Noether Currents for Eulerian Variational Principles in Non Barotropic Magnetohydrodynamics and Topological Conservations Laws

We derive a Noether current for the Eulerian variational principle of ideal non-barotropic magnetohydrodynamics (MHD). It was shown previously that ideal non-barotropic MHD is mathematically equivalent to a five function field theory with an induced geometrical structure in the case that field lines cover surfaces and this theory can be described using a variational principle. Here we use various symmetries of the flow to derive topological constants of motion through the derived Noether current and discuss their implication for non-barotropic MHD.


Introduction
Variational principles for MHD were introduced by previous authors both in Lagrangian and Eulerian form. Vladimirov and Moffatt [1] in a series of papers have discussed an Eulerian variational principle for incompressible MHD. However, their variational principle contained three more functions in addition to the seven variables which appear in the standard equations of incompressible MHD which are the magnetic field B the velocity field v and the pressure P . Yahalom & Lynden-Bell [2] obtained an Eulerian Lagrangian principle for barotropic MHD which will depend on only six functions. The variational derivative of this Lagrangian produced all the equations needed to describe barotropic MHD without any additional constraints. Yahalom [3] have shown that for the barotropic case four functions will suffice. Moreover, it was shown that the cuts of some of those functions [4] are topological local conserved quantities.
Previous work was concerned only with barotropic magnetohydrodynamics. Variational principles of non barotropic magnetohydrodynamics can be found in the work of Bekenstein & Oron [5] in terms of 15 functions and V.A. Kats [6] in terms of 20 functions. Morrison [7] has suggested a Hamiltonian approach but this also depends on 8 canonical variables (see table 2 [7]).
It was shown that this number can be somewhat reduced. In [8,9] it was demonstrated that only five functions will suffice to describe non barotropic magnetohydrodynamics, and that the reduced lagrangian has a distinct geometrical structure including an induced metric.
The theorem of Noether dictates that for every continuous symmetry group of an Action the system must possess a conservation law. For example time translation symmetry results in the conservation of energy, while spatial translation symmetry results in the conservation of linear momentum and rotation symmetry in the conservation of angular momentum to list some well known examples. But sometimes the conservation law is discovered without reference to the Noether theorem by using the equations of the system. In that case one is tempted to inquire what is the hidden symmetry associated with this conservation law and what is the simplest way to represent it.
The concept of metage as a label for fluid elements along a vortex line in ideal fluids was first introduced by Lynden-Bell & Katz [10]. A translation group of this label was found to be connected to the conservation of Moffat's [1] helicity by Yahalom [11] using a Lagrangian variational principle. The concept of metage was later generalized by Yahalom & Lynden-Bell [2] for barotropic MHD, but now as a label for fluid elements along magnetic field lines which are comoving with the flow in the case of ideal MHD. Yahalom & Lynden-Bell [2] has also shown that the translation group of the magnetic metage is connected to Woltjer [12,13] conservation of cross helicity for barotropic MHD. Recently the concept of metage was generalized also for non barotropic MHD in which magnetic field lines lie on entropy surfaces [14]. This was later generalized by dropping the entropy condition on magnetic field lines [15]. In those papers the metage translation symmetry group was used to generate a non-barotropic cross helicity generalization using a Lagrangian variational principle.
Cross Helicity was first described by Woltjer [12,13] and is give by: in which the integral is taken over the entire flow domain. H C is conserved for barotropic or incompressible MHD and is given a topological interpretation in terms of the knottiness of magnetic and flow field lines. Both conservation laws for the helicity in the fluid dynamics case and the barotropic MHD case were shown to originate from a relabelling symmetry through the Noether theorem [2,11,17,18]. Webb et al. [20] have generalized the idea of relabelling symmetry to non-barotropic MHD and derived their generalized cross helicity conservation law by using Noether's theorem but without using the simple representation which is connected with the metage variable. The conservation law deduction involves a divergence symmetry of the action. These conservation laws were written as Eulerian conservation laws of the form D t + ∇ · F = 0 where D is the conserved density and F is the conserved flux. Webb et al. [22] discuss the cross helicity conservation law for non-barotropic MHD in a multi-symplectic formulation of MHD. Webb et al. [19,20] emphasize that the generalized cross helicity conservation law, in MHD and the generalized helicity conservation law in non-barotropic fluids are non-local in the sense that they depend on the auxiliary nonlocal variable σ, which depends on the Lagrangian time integral of the temperature T (x, t). Notice that a potential vorticity conservation equation for non-barotropic MHD is derived by Webb, G. M. and Mace, R.L. [23] by using Noether's second theorem.
Recently the non-barotropic cross helicity was generalized using additional label translation symmetry groups (χ and η translations) [25], this led to additional topological conservation laws the χ and η cross helicities.
Previous analysis depended on Lagrangian variational principles and their Noether currents. Here we introduce a novel approach based on an Eulerian variational principle. We derive the Noether current of the Eulerian variational principle and show how this can be used to derive topological conservation laws using label symmetries.
The plan of this paper is as follows: First we introduce the standard notations and equations of non-barotropic magnetohydrodynamics. Next we introduce the Eulerian variational principle suitable for the non-barotropic case. This is followed by a derivations of the Noether Current and finally we use the Noether current to obtain the generalized non-barotropic cross helicities. Implication for non-barotropic MHD dynamics of the topological conservation laws are discussed.

