Anomalous Hall instability in the Chern-Simons magnetohydrodynamics
AAnomalous Hall instability in the Chern-Simons magnetohydrodynamics
M. Kiamari, M. Rahbardar, a M. Shokri, b and N. Sadooghi ca,c Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran b School of Particles and Accelerators, IPM, P.O. Box 19395-5531, Tehran, Iran
E-mail: [email protected], [email protected],[email protected] (Corresponding author), [email protected]
Abstract:
The Chern-Simons magnetohydrodynamics (CSMHD) is introduced usinga Maxwell-Chern-Simons (MCS) Lagrangian including an axion-like field Θ. The MCSequation of motion derived from this Lagrangian consists of a modified current, includinga chiral magnetic (CM) and an anomalous Hall (AH) current, in addition to the ordinaryOhm current of resistive magnetohydrodynamics (MHD). The former consists of an axialchemical potential, which is given in terms of the temporal comoving derivative of Θ,and the latter arises from the spatial gradient of Θ. As it turns out, the existence of theaxial chemical potential is a nonequilibrium effect that plays no role in the linear stabilityanalysis, whereas the AH current arises as in the first-order linear perturbation of thethermal equilibrium. We analyze the linear stability and causality of the CSMHD in aresistive and chiral medium. We show that the Alfvén modes propagating sufficiently closeto the direction of the magnetic field are unstable but causal. They are also accompaniedby a genuine nonhydro mode. A stable mode in a particular direction can correspond toan unstable mode propagating in the exact opposite direction. The AH instability is amanifestation of a breakdown of the parity. A numerical analysis of the phase velocityconfirms these results. a r X i v : . [ h e p - t h ] F e b ontents B.1 Nonchiral channel 20B.2 Chiral channel 21
The successes of relativistic hydrodynamics [1, 2] in explaining the experimental data fromthe heavy-ion collisions [3–5] have led to overgrowing theoretical attention toward it [6].In particular, the stability and causality of dissipative relativistic hydrodynamics [7–9], iscrucial in numerical simulations. The unreasonable effectiveness of stable and causal for-mulations of dissipative relativistic hydrodynamics outside the equilibrium [10], motivatedthe discovery of hydrodynamics attractors [11]. Relativistic hydrodynamics is a universaltheory in the sense that the underlying microscopic nature of the system appears onlythrough different transport coefficients [12]. The same remark holds for MHD, which cou-ples the dynamics of a conducting fluid with the Maxwell equations [13]. For the fluids inwhich the magnetic Reynolds number is small [14], a resistive formulation of relativisticmagnetohydrodynamic (RMHD) is well motivated. Resistive RMHD can also be formu-lated on universal grounds [15]. However, a macroscopic description of certain phenomena– 1 –ith quantum nature cannot be incorporated into the ordinary formulation of relativis-tic hydrodynamics through a mere modification of transport coefficients. In particular,anomalous transport in the chiral matter and macroscopic effects of spin are so [16–19].In a chiral fluid the number of left- and right-handed fermions is locally unequal. Inthe quark gluon plasma (QGP), such chiral imbalance is a result of the interplay of thechiral anomaly and nontrivial gluon configurations [20, 21]. To describe this fluid usingthe chiral MHD theory, one introduces new terms into the current [16, 22]. The new termsmay at least include a CM vector current and an axial current. These modifications of thecurrent give rise to new terms in the entropy current [17]. The second law of thermody-namics is then employed to find constraints on the new transport coefficients, includingthe anomalous ones [17, 19]. The non-relativistic and relativistic chiral MHD is vastlyinvestigated in the literature [22–26]. In particular, the existence of unstable propagatingmodes has been studied in [22–24].In the present work, we take another approach based on the MCS Lagrangian. TheMCS Lagrangian adds a topological term, including an axion-like field, to the ordinaryMaxwell theory [27]. The axion-like field connects the electromagnetic (EM) fields to thetopological properties of the matter, and gives rise to the equation of axion electrody-namics. Novel phenomena such as the Chiral magnetic effect (CME) [20, 21], the Witteneffect [28], and photon’s topological mass naturally arise from the axion electrodynam-ics [27, 29–31]. One can also extract an extension of the MHD from MCS theory thatincludes anomalous transport without an ad-hoc modification of the current [26, 32, 33].As a result, and in contrast to the chiral MHD, there is no axial current in this so-calledCSMHD. It has been shown that the second law of thermodynamics in MCS theory is aconsequence of the modification of thermodynamic relations in the presence of the axion-like field [32, 33]. Following this observation, we derive the entropy current, and definethe equilibrium state of an electrically neutral fluid. We then perturb the equilibrium toinvestigate the stability and causality of CSMHD. Although in the resistive MHD, theelectric conductivity may not be large enough to suppress the electric field and ensureneutrality, we assume the electric field and vector chemical potential vanish in equilib-rium. Such a power-counting scheme is referred to as the weak electric field regime (see[34, 35]). The axial chemical potential, defined by the time derivative of the axion-likefield, is a non-equilibrium effect, and does not appear in the linear stability treatment.However, the spatial inhomogeneity of the axion-like field at a macroscopic level leads to anovel AH effect at the first order of fluctuations [26, 27]. The combination of the AH andOhm currents are effectively understood through the definition of an effective conduc-tivity, which, as it turns out, does not have a definite sign. We show that its sign andvalue depend on the direction that a mode propagates. The indefiniteness of the effectiveconductivity sign gives rise to unstable but causal Alfvén modes originally introduced in[36], propagating close to the plane transverse to the magnetic field. We also show thatthe Alfvén modes are also manifestly affected by the breakdown of rotational and parity– 2 –ymmetries. In addition to the modification of magnetosonic and Alfvén waves, we findtwo genuine nonhydro (gapped) modes. In contrast to the Alfvén modes, the magnetosonicand their accompanying nonhydro mode are linearly stable for the full range of wavenum-bers, and are not affected by the chirality. Unsurprisingly, the magnetosonic modes aredamped by