropy label for each species. Alsonote, that the vector and scalar potentials are Eulerianvariables (i.e., functions of x ). B. Lagrange-Euler map
In accordance with the above-mentioned Eulerian Clo-sure Principle, we need to ensure that the action Eqs. (1)-(4) can be completely expressed in terms of the desiredset of Eulerian variables, which in turn ensures that theresulting equations of motion will also be completely Eu-lerian, hence representing a physically meaningful model.The connection between the Lagrangian and Eulerianpictures of fluids is the
Lagrange-Euler map . Before look-ing at the mathematical implementation of this map, itis instructive to discuss its meaning. As an example,consider the Eulerian velocity field v ( x, t ) at a particularposition x at time t . The velocity of the fluid at thatpoint will be the velocity of the particular fluid element˙ q ( a, t ) which has started out at position a at time t = 0and then arrived at point x = q ( a, t ) at time t .To implement this idea, we define the Eulerian num-ber density n s ( x, t ) in terms of Lagrangian quantities asfollows: n s ( x, t ) = Z d a n s ( a ) δ ( x − q s ( a, t )) . (5)Using properties of the delta function, this relation canbe rewritten as n s ( x, t ) = n s ( a ) J s (cid:12)(cid:12)(cid:12)(cid:12) a = q − s ( x,t ) , (6)where, J s = det ( ∂q s /∂a ) is the Jacobian determinant.Note that Eq. (6) implies the continuity equation ∂n s ∂t + ∇ · ( n s v s ) = 0 , (7)which corresponds to local mass conservation if we definethe mass density as ρ s = m s n s .The corresponding relation for the Eulerian velocity is v s ( x, t ) = ˙ q s ( a, t ) | a = q − s ( x,t ) , (8)where the dot means differentiation with respect to timeat fixed particle label a . This relation follows from in-tegrating out the delta function in the definition of theEulerian momentum density, M s := m s n s v s , M s ( x, t ) = Z d a n s ( a, t ) δ ( x − q s ( a, t )) m s ˙ q s ( a, t ) . (9)Finally, our Eulerian entropy per unit mass, s s ( x, t ), isdefined by ρ s s s ( x, t ) = Z d a n s ( a ) s s ( a ) m s δ ( x − q s ( a, t )) , (10)completing our set of fluid Eulerian variables for this the-ory, which is { n s , s s , M s } . It is easy to check that theclosure principle is satisfied by these variables.For later use, we quote (without proof) some resultsinvolving the determinant and its derivative ∂q k ∂a j A ik J = δ ij , (11)where A ik is the cofactor of ∂q k /∂a i := q k,i . A convenientexpression for A ik is A ik = 12 ǫ kjl ǫ imn ∂q j ∂a m ∂q l ∂a n , (12)where ǫ ijk (= ǫ ijk ) is the Levi-Civita tensor. UsingEq. (11) one can show that ∂ J ∂q i,k = A ji (13)and using the chain rule ∂∂q k = 1 J A ik ∂∂a i . (14)For further discussions see, e.g., Refs. 3, 5, and 28. C. Varying the two-fluid action
The action of Eq. (1) depends on four dynamical vari-ables: the scalar and vector potentials, φ and A , and thepositions of the fluid elements q s .Varying with respect to φ yields Gauss’s law ∂ k (cid:18) − c ∂A k ∂t − ∂ k φ (cid:19) = 4 πe Z d a n i ( a ) δ ( x − q i ) − πe Z d a n e ( a ) δ ( x − q e ) , where ∂ k := ∂/∂x k , or in more familiar form ∇ · E = 4 πe ( n i ( x, t ) − n e ( x, t )) . (15)Similarly, the variation with respect to A recovers theMaxwell-Ampere law14 π (cid:20) −∇ × ∇ × A + 1 c ∂∂t (cid:18) − c ∂A∂t − ∇ φ (cid:19)(cid:21) − ec Z d a n i [ δ ( x − q e ) n e ˙ q e + δ ( x − q i ) ˙ q i ] = 0or in more familiar form ∇ × B = 4 πJc + 1 c ∂E∂t (16)where the Eulerian current density J is defined as J ( x, t ) = e ( n i v i − n e v e ) . (17)Recall that the other two Maxwell equations are con-tained in the definition of the potentials.Variation with respect to the q s ’s is slightly more com-plex, and we will show a few intermediate steps. Varyingthe kinetic energy term is straight forward and yields − n s ( a ) m s ¨ q s ( a, t ) (18)The j -th component of the interaction term results in e s n s ( a ) (cid:20) − c dA j ( q s , t ) dt + 1 c ˙ q ks ∂A k ( q s , t ) ∂q js − ∂φ ( q s , t ) ∂q js (cid:21) = e s n s ( a ) (cid:20) − c ∂A j ( q s , t ) ∂t − c ˙ q ks ∂A j ( q s , t ) ∂q ks + 1 c ˙ q ks ∂A k ( q s , t ) ∂q js − ∂φ ( q s , t ) ∂q js (cid:21) (19)= e s n s ( a ) (cid:20) E ( q s , t ) + 1 c ˙ q s ( a, t ) × ( ∇ q s × A ( q s , t )) (cid:21) j Note that this expression is purely Lagrangian. The fields A and E are evaluated at the positions q s of the fluidelements and the curl ∇ q s × is taken with respect to theLagrangian position. Also note, since q s = q s ( a, t ), anytotal time derivative of, e.g., A ( q s , t ) will result in twoterms.Variation of the internal energy term yields A ji ∂∂a j ρ s J s ∂U (cid:16) ρ s J s , s s (cid:17) ∂ (cid:16) ρ s J s (cid:17) . (20)Setting the sum of Eqs. (18)-(20) equal to zero andinvoking the usual thermodynamic relations between in-ternal energy and pressure and temperature, p s = ( m s n s ) ∂U s ∂ ( m s n s ) and T s = ∂U s ∂s s (21)results in the well-known (non-dissipative) two-fluidequations of motion m s n s (cid:18) ∂v s ∂t + v s · ∇ v s (cid:19) = e s n s (cid:18) E + 1 c v s × B (cid:19) − ∇ p s (22)Further analysis (see e.g. Refs. 29 and 30) of theseequations usually involves the addition and subtractionof the two-fluid equations and a change of variable trans-formation to V = 1 ρ m ( m i n i v i + m e n e v e ) J = e ( n i v i − n e v e ) ρ m = m i n i + m e n e (23) ρ q = e ( n i − n e ) . The resulting equations are then simplified by, e.g.,making certain assumptions (quasineutrality, v << c ,etc.) and ordering to obtain two new one -fluid equations– one often referred to as the one-fluid momentum equa-tion and the other as the generalized Ohm’s law . III. THE NEW ONE-FLUID ACTION
The first step in building an action functional for fluidmodels is to decide on the relevant Eulerian observablesof the model. Since we want to derive, e.g., the two-fluid model of L¨ust and various reductions, our Eulerianobservables are going to be the set { n, s, s e , V, J, E, B } ,where s = ( m i s i + s e m e ) /m , with m = m e + m i , and n is a single number density variable.Next we have to define our Lagrangian variables andwith them construct the action. Any additional assump-tion (e.g., quasineutrality etc.) and ordering will be im-plemented on the action level. Varying the new actionwill then result in equations of motion that, using prop-erly defined Lagrange-Euler maps, will Eulerianize to,e.g., L¨ust’s equation of motion and the generalized Ohm’slaw. A. New Lagrangian variables
We will start by defining a new set of Lagrangian vari-ables, inspired by Eq. (23) , Q ( a, t ) = 1 ρ m ( a ) ( m i n i ( a ) q i ( a, t ) + m e n e ( a ) q e ( a, t )) D ( a, t ) = e ( n i ( a ) q i ( a, t ) − n e ( a ) q e ( a, t )) ρ m ( a ) = m i n i ( a ) + m e n e ( a ) (24) ρ q ( a ) = e ( n i ( a ) − n e ( a )) . Here Q ( a, t ) can be interpreted as a center of mass posi-tion variable and D ( a, t ) as a local dipole moment vari-able, connecting an ion fluid element to an electron fluidelement. It is then straight-forward to take the time-derivative of Q and D which can be interpreted as thecenter-of-mass velocity ˙ Q ( a, t ) and a Lagrangian cur-rent ˙ D ( a, t ), respectively. Using appropriately definedLagrange-Euler maps, ˙ Q ( a, t ) will map to the Eulerianvelocity V ( x, t ) and ˙ D ( a, t ) to the Eulerian current J ( x, t )as defined by Eq. (23).We will also need the inverse of this transformation, q i ( a, t ) = ρ m ( a ) Q ( a, t ) + m e e D ( a, t ) ρ m ( a ) + m e e ρ q q e ( a, t ) = ρ m ( a ) Q ( a, t ) − m i e D ( a, t ) ρ m ( a ) − m i e ρ q n i ( a ) = ρ m ( a ) + m e e ρ q ( a ) m (25) n e ( a ) = ρ m ( a ) − m e e ρ q ( a ) m . B. Ordering of fields and quasineutrality
Typically, reductions of the full two-fluid model are ob-tained by imposing an auxiliary ordering on the equationsof motion. In order to preserve the variational formula-tion, we perform an ordering directly in the action.To construct the action, we will start with the two-fluidaction of Eq. (1) and change variables to Q and D , but inlight of the fluid models we are interested in, we will firstmake two simplifying assumptions: We order the fieldsin the action so that the displacement current in Eq. (16)will vanish, and we assume quasineutrality. In this sec-tion, we describe this field ordering in detail and discussquasineutrality in the Lagrangian variable context, whichas far as we know has not been done before.The omission of the displacement current is allowed,when the time scale of changes in the field configurationis long relative to the time it takes for radiation to “com-municate” these changes across the system . We usenon-dimensional variables by introducing a characteris-tic scale B for the magnetic field and a characteristiclength scale ℓ for gradients. Times are then normalizedby the Alfv´en time t A = B / √ πρ and the ˙ q s ’s by theAlfv´en speed v A = ℓ/t A , resulting in the following formfor the sum of the field and interaction terms of the La-grangian (4): B π Z dt Z d ˆ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − v A c ∂ ˆ A∂ ˆ t − φ B ℓ ˆ ∇ ˆ φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ˆ ∇ × ˆ A (cid:12)(cid:12)(cid:12) + X s B (cid:20)Z dt Z d ˆ a n ˆ n s ( a ) e s Z d ˆ x δ (ˆ x − ˆ q s ) × (cid:18) v A ℓB c ˆ˙ q s · ˆ A − φ o B ˆ φ (cid:19)(cid:21) , where φ and n are yet to be specified scales for theelectrostatic potential and the densities of both species,respectively. We also require that the two species’ veloc-ities are of the same scale. Requiring the two interactionterms in the Lagrangian to be of the same order resultsin a scaling for φ ; viz., φ ≡ B ℓv A /c . Thus, both partsof the | E | term are of order O ( v A /c ). Neglecting thisterm and varying with respect to ˆ A results inˆ ∇ × ˆ B = 4 πen v A c ℓB (cid:18)Z d a δ (ˆ x − ˆ q i )ˆ n i ( a ) ˆ˙ q i − Z d aδ (ˆ x − ˆ q e )ˆ n e ( a ) ˆ˙ q e (cid:19) , which can be written as B ℓ ˆ ∇ × ˆ B = 4 πj c ˆ J , (26)where j = en v A is a scale for the current.Varying the scaled action with respect to ˆ φ yields0 = Z d ˆ aδ (ˆ x − ˆ q i )ˆ n i − Z d ˆ aδ (ˆ x − ˆ q e )ˆ n e ≡ ∆ˆ n (27)The above equation states that the difference in the twodensities is zero, i.e., the plasma is quasineutral, a prop-erty that holds locally, i.e., n i ( x, t ) = n e ( x, t ). UsingEq. (6), this statement would correspond to the follow-ing in the Lagrangian variable picture: n i ( a ) J i ( a, t ) (cid:12)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) = n e ( a ) J e ( a, t ) (cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) . (28) In the Lagrangian picture we will make the additionalassumption of homogeneity: n i ( a ) = n e ( a ) = constant,which is natural for the plasma we are modeling. It statesthat at t = 0 all fluid elements are identical in the amountof density they carry. Therefore, n i and n e can bereplaced by a constant n . Equation (28) then reducesto a statement about the two Jacobians J i ( a, t ) (cid:12)(cid:12)(cid:12) a = q − i ( x,t ) = J e ( a, t ) (cid:12)(cid:12)(cid:12) a = q − e ( x,t ) , (29)which will play a central role in our development below.Note, that the homogeneity assumption ( n i = n e = n ) does not prohibit us from describing quasineutralplasmas with density gradients. What we would haveto do in this case, would be to pick our labeling scheme,and hence the Jacobian, accordingly, as to reflect theinitial density gradient of the configuration. Thus thereis freedom in this regard beyond what we are assumingnow. C. Action functional
We are now ready to implement the change of vari-ables discussed in Sec. III A. Because of the homogeneityassumption n i ( a ) = n e ( a ) = n , the new variables ofEq. (24) reduce to Q ( a, t ) = m i m q i ( a, t ) + m e m q e ( a, t ) D ( a, t ) = en ( q i ( a, t ) − q e ( a, t )) ρ m ( a ) = mn (30) ρ q ( a ) = 0and the inverse transformation of Eq. (25) to q i ( Q, D ) := q i ( a, t ) = Q ( a, t ) + m e men D ( a, t ) q e ( Q, D ) := q e ( a, t ) = Q ( a, t ) − m i men D ( a, t ) , (31)where we choose the notation q s ( Q, D ) to emphasize thatthe q s should not be thought of as ion/electron trajec-tories any more but as specific linear combinations of Q ( a, t ) and D ( a, t ). In addition, we will need the ionand electron Jacobians, J i ( Q, D ) and J e ( Q, D ), now ex-pressed in terms of Q and D .The resulting action functional has the form: S = − π Z dt Z d x |∇ × A ( x, t ) | + Z dt Z d x Z d a n (cid:26) δ ( x − q i ( Q, D )) (cid:20) ec (cid:18) ˙ Q ( a, t ) + m e men ˙ D ( a, t ) (cid:19) · A ( x, t ) − eφ ( x, t ) (cid:21)(cid:27) + Z dt Z d x Z d a n (cid:26) δ ( x − q e ( Q, D )) (cid:20) − ec (cid:18) ˙ Q ( a, t ) − m i men ˙ D ( a, t ) (cid:19) · A ( x, t ) + eφ ( x, t ) (cid:21)(cid:27) + 12 Z dt Z d a n (cid:20) m i | ˙ Q | ( a, t ) + m i m e me n | ˙ D | ( a, t ) (cid:21) − Z dt Z d a n (cid:20) m i U i (cid:18) m i n J i ( Q, D ) , ( ms − m e s e ) /m i (cid:19) + m e U e (cid:18) m e n J e ( Q, D ) , s e (cid:19)(cid:21) , (32)where recall s = ( m i s i + m e s e ) /m . D. Nonlocal Lagrange-Euler maps
Now we define the Lagrange-Euler maps that connectthe Eulerian observables V and J to the new Lagrangianvariables Q and D . Referring to Sec. II B, one can seethat a Lagrange-Euler map is a relationship betweena Lagrangian quantity and some Eulerian observables,which holds only when it is evaluated on a trajectory x = q s ( a, t ). If we apply the inverse Lagrange-Euler mapsfrom Eqs. (6) and (8) to Eq. (23) and assume quasineu-trality, we get V ( x, t ) = m i m ˙ q i ( a, t ) (cid:12)(cid:12)(cid:12) a = q − i ( x,t ) + m e m ˙ q e ( a, t ) (cid:12)(cid:12)(cid:12) a = q − e ( x,t ) J ( x, t ) = e (cid:18) n J i ( a, t ) ˙ q i ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) − e (cid:18) n J e ( a, t ) ˙ q e ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) n ( x, t ) = m i m (cid:18) n J i ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) (33)+ m e m (cid:18) n J e ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) s ( x, t ) = m i m s i (cid:12)(cid:12)(cid:12) a = q − i ( x,t ) + m e m s e (cid:12)(cid:12)(cid:12) a = q − e ( x,t ) (34) s e ( x, t ) = s e (cid:12)(cid:12)(cid:12) a = q − e ( x,t ) . (35)The definitions of Q ( a, t ) and D ( a, t ) in Eq. (30) sug-gest that their time-derivatives should be associated with V and J , respectively. However, both ˙ Q and ˙ D are non-local objects, since they relate the velocities of electronsand ions which are located at different points in space.This means that neither ˙ Q , nor ˙ D , when evaluated at theinverse maps for a , can Eulerianize to a local velocity or current, since, in general, x = q i ( Q, D ) and x ′ = q e ( Q, D )with x = x ′ or, they are simultaneously evaluated at dif-ferent trajectories. Therefore, we have two different in-verse functions where the Lagrangian