Standard formulation of ideal non-barotropic magnetohydrodynamics
The standard set of equations solved for non-barotropic magnetohydrodynamics are given below: The following notations are utilized: ∂ ∂t is the temporal derivative, d dt is the temporal material derivative and ∇ has its standard meaning in vector calculus. ρ is the fluid density and s is the specific entropy. Finally p(ρ, s) is the pressure which depends on the density and entropy (the non-barotropic case). The justification for those equations and the conditions under which they apply can be found in standard books on magnetohydrodynamics (see for example [16]). The number of independent variables for which one needs to solve is eight ( v, B, ρ, s) and the number of equations (2,4,5,6) is also eight. Notice that equation (3) is a condition on the initial B field and is satisfied automatically for any other time due to equation (2). We will find it useful to introduce the following thermodynamic equations for later use: in the above: ε is the specific internal energy, T is the temperature and w is the specific enthalpy. A special case of equation of state is the polytropic equation of state [26]: K and γ may depend on the specific entropy s. Hence: the last identity is up to a function dependent on s.

Variational principle of non-barotropic magnetohydrodynamics
In the following section we will generalize the approach of [2] for the nonbarotropic case [8,9]. Consider the action: In the specific case of a polytropic equation of state we have according to equation (9): Obviously ν, α, β, σ are Lagrange multipliers which were inserted in such a way that the variational principle will yield the following equations: It is not assumed that ν, α, β, σ are single valued. Provided ρ is not null those are just the continuity equation (4), entropy conservation and the conditions that Sakurai's functions are comoving. Taking the variational derivative with respect to B we see that: Hence B is in Sakurai's form [27] and satisfies equation (3). It can be easily shown that provided that B is in the form given in equation (13), and equations (12) are satisfied, then also equation (2) is satisfied. We notice that the specific form of the magnetic field given in equation (13) appear under different names in the literature. The functions χ and η are sometimes denoted "Euler potentials", "Clebsch variables" and also "flux representation functions" [28]. Equation (13) imply that the magnetic field lines lie on surfaces, the lines may be surface filling but not volume filling. For the time being we have showed that all the equations of non-barotropic magnetohydrodynamics can be obtained from the above variational principle except Euler's equations. We will now show that Euler's equations can be derived from the above variational principle as well. Let us take an arbitrary variational derivative of the above action with respect to v, this will result in: The integral d S · δ vρν vanishes in many physical scenarios. In the case of astrophysical flows this integral will vanish since ρ = 0 on the flow boundary, in the case of a fluid contained in a vessel no flux boundary conditions δ v ·n = 0 are induced (n is a unit vector normal to the boundary). The surface integral d Σ on the cut of ν vanishes in the case that ν is single valued and [ν] = 0 . In the case that ν is not single valued only a Kutta type velocity perturbation [32] in which the velocity perturbation is parallel to the cut will cause the cut integral to vanish.
Provided that the surface integrals do vanish and that δ v A = 0 for an arbitrary velocity perturbation we see that v must have the following form: The above equation is reminiscent of Clebsch representation in non magnetic fluids. A similar expression was obtained by Morrison [7] using an Hamiltonian formalism but in which the s terms is replaced by ψ which is conjugate to s. Let us now take the variational derivative with respect to the density ρ we obtain: In which w = ∂(ερ) ∂ρ is the specific enthalpy. Hence provided that d S · vδρν vanishes on the boundary of the domain and d Σ · vδρ[ν] vanishes on the cut of ν in the case that ν is not single valued 1 and in initial and final times the following equation must be satisfied: Finally we have to calculate the variation with respect to both χ and η this will lead us to the following results: Provided that the correct temporal and boundary conditions are met with respect to the variations δχ and δη on the domain boundary and on the cuts in the case that some (or all) of the relevant functions are non single valued. we obtain the following set of equations: in which the continuity equation (4) was taken into account. By correct temporal conditions we mean that both δη and δχ vanish at initial and final times. As for boundary conditions which are sufficient to make the boundary term vanish on can consider the case that the boundary is at infinity and both B and ρ vanish. Another possibility is that the boundary is impermeable and perfectly conducting. A sufficient condition for the integral over the "cuts" to vanish is to use variations δη and δχ which are single valued. It can be shown that χ can always be taken to be single valued, hence taking δχ to be single valued is no restriction at all. In some topologies η is not single valued and in those cases a single valued restriction on δη is sufficient to make the cut term null. Finally we take a variational derivative with respect to the entropy s: in which the temperature is T = ∂ε ∂s . We notice that according to equation (15) σ is single valued and hence no cuts are needed. Taking into account the continuity equation (4) we obtain for locations in which the density ρ is not null the result: provided that δ s A vanished for an arbitrary δs.

