Mechanisms for Multi-Scale Structures in Dense Degenerate Astrophysical Plasmas
aa r X i v : . [ a s t r o - ph . S R ] J a n Mechanisms for Multi-Scale Structures in DenseDegenerate Astrophysical Plasmas
N.L. Shatashvili • S.M. Mahajan • V.I. Berezhiani
Abstract
Two distinct routes lead to the creation ofmulti–scale equilibrium structures in dense degenerateplasmas, often met in astrophysical conditions. By an-alyzing an e-p-i plasma consisting of degenerate elec-trons and positrons with a small contamination of mo-bile classical ions, we show the creation of a new macroscale L macro (controlled by ion concentration). Thetemperature and degeneracy enhancement effective in-ertia of bulk e-p components also makes the effectiveskin depths larger (much larger) than the standard skindepth. The emergence of these intermediate and macroscales lends immense richness to the process of structureformation, and vastly increases the channels for energytransformations. The possible role played by this mech-anism in explaining the existence of large-scale struc-tures in astrophysical objects with degenerate plasmas,is examined. Keywords sun: evolution; stars: white dwarfs; plas-mas
Dense compressed plasmas, found in astrophysicaland cosmological environments as well as in lab-
N.L. Shatashvili Andronikashvili Institute of Physics, TSU, Tbilisi 0177, Georgia Department of Physics, Faculty of Exact and Natural Sciences,Ivane Javakhishvili Tbilisi State University (TSU), Tbilisi 0179,GeorgiaS.M. Mahajan Institute for Fusion Studies, The University of Texas at Austin,Austin,Tx 78712V.I. Berezhiani Andronikashvili Institute of Physics, TSU, Tbilisi 0177, Georgia School of Physics, Free University of Tbilisi, Georgia oratory experiments investigating interaction of in-tense lasers with high density plasma, are currentlyof great interest. In dense astrophysical objects likewhite and brown dwarfs, neutron stars, magnetars,and cores of giant planets, extreme conditions leadto high density degenerate matter (Sturrock 1971;Shapiro & Teukolsky 1973; Chandrasekhar 1931) ;(Chandrasekhar 1939; Ruderman & Sutherland 1975;Michel 1991; Michel 1982; Koester & Chanmugam 1990;Beloborodov & Thompson 2007). Most astrophysicalplasmas usually contain ions in addition to degener-ate electrons and positrons. Although there is no con-crete evidence, the magnetospheres of rotating neutronstars are believed to contain electron-positron plas-mas produced in the cusp regions of the stars dueto intense electromagnetic radiation. Since protonsor other ions may exist in such environments, three-component electron-positron-ion (e-p-i) plasmas canexist in pulsar magnetospheres (Begelman et al. 1984;Holcomb & Tajima 1989),(Berezhiani & Mahajan 1994,1995).For typical dense plasmas, composed of ions, elec-trons, positrons, and/or holes (in the context of semi-conductors), the lighter species (electrons, positrons,holes) are degenerate while the more massive ions, of-ten, remain nondegenerate (classical).It has been known that the nonlinear phenomena ine-p plasmas develop differently from their counterpartsin the usual electron–ion system. The positron compo-nent could have a variety of origins: 1) positrons canbe created in the interstellar medium due to the inter-action of atoms and cosmic ray nuclei, 2) they can beintroduced in a Tokamak e-i plasma by injecting burstsof neutral positronium atoms ( e + e − ), which are thenionized by plasma; the annihilation time of positron inthe plasma is long compared to typical particle con-finement time (Uddin et al. 2015). The annihilation,which takes place in the interaction of matter (elec- trons) and anti-matter (positrons), usually occurs atmuch longer characteristic time scales compared withthe time in which the collective interaction between thecharged particles takes place (Berezhiani et al. 2015b,and references therein).In the extremely low temperature three componente-p-i plasmas, studied in the context of of pulsar mag-netospheres in (Lominadze et al. 1986; Rizatto 1988) ;(Lakhina & Buti 1991; Halder et al. 2012), thede Broglie wavelength of the charge carriers can be com-parable to the dimension of the systems. Such ultracolde-p-i plasma behaves like a Fermi gas and quantum me-chanical effects are expected to play a significant rolein their linear and nonlinear