resistivity. Although the emergence of the unstable mode has been observedin chiral MHD [22], the source and nature of AH instability discussed in the present paperare different: Whereas in chiral MHD, the instability is induced by the axial charge, inCSMHD, it is a result of the change of topological charges in neighboring domains. We alsodo not employ any approximating constraint on the values of wavenumbers and electricalresistivity.The organization of this paper is as follows. In Sec. 2, we review the equations ofmotion in the MCS theory and its thermodynamics. Then, we fix the hydrostatic configu-ration. In Sec. 3, we implement linear treatment to investigate the stability and causalityof the theory. In Sec. 4, we present a numerical investigation of the phase velocities andthe imaginary parts of the different modes, The paper is concluded in Sec. 5. We use thenatural units in which (cid:126) = c = k = 1. The convention of the metric signature is mostlyminus, namely g µν = diag (+1 , − , − , − In this section, we review the equations of the CSMHD theory and derive the stationarysolution to its EOM in thermal equilibrium. We refer to this solution as the hydrostaticconfiguration of CSMHD. This theory is based on the MCS Lagrangian density [27] L MCS = L Maxwell + L CS , (2.1)with L Maxwell ≡ − F µν F µν − A µ J µ , and L CS ≡ − C A F µν ? F µν . In L CS , Θ( x ) is the axion-like field, and the field strength tensor F µν and its dual ? F µν aregiven by F µν = ∂ µ A ν − ∂ ν A µ , and ? F µν = 12 (cid:15) µναβ F αβ . (2.2)The anomaly coefficient C A before the topological term F µν ? F µν in (2.1) reads C A = N C X f q f e π , (2.3)with P f being the summation over the quark flavors with charge q f and N C the numberof colors. The number of the quark flavors depend on the energy scale of the system– 3 –nder consideration. In (2.1), J µ is the ordinary electromagnetic (EM ) current, and isdetermined by taking the functional derivative of L Maxwell with respect to the EM source A µ . Taking, however, the variation of the full MCS action L MCS with respect to A µ , anadditional term proportional to C A appears in the resulting current J µ , J µ = J µ + C A? F µν P ν , (2.4)where P µ ≡ ∂ µ Θ. The emergence of J µ is a consequence of the spacetime dependencyof Θ, and gives rise to a modification of the homogeneous and inhomogeneous Maxwellequations, ∂ µ? F µν = 0 , and ∂ µ F µν = J ν , (2.5)as well as the energy-momentum conservation relation, ∂ µ T µν Fluid = F νλ J λ − C A (cid:16) F αβ? F αβ (cid:17) P ν . (2.6)Here, T µν Fluid is the fluid energy-momentum tensor (see [26] for some details on the derivationof (2.6)). The inhomogeneous Maxwell equation leads to ∂ µ J µ = 0. Using the homoge-neous Maxwell equation, the Θ dependent part in J µ trivially vanishes, and we are leftwith ∂ µ J µ = 0.As it is argued in [27], the comoving temporal derivative of the axion-like field givesrise to the chiral chemical potential, µ ≡ u µ P µ , (2.7)while, according to [26], its comoving spatial gradient produces the AH current J µ AH = C A (cid:15) µναβ E ν u α P β . (2.8)Decomposing F µν and its dual tensor appearing in (2.4) in terms of the electric and mag-netic field according to [13] F µν = E µ u ν − E ν u µ − (cid:15) µναβ B α u β , ? F µν = B µ u ν − B ν u µ + (cid:15) µναβ E α u β , (2.9)where u µ is the fluid velocity, and using (2.7) the CM current C A µ B µ , and the AH current C A E µν P ν emerge as a part of J µ [26, 27]. Here, E µν is defined as E µν ≡ (cid:15) µναβ E α u β . Before we deduce the hydrostatic configuration, we need to understand the thermody-namics of the MCS theory. To this purpose, we follow the variational method utilized in[37, 38]. We start with the effective action of the fluid, the axion-like field Θ, and theEM fields [39] S = Z d x [ − (cid:15) ( s, n e , Θ) + L MCS ] . (2.10) See Appendix A for the definition of the comoving temporal and spatial derivatives. – 4 –ere, (cid:15) and s are the energy and entropy densities, and n e ≡ u µ J µ . We assume thatthe fluid has no other conserved current but J µ . This is a common assumption at hightemperatures. In our definition of n e , we have replaced J µ with J µ since, as previouslystated, the divergence of the Θ-dependent part of the former is trivial. The derivatives of (cid:15) with respect to its variables are defined as [32] (cid:18) ∂(cid:15)∂s (cid:19) n e , Θ ≡ T, (cid:18) ∂(cid:15)∂n e (cid:19) s, Θ ≡ µ e , (cid:18) ∂(cid:15)∂ Θ (cid:19) n e ,s ≡ R Θ . (2.11)They lead to the first law of thermodynamics,d (cid:15) = T d s + µ e d n e + R Θ dΘ . (2.12)In a thermal equilibrium, T and µ e are identical to the temperature and chemical po-tential, respectively. We apply the variational principle to the action under the followingconstraints: u µ u µ = 1 , ∂ µ J µ = 0 , ∂ µ S µ = 0 . (2.13)Here, S µ is the entropy current. The first constraint comes from the requirement that u µ must be timelike. The second one is a consequence of (2.5), and the third one of theconditions of thermal equilibrium. For later convenience, we decompose S µ and J µ paralleland perpendicular to u µ S µ = su µ + ∆ µν S ν , and J µ = n e u µ + ∆ µν J ν , (2.14)where ∆ µν is defined in Appendix A. As in [37, 38], we introduce an effective Lagrangian,with Lagrange multipliers λ , ξ , and w , that enforces the constraints (2.13) L eff. = − (cid:15) ( s, n e , Θ) + L MCS + λ∂ µ J µ + ξ∂ µ S µ − w ( u µ u µ − . (2.15)Integrating by part, the effective Lagrangian is rewritten as L eff. = − (cid:15) ( s, n e , Θ) − F µν F µν − C A F µν ? F µν − ( n e u µ + ∆ µν J ν ) ( A µ + ∂ µ λ ) − ( su µ + ∆ µν S ν ) ∂ µ ξ − w ( u µ u µ − . (2.16)The variations of the effective Lagrangian with respect to u µ , Θ, n e , and s give rise to wu µ = − n e ( A µ + ∂ µ λ ) − s∂ µ ξ, (2.17) R Θ = − C A F µν ? F µν , (2.18) µ e = − u µ ( A µ + ∂ µ λ ) , (2.19) T = − u µ ∂ µ ξ. (2.20)We note that (2.19) is consistent with the results of [40] with an overall change of sign dueto different metric signature conventions. Plugging (2.20) and (2.19) into (2.17), we find w = T s + µn e . (2.21)– 5 –ssuming at this stage that the fields which are present in the CS Lagrangian are back-ground fields, the full partition function of the theory is given by Z = Z D Φ exp Z β d τ Z d x ( L CS + L [Φ]) ! = exp (cid:18) − VT L CS (cid:19) Z Φ . (2.22)Here, Φ stands generically for all the other fluctuating fields, and Z Φ is their correspondingpartition function. Focusing particularly on the Θ dependent part of the Lagrangian, thecorresponding thermodynamic potential Ω is thus given byΩ( T, µ e , Θ) = − TV ln Z = L CS + · · · , (2.23)gives rise to (cid:18) ∂ Ω ∂ Θ (cid:19) µ e ,T = R Θ . (2.24)Using p = − Ω [41], the definition of L CS from (2.1), and the constraint relation (2.18), theGibbs-Duhem relation is respectively modified as [32, 38]d p = s d T + n e d µ e −R Θ dΘ . (2.25)Combining this expression with (2.12), we recognize w in (2.21) as fluid’s specific enthalpydensity, i.e. w = (cid:15) + p . According to (2.24), w is independent of Θ, and the ordinaryrelation for entropy density is thus not modified, s = 1 T ( (cid:15) + p − n e µ e ) . (2.26) We are now in a position to fix the hydrostatic equilibrium state [8]. We start from thecovariant generalization of the thermodynamic identity (2.26), which reads [8, 34, 35, 42,43] S µ = pβ µ + T µν Fluid β ν − αJ µ , (2.27)where α ≡ µ/T , and β µ ≡ u µ /T . Taking the divergence of S µ and using (2.6) as well as ∂ µ J µ = 0 from (2.13) gives rise to ∂ µ S µ = ∂ µ pβ µ + p∂ µ β µ + (cid:18) F νλ J λ − C A (cid:16) F αβ? F αβ (cid:17) P ν (cid:19) β ν + T µν Fluid ∂ µ β ν − J µ ∂ µ α = ∂ µ pβ µ + p∂ µ β µ + 12 T µν Fluid ( ∂ µ β ν + ∂ ν β µ ) − C A (cid:16) F αβ? F αβ (cid:17) β λ P λ − J µ (cid:20) E µ T + ∂ µ α (cid:21) . (2.28)In thermal equilibrium, the divergence of S µ must vanish. This is obtained if β µ is thesymmetry of the hydrostatic equilibrium state, in the sense that the Lie derivative of everyphysical quantity vanishes [44]. Hence, β is a Killing vector [43] ∂ µ β ν + ∂ ν β µ = 0 . (2.29)– 6 –his immediately eliminates T µν Fluid ( ∂ µ β ν + ∂ ν β µ ), and ∂ µ β µ in (2.28). Noting that the Liederivative of a scalar φ with respect to β µ is simply β µ ∂ µ φ , β λ P λ = β λ ∂ λ Θ also vanishes.This implies, µ = ∂ Θ = 0 in LRF of the fluid. For the last term of (2.28) to vanish it issufficient that E µ = − T ∂ µ α. (2.30)Although, the β -symmetry leads to the time-independency of hydrodynamic variables inthe LRF of the fluid, their spatial gradients are not necessarily vanishing [40]. This is why,we can assume that P = ∇ Θ in P µ = (0 , P ) does not vanish in equilibrium. Let us alsonotice that if there is no chemical potential other than µ present, the electric four-vectorvanishes at equilibrium. Hereafter, we assume that this is the case. The general solutionto the (2.29) reads [43, 45] β µ ( x ) = u µ T + ω µν x ν , (2.31)where ω µν is the thermal vorticity tensor defined as ω µν ≡ − ( ∂ µ β ν − ∂ ν β µ ). Assumingthat in the hydrostatic configuration the thermal vorticity is zero, the solution reduces totime-independent temperature and four-velocity. With the above considerations, for thehydrostatic equilibrium in the LRF, we have (cid:15) = (cid:15) , p = p , s = s , T = T , u µ = (1 , ) ,B µ = (0 , B ) , E µ = 0 , µ = 0 , ∇ Θ = P , Θ = Θ . (2.32)The subscript 0 is used to denote that the quantities are constants. One can check thatthe configuration of (2.32) satisfies (2.6). Hereafter, we assume the following form for J µ J µ = n e u µ + σ e E µ , (2.33)in which σ e is the electric conductivity. Contracting the inhomogeneous MCS equation,i.e. the second equation in (2.5), with β ν , and using (2.32), we find u µ J µ = 0. Using thedefinition of J µ from (2.4), with J µ from (2.33), we arrive at n e, = C A P · B . (2.34)The above equation suggests that in CSMHD, the local charge density can be nonzero inequilibrium. However, a nonvanishing n e, requires a corresponding nonvanishing chemicalpotential µ e . Since we have already assumed that such a chemical potential does not exist,we also need to assume that the local charge density is zero. By this virtue n e, = 0 , and P · B = 0 . (2.35)We should emphasize that in contrast to [32, 38], our setup is dissipative. Out of equilib-rium, the entropy production is governed by the following relation T ∂ µ S µ = σ e E . (2.36) We note that it is legitimate, and sometimes fruitful [9, 46], to consider the fluctuations from a movingobserver’s perspective for which the fluid four-velocity reads u µ = γ (1 , v ). We come back to this issue inSec. 3. – 7 – Collective modes of the CSMHD
In this section, we find the collective mode of the MCS theory in the LRF of the fluid. Tofind the collective modes, we introduce perturbations to the hydrostatic configuration of(2.32) [7, 8]. We then solve the EOM, up to first order in perturbations ∂ µ? δF µν = O ( δ ) , ∂ µ δF µν − δ J ν = O ( δ ) ,∂ µ δT µν Fluid − δ (cid:16) F νλ J λ (cid:17) + C A δ h(cid:16) F αβ? F αβ (cid:17) P ν i = O ( δ ) . (3.1)We assume perturbations around the hydrostatic configuration of the form δ ˜ X ∼ δX exp( − iωt + i k · x ) , for each hydrodyanmic variable X . Plugging the perturbed variables in (3.1), and keepingterms up to the first order in perturbations, gives rise to a system of linear equations as [46] M δX = 0 , (3.2)where M is the matrix of coefficients and δX the unknown perturbative variables. Equa-tion (3.2) is a polynomial equation whose solutions of form ω = ω ( k ) are the so-calledmodes of theory. The modes are called (non)hydro modes , if ω ( k ) is (not) zero for k = .Modes are stable if Im( ω ( k )) <
0, for all values of k , and they are causal if [9, 46]lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re( ω ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ , (3.3)wherein k ≡ √ k . k . Equation (3.2) is not analytically solvable for all modes. To investigatetheir stability, we thus use the Routh-Hurwitz stability criterion [47–49].Employing the MCS equations, it is possible to reduce the number of δX to δX = ( δT, δu x , δu y , δu z ) , (3.4)and, in particular, δp = w T δT, δ(cid:15) = w v s T δT, δn e = 0 . (3.5)Here, v s is the speed of sound, and w = (cid:15) + p . In (3.5), the first and second equationsarise from (2.12) and (2.25). The last relation corresponds to the fact that a nonvanishing µ e, is required for the charge density fluctuation to be physically possible. Before weproceed, it is necessary to understand how the gradient of Θ appears in equilibrium. Todo this, we remind that the scale at which the axion changes is much larger than thescale of the fluctuations. We thus consider P as a constant in the linear analysis. Thesame is also true for the gauge potential A ,µ in equilibrium; if one assumes that A ,µ has nonvanishing second-order derivatives, and hence B has nonvanishing gradients, the (Non)hydro modes are also called (gapped)gapless modes. – 8 –lfvén and magnetosonic excitations disappear. By this virtue, we let Θ = P · x . Further,assuming that B = B ˆ y in the LRF, we haveΘ = P · x + δ Θ , with P ≡ ( P x , , P z ) ,u µ = (1 , − δv ) , with δv = ( δu x , δu y , δu z ) ,B µ = B ( δu y , , − ,
