aa r X i v : . [ h e p - t h ] S e p CERN-PH-TH/2015-182
The spectrum of anomalous magnetohydrodynamics
Massimo Giovannini Department of Physics, Theory Division, CERN, 1211 Geneva 23, SwitzerlandINFN, Section of Milan-Bicocca, 20126 Milan, Italy
Abstract
The equations of anomalous magnetohydrodynamics describe an Abelian plasma whereconduction and chiral currents are simultaneously present and constrained by the second lawof thermodynamics. At high frequencies the magnetic currents play the leading role and thespectrum is dominated by two-fluid effects. The system behaves instead as a single fluid inthe low-frequency regime where the vortical currents induce potentially large hypermagneticfields. After deriving the physical solutions of the generalized Appleton-Hartree equation,the corresponding dispersion relations are scrutinized and compared with the results validfor cold plasmas. Hypermagnetic knots and fluid vortices can be concurrently present atvery low frequencies and suggest a qualitatively different dynamics of the hydromagneticnonlinearities. Electronic address: [email protected]
Introduction
Electrically conducting media are customarily described as a single fluid in the low-frequencybranch of the plasma spectrum. This approach has been extensively applied to the analysis ofhydromagnetic nonlinearities [1] evolving in terrestrial [2] and astrophysical plasmas [3, 4, 5].The same strategy cannot be extended to higher frequencies where the one-fluid descriptionis no longer tenable [6] and the plasma must be treated, at least, as a double fluid. This wellknown aspect of conventional electromagnetic plasmas stems directly from the properties ofthe vector currents which are associated, in the high-frequency limit, with the ions and withthe electrons. When the plasma is globally neutral the total vector current is instead Ohmicin the low-frequency domain.A problem of similar nature occurs in anomalous magnetohydrodynamics [7] describinga charged fluid where axial and vector currents are simultaneously present: while the axialcurrents are not conserved because of the triangle anomaly, the vector currents are even-tually Ohmic. The purpose of this investigation is a systematic discussion of the spectrumof anomalous magnetohydrodynamics (AMHD in what follows). The equations of AMHDdiffer from the ones where only chiral currents are present [8, 9] at finite fermionic density.They generalize the system firstly explored in Ref. [10] accounting for the evolution of thehypermagnetic and hyperelectric fields in the electroweak plasma. Indeed, in the symmetricphase of the electroweak theory the non-screened vector modes of the plasma correspondto the hypercharge which has a chiral coupling to fermions. The axial currents may beassociated with the evolution of the chemical potential or with the presence of an axionlikefield [11, 12] (see also [13]). In both cases the plasma may host parity-odd configurationsof the gauge fields characterized non-vanishing hypermagnetic gyrotropy ~B · ~ ∇ × ~B whichis the hydromagnetic analog of the kinetic gyrotropy (i.e. ~v · ~ ∇ × ~v ) naturally appearing inthe discussion of mean-field dynamos [1, 4]. The dynamical production of hypermagneticknots and Chern-Simons waves during inflation offers a potentially viable mechanism for thegeneration of the baryon asymmetry of the Universe (see last two papers in [10]). In AMHDthe hypermagnetic current is complemented by a vortical current possibly leading to theformation of fluid vortices.The same class of physical systems previously discussed in the electroweak plasma alsoarises in the framework of the so called chiral magnetic effect [14]. Both phenomena are oftenpresented as macroscopic manifestations of triangle anomalies. The model of chiral liquidemerging in the context of AMHD could then be relevant also in the context of the chiralmagnetic effect insofar as axial currents and quark vector currents are concurrently present inthe strongly interacting plasma. In the absence of finite conductivity effects (see e.g. [15, 16])the validity of the second law of thermodynamics is guaranteed by the simultaneous presenceof an hypermagnetic current and of a chiral vortical term. In AMHD the vector currents(eventually responsible of Ohmic dissipation), the chiral currents (determining the anomalouseffects) and the vortical currents (required by the second principle of thermodynamics) are2ll described by the appropriate kinetic coefficients. Whenever possible AMHD will bediscussed in analogy with the spectrum of conventional plasmas. More specifically the planof this investigation is the following. In section 2 we discuss the relativistic problem andderive the general form of the kinetic coefficients. In section 3 the properties of the two-fluidequations are analyzed while section 4 is devoted to the dispersion relations in the high-frequency domain. The one-fluid equations and their implications are presented in section5. Section 6 contains our concluding remarks. To avoid digressions some relevant technicalaspects have been relegated to the appendix. The conservation of the total energy momentum tensor and the evolution of the chiral andvector currents determine the relativistic form of the second law of thermodynamics. If thefour-divergence of the entropy four-vector is to be positive semi-definite (as implied by thegeneralized second law) the chiral and vector currents must contain supplementary kineticcoefficients corresponding to the hypermagnetic and to the vortical currents. In what followsthe dissipative effects are included in the framework of the Landau approach: the total four-velocity coincides then with the velocity of the energy transport defined from the mixedcomponents of the total energy-momentum tensor.
In the simplest situation the total energy-momentum tensor of the system ( T ( tot ) µν in whatfollows) consists of four qualitatively different contributions: the energy-momentum ten-sor of the charged species (denoted by T ( ± ) µν ), the energy-momentum tensor of the chiralspecies (labeled by T ( R ) µν ), the dissipative contribution T ( diss ) µν and the gauge contribution T µν (corresponding to an hypercharge gauge-field strength Y µν ): T ( tot ) µν = T (+) µν + T ( − ) µν + T ( R ) µν + T ( diss ) µν + T µν . (2.1)The covariant conservation of T ( tot ) µν implies, as usual, that ∇ µ T µν ( tot ) = 0 where ∇ µ denotesthe covariant derivative defined from the metric tensor g µν (with signature mostly minus).The chiral and the conduction currents coexist but are not bound to coincide: they obeydifferent equations. More specifically the anomalous current is not covariantly conserved andits evolution can be written as ∇ µ j µR = A R Y αβ e Y αβ , j µR = e n R u µR + ν µR , (2.2) The discussion will be conducted in a general relativistic formulation even if the spectrum of AMHDwill be discussed in flat space-time. Note that e n R and u µR are respectively the concentration and the four-velocity of the chiral species. A R is a numerical