dynamics.Pursuing the consequences of degeneracy in a multi-component plasma, it will be interesting to exploremultiple-scale behavior accessible to such systems. Anobvious goal will be to investigate if degeneracy canaffect, for example, the dynamics of the star collapsewhen its multi-species atmosphere begins to contract.Analysis of multi-scale behavior can also be a guide inpredicting various phenomena in the pre-compact era,or during the life of compact objects.The principal determinant of degeneracy, the den-sity, varies over many orders of magnitude in as-tro/cosmic settings. The rest frame e-p density nearthe pulsar surface is believed to be n ≥ cm − (Gedalin et al. 1998), while in the MeV epoch of theearly Universe , it can be as high as n = 10 cm − (Weinberg 1972). Intense e-p pair creation takesplace during the process of gravitational collapse ofmassive stars (Tsintsadze et al. 2003). It is arguedthat the gravitational collapse of the massive starsmay lead to charge separation with the field strengthexceeding the Schwinger limit resulting in e-p pairplasma creation with estimated density to be n =10 cm − (Han et al. 2012). Superdense e-p plasmamay also exist in GRB sources [ n = (10 − ) cm − ](Aksenov et al. 2010).Dense electron-positron plasma can be produced inlaboratory conditions as well. Indeed, the modernpetawatt lasers systems are already capable of pro-ducing ultrashort pulses with focal intensities I =2 × W/cm (Yanovski et al. 2008). Pulses of evenhigher intensities exceeding I = 10 W/cm are likelyto be available soon in lab or in the Lorentz boostedframes (Dunne 2006; Mourou et al. 2006) ;(Tajima 2014). Interaction of such pulses with gaseousor solid targets could lead to the generation of opticallythin e-p plasmas ( n ∼ − cm − ) denser thansolid state systems (Wang et al. 2014).In the highly compressed degenerate Fermi state(average inter-particle distance smaller than the ther-mal de Broglie wavelength), mutual interactions of the plasma particles become unimportant, and plasma be-comes more ideal as the density increases(Landau & Lifshitz 1980). A plasma may be consid-ered cold (functionally zero temperature) if the thermalenergy of the particles is much lower than the Fermienergy no matter how high the temperature really is(Russo 1988; Cercignani & Kremer 2002). The Fermienergy of degenerate electrons (positrons) is ǫ F = m e c h(cid:0) R (cid:1) / − i , where R = p F /m e c and p F is the Fermi momentum determined by the rest-framedensity p F = m e c ( n/n c ) / . Densities are normal-ized to the critical number-density n c = 5 . × cm − (Akbari-Moghanjoughi 2013).The fluid models are frequently applied to studythe large scale dynamics of relativistic multi-speciesplasmas (Gedalin 1996; Hazeltine & Mahajan 2002).Among such investigations the studies on relaxed (equi-librium) states have attracted considerable attention(Oliveira & Tajima 1995). Constrained minimizationof fluid energy with appropriate helicity invariants hasprovided a variety of extremely interesting equilibriumconfigurations that have been exploited and found use-ful for understanding laboratory as well as astrophysi-cal plasma systems [see e.g. (Oliveira & Tajima 1995;Woltjer 1958; Taylor 1974,1986; Sudan 1979),(Bhattacharjee & Dewar 1982; Dennis et al. 2014) andreferences therein]. Two particularly simple mani-festations of this class of equilibria – Beltrami states – are: 1) The single Beltrami state, ∇ × B = α B¯ , discussed by Woltjer and Taylor (Woltjer 1958;Taylor 1974,1986) in the context of force free singlefluid magnetohydrodynamics (MHD), and 2) a moregeneral Double Beltrami State accessible to Hall MHD– a “two-fluid’ system of ions and inertia-less elec-trons (Steinhauer & Ishida 1997); the latter has beeninvestigated, in depth, by Mahajan and co-workers(Mahajan & Yoshida 1998; Mahajan et al. 2001),(Ohsaki et al. 2001,2002; Shukla & Mahajan 2004a;b),(Mahajan et al. 2002; Mahajan et al. 2005,2006),(Mahajan & Krishan 2005; Yoshida 2011). The con-tent of the Beltrami conditions (derived by constrainedminimization) is the alignment of a general “flow”with its vorticity. This, in turn, forces the totalenergy density of the system to distribute homoge-neously (so called
