Fluid model for relativistic, magnetized plasmas

Transcription

1 PHYSICS OF PLASMAS 15, Fluid model for relativistic, magnetized lasmas J. M. TenBarge, R. D. Hazeltine, and S. M. Mahajan Institute for Fusion Studies, University of Texas at Austin, Austin, Texas 7871, USA Received 4 March 008; acceted 7 May 008; ublished online 7 June 008 Many astrohysical lasmas and some laboratory lasmas are relativistic: Either the thermal seed or the local bulk flow in some frame aroaches the seed of light. Often, such lasmas are magnetized in the sense that the Larmor radius is smaller than any gradient scale length of interest. Conventionally, relativistic magnetohydrodynamics MHD is emloyed to treat relativistic, magnetized lasmas. However, MHD requires the collision time to be shorter than any other time scale in the system. Thus, MHD emloys the thermodynamic equilibrium form of the stress tensor, neglecting ressure anisotroy and heat flow arallel to the magnetic field. Recent work has attemted to remedy these shortcomings. This aer re-examines the closure question and finds a more comlete theory, which yields a more hysical and self-consistent closure. Beginning with exact moments of the kinetic equation, we derive a closed set of Lorentz-covariant fluid equations for a magnetized lasma allowing for ressure and heat flow anisotroy. Basic redictions of the model, esecially of the disersion relation s deendence uon relativistic temerature, are examined. 008 American Institute of Physics. DOI: / I. INTRODUCTION A lasma is relativistic if either the thermal seed the rms seed of the individual articles measured in the fluid rest-frame, or the local bulk flow measured in some relevant frame aroach the seed of light. Such lasmas are ubiquitous in astrohysical henomena e.g., galactic and extragalactic jets, 1 accretion disks of active galactic nuclei, and electron-ositron-ion lasmas in the early universe; 3,4 and in some laboratory fusion exeriments. Often e.g., Refs. 5 7, relativistic lasmas of interest are magnetized, meaning the dynamics are dominated by the magnetic field. The dynamics of such lasmas are tyically described with magnetohydrodynamics MHD, which catures the large-scale electromagnetic features of a magnetized lasma e.g., EB drifts. A relativistic MHD closure has been resented by Anile. 8 Desite the success with MHD at caturing some of the large scale hysics, MHD lasmas are based on the use of a stress tensor whose origin is based on thermodynamic considerations thermal equilibrium rather than electrodynamics, in which electromagnetic forces dominate. Chew, Goldberger, and Low 9 CGL resent an early dearture from the conventional MHD treatment of the stress tensor by allowing gyrotroic ressure; the CGL tensor differentiates between ressures arallel and erendicular to the magnetic field. However, CGL neglects to include heat flow arallel to the magnetic field, which can be raid in low collisionality lasmas. Partly for this reason, the double adiabatic assumtion used by CGL to achieve closure is not valid in many hysical situations. Hazeltine and Mahajan 10 hereafter referred to as I attemted a more hysical relativistic closure with gyrotroic ressure and arallel heat flow. However, close scrutiny of the Hazeltine Mahajan model revealed fundamental deficiencies. The details of the deficiencies are covered in Sec. II. The closure method emloyed in I uses the stress tensor as the constitutive relation for the fluid closure. The form of the stress tensor is derived from exact fluid equations together with orderings characterizing a magnetized lasma. Predictably, such an aroach does not rovide a closed system. Closure is achieved through a reresentative distribution function, consistent with relativity, magnetization, ressure anisotroy, and heat flow. To achieve our closure, we take an aroach arallel to I; weusei as a guide in the search for a more hysical and self-consistent relativistic, magnetized fluid closure. II. CRITIQUE OF HAZELTINE AND MAHAJAN REFS We begin by discussing the covariant fluid closure of I. We refer the reader to I and other related aers to observe the full treatment of the system rather than reiterating both the relativistic closure and the nonrelativistic limit here. Study of the closure resented yielded several major shortcomings of the original model. 1. The first and most ertinent deficiency is aarent from the linearized, nonrelativistic equations of motion. It is manifested by examining the electrostatic resonse of the electron ressure anisotroy, e = e e. Parallel erendicular here refers to being arallel erendicular to the magnetic field. One finds that e m i /m e, leading to grossly exaggerated estimates of the electron anisotroy under the tyical MHD assumtion of vanishing electron inertia. Such anomalous scaling of the ressure anisotroy is not observed in conventional MHD or kinetic MHD. The source of the anomalous scaling of the ressure anisotroy is the use of a single arallel heat flow, Q, rather than searating the arallel heat flow into the arallel flow of arallel heat, q, and the arallel flow of erendicular heat, q. When a single heat flow is used, the evolution of arallel and erendicular ressure are both couled to arallel gradients of the heat flow, and the evolution of the single heat X/008/156/0611/1/$ , American Institute of Physics Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

