Bridging kinetic plasma descriptions and single fluid models
BBridging kinetic plasma descriptions and single fluidmodels
A. Crestetto (cid:63) , F. Deluzet † , D. Doyen ‡ (cid:63) Laboratoire de Math´ematiques Jean Leray UMR 6629,2, rue de la Houssini´ere,F-44322 Nantes Cedex 3, France,[email protected] † Universit´e de Toulouse; UPS, INSA, UT1, UTM,Institut de Math´ematiques de Toulouse,CNRS, Institut de Math´ematiques de Toulouse UMR 5219,F-31062 Toulouse, France,[email protected] ‡ Universit´e de Marne-la-Vall´ee,Laboratoire d’Analyse et de Math´ematiques Appliqu´ees,CNRS, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees UMR 8050,5, boulevard Descartes, Cit´e Descartes - Champs-sur-Marne,F-77454 Marne-la-Valle, France,[email protected] 21, 2020
Abstract
The purpose of this paper is to bridge kinetic plasma descriptions and low frequencysingle fluid models. More specifically, the asymptotics leading to Magneto-Hydro-Dynamic (MHD) regimes starting from the Vlasov-Maxwell system are investigated.The analogy with the derivation, from the Vlasov-Poisson system, of a fluid representa-tion for the ions coupled to the Boltzmann relation for electrons is also outlined. Theaim is to identify asymptotic parameters explaining the transitions from one micro-scopic description to a macroscopic low frequency model. These investigations provideground work for the derivation of multi-scale numerical methods, model coupling orphysics based preconditioning. a r X i v : . [ phy s i c s . p l a s m - ph ] A p r eywords: Plasma, Debye length, MHD, drift limit, fluid limit, Quasi-neutrality, Vlasov-Maxwell, Asymptotic-Preserving scheme.
The aim of this paper is to propose a continuation of the work initiated in [19, 20, 18]focusing on the derivation of asymptotic preserving schemes for kinetic plasma descriptionsin the quasi-neutral limit. The purpose of these numerical methods is to provide a quasi-neutral description of the plasma with no constraints on the simulation parameters related tothe Debye length but with the ability to perform local up-scalings with non neutral plasmadescriptions. This brings a gain in the computational efficiency, since the discretizationparameters can be set according to the physics of interest rather than the small scales(namely the Debye length) described by the model.The methodology introduced in these former achievements is aimed to be generalized hereto more singular limits. In this series of former works, the limit models remain kinetic and thescales of interest are related to the electron dynamics. For instance, the quasi-neutral limitof the Vlasov-Maxwell system investigated in [18] can be interpreted as a kinetic descriptionof the Electron-Magneto-Hydro-Dynamic (E-MHD) [27, 47, 7], accounting for the electroninertia, the massive ions being assumed at rest or slowly evolving. In the present paper, theobjective is to go beyond the kinetic E-MHD with the aim to bridge the Vlasov-Maxwellsystem and Magneto-Hydro-Dynamic (MHD) models. In MHD systems, the scales of interestare defined by the overall plasma dynamic which is governed by the ions, the fast scalesassociated to the electron inertia being filtered out from the equations.This preliminary work is therefore devoted to the derivation of a model hierarchy bridgingeither the Vlasov-Maxwell system and MHD models for magnetized plasmas or, the Vlasov-Poisson and the electron adiabatic response also referred to as Boltzmann relation (see[36, 49, 48] for seminal works and [14] for numerical investigations), for electrostatic frame-works. A wide range of applications of the present investigations can be named, specificallylow variance Particle-In-Cell methods or more generally numerical discretization of kineticmodels implementing a Micro-Macro decomposition of the distribution function. We referfor instance to [9, 11, 24, 37]) for Micro-Macro methods, and to [22] for the moment Guidedmethod; fluid-preconditioned fully implicit methods [4, 5, 6] and Asymptotic-Preserving nu-merical methods [33, 16]. Another application can be envisioned with the hybrid coupling ofParticle-In-Cell methods and MHD descriptions [44, 12] and more generally coupling strate-gies such as the Current-Coupling-Scheme (CCS) and the Pressure-Coupling-Scheme (PCS)(see [42] [51] and the references therein).The aim of this work is to clarify how the asymptotic parameters interact with eachother and define reduced models, but also, to relate these parameters to meaningful physicalquantities. The MHD regime is sometimes derived by letting ε , the vacuum permittivity, goto zero (see for instance [32, 50]) which is referred to as the full Maxwell to the low frequencypre-Maxwell’s equations asymptotic in [26, see section 2.3.3]. It is also common to let theelectron to ion mass ratio go to zero to explain the vanishing of the electron inertia ([26, 35])2n deriving either MHD modelling or the Boltzmann relation. Although the right asymptoticmodels are recovered by this means, these assumptions do not account for changes in thesystem characteristics that may explain for a regime transition: the electron to ion massratio remains constant and the same property holds true for the vacuum permittivity.The outlines of the paper are the following. The plasma kinetic description is introducedin Sec. 2 together with the Maxwell system driving the evolution of the electromagnetic field.A dimensionless form of the system is stated in order to develop an asymptotic analysis andthe derivation of reduced models. A hierarchy of quasi-neutral model is proposed in Sec. 3for the Vlasov-Maxwell system. It encompasses fully kinetic, hybrid as well as single fluid(MHD) plasma descriptions. The electrostatic framework is investigated in Sec. 4. Theelectrostatic limit of the Maxwell system is performed. A hierarchy of models, similar tothat of the electromagnetic framework is derived. Finally, a synthesis of these asymptoticanalysis is proposed in Sec. 5 devoted to conclusions. In this section, the purpose is to unravel a series of asymptotic limits bridging gap betweenthe Vlasov-Maxwell system and a Magneto-Hydro-Dynamic (MHD) model. The difficulty istherefore to identify parameters explaining the transition from one description to the otherand to relate these parameters to specific characteristics of the system. The tools mobilize tomeet this aim are based on the asymptotic analysis of the Vlasov-Maxwell