Euler's equations
We shall now show that a velocity field given by equation (15), such that the equations for α, β, χ, η, ν, σ, s satisfy the corresponding equations (12,17,20,22) must satisfy Euler's equations. Let us calculate the material derivative of v: It can be easily shown that: In which x k is a Cartesian coordinate and a summation convention is assumed. Inserting the result from equations (24,12) into equation (23) yields: In which we have used both equation (15) and equation (13) in the above derivation. This of course proves that the non-barotropic Euler equations can be derived from the action given in equation (10) and hence all the equations of non-barotropic magnetohydrodynamics can be derived from the above action without restricting the variations in any way except on the relevant boundaries and cuts.

Local non-barotropic cross helicity
The function ν, whose material derivative is given in (17), can be multiple valued because only its gradient appears in the velocity (15). However, the discontinuity, [ν], of ν is conserved, since the terms on the right-hand side of (17) describe physical quantities and hence are single valued. A similar equation also holds for barotropic fluid dynamics and barotropic MHD [2,3,4]. We now substitute the expressions for B and v given by (13) and (15) respectively into the formula H C ≡ B · vd 3 x for the cross helicity (see (1)) to obtain where the closed line integral taken along a magnetic field line. Furthermore, dΦ = B · d S = ( ∇χ × ∇η) · d S = dχdη is a magnetic flux element which is co-moving as governed by (2) and d S is an infinitesimal area element. Although the cross helicity is not conserved for non-barotropic flows, inspection of the right-hand side of (27) reveals that it is made of a sum of two terms. One term is conserved, as both dΦ and [ν] are co-moving, and the other is not. This suggests the following definition for the non-barotropic cross helicity It can be written in the more conventional form: in which the topological velocity field is defined as It should be noted that H CN B is conserved even for an MHD not satisfying the Sakurai topological constraint given in (13), provided that we have a field σ satisfying the equation dσ dt = T . This can be verified by direct derivation using only the equation of motion and the sigma equation. Thus the non-barotropic cross helicity conservation law, is more general than the variational principle described by (59) as follows from a direct computation using (2) and (4)- (6). Also note that, for a constant specific entropy s, we obtain H CN B = H C and the non-barotropic cross helicity reduces to the standard barotropic cross helicity. The local form of equation (31) describing the evolution of H CN B per unit volume was described by [19,20].
To conclude we introduce also a local topological conservation law in the spirit of [4] which is the non-barotropic cross helicity per unit of magnetic flux. This quantity which is equal to the discontinuity, [ν], of ν is conserved and can be written as a sum of the barotropic cross helicity per unit flux and the closed line integral of sdσ along a magnetic field line, namely:

Simplified action
The reader of this paper might argue here that the paper is misleading. The author has declared that he is going to present a simplified action for nonbarotropic magnetohydrodynamics instead he added six more functions α, β, χ,η, ν, σ to the standard set B, v, ρ, s. In the following I will show that this is not so and the action given in equation (10) in a form suitable for a pedagogic presentation can indeed be simplified. It is easy to show that the Lagrangian density appearing in equation (10) can be written in the form: In whichˆ v is a shorthand notation for ∇ν + α ∇χ + β ∇η + σ ∇s (see equation (15)) andˆ B is a shorthand notation for ∇χ × ∇η (see equation (13)). Thus L has four contributions: The only term containing v is 2 L v , it can easily be seen that this term will lead, after we nullify the variational derivative with respect to v, to equation (15) but will otherwise have no contribution to other variational derivatives. Similarly the only term containing B is L B and it can easily be seen that this term will lead, after we nullify the variational derivative, to equation (13) but will have no contribution to other variational derivatives. Also notice that the term L boundary contains only complete partial derivatives and thus can not contribute to the equations although it can change the boundary conditions. Hence we see that equations (12), equation (17), equations (20) and equation (22) can be derived using the Lagrangian density: in whichˆ v replaces v andˆ B replaces B in the relevant equations. Furthermore, after integrating the eight equations (12,17,20,22) we can insert the potentials α, β, χ, η, ν, σ, s into equations (15) and (13) to obtain the physical quantities v and B. Hence, the general non-barotropic magnetohydrodynamic problem is reduced from eight equations (2,4,5,6) and the additional constraint (3) to a problem of eight first order (in the temporal derivative) unconstrained equations. Moreover, the entire set of equations can be derived from the Lagrangian densitŷ L.

Elimination of Variables
Let us now look at the three last three equations of (12) [8,9]. Those describe three comoving quantities which can be written in terms of the generalized Clebsch form given in equation (15) as follows: Those are algebraic equations for α, β, σ, which can be solved such that α, β, σ can be written as functionals of χ, η, ν, s, resulting eventually in the description of non-barotropic magnetohydrodynamics in terms of five functions: ν, ρ, χ, η, s. Let us introduce the notation: i ∈ (1, 2, 3). In terms of the above notation equation (36) takes the form: in which the Einstein summation convention is assumed. Let us define the matrix: obviously this matrix is symmetric since A ij = A ji . Hence equation (38) takes the form: Provided that the matrix A ij is not singular it has an inverse A −1 ij which can be written as: In which the determinant |A| is given by the following equation: In terms of the above equations the α i 's can be calculated as functionals of χ i , ν as follows: The velocity equation (15) can now be written as: Provided that the χ i is a coordinate basis in three dimensions, we may write: Inserting equation (45) into equation (44) we obtain: in the above δ in is a Kronecker delta. Thus the velocity v[χ i ] is a functional of χ i only and is independent of ν.

Lagrangian Density and Variational Analysis
Let us now rewrite the Lagrangian densityL[χ i , ν, ρ] given in equation (35) in terms of the new variables: Let us calculate the variational derivative ofL[χ i , ν, ρ] with respect to χ i this will result in: in which the summation convention is not applied if the index is underlined. However, due to equation (44) we may write: Inserting equation (49) into equation (48) and rearranging the terms we obtain: Now by construction v satisfies equation (36) and hence ∂χ k ∂t + v · ∇χ k = 0, this leads to: From now on the derivation proceeds as in equations (18,19,21) resulting in equations (20,22) and will not be repeated. The difference is that now α, β and σ are not independent quantities, rather they depend through equation (43) on the derivatives of χ i , ν. Thus, equations (18,19,21) are not first order equations in time but are second order equations. Now let us calculate the variational derivative with respect to ν this will result in the expression: However, δ ν α k can be calculated from equation (43): Inserting the above equation into equation (52): The above equation can be put to the form: This obviously leads to the continuity equation (4) and some boundary terms in space and time. The variational derivative with respect to ρ is trivial and the analysis is identical to the one in equation (16) leading to equation (17).
To conclude this subsection let us summarize the equations of non-barotropic magnetohydrodynamics: in which α, β, σ, v are functionals of χ, η, s, ν as described above. It is easy to show as in equation (25) that those variational equations are equivalent to the physical equations.