factor that is determined by the specific nature of the chiral speciesand by the coupling to the hypercharge field; note that in the Landau frame ν µR u Rµ = 0.Conversely the conduction current is covariantly conserved, it is a source of the evolutionequations of the gauge fields and it may even contain a dissipative contribution: ∇ µ Y µν = 4 πj ν , j ν = j ν + + j ν − , ∇ µ j µ = 0 , (2.3)where j ν ± = ( q ± e n ± u µ ± + ν µ ± ). The dual field strength e Y µν obeys, as usual, ∇ µ e Y µν = 0.This is the approach followed in [7] which differs from other more conventional approaches[15, 16] where the anomalous current and the conduction current are identified. In thepresent approach the anomalous current is not directly the source of the evolution of thehypercharge.In a general relativistic description Y µν and its dual account for the evolution of thehypercharge field however the gauge field strength can be decomposed into the hyperelectricand hypermagnetic parts denoted, respectively, by E µ and B µ : Y αβ = E α u β − E β u α + E αβρσ u ρ B σ , (2.4)where E αβρσ = √− g ǫ αβρσ and ǫ αβρσ is the four-dimensional Levi-Civita symbol while g isthe determinant of the metric tensor. The total four-velocity of the system follows from( p + ρ ) u µ u ν = X a [ p ( a ) + ρ ( a ) ] u µ ( a ) u ν ( a ) , (2.5)where w = ( ρ + p ) denotes the total enthalpy; the sum in Eq. (2.5) runs over all the speciesof the plasma, both charged and chiral. Close to an equilibrium situation the four-velocity ofthe anomalous species coincides with the bulk velocity of the plasma and, therefore, u µR ≃ u µ .The vorticity four vector can then be defined as: ω µ = e f µα u α ≡ E µαβγ u α f βγ , f βγ = ∇ β u γ − ∇ γ u β . (2.6)From Eqs. (2.4) and (2.6) it follows that the four-divergences of E µ , B µ and ω µ are given by: w ∇ µ ω µ = − ω α ∂ α p − e n E α ω α , (2.7) w ∇ µ B µ = 2 w Y ρσ ω ρ u σ + u µ ∂ α p e Y µα + u µ Y αβ j β e Y µα , (2.8) w ∇ µ E µ = w [4 πj α u α − e Y µρ ω µ u ρ ] + Y βγ u β ∂ γ p + Y βγ u β Y γα j α , (2.9)where w = p + ρ is the enthalpy density of the fluid. Equations (2.7), (2.8) and (2.9) havebeen obtained in the globally neutral case where e n = e n + = e n − and q + = q = − q − but theycan be easily generalized to the case where the plasma is not globally neutral. As soon as we speak of hyperelectric and hypermagnetic fields we are implicitly assuming that theplasma has a finite conduction current so that a preferred frame can be selected where the electric fields aresuppressed. Even if the electric and magnetic fields are non-relativistic concepts, it is practical to introducethe electric and the magnetic components of the gauge field strength in a generally covariant language. .2 First and second principles of thermodynamics Denoting with µ R the chemical potential associated with the anomalous species, the firstprinciple of thermodynamics demands: dE = T dS − pdV + µ R dN R , w = ρ + p = T ς + µ R e n R . (2.10)The fundamental identity E = T S − pV + µ R N R can be divided by a fiducial volumeand the result is the one reported in the second relation of Eq. (2.10) where ς is theentropy density and ρ the total energy density of the system. Combining the two relationsof Eq. (2.10) further thermodynamic relations can be obtained . Since the anomaly-inducedcurrents are protected by topology they are not associated with dissipative effects. Thus theentropy production of the plasma must only come, in the relativistic case, from the viscositycoefficients or from the Ohmic contributions but neither from the chiral currents nor fromthe corresponding diffusive contribution. The absence of dissipative contributions stemmingfrom the anomalous sector demands that the total entropy four-vector must be supplementedby two further coefficients S ω and S B : ς µ = ςu µ − µ R ν µR + S ω ω µ + S B B µ . (2.11)The covariant conservation of the total energy momentum tensor T ( tot ) µν can be written as ∇ µ ς µ − σT Y αβ Y να u ν u β − T µν ( diss ) T ∇ µ u ν = Z , (2.12)where we assumed, for the sake of simplicity, a global charge neutrality of the plasma anda corresponding Ohmic form for the charged species, namely P αµ j µ = σY αν u ν where P αµ = δ αµ − u µ u α is the standard projector. The function Z appearing in Eq. (2.12) is given by Z = ∇ µ (cid:18) S ω ω µ + S B B µ (cid:19) − ν α u β T Y αβ − ∂ β µ R ν βR − A R µ R Y αβ e Y αβ . (2.13)We remark that the specific definition of the entropy four-vector depends on the chemicalpotential of the system. However, since the coefficient A R does not have a definite sign, theanomalous currents may even lead to violation of the second principle of thermodynamicsunless Z vanishes identically. The vortical and the magnetic currents modify also the diffusive contributions denoted,respectively, by ν α and ν αR in Eq. (2.13). Four different coefficients parametrize the relationbetween ( ν α , ν αR ) and ( ω α , B α ): ν α = Λ ω ω α + Λ B B α , ν αR = Λ R ω ω α + Λ R B B α , (2.14) Like, for instance, ς∂ α T + e n R ∂ α µ R = ∂ α p or ∂ α ρ = T ∂ α ς + µ R ∂ α e n R . ω , Λ B ) and (Λ R ω , Λ R B ) all depend on the chemical potential and on the temper-ature. Using Eqs. (2.7), (2.8) and (2.9) the condition Z = 0 together with the explicitexpression of Z (see Eq. (2.13)) becomes: (cid:20) S B − (cid:18) Λ ω T (cid:19)(cid:21) ( ω α B α ) + (cid:20) µ R A R − (cid:18) Λ B T (cid:19)(cid:21) ( E α B α ) − w σ c ω α E β u µ B ν E αβµν S ω + ω α P α + B α Q α = 0 (2.15)where P α and Q α are two differential operators defined respectively, as: P α = ∂ α S ω − w S ω ∂ α p − ∂ α µ R Λ R ω , Q α = ∂ α S B − S B w S ω ∂ α p − ∂ α µ R Λ R B . (2.16)The results of Eqs. (2.13)–(2.15) follow easily if we recall that, by definition, u α ω α , u β E β and u γ B γ are all vanishing.To satisfy the condition expressed by Eq. (2.15) the four-vectors multiplying ω α and B α must vanish together with the coefficients of the terms multiplied by ω α B α and E α B α . Wethen arrive at the following conditions: P α = 0 , Q α = 0 , Λ B = 4 µ R A R , Λ ω = 2 T S B , S ω = 0 . (2.17)If, as established, S ω = 0 then Eq. (2.15) also implies that Λ R ω = 0. All the coefficients weought to determine depend on µ R and on the pressure. Thus the conditions of Eq. (2.17)are equivalent to the following system of equations: (cid:18) ∂ S B ∂p − S B w (cid:19) ∂ α p + (cid:18) ∂ S B ∂µ R − Λ R B (cid:19) ∂ α µ R = 0 , (2.18)where Λ ω = 2 T S B and Λ B = 4 A R µ R T . Using some standard thermodynamic relations(giving the partial derivatives of the pressure and of the rescaled chemical potential withrespect to the temperature) the various kinetic coefficients can be determined, after somealgebra: S B ( µ R , T ) = T a B ( µ R ) , Λ R B = ∂∂µ R (cid:20) T a B ( µ R ) (cid:21) , (2.19)Λ ω ( µ R , T ) = 2 T a B ( µ R ) , Λ B ( µ R , T ) = 4 A R µ R T, (2.20)where a B ( µ R ) is an arbitrary function of the rescaled chemical potential. Note also thatΛ B is fully determined in terms of the coefficient of the anomaly and it is, in practice, onlyfunction of the chemical potential itself since, by definition, µ R T = µ R .In summary, in a globally neutral plasma with an anomalous current, the relativisticsecond law implies that the non-anomalous current must contain magnetic and vorticalcontributions. If the plasma is not hypercharge neutral the form of the kinetic coefficientsis subjected to a higher degree of arbitrariness since a second chemical potential must beintroduced in the analysis (see the appendix of Ref. [7]).6 High-frequency propagation