system. Since thelow frequency plasma modelling is related to a fluid plasma description, the kinetic model isupgraded with collision operators. Therefore, the most refined modelling consists of a Valsovequation for the electrons and the ions, augmented with a collision operator and coupled tothe Maxwell system. Even if the physical model is non collisional or weakly collisional, thetransition towards a fluid limit is accounted for by a collisional process, thanks to a BGKoperator. This choice of collision operator is questionable from a strict modelling view point,nonetheless, the purpose here is to easily derive the fluid limit at a limited computational cost.In this respect the BGK collision operator is a good candidate. First, the whole collisionalprocesses are considered, including both inner and inter species collisions. Nonetheless, onlythe minimal collisional process will be accounted for to derive a MHD regime from thekinetic model. This point will be outlined in the following sections. The introduction ofnon dimensional quantities will naturally reveal dimensionless parameters in the equations.Letting some of these parameters go to zero shapes the hierarchy of models derived for theVlasov-Maxwell system and bridging the gap with MHD models.3 .2 The Vlasov-BGK-Maxwell system
The most refined description of the plasma is constituted by two Vlasov equations, f i and f e being the ion and electron distribution functions ∂ t f i + v · ∇ x f i + qm i ( E + v × B ) · ∇ v f i = Q i , (1) ∂ t f e + v · ∇ x f e − qm e ( E + v × B ) · ∇ v f e = Q e . (2)In these equations, q is the elementary charge, m α is the mass of the species α ( α = e forthe electrons and i for the ions). The BGK collision operator Q α are given by [31] Q α = Q αα + Q αβ , Q αα = ν αα ( M n α ,u α ,T α − f α ) , Q αβ = ν αβ (cid:16) M n α ,u β ,T β − f α (cid:17) , (3) ν αα and ν αβ being the like-particle and inter species collision frequencies which can be definedas [15, 45] ν ii = K n i ( k B T i ) √ √ m i , ν ie = K n e ( k B T i ) √ m e m i , (4a) ν ee = K n e ( k B T e ) √ √ m e , ν ei = K n i ( k B T e ) √ m e , (4b)where, C denotes a constant with a magnitude equal to one, ln(Λ) the Coulomb logarithmand the ions being assumed mono-charged, K = C (cid:18) q π(cid:15) (cid:19) ln(Λ) . (4c)The Maxwellians M n α ,u α ,T α and M n α ,u β ,T β are defined as M n α ,u α ,T α = n α ( x, t ) (cid:18) m α πk B T α ( x, t ) (cid:19) Dv exp (cid:18) − m α | u α ( x, t ) − v | k B T α ( x, t ) (cid:19) , (5a) M n α ,u β ,T β = n α ( x, t ) (cid:18) m α πk B T β ( x, t ) (cid:19) Dv exp (cid:18) − m α | u β ( x, t ) − v | k B T β ( x, t ) (cid:19) , (5b) D v denoting the dimension of the velocity space, k B is the Boltzmann constant. TheMaxwellian parameters are n α , u α and T α the density, mean velocity and temperature asso-ciated to the distribution function f α and defined as n α = (cid:90) Ω v f α dv , n α u α = (cid:90) Ω v v f α dv , γ − n α k B T α = m α (cid:90) Ω v | v − u | f α dv , (6)4ith γ the specific heat ratio whose value depends on the dimensionality of the velocity space D v through γ − D v . (7)The collision operators verify the following conservation properties (cid:90) Q αα m α v n dv = 0 , n = 0 , . . . , , (8a) (cid:90) Q αβ m α v n dv + (cid:90) Q βα m β v n dv = 0 , n = 0 , . . . , . (8b)The temperature and the mean velocity ( u β , T β ) in the inter-specie collision operatorexpression (5b) should be chosen with care in order to guarantee the total momentum andenergy conservation. Indeed the following identities (cid:90) Q αβ m α v dv = ν αβ m α n α ( u β − u α ) , (9a) (cid:90) Q αβ m α v dv = ν αβ (cid:18) D v n α k B (cid:0) T β − T α (cid:1) + 12 m α n α (cid:0) u β − u α (cid:1)(cid:19) (9b)hold true for the operators defined by (5). The trivial choice ( u β , T β ) = ( u β , T β ) do ensurethe plasma total momentum conservation, provided that ν ei m e n e = ν ie m i n i . However inthis case, the plasma total energy is not conserved. We refer to [28] for a seminal work, aswell as [35] and the references therein for recent advances, on the choice of these parameterscompliant with the desired properties (8b) of the inter-specie collision operators.The electromagnetic field ( E, B ) evolution is driven by the Maxwell system:1 c ∂ t E − ∇ x × B = − µ J , (10) ∂ t B + ∇ x × E = 0 , (11) ∇ x · E = ρ(cid:15) , (12) ∇ x · B = 0 , (13)where c is the speed of light, µ the vacuum permeability and (cid:15) the vacuum permittivityverifying µ (cid:15) c = 1. The Maxwell sources are the particle currents and densities ρ = q ( n i − n e ) , (14a) J = q ( n i u i − n e u e ) . (14b)The definition of the collision frequencies as stated by Eqs.(4) relates different time scales.Indeed, because of their different masses, ions and electrons are not equally affected bycollisions. This properties are more clearly emphasized working with dimensionless variablesas proposed in the next section. 5 .3 Scaling of the Vlasov-Maxwell system The equations are written with dimensionless quantities in order to easily identify differentregimes. The scaling is introduced under a priori assumptions that the electronic and ionictemperatures, densities and mean velocities are comparable with a magnitude denoted T , n and u . These scales define the typical Debye length as well as the electron plasma period λ D = (cid:115) (cid:15) k B T q n , τ pe = (cid:114) m e (cid:15) q n . We denote by x and t the characteristic space and time scales of the phenomena observed,which yields to the velocity of interest ϑ = x /t . The magnitude of the thermal velocity forthe specie α is denoted v ,α with v ,α = k B T /m α . Due to the different masses, the thermalveolicity of the electron is not that of the ions. The reference thermal velocity v will bedefined by the ion one v = k B T /m i , hence v ,e = v /ε and v ,i = v , where ε = m e /m i .Finally, the particle current scale is defined as J = qn u . The dimensionless variables aredefined according to x ∗ = xx , t ∗ = tt , v ∗ = vv ,α , f ∗ = fn / ( v ,α ) D v , n ∗ = nn , J ∗ = Jqn u ,E ∗ = EE , B ∗ = BB , the collision frequencies verifying ν ee, = ν ei, = 1 ε ν ii, , ν ie, = εν ii, , ε = (cid:114) m e m i . (15)On the fastest time scales, the electron distribution function relaxes towards a Maxwellian.On the same time scale, the electron mean velocity and temperature relax towards thatof those of the ions. The relaxation of the