Bernoulli condition ). The com-bined Beltrami-Bernoulli conditions define an equi-librium state that fits the notion of relaxed state ,and constitutes a nontrivial helicity-bearing state.The characteristic number of a state is determinedby the number of independent single Beltrami sys-tems needed to construct it. For adequate descrip-tion these states were named
Beltrami-Bernoulli States (BB) (Mahajan & Yoshida 1998; Mahajan et al. 2001;
Berezhiani et al. 2015a); the latter ref. was also thefirst to study the effects of degeneracy (requiring Fermi-Dirac statistics) on BB states.This paper, though worked out on the lines of(Berezhiani et al. 2015a), registers a major departureleading to the most important result of this article –by studying the BB states in an e-p-i (small dynamicion contamination added to a primarily e-p plasma),we will demonstrate the creation of a new macroscopiclength scale L macro lying between the system size andthe relatively small intrinsic scales (measured by theskin depths) of the system.The new BB equilibrium is defined by: two rel-ativistic Beltrami conditions (one for each dynamicdegenerate species), one non-relativisc Beltrami con-dition for ion fluid, an appropriate Bernouli condi-tion, and Ampere’s law to close the set. This set ofequations will lead to what may be called a quadru-ple Beltrami system [for multi-Beltrami systems see(Mahajan & Lingam 2015)]. The ions, though a smallmobile component, play an essential role, they createan asymmetry in the electron-positron dynamics (tomaintain charge neutrality, there is a larger concen-tration of electrons than positrons) and that asymme-try introduces a new and very important dynamicalscale. This scale, though present in a classical non-degenerate plasma, turns out to be degeneracy depen-dent and could be vastly different from its classicalcounterpart. The significance of this scale in under-standing the physics of relevant systems will be ex-plored. Presence of mobile ions leads to ”effective mass”asymmetry in electron and positron fluids, which, cou-pled with degeneracy-induced inertia, manifests in theexistence of Quadruple Beltrami fields. Illustrative ex-amples and application for concrete astrophysical sys-tems will be suggested. Charge neutrality in an e-p-i plasma of degenerate elec-trons ( − ), positrons (+) and a small mobile ion com-ponent, forces the following density relationships N − = N +0 + N i = ⇒ N +0 N − = 1 − α (1)with α = N i N − , where α labels the excess electron content. In thispaper we will study only the α ≪ e(p) dynamics will be described by the relativistic degenerate fluid equations(Berezhiani et al. 2015a; Berezhiani et al. 2015b, andreferences therein): the continuity ∂N ± ∂t + ∇ · ( N ± V ± ) = 0 , (2)and the equation of motion ∂∂t (cid:0) G ± p ± (cid:1) + m ± c ∇ (cid:0) G ± γ ± (cid:1) = q ± E + V ± × Ω ± (3)where p ± = γ ± m ± V ± is the hydrodynamic mo-mentum, n ± = N ± /γ ± is the rest-frame parti-cle density ( N ± denotes laboratory frame num-ber density), q ± ( m ± ) is the charge (mass) of thepositron (electron) fluid element, V ± is the fluidvelocity, and γ ± = (cid:0) − V ± /c (cid:1) − / . Notice thatthe degeneracy effects manifest through the ”effec-tive mass” factor G ± = w ± /n ± m ± c , where w ± is an enthalpy per unit volume. The generalexpression for enthalpy w ± for arbitrary densityand temperature (for a plasma described by localDirac-Juttner equilibrium distribution function) can befound in (Cercignani & Kremer 2002). For a fully de-generate (strongly degenerate) e(p) plasma, however,this very tedious expression smoothly transfers to theone with just density dependence, i.e, w ± ≡ w ± ( n )(Berezhiani et al. 2015a). In fact w ± /n ± m e c = (cid:0) R ± ) (cid:1) / , where R ± [= p F ± /m ± c with p F ± being the Fermi momentum] have been defined earlier.The mass factor, then, is simply determined by theplasma rest frame density, G ± = [1 + ( n ± /n c ) / ] / for arbitrary n ± /n c .On taking the curl of these equations, one can castthem into an ideal vortex dynamics (Mahajan 2003 andreferences therein) ∂∂t Ω ± = ∇× ( V ± × Ω ± ) , (4)in terms of the generalized (canonical) vorticities Ω ± =( q ± /c ) B + ∇× ( G ± p ± ) .It is, perhaps, the right juncture to re-emphasize thatthe so called plasma approximation for a degenerate e(p) assembly is valid if their average kinetic energy ( ∼ ǫ ± F ) is larger than the interaction energy ( ∼ e ( n ± ) / ).This condition is fulfilled for a sufficiently dense fluidwhen n ± ≫ (2 m − e / (3 π ) / ~ ) = 6 . · cm − ;such a condition would imply R ± ≫ . · − (Berezhiani et al. 2015b). The low frequency dynamics is, now, closed withAmpere’s law ∇ × B = 4 πec (cid:2) (1 − α ) N + V + − N − V − ) + α N i V i (cid:3) , (5)another relation between V i , V ± and B . Notice thatthe small static/mobile ion population, represented by α and V i , creates an asymmetry between the currentscontributed by the electrons and positrons. This willbe the source of a new scale-length that turns out tobe much larger than the intrinsic electron and positronscale lengths (skin depths).In this paper, we explore the combined effects ofasymmetry and degeneracy on a special class of e-p-iequilibria known as the Beltrami-Bernoulli (BB) states.We expect to find large-scale structures originating inthe ion-induced asymmetry. The e-p-i plasma systemis, in some sense, more advanced and complete thanthe electron-ion system studied in a recent paper whereit was shown that the electron degeneracy transformedBB states may be pertinent to advance our understand-ing of the evolution of certain astrophysical objects(Berezhiani et al. 2015a). Before we write down the equations for the BB states,it is useful to express all physical quantities in nor-malized dimensionless form. In this paper, the den-sity is normalized to N − (the corresponding rest-frame density is n − ); the magnetic field is normal-ized to some ambient measure | B | ; all velocities aremeasured in terms of the corresponding Alfv´en speed V A = V − A = B / q πn − m − G − ; all lengths [times]are normalized to the ”effective” electron skin depth λ eff [ λ eff /V A ] , where λ eff ≡ λ − eff = 1 √ cω − p = c s m − G − πn − e ; (6) G ± ( n ± ) = [1 + ( R ± ) ] / with (7) R ± = (cid:18) n ± n c (cid:19) / ; R +0 = (1 − α ) / R − . (8)The intrinsic skin depths, the natural length scales ofthe dynamics, are generally much shorter compared to the system size. For the degenerate electron fluid, theeffective mass goes from G − ( n − ) = 1+ ( n − n c ) / in thenon-relativistic limit ( R − ≪ G − ( n − ) = (cid:16) n − n c (cid:17) / in the ultra-relativistic regime ( R − ≫ G ± (Berezhiani et al. 2015a)): The Beltrami conditions B ± ∇ × ( G ± γ ± V ± ) = a ± n ± G ± ( G ± γ ± V ± ) , (9)aligning the Generalized vorticities along their velocityfields, and the Bernoulli conditions ∇ ( G ± γ ± ± ϕ ) = 0 = ⇒ (10) G + γ + + G − γ − = const . (11)In the latter, ϕ = 0 due to the asymmetry, but gravityis ignored.The separation constants a ± are related to the totalsystem energy, and the generalized helicities h ± = Z ( curl − Ω ± ) · Ω ± d r . (12)This set, coupled with Ion fluid Beltrami Condition: B + ζ ∇ × V i = α a i n i V i , where ζ = (cid:20) G − m − m i (cid:21) − (13)together with Ampere’s law Eq.(5) defines the BB sys-tem for an e-p-i plasma – a degenerate e-p system madesomewhat asymmetric by a small fraction of mobile ions( α ≪ , | V i | ≪ | V ± | ).Notice that there are, in fact, two asymmetry-introducing mechanisms in the e-p-i system: differ-ent effective inertias for the positively and negativelycharged particles of the bulk species is one, while thesmall contamination from mobile ions ( α = 0 , V i = 0)constitutes the other. Each one of these is responsi-ble for creating a net “current”. The structure for-mation mechanism explored in (Mahajan et al. 2009;Berezhiani et al. 2010) (Steinhauer & Ishida 1997),(Mahajan et al. 2001; Mahajan et al. 2002),(Ohsaki et al. 2001,2002)), originates, for instance,in the effective inertia difference. Asymmetry be-tween the plasma constituents increases the numberof conserved helicities, and eventually translates into a higher index Beltrami state. Later, we will explicitlyshow that for the degenerate e-p-i system, the Bel-trami part of BB state is a Quadruple Beltrami state.When the second asymmetry mechanism is neglected( α → , V i →
0) a
Triple Beltrami
