A Hamiltonian gyrofluid model based on a quasi-static closure
aa r X i v : . [ phy s i c s . p l a s m - ph ] J u l Under consideration for publication in J. Plasma Phys. A Hamiltonian gyrofluid model based on aquasi-static closure
E. Tassi , T. Passot and P.L. Sulem Universit´e Cˆote d’Azur, CNRS, Observatoire de la Cˆote d’Azur, Laboratoire J.L. Lagrange,Boulevard de l’Observatoire, CS 34229, 06304 Nice Cedex 4, France(Received xx; revised xx; accepted xx)
A Hamiltonian six-field gyrofluid model is constructed, based on closure relationsderived from the so-called ”quasi-static” gyrokinetic linear theory where the fields areassumed to propagate with a parallel phase velocity much smaller than the parallelparticle thermal velocities. The main properties captured by this model, primarily aimedat exploring fundamental problems of interest for space plasmas such as the solar wind,are its ability to provide a reasonable agreement with kinetic theory for linear low-frequency modes, and at the same time to ensure a Hamiltonian structure in the absenceof explicit dissipation. The model accounts for equilibrium temperature anisotropy, ionand electron finite Larmor radius corrections, electron inertia, magnetic fluctuationsalong the direction of a strong guide field, and parallel Landau damping, introducedthrough a Landau-fluid modeling of the parallel heat transfers for both gyrocenterspecies. Remarkably, the quasi-static closure leads to exact and simple expressions forthe nonlinear terms involving gyroaveraged electromagnetic fields and potentials. One ofthe consequences is that a rather natural identification of the Hamiltonian structure ofthe model becomes possible when Landau damping is neglected. A slight variant of themodel consists of a four-field Hamiltonian reduction of the original six-field model, whichis also used for the subsequent linear analysis. In the latter, the dispersion relations ofkinetic Alfv´en waves and the firehose instability are shown to be correctly reproduced,relatively far in the sub-ion range (depending on the plasma parameters), while thespectral range where the slow-wave dispersion relation and the field-swelling instabilitiesare precisely described is less extended. This loss of accuracy originates from the breakingof the condition of small phase velocity, relative to the parallel thermal velocity of theelectrons (for kinetic Alfv´en waves and firehose instability) or of the ions (in the case ofthe field-swelling instabilities).
1. Introduction
Modeling the dynamics of collisionless (or weakly collisional) plasmas at scales compa-rable to or smaller than the ion Larmor radius is an important issue both for laboratoryand astrophysical plasmas. At the level of a kinetic description, a valuable tool is given bythe gyrokinetic theory which provides a reduction of the Vlasov-Maxwell (VM) equationsby focusing on phenomena with a characteristic time scale large compared with the iongyro-period. This approach, which eliminates the dependency on the gyration angle,typically adopts, as dynamical variables, the distribution functions of the gyrocentersrather than those of the particles. Within the gyrokinetic framework, a subset of modelsconsists of the so-called δf -gyrokinetic models, which assume the gyrocenter distributionfunctions of the various particle species, to be close to those of an equilibrium state. Inspite of this reduction, numerical simulations of three-dimensional gyrokinetic equationsin a turbulent regime (even in the δf framework) require huge computational resources,which justifies the development of simpler (although less complete) descriptions basedon gyrofluid equations governing the evolution of a finite number of moments of thegyrocenter distribution functions. The relation between gyrocenter and particle momentsis well-defined and can usually be computed perturbatively. As in the case of the fluidhierarchy derived from the VM equations, a gyrofluid hierarchy of equations needs tobe closed, in order to obtain a gyrofluid model with a finite number of dynamicalvariables. An important condition to prescribe at the level of closure assumptions is thepreservation, in the absence of dissipation, of the Hamiltonian character of the parentgyrokinetic equations. This guarantees that in the reduction from a gyrokinetic to agyrofluid system, not only no uncontrolled dissipation of the total energy is introducedbut also that further invariants (Casimir invariants) of the system exist, and thatthe dynamics takes place on hyper-surfaces in phase space where the values of theseinvariants are constant. Another constraint is the consistency with the linear gyrokinetictheory. In particular, this requires retaining the influence of resonant effects such asLandau damping. A closure accounting for Landau damping in the δf approach typicallyintroduces dissipation and thus prevents the model from being Hamiltonian. Such formof dissipation is, however, voluntarily added and the main requirement is that the modelpossesses a Hamiltonian structure when the dissipative terms are removed.In this spirit, the main goal of the present paper is to construct a gyrofluid modelpossessing the above mentioned properties, and primarily addressed to study phenom-ena relevant for collisionless space plasmas. Motivated by measurements of sub-protonfluctuations in the solar wind (see Sahraoui et al. (2010) and Alexandrova et al. (2013)or Bruno & Carbone (2013) for reviews), reduced fluid models have already been derivedand numerically integrated to explore the dynamics of space plasmas. Kinetic Alfv´en wave(KAW) turbulence was for example addressed in Boldyrev & Perez (2012). A more gen-eral Hamiltonian reduced gyrofluid model (Passot et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. (2018) for the generation of the solar wind compressive fluctuations, turnsout to occur in a wider range of beta parameters in the presence of temperature anisotropy(Tenerani et al. δf gyrokineticequations presented in Kunz et al. (2015) where, for the sake of simplicity, we assume anelectron-proton plasma with an equilibrium state described by bi-Maxwellian distributionfunctions with no mean drift velocity. Such a system fully satisfies our requirements.Indeed, it is a δf gyrokinetic model mainly conceived for pressure-anisotropic astrophys-ical plasmas and, as such, it specifically accounts for equilibrium temperature anisotropy(unlike most of gyrokinetic models which consider a generic equilibrium distributionfunction or specialize to the case of a Maxwellian equilibrium). Also, it accounts forparallel magnetic perturbations and it has been shown to possess (at least in the limit ofinterest for our derivation) a Hamiltonian structure (Tassi 2019). As far as the numberof moments to be retained in the gyrofluid model is concerned, the inclusion of Landaudamping requires to retain at least the first three moments for each particle species.This is why, under the assumption of a two-species plasma, we opt for the derivationof a six-field gyrofluid model evolving three moments, including parallel temperaturefluctuations, for each species. Nevertheless, a four-field Hamiltonian reduced versionwill also be presented and applied. Another novelty with respect to already existingHamiltonian gyrofluid models, is the adoption of a closure relation, referred to as quasi-static , derived from linear gyrokinetic theory in the limit of slowly-evolving fields. Moreprecisely, according to such closure, all gyrofluid moments that are not determined bygyrofluid evolution equations, are fixed according to their expression obtained from thegyrokinetic linear theory in the limit | ω {p k z v th k s q| !
1, where ω is the frequency of amode, k z the component of its wave vector along the direction of the guide field, and v th k s is the thermal speed of the species s , associated with the equilibrium temperaturealong the direction of the guide field. As such, this closure is suitable for fields slowly-evolving (i.e. quasi-static) with respect to particles travelling at the parallel equilibriumthermal speed v th k s . The derivation of this closure relation will be presented in AppendixA (see in particular Eq. (A 21) to find the expressions for the various gyrofluid momentsaccording to the quasi-static closure). We anticipate, however, two remarkable propertiesthat this closure possesses. The first one is that the quasi-static closure relation turnsout to be compatible with a Hamiltonian structure. The second one is that it allows forexact expressions, in terms of canonical Poisson brackets, for all the nonlinear termsin the gyrofluid equations, and in particular for those involving only gyroaveragedelectromagnetic fields or potentials. This is not the case, to the best of our knowledge,for the previously derived reduced gyrofluid models.The model assumes the presence of a strong magnetic guide field and evolves, for bothelectrons and ions, gyrocenter density fluctuations as well as velocity and temperaturefluctuations referred to the direction parallel to the guide field. A dissipative variant of themodel accounting for parallel Landau damping is then formulated through a Landau-fluidmodeling of the parallel heat fluxes (Hammett & Perkins 1990; Hammett et al. β e,i parameters, where β e,i indicates, for each species ( e for electrons and i for ions), the ratio between the equilibriumkinetic pressure and the magnetic pressure exerted by the guide field. In this respect,this gyrofluid model differs from most of the presently available gyrofluid models, whichrequire β e,i !