0) + (0 , − δB ) , with δB ≡ ( δB x , δB y , δB z ) ,E = (0 , − δE ) , with δE ≡ ( δE x , δE y , δE z ) , k = k (sin θ cos φ, cos θ, sin θ sin φ ) , with k = √ k . k . (3.6)Here, θ is the polar angle with the zenith direction taken parallel to B , and φ is the az-imuthal angle defined in the xz plane. Plugging (3.6) into the homogeneous MCS equation[first equation in (2.5)], and keeping terms up to the first-order, gives rise to δB x = − kω (cid:18) ( B δu x − δE z ) cos θ + δE y sin φ sin θ (cid:19) ,δB y = k sin θω (cid:18) ( B δu x − δE z ) cos φ + ( δE x + B δu z ) sin φ (cid:19) ,δB z = − kω (cid:18) ( B δu z + δE x ) cos θ − δE y cos φ sin θ (cid:19) . (3.7)Plugging the above results into (2.4), the total current J µ is found to be J = C A kω (cid:18) P × δE + ( P · δv − iωδ Θ) B (cid:19) · ˆ k (cid:107) + (cid:16) ˆ k ⊥ × P (cid:17) · δE , J x = − σ e δE x − C A ( P × δE ) · ˆ x , J y = − σ e δE y − C A (cid:18) ( P × δE ) · ˆ y + ( P · δv − iωδ Θ) B (cid:19) , J z = − σ e δE z − C A ( P × δE ) · ˆ z . (3.8)Here, ˆ k ⊥ and ˆ k (cid:107) are the perpendicular and parallel parts of the wavenumber unit vector ˆ k ≡ k /k with respect to B , ˆ k ⊥ ≡ (cos φ, , sin φ ) sin θ , and ˆ k (cid:107) ≡ (0 , ,
0) cos θ. As a next step, we solve the inhomogeneous MCS equation [second equation in (2.5)] to re-duce the unknown perturbations to the four given in (3.4). The results are too cumbersometo be reproduced here. Plugging then the resulting expressions into the energy-momentumconservation relation (2.6), leads to an equation of form (3.2). For this equation to besolvable, the determinant of M must vanish. We decompose the determinant into twochannels: det( M ) = s w D NC D C , (3.9)– 9 –here the channels read D NC = B σ e (cid:16) ω − v s k (cid:17) (cid:16) ω − k (cid:17) − w ω (cid:16) iω − σ e ω − ik (cid:17) (cid:16) ω − v s k (cid:17) − B v s k σ e (cid:16) ω − k (cid:17) cos(2 θ ) , (3.10) D C = 8 cos θ B σ e k (cid:18) − sin θ (cid:19) + (cid:20) w ω (cid:16) iω − σ e ω − ik (cid:17) − B σ e (cid:18) ω − k (cid:19)(cid:21) +8 C A (cid:16) w + B (cid:17) (cid:18) ω − B B + w k cos θ (cid:19) (cid:16) ˆ k ⊥ × P (cid:17) · ˆ y . (3.11)The first channel, i.e. D NC , is independent of the CS term in the Lagrangian (2.1), andwe call it nonchiral. By the same virtue, we refer to the second one, i.e. D C , as the chiralchannel. In particular, P appears only in D C . Before we proceed, we may benchmark our method with the known results of iMHD in[36]. To do so, we expand det( M ) from (3.9) in powers of σ e , and keep the highest orderterm. This gives rise to (cid:18) B B + w (cid:16) ω − k (cid:17) (cid:16) ω − v s k cos θ (cid:17) + w B + w ω (cid:16) ω − v s k (cid:17) (cid:19) × (cid:18) B B + w (cid:16) ω − k cos θ (cid:17) + w B + w ω (cid:19) = 0 . (3.12)The first two eigenfrequencies are the relativistic Alfvén modes ω A , ± = ± v a k cos θ , (3.13)with the Alfvén speed v a , defined by v a ≡ B B + w . (3.14)We identify four other eigenfrequencies as the frequencies of slow ( ω sms , ± ) and fast ( ω fms , ± )magnetosonic modes ω sms , ± = ± v s k √A − B , ω fms , ± = ± v s k √A + B , (3.15)with A ≡ (cid:18) − v a sin θ + v a v s (cid:19) , B ≡ vuut A − v a cos θv s ! . (3.16)In the absence of the magnetic field, all eigenfrequencies vanish, except those of the fastmagnetosonic modes. The latter reduces to the sound mode in the perfect fluid, ω A , ± = 0 , ω sms , ± = 0 ω fms , ± = ± v s k . Even in the presence of the magnetic field, the Alfvén and slow magnetosonic modes donot propagate in the plane perpendicular to the magnetic field, i.e. θ = π/ ω A , ± = 0 , ω sms , ± = 0 , ω fms , ± = ± v s k s − v a + (cid:18) v a v s (cid:19) . – 10 – .2 Nonchiral channel We now analyze the nonchiral channel. This channel is a polynomial of order five, andaccording to Abel’s impossibility theorem [50], an exact solution for the correspondingeigenfrequencies cannot be obtained. The nature of these modes can, however, be revealedusing a long-wavelength expansion ω sms , ± = ± v s k √A − B − i (1 − v a ) (1 − v s ( A − B )) (1 − A + B )4 σ e B k + O (cid:16) k (cid:17) ,ω fms , ± = ± v s k √A + B − i (1 − v a ) (1 − v s ( A + B )) ( A − B )4 σ e B k + O (cid:16) k (cid:17) ,ω NC , nh = − iσ e − v a + i (1 − v a ) (1 + v s (1 − A )) σ e k + O (cid:16) k (cid:17) . (3.17)The first two eigenfrequencies belong to the slow and fast magnetosonic nonchiral (NC)eigenfrequencies modified by the presence of finite electrical conductivity. As (3.17) sug-gests, the resistivity 1 /σ e damps the magnetosonic modes. The third NC eigenfrequencyis genuine and contains a nonhydro part. Due to the positivity of the electrical conduc-tivity, which is required by the second law of thermodynamics [12], the nonhydro mode isstable. A thorough examination of its linear stability using the Routh-Hurwitz criteria ispresented in Appendix B. The analysis proves that the nonchiral channel is linearly stable.To check whether this channel is causal, we use the asymptotic causality condition(3.3). Expanding D NC in terms of k , and keeping the highest order term gives rise to D NC ∼ ik v g (cid:16) v g − v s (cid:17) (cid:16) v g − (cid:17) w , (3.18)where v g is the group velocity, which in k → ∞ is given by v g ∼ ω/k . Setting (3.18) equalto zero, it turns out that the asymptotic group velocity does not exceed the speed of light.This indicates that the chiral channel is causal. In what follows, we investigate the chiral channel. To do this, let us first emphasize that theeigenfrequencies of this channel vanish in the direction of the magnetic field, i.e. θ = π/ P = P (cos( φ − ∆) , , sin( φ − ∆)) = ⇒ (cid:16) ˆ k ⊥ × P (cid:17) · ˆ y = P sin ∆ sin θ . (3.19)Here, ∆ is the angle between the vector P and the transverse wavenumber vector ˆ k ⊥ .Assuming 0 ≤ ∆ < π , we can constraint P to be positive. Although this channel ispolynomial of order three, its exact solution is too complicated to be useful. Therefore,similarly to the nonchiral case, we perform a long-wavelength expansion to obtain ω A , ± = ± v a k cos θ − i (1 − v a )(1 − v a cos θ )2 ( σ e − C A P sin ∆ tan θ ) k ± (1 − v a ) (1 − v a cos θ ) (1 − v a cos θ )8 v a cos θ ( σ e − C A P sin ∆ tan θ ) k + O (cid:16) k (cid:17) ,ω C , nh = − i ( σ e − C A P sin ∆ tan θ )1 − v a − i (1 − v a )(1 − v a cos θ )( σ e − C A P sin ∆ tan θ ) k + O (cid:16) k (cid:17) . (3.20)– 11 – comparison with (3.17), suggests the definition of an effective conductivity which mixesthe Ohmic and the AH conductivities, σ eff . ≡ σ e − C A P sin ∆ tan θ . (3.21)Let us assume 0 ≤ θ < π/