Lagrangian Density in Explicit Form
Let us put the Lagrangian density of equation (47) in a slightly more explicit form. First us look at the term v 2 : in the above we use equation (46) and equation (39). Next let us look at the expression: Inserting equation (57) and equation (58) into equation (47) leads to a Lagrangian density of a more standard quadratic form: We now define the metric g jn = A −1 jn and obtain the geometrical Lagrangian: The Lagrangian is thus composed of a geometric kinetic term which is quadratic in the temporal derivatives, a "gyroscopic" terms which is linear in the temporal derivative and a potential term which is independent of the temporal derivative.

Noether Current
Let us assume that all the equations of motion and boundary conditions of non barotropic MHD are satisfied. In this case we have according to equation (35): For the current purpose it does not matter if α, β and σ are independent variational variables or depend on other variational variables through equation (43). Now suppose that the variations δν, δχ, δη, δs are symmetry variations such that δA = 0. In that case one obtains a conserved Noether current: As the variations in the specific entropy s will generally vary the specific internal energy term in the lagrangian we do not expect non trivial entropy symmetry transformation and the action will only be invariant for δs = 0, hence:

Lagrangian and Eulerian variations
The value of a function f can be modified by evaluating it at a different point in space, the difference between the new and old values would be: in which x is a coordinate vector and ξ is a displacement vector, the equality is correct to first order in ξ. Alternatively we can modify the value of a function by changing it to a different function f ′ , in this case the difference between the new and old values would be: for a small δf this just the standard variation of variational analysis or an Eulerian variation. Finally we can do both, in the last case the difference between the new and old values would be: Hence: Keeping only first order terms we obtain: in which ∆ is a Lagrangian variation. Now suppose that a specific function is connected to a fluid element in such a way that its value in space is determined only by fluid element location. And suppose that the fluid element is displaced as dictated by the flow. Such a function would be denoted a label of the flow and its material derivative would vanish. Moreover, for a label: in order to change the value of a label in a certain point in space the fluid element must be displaced and another (with a different label value) must take its place. If follows from equation (68) that for a label: Now suppose we have a set of three labelsχ i such that: in which we use the Einstein summation convention and x k are Cartesian coordinates. The inverse of the matrix ∂χi ∂x k is ∂x k ∂χi as: δ j k is Kronecker's delta. It thus follows that one can calculate the displacement vector ξ as follows:

Noether current for label symmetries
We now study the form of the Noether current equation (63) for the case of label symmetry transformations. It is clear from equation (12) that χ, η can be taken to be labels. Hence we can write the conserved Noether current defined in equation (63) as: Or using equation (15) as: We use the topological velocity defined in equation (30): and write: Suppose now that we are considering label symmetry transformations with the infinitesimal form:χ i + δχ i this type of transformations will induce a transformation on other functions (ν as an example) which could be thought of as functions of labels a transformation of the form: Hence by equation (68): it follows that for an induced infinitesimal label transformation any function will transform as a label. From equation (79) it follows that the Noether current will take the following form for a symmetry label transformation: This Noether current form is identical to equation (47) of [15] and equation (14) of [25], which were derive from a Lagrangian variational principle. We note, however, that this form is limited to the case of label transformations and the general form given in equation (62) allows us to exploit larger symmetry groups. Next we will study some symmetry transformations of the action A, in order to do this we shall first introduce the Load and Metage quantities.