2. The gyrokinetic parent model
In order to derive a gyrofluid model only based on a quasi-static closure assumption,we consider as starting point the following set of gyrokinetics equations, which corre-sponds to the system provided by Eqs. (C58), (C60), (C66)-(C68) of Kunz et al. (2015)when collisions and equilibrium velocities are neglected and bi-Maxwellian distributionfunctions are chosen as equilibrium distribution functions for all the particle species. Forsimplicity, we specialize to the case of a plasma consisting of two species: electrons andone species of single ionized particles. The equations of the resulting gyrokinetic modelare given by: B r g s B t ` cB « J s r φ ´ v } c J s r A k ` µ s B q s J s r B k B , r g s ff ` v } BB z ˜r g s ` q s T k s F eq s ˜ J s r φ ´ v } c J s r A k ` µ s B q s J s r B k B ¸¸ “ , (2.1) ÿ s q s ż d W s J s r g s “ ÿ s q s T K s ż d W s F eq s ` ´ J s ˘ r φ ´ ÿ s q s ż d W s µ s B T K s F eq s J s J s r B k B , (2.2) ÿ s q s ż d W s v } J s ˜r g s ´ q s T k s v } c F eq s J s r A k ¸ “ ´ c π ∇ K r A k ` ÿ s q s m s ż d W s F eq s ˜ ´ Θ s v } v th k s ¸ p ´ J s q r A k c , (2.3) ÿ s β K s n ż d W s µ s B T K s J s r g s “ ´ ÿ s β K s n q s T K s ż d W s µ s B T K s F eq s J s J s r φ ´ ˜ ` ÿ s β K s n ż d W s F eq s ˆ µ s B T K s J s J s ˙ ¸ r B k B . (2.4)The index s P t e, i u adopted above indicates the particle species, so that quantitieslabelled with s refer to the electron species when s “ e and to the ion species when s “ i .In the system (2.1)-(2.4), the function r g s is defined by r g s p x, y, z, v } , µ s , t q “ r f s p x, y, z, v } , µ s , t q ` q s T k s v } c F eq s p v } , µ s q J s r A k p x, y, z, t q , (2.5)where r f s is the perturbation of the gyrocenter distribution function for particles of species s , q s is the charge of these particles (so that q e “ ´ e and q i “ e , with e indicating theproton charge) and m s their mass. Furthermore, c denotes the speed of light. The bi-Maxwellian equilibrium distribution function is given by F eq s p v } , µ s q “ ´ m s π ¯ { n T { k s T K s e ´ msv } T k s ´ µsB T K s , (2.6)where n is the uniform and constant equilibrium density, T k s and T K s are respectivelythe equilibrium temperatures of the s -th particle species parallel and in a plane per-pendicular to an equilibrium magnetic guide field of amplitude B , directed along the z direction of a Cartesian coordinate frame t x, y, z u . We suppose that the spatial domainof the system corresponds to a box D “ tp x, y, z q : ´ L x ď x ď L x , ´ L y ď y ď L y , ´ L z ď z ď L z u , with L x , L y and L z positive constants. All quantities of the system which dependon the spatial variables x , y and z are supposed to satisfy periodic boundary conditions onthe domain D , so that they can be expanded in Fourier series. We indicated with v } P R the velocity coordinate parallel to the guide field and with µ s “ m s v K {p B q P r , `8q the magnetic moment of the particle of species s in the unperturbed guide field, where v K corresponds to the velocity coordinate perpendicular to the guide field. We assumethat all functions depending on v } decay to zero as v } Ñ ˘8 . Functions dependingon µ s are assumed to tend to zero as µ s Ñ `8 and to be bounded at µ s “
0. Thecoordinate t P r , `8q refers to time. The expressions J s and J s are related to thestandard gyroaverage operators for the species s in Fourier space. Their definition can beintroduced explicitly in the following way: adopting the notation x to indicate a point ofcoordinates p x, y, z q P D and, similarly, k to indicate a point p k x , k y , k z q P D , where D isthe lattice defined by D “ tp πl {p L x q , πm {p L y q , πn {p L z qq : p l, m, n q P Z u , we canconsider a function f : D ˆ r , `8q Ñ R , periodic over D , so that it admits the Fourierrepresentation f p x , t q “ ř k P D f k p t q exp p i k ¨ x q . The action of the operators J s and J s on the function f is defined by J s f p x , t q “ ÿ k P D J p a s q f k p t q exp p i k ¨ x q , (2.7) J s f p x , t q “ ÿ k P D J p a s q a s f k p t q exp p i k ¨ x q , (2.8)where J and J indicate the zeroth and first order J Bessel functions, respectively, a s “ k K ρ K s is the perpendicular Larmor radius associated with the species s , with k K and ρ K s corresponding to the perpendicular wave number and the Larmor radius of the particle ofspecies s . The former is defined as k K “ b k x ` k y with k x “ πl {p L x q , k y “ πm {p L y q for l, m P Z , while the latter is given by ρ K s “ v K { ω cs , where ω cs “ eB {p m s c q is thecyclotron frequency referred to the guide field and related to the particle of species s .The leading order expression (up to second order terms in the perturbations) for themagnetic field is given by B p x, y, z, t q “ ∇ r A k p x, y, z, t q ˆ ˆ z ` p r B k p x, y, z, t q ` B q ˆ z, (2.9)where ˆ z is the unit vector along the z direction, r A k (referred to as magnetic fluxfunction) corresponds to the z component of the magnetic vector potential and r B k is the perturbation of the magnetic guide field, also referred to as parallel magneticperturbation or parallel magnetic fluctuations. We remark that the guide field is as-sumed to be spatially homogeneous. This assumption is valid for a local descriptionof space plasmas such as the solar wind, where the background magnetic field varieson scales so large that, in the local description, it can be assumed to be homoge-neous. The situation would be different, for instance, in the case of tokamak plas-mas. Indeed, gyrofluid models more oriented towards tokamak applications (as, for in-stance those of Snyder & Hammett (2001); Madsen (2013); Brizard (1992); Scott (2010);Waelbroeck & Tassi (2012); Keramidas Charidakos et al. (2015)) take into account back-ground magnetic inhomogeneities. The set of electromagnetic quantities involved in thesystem is completed by the electrostatic potential r φ “ r φ p x, y, z, t q . In Eqs. (2.2)-(2.4) weadopted the symbol d W s “ p πB { m s q dµ s dv } to indicate the volume element in spacevelocity.The parameters Θ s and β K s are defined by Θ s “ T K s T k s , β K s “ π n T K s B (2.10)and measure, for each species s , the equilibrium temperature anisotropy and the ratiobetween equilibrium kinetic and magnetic pressure, respectively.Equation (2.1) is the gyrokinetic equation related to the species s , whereas Eqs. (2.2)-(2.4) relate the gyrocenter distribution functions to electromagnetic quantities in thenon-relativistic limit. In particular, Eq. (2.2) corresponds to the quasi-neutrality relation,whereas Eqs. (2.3) and (2.4) descend form Amp`ere’s law projected along directionsparallel and perpendicular to the guide field, respectively.The gyrokinetic model (2.1)-(2.4) is valid for small perturbation of the equilibriumdistribution function ( δf approximation) and weak variations of the fields along thedirection of the guide field, the equilibrium temperature anisotropy Θ s and the parameter β K s of all the species, being kept finite in this asymptotics. Further details aboutthe derivation the regime of validity of the model and its derivation can be found inKunz et al. (2015). Its Hamiltonian structure, on the other hand, is presented in Tassi(2019).
3. The gyrofluid model
We define the following gyrofluid moments: r N s “ ż d W s r f s , r U s “ n ż d W s v } r f s , (3.1) r T k s “ T k s n ż d W s ˜ v } v th k s ´ ¸ r f s , r T K s “ T K s n ż d W s ˆ µ s B T K s ´ ˙ r f s , r Q k s “ T k s v th k s ż d W s ˜ v } v th k s ´ v } v th k s ¸ r f s . (3.2)For each particle species, the fields r N s and r U s represent the fluctuations of the gyrocenterdensities and parallel fluid velocities, respectively. On the other hand, r T k s and r T K s correspond to the fluctuations of the gyrofluid temperatures defined with respect tothe parallel and perpendicular gyrocenter velocities, respectively, whereas r Q k s indicatesthe gyrocenter parallel heat flux fluctuations. In defining the parallel temperature andheat flux fluctuations, we introduced the constant v th k s “ b T k s { m s , corresponding tothe parallel thermal velocity associated with the species s .We intend to derive a gyrofluid model by taking moments of the gyrokinetic equation(2.1) and by imposing a closure relation derived from the quasi-static linear theory.In particular, the gyrofluid model, accounting for equilibrium temperature anisotropy,should be able, in the limit of vanishing finite Larmor radius effects, to reproduce thefield-swelling instability criterion of Basu & Coppi (1984). We restrict to the evolutionof the first three moments referring to the parallel direction. Therefore, the resultinggyrofluid model should evolve the following six fields: r N e , r N i , r U e , r U i , r T k e and r T k i . Also,we show that the model conserves the total energy and, moreover, that it possesses anoncanonical Hamiltonian structure, as is the case for the parent gyrokinetic model (Tassi2019). We refer to such model as to the 6-field gyrofluid (GF6) model.For the sake of the comparison, carried out in Sec. 4, of the linear gyrofluid theorywith the