2 0611- TenBarge, Hazeltine, and Mahajan Phys. Plasmas 15, flow is driven by arallel ressure gradients. Using searate heat flows results in the exected evolution of the ressures and heat flows, namely, d /dt q, d /dt q, dq /dt, and dq /dt. Though searating the two forms of arallel heat is relatively common in the literature see, e.g., Refs. 9, 13, and 14, the distinction between the heat flows does not aear in the stress tensor, which forms the constitutive relation for a fluid lasma closure. Therefore, I attemts a closure involving the single heat flow, which corresonds to the sum, Q =q +q. Including searate heat flows involves modifying the distribution used in I and using higher-order moment equations to obtain evolution equations for the two heat flows. However, the stress tensor is not changed. On a related note, I does a very oor job of redicting the onset of the mirror instability. The source of this error is the unusual couling of the ressures and heat flows noted above. The use of a relativistic bi-maxwellian accurate to first order in the ressure anisotroy rovides a better estimate of the mirror instability but still does not agree fully with kinetic MHD. However, a relativistic bi-maxwellian retaining second-order ressure anisotroy terms catures the correct mirror instability. Note that, keeing accuracy to this order is reasonable since the fourth-rank moment energyweighted stress will naturally have terms second-order in the anisotroy.. Examining the thermodynamics of I leads to a thermodynamic temerature of the following form: T = + 3 n + 4 3, where is the scalar ressure and is the ressure anisotroy. This form makes thermodynamic calculations awkward and can lead to confusion with the more tyical definition of the thermodynamic temerature, T= /n. Also in I, the enthaly density, h, is defined to be h = u +, where u is the internal energy density. Tyically, enthaly is defined to be h=u+. Again, there is nothing inherently incorrect with this definition, but it can also lead to confusion. These shortcomings are addressed here by modifying the distribution in I to aroximate a nonrelativistic bi- Maxwellian exanded for small ressure anisotroy with only first-order terms retained, and by making a small modification to the 0,0 comonent of the stress tensor. 3. Aroximate arallel and erendicular rojection oerators were used in I as annihilators of the gyroscale ortions of the exact moment equations to derive evolution equations for the arameters of the fluid system. Use of these oerators leads to nearly redundant evolution equations which only agree in the nonrelativistic limit. Thus, the redundancy leads to surious instabilities in the moderate to ultrarelativistic temerature regimes of linear theory. This issue is solved by relacing the rojection oerators with more fundamental annihilators and discussed further in Sec. IV B. 4. The form of the relativistic heat flux evolution equation rovided in I omits relevant terms from the gyrohase deendent ortion due to an ordering error. Also, the nonrelativistic form of the closure resented in Hazeltine and Mahajan 11,1 contains algebraic errors which, when combined with the omission noted above, lead to an incorrect evolution equation for the arallel heat flux. III. RELATIVISTIC PLASMA CONCEPTS Here, we review some basic roerties of relativistic electromagnetic theory, define what it means for a lasma to be magnetized, discuss some of the consequences of magnetization, and resent the moments used in our theory. Because the majority of this material was covered in I, the resent treatment is brief. We use the Einstein summation convention throughout, with Greek indices running from 0 to 3 and Roman indices from 1 to 3. Boldface tye tyically reresents the threevector ortion of a four-vector, for instance an arbitrary fourvector C may be written as C =C 0,C. All seeds are normalized to the seed of light, so that c = 1. We use =diag 1,1,1,1 as the signature for our Minkowski tensor. A. Magnetized lasma We make use of the following Lorentz scalars formed from the Faraday tensor, F, and its dual, F, 1 F F = B E W, 1 1 F F = E B W. The latter relation is of significant imortance because, or equivalently E, will be a small arameter of our theory. Two conditions must be satisfied for our lasma to be considered magnetized: 1. The two electromagnetic field invariants must satisfy W 0, The thermal gyroradius must be small comared to any gradient scale length, 1, where is the ratio of the thermal gyroradius of any lasma secies to any gradient scale length. We assume the ordering for convenience. We will imlicitly use this definition of a magnetized lasma throughout the following analysis. B. Quasirojectors As is tyical in a magnetized lasma, notions of arallel and erendicular to the field lay imortant roles. Thus, we need a covariant meaning for arallel and erendicular. Such a meaning is rovided by 4 5 Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