Load and Metage
The following section follows closely a similar section in [2,14,15,24]. Consider a thin tube surrounding a magnetic field line, the magnetic flux contained within the tube is: and the mass contained with the tube is: in which dl is a length element along the tube. Since the magnetic field lines move with the flow by virtue of equation (2) and equation (4) both the quantities ∆Φ and ∆M are conserved and since the tube is thin we may define the conserved magnetic load: in which the above integral is performed along the field line. Obviously the parts of the line which go out of the flow to regions in which ρ = 0 have a null contribution to the integral. Notice that λ is a single valued function that can be measured in principle. Since λ is conserved it satisfies the equation: This can be viewed as a manifestation of the frozen-in law of B ρ . By construction surfaces of constant magnetic load move with the flow and contain magnetic field lines. Hence the gradient to such surfaces must be orthogonal to the field line: Now consider an arbitrary comoving point on the magnetic field line and denote it by i, and consider an additional comoving point on the magnetic field line and denote it by r. The integral: is also a conserved quantity which we may denote following Lynden-Bell & Katz [10] as the magnetic metage. µ(i) is an arbitrary number which can be chosen differently for each magnetic line. By construction: This can be viewed as another manifestation of the frozen-in law of B ρ . Also it is easy to see that by differentiating along the magnetic field line we obtain: Notice that µ will be generally a non single valued function, we will show later in this paper that symmetry to translations in µ; will generate through the Noether theorem the conservation of the magnetic cross helicity. At this point we have two comoving coordinates of flow, namely λ, µ obviously in a three dimensional flow we also have a third coordinate. However, before defining the third coordinate we will find it useful to work not directly with λ but with a function of λ. Now consider the magnetic flux within a surface of constant load Φ(λ). The magnetic flux is a conserved quantity and depends only on the load λ of the surrounding surface. Now we define the quantity: Obviously χ satisfies the equations: Let us now define an additional comoving coordinate η * since ∇µ is not orthogonal to the B lines we can choose ∇η * to be orthogonal to the B lines and not be in the direction of the ∇χ lines, that is we choose η * not to depend only on χ. Since both ∇η * and ∇χ are orthogonal to B, B must take the form: However, using equation (3) we have: Which implies that A is a function of χ, η * . Now we can define a new comoving function η such that: In terms of this function we obtain the Sakurai (Euler potentials) presentation: And the density is now given by the Jacobian: It can easily be shown using the fact that the labels are comoving that the above forms of B and ρ satisfy equation (2), equation (3) and equation (4) automatically.
We can now write a Lagrangian density in terms of the labels, in which ρ is no longer an independent variational variable but rather a quantity dependent on µ through equation (95). The Lagrangian density of equation (35) takes the form: ∂(x, y, z) , s) Notice however, that η is defined in a non unique way since one can redefine η for example by performing the following transformation: η → η + f (χ) in which f (χ) is an arbitrary function. The comoving coordinates χ, η serve as labels of the magnetic field lines. Moreover the magnetic flux can be calculated as: In the case that the surface integral is performed inside a load contour we obtain: There are two cases involved; in one case the load surfaces are topological cylinders; in this case η is not single valued and hence we obtain the upper value for Φ(λ). In a second case the load surfaces are topological spheres; in this case η is single valued and has minimal η min and maximal η max values. Hence the lower value of Φ(λ) is obtained. For example in some cases η is identical to twice the latitude angle θ. In those cases η min = 0 (value at the "north pole") and η max = 2π (value at the "south pole").
Comparing the above equation with equation (89) we derive that η can be either single valued or not single valued and that its discontinuity across its cut in the non single valued case is [η] = 2π.
The triplet χ, η, µ will suffice to label any fluid element in three dimensions. But for a non-barotropic flow there is also another possible label s which is comoving according to equation (6). The question then arises of the relation of this label to the previous three. As one needs to make a choice regarding the preferred set of labels it seems that the physical ones are χ, η, s in which we use the surfaces on which the magnetic fields lie and the entropy, each label has an obvious physical interpretation. In this case we must look at µ as a function of χ, η, s. If the magnetic field lines lie on entropy surface then µ regains its status as an independent label. The density can now be written as: Now as µ can be defined for each magnetic field line separately according to equation (86) it is obvious that such a choice exist in which µ is a function of s only. One may also think of the entropy s as a functions χ, η, µ. However, if one change µ in this case this generally entails a change in s and the symmetry described in equation (86) is lost in the Action. In what follows we shall ignore the status of s as a label and consider it as a variational variable which only attains a status of a label at the variational extremum.

Metage translations
In what follows we consider the transformation (see also equation (86)): Hence a is a label displacement which may be different for each magnetic field line, as the field line is closed one need not worry about edge difficulties. This transformation satisfies trivially the conditions (100,102). If we take the infinitesimal symmetry transformation δµ = a, δχ = δη = 0 we can calculate the associated fluid element displacement with this relabelling using equation (73) and equation (88).
Inserting equation (109) into equation (80) we obtain the conservation law: In the simplest case we may take δµ to be a small constant, hence: Where H CN B is the non barotropic global cross helicity [19,29,30] defined as: We thus obtain the conservation of non-barotropic cross helicity using the Noether theorem and the symmetry group of metage translations. Of course one can perform a different translation on each magnetic field line, in this case one obtains: Now since δµ is an arbitrary (small) function of χ, η it follows that: is a conserved quantity for each magnetic field line. Along a magnetic field line the following equations hold: in the aboveB is an unit vector in the magnetic field direction an equation (88) is used. Inserting equation (115) into equation (114) we obtain: which is just the circulation of the topological velocity along the magnetic field lines. This quantity can be written in terms of the generalized Clebsch representation of the velocity equation (15) as: [ν] is the discontinuity of ν. This was shown to be equal to the amount of non barotropic cross helicity per unit of magnetic flux in equation (32) [29,30].