linear gyrokinetic theory we also consider an extension of GF6 accounting fora Landau-fluid closure, analogously to that discussed in Hammett & Perkins (1990);Hammett et al. (1992); Snyder et al. (1997); Passot & Sulem (2007, 2015); Tassi et al. (2018). This variant of the model, which we denote as GF6L, differs from GF6 for theexpression of the parallel heat flux fluctuations r Q k s . Therefore, in order to avoid someredundancy in the exposition, in the following we present the model equations leaving r Q k s unspecified and we will subsequently indicate the corresponding expressions for r Q k s leading to GF6 and GF6L, respectively. The closure leading to GF6L, in particular, willbe given in Sec. 3.2.A further variant of the model, denoted as GF4, will also be considered in Sec. 4.This model evolves only the four fields r N e , r N i , r U e and r U i and represents a minimalHamiltonian model, derived from the quasi-static closure, capable to reproduce the field-swelling instability criterion. This variant of the model will be introduced in Sec. 3.2.The six-field system, both in the Hamiltonian (GF6) and dissipative (GF6L) versions,can be written, in a dimensionless form, as B N s B t ` r G s φ ` sgn p q s q τ K s G s B k , N s s ´ r G s A k , U s s ` B U s B z “ , (3.3) BB t ˆ m s m i U s ` sgn p q s q G s A k ˙ ` „ G s φ ` sgn p q s q τ K s G s B k , m s m i U s ` sgn p q s q G s A k (3.4) ´ τ K s Θ s r G s A k , N s ` T k s s` BB z ˆ sgn p q s q G s φ ` τ K s G s B k ` τ K s Θ s p N s ` T k s q ˙ “ , B T k s B t ` r G s φ ` sgn p q s q τ K s G s B k , T k s s´ r G s A k , U s ` Q k s s ` BB z p U s ` Q k s q “ , (3.5) ÿ s ˆ sgn p q s q G s N s ` p ´ Θ s q Γ s φτ K s ` p Θ s G s ´ q φτ K s ` sgn p q s qp ´ Θ s qp Γ s ´ Γ s q B k ` sgn p q s q Θ s G s G s B k ˘ “ , (3.6) ∇ K A k “ β K e ÿ s ˆ m i m s ˆ ´ Θ s ˙ p ´ Γ s q A k ´ sgn p q s q G s U s ˙ , (3.7) B k “ ´ ÿ s β K s ˆ G s N s ` p ´ Θ s qp Γ s ´ Γ s q sgn p q s q φτ K s ` Θ s G s G s sgn p q s q φτ K s ` p ´ Θ s qp Γ s ´ Γ s q B k ` Θ s G s B k ˙ . (3.8)In Eqs. (3.3)-(3.8), we adopted the following normalized quantities N s “ r N s n , U s “ r U s c s K , (3.9) T k s “ r T k s T k s , Q k s “ r Q k s n T k s c s K , (3.10) φ “ e r φT K e , A k “ r A k B ρ s , B k “ r B k B , (3.11) x “ ¯ xρ s , y “ ¯ yρ s , z “ ¯ zρ s , t “ ω ci ¯ t, (3.12)where the quantities with overbars in Eq. (3.12) are the dimensional spatial and timecoordinates. In Eqs. (3.9)-(3.12) the quantities : c s K “ c T K e m i , ρ s “ c s K ω ci (3.13)were introduced, which indicate the sound speed and the sonic Larmor radius, respec-tively, based on the perpendicular equilibrium temperature.The parameter τ K s “ T K s T K e , (3.14)for s “ e, i , on the other hand, measures the ratio of the equilibrium perpendiculartemperatures.In Eqs. (3.3)-(3.8) we introduced the operator b s “ ´ ∇ K ρ th K s , with ∇ K denoting theLaplacian relatively to the transverse variables and ρ th K s “ ω cs c T K s m s (3.15)indicating the perpendicular thermal Larmor radius associated with the species s .The canonical bracket r , s , on the other hand, is defined as r f, g s “ B x f B y g ´ B y f B x g ,for two functions f and g .The gyroaverage operators G s , G s , Γ s , Γ s in Eqs. (3.3)-(3.8) can in turn beexpressed in terms of the operators G p b s q “ e ´ b s { , G p b s q “ e ´ b s { , (3.16) Γ p b s q “ I p b s q e ´ b s , Γ p b s q “ I p b s q e ´ b s , (3.17)which have to be intended as Fourier multipliers whose symbols are obtained by replacing b s by k K ρ th K s . We make this statement more precise, as an example, in the case of theoperator G i , referred to the ion species. The expression G i f is defined by G i f p x , t q “ ř k P D G p b i q f k p t q exp p i k ¨ x q “ ř k P D exp p´ k K ρ th K i { q f k p t q exp p i k ¨ x q , for a function f periodic in space. Analogous expressions are valid for the other gyroaverage operators.In Eq. (3.17), in particular, the symbols I and I indicate the modified Bessel functions I of order zero and one, respectively.The expressions for the operators G s and G s given in Eq. (3.16) correspond tothose present in Brizard (1992) and follow from assuming that the perturbation of thedistribution function can be written as r f s p x, y, z, v } , µ s , t q “ F eq s p v } , µ s q `8 ÿ m,n “ ? m ! H m ˜ v } v th k s ¸ L n ˆ µ s B T K s ˙ f mn s p x, y, z, t q , (3.18)where H m and L n indicate the Hermite and Laguerre polynomials, respectively, of order m and n , with m and n non-negative integers. The functions f mn s are coefficients of theexpansion and are related to the moments of r f s , with respect to Hermite polynomialsin v } { v th k s and Laguerre polynomials in µ s B { T K s . Indeed, from the orthogonalityproperties of Hermite and Laguerre polynomials, the following relation holds: f mn s “ n ? m ! ż d W s H m ˜ v } v th k s ¸ L n ˆ µ s B T K s ˙ r f s . (3.19) : Note that, according to a customary notation, in the symbols c s K and ρ s , the subscript s is to indicate sonic quantities and not the particle or gyrocenter species. f mn s : f s “ N s , f s “ d Θ s τ K s c m s m i U s , (3.20) f s “ ? T } s , f s “ ´ T K s , (3.21) f s “ ? d Θ s τ K s c m s m i Q k s , (3.22)where, in Eq. (3.21), we introduced the normalized gyrocenter perpendicular temperaturefluctuations T K s “ r T K s { T K s .As above anticipated, in the system (3.3)-(3.8) we temporarily left unspecified theexpression for the parallel heat flux fluctuations Q } s . In order to obtain the model GF6,the infinite hierarchy of gyrofluid equations following from the parent gyrokinetic model(2.1)-(2.4) is closed by imposing relations obtained by computing the gyrocenter momentsother than N s , U s and T } s from a linearization of the parent gyrokinetic system about ahomogeneous equilibrium, in the quasi-static limit. In the case of Q k s , this leads (considerEqs. (3.22) and (A 21)) to Q k s “ , s “ e, i. (3.23)The model GF6 is thus obtained by inserting the relation (3.23) into the system (3.3)-(3.8). Details on the derivation of the closure relations originated from the quasi-staticassumption can be found in Appendix A. A remarkable property of this closure is thatit leads to the annihilation of all the contribution of the higher order moments in thegyrofluid equations.We find it useful to provide also a reformulation of the evolution equations (3.3)-(3.5),which should help putting in evidence the physical nature of the terms contributing tothe evolution of the various fields. Eqs. (3.3)-(3.5) can indeed be rewritten as B N s B t ` u K s ¨ ∇ N s ` ∇ k s U s “ , (3.24) m s m i B U s B t ` m s m i u K s ¨ ∇ U s ´ sgn p q s q E k s ` ∇ k s P s “ , (3.25) B T k s B t ` u K s ¨ ∇ T k s ` ∇ k s U s “ ´ ∇ k s Q k s , (3.26)where the parallel gradient operator ∇ k s is defined, for each species s , by ∇ k s f “ ´r G s A k , f s ` B f B z , (3.27)for a function f . From the formulation (3.24)-(3.26) it emerges that the gyrocenterdensity, parallel momentum and parallel temperature fluctuations are all advected, inthe perpendicular plane, by the incompressible velocity field u K s “ ˆ z ˆ ∇ ` G s φ ` sgn p q s q τ K s G s B k ˘ . (3.28)Such velocity field includes a first contribution, associated with G s φ , which correspondsto the usual E ˆ B drift (based on the gyroaveraged electrostatic potential), ubiqui-tous in low- β gyrofluid models such as those discussed in Waelbroeck & Tassi (2012),Snyder & Hammett (2001), Keramidas Charidakos et al. (2015) and Waelbroeck et al. (2009). When higher β values are allowed, however, the perpendicular advection acquires1a further contribution due to the parallel magnetic perturbations, as it transpires from Eq.(3.28). We remark that, similarly to the E ˆ B contribution, also the latter contributiondoes not vanish in the limit b s Ñ u K s ,evolves due to the term ´ sgn p q s q E k s , where E k s is defined by E k s “ ´ B G s A k B t ´ r G s φ, G s A k s ´ B G s φ B z . (3.29)Such term represents the force exerted by the gyroaveraged electric field, along thedirection of the magnetic field. A further source for the parallel momentum is due tothe term ∇ k s P s , where P s “ τ K s ˆ G s B k ` Θ s p N s ` T k s q ˙ . (3.30)This terms is associated with the parallel component of the divergence of an anisotropicpressure tensor.The parallel temperature equation (3.26) has, on its left-hand side, the same structureof the continuity equation (3.24). We just remark the presence of the coefficient 2multiplying ∇ k s U s . This coefficient, of course, follows directly from taking the sec-ond order moment, in Hermite polynomials for the normalized parallel velocity, of thegyrokinetic equation (2.1), as discussed in Appendix B. However, as pointed out byKeramidas Charidakos et al. (2015), in the presence of background magnetic curvaturesuch coefficient, in general, has to be adjusted in order to obtain a Hamiltonian structure.Finally, we consider the term on the right-hand side of Eq. (3.26), associated with theparallel heat flux. If the expression for Q k s is chosen according to the quasi-static closure,i.e. imposing the relation (3.23), this term vanishes and the system, as will be shown inSec. 3.1, is Hamiltonian. On the other hand, if the Landau fluid closure (3.35) is chosen,this term acts as a sink and the system is not energy-conserving.3.1. Hamiltonian structure of GF6