3 Fluid model for relativistic, magnetized lasmas Phys. Plasmas 15, e F F W, b e. e and b become aroximate erendicular and arallel rojection oerators in the magnetized limit. In a frame in which the transverse electric field vanishes a subset of the instantaneous rest-frame R, the action of e and b on an arbitrary four-vector C =C 0,C is given by b C R = C 0,C, 8 e C R = 0,C. and have the tyical three-dimensional meaning: C =BB C/B =bb C, C =C C, where b is the standard abbreviation bb/b. Gradients of the rojection oerators will be used imlicitly later in our analysis. Thus, we resent their forms. To do so, we begin by recalling the Maxwell stress tensor = F F 1 4 F F and observe e = + W, b = W. Maxwell s Eqs. 13 and 14 resented in the following section imly = F J, where = / x. Thus, it is straightforward to show b = F W J + 1 b log W, e = F W J + 1 e log W. C. Closing Maxwell s equations Since lasmas are strongly couled to the electromagnetic field, we must consider a closure involving Maxwell s equations. The couling of the electromagnetic field to a lasma enters a fluid descrition through the second-moment equation, which constitutes the conservation of energy-momentum. 15 In relativistic form, the secondmoment equation takes the form T F J =0, 1 where T reresents the total summed over all secies energy-momentum tensor for the lasma and J is the current density four-vector. Thus, the second-moment equation is used as a constitutive relation for magnetized lasmas, roviding closure to Maxwell s equations, F = J, 13 F + F + F =0. 14 It remains to comute the current density in a magnetized lasma. Equation 1, when comosed with F, rovides two comonents of the current density e J = F W T. 15 There are two indeendent comonents because the erendicular quasirojector has a two-dimensional null sace. Charge conservation J =0 and quasineutrality J U = rovide the two remaining comonents of the current density, where U =,V is the local 4-velocity of the fluid, with =1 V 1 the relativistic dilation factor. That Eq. 17 rovides a good reresentation of quasineutrality will be resented in Sec. III D. We conclude that knowing the lasma stress tensor, and thus the current density, is sufficient to close Maxwell s equations. D. Moments Our analysis involves moments u to and including the fourth rank. We exress each moment in terms of the distribution function fx,, where reresents the fourmomentum, = d3 0 f, T = d3 0 f, M = d3 0 f, R = d3 0 f Here, d 3 / 0 reresents the invariant momentum-sace volume, where 0 = m +. is the four-vector fluid article-flux density, T is the stress-energy tensor, M is tyically referred to as the stress flow tensor, and R will be referred to as the energy-weighted stress tensor. The exact moments of the collisionless kinetic equation associated with the four requisite moments for our analysis reresent article conservation, momentum evolution, stressflow evolution, and energy-weighted stress evolution, =0, 3 Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

4 TenBarge, Hazeltine, and Mahajan Phys. Plasmas 15, T = ef, 4 = n R 1,V + V E, 30 M = ef T + F T, R = ef M + F M + F M. 5 6 Note in the second and higher moment equations, the left-hand side involves the macroscoic scale, while the right-hand side deals with the short gyroscale. Thus, the small-gyroradius limit is obtained formally by allowing the charge to become arbitrarily large, e. No restrictions on the size of higher order moments is assumed. Our analysis does not require higher moments because we only need those corresonding to the scalar coefficients aearing in the energy-momentum tensor. This tensor rovides the framework for the closure of the lasma- Maxwell system. At this oint, we restrict our analysis to a lasma with a single ion secies in the interest of simlicity. We define the Lorentz scalar 0 R =d 3 f R to be the rest-frame density, n R, and define the fluid velocity of a secies to be U = /n R. 7 In order to satisfy quasineutrality, we require, to leading order, the electrons and ions have the same rest-frame densities, and reside in the same aroximate rest-frame to avoid arbitrarily large current densities; we do not restrict lasma flow, however. Equation 17 then follows as the leading order exression of quasineutrality. E. Gyro-ordering We must now determine evolution equations for the four comonents of the flux density. First, we note that all moments can be exanded in the form = 0 + 1, where the arenthetical subscrit refers to the order of the term with resect to the gyroradius. Thus, Eq. 4 rovides F 0 0 =0, 8 where we distinguish the lowest order Faraday tensor, F 0 F E =0, from its first-order