The kinetic coefficients of Eqs. (2.19) and (2.20) can be redefined, for practical reasons, as: c ω ( µ R , T ) = 8 πT a B ( µ R ) , c B ( µ R , T ) = 16 π A R µ R T,c RB ( µ R , T ) = ∂∂µ R (cid:20) T a B ( µ R ) (cid:21) , c Rω ( µ R , T ) = 0 . (3.1)The hypermagnetic and hyperelectric fields denoted, in what follows, by ~B and ~E obey thefollowing set of equations ~ ∇ · ~E = 4 πq ( e n + − e n − ) , ~ ∇ × ~E = − ∂ t ~B, (3.2) ~ ∇ × ~B = 4 πq ( e n + ~v + − e n − ~v − ) + c ω ~ω − c B ~B + ∂ t ~E, (3.3)where ~B is divergenceless (i.e. ~ ∇ · ~B = 0). The coefficients c ω ( µ R , T ) and c B ( µ R , T ) multiply,respectively, the vortical and the magnetic currents of Eq. (3.3). The three-vector where ~ω defines the total vorticity ~ω = ( e ρ + ~ω + + e ρ − ~ω − ) / ( e ρ + + e ρ − ) and should not be confused with thefrequency (denoted by Ω in what follows). The energy densities of the charged species aredenoted by e ρ ± . To establish a direct connection with the spectrum of conventional plasmasin the limit of vanishing kinetic coefficients we shall preferentially consider the situationwhere the charged species are massive, namely e ρ ± = e n ± ( m ± + 3 T ± /
2) and p ± = e n ± T ± with T ± /m ± ≪ c Rω ( µ R , T ) and c RB ( µ R , T ) of Eq. (3.1) affect directly theevolution of the concentrations: ∂ t e n + + ~ ∇ · ( e n + ~v + ) + 1 q ~ ∇ · ( c ω λ + ~ω + ) − q ~ ∇ · ( c B λ + ~B ) = 0 , (3.4) ∂ t e n − + ~ ∇ · ( e n − ~v − ) − q ~ ∇ · ( c ω λ − ~ω − ) + 1 q ~ ∇ · ( c B λ − ~B ) = 0 , (3.5) ∂ t e n R + ~ ∇ · ( e n R ~v R ) + ~ ∇ · ( c R ω ~ω ) − ~ ∇ · ( c R B ~B ) = − A R ~E · ~B, (3.6)where λ ± = e ρ ± / ( e ρ + + e ρ − ). Concerning Eqs. (3.2)–(3.3) and Eqs. (3.4)–(3.6) few commentsare in order. If e n + = e n − a second chemical potential µ Y (corresponding to the hypercharge)can be introduced in Eq. (2.10). The global hypercharge neutrality of the plasma implies µ Y = 0. The peculiar velocities determining the currents obey the following set of equations: ∂ t ~v − + ( ~v − · ~ ∇ ) ~v − = − q e n − e ρ − [ ~E + ~v − × ~B ] + Γ c ( ~v + − ~v − ) − ~ ∇ p + ρ + , (3.7) ∂ t ~v + + ( ~v + · ~ ∇ ) ~v + = q e n + e ρ + [ ~E + ~v + × ~B ] + Γ c e ρ − e ρ + ( ~v − − ~v + ) − ~ ∇ p − ρ − , (3.8) ∂ t ~v R + ( ~v R · ~ ∇ ) ~v R = 0 , (3.9)where the pressure gradients shall be eventually neglected; Γ c denotes the collision frequencydetermining the generalized conductivity in the single fluid limit.7 .2 Linearization of the two-fluid equations Equations (3.2)–(3.3), (3.4)–(3.6) and (3.7)–(3.9) will now be linearized in the presence ofthe weak background magnetic field ~B with the aim of deriving the dispersion relations.The background field will be considered homogeneous: this means that the variation of ~B occurs over typical length-scales much larger than 1 /c B . The fluctuations of the variousquantities will be introduced as follows: e n ± ( t, ~x ) = n + δ e n ± ( t, ~x ) , e n R ( t, ~x ) = n + δ e n R ( t, ~x ) , ~B ( t, ~x ) = ~B + δ ~B ( t, ~x ) , (3.10)while for the other quantities (i.e. ~v ± ( t, ~x ) = δ~v ± ( t, ~x ), ~v R ( t, ~x ) = δ~v R ( t, ~x ) and ~E ( t, ~x ) = δ ~E ( t, ~x )) the fluctuations coincide with the field itself. In Eq. (3.10) n and n are, respec-tively, the uniform background charge and the uniform chiral concentration. The homoge-neous value of the chemical potential is related to n and the kinetic coefficients will alsobe homogeneous. In the case of approximate thermal equilibrium the chemical potentialcan be related to the concentration as µ R = µ e n R /ς where ς denotes the entropy densityat equilibrium and where µ is a numerical constant. Therefore if e n R is perturbed arounda homogeneous background the kinetic coefficients will also be, in the first approximationhomogeneous. Thanks to Eq. (3.10) the perturbed version of the evolution of the concen-trations can be written as: δ e n ′± + n ( ~ ∇ · δ~v ± ) = 0 , δ e n ′ R + n ( ~ ∇ · δ~v R ) = − A R δ ~E · ~B , (3.11)where the prime denotes a derivation with respect to the time coordinate t . Since the kineticcoefficients are homogeneous in the first approximation, their contribution disappears fromEq. (3.11). With the same notations Eqs. (3.7), (3.8) and (3.9) imply instead: δ~v ′± = ± qm ± (cid:20) δ ~E + δ ~v ± × ~B (cid:21) , δ~v ′ R = 0 , (3.12)where Γ c has been neglected but it will become relevant at low frequencies, as we shall seelater. Finally, after inserting Eq. (3.10) into Eqs. (3.2) and (3.3) we obtain: ~ ∇ · δ ~E = 4 πq ( δ e n + − δ e n − ) , ~ ∇ · δ ~B = 0 , ~ ∇ × δ ~E = − δ ~B ′ , (3.13) ~ ∇ × δ ~B = δ ~E ′ + 4 π q n ( δ~v + − δ~v − ) − c B δ ~B + c ω (cid:20) λ + ~ ∇ × δ~v + + λ − ~ ∇ × δ~v − (cid:21) . (3.14)From Eqs. (3.12) the equation obeyed by δ~ω ± can also be deduced and they are δ~ω ′± = ± q [ − δ ~B ′ + ~ ∇ × ( δ~v ± × ~B )] /m ± . Recalling the standard vector identities the equation for In section 4 we shall specifically discuss also the opposite limit where ~B varies appreciably over typicallengths L < /c B and we shall see that, in this case, the background solution belongs to the class of Beltramifields. In particular we recall that ~ ∇ × ( ~a × ~b ) = [ ~a ( ~ ∇ · ~b ) − ~b ( ~ ∇ · ~a ) + ( ~b · ~ ∇ ) ~a − ( ~a · ~ ∇ ) ~b ]. ~ω ± can also be expressed as: δ~ω ′± = ± qm ± (cid:20) − δ ~B ′ − ~B ( ~ ∇ · δ~v ± ) + ( ~B · ~ ∇ ) δ~v ± (cid:21) . (3.15)From Eqs. (3.12) and (3.14) the relevant dispersion relations and the associated refractionindices can be obtained by treating separately the motions parallel and perpendicular to themagnetic field direction. While in conventional plasmas the Appleton-Hartree dispersion relation has been extensivelydiscussed in the literature [6, 17], the AMHD equations linearized in the two-fluid limitcontain vortical and magnetic currents. The Laplace transform of Eq. (3.14) implies thefollowing equation( ~ ∇ × δ ~B ) Ω = − i Ω ε s (Ω) · δ ~E Ω − c B δ ~B Ω + i c ω ~ ∇ × [ ε v (Ω) · δ ~E Ω ] , (3.16)where Ω is the frequency (not to be confused with the total vorticity). In Eq. (3.16) ε s (Ω)and ε v (Ω) denote, respectively, the standard and the vortical components of the dielectrictensor. The explicit form of ε s (Ω) and ε v (Ω) can be found in appendix A; taking then thecurl of Eq. (3.13) and using Eq. (3.16) we obtain the following equation: ~ ∇ × ( ~ ∇ × δ ~E Ω ) = Ω ε s (Ω) · δ ~E Ω − c B ~ ∇ × δ ~E Ω − Ω c ω ~ ∇ × [ ε v (Ω) · δ ~E Ω ] . (3.17)We can now go to Fourier space and write Eq. (3.17) as: − ~k × ~k × δ ~E ~k Ω = Ω ε s (Ω) · δ ~E ~k Ω − i c B ~k × δ ~E ~k Ω − i c ω Ω ~k × [ ε v (Ω) · δ ~E ~k Ω ] . (3.18)We can therefore introduce the refractive index n satisfying n = k/ Ω where k = | ~k | ;choosing the coordinate system as ~k = (0 , n Ω sin θ, n