ionic distribution function towards the localequilibrium is slower, by a factor ε − = (cid:112) m i /m e . Finally, the ions are almost unaffectedby the collisions against the electrons. The relaxation of the ionic distribution functiontowards that of the electrons define the largest time scale, by a factor ε − compared to thethe relaxation towards the thermodynamical equilibrium.The dimensionless ionic and electronic Vlasov equations can be rewritten as (keeping thesame notations for dimensionless variables): ξ∂ t f i + v · ∇ x f i + η ( E + βξ v × B ) · ∇ v f i = ξκ (cid:16) ν ii ( M n i ,u i ,T i − f i ) + εν ie (cid:0) M n i ,u e ,T e − f i (cid:1) (cid:17) , (16) ξε∂ t f e + v · ∇ x f e − η ( E + βεξ v × B ) · ∇ v f e = ξκ (cid:16) ν ee ( M n e ,u e ,T e − f e ) + ν ei (cid:0) M n e ,u i ,T i − f e (cid:1) (cid:17) , (17)6ogether with the dimensionless Maxwell system writting λ η ( α ∂E∂t − β ∇ x × B ) = − α Mξ J , (18a) β∂ t B + ∇ x × E = 0 , (18b) λ η ∇ x · E = n i − n e , (18c) ∇ x · B = 0 , (18d) J = n i u i − n e u e . (18e)This system is written thanks to the following dimensionless parameters ε = m e m i the ratio of the electronic and ionic masses ,λ = λ D x the scaled Debye length ,M = u v the ionic Mach number, with v = (cid:114) k B T m i the ionic speed of sound ,ξ = ϑ v the ratio of the typical velocity to the ionic speed of sound ,α = ϑ c the ratio of the typical velocity to the speed of light ,η = qx E k B T the ratio of the electric and plasma internal energies ,β = ϑ B E the induced electric field relative to the total electric field ,κ − = ν ii, t the number of ion-ion collisions during the typical time . (19)The dimensionless Maxwellians are defined by M n e ,u α ,T α = M n e ,u α ,T α = n e ( x, t ) (cid:18) πT α ( x, t ) (cid:19) Dv exp (cid:18) − | M εu α ( x, t ) − v | T α ( x, t ) (cid:19) , (20a) M n i ,u α ,T α = M n i ,u α ,T α = n i ( x, t ) (cid:18) πT α ( x, t ) (cid:19) Dv exp (cid:18) − | M u α ( x, t ) − v | T α ( x, t ) (cid:19) . (20b)Some comments can be stated regarding the meaning of these parameters and the scalingrelations.The typical mean velocity and temperature are assumed to be the same for the electronsand the ions. Accordingly, the relaxation of the electron mean velocity and temperaturetowards that of the ions may be assume to marginally contribute to the evolution of thesystem. This assumption is therefore consistent with the investigation of resistive-less plasmamodellings and the neglect of the inter-species collisions.The parameter ξ is intended to provide a measure of how the electronic and ionic dynamicsare resolved. The choice ξ = 1 means that the system is assumed to evolve at a speed7omparable to the ionic thermal velocity v , while ξε = 1 performs a rescaling of this typicalvelocity to the electron microscopic velocity. Setting ξ = M relates the typical speed of thesystem to the ionic mean velocity u . Actually, the Mach number measures the gap betweenthe microscopic (thermal) and macroscopic velocity scales.The scaling relation η = 1 is generally assumed in single fluid plasma representation.The plasma internal energy is then on a par with the electric energy. This equilibriumis fundamental in the derivation of the Boltzmann relation. The identity βM = ξ is alsocommon in single-fluid plasma models. This amounts to assume that the induced electric fieldscales as the product of the plasma mean velocity and the typical magnetic field E = u B .In other words, the magnetic field is essentially transported with the plasma flow. This laterassumption is in line with the Alfven’s frozen theorem [41, 26, 13, 43] characteristic of idealMHD models: the magnetic field is frozen into the plasma and transported by its flow.The derivation of reduced models consists in identifying small dimensionless parametersand let them go to zero. The smallness of the scaled Debye length refers to a typical spacescale much larger than the physical Debye length. This means that it assumed that the chargeseparations occurring on space scales comparable to the Debye length are not important toexplain the evolution of the system. Sending the scaled Debye length to zero performs alow frequency filtering into the equation deriving thus a quasi-neutral model. In the contextof the derivation of numerical methods, the typical length relates to the mesh size. Thisoutlines the advantage of reduced models: the low frequency filtering operated by vanishingsmall parameters permits to derive numerical methods with discretization parameters (meshsize and time step) unconstrained by the small scales filtered out from the original equations. The aim here is to reduce the number of free dimensionless parameters deriving by thismeans different reduced models well suited for the description of low frequency phenomena.As depicted in Fig. 1, the starting point of this hierarchy of models implements the minimalupgrades of the Vlasov-Maxwell system to recover a MHD regime. Precisely the only interspecies collisions are taken into account in the initial model in order for the distributionfunction to relax towards the local equilibrium. This yields ξ∂ t f i + v · ∇ x f i + η ( E + βξ v × B ) · ∇ v f i = ξκ ν ii ( M n i ,u i ,T i − f i ) , (21a) ξε∂ t f e + v · ∇ x f e − η ( E + βεξ v × B ) · ∇ v f e = ξκ ν ee ( M n e ,u e ,T e − f e ) , (21b)for the evolution of the ions and electrons coupled to the dimensionless Maxwell systemdefined by Eqs. (18). 8rom the scaling relations stated by Eq. (15), discarding the inter-species collisions makesense for the ions. Due to their large mass, the ions are almost unaffected by encounterswith electrons. For the electrons, this assumption is not in line with the scaling of the likeand inter-species collision frequencies. However, the purpose here, is to propose a physicallymeaningful framework to clarify the foundation of a numerical method bridging the gapbetween a kinetic description of a weakly (or non) collisional magnetized plasma with aMHD regime. The interspecies collisions give rise to the resistivity in the macroscopic systemwhich is not the targeted class of modelling for this work. Massless Hall-MHDVlasov-BGK-Maxwell Euler-MaxwellHybrid Hall-MHD Electron-MHDHall-MHDKinetic Electron-MHD M → κ → α, λ → Mε ) → κ → κε ) , ( Mε ) → α, λ → Fluid HierarchyKinetic Hierarchy
Figure 1: Fluid and kinetic (quasi-neutral) model hierarchies derived from the Vlasov-BGK-Maxwell system.