counterart F 1 F F 0 E. Recalling the action of the Faraday tensor on a four-vector, Eq. 8 imlies 0 0 E + 0 B =0, 9 which reroduces the familiar MHD Ohm s law, E+VB =0. As such, Eq. 8 fixes the two erendicular comonents of the flow. The article conservation law, Eq. 3, fixes another of the comonents. At this oint, we dro the ordering subscrits and use and U to refer to the zeroth-order fields from this oint on. Similarly, we dro the ordering subscrit from the Faraday tensor where it is nonessential. We can now write the flow in the form where V E =Eb/B, V =bb V, and is evaluated at the lowest order flow velocity. Before moving on, we note that Eq. 4 has become T 0 = ef ef taking gyro-ordering of the moments and the Faraday tensor into account. The remainder of this aer is devoted to comuting the stress tensor. Conventional MHD avoids this issue by assuming the stress tensor has the thermodynamic equilibrium form T = + hu U, 3 where is the ressure and h is the enthaly density. This form only ertains to the highly collisional regime in which thermal relaxation occurs more raidly than any other rocess of interest. Thus, our analysis can be viewed as taking lace in a regime of much lower collisionality. We ignore collisions altogether and comute the stress tensor subject to electromagnetic forces alone. IV. COVARIANT EVOLUTION EQUATIONS A. Magnetized stress We use the magnetized limit of Eq. 5 to find F T + F T =0. 33 We use indicial symmetry of the stress tensor, antisymmetry of the Faraday tensor, along with roerties of the rojection oerators to conclude the stress tensor must have the following form: T = + hu U k k e + Q k U + U k, 34 where = + /3, =, Q, and h are Lorentz scalars corresonding to ressure, ressure anisotroy, total arallel heat flow, and enthaly density, resectively. We differentiate between the arallel flow of arallel heat, q, and the arallel flow of erendicular heat, q, with Q =q +q and Q =/ 5q 3/ 5q. It is imortant to note that this distinction does not enter at this order in the moment equations. The total arallel heat flow is the only distinct comonent that aears in the stress tensor. This stress tensor differs from that in I rimarily in notation. Here, the enthaly resented corresonds to the standard thermodynamic definition, h=u+, where u=t R 00 is the energy density. In I, h =u+. k must satisfy e k =0 to satisfy force balance and U k =0 to reserve the significance of and. These constraints on k leave free only one comonent, which corresonds to the Lorentz boosted unit vector b. Thus, k = F U = W W B B W V,b + B W V V E + E W E. 35 Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

5 Fluid model for relativistic, magnetized lasmas Phys. Plasmas 15, Two evolution equations are rovided by Eq. 31, once we identify an annihilator of the 1 term. Aroriate choices in the magnetized limit are k and U, since k U F. We find k T B = en R E 36 W and U T =0. 37 These equations advance the arallel momentum and total scalar ressure/energy, resectively. B. Subtleties of annihilator choice The evolution equations in I make use of the rojection oerators as annihilators of the gyroscale deendent ortions of the moment Eqs This annihilator choice leads to subtle inconsistencies in the derived evolution equations, resulting from imlicit, redundant use of evolution equations. Further, the inconsistencies cause surious instabilities to develo in the linear theory for moderate to ultrarelativistic temeratures. Consider first the arallel rojection oerator. We can write the oerator in terms of U and k as b =k k U U +O. Similarly, we can write e = +U U k k +O. If we oerate on the third rank moment Eq. 0, with U U, the resulting equation would be an evolution equation for the total energy which agrees with that found at the second rank, Eq. 37, only in the nonrelativistic limit. Therefore, oerating with b would result in the imlicit usage of a redundant energy equation that disagrees with the lower rank derived equation in all but the nonrelativistic limit. The redundancy continues to higher order and with usage of the erendicular oerator. Thus, we avoid using the rojection oerators as annihilators in favor of more fundamental tensors in our system, U and k. C. Magnetized stress flow The exression for M given in I does not allow for searate arallel and erendicular heat flows. The three auxiliary arameters aearing in the stress flow tensor emloyed in I only ermit deendence of the stress flow on,, and Q. We modify the model for the stress flow t include an additional auxiliary arameter m 4 in what follows to ermit the freedom of having two arallel heat flows. In the magnetized limit, the fourth-rank conservation law determines the form of the stress-flow tensor F M =0, 38 where the suer subscrit arentheses indicate indicial symmetrization over noncontracted indices, U U + U + U. We are also constrained by the definition of the stress-flow Eq. 0 and article flux Eq. 18. From the definitions, it can be seen that contracting two indices