Ω cos θ ) we can obtain from Eq. (3.18)the following Appleton-Hartree matrix: [1 − ε n + c ω ε n c ( θ )] − i [ ε n + c B n Ω c ( θ ) + c ω n ε c ( θ )] i c B n Ω s ( θ ) i [ ε n + c B nω c ( θ ) + c ω n ε c ( θ )] [ c ( θ ) − ε n + ε (Ω) n c ω c ( θ )] − s ( θ ) c ( θ ) − i c B n Ω s ( θ ) − i c ω n ε (Ω) − s ( θ ) c ( θ ) + c ω n ε s ( θ ) [ s ( θ ) − ε k ( ω ) n ] . (3.19)The above matrix reduces to the standard form of the Appleton-Hartree matrix in the limit c ω → c B → The refractive index cannot be confused with the concentrations denoted by e n ± and e n R ; their homoge-neous values n and n carry specific subscripts so that the notations are clearly established. Dispersion relations
The determinant of the Appleton-Hartree matrix obtained in Eq. (3.19) leads to the followingexpression:sin θ ( ε k − n )[ n ( ε L + ε R ) − ε R ε L ] − θε k ( n − ε L )( n − ε R )+2 n [ c B f B ( ε, Ω , n, θ ) + c B g B ( ε, Ω , n, θ ) + c ω f ω ( ε, Ω , n, θ ) + c ω g ω ( ε, Ω , n, θ ) c B c ω h ( ε, Ω , n, θ ) + c B c ω h ( ε, Ω , n, θ ) + c ω c B h ( ε, Ω , n, θ )] = 0 . (4.1)Equation (4.1) is written in terms of the 7 functions explicitly reported in Eq. (A.7) ofappendix A. These functions have a specific dependence upon the dielectric tensors; with acollective notation such a dependence has been indicated by ε . The notations followed inEq. (4.1) imply that c B multiplies f B , c ω multiplies f ω ; g B and g ω multiply, respectively, c B and c ω ; the three functions h , h and h multiply instead the mixed products. Finally bothin Eqs. (4.1) and in Eq. (A.7) we have introduced ε L = ( ε + ε ) and ε R = ( ε − ε ) givenby: ε L (Ω) = 1 − Ω p (Ω + Ω B − )(Ω − Ω B + ) , ε R (Ω) = 1 − Ω p (Ω + Ω B + )(Ω − Ω B − ) , (4.2)where Ω p = (Ω p + + Ω p − ). When c B = c ω = 0 the magnetic and the vortical currents areabsent from the two-fluid AMHD equations and Eq. (4.1) implies the standard result [6]:sin θ (cid:18) n − ε k (cid:19)(cid:20) (cid:18) ε L + 1 ε R (cid:19) − n (cid:21) = cos θ (cid:18) ε L − n (cid:19)(cid:18) ε R − n (cid:19) . (4.3)The dispersion relations for a wave propagating parallel (i.e. θ = 0) and perpendicular (i.e. θ = π/
2) to the magnetic field direction can be obtained easily derived from Eq. (4.3). If θ = 0 Eq. (4.3) reduces to ( n − ε R )( n − ε L ) = 0 while for θ = π/ n − ε k )[ n ( ε L + ε R ) − ε L ε R ] = 0. These dispersion relations give therefore the conventionalresults which will be generalized hereunder. In the absence of magnetic field there is no preferred direction and the dispersion relationsfollow from Eqs. (4.3) and (A.7) by setting all the Larmor frequencies to zero. In this case ε R = ε L = ε k and the dispersion relations stem from the following two conditions, namely: ε k (Ω) = 0 , ( n − ε k )Ω ∓ nc B = 0 . (4.4) Along θ = 0 we thus obtain usual dispersion relations for the two circular polarizations of the electro-magnetic wave, i.e. n = ε R and n = ε L , while along θ = π/ n = ε k ) and “extraordinary” (i.e. n = 2 ε R ε L / ( ε R + ε L )) plasma waves. c ω is absent from Eq. (4.4) since the two-fluid effects cancelin the total vorticity. This cancellation is either exact (as in the case of the free-field propa-gation) or approximate (as we shall see later in the presence of the magnetic field). Indeed,as it can be explicitly verified from Eqs. (A.1), (A.5) and (A.6), ǫ v (Ω) → B → B → ε k (Ω) = 0 implies Ω = Ω p . This wave does not propagate sinceits group velocity vanishes and these are nothing but the electrostatic plasma oscillations[6]. The solution of the second equation in Eq. (4.4) is instead n = ± c B
2Ω + s − Ω p Ω + c B . (4.5)Equation (4.5) implies also Ω = Ω p + k ∓ kc B ; these modes are propagating but onlyaffected by the magnetic current, as previously remarked. The birefringent nature of thedispersion relations will be discussed a bit later since this free-field effect may interfere withthe presence of the background magnetic field.If c B → c ω → n → ε k (Ω) = 0 or ε L (Ω) = 0 or even ε R (Ω) = 0. The frequencies arisingfrom the previous conditions are cut-offs because, for given equilibrium conditions, theydefine frequencies above or below which the wave ceases to propagate at any angle ( k → v p = Ω /k → ∞ ). This is what happens, in particular, with the dispersionrelation of Eq. (4.5). Let us finally remark that the remaining two cut-offs stemming fromthe conditions ε L (Ω) = 0 and ε R (Ω) = 0 in Eqs. (4.2) are given, respectivey, by:Ω R = q Ω p + (Ω B + + Ω B − ) / − (Ω B + + Ω B − ) / , (4.6)Ω L = q Ω p + (Ω B + + Ω B − ) / B + + Ω B − ) / . (4.7) Taking the limit θ → ε k { n c B + [ n + n c ω ( ε + ε ) − ε R ]Ω }{ n c B − [ n + n c ω ( − ε + ε ) − ε L ]Ω } = 0 . (4.8)If ε k (Ω) = 0 we go back to the case of electrostatic oscillations. Therefore, assuming ε k (Ω) =0, Eq. (4.8) implies that the standard dispersion relations are modified as: n = 12Ω (cid:20) − c B − c ω ( ε + ε )Ω ± q ε R Ω + [ c B + c ω Ω( ε + ε )] (cid:21) , (4.9) n = 12Ω (cid:20) c B + c ω ( ε − ε )Ω ± q ε L Ω + [ c B + c ω Ω( ε − ε )] (cid:21) . (4.10) The positive square root has been chosen in Eq. (4.5) in order to get Ω >
0; we consider only positiveΩ since solutions with Ω < L -mode and R -mode are given, respectively,by: Ω ε R (Ω) = k + k [ c B + c ω ( ε + ε )Ω] , (4.11)Ω ε L (Ω) = k − k [ c B + c ω ( ε − ε )Ω] . (4.12)In the high-frequency limit (i.e. formally Ω → ∞ ) we have that c ω ( ε ± ε )Ω → c ω ( ε + ε )Ω = q c ω ( m + + m − ) (cid:20) ΩΩ B − − Ω + ΩΩ B + + Ω (cid:21) , (4.13) c ω ( ε − ε )Ω = q c ω ( m + + m − ) (cid:20) ΩΩ − Ω B + − ΩΩ + Ω B − (cid:21) . (4.14)The results of Eqs. (4.11)–(4.12) and (4.13)–(4.14) demonstrate, once more, that in thehigh-frequency limit of the spectrum the magnetic current dominates against the vorticalcurrent. For intermediate frequencies (i.e. as soon as we reduce Ω) the terms containing thenatural frequencies of the plasma come then into play so that for the R and L modes thecorresponding dispersion relations become:Ω = k + kc B + Ω p Ω(Ω − Ω B − ) , R − mode , Ω = k − kc B + Ω p Ω(Ω + Ω B − ) , L − mode . (4.15)As in the standard case, the phase velocity of the R -mode is greater than that of the L -mode.In Eq. (4.15) we assumed m + > m − and therefore Ω B + < Ω B − . In the limit k → R -mode cut-off occurs above Ω p while the L -mode cut-off occurs below Ω p (i.e., recallingEqs. (4.6) and (4.7), Ω → Ω R and Ω → Ω L ). In the low-frequency limit ε R and ε L coincideto leading order in (Ω / Ω B + ) and in (Ω / Ω B − ) sincelim Ω → ε R (Ω) = lim Ω → ε L (Ω) → p Ω B + Ω B − = 1 + 1 v A , v A = B q πn ( m + + m − ) , (4.16)where v A denotes the Alfv´en velocity of the system. In the low-frequency limit the dispersionrelations for the R -mode and for the L -mode are, respectively,Ω = v A v A (cid:26) k + k (cid:20) c B + q c ω m (cid:18) ΩΩ B + + ΩΩ B − (cid:19)(cid:21)(cid:27) , (4.17)Ω = v A v A (cid:26) k − k (cid:20) c B − q c ω m (cid:18) ΩΩ B + + ΩΩ B − (cid:19)(cid:21)(cid:27) , (4.18)since v A ≪ v A / (1 + v A ) ≃ v A .12aving determined the dispersion relations in the case of parallel propagation, the Fara-day rotation rate can be easily determined with the standard procedure. The generalizedFaraday rotation angle experienced by the linearly polarized radiation travelling parallel tothe magnetic field direction can be obtained as∆Φ = Ω2 (cid:26) c B Ω + c ω ε + s ε L + (cid:20) c B