quantities are tobe evaluated, namely, a = q − i ( x, t ) and a = q − e ( x ′ , t )which should be thought of as the inverse functions of x = q i ( Q, D ) and x ′ = q e ( Q, D ). To make this work, we define our Lagrange-Euler maps with x = x ′ as V ( x, t ) = m i m (cid:18) ˙ Q ( a, t ) + m e men ˙ D ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) (36)+ m e m (cid:18) ˙ Q ( a, t ) − m i men ˙ D ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) J ( x, t ) = en J i ( a, t ) (cid:18) ˙ Q ( a, t ) + m e men ˙ D ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) − en J e ( a, t ) (cid:18) ˙ Q ( a, t ) − m i men ˙ D ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) . Due to Eq. (29), the two Jacobian determinants are equal(as long as they are evaluated at the respective inversefunctions) and can be replaced by a common Jacobiandeterminant, J .The maps just defined are straight-forward to applyfor mapping an Eulerian statement to a Lagrangian one,but for our purpose, we have to invert them. To keepcareful track of the two inverse functions, we first invertthe intermediate relations V ( x, t ) + m e men ( x, t ) J ( x, t )= (cid:18) ˙ Q ( a, t ) + m e men ˙ D ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) (37) V ( x, t ) − m i men ( x, t ) J ( x, t )= (cid:18) ˙ Q ( a, t ) − m i men ˙ D ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) , (38)where we have used Eq. (6). The inverse Lagrange-Eulermaps are now given by˙ Q ( a, t ) = m i m (cid:18) V ( x, t ) + m e men ( x, t ) J ( x, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = q i ( Q,D ) + m e m (cid:18) V ( x ′ , t ) − m i men ( x ′ , t ) J ( x ′ , t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x ′ = q e ( Q,D ) ˙ D ( a, t ) = en (cid:18) V ( x, t ) + m e men ( x, t ) J ( x, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = q i ( Q,D ) − en (cid:18) V ( x ′ , t ) − m i men ( x ′ , t ) J ( x ′ , t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x ′ = q e ( Q,D ) . (39)Note that the construction of the maps of Eqs. (36) and(39) can be done with any invertible linear combinationof the time derivatives of our Lagrangian variables. Theonly restriction is that the action should comply withthe Eulerian Closure Principle, i.e., it should be express-ible entirely in terms of the Eulerian observables. It isstraightforward to show that this is true in our case. E. Lagrange-Euler maps without quasineutrality
Had we not assumed quasineutrality, we would haveto proceed differently: Eq. (23) implies that the properLagrangian variables that would Eulerianize to velocityand current would be V ( x, t ) = m i (cid:16) n i J i ˙ q i ( a, t ) (cid:17)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) m i (cid:16) n i J i (cid:17)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) + m e (cid:16) n e J e (cid:17)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) + m e (cid:16) n e J e ˙ q e ( a, t ) (cid:17)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) m i (cid:16) n i J i (cid:17)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) + m e (cid:16) n e J e (cid:17)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) ,J ( x, t ) = e (cid:18) n i J i ˙ q i ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) − e (cid:18) n e J e ˙ q ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) . The above equations suggest that without quasineutral-ity, the definitions for ˙ Q , ˙ D , etc. should be modified tothe following:˙ Q ( a, t ) = 1 ρ m ( a ) (cid:0) m i J e n i ( a ) ˙ q i ( a, t )+ m e J i n e ( a ) ˙ q e ( a, t ) (cid:1) ˙ D ( a, t ) = e (cid:0) J e n i ( a ) ˙ q i ( a, t ) − J i n e ( a ) ˙ q e ( a, t ) (cid:1) ρ m ( a ) = m i J e n i ( a ) + m e J i n e ( a )where ˙ Q/ ( J i J e ) maps to V ( x, t ) and ˙ D/ ( J i J e ) to J ( x, t ). In this case, however, both ˙ Q and ˙ D are im-plicitly defined, since J i and J e depend on them. This problem is absent when only manipulating the Eulerianequations of motion. It might suggest though that whenquasineutrality does not hold, the one-fluid descriptionmight not be appropriate. This can also be seen in themost general case derived by L¨ust in Ref. 29. The result-ing equations of motion in V and J still contain termsexplicitly referring to ion/electron quantities, e.g., n i and n e . From a variational point of view, it is not obvi-ous how to apply the Eulerian Closure Principle withoutquasineutrality. It seems that in order to preserve it, onewould need to distinguish between integrations over ionand electron labels, so that the d a could be related tothe proper J s . F. Derivation of the continuity and entropy equations