of the stress-flow reduces to the momentum flux M = m n R U. 39 Given the above two constraints and assuming the only fourvectors aearing in the stress-flow are U and k, the stressflow must have the form M = m n R U U U + m k M k, 40 k where M 1 = U +6U U U, 41 M = b U +4U U U, M 3 = k +6U U k, M 4 = b k 3 k, and the m k are scalars to be determined later. The M k also satisfy M k =0, so that the second constraint above is satisfied. We construct an evolution equations for the magnetized stress-flow by finding annihilators for the right-hand side of Eq. 5. Two such equations are: k k M =e B Q E, 45 W U k + U k M = e B W E h These equations can be considered to advance the arallel ressure and total arallel heat flow, resectively. We note that we cannot evolve the two arallel heat flows individually at this order. This is because evolving the searate heat flows requires a timelike 0-comonent derivative of the elements of the stress flow containing each arallel heat flow. It will become clear after evaluating the m k that such searation is not ossible in this order. The m k aearing in the stress flow can be taken to be auxiliary arameters of our system. Thus, we will need to exress them in terms of the dynamical variables aearing in our system. As such, it is convenient to examine the instantaneous rest frame comonents of the stress flow in terms of the m k, which are listed in Aendix A. D. Magnetized energy-weighted stress We construct the energy-weighted stress tensor in much the same way as the three revious tensors. We begin with the constraint rovided by the fifth-rank conservation law in the magnetized limit F R =0. 47 Our second constraint follows from the definitions of the energy-weighted stress Eq. 1 and the stress Eq. 19 when contracting two indices of the energy-weighted stress R = m T. 48 Our third constraint follows from contracting all four indices of the energy-weighted stress, Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

6 TenBarge, Hazeltine, and Mahajan Phys. Plasmas 15, R = m, 49 where =T = u+3. Unlike the third rank tensor whose indicial symmetrization is straightforward, the fourth rank tensor will have unique symmetrizations based on each tensor s construction, which are given in Aendix B. The following exression gives the simlest fourth-rank tensor that satisfies the above constraints, without introducing new indeendent variables: where R = m U U T +8U U U U Q k U U U + r k R k, k R 1 = +6 U U +48U U U U, R = b +8b U U + U U U U U U, 5 R 3 = U k 8U k k k, R 4 = b U k 6U k k k, R 5 = e b +e U U +b U U +16U U U U. 55 It can be seen that the R k =0 so that R = m T and R =m. The extra terms multilying m in Eq. 50 account for over counting certain elements of T due to symmetry conditions on R. Again, we construct evolution equations for the energyweighted stress by identifying annihilators of the right-hand side of Eq. 6, k k k R =3e B W E m 1 + m. 56 This can be viewed as evolving the arallel comonent of the arallel heat flow. As in the stress flow tensor, the r k can be viewed as auxiliary arameters. Thus, we need to exress them in terms of the rest-frame comonents of the energy-weighted stress. Such exressions are rovided in Aendix A. We now have evolution equations for n R,,, Q, q, and the three vector comonents of. We will take these to be our set of dynamical variables. We consider the enthaly, h, to be an auxiliary arameter in much the same way we treat the m k and r k as auxiliary arameters. Thus, our fluid system is nearly closed; however, we still need to evaluate the auxiliary arameters in terms of the dynamical variables. For this, we need a distribution function. V. DISTRIBUTION FUNCTION A. Choosing a distribution Since we have auxiliary arameters not yet related to our dynamical variables, we require a distribution function to close our fluid system. Any lowest order distribution chosen must be gyrotroic, solve the drift-kinetic equation, and reroduce the stress tensor, Eq. 34. Satisfying the first requirement is straightforward. The second is difficult to imlement in a fluid treatment and tyically abandons the fluid oint of view in favor of kinetic MHD, making the drift-kinetic equation art of the closure. 16,17 The third requirement restricts us to any of a class of distributions that reroduce the stress tensor. Therefore, we choose a reresentative distribution from the equivalence class of distributions reroducing the stress tensor, caable of also reresenting the fluid equations of motion. The arameters in the distribution are roortional to the dynamical variables of the fluid system and evolve according to the fluid equations. We use such a arameterized distribution in lace of the drift-kinetic equation to close our system. B. Exlicit form After examining revious literature, 13,14 it became clear in nonrelativistic theory, a bi-maxwellian or twotemerature Maxwellian is a good choice for caturing features of kinetic theory in a fluid aroach. As such, our distribution can be considered