The quasi-static closure (3.23) leading to GF6, allows the resulting model to becast in Hamiltonian form. In particular, it can be verified by direct computation that,when the electromagnetic fluctuations φ , A k and B k can be expressed in terms of thedynamical variables N e , N i , M e and M i (where we introduced the short-hand notation M s “ p m s { m i q U s ` sgn p q s q G s A k to indicate the parallel canonical momenta), by makinguse of the relations (3.6)-(3.8), the evolution equations (3.3)-(3.5) complemented by Eq.(3.23), can be written in the Hamiltonian form B N s B t “ t N s , H u , B M s B t “ t M s , H u , B T k s B t “ t T k s , H u , s “ e, i. (3.31)2In Eq. (3.31) H is the Hamiltonian functional H p N e , N i , M e , M i , T k e , T k i q “ ÿ s ż d x ˆ τ K s Θ s N s ` m i m s M s ` p sgn p q s q G s φ ` τ K s G s B k q N s ´ sgn p q s q m i m s G s A k M s ` τ K s Θ s T k s ¸ , (3.32)and t , u is a noncanonical Poisson bracket given by t F, G u “ ´ ÿ s ż d x „ sgn p q s q ˆ N s ˆ r F N s , G N s s ` τ K s Θ s m s m i r F M s , G M s s ` r F T k s , G T k s s ˙ ` M s pr F M s , G N s s ` r F N s , G M s s ` pr F M s , G T k s s ` r F T k s , G M s sqq` T k s ˆ τ K s Θ s m s m i r F M s , G M s s ` r F N s , G T k s s ` r F T k s , G N s s ` r F T k s , G T k s s ˙˙ ` F N s B G M s B z ` F M s B G N s B z ` F T k s B G M s B z ` F M s B G T k s B z , (3.33)for two functionals F and G . For details about the noncanonical Hamiltonian formulationof fluid models one can refer, for instance, to Morrison (1998). In Eq. (3.33) the subscriptson the functionals indicate functional derivatives. In order to verify the formulation (3.31)it is convenient to remark that, from Eq. (3.32), one obtains H N s “ τ K s Θ s N s ` sgn p q s q G s φ ` τ K s G s B k , H M s “ U s , H T k s “ τ K s Θ s T k s . (3.34)In order to derive the relations (3.34) we made use of the formal symmetry of the opera-tors G s and G s , i.e. ş d x f G s g “ ş d x gG s f and ş d x f G s g “ ş d x gG s f fortwo functions f and g , as well as of the formal symmetry of the linear operators in termsof which one can express φ, B k and A k in terms of N s and M s through Eqs. (3.6)-(3.8).We did not provide the explicit expression for such operators which can, however, beobtained considering the representation in Fourier series of the fields involved, followingthe procedure discussed in Tassi (2019).The Hamiltonian functional (3.32) is a conserved quantity for the dynamics andcorresponds to the total energy.In Eq. (3.33), the sum of all the terms with s “ e is a Poisson bracket in itsown right. Similarly, all the terms with s “ i form a Poisson bracket. The sum ofthese two contributions is a direct sum of Poisson brackets which is in turn a Poissonbracket verifying in particular the Jacobi identity. The Poisson brackets referring tothe electron and ion quantities correspond to those already discussed in other Hamil-tonian reduced fluid models. The reader can in particular refer to Tassi (2015) andKeramidas Charidakos et al. (2015) for the verification of the Jacobi identity for bracketsof such form and for a discussion of the corresponding Casimir invariants. The modelGF6, in the two-dimensional limit when the dependence on the z coordinate is suppressed,can also be cast in the form of a system of advection of equations for Lagrangianinvariants, as is the case for several other reduced fluid and gyrofluid models (see,e.g. Keramidas Charidakos et al. (2015); Waelbroeck et al. (2009); Waelbroeck & Tassi(2012); Grasso & Tassi (2015); Tassi (2015, 2019); Schep et al. (1994)). Remark:
The gyroaverage operators G with a form different from (3.16) have been3proposed in the literature (see, e.g. Dorland & Hammett (1993); Snyder & Hammett(2001); Scott (2010)), when the expansion (3.18) is not assumed, and are frequentlyadopted. In particular, G p b s q “ Γ { p b s q was shown to provide better agreement withthe linear theory at large b s (Dorland & Hammett 1993). We point out, however, thatmany important features of the model (3.3)-(3.8), such as the total energy conservationand the Hamiltonian structure, are guaranteed whatever the form of the operators G and G of Eq. (3.16) is, provided these operators are linear and formally symmetric, inthe sense defined above. This is in particular the case with G p b s q “ Γ { p b s q . Theseissues are also discussed by Mandell et al. (2018).3.2. Variants of the model
Six-field model with Landau closure (GF6L)
The variant GF6L of the six-field gyrofluid model, accounting for Landau damping,corresponds to the system (3.3)-(3.8) with Q k s given by Q k s “ ´ α s L T k s , s “ e, i. (3.35)In Eq. (3.35) we introduced the constant α s “ p { π q { p m i { m s q { . The operator L holdsfor the Landau damping operator. Its modeling in the nonlinear regime is discussed inTassi et al. (2018). In the linear approximation, it reduces to the negative of the Hilberttransform in the direction of the ambient magnetic field (here taken in the z direction).The presence of this Landau operator in reduced fluid models breaks the Hamiltonianstructure by violating energy and Casimir conservation (Tassi et al. (2018); Grasso et al. (2020)). Its purpose, on the other hand, is to introduce terms that allow the lineardispersion relation of the gyrofluid model to reproduce that of the parent gyrokineticmodel.3.2.2. Four-field model with quasi-static closure (GF4)
The second variant GF4 is obtained by retaining the evolution equations for N s and M s and imposing the quasi-static closure on the parallel temperature fluctuations T k s .Considering Eqs. (3.21) and (A 21), this amounts to setting T k s “ , s “ e, i. (3.36)4The resulting model reads B N s B t ` r G s φ ` sgn p q s q τ K s G s B k , N s s ´ r G s A k , U s s ` B U s B z “ , (3.37) BB t ˆ m s m i U s ` sgn p q s q G s A k ˙ ` „ G s φ ` sgn p q s q τ K s G s B k , m s m i U s ` sgn p q s q G s A k (3.38) ´ τ K s Θ s r G s A k , N s s ` BB z ˆ sgn p q s q G s φ ` τ K s G s B k ` τ K s Θ s N s ˙ “ , ÿ s ˆ sgn p q s q G s N s ` p ´ Θ s q Γ s φτ K s ` p Θ s G s ´ q φτ K s ` sgn p q s qp ´ Θ s qp Γ s ´ Γ s q B k ` sgn p q s q Θ s G s G s B k ˘ “ , (3.39) ∇ K A k “ β K e ÿ s ˆ m i m s ˆ ´ Θ s ˙ p ´ Γ s q A k ´ sgn p q s q G s U s ˙ , (3.40) B k “ ´ ÿ s β K s ˆ G s N s ` p ´ Θ s qp Γ s ´ Γ s q sgn p q s q φτ K s ` Θ s G s G s sgn p q s q φτ K s ` p ´ Θ s qp Γ s ´ Γ s q B k ` Θ s G s B k ˙ (3.41)and corresponds to taking Eqs. (3.3), (3.4), (3.6), (3.7) and (3.8) of GF6 with T k s “ H p N e , N i , M e , M i q “ ÿ s ż d x ˆ τ K s Θ s N s ` m i m s M s ` p sgn p q s q G s φ ` τ K s G s B k q N s ´ sgn p q s q m i m s G s A k M s ˙ (3.42)and by the Poisson bracket t F, G u “ ´ ÿ s ż d x „ sgn p q s q ˆ N s ˆ r F N s , G N s s ` τ K s Θ s m s m i r F M s , G M s s ˙ ` M s pr F M s , G N s s ` r F N s , G M s sqq ` F N s B G M s B z ` F M s B G N s B z . (3.43)If one neglects electron FLR effects (i.e. b e Ñ B k “ Θ e “ Θ i “
1) and sets G p b i q “ Γ { p b i q (i.e. takes the alternative form of the ion gyroaverage operatormentioned in the above remark in Sec. 3.1), GF4 reduces to the Hamiltonian gyrofluidmodel of Waelbroeck & Tassi (2012), the latter taken in the limit of vanishing magneticcurvature and equilibrium density gradients. We note, however, that, once that the quasi-static closure relations are determined, as in Eq. (A 21), all the terms in GF4 (and likewisefor GF6), are determined exactly. In particular no approximations of the gyroaverageoperators (unlike, for instance, in Waelbroeck & Tassi (2012) and Scott (2010)) arecarried out. Terms involving gyroaverage operators are determined exactly also in Brizard(1992), but without making use of the quasi-static closure. As a result, in Brizard (1992),nonlinear terms involving more than one gyroaverage operator do not result in having5the single canonical bracket structure (as is the case in GF4, for instance, with the termsgn p q s qr G s φ, G s A k s appearing in the second line of Eq. (3.38)) and which is crucial fordetermining the Lie-Poisson Hamiltonian structure. This property, which follows from thequasi-static closure, differentiates the models presented in our paper also from its closestpredecessors, i.e. the gyrofluid models constructed with the technique recently presentedin Tassi (2019). The latter gyrofluid models, in fact, are also Hamiltonian and account forequilibrium temperature anisotropy, but adopt a different closure. Namely, all gyrofluidmoments involving finite powers of the magnetic moment µ s (e.g. the perpendiculartemperature fluctuations) are set equal to zero. This allows for a Hamiltonian structurebut the terms involving gyroaverage operators are not all determined exactly from takingthe moments of the gyrokinetic equations. For this reason, we think that, for situationswhere the quasi-static assumption | ω {p k z v th k s q| !