2Ω + c ω ε + ε ) (cid:21) − s ε R + (cid:20) c B
2Ω + c ω ε − ε ) (cid:21) (cid:27) ∆L , (4.19)where ∆L is the distance travelled by the signal in the direction parallel to the magneticfield direction. It is interesting to compare the contribution of the terms depending upon c B and those depending upon the background magnetic field intensity, i.e. the terms appearingin the squared brackets. Recalling the expressions of ( ε R , ε L ) we have that Ω B + ≪ Ω B − and Ω p + ≪ Ω p − (always assuming m + ≫ m − ). In this case ∆Φ / ∆ L interpolates betweenthe standard result (Ω B − / p − / Ω) (valid when c B →
0) and the constant rotation rate c B / B → c ω is subdominant at high frequencies and can be neglected. By setting θ → π/ n ( ε L + ε R )Ω − n { c B ( ε L + ε R ) + c B c ω [ ε ( − ε L + ε R ) + ε ( ε L + ε R )]Ω+[ ε k ε R + ε L ( ε k + 2 ε R )]Ω } + 2 ε L ε R Ω ε k = 0 . (4.20)The solution of Eq. (4.20) can be obtained by first solving in terms of n . The result is n = J ( ε, Ω) ± q M ( ε, Ω)2( ε R + ε L ) Ω , (4.21) J ( ε, Ω) = c B ( ε L + ε R ) + c B c ω [ ε ( − ε L + ε R ) + ε ( ε L + ε R )]Ω+ [ ε k ε R + ε L ( ε k + 2 ε R )]Ω , (4.22) M ( ε, Ω) = − ε k ε R ε L ( ε L + ε R )Ω + { c B ( ε L + ε R )+ c B c ω [ ε ( ε R − ε L ) + ε ( ε L + ε R )]Ω + [ ε k ε R + ε L ( ε k + 2 ε R )]Ω } . (4.23)Equation (4.21) in the limit c ω → c B → n = ε k ) and to the extraordinary mode (i.e. n = 2 ε R ε L / ( ε R + ε L )) if wechoose the minus. In the high-frequency limit the terms multiplying the vortical current arealways negligible as already remarked in the case of the parallel propagation. The phenomenarelated to the oblique propagation will not be specifically discussed.13 .4 Spectrum around a hypermagnetic knot Introducing the three mutually orthogonal unit vectors ˆ a ( z, p ), ˆ b ( z, p ) and ˆ z defined in ap-pendix B, we can consider the modes of fluctuation of the hypermagnetic field around a fullyinhomogeneous background ~B ( t, ~x ), namely: ~B ( t, ~x ) = ~B ( t, ~x ) + δ ~B ( t, ~x ) . (4.24)Since the background solution is not uniform we can align ~B along ˆ a ( z, p ) and write that ~B ( z ) = B ˆ a ( z, p ). The background equations are solved by setting p = − c B (since ~ ∇× ~B = p ~B ). As in the homogeneous case the velocites vanish on the background solution. For L < /c B the background field is homogeneous and the previous analyses apply. For typicallength-scales larger than the scale of spatial variation of hypermagnetic knot (i.e. L ≫ /c B )there are two separate possibilities for the perturbed velocity field: either δ~v k ~B or δ~v ⊥ ~B .These two cases will now be separately examined.The case of parallel propagation mirrors exactly the one already discussed in the case ofuniform field. If we assume that δ~v k ~B the dispersion relations follow from { [ k − Ω ε k (Ω)] − c B k } ε k (Ω) = 0 . (4.25)The parallel dielectric tensor is ε k (Ω) = 1 − Ω p / [Ω(Ω + i Γ c )] where the correction comingfrom the collision rate has been added for immediate convenience. The solution ε k (Ω) = 0gives, as before, the electrostatic wave. The solution of { [ k − Ω ε k (Ω)] − c B k } = 0 gives,respectively, a high-frequency and a low-frequency branch. The high-frequency branch hasthe same dispersion relation of the free-field case, namely Ω ≃ Ω p + k ∓ kc B . The low-frequency branch is instead derived from the explicit form of the dispersion relation writtenas: Ω + i Γ c = Ω k (Ω + i Γ c ) − Ω p Ω k ± c B k (Ω + i Γ c ); (4.26)neglecting the first term at the right-hand side of the previous equation (which is unimportantat low frequencies) we have that:Ω = − i Γ c (1 ∓ c B /k )1 + Ω p /k ∓ c B /k . (4.27)The low-frequency mode, in which the conducting current dominates over the displacementcurrent, has no counterpart in vacuum. In the low-frequency mode, a small electric fieldproportional to Γ c exist to give the necessary current parallel to the magnetic field. Inthe limit Γ c → δ ~B ( t, z ) = δB ( t )ˆ b ( z, p ) + δB ( t )ˆ z, δ~v ( ± ) ( t, z ) = δv ( ± )1 ( t )ˆ b ( z, p ) + δv ( ± )2 ( t )ˆ z. (4.28)14or a generic velocity fluctuation orthogonal to ~B we have δ~v × ~B = [ B ( δ~v · ˆ z )ˆ b − B ( δ~v · ˆ b )ˆ z ];the solutions for δv ( ± )1 ( t ) and δv ( ± )2 ( t ) can then be expressed as: δv ( ± )1 (Ω) = qm ± (Ω B ± − Ω ) (cid:20) ± i Ω δE + Ω B ± δE (cid:21) ,δv ( ± )2 (Ω) = qm ± (Ω B ± − Ω ) (cid:20) ± i Ω δE − Ω B ± δE (cid:21) . (4.29)The dispersion relations in this case are given by ε R ε L − c B c ω (cid:20) ε L ( ε − ε ) + ε R ( ε + ε ) (cid:21) = 0 . (4.30)In the high-frequency limits defined by Eqs. (4.13)–(4.14), Eq. (4.30) is satisfied if ε R ε L = 0which is verified when either ε L or ε R are vanishing. Equation (4.30) leads to vanishing groupvelocity in the high-frequency regime: the corresponding modes are then not propagating.The proper frequencies defined by these equations have been already derived in Eqs. (4.11)and (4.12). The two-fluid equations can now be combined with the purpose of deriving the effective singlefluid description valid for sufficiently large length-scales and for frequencies much smallerthan Ω p and Ω B ± . The one-fluid variables are the total current ~J = q ( n + , ~v + − n − ~v − ), thebulk velocity of the plasma ~v = ( m + ~v + + m − ~v − ) / ( m + + m − ) and the total mass density ρ m = ( m + n + + m − n − ). In the globally neutral case ~J and ρ m become, respectively, ~J = n ( ~v + − ~v − ) and ρ m = n ( m + + m − ). Summing-up Eq. (3.7) (multiplied by m + ) and Eq.(3.8) (multiplied by m − ) the evolution equation for the bulk velocity of the plasma is ρ m (cid:20) ∂ t ~v + ~v · ~ ∇ ~v (cid:21) = ~J × ~B − ~ ∇ P + η ∇ ~v, (5.1)where the shear viscosity contribution, labeled by η , has been added for convenience .Taking the difference of Eqs. (3.7) and (3.8) (and assuming m + > m − ) the generalizedOhm’s law can be written as: ∂ t ~J + Γ c ~J ≃ Ω P − π (cid:18) ~E + ~v × ~B + ~ ∇ p − qn − ~J × ~Bqn (cid:19) , (5.2)where global neutrality has been assumed. Note that we have also kept the thermoelectricterm (depending on the pressure gradient of the lightest charge carriers) and the Hall term.Since we shall mainly consider the case of homogeneous pressures the thermoelectric termwill be neglected; the Hall term is a higher order contribution, as we shall argue. If the total pressure does not vanish Eq. (5.1) is modified as follows ∂ t [ w ~v ] + ( ~v · ~ ∇ )[ w ~v ] + ~v ~ ∇ · [ w~v ] = − ~ ∇ P + ~J × ~B + η ∇ ~v where w , as already discussed, is the enthalpy density.