the relativistic analog of the nonrelativistic bi-maxwellian. Our distribution has the form fx, = f M 1+ˆ + + * e + + * b + k k k k + *e e + **k k e + Q b 1+Qˆ e + Qk k, 57 where f M is a relativistic Maxwellian. The scalars describe ressure anisotroy, while the Q scalars measure heat flow. Thus, our distribution can be arametrized by our dynamical variables, n R,,, q, and q. The form of our distribution mirrors that found in I only in the first three terms and the last term multilying the square brackets. Note that we do not simly write the distribution in the standard nonrelativistic form with the directional temerature deendence in the exonent. If we were to make such an attemt, evaluating moments of the distribution would become intractable. Recall that a relativistic Maxwellian has the following form: f M x, = N M e U P /T, where P = +ea is the canonical momentum, U P defines the invariant energy, Tx the scalar temerature, and N M x the scalar normalization factor. In the rest frame, we have f MR = N M e P0 /T. Moments of the rest frame Maxwellian have the form Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

7 Fluid model for relativistic, magnetized lasmas Phys. Plasmas 15, ds 1+s sn e 1+s 1 3 n 1K n = n, where K n is the nth MacDonald function, s= /m, and =m/t. We can now comute the normalization factor N M = n R e /T 4m TK, where =A 0 is the electrostatic otential. Thus, the rest frame Maxwellian is n R e 0 /T f MR = 4m TK. 58 Returning to evaluating the arameters of our distribution, we comare our distribution to the nonrelativistic bi- Maxwellian exanded for small ressure anisotroy to determine */, */, */, and **/. Doing so yields fx, = f M 1+ˆ + e b + e +4 b + 4k k k k + e e 4k k e + Q b 1+Qˆ + e + Qk k. For reference, exanding a bi-maxwellian for small ressure anisotroy yields fx,v = N 1/ ex mn v + v = N 3/e mnv mn mn 6 v v + 6 v +4v mn C. Scalar moments We choose ˆ and Qˆ to ensure that the rest-frame density is Maxwellian and the rest-frame flow velocity vanishes. and are chosen so that =nt=t 33 R +T 11 R /3 and =T 33 R T 11 R. is chosen by matching the nonrelativistic limit =m/t of R 1133 R to its bi-maxwellian counterart, m /n. Q 3 is chosen to satisfy T 03 R =Q, and Q is chosen by matching the nonrelativistic limits of the elements of the stress flow tensor involving heat flow to their bi-maxwellian counterarts, i.e., M 003 R =mq =M 333 R +M 113 R =mq +mq. Thus, in the rest-frame, the distribution function becomes f R x, = f 1 MR K 4 4 K m 4 + m 4 4 q 3 K Q m m + 1 K 4 6 K m m 7 m 4 + n R K K m + q m, 60 where and refer to ressure and ressure anisotroy, while and refer to arallel and erendicular momenta. Exlicitly, the scalar comonents of the distribution are = mn R /, ˆ = 1 K 4 6 K, = 1 K, 6 mn R v 4 + v 4 4v v, where N is the normalization factor, v is the article velocity, and refer the arallel and erendicular ressure, and and refer to the scalar ressure and ressure anisotroy. In the instantaneous rest-frame with coordinates oriented such that B=0,0,B, our distribution reduces to f R x, = f MR1+ˆ + m + + m Q Qˆ + m m + Q m, m where and here refer to arallel and erendicular comonents of momenta. = 1 K 4 18, = 7, Qˆ = K Q q 1, Q 3 = q q Q =, 3 q mn R K K, where K= / K K 4 / Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

8 TenBarge, Hazeltine, and Mahajan Phys. Plasmas 15, VI. CLOSED FLUID EQUATIONS A. Covariant closure summary We have chosen n R, V,,, q, and q as the dynamical variables of our collisionless, small gyroradius fluid system. The covariant evolution equations for the chosen dynamical variables of our system are =0, k T = en R B W E, U T =0, k k M =e B W Q E, U k + U k M = e B W E h + 3, k k k R =3e B W E m 1 + m, where the flux, stress T, stress flow M, and energyweighted stress R are given by Eqs. 18 1, resectively, and the m k are given in Aendix A. Therefore, Eqs constitute a closed covariant set of fluid equations. B. Three-vector form It is often convenient to exress fluid equations in threevector form, sacrificing exlicit Lorentz covariance. As such, we resent the three-vector form of our closed fluid system here. We begin by noting the following identities: U = d dt = d d, k = d ds, where d/dt is the conventional convective derivative and reresents the roer time. The exlicit forms of Eqs. 70 and 71 can be exressed as hk dv d + d ds + dq d Q d log n R + Q k dv d ds 1 log W d + 6 ds 3 k B = en R E, 75 W d d h + hd log n R dq d ds dv k 3 ds 1 log W d Q k dv 6 d d Q k =0, where 76 k = 1 W 1 B d dt V E b V 1 d log W. 77 ds From the third moment equations, we have from Eqs. 7 and 73 d d m 1 + m + m 1 + m k dv ds d log n R d +m m 4 k +1m 3 k dv d m d log W 4 ds +3 d dsm m 4 =e B W Q E, d d5m m 4 + d ds m 1 + m + 5m 1 +3m k dv d 1 m d log W +7m ds 3 m 4k dv ds 6m 3 d log n R d = e B W E h m d log W 4 + m n R k dv d d Turning to the energy-weighted stress, we have from Eq. 74 d d 5r 3 +3r 4 + d ds 3r 1 +6r +18r 1 k dv d + r 30k dv d 3 d ds log W 5r 3 +3r 4 d d log n R +3k dv ds+ r 56k dv d W + 3 d ds log +3m k dv d =3e B W E m 1 + m, where the r k are given in Aendix A. C. Nonrelativistic limit 80 We now resent the fully nonrelativistic NR, = m / T 1 and V V 1/, form of our closed system. Since labelling the rest-frame is somewhat inaroriate in this limit, we use nn R. We also use the common notation b. In the NR limit, Eqs. 69, 75, 76, and become after some maniulation dn + n V =0, dt 81 Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