2. For 0 ≤ ∆ < π , the AH current is in the opposite directionof the Ohm one. Since tan θ is unbounded, for any value of P , there exists a critical valuefor θ such that for polar angles larger than that σ eff . becomes negative. Consequently, thenonhydro mode ω C , nh becomes unstable, and the Alfvén modes ω A , ± are amplified. On theother hand, for π ≤ ∆ < π , the AH current enhances the Ohm current. Therefore, thenonhydro mode is stable, and the Alfvén modes are damped. The same remarks hold for π/ < θ ≤ π , with modes in 0 ≤ ∆ < π interval being stable and the ones in π ≤ ∆ < π being stable. In summary, a mode propagating in some angle θ might be stable while itsmirrored one in the opposite angle is unstable. This is a manifestation of parity symmetrybreaking caused by the CP violating Chern-Simons term. The Routh-Hurwitz analysis inAppendix B confirms these remarks. π π π π - - π π π πθ C A P s i n Δ / σ e - - - Figure 1 . Depiction of σ eff . cos θ/σ e . The horizontal axes is the polar angle θ , and the verticalone is C A P sin ∆ /σ e . As the figure suggests, for any value of P there exists a range of θ and ∆for which σ eff . cos θ < As in the nonchiral case, we check the causality of the chiral channel using (3.3). Theleading term for the short wavelength expansion reads D C ∼ ik v g (cid:16) v g − (cid:17) w , (3.22)which implies that the chiral channel is causal. The causality of the chiral channel hascrucial consequences. First, it implies that the linear stability of the system in a moving– 12 – igure 2 . The schematic view of the unstable regions in space. The cone is defined as θ = θ c ,wherein θ c is the angle at which the effective conductivity (3.21) vanishes. The upper quarter isthe interval θ c ≤ θ < π/ π/ < θ ≤ π − θ c (blue). The chiralchannel is unstable within these two quarters and stable outside them. frame is similar to the LRF. Second, AH instability is not fictitious. One may be temptedto write a relaxation equation for the current to remove the instability. However, such anapproach does not work. Let us remind that the relaxation time approach is essentiallyemployed to avoid instantaneous propagation of signals. The CSMHD modes are, however,causal. Hence a relaxation time approach seems to be useless. We nevertheless use thefollowing ansatz to check whether it can cure the instability problem of this mode τ J ∆ µν D J ν + J µ = σ e E µ . (3.23)Let us note that since the axionic part of the current is dissipationless, a relaxation equationcan only be written for the Ohmic part of the current. An explicit computation of themodes using (3.23) confirms that the AH instability cannot be removed by this relation. In this section, we present numerical results for the collective modes. First, we depict thephase velocities v ph ≡ Re( ω/k ) for different modes. We then plot the imaginary part ofeigenfrequencies for the nonchiral and chiral channels. Since our results are independentof the electrical conductivity, we make certain quantities dimensionless by dividing themby σ e ω ? ≡ ω/σ e , k ? ≡ k/σ e , P ? ≡ C A P/σ e . (4.1) This is confirmed by explicit computations, which are not reproduced in the present work. – 13 – - - - ( c.1 ) v a / v s = k * = - - ( b.1 ) v a / v s = k * = - - - - ( a.1 ) v a / v s = k * = - - ( c.2 ) v a / v s = k * = ( b.2 ) v a / v s = k * = - - ( a.2 ) v a / v s = k * = - - - - ( c.3 ) v a / v s = k * = - - ( b.3 ) v a / v s = k * = - - - - ( a.3 ) v a / v s = k * = Slow magentosonic Fast magnetosonic Alfven
Figure 3 . The figures represent the phase velocity Re( ω ) /k in the plane of B - P . The figuresare for different values of k/σ e (columns 1, 2, and 3) and v a /v s (rows a, b, and c). For allfigures, P = σ e / (cid:16) √ C A (cid:17) . The horizontal orange gridline in each figure corresponds to thedirection of B , while the vertical dashed gray gridline is in the direction of P . The dash-dotted circle demonstrates the speed of light. The blue and red curves correspond to slow andfast magnetosonic modes in (3.17), respectively. The green curves are the phase velocity of theAlfvén modes in (3.20). The Alfvén modes are symmetric. As it turns out, for any particular choice of parameters, there exists two critical polarangles θ c and π − θ c for which the effective conductivity vanishes θ c ≡ arctan (cid:18) P ? sin ∆ (cid:19) . (4.2)– 14 – - - - ( c.1 ) v a / v s = k * = - - ( b.1 ) v a / v s = k * = - - - - ( a.1 ) v a / v s = k * = - - ( c.2 ) v a / v s = k * = ( b.2 ) v a / v s = k * = - - ( a.2 ) v a / v s = k * = - - - - ( c.3 ) v a / v s = k * = - - ( b.3 ) v a / v s = k * = - - - - ( a.3 ) v a / v s = k * = Slow magentosonic Fast magnetosonic Alfven
Figure 4 . The figures represent the phase velocity Re( ω ) /k in the plane perpendicular to P . For the upper half-plane, ∆ = π/
2, and in the lower one ∆ = 3 π/
2. The figures are fordifferent values of k/σ e (columns 1, 2, and 3) and v a /v s (rows a, b, and c). For all figures P = σ e / (cid:16) √ C A (cid:17) . The horizontal orange gridline in each figure corresponds to the direction of B , while the vertical dashed gray gridline is in the direction perpendicular to both B and P .The dash-dotted circle demonstrates the speed of light. The blue and red curves correspond toslow and fast magnetosonic modes in (3.17), respectively. The green curves are the phase velocityof the Alfvén modes in (3.20). Even for small values of v a /v s , the Alfvén modes reach the speedof light in a certain direction. – 15 –hese angles divide the space into stable and unstable regions. A schematic picture ofthis division is presented in Fig. 2. The chiral channel is unstable inside the green upper( θ c < θ < π/
2) and blue lower (3 π/ < θ < π − θ c ) quarters. The remained symmetriesof the space allow us to choose a particular vertical slice, which is the B - z plane, and aparticular horizontal one, which is the B - P plane. In the B - z plane, ∆ = π/ π/ B - P plane ∆ = 0 for both halves. We note that theabsolute value of ∆ is not significant in our analysis, because it can be absorbed into P ? ,but the sign of sin ∆ matters. For simplicity, we call the different modes of (3.20) negativeAlfvén ( ω A , − ), positive Alfvén ( ω A , + ), and chiral nonhydro ( ω C , nh ) modes. i) Phase velocities We use polar plots to depict the phase velocities. To do so, we need to transform fromspherical coordinates to polar coordinates in B - z and B - P planes. In the B - z , we definethe polar angle as ϕ ≡ sgn (sin ∆) θ . Positive (negative) ϕ corresponds to the upper (lower) half of the B - z plane. Since ∆ = 0for the B - P plane, the upper and lower half-planes are similar. In each figure of Figs. 3-8,the absolute value of the phase velocity at any particular value of ϕ is equal to the radiusof the corresponding curve. The sign of the velocity is not shown, and the sign of plot ticksare just indicators of the corresponding quarter. The phase velocities in the B - P planeare depicted in Fig. 3. The modes behave similarly