4. Comparison with the linearized gyrokinetic parent model
KAWs dispersion relation
We first discuss the KAW dispersion relation as predicted by the gyrofluid models,with and without Landau damping (GF6L and GF4 respectively), in comparison withthe predictions of the linearized parent model (Kunz et al. et al. (2012). This latter dispersionrelation involves the plasma response function R p ζ s q of the particles of species s , whichis related to the corresponding plasma dispersion function Z p ζ s q by R p ζ s q “ ` ζ s Z p ζ s q with ζ s “ ω {p { | k z | v th } s q . Different Pad´e approximants R ij p ζ e q (for which we follow thenotations of Hunana et al. (2019)) are used to estimate the electron response function.In order to test the validity of the quasi-static assumption that affects the form ofthe FLR terms, independently from the effects resulting from Landau damping, we areled to compare the prediction of the 4-field gyrofluid model GF4 with the gyrokineticdispersion relation obtained by choosing for the electron plasma response function, thefunction R p ζ e q “ {p ´ ζ e q (model denoted GKNL). This choice directly results fromthe assumption T } e “ : . Predictions of the GF6L model will, on the other hand, be comparedwith the full gyrokinetic dispersion relation, referred to as GK. In the latter description,we use for the electron Pad´e approximant the function R p ζ e q , which shows an excellentagreement with the exact response function. For the ions, we use in all the cases thePad´e approximant R p ζ i q “ {p ´ ζ i ´ iπ { ζ i q (or R p ζ i q in the absence of Landaudamping). Higher order Pad´e approximants give almost identical results.Because we are making use of several acronyms to refer to the different adoptedmodels, we summarized them in Table 1. In all the examples presented in this Section, Θ i “
1. Introducing the ion to electron parallel temperature ratio at equilibrium τ } “ T } i { T } e “ τ K i Θ e { Θ i and denoting by α the angle between the wave vector andthe ambient magnetic field (propagation angle), we compare, in Fig. 1 (left), the (real)normalized KAW frequency ω {p k z c A q for β K e “ τ } “ Θ e “ α “ ˝ as a functionof kd i (where k “ a k K ` k z and d i is the ion inertial length defined by d i “ c A { ω ci ,with c A “ a B {p πm i n q the Alfv´en velocity), calculated using GF4 (black diamond : Note that, choosing a closure that sets equal to zero a higher even order moment in thehierarchy of parallel moments would only improve the matching with kinetic theory at very large ζ e but not in the zero- ζ e limit. Differently, closures at an odd order are not consistent with thequasi-static assumption, as they rather correspond to an adiabatic regime. Table 1.
Acronyms and corresponding models.Acronym ModelGF6 Hamiltonian six-field gyrofluid model with quasi-static closure(Eqs. (3.3)-(3.8) with Q k s given by Eq. (3.23))GF6L Six-field gyrofluid model with Landau closure(Eqs. (3.3)-(3.8) with Q k s given by Eq. (3.35))GF4 Hamiltonian four-field gyrofluid model with quasi-static closure(Eqs. (3.37)-(3.41))GK Full gyrokinetic dispersion relation (from Kunz et al. (2018))GKNL Gyrokinetic dispersion relation with no Landau dampingBC84 Asymptotic model by Basu & Coppi (1984) (Eq. (C 10))DK Full drift-kinetic dispersion relation (Eq. (C 7))DKNL Drift-kinetic dispersion relation with no Landau damping symbols), with the GKNL prediction (green solid line). An excellent agreement is found,the slight deviation appearing as kd i ą
40 probably resulting from the failure of thequasi-static approximation used in the calculation of all the moments starting fromthe temperature. For the same plasma parameters, the case with Landau damping isdisplayed as a red solid line for GK and blue diamond symbols for GF6L, with a goodagreement up to scales kd i Æ
20. The damping rate is displayed with the same symbolsin Fig. 1 (middle). Interestingly, the prediction of GF6 (not shown) departs from GKNLsignificantly at all the scales. Indeed, in GF6, the parallel temperature fluctuations thatobey a dynamical equations with zero heat flux do not approach a quasi-static dynamics.Such a non-dissipative odd-order closure rather fits an adiabatic regime (see footnotein this Section). Landau damping is requested to ensure convergence to a quasi-staticregime. In that sense, GF4 is preferable to GF6 for addressing a non-dissipative problem,the interest of the latter model being mainly to provide a framework where Landaudamping can be supplemented to an otherwise Hamiltonian description.4.2.
Firehose instability
Figure 1 (right) displays for β K e “ . τ } “ ´ , α “ ˝ , Θ e “ .
16, the growthrate of the firehose instability as a function of kd i predicted by GF4 and GF6L (blackand blue diamond symbols respectively), together with the GKNL and GK dispersionrelations (green and red solid lines). In all the cases, the agreement is excellent, thequenching of the instability being in particular well reproduced.The agreement found between the KAWs dispersion relations predicted by the gyrofluidmodels and the gyrokinetic theory, even when limited to scales such that kd i Æ
20, anddespite a large value of ζ i “ b Θ e {p β K e τ } q ω {p| k z | c A q , can be attributed to the factthat, at least within the linear theory, ion acoustic and kinetic Alfv´en waves remain7 Figure 1.
Left: Normalized real part of the KAW frequency ω {p k z c A q for β K e “ τ } “ Θ e “ α “ ˝ , as a function of kd i , calculated using GF4 (black diamond symbols) and GF6L(blue diamond symbols) models (respectively without and with Landau damping) and withthe gyrokinetic dispersion relations GKNL (green solid line) and GK (red solid line). Middle:Damping rate from the GK theory (red solid line) and from the GF6L model (blue diamondsymbols). Right: Normalized growth rate of the firehose instability as a function of kd i , for β K e “ . τ } “ ´ , Θ e “ . α “ ˝ , as predicted by GF4 and GF6L models, togetherwith GKNL and GK dispersion relations, using the same graphic conventions. essentially decoupled. The influence of the ion closure relation on the KAW propertiesthus remains limited. Deviations from the gyrokinetic theory at small scales are mostlydue to the fact that when reaching these scales ζ e “ a m e { m i a Θ e { β K e ω {p| k z | c A q becomes non-negligible.4.3. SW dispersion relation and field swelling instability
Figure 2 concerns a similar comparison in the case of the field swelling instabilitydiscussed in Appendix C, for β K e “ τ } “ α “ ˝ , Θ e “ . ζ i , which is not small enough (in this case, ζ e remains reasonably small). Weshow in Fig. 2 (right) with Θ e “ .
41, again with β K e “ α “ ˝ , that a muchbetter agreement can be found when ions are hotter ( τ } “ ζ i . The case with cold ions ( τ } “ ´ ), for which the ion dynamics is decoupled, isdisplayed in Fig. 3, showing an even better agreement. The left panel displays the realpart of the slow wave for Θ e “
1, obtained with the GKNL model (solid green line)or the GF4 model (diamond symbols). The right panel shows the growth rate of thefield-swelling instability for Θ e “ .
01 (keeping unchanged the other parameters).
5. Conclusion
We derived a 6-field Hamiltonian gyrofluid model, referred to as GF6, retaining thegyrocenter density, the parallel velocity and temperature fluctuations for each species,under the sole assumption that all the other gyrocenter moments are calculated fromthe quasi-static linear kinetic theory. Such an assumption on the closure turns out toyield exact expressions for all the terms of the model, without, in particular, requiringapproximated expressions for the terms involving gyroaverage operators. Nonlinear termsinvolving more than one gyroaverage operator, in particular, appear in the form of asingle canonical bracket, which naturally lets the model fit in the class of Hamiltonianmodels with a Lie-Poisson structure. The model accounts for equilibrium temperature8
Figure 2.
Normalized growth rate of the field swelling instability versus kd i for α “ ˝ and β K e “
1, Left: Predictions of GF4 and GF6L, compared with those of GKNL and GKrespectively, for τ } “ Θ e “ .
2. Right: Prediction of GF4 compared with that of GKNL,for τ } “
50 and Θ e “ .
41 . Same graphic conventions as in Fig. 1 are used.
Figure 3.
Left: Normalized slow-wave frequency versus kd i in regimes where the ions are cold( τ } “ ´ ), with β K e “ α “ ˝ , as predicted by the GF4 model and the GKNLdispersion relation. Right: normalized growth rate of the swelling instability for τ } “ ´ and Θ e “ .