9 Fluid model for relativistic, magnetized lasmas Phys. Plasmas 15, mnb dv dt + + log B = ene, 8 log d B dt n 3 + q +q q log B =0, 83 log d + dt Bn q q log B =0, log d q q B 3 dt n m n =0, log d q q + dt n m n mn log B =0. 86 The NR limit of our closure coincides with a bi- Maxwellian MHD closure in which gyroviscous comonents of the stress tensor are retained as resented by Ramos. 13 As such, the system roduces disersion relations whose numerical coefficients coincide with those obtained through kinetic theory, and the system correctly redicts the onset of the mirror and firehose instabilities. D. Linear redictions Having comleted our closure, we now examine some basic redictions of the linearized relativistic system. Linearizing Eqs. 15 and about an isotroic equilibrium with no heat flow, equal electron and ion equilibrium temeratures, and nonrelativistic flow seed yields a lengthy set of equations resented fully in Aendix A. We resent the following linearized version of Eq. 7 to comare with the nonrelativistic limit of the same equation as an examle: v sf s +3 s K s n n + sf s + s K s v s + 3 K 4 s s s v s K 4 s + s K s cosq s 31 K 4 s 5 s K s cos Q s + s s K s kˆ v =0, n 3v s n + sv 6 5 cosq s cosq s s kˆ v =0, where v=/k, kˆ =k /k, cos=k /k, v A =B / 0 m i +m e n, subscrit s denotes secies, suerscrit T denotes a sum over the secies, and f s = s + s K 1 s s K s 1 K s. Using the full set of linear equations, we lot the hase velocity squared versus i i.e., the inverse temerature in Fig. 1 for v A =10 6, =30, and m i /m e =1833. Also lotted in Fig. 1 as the dashed lines are the linearized version of the FIG. 1. Phase velocity squared vs i =m i /T for the general linearized evolution equations solid and their nonrelativistic limit dashed. v A =10 6, =30, and m i /m e =1833. nonrelativistic Eqs From lowest to highest hase seed for large, we have the slow magnetosonic, two ion acoustic, shear Alfvén, fast magnetosonic, and two electron acoustic modes. For the lotted arameters, the electron modes are the first to show significant deviation for increasing temerature at roughly 100 kev. At this temerature, the nonrelativistic theory begins to redict suerluminal hase velocities for the electron acoustic modes. Also of note, the hase velocity of the shear and slow magnetosonic Alfvén modes behave quite differently in the ultrarelativistic regime. In this regime, the correct disersion relation is v i + e v 8 A, where the nonrelativistic theory would simly state v v A. VII. SUMMARY Maxwell s equations are closed in a magnetized lasma when the four-vector current can be exressed in terms of the stress tensor T = T, secies where T is the stress tensor of the individual lasma secies. This closure rocedure is given by Eq. 15 and later equations. Thus, a closed fluid descrition of lasma dynamics relies on equations that fix the evolution of the stress tensor of each lasma secies. For this reason, the stress tensor is said to rovide the constitutive relation for a lasma fluid closure. We obtain our descrition of the stress tensor, Eq. 34, via electromagnetic constraints rather than the simler MHD thermodynamic arguments T = + hu U k k e + Q k U + U k, 89 where b and e are aroximate rojection oerators in- Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

10 TenBarge, Hazeltine, and Mahajan Phys. Plasmas 15, troduced in Sec. III. The fluid 4-velocity, U, and the heat flow, Q k =q +q k, are constrained by F U =0, 90 M R 000 = m n R +3m 1 + m, M R 003 =5m m 4, e k =0, 91 M R 011 = M R 0 = m 1, U k =0. 9 Equation 90 rovides the first of our evolutionary constraints by reroducing the familiar EB drift. We still need evolution equations for the two remaining free comonents of the flow,, which are the rest-frame density, n R, and the arallel flow, V. Also, from the stress tensor, we need to evolve = + /3, =, h, and the two comonents of the rest-frame heat flow, q and q. Quasineutrality, Eq. 17, requires that n R be the same for all secies, while the other quantities in the stress tensor are free to vary from secies to secies. Thus, we choose the following six arameters n R, V,,, q, and q as our dynamical variables. The evolution equations for the six dynamical variables of our system in various forms are given in Sec. VI. At this oint in the closure, we have ten scalar auxiliary arameters which are not fixed. These are the enthaly density, h, the four scalar arameters, m k, of the stress flow, and the five scalar arameters, r k, of the energy-weighted stress. We exress these auxiliary arameters via a reresentative distribution, which is arameterized by our dynamical variables. Thus, the distribution evolves according to Eqs , and our auxiliary arameters can be exressed in terms of the dynamical variables, as resented in Aendix A. Our closure rovides a more accurate hysical descrition of relativistic, magnetized fluid lasmas than reviously resented by Hazeltine and Mahajan. 