State follows(Bhattacharjee et al. 2003; Mahajan & Lingam 2015).It should also be mentioned that initial asymmetryin densities ( α = 0) can also contribute to an ”effec-tive inertia” difference in the electron-positron plasmamaking G − = G + ; the index of the Beltrami system,however, is contingent on the simultaneous presence ofboth asymmetries (see Appendix A). An appropriate but tedious manipulation of the set Eq.-s (5)-(13), carried out in Appendices A and B, leadsus to an explicit quadruple Beltrami equation obeyedby the Ion Fluid Velocity V i (the Beltrami index ismeasured by the highest number of curl operators).Written schematically as ∇ × ∇ × ∇ × ∇ × V i − b ′ ∇ × ∇ × ∇ × V i ++ b ′ ∇ × ∇ × V i − b ′ ∇ × V i + b ′ V i = 0 . (14)Equation (14) was derived in the incompressible ap-proximation, and for γ + ∼ γ − ≡
1. The b ′ coeffi-cients are defined in Appendix B. Incompressibility as-sumption is expected to be adequate for outer layersof compact objects, though, compressibility effects canbe significant e.g. in the atmospheres of pre-compactstars (Berezhiani et al. 2015a). The ion fluid velocityand the magnetic field are related to the e-p plasmaaverage bulk fluid velocity V = 12 [(1 − α ) V + + V − ] , (15)through V == η (cid:0) βG +0 ∇ × ∇ × B − [ a + (1 − α ) β − a − ] ∇ × B (cid:1) ++ η ([1 + (1 − α ) β ] B ) − (16) − αβ ∇ × V i + α a + (1 − α ) β − a − ] V i with η ≡ [ a + (1 − α ) β + a − ] − and β = G − /G +0 . Once Eq.(14) is solved for V i , the vector fields B and V can be determined from Eqs.(13)-Eq.(16). We have chosen to work, here, with the more familiar e-p plasmabulk velocity V rather than the normalized momenta P ± = G ± ( n ± ) V ± . The e-p-i system is symmetric in B , V , V i in the sense that either of them obeys a quadru-ple curl equation. The curl curl curl (Triple Beltrami)equation and its solutions describing a compressible de-generate pure e-p plasma are given in Appendix C.In Appendix B we give some illustrative examples ofQuadruple [Triple] Beltrami states for degenerate e-p-iplasmas interesting for astrophysical context. Since theeffect of compressibility in degenerate e-i plasma wasstudied in (Berezhiani et al. 2015a) we do not discussit here and, instead, concentrate on emphasizing theeffects of asymmetry stemming from the dynamic ioncontamination.The quadruple Beltrami (14) can be factorized as(details in Appendix B)( curl − µ )( curl − µ )( curl − µ )( curl − µ ) V i = 0 , (17)where µ i -s define the coefficients in Eq.(14) and are thefunctions of α, β, n − and the degeneracy-determinedmass factor G +0 . The general solution of Eq.(17) isa sum of four Beltrami fields F k (solutions of BeltramiEquations ∇ × F k = µ k F ) while eigenvalues ( µ k ) ofthe curl operator are the solutions of the fourth orderequation µ − b ′ µ + b ′ µ − b ′ µ + b ′ = 0 . (18)An examination of the various b ′ coefficients of (18),displayed in detail in (A10,A11) for the most relevantlimit α ≪ b ′ , b ′ , and b ′ , do get somewhat modified by α ≪ b ′ that is most profoundly affected; being proportionalto α , it tends to become small, i.e, the correspond-ing scale length becomes large as α approaches zero;the scale length becomes strictly infinite for α = 0,and disappears reducing (18) to a triple Beltrami sys-tem. Thus the ion contamination-induced asymmetrymay lead to the formation of macroscopic structuresthrough creating an intermediate/large length scale,much larger than the intrinsic scale skin depths, andless than the system size. The possible significanceand importance of this somewhat natural mechanism (asmall ion contamination is rather natural) for creatingMacro-structures in astrophysical objects, could hardlybe overstressed. It is important to note that this mech-anism operates for all levels of degeneracy (the rangeof R − was irrelevant). This new macroscopic scale can be “determined” bydominant balance arguments; as the scale gets larger, |∇| gets smaller, and the dominant balance will bebetween the last terms of (18), yielding [we remind thereader, that all lenghts are normalized to the λ eff , and ζ ≫ L macro = | b ′ || b ′ | = Aα (19)where A = ζ | ( a + − a − )[1 − αζ ( G +0 )] + αζ a i [( G +0 ) − a + a − ] || a i ( a + − a − ) − a + a − | (20)is a somewhat complicated function of the plasma pa-rameters.Assuming that the densities of the e-p-i plasmas ofinterest are such that αG +0 /ζ = α β ( G +0 ) m − m i ≤ α ≪ e(p) component density range is within (10 − ) cm − ] , we can simplify A when both a + ≪ a i and a − ≪ a i .