to those of the resistive MHD [51]. Forsmall values of k ? , the slow magnetosonic and Alfvén modes have similar phase velocities,and the fast magnetosonic modes are the fastest ones. This behavior is not surprisingbecause the limit k (cid:28) σ e is the iMHD limit. As k ? increases, the Alfvén modes obtainphase velocities closer to the fast magnetosonic ones. In the nonchiral channel, the phasevelocity vanishes for cos θ = 0, as it can be analytically found from (3.10). On the otherhand, a similar general statement cannot be expressed for the phase velocity at sin θ = 0.The special case of k ? = 1, for which the phase velocity of slow magnetosonic modesbecomes equal to the speed of sound, is interesting. The nonhydro mode of the nonchiralchannel does not propagate, i.e. its phase velocity is always zero. The modes in the B - P plane are symmetric, and we do not represent any other figure in this plane.We represent the phase velocities in the B - z plane with some details. In Fig. 4, thephase velocities of all modes are drawn. In the stable polar region, the phase velocitiesare similar to those of the B - P plane. However, in the unstable region, the Alfvén modesbehave drastically different. Even for small values of k ? , there exists a region for whichthe Alfvén modes propagate with the speed of light. We can understand this behavior byinspecting the first k -dependent term in Alfvén phase velocity which is found from (3.20), v ph , Alfvén = ± v a cos θ ± (1 − v a ) (1 − v a cos θ ) (1 − v a cos θ )8 v a cos θσ . k + O (cid:16) k (cid:17) . (4.3)– 16 –hen σ eff . becomes very small, the phase velocity increases. But one should keep in mindthat the higher order terms are absent in (4.3), and the phase velocity does not actuallytend to infinity as this relation suggests. Also, as (4.3) suggests, this region widens as k ? increases. For sufficiently large k ? , the Alfvén modes obtain the speed of light. Incontrast to the magnetosonic modes, the Alfvén ones are asymmetric under the mirrorsymmetry with respect to the direction of B : The chiral channel is not symmetric undertransformation of θ → π − θ , while the nonchiral one is. Increasing v a /v s , which for a fixedtemperature corresponds to stronger magnetic fields, has the same effect as in B - P plane.In the nonchiral channel, the nonhydro mode has nonzero phase velocity. The negativeand positive Alfvén modes overlap with each other and the nonhydro one. Therefore,to better understand the behavior of the chiral channel, we draw the phase velocitiesseparately. The phase velocity of the nonhydro mode cannot be understood using thelong-wave expansion (4.3). However, we can rely on numerical inspection to understandthe peculiar behavior of the chiral channel. We start with the chiral channel’s modes inthe upper half-plane. In the first stable region, namely 0 < θ < θ c , only the Alfvén modespropagate. The phase velocities have opposite signs but equal values. The velocities ofthese hydro modes are enhanced by increasing k ? (Fig. 5) and v a /v s (Fig. 7). The velocitiestend to the speed of light as we get closer to the critical angle. At the critical angle, thenegative Alfvén mode is replaced by the nonhydro one. Both modes propagate with thespeed of light. For the nonhydro mode, this only happens exactly at the critical angle andis not captured in Fig. Fig. 6. Then we enter the upper unstable region, i.e. θ c < θ < π/ k ? . For small values of k ? , the negative Alfvén mode obtain positivevelocities, while the positive Alfvén mode is replaced by the nonhydro mode, which has anegative velocity. As k ? is increased, the nonhydro mode is suppressed, and the positiveand negative Alfvén modes obtain velocities with the right signs. The same remarks holdfor the second quarter of the upper half-plane. The modes behave similarly in the lowerhalf-planes, with negative and positive Alfvén modes swapped. ii) Imaginary parts In Figs. 9 and 10 the k ? dependence of the imaginary parts of all modes in both channelsare plotted. As it turns out, they become almost constant after a particular value of k ? .In particular, the imaginary parts of the nonchiral channel modes, presented in Fig. 9, arealways negative. This confirms our proof presented in Appendix B. For the chiral channel,the imaginary parts become positive in the unstable regions. Let us also notice that inFig. 10 the modes in the exact opposite direction have negative imaginary parts. Theimaginary parts of the modes at the two critical angles are also depicted in Fig. 10. Allimaginary parts vanish in the direction of the critical angle, while in the exact oppositedirection they have nonvanishing negative values. Therefore at the critical angle, the chiralchannel is still stable. – 17 – Concluding remarks
In the present work, we performed an analysis of the linear stability of a resistive CSMHD.We started with the MCS Lagrangian that produces the CM current through the comovingtemporal derivative of an axion-like field. After reproducing the results of [32] for theMCS thermodynamics, we identified the global equilibrium state in CSMHD by applyinga standard entropy current analysis. We showed that the axial chemical potential µ vanishes in global equilibrium, but the spatial gradient of the axion P = ∇ Θ can giverise to a nonzero electrical charge density. To proceed, we chose the conjugate chemicalpotential of the electrical charge density µ e to be zero in the equilibrium. This choiceis equivalent to the power counting scheme in which the magnetic field is of order O (1),while the electric field is of order O ( ∂ ). Hence, in this weak electric field regime [35], theelectric field vanishes in the thermodynamical equilibrium. As a consequence, the electriccharge density vanishes, and the spatial gradient of the axion-like field is constrainedto be perpendicular to the magnetic field. With the hydrostatic configuration fixed, weintroduced linear perturbation to find the collective modes. We found that there existthree extra modes in CSMHD, in addition to the six ones of iMHD. These nine modesare divided into two channels: Five in a nonchiral or non-axionic channel and four ina chiral or axionic one. The nonchiral channel consists of slow and fast magnetosonicmodes, which are damped by the nonzero electrical resistivity. This channel also possessesa nonhydro (gapped) mode. Using the Routh-Hurwitz criteria and asymptotic causalitycondition, we showed that the nonchiral channel is linearly stable and causal. The chiralchannel includes the modified Alfvén modes and a nonhydro (gapless) mode. The stabilityof these modes is controlled by a combination of the Ohm and AH conductivities, whichcan be considered as