Before we derive the equations of motion for severaldifferent models in the next section, we derive here thecontinuity equation, which all of the models below havein common, and the entropy equations.Due to the identity of the Jacobians from Eq. (29), theequation for n (Eq. (33)) reduces to n ( x, t ) = (cid:18) n J i ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − i ( x,t ) = (cid:18) n J e ( a, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) , (40)where q − s are still the inverse functions of q s ( Q, D ). In-verting the equation for the ions and taking the timederivative yields dndt (cid:12)(cid:12)(cid:12)(cid:12) x = q i ( Q,D ) = ddt n J i ( a, t ) = − n J i ( a, t ) ∂ J i ∂t . To Eulerianize the equation above, we use the well-knownrelations d/dt = ∂/∂t + v · ∇ and ∂ J /∂t = J ∇ · v . Thekey here is to use the correct Eulerian velocity, in thiscase the ion velocity in terms of V and J . The result is ∂n∂t + (cid:16) V + m e men J (cid:17) · ∇ n = − n ∇ · (cid:16) V + m e men J (cid:17) which can be further reduced to ∂n∂t + ∇ · ( nV ) + m e me ∇ · J = 0 . However, we already know from Eq. (26) that ∇ · J =0. Therefore, no matter which equality we choose inEq. (40), the continuity equation will be the same, ∂n∂t + ∇ · ( nV ) = 0 (41)Similarly, from Eq. (34) we obtain ∂s∂t + V · ∇ s = 0and from Eq. (35) ∂s e ∂t + (cid:16) V − m i men J (cid:17) · ∇ s e = 0 , or to leading order in m e /m i ∂s e ∂t + (cid:18) V − en J (cid:19) · ∇ s e = 0 . IV. DERIVATION OF REDUCED MODELS
If we vary the action functional (32) with respect to Q and D and subsequently apply the Lagrange-Eulermap we recover the momentum equation and generalizedOhm’s law of L¨ust (in the non-dissipative limit): nm (cid:18) ∂V∂t + ( V · ∇ ) V (cid:19) (42)= −∇ p + J × Bc − m i m e me ( J · ∇ ) (cid:18) Jn (cid:19) E + V × Bc = m i m e me n (cid:18) ∂J∂t + ( J · ∇ ) V − ( J · ∇ ) (cid:18) Jn (cid:19)(cid:19) + m i m e me ( V · ∇ ) (cid:18) Jn (cid:19) + ( m i − m e ) menc ( J × B ) − m i men ∇ p e + m e men ∇ p i . (43)We will not show this lengthy, although straightforward,calculation here, but instead show the detailed derivationof extended MHD in the next section which requires onemore ordering in the action of Eq. (32). A. Extended MHD
At this point we will make one more simplification: Wedefine the mass ratio µ = m e /m i and order the actionfunctional keeping terms up to first order in µ . Up tofirst order, the change of variables is q i ( Q, D ) = Q ( a, t ) + µen D ( a, t ) q e ( Q, D ) = Q ( a, t ) − − µen D ( a, t ) (44) and the action takes on the form S = − π Z dt Z d x |∇ × A ( x, t ) | + Z dt Z d x Z d a n (cid:26) δ ( x − q i ( Q, D )) × (cid:20) ec ˙ Q ( a, t ) + µcn ˙ D ( a, t ) · A ( x, t ) − eφ ( x, t ) (cid:21)(cid:27) + Z dt Z d x Z d a n (cid:26) δ ( x − q e ( Q, D )) × (cid:20) − ec ˙ Q ( a, t ) + (1 − µ ) cn ˙ D ( a, t ) · A ( x, t ) + eφ ( x, t ) (cid:21)(cid:27) + 12 Z dt Z d a n m i (cid:18) (1 + µ ) | ˙ Q | ( a, t ) + µe n | ˙ D | ( a, t ) (cid:19) − Z dt Z d a n (cid:18) U e (cid:18) n J e ( Q, D ) , s e (cid:19) + U i (cid:18) n J i ( Q, D ) , s i (cid:19) (cid:19) . (45)where for convenience we have replaced the U s , the in-ternal energy per unit mass, by U s , the internal energyper particle. The pressure is obtained from the latteraccording to p s = n ∂ U s /∂n .Varying the action with respect to Q k yields0 = − n m i (1 + µ ) ¨ Q k ( a, t ) − ∂ k p + n (cid:20) ec (cid:18) ˙ Q j ( a, t ) + µen ˙ D j ( a, t ) (cid:19) ∂A j ( x, t ) ∂x k − e∂ k φ ( x, t ) − ec ddt A k ( x, t ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = q i ( Q,D ) + n (cid:20) − ec (cid:18) ˙ Q j ( a, t ) − (1 − µ ) en ˙ D j ( a, t ) (cid:19) ∂A j ( x, t ) ∂x k + e∂ k φ ( x, t ) + ec ddt A k ( x, t ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = q e ( Q,D ) . (46)The variation of the internal energy term proceeds byvarying q s through Eqs. (44), giving δq s = δQ and usingthese expressions in the variation of the Jacobians J s .We have given the Eulerian result since the Lagrangianone has two terms of the form of Eq. (20), and it is cum-bersome to carry this through the rest of the calculation.(See Ref. 1 for a treatment that orders the Eulerian equa-tions directly.) Consistent with Dalton’s law, the totalsingle fluid pressure is p = p i + p e and both these pres-sures come in entirely at the zeroth order of µ . Note,that the two time derivatives of A do not cancel, becausethey are advected by different flows, or, since we are stillin the Lagrangian framework, they are evaluated at dif-ferent arguments.To find the Eulerian equations of motion, we start withEq. (39) (up to first order in µ ) and impose locality, i.e., x = x ′ , such that ˙ Q maps to V ( x, t ) and ˙ D to J ( x, t ).However, the time derivatives of ˙ Q and ˙ D have to betreated with care as they each consist of two terms thatare advected with different velocities. We will show howto Eulerianize the equations of motion in detail.The ¨ Q in the first term of Eq. (46) can be re-writtenas ¨ Q ( a, t ) (47)= (1 − µ ) ddt (cid:18) V ( x, t ) + µen ( x ) J ( x, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = q i ( Q,D ) + µ ddt (cid:18) V ( x, t ) − (1 − µ ) en ( x ) J ( x, t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = q e ( Q,D ) = (1 − µ ) (cid:18) ∂V∂t + ∂q i ∂t · ∇ V + µen ∂∂t (cid:18) Jn (cid:19) (48)+ µe ∂q i ∂t · ∇ (cid:18) Jn (cid:19)(cid:19) + µ (cid:18) ∂V∂t + ∂q e ∂t · ∇ V − (1 − µ ) en ∂∂t (cid:18) Jn (cid:19) − (1 − µ ) e ∂q e ∂t · ∇ (cid:18) Jn (cid:19)(cid:19) . From Eqs. (44), we can find explicit expressions for thetime derivatives of the q s ( Q, D ), ∂q i ∂t = ˙ Q + µen ˙ D −→ V + µen J (49) ∂q e ∂t = ˙ Q − − µen ˙ D −→ V − − µen J . (50)Inserting these expression into Eq. (48), we find aftersome algebra that¨ Q ( a, t ) −→ ∂V∂t +( V ·∇ ) V + µ (1 − µ ) ne ( J ·∇ ) (cid:18) Jn (cid:19) . (51)Next we Eulerianize the interaction terms of Eq. (46)using Eq. (39) (up to first order in µ ) and Eq. (44). Theresult is nec (cid:20)(cid:16) V j + µen J j (cid:17) ∂A j ∂x k − c∂ k φ − ∂A k ∂t − ∂q i ∂t · ∇ A k (cid:21) + nec (cid:20)(cid:18) − V j + (1 − µ ) en J j (cid:19) ∂A j ∂x k + c∂ k φ + ∂A k ∂t + ∂q e ∂t · ∇ A k (cid:21) , (52)which, after substitution using Eqs. (49) and (50), yields1 c (cid:18) J j ∂A j ∂x k − J j ∂A k ∂x j (cid:19) = ( J × ( ∇ × A )) k c . (53)The full Eulerian version of the equation of motion forthe velocity of Eq. (46), also referred to as the momentum equation is nm (cid:18) ∂V∂t + ( V · ∇ ) V (cid:19) = −∇ p + J × Bc (54) − m e e ( J · ∇ ) (cid:18) Jn (cid:19) . Note, it was shown in Ref. 1 that the last term of Eq. (54)is necessary for energy conservation.Next, varying the action with respect to D k yields0 = m i µn e ¨ D k ( a, t ) + (1 − µ ) en ∂ k p e − µen ∂ k p i + µ (cid:20)(cid:18) − c ddt A k ( x, t ) − ∂ k φ ( x, t ) + 1 c (cid:16) ˙ Q j ( a, t )+ µen ˙ D j ( a, t ) (cid:19) ∂A j ( x, t ) ∂x k (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x = q i ( Q,D ) + (1 − µ ) (cid:20)(cid:18) − c ddt A k ( x, t ) − ∂ k φ ( x, t ) + 1 c (cid:16) ˙ Q j ( a, t ) − (1 − µ ) en ˙ D j ( a, t ) (cid:19) ∂A j ( x, t ) ∂x k (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x = q e ( Q,D ) . (55)This time the Jacobians of the internal energies are variedusing δq e = − (1 − µ ) δD/ ( en ) and δq i = µδD/ ( en ),which again follow from Eqs. (44). Note, it is for thisreason that only the electron pressure appears to leadingorder in Ohm’s law for extended MHD.Eulerianizing the ¨ D term in Eq. (55) yields m i µn e ¨ D ( a, t ) = m i µe n (cid:18) ∂J∂t + ( J · ∇ ) V − ( J · ∇ ) (cid:18) Jn (cid:19)(cid:19) + m i µe ( V · ∇ ) (cid:18) Jn (cid:19) + m i µe n J ( V · ∇ ) n (56)where we have used the continuity equation Eq. (41) toeliminate the time derivative of n and kept leading orderin µ terms. The interaction terms in Eq. (55) reduce to E + V × ( ∇ × A ) c − (1 − µ ) enc J × ( ∇ × A ) . (57)In Eqs. (56) and (57) we see the presence of some termsinvolving µ , in front of ¨ D and J × B , respectively. How-ever, in the latter case it occurs in the factor (1 − µ )and since our ordering is µ <<
1, we can drop the µ -dependence in Eq. (57), to lowest order. However, inEq. (56), we cannot throw out all the terms that dependon µ since the factor µm i / ( ne ) cannot be cast into adimensionless form, and hence one cannot invoke the or-dering µ << .The Eulerian version of the equation of motion of thecurrent Eq. (55) (after keeping zeroth order in µ ), also0known as generalized Ohm’s law , is then E + V × Bc = m e e n (cid:18) ∂J∂t + ( J · ∇ ) V − ( J · ∇ ) (cid:18) Jn (cid:19)(cid:19) + ( J × B ) enc − ∇ p e en + m e e ( V · ∇ ) (cid:18) Jn (cid:19) + m e e n J ( V · ∇ ) n . (58)The last two terms on the right hand side of Eq. (58)can be combined to give (cid:0) m e / ( e n ) (cid:1) ( V · ∇ ) J and since ∇ · J = 0, we can add a V ( ∇ · J ) term without changingthe result, and combine most terms in the divergence ofthe tensor V J + JV to obtain the following equation: E + V × Bc = m e e n (cid:18) ∂J∂t + ∇ · ( V J + JV ) (cid:19) − m e e n ( J · ∇ ) (cid:18) Jn (cid:19) + ( J × B ) enc − ∇ p e en . (59)Equations (54) and (59) constitute the extended MHDmodel. B. Hall MHD
Hall MHD is a limiting case, for which previous workof an action functional nature exists . Here we ob-tain the actional functional by expanding and retainingonly terms up to zeroth order in µ , i.e., if we neglect theelectron inertia ( m e → S = − π Z dt Z d x |∇ × A ( x, t ) | + Z dt Z d a n (cid:20) m | ˙ Q | ( a, t ) − U i (cid:18) n J i ( Q ) , s i (cid:19) − U e (cid:18) n J e ( Q, D ) , s e (cid:19) (cid:21) + Z dt Z d x Z d a n (cid:26) δ (cid:18) x − Q ( a, t ) − en D ( a, t ) (cid:19) × (cid:20) − ec (cid:18) ˙ Q ( a, t ) − en ˙ D ( a, t ) (cid:19) · A ( x, t ) + eφ ( x, t ) (cid:21)(cid:27) + Z dt Z d x Z d a n (cid:26) δ ( x − Q ( a, t )) × h ec ˙ Q ( a, t ) · A ( x, t ) − eφ ( x, t ) i (cid:27) . (60)and Eqs. (44) become q i ( Q, D ) = Q ( a, t ) q e ( Q, D ) = Q ( a, t ) − D ( a, t ) / ( en ) (61)Observe we have also replaces m i by m in the kineticenergy term, which is correct to leading order in µ . The inverse maps required for Eulerianizing the equa-tions of motion reduce to˙ Q ( a, t ) = V ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) x = q i = Q ˙ D ( a, t ) = en V ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) x = q i = Q (62) − en (cid:18) V ( x ′ , t ) − J ( x ′ , t ) en ( x ′ , t ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x ′ = q e ( Q,D ) . Following the procedure outlined in the previous sectionfor extended MHD, we arrive at what is commonly re-ferred to as Hall MHD, nm (cid:18) ∂V∂t + ( V · ∇ ) V (cid:19) = −∇ p + J × Bc (63) E + V × Bc = J × Bnec − ne ∇ p e , (64)which are the usual forms of the momentum equation andOhm’s law for Hall MHD. C. Electron MHD