(i) When a + and a − are not equal the simplifiedexpression for dimensionless L macro ∼ ζα (cid:12)(cid:12)(cid:12)(cid:12) a i + αζ ( G +0 ) − a + a − a + − a − (cid:12)(cid:12)(cid:12)(cid:12) (21)for a i ≤ ζ satisfies L macro ≫ a + ∼ a − = a = ( G +0 ) / L macro ∼ a i a | ( G +0 ) − a | ≫ a i ≫ a .Without ion contamination ( α = 0), the degener-ate e-p system is still capable of creating length scaleslarger than the non-relativistic skin depths through thedegeneracy-enhanced inertia of the light particles. No-tice that even with equal effective masses ( G − = G + ≃ G ( n ) at equal electron-positron temperature), inertiachange due to degeneracy can cause asymmetry in e(p) fluids [see (Mahajan & Lingam 2015)].This, perhaps, is the right juncture to summarize thescale hierarchy encountered in this paper:1) For a pure electron-positron plasma, the equilib-rium is triple Beltrami with the following fundamentalthree scales; system size L , and the two intrinsic scales(electron and positron skin depths).2) The e-p skin depths, microscopic in a non degener-ate plasma, can become much larger due to degeneracyeffects and could be classified as meso-scales, l meso [seeAppendix C, Eq.(C5)]. 3) When a dynamic low density ion species is added,the equilibrium becomes quadruple Bertrami with a newadditional scale, L macro . Although the exact magni-tude of this scale is complicated [Eq. (21)], its origin isentirely due to the ion contamination; this scale disap-pears as the ion concentration α goes to zero. Boththe larger ion mass and low density contribute towardsboosting L macro [see Appendix A].4) The meso-scale l meso cannot become very largebut for some special constraints on the Bertrami pa-rameters, for instance, if a − = a + and both a ± ≪ In the present paper we derived
Quadruple [ Triple ] Bel-trami relaxed states in e-p-i plasma with classical ions,and degenerate electrons and positrons. Such a mix isoften met in both astrophysical and laboratory condi-tions.The presence of the mobile ion component has astriking qualitative effect; it converts, what would havebeen, a triple
Beltrami state to a new quadruple
Bel-trami state. In the process, it adds structures at abrand new macroscopic scale L macro (absent whenion concentration is zero) that is much larger than theintrinsic skin depth ( λ = c q m − πn − e ) of the lighter com-ponents.Though primarily controlled by the mobile ion con-centration, L macro also takes cognizance of the elec-tron and positron inertias that could be considerablyenhanced by degeneracy. In fact even in the ab-sence of ions ( L macro → infinity), the Beltrami statescould be characterized by what could be called meso-scales – the temperature and degeneracy-boosted effec-tive skin depths λ ± eff larger than λ [according to (6) λ eff ± /λ = q G ± > < q G ± < . − ) cm − ] . At the same time it hasto be emphasized here that for larger scale to exist wedo need an entirely different mechanism – a dynamicion-species with a much lower density and higher restmass (justified by observations for many astrophysi-cal objects plasmas) – this scale corresponds to theion skin depth enhanced, dramatically, by low density[ λ i = ( α m − /m i ) − / λ ≫ λ ].The creation of these new intermediate scales (be-tween the system size, and λ ) adds immensely to therichness of the structures that such an e-p-i plasma cansustain; many more pathways become accessible for en-ergy transformations. Such pathways could help us bet-ter understand a host of quiescent as well as explosiveastrophysical phenomena - eruptions, fast/transient outflow and jet formation, magnetic field generation,structure formation, heating etc. At the same time, re-sults found in present manuscript indicate that whenthe star contracts, for example, its outer layers keepthe multi-structure character although density in thestructures, as shown in (Berezhiani et al. 2015a), be-comes defined by lighter components degeneracy pres-sure. Future studies will include a detailed investiga-tion of present model to explore the evolution of multi-structure stellar outer layers while contracting, cooling. Authors acknowledge special debt to the Abdus SalamInternational Centre for Theoretical Physics, Trieste,Italy. The work of S.M.M. was supported by USDOEContract No. DEFG 03-96ER-54366.
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A Appendix - Derivation of Quadruple Beltrami Equation