To identify easily a fluid regime, the distribution function is decomposed into a Maxwellian M n α ,u α ,T α and a deviation from this Maxwellian κ g α according to f α = M n α ,u α ,T α + κ g α , (22a)the deviation verifying (cid:10) v n g α (cid:11) = (cid:90) Ω v v n g α dv = 0 , n = 0 , , . (22b)With this decomposition, the Vlasov-Boltzmann equations (21) can be recast into a hydro-dynamic set of equations with kinetic corrections, depending on the moment of the deviation9 α , yielding ξM ∂ t n i + ∇ x · ( n i u i ) = 0 (23a) M (cid:18) ξM ∂ t ( n i u i ) + ∇ x · ( n i u i ⊗ u i ) (cid:19) + ∇ x p i − η n i (cid:0) E + βMξ u i × B (cid:1) = − κ ∇ x · (cid:104) v ⊗ vg i (cid:105) , (23b) ξM ∂ t W i + ∇ x · (cid:0) ( W i + p i ) u i (cid:1) − ηn i E · u i = − κM ∇ x · (cid:28) | v | vg i (cid:29) . (23c)with W i = ( M ) n i u i + p i γ − , p i = n i T i , (23d)for the ions, and an equivalent system for the electrons, ξM ∂ t n e + ∇ x · ( n e u e ) = 0 (24a)( M ε ) (cid:18) ξM ∂ t ( n e u e ) + ∇ x · ( n e u e ⊗ u e ) (cid:19) + ∇ x p e + η n e (cid:0) E + βMξ u e × B (cid:1) = − κ ∇ x · (cid:104) v ⊗ vg e (cid:105) , (24b) ξM ∂ t W e + ∇ x · (cid:0) ( W e + p e ) u e (cid:1) + ηn e E · u e = − κM ε ∇ x · (cid:28) | v | vg e (cid:29) . (24c)with W e = ( M ε ) n e u e + p e γ − , p e = n e T e . (24d)These two systems are coupled to a set of equations (the Maxwell system (18)) driving thechanges in the electromagnetic field. Omitting the collisions, the fastest velocity in this system is the propagation of waves at thespeed of light described by the Maxwell system. The Debye length as well as the plasmaperiod also define small space and time scales for large plasma densities. The quasi-neutrallimit is defined by the following scaling relations:( α, λ ) → , α ∼ λ . (25)This amounts to assume that the scaled Debye length is small compared to the typical lengthand that the system evolves at a speed lower than the speed of light. By this means, the smallscales related to these parameters are filtered out of the equations. The last hypothesis α ∼ λ is essential to recover the low frequency Ampere’s law, derived by neglecting the displacementcurrent. This equation being common to Magneto-Hydro-Dynamic models, the quasi-neutral10imit encompasses this two assumptions. With the vanishing of this generalised dimensionlessDebye length ( λ, α ) →
0, the Maxwell system degenerates into β ∇ x × B = Mξ J , (26a) β∂ t B + ∇ x × E = 0 , (26b) n i = n e , (26c) ∇ x · B = 0 . (26d)From Gauss’s law, the property of the electronic density to match that of the ions is re-covered, which genuinely enforces the quasi-neutrality of the plasma. The electric field hasno contribution in both these degenerate Gauss and the Ampere equations. The remainingoccurrence of the electric field is limited to the Faraday equation (26b). Therefore, this set ofequations is not well suited for the computation of the electric field. Indeed, the electrostaticcomponent of the electric field can be arbitrarily chosen in Eqs. (26).In the quasi-neutral limit, the electric field is provided by the particle current J ratherthan displacement current ( ∂E/∂t originally present in Ampere’s law). To close the system,the dependence of J with respect to E shall be explained to restore uniqueness of the electricfield. This is related to the model describing the plasma. The aim here is to follow the microscopic dynamic of the electrons. The velocity of interestis the kinetic velocity of the electrons. This amounts to set ϑ = v /ε or equivalently ξε = 1,yielding ∂ t f i + ε ( v · ∇ x f i + η ( E + εβv × B ) · ∇ v f i ) = 1 κ ν ii ( M n i ,u i ,T i − f i ) , (27a) ∂ t f e + v · ∇ x f e − η ( E + βv × B ) · ∇ v f e = 1 εκ ν ee ( M n e ,u e ,T e − f e ) , (27b)The collisions are assumed to be ineffective on the characteristic time scale: κε (cid:29) ε (cid:28) λ = α ) →
0, the system at hand here is recast into (see [18]) ∂ t f e + v · ∇ x f e − η ( E + βv × B ) · ∇ v f e = 0 , (28) β ∇ x × B = ( M ε ) J , (29) β∂ t B + ∇ x × E = 0 , (30) n e = n i = n , (31) ∇ x · B = 0 . (32)11irst, note that the formal time derivative of the Faraday equation (30) together with thecurl of Ampere’s law yields ∇ x × ∇ x × E = − ( M ε ) ∂ t J , (33)which outlines that the electric field is known up to the gradient of a potential in this system.In [21, 18] the ill-posed nature of this equation is corrected by explaining the relation betweenthe current density and the electric field. The first moment of Eq. (28) yields the conservationof the electronic momentum, with(
M ε ) ∂∂t ( nu e ) = −∇ x · S − η (cid:0) nE + β ( M ε ) nu e × B (cid:1) = 0 , (34a)with n = n e = n i and S = (cid:90) ( v ⊗ v ) f e dv . (34b)Inserting this identity into Eq. (33) gives ηnE + ∇ x × ∇ x × E = ηβ ( M ε ) J × B − ∇ x · S . (35)This equation is well posed in the quasi-neutral limit ( n >
0) and can be used for thecomputation of the electric field. This yields the following definition of the quasi-neutralmodel ∂ t f e + v · ∇ x f e − ( E + v × B ) · ∇ v f e = 0 , (36a) nE + ∇ x × ∇ x × E = ( M ε ) J × B − ∇ x · S , (36b) ∂ t B + ∇ x × E = 0 , (36c) ∇ x · B = 0 . (36d)This model is written under the assumption η = 1 meaning that the plasma thermal energyis on a par with the electric energy together with β = 1.Note that the electric field provided by Eq. (36b) enforces a divergence free particlecurrent, or more precisely ∂ t ( ∇ · J ) = 0. This yields, thanks to the continuity equation: ∂ ρ∂t = 0 . This proves the consistency of this model with the quasi-neutrality assumption (matching ofthe electronic and ionic densities) as soon as the initial data are compliant with this regime.The evolution of the ions may be taken into account thanks to another Vlasov equation.This brings corrections in deriving Eq. (36b) accounting for the contribution of the ionicparticle current.The characteristics of this models are similar to the so-called Electron MHD: the timescale of interest is that of the electrons, the ions merely creating a motionless background forthe fast electron flows [34]. In particular, this modelling accounts for the inertia of electrons.A noticeable difference with the Electron-MHD (see Sec. 3.4) lies in the kinetic descriptionof the plasma. An asymptotic-Preserving method is proposed in [18] to bridge this quasi-neutral model and the Vlasov-Maxwell system. The properties of this quasi-neutral plasmadescription are investigated in [50] by means of a linear stability analysis.12 .2.2 A hybrid formulation of the Hall-MHD