Electron MHD is another limiting case wherewe neglect the ion motion completely. This theory isused to describe the short time scale motion of the elec-trons in a neutralizing ion background. Since the ionsare immobile, we require ˙ q i = 0 and q i = q i ( a ). Also, werequire that there be no electric field and, consequently,we neglect φ from the action. In this case, using the Q , D formulation of the previous sections is redundantsince there is only a single fluid. From ˙ q i = 0 we find˙ D = − ( en m/m e ) ˙ Q . (The same relation holds between Q and D up to an additive constant which represents theconstant position of the ion). In addition, the Lagrange-Euler map takes on the simple form v e ( x, t ) = (cid:18) µ (cid:19) ˙ Q ( a, t ) (cid:12)(cid:12)(cid:12)(cid:12) a = q − e ( x,t ) , (65)where q e ( a, t ) = (1 + 1 µ ) Q ( a, t ) . (66)The remaining terms in the action are S = − π Z dt Z d x |∇ × A ( x, t ) | + Z dt Z d a n (cid:20) m e | ˙ q e | ( a, t ) − U e (cid:18) n J e ( Q ) (cid:19)(cid:21) − Z dt Z d x Z d a δ ( x − q e ) en c ˙ q e · A ( x, t ) , (67)which is essentially the same action as that of Ref. 16. Itis also straight-forward to express this action in terms of Q using Eqs. (65) and (66).1Upon varying the action (either in terms of q e or Q )and Eulerianizing the following equation of motion andconstraint are obtained: m e (cid:18) ∂v e ∂t + v e · ∇ v e (cid:19) + ec ∂A∂t = ec ( v e × B ) − ∇ p e n ∇ × B = − πc env e , which are the usual equations of electron MHD. V. NOETHER’S THEOREM
In this section we will investigate the invariants of theaction functionals for the quasineutral L¨ust equations ofEq. (32) and the extended MHD system of Eq. (45) us-ing Noether’s theorem. Note that both actions can beexpressed either in terms of (
Q, D ) or in terms of ( q i , q e ),which are related through a simple linear transformation,e.g., Eq. (30). Furthermore, both sets of variables obeythe Eulerian Closure Principle. Hence, it is equivalent towork with an action expressed in terms of either set ofvariables. For convenience, we shall work with the lat-ter set, as the Euler-Lagrange maps are easier to apply.Noether’s theorem states that if an action is invariantunder the transformations q ′ s = q s + K s ( q s , t ) ; t ′ = t + τ ( t ) , (68)i.e., S = Z t t dt Z d z L ( q s , ˙ q s , z, t )= Z t ′ t ′ dt ′ Z d z ′ L ( q ′ s , ˙ q ′ s , z ′ , t ′ ) , then there exist constants of motion given by C = Z d z (cid:20) τ (cid:18) ∂ L ∂ ˙ q s · ˙ q s − L (cid:19) − K s · ∂ L ∂ ˙ q s (cid:21) , (69)where the index s represents the number of independentvariables q in the system. Our actions are mixedLagrangian and Eulerian, so the variable z can denote a or x .
1. Time translation
It is straight-forward to see that the action is invariantunder time translation with K s = 0; τ = 1 . The corresponding constant of motion, the energy, isfound to be E = Z d x " |∇ × A | π + X s Z d a (cid:18) n m s | ˙ q s | + n U s (cid:18) n J s , s s (cid:19)(cid:19)(cid:21) . Using suitable Lagrange-Euler maps to express our an-swer in terms of the Eulerian variables { n, V, J } , we ob-tain E = Z d x (cid:20) | B | π + n U i + n U e + mn | V | m e m i mne | J | (cid:21) (70)for the quasineutral L¨ust model and E = Z d x (cid:20) | B | π + n U i + n U e + mn | V | m e ne | J | (cid:21) (71)for the extended MHD model. Note that the twoenergies are different since the extended MHD modelincludes the mass ratio ordering.
2. Space translation
Space translations correspond to K s = k ; τ = 0 , where k is an arbitrary constant vector. Under spacetranslations, the constant of motion is the momentum,which is found to be P = k · Z d a ( n m i ˙ q i + n m e ˙ q e )+ k · Z d x ec A (cid:26)Z n [ δ ( x − q i ) − δ ( x − q e )] d a (cid:27) , Using the Lagrange-Euler maps one can show that P = k · Z d x nmV is the conserved quantity. Note that k is entirely arbi-trary, and hence we see that the total momentum P = Z d x ρV (72)is conserved. This is also evident from the correspondingdynamical equation for V .
3. Rotations
The actions are also invariant under rotations whichcorrespond to K s = k × q s ; τ = 0 , L = k · Z d x nm r × V , and since we know that k is arbitrary, we conclude thatthe angular momentum given by L = Z d x ρ r × V (73)is a constant of motion.