The Ampere’s law generally can be written in dimensionless variables as: ∇ × B = 12 (cid:20) α N i N i V i + (1 − α ) N + N +0 V + − N − N − V − ) (cid:21) (A1)and(i) if α = 1 (e-i plasma, quasineutralisty reads as N i = N − = N ) we have: V − ≡ V e = V i − N ∇ × B leading to Double Beltrami (DB) states in e-i plasma with degenerate electrons (Berezhiani et al. 2015a);(ii) while when α = 0 (purely symmetric e-p plasma, quasineutralisty reads as N + = N − = N ) we have: V − − V + = − N ∇ × B that shall lead to higher Beltrami states when inertia effects in electron and positron fluids are taken into account[similar effect was discussed for relativistic non-degenerate plasmas in (Yoshida & Mahajan 1999; Iqbal et al. 2008;Bhattacharjee et al. 2003; Mahajan & Lingam 2015)].Observations show that ion fluid fraction can be small ( α ≪ V i ≪ V − , V + ] and, hence, one can imagine that the mobilityof ions can be ignored in most of the cases (Oliveira & Tajima 1995) except when α = 1 where, as it was shown in(Mahajan et al. 2001; Mahajan et al. 2002; Mahajan et al. 2005,2006), flow effects can be crucial in creating thestructural richness in astrophysical environments as well as in the heating/cooling processes, Generalized Dynamotheory and flow acceleration phenomena. The case of α = 1 - pure e-i plasma with degenerate electrons wasalready studied in (Berezhiani et al. 2015a) and it was shown that when ignoring inertia effects in electron fluidthe Double Beltrami states are accessible in the system. Hence, it is expected, that when ion fluid velocity is notnegleted in e-p-i plasma with degenerate electrons and positrons number of relaxed states can be either 2 (whenignoring degenerate e(p) fluids inertia effects although G − = G + ) or 4 (when degenerate fluids inertia effects aretaken into account); at the same time neglecting the ion flow effects ( α → , V i →
0) we shall obtain the SingleBeltrami state in former situation and the Triple Beltrami States in latter case – this problem is a scope of ourstudy below [see (Mahajan & Lingam 2015) and its results].Let us now show how Beltrami states may acquire new structures due to degeneracy or/and the small fractionof mobile ions. We will study an incompressible e-p-i plasma with the simplifying assumption γ + ∼ γ − ≡ G + + G − = const . The Ampere’s law (A1), in dimensionlessvariables, is written as ∇ × B = 12 [ α V i + (1 − α ) V + − V − ] . (A2)In terms of the e-p plasma bulk flow average velocity V = 12 [(1 − α ) V + + V − ] (A3)one can express the Generalized Momenta for positron and electron fluids as [ P ± = G ± ( n ± ) V ± ] : P + = G +0 − α ( V + ∇ × B − α V i ) ; P − = G − ( V − ∇ × B + α V i ) , (A4) Introducing β ≡ G − G +0 and using Eqs. (A4) in Eqs. (9), straightforward algebra leads to: V = η (cid:0) β G +0 ∇ × ∇ × B − [ a + (1 − α ) β − a − ] ∇ × B + [1 + (1 − α ) β ] B (cid:1) −− α β ∇ × V i + α a + (1 − α ) β − a − ] V i (A5)with η ≡ [ a + (1 − α ) β + a − ] − . We have to add the Ion flow Beltrami condition (13) written for incompressible case as B + ζ ∇ × V i = α a i V i (A6)to close the system of equations for incompressible e-p-i degenerate plasma.Plugging the Eq. (A5) into the Eq.-s (A4) and then using them in Eq.-s (9) we get2 β ( G +0 ) ∇ × ∇ × ∇ × B − G +0 α ∇ × ∇ × B + α ∇ × B − α B = (A7)= α ( G +0 ) ∇ × ∇ × V i − α ( G +0 ) α ∇ × V i − α (1 − α ) a + a − V i , where α = [ a + (1 − α ) β − a − ] , α = [1 + (1 − α ) β ] G +0 − β (1 − α ) a + a − , α = (1 − α ) ( a + + a − ) . (A8)The equation (A7) for immobile ions ( V i ≡
0) will eventually give the so called ”Triple Beltrami” equation for themagnetic field B (i.e. l.h.s. of Eq.(A7) ≡ α ≪ ⇒ (1 − α ) →
1; [1 + (1 − α ) β ] → V i :( G +0 ) ∇ × ∇ × ∇ × ∇ × V i − b ( G +0 ) ∇ × ∇ × ∇ × V i + b ( G +0 ) ∇ × ∇ × V i − b ∇ × V i + b V i = 0 , (A9)where b = ( a + − a − ) + 12 a i ( G +0 ); b = [( G +0 ) − a + a − ] + αζ a i ( a + − a − ) + αζ ( G +0 ); (A10) b = ( a + − a − )[1 − αζ ( G +0 )] + αζ a i [( G +0 ) − a + a − ]; b = αζ [ a i ( a + − a − ) − a + a − ] . (A11)Notice, that in above relations the terms with coefficient ( α/ζ ) will vanish for either α → m i → ∞ (which means that ions are immobile!). In such case we arrive to TripleBeltrami Equations for B and V as mentioned above. Also, it is interesting to note that due to mobile ions(i.e. when ( α/ζ ) = 0) the large scale is automatically there due to b = 0 in Eq.(A9). We will show this indetail in the Appendix B.Solving the Eq.(A9) for V i and plugging it into (A6) we will get the equation for B ; for the pure incompressiblee-p plasma it is better to use Eq.(A5) directly to find the magnetic field B . B Appendix - Scale separation in degenerate e-p-i Plasma – Quadruple Beltrami Structures