Hybrid modelling [52, 54, 51] refers to a class of plasma models where the ions are describedby a kinetic equation while the fluid limit is assumed for the electrons. This is in line withthe scaling relations of the collision frequencies stated by Eqs. (15). The relaxation of theelectronic distribution function towards the local equilibrium is indeed faster than for theions. The aim of these modelling is to filter out of the equations the fast scales carried bythe electron dynamics. Therefore, a zero inertia regime is also assumed for the electronstogether with the fluid limit and the quasi-neutrality of the plasma.The typical velocity selected here is the microscopic (thermal) velocity of the ions. Thistranslates into the identity ξ = 1 resulting in the following system for the plasma: ∂ t f i + v · ∇ x f i + η ( E + βξ v × B ) · ∇ v f i = 1 κ ν ii ( M n i ,u i ,T i − f i ) , (37a) ∂ t f e + 1 ε (cid:18) v · ∇ x f e − η ( E + βεξ v × B ) · ∇ v f e (cid:19) = 1 εκ ν ee ( M n e ,u e ,T e − f e ) , (37b)The fluid limit for the electrons is selected assuming ( εκ ) (cid:28) λ = α (cid:28)
1. To overcome the degeneracy of the Maxwell system in the quasi-neutral limit, the electronic momentum is harnessed to provide the so-called generalisedOhm’s law. The electronic system can be recast into(
M ε ) (cid:18) M ∂ t ( n e u e ) + ∇ x · ( n e u e ⊗ u e ) (cid:19) + ∇ x p e + η (cid:16) n e E + ( βM ) u e × B (cid:17) = − ( κ ( M ε )) ∇ x · σ e , M ∂ t W e + ∇ x · (cid:16) ( W e + p e ) u e (cid:17) + ηn e E · u e = − κM ε ∇ x · (cid:16) ( M ε ) σ e · u e + µ e ∇ x T e (cid:17) ;with, owing to the quasi-neutrality assumption, n e = n i = n .The dynamic described by these equations is stiff, this is due to the smallness of ( ξε )in this regime: the thermal velocity of the ions (defined as the typical velocity) is smallcompared to that of the ions. Therefore, the electrons are in a low Mach regime. Assuming( M ε ) (cid:28) ∇ x ( nT e ) + η n (cid:16) E + ( βM ) u e × B (cid:17) = 0 , (38a) ∇ x T e = 0 . (38b)The classical massless approximation for the electrons is recovered with the generalisedOhm’s law and a homogeneous electronic temperature. The definition of the mean velocity u e is derived from the particle current density J = n ( u i − u e ) together with Amp`ere’s law(29), yielding u e = u i − βM ∇ x × Bn , (39)13he hybrid plasma modelling writes (assuming η = β = 1) ∂ t f i + v · ∇ x f i + ( E + v × B ) · ∇ v f i = 0 , (40a) E = − M u × B + M ∇ x × Bn × B − T e ∇ x nn , (40b) ∂B∂t + ∇ x × E = 0 , ∇ x · B = 0 , (40c)with n = (cid:90) f i dv , nu = (cid:90) vf i dv . (40d)The derivation of a similar model is proposed in [1] with numerical investigations in [17]. This model is obtain by letting κ → ξM ∂ t n e + ∇ x · ( n e u e ) = 0 (41a) (cid:18) ξM ∂ t ( n e u e ) + ∇ x · ( n e u e ⊗ u e ) (cid:19) + 1( M ε ) (cid:18) ∇ x p e + η n e ( E + βMξ u e × B ) (cid:19) = 0 , (41b) ξM ∂ t W e + ∇ x · (cid:0) ( W e + p e ) u e (cid:1) + ηn e E · u e = 0 . (41c)A similar system is derived for the ions however with ε = 1 and η replaced by − η . Thesetwo sets of conservation laws are coupled to the Maxwell system (18). Performing the quasi-neutral limit in this system ( λ = α ) → ξε = 1 yields the quasi-neutral bi-fluid Euler-Maxwell system also referred to as Electron-MHD system. This model is similar to that of Sec. 3.2.1 but with a fluid description for theplasma. It is implemented and investigated numerically in the framework of Asymptotic-Preserving methods in [21]. The Hall-MHD regime (see [38, 53, 43]) is recovered from the assumptions of the precedentsection but with a typical velocity equal to the plasma mean flow yielding ξ = M . The fastelectronic dynamics is filtered out from the equations to provide a low frequency modellingfor the plasma driven by the evolution of the massive ions. The plasma velocity, denoted u is defined as that of the heavy species u = u i . The other parameters obey the classicalscaling relations of MHD models: η = 1 and β = 1.14n the drift regime ( M ε → E e = p e / ( γ − , (42a)with the electronic momentum and energy verifying ∇ x p e = − n ( E + u e × B ) , (42b) ∂ t E e + ∇ x · (cid:0) ( E e + p e ) u e (cid:1) = − nE · u e . (42c)The generalysed Ohm’s law (42b) is harnessed to compute the electric field. The electronicvelocity u e is substituted by u e = u − J/n .The electric field is computed thanks to the generalized Ohm’s law, giving rise to E = − u × B + ∇ x × Bn × B − ∇ x p e n . (43)The first term of the right hand side of the equation (43) is the classical frozen field term,explaining the convection of the magnetic field together with the plasma . The second andthird contributions are the so-called Hall and diamagnetic terms. Inserting this definition inthe Faraday law (26b), the magnetic induction equation can be constructed, with ∂ t B + ∇ x · ( u ⊗ B − B ⊗ u ) = −∇ x × (cid:18) ∇ x × Bn × B (cid:19) + ∇ x × (cid:18) ∇ x p e n (cid:19) . (44)Finally the plasma mass density, momentum and total pressure p and energy W definedby p = p i + p e = n ( T i + T e ) , (45a) W = W i + E e = ( M ) nu + pγ − , (45b)verify ∂ t n + ∇ x · ( nu ) = 0 (45c) ∂ t ( nu ) + ∇ x · ( nu ⊗ u ) + 1 M ∇ x p = 1 M J × B (45d) ∂ t W + ∇ x · (cid:0) ( W + p ) u (cid:1) + ∇ x · (cid:16) ( E e + p e )v H (cid:17) = E · J , (45e)the Hall velocity v H , which can be interpreted as the electron velocity in the ion frame, isdefined by v H = − Jn . (46)The ideal MHD equations are classically written under a conservative form using the systemtotal pressure and energy p TOT = p + B , W TOT = W + B , (47a)15riting the system as ∂ t n + ∇ x · ( nu ) = 0 (47b) ∂ t ( nu ) + ∇ x · (cid:16) nu ⊗ u − M B ⊗ B (cid:17) + 1 M ∇ x p TOT = 0 (47c) ∂ t W TOT + ∇ x · (cid:16) ( W TOT + p TOT ) u − ( B · u ) B (cid:17) = −∇ x · (cid:16) ( E e + p e )v H (cid:17) , (47d) ∂ t B − ∇ x · (cid:16) B ⊗ ( u + v H ) − ( u + v H ) ⊗ B (cid:17) = ∇ x × (cid:18) ∇ x p e n (cid:19) . (47e)This set of equations is supplemented with the electronic energy conservation (42c).The ideal MHD equations are recovered from this system assuming an ideal Ohm’s lawwhere the current density is assumed small compared to the ion mean velocity and thereforeneglected. However, in this simplified framework (omitting the unlike particle collisions),there are no mechanisms preventing the electron mean velocity to depart from that of theions. Consequently, the generalized Ohm’s law incorporates the Hall velocity in complementto the so called ideal Ohm’s law. The effect of the resistivity should be consider to derivethe ideal mHD regime.The drift approximation operated for the electrons amounts to vanishing the electronicMach numbers ( M ε ). The scale separation introduced by the small electron to ion mass ratio ε is not always sufficient to consider this limit independently to vanishing ionic Mach numbers M →
0. For low ionic Mach number a low frequency filtering may be operated performingthe limit of vanishing electronic Mach numbers jointly with the ionic Mach numbers. Thisasymptotic defined the Massless MHD regime [3].