a novel effective conductivity. In contrast to the Ohm conductivity,effective conductivity becomes negative for the modes propagating sufficiently close to thedirection of the magnetic field. Consequently, the chiral channel is unstable in this region.However, this channel is causal, and therefore the instability is physical. We also performeda numerical inspection of phase velocities and imaginary parts of different modes. As ourresults show, there is a critical angle that separates stable and unstable regions of thespace for the chiral channel. In the direction of this critical angle, the Alfvén waves travelwith the speed of light without becoming unstable.The current work has a theoretical nature, in which we explored the stability andcausality of the modes propagating in a chiral medium. Although the CM current thatarises from the MCS theory has a physical explanation, this theory is not the only approachto the CME. To the best of our knowledge, other consequences of the MCS theory arenot well understood in the context of the QGP physics. In particular, in contrast to thecondensed matter physics [52], we are unaware of a physical explanation for the occurrenceof the AH effect in the QGP. It might be interesting to investigate the possible mechanisms In [26], we have presented another application of the presence of the AH current within CSMHD. – 18 –hat give rise to the AH effect in different states of strongly interacting quark matter.We close this paper by suggesting two possible directions that extend this work. Inthe present work, we have assumed that the electric chemical potential is zero, which isequivalent to the assumption of electric field being of order O ( ∂ ). A possible extensionwould be to consider the strong electric field regime, in which the electric field is of order O (1) and the electric chemical potential is nonzero in the equilibrium. Another interestingextension is to assume the equilibrium state to be in a rigid rotation. The work in bothdirections is in progress. A Notations, conventions and useful formulae
The energy-momentum tensor of the perfect fluid is given by [1, 2, 12] T µν Fluid(0) = εu µ u ν − p ∆ µν . (A.1)Here, ε is the energy density, p the pressure, and u µ the fluid four-velocity normalized as u µ u µ = 1. These so-called hydrodynamic variables have unique definitions for the perfectfluid [12]. Consequently, the LRF is unambiguously defined by u µ = (1 , ). In (A.1),∆ µν ≡ g µν − u µ u ν projects vectors and tensors in the direction orthogonal to u µ . Thecomoving temporal D and spatial derivatives ∇ ⊥ µ read D ≡ u µ ∂ µ , ∇ ⊥ µ ≡ ∆ νµ ∂ ν . (A.2)As any antisymmetric tensor of rank two, F µν and ? F µν can be decomposed with respectto the timelike vector u µ [13] F µν = E µ u ν − E ν u µ − (cid:15) µναβ B α u β , ? F µν = B µ u ν − B ν u µ + (cid:15) µναβ E α u β , (A.3)where the EM four-vectors are defined as E µ ≡ F µν u ν , B µ ≡ (cid:15) µναβ F να u β . (A.4)One should bear in mind that only for the comoving observer, say in the LRF, the abovefour-vectors coincide with the physical electric and magnetic fields, i.e. E µ = (0 , E ) and B µ = (0 , B ). By this virtue E ≡ q − E µ E µ = | E | LRF , and B ≡ q − B µ B µ = | B | LRF . (A.5)We note that while E and B are Lorentz invariants, | E | and | B | are not. B Routh-Hurwitz stability analysis
In this appendix, we apply the Routh-Hurwitz stability criteria [49] to channels found inSec. 3. – 19 – .1 Nonchiral channel
For simplicity, we rewrite the nonchiral channel (3.10) as D NC ≡ − v a w D NC = σ e v a ( ω − k ) (cid:16) ω − v s k (cid:17) − ω (1 − v a ) (cid:16) ω − v s k (cid:17) (cid:16) ik + ω ( σ e − iω ) (cid:17) − v s k n e v a cos(2 θ ) (cid:16) ω − k (cid:17) . (B.1)To apply the Routh-Hurwitz criteria, we perform the substitution ω → iζ [9]. Conse-quently, D NC is transformed into a 5th order polynomial in ζ , D NC = X i =0 a i ζ i . We employ, at this stage, the Routh-Hurwitz criteria to find whether the real part of ζ ispositive. The Routh table readsR NC = a a a a a a b b c c = a d e = a . (B.2)The coefficients a i read a = 2 v s k σ e v a cos θ , a = 2 v s (1 − v a ) k , a = 2 k σ e h v s + v a (cid:16) − v s sin θ (cid:17)i ,a = 2 (cid:16) v s (cid:17) (cid:16) − v a (cid:17) k , a = 2 σ e , a = 2 (cid:16) − v a (cid:17) . All of the above coefficients are positive. Therefore, according to the criteria, all otherelements in the first column of the Routh table (B.2) must also be positive to ensureRe( ζ ) >
0. The next two coefficients are b = a a − a a a = 2 k (cid:16) − v a (cid:17) (cid:16) − v a + v a v s sin θ (cid:17) ,b = a a − a a a = 2 k v s (cid:16) − v a (cid:17) (cid:16) − v a cos θ (cid:17) . The positivity of the above coefficients is obvious. We now turn to c , c = a b − a b b = 2 k σ e − v a + v s v a sin θ (cid:16) − v a (cid:17) (cid:16) − v s sin θ (cid:17) + v s sin θ (cid:16) − v a sin θ (cid:17) . (B.3)The positivity of the terms outside the brackets is apparent. The expression inside thebracket can be assumed as a second-order polynomial in v s , whose discriminant is negative∆ = − (1 − v a ) sin (2 θ ) < . – 20 –ince (1 − v a ) > c is also positive. By the same virtue, d is positive d = b c − b c c = b c − b a c = 2 v s (cid:16) − v s (cid:17) (cid:16) − v a (cid:17) k cos ( θ ) (cid:16) − v a (cid:17) (cid:16) − v s sin θ (cid:17) + v s sin θ (cid:16) − v a sin θ (cid:17) − > . (B.4)We conclude that all elements of the first column of the Routh table (B.2) have the samesign. The nonchiral channel is thus stable. B.2 Chiral channel
As for the nonchiral case, we rewrite the chiral channel as D C = 1 − v a w = − σ eff . cos θ (cid:16) ω − k v a cos θ (cid:17) + iω cos θ (cid:16) − v a (cid:17) (cid:16) ω − k (cid:17) . (B.5)Performing substitution ω → iζ gives rise to a third order polynomial in ζ D C = X i =0 a c,i ζ i , The coefficients read a c, = k v a σ eff . cos θ , a c, = k (cid:16) − v a (cid:17) cos θ , a c, = σ eff . cos θ , a c, = (cid:16) − v a (cid:17) cos θ . (B.6)We do not need to reproduce the whole Routh table to realize that the chiral channel isunstable for regions of θ . For the coefficients to have the same sign, it is required that σ eff . cos θ > . As stated in Sec. 3, such a condition cannot be satisfied for all values of θ . This is visualizedin Fig. 1. We conclude that the modes of the chiral channel are always unstable within aregion around the direction transverse to the magnetic field. Acknowledgments
This work is supported by Sharif University of Technology’s Office of Vice President forResearch under Grant No: G960212/Sadooghi. In particular, M. K. thanks this office forfinancial support. M. S. thanks D. Rischke for valuable discussions.