01. Same conventions as in Fig. 1 are used. anisotropy and also retains both ion and electron FLR corrections, electron inertia andparallel magnetic fluctuations. In a variant of the model (GF6L) parallel Landau dampingis retained through a Landau-fluid modelization of the gyrocenter parallel heat fluxes.A second variant of the model (GF4) is obtained by prescribing parallel isothermality,which still falls in the frame of the quasi-static closure and allows for a Hamiltonianformulation. The comparison of the dispersion relations of KAWs and SWs predictedfrom GF6L or GF4, with those derived from the parent gyrokinetic theory where theplasma response function is replaced by a Pad´e approximant, provides an estimate of themaximal transverse wavenumber beyond which the phase velocity of the corresponding9wave is too large compared with the electron (in the former case) or ion (in the latter case)parallel thermal velocity for consistency with a closure condition based on a quasi-staticassumption. It turns out that the agreement extends to transverse scales significantlysmaller than the ion Larmor radius in the case of KAWs, mostly because, at least atthe linear level, SWs and KAWs are essentially decoupled, making the influence of theion closure relation on the KAW properties relatively limited. This situation contrastswith the case of the SWs for which the dispersion relation is accurately reproducedonly at scales larger than a significant fraction of the ion Larmor radius. Under theseconditions, the model reproduces the instabilities induced by temperature anisotropy,such as firehose or field-swelling instabilities. It should nevertheless be noted that, as itassumes small perturbations of an equilibrium state, the model does not permit evolutionof the mean temperatures, an effect usually considered as contributing efficiently to thesaturation of these instabilities. The subcritical nonlinear regime is however expectedto be accurately described. The model will in particular be most useful for studying thecoupling of KAWs with SWs which can generate large-scale parametric decay instabilitiesat small β e , a regime especially relevant in the regions of the solar wind relatively closeto the Sun explored by space missions such as Parker Solar Probe or Solar Orbiter.In general, to the best of our knowledge, our gyrofluid model is the only one, at thepresent moment, possessing the following features, which could make it a valuable tool forlocal investigations of basic plasma phenomena of interest for space plasmas: it accountsfor equilibrium temperature anisotropies as well as parallel magnetic perturbations; itreproduces, in a rather wide range of values of parameters, compatible with the quasi-static closure, quantitative features of known kinetic linear dispersion relations; the modelequations, and in particular the terms involving FLR corrections, are calculated exactly,unlike other gyrofluid models which adopt truncations or approximations of such terms;it possesses a Hamiltonian structure.0 Appendix A. Derivation of closure relations from the gyrokineticlinear theory
We consider the linearization of the gyrokinetic system (2.1)-(2.4) about the equilib-rium state r g s “ r f s “ r φ “ r A k “ r B k “
0. The resulting linearsystem can be written in the form BB t ˜ r f s ` q s T k s v } c F eq s J s r A k ¸ ` v } BB z ˜ r f s ` q s T k s F eq s ˜ J s r φ ` µ s B q s J s r B k B ¸¸ “ , (A 1) ÿ s q s ż d W s J s r f s “ ÿ s q s T K s ż d W s F eq s ` ´ J s ˘ r φ ´ ÿ s q s ż d W s µ s B T K s F eq s J s J s r B k B , (A 2) ÿ s q s ż d W s v } J s r f s “ ´ c π ∇ K r A k ` ÿ s q s m s ż d W s F eq s ˜ ´ Θ s v } v th k s ¸ p ´ J s q r A k c , (A 3) ÿ s β K s n ż d W s µ s B T K s J s r f s “ ´ ÿ s β K s n q s T K s ż d W s µ s B T K s F eq s J s J s r φ ´ ˜ ` ÿ s β K s n ż d W s F eq s ˆ µ s B T K s J s ˙ ¸ r B k B , (A 4)where we adopted the same notation with the tilde symbol, that we used in Eqs.(2.1)-(2.4), to indicate the dynamical variables r f s of the linearized system and the fieldperturbations r φ, r A k and r B k .We introduce the following Fourier series representation: r f s p x , v } , µ s , t q “ ÿ k P D r f s k p v } , µ s q e i p k ¨ x ´ ωt q , r φ p x , t q “ ÿ k P D r φ k e i p k ¨ x ´ ωt q , (A 5) r A k p x , t q “ ÿ k P D r A k k e i p k ¨ x ´ ωt q , r B k p x , t q “ ÿ k P D r B k k e i p k ¨ x ´ ωt q , (A 6)with ω P C indicating the complex frequency.For any given k P D , from Eq. (A 1) we obtain the relation r f s k “ r ζ s p v } { v th k s q ´ r ζ s v } c q s T k s F eq s J p a s q r A k k ´ p v } { v th k s q ´ r ζ s v } v th k s F eq s ˜ q s T k s J p a s q r φ k ` µ s B T k s J p a s q a s r B k k B ¸ , (A 7)where r ζ s “ ω {p k z v th k s q .We consider now the quasi-static limit | r ζ s | !
1. In this limit, the relation (A 7) reducesto r f s k “ ´ F eq s ˜ q s T k s J p a s q r φ k ` µ s B T k s J p a s q a s r B k k B ¸ . (A 8)1In the following, we make use of the relation (A 8), derived from the linear theory, inorder to determine the closure relations to insert in the hierarchy of nonlinear gyrofluidequations. For this purpose, we adopt the Hermite-Laguerre expansion of Eq. (3.18) forthe perturbation of the distribution function in the linearized system and we write r f s p x , v } , µ s , t q “ F eq s p v } , µ s q `8 ÿ m,n “ ? m ! H m ˜ v } v th k s ¸ L n ˆ µ s B T K s ˙ f mn s p x , t q (A 9)with f mn s p x , t q “ ÿ k P D f mn s k e i p k ¨ x ´ ωt q (A 10)in Fourier representation. From Eqs. (A 9), (A 10), using the orthogonality relations forHermite and Laguerre polynomials, one obtains f mn s k “ n ? m ! ż d W s r f s k H m ˜ v } v th k s ¸ L n ˆ µ s B T K s ˙ . (A 11)Inserting the relation (A 8) into Eq. (A 11) and using the orthogonality relations forHermite polynomials, one has f mn s k “ ´ δ m ˜ G n q s T k s r φ k ` Θ s G n r B k k B ¸ , (A 12)where the operators G n and G n are defined by G n “ πB m s ż dµ s f eq s p µ s q L n ˆ µ s B T K s ˙ J p a s q , (A 13) G n “ πB m s ż dµ s f eq s p µ s q L n ˆ µ s B T K s ˙ µ s B T K s J p a s q a s , (A 14)with f eq s p µ s q “ m s πT K s e ´ µsB T K s . (A 15)Explicit expressions for the operators G n and G n can be found by computing theintegrals in Eqs. (A 13) and (A 14), which yields G n p b s q “ e ´ b s { n ! ˆ b s ˙ n , n ě , (A 16) G p b s q “ e ´ b s { , G n p b s q “ ´ e ´ b s { ˜ˆ b s ˙ n ´ p n ´ q ! ´ ˆ b s ˙ n n ! ¸ , n ě . (A 17)In order to obtain the expressions (A 16)-(A 17) use was made of the orthogonality ofLaguerre polynomials as well as of the relations (Szeg¨o 1975) J p a s q “ e ´ b s { `8 ÿ n “ L n ´ µ s B T K s ¯ n ! ˆ b s ˙ n , (A 18)2 J p a s q a s “ e ´ b s { `8 ÿ n “ L p q n ´ µ s B T K s ¯ p n ` q ! ˆ b s ˙ n , (A 19)2where L p q n is a generalized Laguerre polynomial. Making use of the Fourier representa-tions (A 10) , (A 5) and (A 6) for for f mn s , r φ and r B k , respectively, one can deduce fromEq. (A 12) the relation f mn s p x , t q “ ´ δ m ˜ G ns q s T k s r φ ` Θ s G ns r B k B ¸ , (A 20)or, equivalently, f mn s p x , t q “ ´ δ m Θ s ˆ sgn p q s q G ns φτ K s ` G ns B k ˙ , (A 21)where we also made use of the normalization (3.11) for φ and B k . The operators G ns and G ns are defined, consistently with the definition of G s and G s given in Sec. 3, by G ns f p x , t q “ ř k P D G n p b s q f k p t q exp p i k ¨ x q and G ns f p x , t q “ ř k P D G n p b s q f k p t q exp p i k ¨ x q , for a function f and n ě t is provided by the factor e ´ iωt , but whenthe relations (A 21) are used as closures for the nonlinear models, the dependence on t is of course left arbitrary).The relations (A 21) descending from the quasi-static assumption, are adopted asclosure relations in GF6 and GF6L for all the moments involved in the model, except for N s , M s , T k s , which are derived by solving the evolution equations (3.3)-(3.5). The parallelheat flux fluctuations Q k s , on the other hand, are determined, as already mentioned,again by a quasi-static closure for GF6 (Eq. (3.23) which follows from Eq. (A 21) for m “ , n “