The electrostatic regime is recovered from the dimensionless Maxwell system (18) by letting α go to zero. This assumption shall be interpreted as a typical velocity negligible comparedto the speed of light. From Amp`ere’s law λ η ∂E∂t + Mξ J = βλ ηα ∇ x × B , (48)the limit α → ∇ × B = 0. Nonetheless, the right hand side of Eq. (48) remainsan undetermined form. Therefore the Ampere’s law is not well suited for the computationof the electric field in the electrostatic limit. However, subjected to convenient boundaryconditions, the property ∇ x × B = 0 together with ∇ x · B = 0 yields ∂ t B = 0. Inserting thisproperty into the Faraday equation (18b) provides an electrostatic electric field: ∇ × E = 0.Furthermore, the undetermined form in Eq. (48) is divergence free. Therefore, computingthe divergence of Ampere’s law provides λ η ∂ ∇ x · E∂t = − Mξ ∇ x · J , ∇ x × E = 0. Notethat, owing to the continuity equation ∂ρ∂t + Mξ ∇ x · J = 0 , originating from the conservation of the particle densities (23a) and (24a), the divergence ofAmp`ere’s law is equivalent to the time derivative of Gauss law, with ∂∂t (cid:0) λ η ∇ x · E − ρ (cid:1) = 0 . Therefore, in the electrostatic regime, the Gauss equation is used to compute the electricfield.
This analysis is carried out under the assumption of a vanishing magnetic field ( B = 0). Theplasma description consider in the sequel is therefore ξ∂ t f i + v · ∇ x f i + ηE · ∇ v f i = ξκ (cid:16) ν ii ( M n i ,u i ,T i − f i ) (cid:17) , (49a) ξε∂ t f e + v · ∇ x f e − ηE · ∇ v f e = ξκ (cid:16) ν ee ( M n e ,u e ,T e − f e ) (cid:17) (49b)coupled to the Gauss equation − λ η ∆ φ = n i − n e , (49c) φ being the electrostatic potential, with E = −∇ φ and n α = (cid:82) f α dv . The quasi-neutralityof the plasma is recovered for vanishing scaled Debye length λ →
0. In this regime, similarlyto the electromagnetic case, an equation needs to be manufactured from the motion of theparticles, to compute the electric field. This is classically obtained thanks to the equationof the electronic momentum conservation. The electric field is computed in order for thisconservation to be satisfied. In [20] an equivalent approach is proposed. It consists in usingthe time derivatives of Gauss law to produce − λ η ∂ ∂t ∆ φ = ∂ ∂t ( n i − n e ) , (50)In the quasi-neutral limit ( λ → ∂ ∂t ( n i − n e ) = 0 . From the system (24), the following identity is recovered (cid:18) ξM (cid:19) ∂ n e ∂t = −∇ x · (cid:16) ∇ x · ( n e u e ⊗ u e ) (cid:17) − M ε ) ∇ x · (cid:16) ∇ x P e + ηn e E (cid:17) . (51) P e = p e I d + κ (cid:104) v ⊗ vg e (cid:105) , (cid:104) v ⊗ vϕ (cid:105) = (cid:90) ϕ dv . (52)17esuming the scaling relations of Sec. 3.2.1: ξ = 1 /ε and κε >
1, assuming that the ions areat rest, the evolution of the charge density n i − n e is governed by Eq. (51) with ∂ n e ∂t = − ( M ε ) ∇ x · (cid:16) ∇ x · ( n e u e ⊗ u e ) (cid:17) − ∇ x · (cid:16) ∇ x P e + ηn e E (cid:17) . (53)The evolution of the density is barely independent of the mean flow velocity but relies onthe balance between the pressure and the electric forces. The equation providing the electricpotential φ is obtained by inserting this relation into Eq. (50) and passing to the limit λ → ∂ t f e + v · ∇ x f e + η ∇ x φ · ∇ v f e = 1 εκ (cid:16) ν ee ( M n e ,u e ,T e − f e ) , (54a) − η ∇ x · ( n e ∇ x φ ) = −∇ x · (cid:16) ∇ x · P e − ( M ε ) ∇ x · ( n e u e ⊗ u e ) (cid:17) . (54b)According the values of κ , the collision term in Eq. (54a) may be disregarded, definingtherefore a non collisional kinetic description. Contrariwise, letting κ →
0, a fluid descriptionfor the electrons may be derived, with1(
M ε ) ∂ t n e + ∇ x · ( n e u e ) = 0 (55a)1( M ε ) ∂ t ( n e u e ) + ∇ x · ( n e u e ⊗ u e ) + 1( M ε ) ( ∇ x p e − ηn e ∇ x φ ) = 0 , (55b)1( M ε ) ∂ t W e + ∇ x · (cid:0) ( W e + p e ) u e (cid:1) − ηn e ∇ x φ · u e = 0 . (55c)Note that a similar equation to (53), but with P e = p e I d, may be worked out of the conserva-tion of the electronic density (55a) and momentum (55b). This outlines that the electronicdynamics, in particular the electronic speed of sound, is resolved in this model. Compara-ble models are implemented and numerical experiences in the context of the Asymptotic-Preserving methods in [10] for fluid plasma description and [40] for kinetic equations. The typical velocity is chosen to be that of the ions with either ξ = 1 for the kineticdescriptions of the ions and ξ = M for macroscopic models.The hybrid modelling investigated in Sec. 3.2.2 is defined by the scaling relations ξ = 1,( εκ ) (cid:28) η = 1. The equilibria stated by Eqs. (38) yields: T e ∇ x n e = n e ∇ x φ , (56)with an homogeneous electronic temperature. This equation is integrated to provide theso-called Boltzmann relation n e = n exp (cid:18) − φT e (cid:19) , (57)18 being a constant (independent of the space variable x ) that should be determined fromadequate conditions [30]. Due to the Boltzmann relation, the quasi-neutral limit is notsingular any more. Indeed plugging the Boltzmann relation (57) into the Gauss equationyields − λ ∆ φ = n i − n exp (cid:18) − φT e (cid:19) . (58)This equation is not degenerate for the computation of the electric potential for vanishing λ . Indeed, the non linear part of the equation provides a means of computing φ in thequasi-neutral limit. This property is thoroughly investigated in [23].The hybrid electrostatic model may be recast into ∂ t f i + v · ∇ x f i − ∇ x φ · ∇ v f i = 1 κ (cid:16) ν ii ( M n i ,u i ,T i − f i ) (cid:17) , (59a) φ = − T e ln (cid:18) n i n (cid:19) . (59b)Letting κ → κ/ ( M ε ) (cid:28) ξ = M yields the quasi-neutral fluid model ∂ t n i + ∇ x · ( n i u i ) = 0 (60a) ∂ t ( n i u i ) + ∇ x · ( n i u i ⊗ u i ) + 1 M ( ∇ x p i + n i ∇ x φ ) = 0 , (60b) ∂ t W i + ∇ x · (cid:0) ( W i + p i ) u i (cid:1) + n i ∇ x φ · u i = 0 , (60c) φ = − T e ln (cid:18) n i n (cid:19) . (60d)In the models (59) and (60) following the evolution of the plasma at the ionic scale, thefast electronic dynamics introduced by the electron inertia is filtered out of the equations byperforming the low frequency limit ( M ε ) → In this paper, we propose an asymptotic analysis bridging kinetic plasma descriptions cou-pled to the Maxwell system and single plasma modelling. Two frameworks are investigated.The first one is devoted to electromagnetic fields. The plasma is represented by a hierarchyof models starting with the bi-kinetic Vlasov-Maxwell system while ending with the singlefluid Hall-Magneto-Hydro-Dynamic model. The second framework is dedicated to electro-static fields. In this context, the asymptotic analysis permits to derive a hierarchy of modelsbridging the bi-kinetic Vlasov-Poisson system to a single fluid representation consisting of afluid system for the ions coupled to the Boltzmann relation for the electrons. The investiga-tions proposed within this document unravel different asymptotic parameters explaining thetransition from one model to the other. The effort conducted in the present work consists19n relating these asymptotic parameters to characteristics of the system. This means thatthe transition from one model to the other may be explained by a change in the plasmacharacteristics or the typical scales at which the plasma is observed.This last notion is important in the perspective of designing a numerical method. Indeed,the discretization of these equations requires the use of a mesh interval as well as a time step.These two numerical parameters define the typical space and time scale, therefore a velocityscale as well, the numerical method is aimed at capturing. This is related to the parameter ξ used for the asymptotic analysis. Regarding the quasi-neutral modellings investigatedwithin this document, different choices are operated for this parameter. The fastest scalesare related to the electron thermal velocity when the fast electron dynamics is intended tobe captured by the model. This is for instance the value selected for Electron-MHD regimes,either in the fluid or kinetic frameworks. For hybrid or single fluid plasma representations,the velocity scale is reduced to that of the mean flow of the plasma defined by the massiveions. The organisation of Sec. 4 is aimed at emphasizing this feature.The second parameter, already established in precedent work (see [18, 21, 20]), is thegeneralized scaled Debye length λ . It actually encompasses the scaled Debye length and theratio of the typical velocity to the speed of light. Vanishing the generalised Debye lengthamounts to filter out from the equations the small scales attached to the charge separationas well as those related to the propagation of electromagnetic waves at the speed of light.The quasi-neutral limit is therefore a low frequency limit. Quasi-neutrality breakdowns maybe explained by a refinement of the typical length scale or, for instance, a decrease of theplasma density. This changes are well accounted for by the asymptotic parameters set up toperform the analysis.The vanishing of the electron inertia is related to a low Mach regime ( M ε ) (cid:28)
1. Insingle fluid plasma representation, the fast electron dynamic is dropped out of the equationsto perform a low frequency filtering, the system being assumed to evolve at a lower speedattached to the massive ions. Nonetheless, the nature of the flow may be subjected tosignificant changes explaining that the particle inertia becomes significant again to accountfor the system evolution. This is illustrated in studies of plasma flows in sheaths, withsupersonic particles, while the mean plasma velocity is small compared to the speed ofsound in the plasma bulk [8, 46, 29, 25, 39]. Accounting for this phenomena is possibleselecting the appropriate typical velocity to resolve or filter the fast electron dynamic.Finally the fluid assumption is classically related to a vanishing of the Knudsen number κ (cid:28) ξ , λ , M ε , κ )define a hierarchy of reduced models bridging kinetic plasma descriptions coupled to theMaxwell system to quasi-neutral plasma representations including kinetic, hybrid and singlefluid modellings. These later models, namely the Hall-MHD and the Boltzmann relationare widely used to design efficient numerical methods. The asymptotic analysis conductedwithin this document draws the guidelines for the derivation of numerical methods imple-menting local up-scalings, therefore widening the use of numerical methods discretizing thesereduced models. 20 cknowledgments This work has been supported by “F´ed´eration de Fusion pour la Recherche par ConfinementMagn´etique” (FrFCM) in the frame of the project ”BRIDIPIC: BRIDging Particle-In-Cellmethods and low frequency numerical models of plasmas”.A. Crestetto acknowledges support from the french “Agence Nationale pour la Recherche(ANR)” in the frame of the projects MoHyCon ANR-17-CE40-0027-01 and MUFFIN ANR-19-CE46-0004.