4. Galilean boosts
When discussing boosts, we have to consider that theaction may remain invariant even when the followingholds S = Z t t dt Z d z L ( q s , ˙ q s , z, t )= Z t ′ t ′ dt ′ Z d z ′ ( L ( q ′ s , ˙ q ′ s , z ′ , t ′ ) + ∂ µ λ µ ) , because the second term vanishes identically. In all theprevious derivations of the constants of motion, the in-finitesimal transformations did not involve time explic-itly. A boost, though, corresponds to K s = ut ; τ = 0 , where u is an arbitrary constant velocity. For a Galileanboost in a one-fluid model, the corresponding invariantquantity is given by B = Z d a mn ( q − ˙ qt ) , and since we have two different species, this generalizesto B = X s Z d a m s n s ( q s − ˙ q s t ) . Using the corresponding Lagrange-Euler maps, the Eu-lerianized expression is given by B = Z d x ρ ( x − V t ) . (74) VI. CONCLUSIONS
In this paper, we derived several fluid models from ageneral two-fluid action functional. All approximations,ordering schemes, and changes of variables were done inthe action functional before Hamilton’s principle was in-voked. We defined a new set of Lagrangian variables, and under the assumption of quasineutrality, we constructeda new set of nonlocal Lagrange-Euler maps assuring thatour Lagrangian equations of motion can be Eulerianized.Lastly, we derived several conservation laws for thesemodels using Noether’s theorem.The novel nonlocal Lagrange-Euler map of this paper isof particular general importance. Usual Lagrange-Eulermaps (also known as momentum maps) entail the advec-tion of various quantities by a single velocity field andthis can be traced to the algebraic structure of the Pois-son bracket written in terms of Eulerian variables (seee.g. Ref. 3). For single fluid models like MHD the Pois-son bracket has semi-direct product structure, whichoccurs in a variety of fluid contexts (e.g. Refs. 41–43).However, many systems do not possess this semi-directproduct structure (e.g. Refs. 44–47) and indeed a generaltheory of algebraic extensions was given in Ref. 48. Itis the selection of the set of observables and the ECPthat give rise to the general algebras underlying Pois-son brackets. Detailed construction of general algebrasof Ref. 48 will be reported in future publications, alongwith derivations of other single fluid models includinggyroviscosity . ACKNOWLEDGMENTS
IKC, ML, PJM, and RLW received support from theU.S. Dept. of Energy Contract K. Kimura and P. J. Morrison, Phys. Plasmas (to appear),arXiv preprint arXiv:1406.2745 (2014). C. Tronci and E. Tassi and E. Camporeale and P. J. Morrison,Plasma Phys. Cont. Fusion , 095008 (2014). P. J. Morrison, Reviews of Modern Physics , 467 (1998). P. J. Morrison, Physics of Plasmas , 8102 (2005). P. J. Morrison, in
American Institute of Physics Conference Se-ries , Vol. 1188 (2009) pp. 329–344. P. J. Morrison, M. Lingam, and R. Acevedo, Phys. Plasmas (to appear), arXiv preprint arXiv:1405.2326 (2014). A. Taub, in
Symposium in Applied Mathematics of AmericanMathematical Society , Vol. 1 (1949) pp. 148–157. J. Herivel, Math. Proc. Cambridge Philos. Soc , 344 (1955). P. Penfield Jr, Physics of Fluids , 1184 (1966). F. Low, Proceedings of the Royal Society of London. Series A.Mathematical and Physical Sciences , 282 (1958). P. Sturrock, Annals of Physics , 306 (1958). P. Butcher, Philosophical Magazine , 971 (1953). J. Dougherty, Journal of Plasma Physics , 331 (1974). W. Newcomb, Nuclear Fusion , 451 (1962). P. Penfield Jr and H. A. Haus, Physics of Fluids , 1195 (1966). V. Ilgisonis and V. Lakhin, Plasma Physics Reports , 58(1999). N. Padhye and P. J. Morrison, Plasma Physics Reports , 869(1996). N. Padhye and P. J. Morrison, Physics Letters A , 287 (1996). G. Webb and G. Zank, Journal of Physics A: Mathematical andGeneral , 545 (2007). G. Webb, G. Zank, E. K. Kaghashvili, and R. Ratkiewicz, Jour-nal of Plasma Physics , 811 (2005). T. Andreussi, P. J. Morrison, and F. Pegoraro, Plasma Physicsand Controlled Fusion , 5001 (2010). T. Andreussi, P. J. Morrison, and F. Pegoraro, Bulletin of theAmerican Physical Society (2012). K. Els¨asser, Physics of Plasmas , 3161 (1994). Y. Kawazura and E. Hameiri, Physics of Plasmas , 082513(2012). T. Andreussi, P. J. Morrison, and F. Pegoraro, Physics of Plas-mas , 092104 (2013). J. Squire, H. Qin, W. Tang, and C. Chandre, Physics of Plasmas , 2501 (2013). S. M. Moawad, J. Plasma Phys. , 873 (2013). R. Salmon, Annual Review of Fluid Mechanics , 225 (1988). R. L¨ust, Fortschritte der Physik , 503 (1959). J. Goedbloed and S. Poedts,
Principles of Magnetohydrodynam-ics (Cambridge University Press, Cambridge, U.K., 2004). First presented in A. Wurm and P.J. Morrison, Action principlederivation of one-fluid models from two-fluid actions, Bulletin ofthe Am. Phys. Soc, Vol. 54, Nr. 4 (2009). P. Roberts,
An introduction to magnetohydrodynamics (Long-mans, Green and Co ltd, London, 1967). Note, there are typos in Eqs. (2.9) and (2.10) of Ref. 29 that pre-vent the term N from vanishing when imposing quasineutrality. R. G. Littlejohn, Journal of Plasma Physics , 111 (1983). M. Hirota, Z. Yoshida, and E. Hameiri,Physics of Plasmas , 022107 (2006). Z. Yoshida and E. Hameiri, Journal of Physics A: Mathematical and Theoretical , 335502 (2013). M. Lighthill, Phil. Trans. Roy. Soc. , 397 (1960). S. I. Braginskii, in
Reviews of Plasma Physics , Vol. 1, edited byM. Leontovich (Consultants Bureau, New York, 1965) pp. 205–311. L. I. Rudakov, C. E. Seyler, and R. N. Sudan, Comments PlasmaPhys. Cont. Fusion , 171 (1991). P. J. Morrison and J. M. Greene, Phys. Rev. Lett. , 790 (1980). G. Rosensteel, Ann. Phys. , 230 (1988). D. D. Holm and B. A. Kupershmidt, Physica D , 347 (1983). J. E. Marsden and P. J. Morrison, Contemp. Math. , 133(1984). R. D. Hazeltine, C. T. Hsu, and P. J. Morrison, Phys. Fluids , 3204 (1987). D. Grasso, F. Califano, F. Pegoraro, and F. Porcelli, Phys. Rev.Lett. , 5051 (1994). E. Tassi, P. J. Morrison, F. L. Waelbroeck, and D. Grasso,Plasma Phys. and Control. Fusion , 085014 (2008). P. J. Morrison, E. Tassi, and N. Tronko, Phys. Plasmas ,042109 (2013). J.-L. Thiffeault and P. J. Morrison, Physica D , 397 (205–244).49