0) or by the Landau closure (3.35) for GF6L. The closure (3.36) adopted forGF4, is again a quasi-static closure following from Eq. (A 21) when m “ n “ Appendix B. Derivation of the model equations
The gyrofluid system (3.3)-(3.8) descends from the parent gyrokinetic system (2.1)-(2.4) upon applying to the perturbations of the distribution functions the expansion(3.18). In order to obtain a closed system with a finite number of equations, suchexpansion is constrained in the following way. The moments f s , f s and f s (or,equivalently, by virtue of Eqs. (3.20)-(3.21), the gyrofluid densities, parallel velocitiesand temperatures N s , U s and T k s ), for each species s , get determined by evolutionequations obtained by making the product of all the terms of the gyrokinetic equation(2.1) with the zero, first and second order Hermite polynomial in the variable v } { v th k s and integrating over the velocity volume element d W s . For GF6L, the parallel heat flux Q k s gets determined by the relation (3.35). All the other moments, on the other hand,are assumed to be given by the relations (A 20), or, in normalized form, by Eq. (A 21),obtained from the linear theory in the quasi-static limit. With this prescription, theexpansion (3.18) becomes r f s p x , v } , µ s , t q “ F eq s p v } , µ s q ˜ r N s n p x , t q ` v } v th k s r U s v th k s p x , t q` ˜ v } v th k s ´ ¸ r T k s T k s p x , t q ` ˜ v } v th k s ´ v } v th k s ¸ r Q k s n T k s v th k s p x , t q´ `8 ÿ n “ L n ˆ µ s B T K s ˙ ˜ G ns q s T k s r φ p x , t q ` Θ s G ns r B k B p x , t q ¸¸ , (B 1)3where we made use of the fact that H p v } { v th k s q “ H p v } { v th k s q “ v } { v th k s , H p v } { v th k s q “ v } { v th k s ´ H p v } { v th k s q “ v } { v th k s ´ v } { v th k s .Inserting the expansion (B 1) into Eqs. (2.5) and (2.1), and integrating over d W s oneobtains BB t r N s n ` cB ˜« G s r φ, r N s n ff ´ `8 ÿ n “ « G ns r φ, q s T k s G ns r φ ` Θ s G ns r B k B ff ´ `8 ÿ n “ « T K s q s G ns r B k B , q s T k s G ns r φ ` Θ s G ns r B k B ff ` T K s q s « G s r B k B , r N s n ff (B 2) ´ « G ns r A k , r U s c ff¸ ` B r U s B z “ , where we made use of the definitions (A 13) and (A 14) and of the orthogonality ofHermite polynomials. We remark, at this point, that the sum of the last term in the firstline of Eq. (B 2) with the first term on the second line of Eq. (B 2) yields zero, becauseof the antisymmetry of the canonical bracket r , s . We thus conclude that the quasi-staticclosure (A 20) has the remarkable property of annihilating, in the continuity equation,all the contributions associated with the moments f n s , for n ě
1. By virtue of thiscancellation, from Eq. (B 2) we obtain BB t r N s n ` cB « G s r φ ` T K s q s G s r B k B , r N s n ff ´ B ” G s r A k , r U s ı ` B r U s B z “ . (B 3)Applying to Eq. (B 3) the normalization (3.9)-(3.12), one obtains Eq. (3.3).In order to derive Eq. (3.4) we point out first that, with the help of the identities J p a s q “ e ´ b s { `8 ÿ n “ L n ´ µ s B T K s ¯ n ! ˆ b s ˙ n , (B 4)2 J p a s q a s “ e ´ b s { `8 ÿ n “ L p q n ´ µ s B T K s ¯ p n ` q ! ˆ b s ˙ n , (B 5) ż `8 dx e ´ x L n p x q L m p x q “ δ mn , (B 6) L p q m p x q “ ´ ddx L m ` p x q “ ´ m ` x L m ` p x q ` m ` x L m p x q , (B 7)4we obtain the relation (see also Brizard (1992)) cB ż d W s F eq s « J s r φ, q s T k s v } v th k s c J s r A k ff (B 8) “ n v th k s B ÿ k , k P D ż d ˆ µ s B T K s ˙ e ´ µ s B { T K s ˆ « q s T k s J ˆ k K ω cs c µ s B m s ˙ r φ, J ˜ k K ω cs c µ s B m s ¸ q s T k s r A k ff e i p k ` k q¨ x (B 9) “ n v th k s B ÿ k , k P D `8 ÿ m,n “ ż d ˆ µ s B T K s ˙ e ´ µ s B { T K s ˆ »– q s T k s e ´ b s { L m ´ µ s B T K s ¯ m ! ˆ b s ˙ m r φ, e ´ b s { L n ´ µ s B T K s ¯ n ! ˜ b s ¸ n r A k fifl e i p k ` k q¨ x “ n v th k s B `8 ÿ n “ « q s T k s G ns r φ, G ns r A k ff , (B 10)and, by an analogous procedure, the relation cB ż d W s F eq s « µ s B q s J s r B k B , q s T k s v } v th k s c J s r A k ff “ n Θ s v th k s B `8 ÿ n “ « G ns r B k B , G ns r A k ff . (B 11)Upon multiplying Eq. (2.1) by p { n q v } { v th k s and integrating over d W s one obtains,adopting the expansion (B 1) as well as the relations (A 13), (A 14), (B 10) and (B 11),the following equation BB t ˜ r U s v th k s ` q s T k s v th k s c G s r A k ¸ ` cB « G s r φ ` T K s q s G s r B k B , r U s v th k s ff ` v th k s B `8 ÿ n “ « q s T k s G ns r φ ` Θ s G ns r B k B , G ns r A k ff ´ v th k s B « G s r A k , r N s n ` r T k s T k s ff ` v th k s B `8 ÿ n “ « G ns r A k , q s T k s G ns r φ ` Θ s G ns r B k B ff (B 12) ` v th k s BB z ˜ r N s n ` q s T k s G s r φ ` Θ s G s r B k B ` r T k s T k s ¸ “ . Also in this case, the quasi-static closure leads to a remarkable cancellation. Indeed,among the nonlinear terms involving only electromagnetic quantities (i.e. r φ , r A k and r B k )all those containing gyroaverage operators G ns and G ns , with n ě
1, vanish. As a5result, Eq. (B 12) reduces to BB t ˜ r U s v th k s ` q s T k s v th k s c G s r A k ¸ ` cB « G s r φ ` T K s q s G s r B k B , r U s v th k s ff ` v th k s B « q s T k s G s r φ ` Θ s G s r B k B , G s r A k ff ´ v th k s B « G s r A k , r N s n ` r T k s T k s ff (B 13) ` v th k s BB z ˜ r N s n ` q s T k s G s r φ ` Θ s G s r B k B ` r T k s T k s ¸ “ . Equation (3.4) is then obtained from Eq. (B 13) after applying the normalization (3.9)-(3.12).Equation (3.5) is obtained upon multiplying Eq. (2.1) by p { n qp v } { v th k s ´ q andintegrating over d W s . Making use of the expansion (B 1) and of the orthogonality ofHermite polynomials one obtains BB t r T k s T k s ` cB ˜« G s r φ, r T k s T k s ff ` T K s q s « G s r B k B , r T k s T k s ff ´ « G s r A k , r U s c ff ´ « G s r A k , r Q k s n T k s c ff¸ ` v th k s BB z ˜ r U s v th k s ` r Q k s n T k s v th k s ¸ “ . (B 14)Adopting the normalization (3.9)-(3.12), Eq. (3.5) follows from Eq. (B 14).Equations (3.6), (3.7) and (3.8) follow from Eq. (2.2), (2.3) and (2.4), respectively, uponinserting the expansion (B 1), evaluating the integrals and applying the normalization(3.9)-(3.12). With regard to the evaluation of the integrals and the derivation of theequations in the form (3.6), (3.7) and (3.8), we remark that, in addition to Eqs. (A 13)-(A 14), the following relations are of use:1 n ż d W s F eq s J p a s q “ Γ p b s q “ `8 ÿ n “ G n p b s q , (B 15)1 n ż d W s F eq s µ s B T K s J p a s q J p a s q a s “ Γ p b s q ´ Γ p b s q “ G p b s q ` `8 ÿ n “ G n p b s q G n p b s q , (B 16)1 n ż d W s F eq s ˆ µ s B T K s J p a s q a s ˙ “ p Γ p b s q ´ Γ p b s qq “ G p b s q ` `8 ÿ n “ G n p b s q . (B 17) Appendix C. Field-swelling instabilities
C.1.