15n the globally neutral case the single fluid equations stipulate that ~E , ~B and ~J are allsolenoidal (i.e. ~ ∇ · ~E = ~ ∇ · ~B = ~ ∇ · ~J = 0). A fourth possible solenoidal vector is the bulkvelocity of the plasma ~v . Indeed, since the evolution of ρ m and ρ q = q ( n + − n − ) is given by ∂ t ρ m + ~ ∇ · ( ρ m ~v ) = 0 , ∂ t ρ q + ~ ∇ · ~J = 0 , (5.3)the incompressible closure ~ ∇ · ~v = 0 will be adopted; consistently with the incompressibleclosure ρ m will be considered homogeneous, at least in the first part of this section. Afull discussion of other possible closure (such as the ones conventionally adopted in conven-tional plasmas) is desirable but beyond the scoped of this analysis. The remaining one-fluidequation containing the vortical and the magnetic currents, can be written as: ~ ∇ × ~B − ∂ t ~E = 4 π ~J + c ω ~ω − c B ~B, ~ω = ~ ∇ × ~v. (5.4)Since the one-fluid description involves the lowest branch of the spectrum we can neglectthe displacement current that becomes relevant only for the electromagnetic propagation.For the same reason we can neglect the time derivative in Eq. (5.2), i.e. ∂ t ~J ≪ Ω p ~E .Consequently Eqs. (5.2) and (5.4) in the low-frequency branch of the spectrum become ~ ∇ × ~B = 4 π ~J + c ω ~ω − c B ~B, ~E = ~J /σ − ~v × ~B. (5.5)Recalling that ~ ∇× ~E = − ∂ t ~B , Eq. (5.5) can be used to obtain an equation that is reminiscentof the magnetic diffusivity equation, namely ∂ t ~B = ~ ∇ × ( ~v × ~B ) + ∇ ~B πσ + 14 πσ ~ ∇ × ( c ω ~ω ) − πσ ~ ∇ × ( c B ~B ) . (5.6)Introducing the vorticity ~ω into Eq. (5.1) and dividing both sides of the equation by ρ m we obtain ∂ t ~v + ~ω × ~v = ~J × ~Bρ m − ~ ∇ (cid:20) Pρ m + v (cid:21) + ν kin ∇ ~v. (5.7)Taking the curl of Eq. (5.7) the evolution equation of the vorticity becomes: ∂ t ~ω = ~ ∇ × ( ~v × ~ω ) + ~ ∇ × ( ~J × ~B ) ρ m + ν kin ∇ ~ω. (5.8)The most interesting solutions of the one-fluid equations will involve the situations wherethe vortical and the magnetic currents play the dominant role. However, before turning theattention on these classes solutions it is useful to remark that the equilibrium solutions ofthe plasma at rest (i.e. ~v = 0) are simply given by ~ ∇ P = ~J × ~B, π ~J = (cid:18) ~ ∇ × ~B − c ω ~ω + c B ~B (cid:19) . (5.9) Note that the global neutrality implies ρ q = 0 and Eq. (5.3) demands ~ ∇ · ~J = 0 in full agreement withthe solenoidal nature of the total Ohmic current. π ~ ∇ P = ( ~ ∇ × ~B ) × ~B, ( ~B · ~ ∇ ) P = 0 , ( ~J · ~ ∇ ) P = 0 . (5.10)The relations of Eq. (5.10) are not explicitly modified by the presence of vortical andmagnetic currents. The two last conditions in Eq. (5.10) define the so-called magneticsurfaces: the pressure gradient vanishes along the lines of magnetic force and along thecurrent lines. We conclude that neither the vortical not the magnetic current affect directlythe equilibrium solutions. The single fluid AMHD equations admit various solutions that have no counterpart in thecase of ordinary MHD. Consider first the situation where the hypermagnetic magnetic fieldand the velocity are parallel and have non-vanishing magnetic and kinetic gyrotropy, i.e. ~v × ~B = 0 , ~v · ~ ∇ × ~v = p v ( t ) v , ~B · ~ ∇ × ~B = p B ( t ) B . (5.11)The simplest way to realize the situation described by Eq. (5.11) is to require that ~v and ~B are both Beltrami-like fields (see appendix B for this terminology) characterizedby ~ ∇ × ~v = p v ( t ) ~v and ~ ∇ × ~B = p B ( t ) ~B . Moreover, since ~v × ~B = 0, it is natural to requirethat p v ( t ) = p B ( t ). From Eq. (5.5) the total current ~J can be easily determined; the Ohmicelectric field is then given by: ~E = p B ( t ) + c B ( t )4 πσ ~B − c ω ( t )4 πσ ~ω. (5.12)From Eq. (5.6) the hypermagnetic field is obtained by solving the following equation ∂ t ~B = − p B ( t )[ p B ( t ) + c B ( t )]4 πσ ( t ) ~B + c ω ( t )4 πσ ( t ) p B ( t ) ~ω, (5.13)where ~ω ( t, z ) is the solution of Eq. (5.8). Thanks to the symmetries of the problem thesolution of this equation is given by ~ω ( t, z ) = ~ω ( z ) exp [ − R t p B ( t ′ ) ν kin ( t ′ ) dt ′ ], where ω ( z )is the initial vorticity which can also be written as ~ω ( z ) = p B (0) ~v ( z ). Equation (5.13) canthen be solved in general terms. However, recalling that c B ( t ) and c ω ( t ) are explicit functionsof time but they depend on the rescaled chemical potential since p B ( t ) is arbitrary we canchoose p B ( t ) = − c B ( t ). In this case, the solution of Eq. (5.13) shares the same properties ofthe general solution but it is mathematically simpler: ~B ( t, z ) = ~ω ( z ) Z t dt ′ c B ( t ′ ) c ω ( t ′ )4 πσ ( t ′ ) e − R t ′ c B ( t ′′ ) ν kin ( t ′′ ) dt ′′ . (5.14) In this discussion we shall keep the time-dependence in the kinetic coefficients even if, strictly speaking, c B and c ω may depend on the temperature and of the chemical potential but they are constant in time.However, in curved backgrounds a mild breaking of conformal invariance may induce a time dependencewhich is, however, not central to the present analysis. c ω and c B are bothconstant. Equation (5.14) can then be solved and the result is ~B ( z, t ) = − ~v ( z ) c ω π ν kin σ (cid:20) − e − ν kin c B t (cid:21) , (5.15)where we used that ~ω ( z ) = − c B ~v ( z ) when c B is constant in time and c B (0) = c B . This resultis also valid for a relativistic equation of state (i.e. w = 4 ρ/
3) provided the incompressibleclosure is consistently adopted and can be easily generalized to curved backgrounds. Whilethese generalizations are not germane to our theme it is worth to emphasize that in thelimit t → ∞ the suppression of the magnetic field is controlled by 4 πσν kin which is nothingbut the Prandtl number given as the ratio of the magnetic and of the kinetic Reynoldsnumber [2]. The Prandtl number is roughly independent on the temperature. For instanceat the electroweak epoch [10] we would have that ν kin ≃ / ( α ′ T ) while σ ≃ T /α ′ where α ′ = g ′ / (4 π ). Recalling the results of Eqs. (2.19) and (2.20) we therefore have thatlim t →∞ ~B ( t, z ) = c ω ( T )4 π α ′ ~v , c ω ( T ) = 2 T a ( µ R ) , (5.16)where a B ( µ R ) is the usual arbitrary function of the rescaled chemical potential µ R = µ R /T (see Eqs. (2.19) and (2.20)). The suppression due to the conductivity is therefore eliminatedand what is left is a milder suppression O ( α ′ ). In the same limit the hypermagnetic currentturns out to be more suppressed than the vortical current. As long as the inverse of thePrandtl number scales as α ′ the previous discussion is generally valid and this is whathappens in the case of the electroweak plasma [10] (see also [18] for specific estimates of theconductivity in the electroweak phase ). If the hypermagnetic field and the bulk velocity of the plasma are orthogonal (i.e. ~v · ~B = 0),employing the helical basis of appendix B the hypermagnetic field and the velocity field canbe written as: ~B ( t, z ) = B ( t )ˆ z + B ( t )ˆ a ( z, p ) , ~v ( z, t ) = v ( t )ˆ b ( z, p ) . (5.17)From Eq. (5.5) the AMHD current becomes: ~J ( t, z ) = B ( t )4 π [ c B + p ]ˆ a ( z, p ) − c ω π p v ( t )ˆ b ( z, p ) + c B B ( t )4 π ˆ z. (5.18)By analyzing the structure of the evolution equation of the magnetic field and of the vorticityit emerges that the system is consistent provided ∂ t B = 