Transformations of magnetic surfaces
Consider the following transformations: in which δη, δχ are considered small in some sense. Inserting the above quantities into equation (102) and keeping only first order terms we arrive at: This equation can be solved as follows: in which δf = δf (χ, η) is an arbitrary small function. In this case we obtain a particle displacements of the form: A special case that satisfies equation (120) is the case of a constant δχ and δη, those two independent displacements lead to two new topological conservation laws: Where the new non barotropic global cross helicities are defined as: We will find it useful to introduce the abstract "magnetic fields" as follows: In terms of which we obtain the new helicities in a more conventional form: It is more plausible that those symmetries and conservation laws hold for magnetic field lines which lie on topological torii. In this case η is non single valued [2] and thus the translation in this direction resembles moving fluid elements along closed loops. Both those helicities suffer a topological interpretation in terms of the knottiness of the abstract magnetic field lines and the flow lines. Finally we remark that for barotropic MHD v t can be replaced with v.

Direct Derivation
Before continuing to discuss the possible applications of the topological constants of motion, we shall demonstrate that the generalized cross helicities are indeed constant without relying on the Noether theorem.

Direct derivation of the constancy of non barotropic cross helicity
Taking the temporal derivative of the non barotropic cross helicity given in equation (112) we obtain: in the above d dt is an ordinary temporal derivative, and we use the notation: ∂ t ≡ ∂ ∂t . Using equation (2) it follows that: in which we have used a standard identity of vector analysis and the definition: v t is defined in equation (30). Next we calculate: Taking into account Euler equation (5) and the standard thermodynamic identities of equation (7): Hence: Taking into account equation (22) it follows that: And taking into account equation (6) it follows that: Inserting equation (132), equation (133) and equation (134) into equation (130) it follows that: Hence: Combining equation (128) with equation (136) we arrive at the result: in which we take into account equation (3). Inserting equation (137) into equation (127) and using Gauss theorem we obtain a surface integral: The surface integral encapsulates the volume for which the non barotropic cross helicity is calculated. If the surface is taken at infinity the magnetic fields vanish and thus: which means that H CN B is a constant of motion. We notice the complexity of the direct derivation with respect to the elegance and simplicity of the Noether theorem approach. However, obtaining the same result using different methods strengthens our confidence that no mathematical error was accidentally introduced.

Direct derivation of the constancy of non barotropic χ cross helicity
Taking the temporal derivative of the non barotropic χ cross helicity given in equation (126) we obtain: Let us calculate ∂ t B χ where B χ is defined in equation (125): It follows that: Using equation (12) and equation (87) we obtain: in which we used standard vector analysis identities. It thus follows that: Next we calculate: Taking into account equation (131): in which the current density is given by: Now: However: in which we used equation (13) and equation (88). Inserting equation (150) into equation (146) will yield: Inserting equation (151), equation (133) and equation (134) into equation (145) it follows that: Combining equation (144) with equation (153) we arrive at the result: in which we take into account equation (141). Inserting equation (154) into equation (140) and using Gauss theorem we obtain a surface integral: The surface integral encapsulates the volume for which the χ non barotropic cross helicity is calculated and an additional surface integral is performed along the cut of η, in case that η is not single valued (see equation (98)). If the surface is taken at infinity the magnetic fields and current densities vanish and thus: hence for spherical topologies of magnetic field lines or for a current density J parallel to the cut we obtain: which means that H CN Bχ is a constant of motion. We notice the complexity of the direct derivation with respect to the elegance and simplicity of the Noether theorem approach. However, obtaining the same result using different methods strengthens our confidence that no mathematical error was accidentally introduced.