Here we give some illustrative examples of Quadruple [Triple] Beltrami states for degenerate e-p-i [e-p] plasmasinteresting for astrophysical context.Introducing b ′ = b G +0 ; b ′ = b G +0 ; b ′ = b ( G +0 ) ; b ′ = b ( G +0 ) (B1)the Eq. (A9) reads as: ∇ × ∇ × ∇ × ∇ × V i − b ′ ∇ × ∇ × ∇ × V i + b ′ ∇ × ∇ × V i − b ′ ∇ × V i + b ′ V i = 0 , (B2)Equation (B2) can be written as( curl − µ )( curl − µ )( curl − µ )( curl − µ ) V i = 0 , (B3)where µ k -s define the coefficients in Eq. (B2) as b ′ = µ + µ + µ + µ , b ′ = µ µ + µ µ + µ µ + µ µ + µ µ + µ µ ,b ′ = µ µ µ + µ µ µ + µ µ µ + µ µ µ , b ′ = µ µ µ µ . (B4)The general solution of Eq.(B3) is a sum of four Beltrami fields F k (solutions of Beltrami Equations ∇ × F k = µ k F ) while eigenvalues ( µ k ) of the curl operator are the solutions of the fourth order equation µ − b ′ µ + b ′ µ − b ′ µ + b ′ = 0 . (B5)Since b ′ → α ≪ m − ≪ m i = ⇒ that the Eq.(B5) can be reduced to µ ( µ − µ ) ( µ − µ ) ( µ − µ ) = 0 . (B6)solving of which gives µ ≃ µ − b ′ µ + b ′ µ − b ′ = 0 . (B7)There are 3 possible simple, analytically tractable, scenarios:(i) If ( a + − a − )[1 − αζ ( G +0 )] = − αζ a i [( G +0 ) − a + a − ] (B8)then b ′ ≃ µ ( µ + a ) ( µ − a ) = 0 , (B9)solving of which we find that µ ≃ a + and a − as: µ = − a ; µ = a , where a = 12 b ′ ∓ q ( b ′ ) − b ′ and a = 12 b ′ ± q (b ′ ) − ′ . (B10) In pure e-p plasma above conditions reduce to a + ∼ a − ( a = a + while a = a − ) and eventually there are only3 scales in total (defining equation is Triple Beltrami ).(ii) If α and β and other parameters are such that both b ′ ≃ b ′ ≃ b ′ = µ µ = 0 . Here againone of the roots is zero (let it be µ = 0 ), while other two satisfy the relations: µ + µ = 0 ; µ µ = b ′ . (B11)Hence, also in this case length-scales are vastly separated (two scales are of similar range (short scales) and oneis significantly large-scale): | µ | = | µ | ; µ ≃ a ± must be such that b ′ < a + = a − = a > ( G/n ) / .(iii) If α and β and other parameters are such that all b ′ ≃ b ′ ≃ b ′ ≃ b ′ ≃
0, then all the scale parameters µ , µ , µ , µ become close to zero – no separationof length scales. For pure e-p plasma this is the case when a + = a − = a = ( G/n ) / .Thus, we can conclude, that for a rather big range of parameters there is a guaranteed scale separation in e-p-iplasma with degenerate lighter components. C Appendix - Scale separation in compressible degenerate e-p Plasma – Triple Beltrami Structures
Note that with no fraction of ions [ α = 0 , β = 1 ] there is no charge separation and the scalar potential ϕ ≡ n − = n + = n , B = ∇ × A ; E = − c ∂∂t A and the assumption of T ± → G − = G + = G ( n ) . The existence ofsoliton-like electromagnetic (EM) distributions in such a fully degenerate electron-positron plasma was studied in(Berezhiani et al. 2015b).Then, for pure compressible e-p plasma , if ∇ [ G ± ( n ± )] is at a much slower rate than the spatial derivativesof B and V ± , we can write instead of (A5) following relation [with corresponding η = 1 / ( a + + a − ) ]: V = η (cid:18) Gn ∇ × n ∇ × B − a + − a − n ∇ × B (cid:19) + η B n (C1)and instead of (A7) we obtain: ∇ × (cid:18) Gn (cid:19) ∇ × (cid:18) n (cid:19) ∇ × B − κ ∇ × (cid:18) n (cid:19) ∇ × B + κ ∇ × (cid:18) Gn − a + a − (cid:19) B − κ B = 0 , (C2)where κ = ( a + − a − ); κ = 1 G ; κ = a + − a − G . (C3)After simple algebra we arrive to the defining equation for V = ( V + + V − ): (cid:18) Gn (cid:19) ∇ × (cid:18) n (cid:19) ∇ × ∇ × V − κ (cid:18) n (cid:19) ∇ × ∇ × V + κ ∇ × (cid:18) Gn − a + a − (cid:19) (cid:16) nG (cid:17) V − κ V = 0 . (C4)Solution of Eq.-s (C2) and (C4) is possible following the scenarios given in Appendix B, solutions will besimilar to those given after Eq.(B7), just density dependent. Estimation for the large scale l meso in case of puredegenerate e-p plasma, derived from the dominant balance, gives: l meso = | κ || κ | | ( G/n ) − a + a − | = 2 | ( G/n ) − a + a − || a + − a − | ≫ a + = a − = a = (cid:18) G ( n ) n (cid:19) / ..