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Alfven (+)
Figure 5 . The figures represent the phase velocity Re( ω ) /k of the Alfvén modes in the planeperpendicular to P . For the upper half-plane, ∆ = π/
2, and in the lower one ∆ = 3 π/
2. Thefigures are for different values of k/σ e (columns 1, 2, and 3) and P ? = C A P/σ e (rows a andb). Each row is divided into I and ii parts, with i (ii) demonstrating the negative (positive)Alfvén modes of (3.20). v a /v s = 0 . B , while the vertical dashed gray gridline is in the directionperpendicular to both B and P . The dash-dotted circle demonstrates the speed of light. Thepositive and negative Alfvén modes propagate with the speed of light in opposite polar directions.This direction gets closer to the direction of B for larger values of P ? . The asymmetry decreasesfor larger values of k/σ e . – 25 – - - - ( b.1 ) P * = k * = - - - - ( a.1 ) P * = k * = - - ( b.2 ) P * = k * = - - ( a.2 ) P * = k * = - - - - ( b.3 ) P * = k * = - - - - ( a.3 ) P * = k * = Figure 6 . The figures represent the phase velocity Re( ω ) /k of the chiral nonhydro mode in theplane perpendicular to P . For the upper half-plane, ∆ = π/
2, and in the lower one ∆ = 3 π/ k/σ e (columns 1, 2, and 3) and P ? = C A P/σ e (rows a andb). For all figures, v a /v s = 0 .
5. The horizontal orange gridline in each figure corresponds to thedirection of B , while the vertical dashed gray gridline is in the direction perpendicular to both B and P . The dash-dotted circle demonstrates the speed of light. – 26 – - - - ( b.ii.1 ) P * = v a / v s = - - ( b.i.1 ) P * = v a / v s = - - ( a.ii.1 ) P * = v a / v s = - - - - ( a.i.1 ) P * = v a / v s = - - ( b.ii.2 ) P * = v a / v s = ( b.i.2 ) P * = v a / v s = ( a.ii.2 ) P * = v a / v s = - - ( a.i.2 ) P * = v a / v s = - - - - ( b.ii.3 ) P * = v a / v s = - - ( b.i.3 ) P * = v a / v s = - - ( a.ii.3 ) P * = v a / v s = - - - - ( a.i.3 ) P * = v a / v s = Alfven (-)
Alfven (+)
Figure 7 . The figures represent the phase velocity Re( ω ) /k of the Alfvén modes in the planeperpendicular to P . For the upper half-plane, ∆ = π/
2, and in the lower one ∆ = 3 π/
2. Thefigures are for different values of v a /v s (columns 1, 2, and 3) and P ? = C A P/σ e (rows a andb). Each row is divided into I and ii parts, with i (ii) demonstrating the negative (positive)Alfvén modes of (3.20). k/σ e = 0 . B , while the vertical dashed gray gridline is in the directionperpendicular to both B and P . The dash-dotted circle demonstrates the speed of light. Thepositive and negative Alfvén modes propagate with the speed of light in opposite polar directions.This direction gets closer to the direction of B for larger values of P ? . – 27 – - - - ( b.1 ) P * = v a / v s = - - - - ( a.1 ) P * = v a / v s = - - ( b.2 ) P * = v a / v s = - - ( a.2 ) P * = v a / v s = - - - - ( b.3 ) P * = v a / v s = - - - - ( a.3 ) P * = v a / v s = Figure 8 . The figures represent the phase velocity Re( ω ) /k of the chiral nonhydro mode in theplane perpendicular to P . For the upper half-plane, ∆ = π/
2, and in the lower one ∆ = 3 π/ v a /v s (columns 1, 2, and 3) and P ? = C A P/σ e (rows a,and b). k/σ e = 0 . B , while the vertical dashed gray gridline is in the direction perpendicular toboth B and P . The dash-dotted circle demonstrates the speed of light. – 28 – - - - k * I m ( ω * ) Slow magnetosonic Fast magnetosonicNonhydro
Figure 9 . The figure represents Im( ω ) /σ e vs. k/σ e for the nonchiral channel (3.10). As shownin the text, Im( ω ) is always negative and the nonchiral channel is stable. The fast magnetosonicmodes are suppressed more strongly than the slow ones. Im( ω ) becomes almost constant after asufficiently large value of k . – 29 – b.1 ) θ = π / - - - - k * I m ( ω * ) ( a.1 ) θ = π / - - - - I m ( ω * ) ( b.2 ) θ = π / - - - - k * ( a.2 ) θ = π / - - - - (-) Alfven ( θ ) (-) Alfven ( π - θ )(+) Alfven ( θ ) (+) Alfven ( π - θ ) Nonhydro ( θ ) Nonhydro ( π - θ ) Figure 10 . The figure represents Im( ω ) /σ e vs. k/σ e for the chiral channel (3.11): (a.1) representsa value of θ within the upper unstable region of Fig. 2; (a.2) represents a value of θ within theupper stable region of Fig. 2; (b.1) represents the angle θ c at which σ eff . = 0 within the upperregion of Fig. 2; (b.2) represents the angle θ c at which σ eff . = 0 within the lower region of Fig. 2.The mode propagating in exactly opposite direction are drawn with the same color but dashedlines. The channel is still stable at this critical angle. The Im( ω ) becomes almost constant aftera sufficiently large value of k ? ..