Direct derivation of the constancy of non barotropic η cross helicity
Taking the temporal derivative of the non barotropic η cross helicity given in equation (126) we obtain: Let us calculate ∂ t B η where B η is defined in equation (125): It follows that: Using equation (12) and equation (87) we obtain: in which we used standard vector analysis identities. It thus follows that: Next we calculate: Taking into account equation (131): Now: However: It thus follows that: in which we used equation (13) and equation (88). Inserting equation (167) into equation (164) will yield: Inserting equation (168), equation (133) and equation (134) into equation (163) it follows that: Combining equation (162) with equation (170) we arrive at the result: in which we take into account equation (159). Inserting equation (171) into equation (158) and using Gauss theorem we obtain a surface integral: The surface integral encapsulates the volume for which the η non barotropic cross helicity is calculated. If the surface is taken at infinity the magnetic fields and current densities vanish and thus: which means that H CN Bη is a constant of motion. We notice the complexity of the direct derivation with respect to the elegance and simplicity of the Noether theorem approach. However, obtaining the same result using different methods strengthens our confidence that no mathematical error was accidentally introduced.

Possible Application
In his important review paper "Physics of magnetically confined plasmas" A. H. Boozer [31] states that: "A spiky current profile causes a rapid dissipation of energy relative to magnetic helicity. If the evolution of a magnetic field is rapid, then it must be at constant helicity." Usually topological conservation laws are used in order to deduce lower bounds on the "energy" of the flow. Those bounds are only approximate in non ideal flows but due to their topological nature simulations show that they are approximately conserved even when the "energy" is not. For example it is easy to show that the "energy" is bounded from below by the non-barotropic cross helicity as follows (see [24]): the second equation is a result of the Cauchy-Schwartz inequality. In this sense a configuration with a highly complicated topology is more stable since its energy is bounded from below. It is a simple thing to show that similar bounds occur also for the χ and η helicities: Hence the kinetic energy is bounded by three differen bounds and so it the "total" energy. The importance of each of those bounds is dependent on the flow.

Conclusion
We have derived a Noether current from an Eulerian variational principle on non-barotropic MHD, this was shown to lead to to the conservation of nonbarotropic cross helicity. The connection of the translation symmetry groups of labels to both the global non barotropic cross helicity conservation law and the conservation law of circulations of topological velocity along magnetic field lines was elucidated. The latter were shown to be equivalent to the amount of non barotropic cross helicity per unit of magnetic flux [29,30,24]. Further more we have shown that two additional cross helicity conservation laws exist the χ and η cross helicities. Those lead to new bounds on MHD flows in addition to the bounds of the standard non-barotropic cross helicity discussed in [30] for ideal non-barotropic MHD. The importance of constants of motion for stability analysis is also discussed in [39]. The significance of those constraints for nonideal MHD and for plasma physics in general remains to be studied in future works. It is shown that non-barotropic MHD can be derived from a variational principle of five functions. The formalism is given in a Lagrangian presentation with a geometrical structure.
Possible applications include stability analysis of stationary MHD configurations and its possible utilization for developing efficient numerical schemes for integrating the MHD equations. It may be more efficient to incorporate the developed formalism in the framework of an existing code instead of developing a new code from scratch. Possible existing codes are described in [33,34,35]. Applications of this study may be useful to both linear and non-linear stability analysis of known barotropic MHD configurations [36,37,38,39,40,41]. As for designing efficient numerical schemes for integrating the equations of fluid dynamics and MHD one may follow the approach described in [42,43,32,44].
Another possible application of the variational method is in deducing new analytic solutions for the MHD equations. Although the equations are notoriously difficult to solve being both partial differential equations and nonlinear, possible solutions can be found in terms of variational variables. An example for this approach is the self gravitating torus described in [45].
One can use continuous symmetries which appear in the variational Lagrangian to derive new conservation laws through the Noether theorem. An example for such derivation which still lacks physical interpretation can be found in [46]. It may be that the Lagrangian derived in [3] has a larger symmetry group. And of course one anticipates a different symmetry structure for the non-barotropic case.
Topological invariants have always been informative, and there are such invariants in MHD flows. For example the two helicities have long been useful in research into the problem of hydrogen fusion, and in various astrophysical scenarios. In previous works [2,4,11] connections between helicities with symmetries of the barotropic fluid equations were made. The Noether current here derived may help us to identify and characterize as yet unknown topological invariants in MHD .