0 (i.e. constant magnetic field along Similar kinds of considerations s can also be developed in the case of a strongly interacting plasma aslong as the same scaling occurs [19]. z ) and provided c ω = 0. In this case the coupled evolution of the vorticity and of the magneticfield obeys dωdt = pc B B πρ m B − B πρ m p [ c B + p ] B , dB dt = ωB − pB πσ [ c B + p ] , (5.19)where ω ( t ) = p v ( t ) and ω ( z, t ) = ω ( t )ˆ b ( z, p ). The equations can be diagonalized with aspecific choice of the coordinate system. The simplest and most convenient one is p = − c B ;in this case the two equations can be combined by differentiating once Eq. (5.19). The resultis d B dt + c B v A B = 0 , v A = B √ πρ m , (5.20)where v A is the Alfv´en velocity. This solution has been swiftly presented in Ref. [7] andrecently rediscovered in [24]. Equation (5.20) describes the AMHD analog of the non-linearAlfv´en wave. The anomalous Alfv´en wave has been already discussed in section 3 as alow-frequency limit of the two-fluid equations. So far we considered small fluctuations of the chiral concentration around an otherwisehomogeneous value denoted by n in section 2. In the opposite case the AMHD equationsimply a specific relation between the concentration (or the chemical potential) and thetopological properties of the hypermagnetic fields. To illustrate this point we will showthat close to equilibrium the chemical potential is determined not only by the magneticgyrotropy but also by the total vorticity of the plasma. Hypermagnetic field configurationswith non-vanishing gyrotropy have been used to model the generation of the baryon orlepton asymmetry [10] (see also [25, 26, 27]). Consider, therefore, the evolution equation ofthe chemical potential which can be written as ∂ t µ R + Γ µ R = − µ ς A R ~E · ~B, (5.21)where Γ is the perturbative rate of he chirality flip processes (in the case of [10] it is deter-mined by the scattering of right electrons with the Higgs and gauge bosons and with thetop quarks because of their large Yukawa coupling). In Eq. (5.21) we also used the follow-ing general relation relation µ R = µ n R /ς where µ is a numerical factor depending on thespecific features of the plasma while ς is, as usual, the entropy density.To compute µ R in the proximity of an equilibrium situation we need to deduce thehyperelectric field. Recalling then Eq. (5.5), the hyperelectric field ca be related to the totalOhmic current so that Eq. (5.21) will become ∂ t µ R + Γ µ R − T a B ( µ R ) µ σς A R ~ω · ~B + 16 µ T µ R σς A R B = − µ πσς ~B · ~ ∇ × ~B. (5.22)19e shall now choose a B ( µ R ) = µ R ; if a B ( µ R ) = µ R in the evolution equation of the concen-tration we should add a further term proportional to ~ ∇ · [ c RB ( µ R ) ~B ]. This term vanishes inthe case a B ( µ R ) ∝ µ R : c RB ( µ R ) contains the derivative of a B ( µ R ) with respect to µ R and it istherefore constant. In more general situations Eq. (5.22) will just contain a supplementarycontribution of the type ~B · ~ ∇ µ R .The rescaled chemical potential enters the infinitely conducting limit (see appendix B)and the smallness of the particle asymmetries is the rationale for the minuteness of therescaled chemical potentials in approximate thermal equilibrium. At equilibrium, we candetermine µ R from Eq. (5.22) and the result is given by: µ R = − (cid:18) µ A R π ς σ (cid:19) ~B · ~ ∇ × ~B [Γ + Γ B − Γ ω ] , (5.23)While Γ is the perturbative chirality flip rate, the other terms can be understood as ratesstemming from the hypermagnetic current and from the vortical current and they areΓ B = 16 µ ς σ A R T B , Γ ω = 8 T µ π σ ς A R ~ω · ~B. (5.24)In the case of right electrons (see [10]) A R = − g ′ y R / (64 π ) where g ′ denotes the gaugecoupling and y R = − µ R = µ n R /ς and µ = 87 π N eff / N eff is the effective numberof relativistic degrees of freedom of the system . If the plasma is hypercharge neutral thevalue of the chemical potential can be estimated from the asymmetry in the case where allthe standard model charges are in complete thermal equilibrium. If all the asymmetry isattributed to the right electrons (which is, in some sense, the most favourable situation)then µ R = (87 π / N eff ( n R /ς ) where N eff = 106 .
75. With these specifications Eq. (5.23)becomes µ R = − α ′ π σ T ~B · ~ ∇ × ~B [Γ + Γ B − Γ ω ] , Γ B = 783 α ′ π σ B T , (5.25)where α ′ = g ′ / (4 π ). Equation (5.25) coincides with the previous results (see e.g. Eq. (6.15)of the last paper of [10] and see also [25, 26, 27]) in the limit Γ ω →
0. The results of Eq.(5.25) show that the final value of the chemical potential depends on the properties of theflow entering the definition of Γ ω . In summary we can say that the magnetic currents andthe vortical currents can affect a number of processes such as the formation of the baryonasymmetry or the dynamics of the electroweak phase transition. Similar kinds of effects canbe expected in the case of strongly interacting plasmas where the magnetic gyrotropy canalso determine the properties of the chemical potential. There have been a number of suggestions for possible roles that the abelian hypermagnetic Chern-Simonsterm might play in cosmology. One of them is related to the observation that right-handed electrons, whichdo not take part in weak interactions and also have a very small Yukawa coupling, are practically decoupledfrom the thermal ensemble above temperatures of about 10 TeV. Concluding remarks
The dispersion relations of anomalous magnetohydrodynamics are affected by the vorticaland the hypermagnetic currents. The vortical currents do not impact on the high-frequencybranch of the spectrum but the opposite is true at lower frequencies where new solutionsdescribe the simultaneous presence of hypermagnetic knots and fluid vortices. These parity-odd configurations carry, respectively, hypermagnetic and kinetic gyrotropy. The physicalproperties of the system roughly interpolate between the features of conventional chiralliquids and the results valid for cold electromagnetic plasmas. While chiral currents areanomalous and do not contribute to entropy production, vector currents are associated withthe generalized Joule heating.When chiral and Ohmic currents are simultaneously present the second law of thermody-namics constrains the kinetic coefficients. The hypermagnetic, vortical and Ohmic currentsaffect the evolution of the gauge fields and determine the hyperelectric field of the plasma.In anomalous magnetohydrodynamics the perfectly conducting limit is well posed and thehypermagnetic helicity of the knots is strictly conserved, as it happens in the case of conven-tional plasmas. The hypermagnetic currents are then completely washed out in the perfectlyconducting limit and strongly suppressed when the conductivity is large but finite. Close tothermal equilibrium the concentration of the chiral species and the corresponding chemicalpotential will depend not only on the hypermagnetic gyrotropy but also and on the vorticalcurrents.In summary the evolution equations of anomalous magnetohydrodynamics offer a minimaltheoretical framework where the interplay between conduction currents and chiral currentscan be quantitatively analyzed. It is therefore fair to say that the results derived herecomplement and extend some of the present and earlier strategies aimed at an improvedunderstanding of chiral liquids when generalized Ohmic effects cannot be neglected in theevolution of hypermagnetic and hyperelectric fields at finite fermionic density.21