10 The system allows detailed study of various astrohysical and laboratory lasmas at a more realistic level than MHD. Also, in the nonrelativistic limit, our closure reduces to a set of equations resented by Ramos 13 obtained via a bi-maxwellian closure in which gyroviscous terms of the stress tensor are retained. In forthcoming aers, we will exlore the thermodynamic roerties of an imerfect relativistic lasma through the inclusion of collisions. ACKNOWLEDGMENTS This work was suorted by the U.S. Deartment of Energy. APPENDIX A: AUXILIARY PARAMETERS Here, we list the nonvanishing rest-frame moments of the third and fourth rank in terms of the auxiliary arameters m k and r k and exress the auxiliary arameters of our system in terms of the dynamical variables. We orient our rest-frame such that B=0,0,B. The nonvanishing comonents of the third rank moments are M R 033 = m 1 + m, M R 113 = M R 3 = m 3 3 m 4, M R 333 =3m 3 + m 4. And for the fourth rank, we have R R 0000 = m u r 1 +10r +4r 5, R R 0011 = R R 00 = m +5r 1 + r + r 5, R R 0033 = m +5r 1 +8r +r 5, R R 0003 = m Q 3r 3 3r 4, R R 0113 = R R 03 = r 3, R R 0333 = 5r 3 3r 4, R R 1111 = R R =3R R 11 =3r 1, R R 1133 = R R 33 = r 1 + r + r 5, R R 3333 =3r 1 +6r. The auxiliary arameters of our system are determined by evaluating the rest frame moments above via our distribution function, Eq. 60, m 1 = m K 1 3 K K 4 6 K 4 K K 9 K 18 K, m = m K 4 1 K 4 K 5 3 K K, m 3 = mq 1 K 4 1+Q 3 K +3Q, m 4 = mq K 4 1 Q K +3Q. r 1 = m n R K 3 K K 4 6 K K 9 K K, K 5 Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

11 Fluid model for relativistic, magnetized lasmas Phys. Plasmas 15, r = 1 m K 4 n R r 3 = m + 3 K 4 K 4 K Q K 5 K +3QK 4, K + K 5 K, r 4 = m K 4 3 K Q 4K 5+3Q K 5 +3QK 4, r 5 = 1 m K 5, n R K where K s = s K s K 5 s K 4 s. We can also now exress the enthaly density in terms of our dynamical variables by looking at T 00 R =h, h = mn R K. APPENDIX B: FOURTH RANK SYMMETRIZATION The construction of R will involve tensors of the three following forms, aside from fully asymmetric and fully symmetric: 1. Symmetric times asymmetric, i.e., U k. This form will have 1 terms in the symmetrization A 1 = A + A + A + A + A + A + A + A + A + A + A + A.. Symmetric times symmetric, i.e., U U. This form will have six terms, A = A + A + A + A + A + A. For the secial case of a fourth rank comosed of the two identical symmetric second rank tensors, i.e.,, only the first three terms contribute to symmetrization. 3. Third rank symmetric times a four-vector, i.e., k U U U. This form has four terms, A 3 = A + A + A + A. APPENDIX C: LINEARIZED EVOLUTION EQUATIONS Here, we resent the full set of linearized Eqs. 15 and about an isotroic equilibrium with no heat flow, equal electron and ion equilibrium temeratures, and nonrelativistic flow seed, v n n cosv kˆ v =0, C1 s s K s vv cos s 3 cos s + v Q s = iq sn k E, C s f s v n n + 1 sf s v s + cosq s =0, C3 v sf s +3 s K s n n + sf s + s K s v s + 3 K 4 s s s v s 31 K 4 s 5 s K s cos Q s + K 4 s s K s cosq s s + kˆ s v =0, K s s s +5 s K s vv + s f s cos n n sf s + s K s cos s 3 K 4 s s s cos s C4 K 4 s +5 s K s v Q s = iq sn k s s K s E, C5 6 s K s + svv +f s + s K s cos n n f s + s K s cos s 4 K 4 s 3 s cos s v + 4 s K s s K s K Q s 5 K 5 s vk 5 s Q s 3 s K s = iq sn s k K s E, i i K i + e e K e v i + e v A cos v C6 i + e v A kˆ kˆ v vkˆ T + 1 v kˆ T 3 n =0. C7 1 A. Ferrari, ARAA 36, F. C. Michel, Rev. Mod. Phys. 54, V. I. Berezhiani and S. M. Mahajan, Phys. Rev. E 5, Y. B. Zeldovich and I. Novikov, Relativistic Astrohysics University of Chicago Press, Chicago, M. A. Aloy, E. Müller, J. M. Ibanez, J. M. Marti, and A. MacFayden, Aita J. 531, L Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js

12 TenBarge, Hazeltine, and Mahajan Phys. Plasmas 15, M. Camenzind, Aeros. Am. 156, A. I. MacFayden, S. E. Woolsey, and A. Hager, Aita J. 550, A. M. Anile, Relativistic Fluids and Magneto-Fluids: With Alications in Astrohysics and Plasma Physics Cambridge University Press, Cambridge, G. F. Chew, M. L. Goldberger, and F. E. Low, Proc. R. Soc. London, Ser. A 36, R. D. Hazeltine and S. M. Mahajan, Aita J. 567, R. D. Hazeltine and S. M. Mahajan, Phys. Plasmas 9, R. D. Hazeltine and S. M. Mahajan, Phys. Plasmas 9, J. J. Ramos, Phys. Plasmas 10, P. B. Snyder, G. W. Hammett, and W. Dorland, Phys. Plasmas 4, E. G. Tsikanshvili, J. G. Lominadze, A. D. Rogava, and J. J. Javakishvili Phys. Rev. A 46, M. D. Kruskal and C. R. Oberman, Phys. Fluids 1, M. N. Rosenbluth and N. Rostoker, Phys. Fluids, Downloaded Ar 009 to Redistribution subject to AIP license or coyright; see htt://o.ai.org/o/coyright.js