Dispersion relation at the MHD scales
In this Appendix, we first provide a simple derivation of the dispersion relation forfast and slow modes at MHD scales in the presence of temperature anisotropy, startingfrom the kinetic-MHD equations (see Eqs. (37), (38b), (44 a,b,c) (46)-(48) from Kulsrud(1983), or equivalently Eqs. (1)-(8) from Snyder et al. (1997)).6The transverse velocity can be decomposed into compressible and solenoidal parts bywriting u K “ ´ ∇ K χ c ` ∇ K ˆ p χ s p z q . (C 1)One immediately gets (see e.g. Eqs. (48),(52) and (56) of Passot & Sulem (2006) whereFLR corrections are neglected) ´B t B xx χ c ` B xx ´ p K ρ ` c A B } B ˘ ` ` c A ` p K ´ p } ρ ˘ B zz B } B “ , (C 2) B t B } B ´ B xx χ c “ , (C 3)where p K and p } denote the total (ion plus electron) perpendicular and parallel pressurefluctuations, with the subscript 0 referring to the equilibrium values. Furthermore, ρ denotes the equilibrium plasma density.In Eq. (C 2), the perpendicular pressure fluctuations are given by the drift-kinetictheory, as found e.g. in Eq. (27) of Snyder et al. (1997), in the form p K r ρ “ T K r m i ˆ p ´ Θ r R p ζ r qq B } B ´ R p ζ r q q r ψT } r ˙ , (C 4)together with n r n “ p ´ Θ r R p ζ r qq B } B ´ R p ζ r q q r ψT } r , (C 5)where q r “ e for ions and ´ e for electrons, respectively. The potential ψ is defined interms of the parallel electric field by E z “ ´B z ψ . These expressions for the perpendicularpressure and density perturbations identify with the large-scale limit of formulas givenin Appendix B of Passot & Sulem (2007). Quasineutrality requires the equality of theelectron ( n e ) and ion ( n i ) number-density fluctuations, which prescribes eψT k e “ τ } Θ e R p ζ e q ´ Θ i R p ζ i q τ } R p ζ e q ` R p ζ i q B } B . (C 6)Plugging the expressions for the pressures in Eqs. (C 2)-(C 3) thus leads to ω k c A “ ´ k z k β } e p ´ Θ e ` τ } p ´ Θ i qq` k K k β K e ” p ´ Θ e R p ζ e qq ` τ K i p ´ Θ e R p ζ e q ´ Θ i R p ζ i qq` τ } p ` τ K i q R p ζ e q ´ Θ e R p ζ e q ´ Θ i R p ζ i q τ } R p ζ e q ` R p ζ i q ¯ı . (C 7)We note that ζ i “ ω τ } ω s with ω s “ k z c s } where c s } “ d T k e m i is the sound speed basedon the parallel electron temperature.C.2. Link with the fluid theory
In order to make a link with fluid theory as performed in Basu & Coppi (1984), wenote that keeping all the terms in the ion density and parallel velocity equations (inparticular the time derivatives), is equivalent to expanding R p ζ i q for ζ i large, i.e. in theadiabatic limit where R p ζ i q « ´ {p ζ i q . It thus follows that the ions cannot be hot, atleast outside an angular boundary layer near the transverse direction.7In addition to adiabatic ions, let us also assume a quasi-static limit for the electrons,which leads to assume ζ e Ñ R p ζ e q «
1. Equation (C 7) then rewrites ω k c A “ ´ k z k β } e p ´ Θ e ` τ } p ´ Θ i qq` k K k β K e ” p ´ Θ e q ` τ K i ˆ ´ Θ e ` Θ i ζ i ˙ ` τ } p ` τ K i q ´ Θ e ` Θ i {p ζ i q τ } ´ {p ζ i q ¯ı . (C 8)For cold ions ( τ } “ τ K i “ ˆ ω ω s ´ ˙ „ ´ ω k v A ´ k z k p β } e ´ β K e q ` k K k Θ e p β } e ´ β K e q ` k K k Θ e β K e ω ω s “ , (C 9)which also rewrites in the form of Eq. (44) of Basu & Coppi (1984) ˆ ω ω s ´ ˙ „ ´ ω k v A ` k K k Θ e p β } e ´ β K e q ´ k z k p β } e ´ β K e q ` k K k Θ e β K e “ . (C 10)The slow mode is obtained when ω „ ω s , while the fast mode corresponds to ω " ω s .C.3. The field swelling instabilities
C.3.1.
Slow mode
As discussed in Basu & Coppi (1984), it follows from Eq. (C 10) that the slow modebecomes unstable when 1 ă Θ e β K e ` β K e ă . (C 11)As Θ e is increased from 1, the phase velocity decreases and becomes zero at 1 ` { β K e .As Θ e is further increased, a pair of purely imaginary complex conjugate roots appears,leading to the so-called slow-mode swelling instability.C.3.2. Fast mode
According to the theory of Basu & Coppi (1984) that assumes ζ e very small, the fastmode becomes unstable when Θ e ą p ` β K e q β K e . (C 12)The validity conditions require in particular that p k z { k K q ą p m e { m i qp { β } e q in order toensure that ζ e is small enough.C.4. Instability growth rate
Le us now consider the full dispersion relation, given by Eq. (C 7), with a generalelectron response function. The instabilities are illustrated in Fig. 4 which displays,as a function of the perpendicular to parallel electron temperature anisotropy Θ e , the8 Figure 4.
Imaginary part of r “ ω { kc A for the unstable mode, solution of Eq. (C 7), forpropagation angles α “ ˝ (left) and 80 ˝ (middle) as a function of the perpendicular to parallelelectron temperature anisotropy Θ e in the case β K e “ R p ζ i q “ ´ {p ζ i q ) for various approximations of R p ζ e q : BC84 (black solid line) uses R p ζ e q “
1, DK (red solid line) uses R p ζ e q and DKNL (green solid line) uses R p ζ e q (noelectron Landau damping). Superimposed diamond symbols refer to predictions of GF4 modeltaken in the large-scale limit. Right panel corresponds to DKNL and GF4 models with α “ ˝ in the case τ K i “ imaginary part of r “ ω { kc A for the unstable mode, solution of Eq. (C 7) with β K e “ α “ ˝ (left) and α “ ˝ (middle),when using four different approximations for the plasma response functions. The firstone, referred to as BC84 (black solid line) uses R p ζ i q “ ´ {p ζ i q and R p ζ e q “ R p ζ e q “ {p ´ ζ e ´ iπ { ζ e q (the use of R leads to almost identical results) and for the ions thesame large- ζ i limit as in the BC84 model, since the ions are cold. The third model, calledDKNL (green solid line) differs from the previous one by the fact that electron Landaudamping is suppressed (using R p ζ e q “ {p ´ ζ e )). Superimposed diamond symbolsrefer to the prediction of GF4, taken in the large-scale limit. It is legitimate to considerthis model, derived in the limit ζ i Ñ
0, in the regime of cold ions, because in this casethe dynamics is insensitive to the closure assumption. The right panel corresponds to thecase with τ K i “ α “ ˝ . It shows that the GF4 model is still approximately valideven in a situation where ζ i is not small.Figure 5 displays, as a function of Θ e , the positive real part of the three roots of Eq.(C 7) (referred to as min , int and max in increasing order of magnitude, displayed inturquoise, magenta and brown colors respectively), in the case of DKNL model (solidlines) or for the DK model (dash-dotted lines), with the predictions of the GF4 modelsuperimposed as diamond symbols, again for 89 ˝ (left) and 80 ˝ (right) propagation anglesand cold ions. For the 89 ˝ angle (which falls outside the range of admissible angles forthe BC84 model), the destabilization of the slow mode for Θ e ą Θ e ą Θ e ą
4. The DKNL model displays a very different behavior, whereby the fastmode (brown solid line in the left panel of Fig. 5) remains almost unchanged for thewhole range of values of Θ e . The slow mode (turquoise solid line in the same panel)disappears for Θ e ą Θ e ą
4. The intermediate root for Θ e ă Figure 5.
Positive real part of the three roots (denoted min (turquoise), int (magenta) andmax(brown) in increasing order of magnitude for DKNL (solid lines) and DK (dash-dottedlines), together with predictions of GF4 (diamond symbols), versus Θ e , for α “ ˝ (left) and α “ ˝ (right), in the case of cold ions. is here the only extra mode for the present choice of the Pad´e approximant ( R p ζ e q ,leading to what we called DKNL)). The instability that continues to exist for Θ e ą Θ e between 2 and 4 (see Fig. 5 left, dash-dotted line). We also note that theuse of the R Pad´e does not change the roots associated with the slow and fast modes,but only those associated to the extra damped plasma modes (not shown). We concludethat in an angular boundary layer close to 90 ˝ , the fast mode is always stable, and theinstability that continues to exist for Θ e ą α approaches 90 ˝ .For α “ ˝ , the value of ζ e is sufficiently small for the approximation R p ζ e q “ Θ e ă Θ e ą
4, the real frequency of the fastmode vanishes and this mode becomes unstable, as predicted in Basu & Coppi (1984).The slow mode reappears on this intermediate branch with a very good match betweenthe GF4 and DK as well as DKNL predictions.Complementary information is presented in Fig. 6 which displays the growth rate ofthe unstable mode as a function of the angle α for the three models described abovefor Θ e “ Θ e “ . Θ e “ ˝ and 90 ˝ but its predictions deviate for smaller angles. For Θ e “ . k Figure 6.
Growth rate of the unstable mode as a function of the propagation angle α for theBC84 (black), DK (red), DKNL (green) and GF4 (diamond symbols) models in the case Θ e “ Θ e “ . Θ e “
3, BC84 and DKNL curves are superimposed. up to angles α “ ˝ . This fast-mode instability is only recovered with DKNL (and DK)at oblique angles, the deviation with BC84 starting to be significant for α ą ˝ . Forthis value of Θ e , the slow mode is always stable in the GF4 model. Remarks: ‚ Influence of warm ions : If one considers the fast mode at an angle close (but notequal) to 90 degrees, one can assume warm ions and at the same time R p ζ i q “
0. Takingalso R p ζ e q “
1, and using Eq. (C 7), one gets the dispersion relation (Eq. 20) obtainedby Pokhotelov & Onishchenko (2014) who show that ions are stabilizing. ‚ The case with ion temperature anisotropy:
With finite (but isotropic) ion tem-peratures, other modes are present, but the one which becomes first unstable whenelectron temperature anisotropy is increased is still the slow mode. The case where theinstability is driven by ion temperature anisotropy (so-called classical mirror instabilitywith isotropic electrons) is in contrast different since the instability originates from theextra mode associated with the finite ion temperature fluctuations and not from the slowmode which continues to exist and to be stable above the mirror threshold. An interestingpoint is that the mirror mode, usually thought of being non-propagating, originates fromone of the damped propagating ”ion temperature modes”. These modes only becomenon-propagating for a large enough ion temperature anisotropy.
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