Generalized Appleton-Hartree equation
A.1 Explicit form of ε s and ε v We are going to give, in what follows the explicit form of the dielectric tensors appearing insection 3. The matrix form of ε s (Ω) and ε v (Ω) is given by: ε s (Ω) = ε (Ω) i ε (Ω) 0 − iε (Ω) ε (Ω) 00 0 ε k (Ω) , ε v (Ω) = ε (Ω) − i ε (Ω) 0 iε (Ω) ε (Ω) 00 0 0 , (A.1)where ε (Ω), ε (Ω), ε (Ω), ε (Ω) and ε k (Ω) are defined as: ε (Ω) = 1 − Ω p + Ω − Ω B + − Ω p − Ω − Ω B − , (A.2) ε (Ω) = (cid:18) Ω B − Ω (cid:19) Ω p − Ω − Ω B − − (cid:18) Ω B + Ω (cid:19) Ω p + Ω − Ω B + . (A.3) ε k (Ω) = 1 − Ω p + Ω − Ω p − Ω , (A.4) ε (Ω) = q Ω( m + + m − ) (cid:20) B − − Ω − B + − Ω (cid:21) , (A.5) ε (Ω) = q ( m + + m − ) (cid:20) Ω B − Ω B − − Ω + Ω B + Ω B + − Ω (cid:21) . (A.6)Both ε (Ω) and ε (Ω) have dimensions of an inverse frequency squared; ε (Ω), ε (Ω) and ε k (Ω) are instead dimensionless. The frequencies appearing in Eqs. (A.2)–(A.4) and (A.5)–(A.6) are the plasma and the Larmor frequencies associated with the charge carriers ofboth signs, i.e. Ω p ± = √ πq n /m ± and Ω B ± = qB /m ± . To compare the dispersionrelations with the standard situation of cold plasmas we must bear in mind that the ratiosof the plasma and Larmor frequencies are related to the inverse ratio of the masses, i.e.Ω p + / Ω p − = Ω B + / Ω B − = m − /m + . A.2 The seven function
The generalized form of the Appleton-Hartree equation (see Eq. (4.1)) depends on 7 functionswhose explicit form is given by: f B ( ε, Ω , n, θ ) = ε k n Ω cos θ + ( ε L + ε R )2 n Ω sin θ,g B ( ε, Ω , n, θ ) = ε k ( ε L − ε R ) n Ω cos θ,f ω ( ε, Ω , n, θ ) = cos θn (cid:26) ( ε − ε )( ε + ε ) ε k + n (cid:20) sin θε − ( ε − ε ) sin θ (cid:21)(cid:27) , ω ( ε, Ω , n, θ ) = cos θ n (cid:26) ε k (cid:20) − n ε + ε ( ε L − ε R ) + ε ( ε L + ε R ) (cid:21) − n ε ε k cos θ + n ε ( ε L − ε R ) sin θ + n (cid:20) n ε − ε ( ε L − ε R ) − ε ( ε L + ε R ) (cid:21) sin θ (cid:27) ,h ( ε, Ω , n, θ ) = − ε n Ω cos θ sin θ,h ( ε, Ω , n, θ ) = 12 n Ω (cid:26) ε ε k cos θ + sin θ (cid:20) ε ( ε L + ε R ) + ε ( − ε L + ε R ) sin θ (cid:21)(cid:27) ,h ( ε, Ω , n, θ ) = − ε ε sin 2 θ (1 + sin θ )2 n Ω . (A.7)The functions reported in Eq. (A.7) determine, through Eq. (4.1), the form of the dispersionrelations when the hypermagnetic and the vortical currents are simultaneously present inthe anomalous magnetohydrodynamics equations. B Hypermagnetic knots and Beltrami fields
In the resistive approximation, the hyperelectric and the hypermagnetic fields are not exactlyorthogonal and the nature of this misalignment is crucial both for the generation of thebaryon asymmetry and for the chiral magnetic effect. In AMHD the induced hyperelectricfield stems directly from the approximate form of the Ohm’s law and it vanishes exactly, inthe plasma frame, when the conductivity goes formally to infinity. In the same limit thecontribution of the chemical potential to the anomalous hypermagnetic diffusivity equationgets always erased. At finite conductivity the anomalous contribution can be often rephrasedin terms of the magnetic gyrotropy [2] which defines hypermagnetic knot solutions [10].
B.1 Hypermagnetic knots
The configurations minimizing the hypermagnetic energy density with the constraint thatthe helicity be conserved coincide, in the perfectly conducting limit, with the ones obtainablein ideal magnetohydrodynamics where the anomalous currents are neglected [1, 2, 3, 4].In the perfectly conducting limit Eq. (5.6) leads to ∂ t ~B = ~ ∇ × ( ~v × ~B ) + O ( µ R /σ ) whichis qualitatively similar to the result of conventional magnetohydrdynamics. Defining thevector potential in the Coulomb gauge, the magnetic diffusivity equation becomes, up tosmall corrections, ∂ t ~A = ~v × ( ~ ∇ × ~A ). The analysis of Ref. [22] can then be exploited. Themagnetic energy density shall then be minimized in a finite volume under the assumptionof constant magnetic helicity by introducing the Lagrange multiplier p B . By taking thefunctional variation of G = R V d x {| ~ ∇ × ~A | − p B ~A · ( ~ ∇ × ~A ) } , with respect to ~A and byrequiring δ G = 0, the configurations extremizing G are such that ~ ∇× ~B = p B ~B . In performingthe functional variation we assumed that V is the fiducial volume of a closed system.The configurations ~ ∇ × ~B = p B ~B have been used to describe hypermagnetic knots (see[10], third and fourth papers); in this case q has dimensions of an inverse length and sets the23cale of the hypermagnetic knot which is related to Chern-Simons waves. The configurationswith constant p B represent the lowest state of magnetic energy which a closed system mayattain also in the case where anomalous currents are present, provided the ambient plasmais perfectly conducting. B.2 Gyrotropic bases
The knotted solutions can be expanded in an appropriate gyrotropic basis. Let us thenconsider a vector field ~a fields satisfying ~a × ( ~ ∇ × ~a ) = 0. The simplest realization ofthese Beltrami fields is provided by the eigenvectors of the curl operator but more generalsituations are know and have been extensively examined in the literature. Two gryrotropicand orthonormal bases of opposite parity are given by (ˆ a , ˆ b , ˆ z ) and by (ˆ c , ˆ d , ˆ z )ˆ a ( z, p ) = { cos pz, − sin pz, } , ˆ b ( z, p ) = { sin pz, cos pz, } , (B.1)ˆ c ( z, p ) = { cos pz, sin pz, } , ˆ d ( z, p ) = {− sin pz, cos pz, } , (B.2)As anticipated the bases of Eqs. (B.1) and (B.2) are orthonormal. Indeed we have ˆ a · ˆ b =ˆ a · ˆ z = ˆ b · ˆ z = 0 and (ˆ a × ˆ b ) · ˆ z = 1 (and similarly for ˆ c , ˆ d and ˆ z ). The unit vectors of Eqs.(B.1) and (B.2) are normalized eigenvectors of the curl operator with eigenvalues + p and − p .In ordinary MHD knotted solutions can be constructed from Beltrami fields by postulat-ing a solenoidal (static) current and by neglecting the displacement current. In anomalousmagnetohydrodynamics these simple constructions cannot be immediately extended becauseof the magnetic and vortical currents. The knot solutions obtainable by extremizing thefunctional G correspond to uniform magnetic fields well inside the core of the knot. Thisconclusion is evident if we use the basis of Eqs. (B.1) and (B.2). For instance in the limit pz < ~B ( z, p ) = B ˆ a ( z, p ) → B ˆ x is practically uniform and directedalong the ˆ x axis. The connections between Beltrami fields, force-free solutions in ordinaryMHD equilibrium and electromagnetic waves propagation have been explored in a number ofpapers [22, 23, 20, 21] starting from the classic works of Fermi and Chandrasekhar [28]. It isalso possible to obtain hypermagnetic knot solutions with finite helicity and finite gyrotropywhich do not satisfy the relation of Beltrami fields. These solutions have been studied in anumber of interesting frameworks (see last two papers of Ref. [10] and also [29, 30]).24 eferences [1] H. K. Moffat, Magnetic field generation in electrically conducting fluids , (CambridgeUniversity Press, Cambridge 1978).[2] D. Biskamp,
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