A Micro-Macro decomposition, computation of the vis-cous terms.
A.1 Introduction, definitions, elementary properties
The analysis carried out in this section are developed in the electrostatic framework andspecified for the electrons. The extensions for either electromagnetic fields or the ions arestraightforward and are therefore omitted for conciseness.We first introduce the projector onto the Maxwellian For any smooth function ϕ , theprojector onto the Maxwellian denoted M e with M e = n e ( x, t ) 1 (cid:0) πT e ( x, t ) (cid:1) D v / exp (cid:32) ( v − ( M ε ) u e ( x, t )) T e ( x, t ) (cid:33) . (61a)For any smooth function ϕ , the projector onto M e , denoted Π M e , is defined as (see [2, 9])Π M e ( ϕ ) = (cid:34) (cid:104) ϕ (cid:105) + (cid:0) v − ( M ε ) u e (cid:1) T e · (cid:10)(cid:0) v − ( M ε ) u e (cid:1) ϕ (cid:11) + 2 D v (cid:32) | v − ( M ε ) u e | T e − D v (cid:33) (cid:42)(cid:32) | v − ( M ε ) u e | T e − D v (cid:33) ϕ (cid:43) (cid:35) M e n e . (61b)21here (cid:104) ϕ (cid:105) = (cid:82) ϕ dv . For k = 1 , . . . , D v , we have the following properties( I d − Π M e )( ∂ t M e ) = ( I d − Π M e )( E · ∇ v M e ) = 0 , (62a)( I d − Π M e ) (cid:18) n e M e v k ∂ x k n e (cid:19) = 0 ; (62b)( I d − Π M e ) (cid:0)(cid:0) v − ( M ε ) u e (cid:1) · ∂ x k u e v k M e (cid:1) = (cid:16) − (cid:0) v − ( M ε ) u e (cid:1) D v ∂ x k u e,k + (cid:0) v k − ( M ε ) u e,k (cid:1)(cid:0) v − ( M ε ) u k (cid:1) · ∂ x k u e (cid:17) M e ; (62c)( I d − Π M e ) (cid:18)(cid:16) ( v − ( M ε ) u e ) T e − D v T e (cid:17) M e v k ∂ x k T e (cid:19) = M e (cid:32) (cid:0) v − ( M ε ) u e (cid:1) T e − D v + 22 T e (cid:33) (cid:16) v k − ( M ε ) u e,k (cid:17) ∂ x k T e . (62d)Furthermore, if g e satisfies the Micro-Macro decomposition (22), the following identitiesholds true Π M e ( g e ) = Π M e ( ∂ t g e ) = 0 (62e) A.2 Computation of the deviation to the Maxwellian
The aim here is to characterize g e or more specifically an approximation to the first orderin κ . Inserting the micro-macro decomposition into the Vlasov-Boltzmann equation (49b)yields ξε∂ t M e + v · ∇ x M e − ηE · ∇ v M e + κ ( ξε∂ t g e + v · ∇ x g e − ηE · ∇ v g e ) = L M e g e , where L M e g e = − ξν ee g e . This provides, using properties (62), L M e g e = ( I d − Π M e ) ( v · ∇ x M e ) + κ (cid:16) ξε∂ t g e + ( I d − Π M e ) (cid:0) v · ∇ x g e − ηE · ∇ v g e (cid:1)(cid:17) . It follows g e = ( L M e ) − (cid:16) ( I d − Π M e ) ( v · ∇ x M e ) + O ( κ ) (cid:17) , (63a)with v k ∂ x k M e = (cid:32) ∂ x k n e n e + M εT e (cid:16) v − ( M ε ) u e (cid:17) · ∂ x k u e + (cid:16) ( v − ( M ε ) u e ) T e − D v T e (cid:17) ∂ x k T e (cid:33) v k M e . (63b)22rom properties (62), we can state the expression of the deviation to the Maxwellian g e = − M e ξν ee D v (cid:88) k =1 (cid:32) M εT e (cid:16) − (cid:0) v − ( M ε ) u e (cid:1) D v ∂ x k u e,k + (cid:0) v k − ( M ε ) u e,k (cid:1)(cid:0) v − ( M ε ) u e (cid:1) · ∂ x k u e (cid:17) + (cid:32) (cid:0) v − ( M ε ) u e (cid:1) T e − D v + 22 T e (cid:33) (cid:16) v k − ( M ε ) u e,k (cid:17) ∂ x k T e (cid:33) + O (cid:18) κξ (cid:19) . (64) A.3 Computation of the viscous terms
The viscous terms are defined by (cid:104) v ⊗ vg e (cid:105) = − M εξ σ e + O (cid:18) κξ (cid:19) , (cid:28) | v | vg e (cid:29) = − ξ (cid:16) ( M ε ) σ e · u e + q e (cid:17) + O (cid:18) κξ (cid:19) . Following the characterization (64) of g e we can write σ e = − T e ν ee (cid:32) (cid:42) − (cid:0) v − ( M ε ) u e (cid:1) D v M e (cid:0) v ⊗ v (cid:1)(cid:43) D v (cid:88) l =1 ∂ x l u e,l + D v (cid:88) l =1 (cid:68) ( v ⊗ v ) M e (cid:0) v l − ( M ε ) u e,l (cid:1) D v (cid:88) k =1 (cid:0) v k − ( M ε ) u e,k (cid:1) ∂ x l u e,k (cid:69)(cid:33) ,q e = − ν ee (cid:42) | v | v M e (cid:32) (cid:0) v − ( M ε ) u e (cid:1) T e − D v + 22 T e (cid:33) (cid:16) v − ( M ε ) u e (cid:17) · ∇ x T e (cid:43) , Inserting in these definitions, the following identities (cid:42) | v | v ⊗ v ) M e (cid:32) (cid:0) v − ( M ε ) u e (cid:1) T e + D v + 22 T e (cid:33) (cid:16) v − ( M ε ) u e (cid:17) · ∇ x T e (cid:43) = 0 , − T e ν ee (cid:32) (cid:42) (cid:0) v − ( M ε ) u e (cid:1) D v (cid:0) v ⊗ v (cid:1) M e (cid:43) D v (cid:88) l =1 ∂ x l u e,l − D v (cid:88) l =1 (cid:68) ( v ⊗ v ) M e (cid:0) v l − ( M ε ) u e,l (cid:1) D v (cid:88) k =1 (cid:0) v k − ( M ε ) u e,k (cid:1) ∂ x l u e,k (cid:69)(cid:33) = ( M ε ) σ e · u e ;we obtain σ e = − ν ee ( n e T e ) (cid:0) ∇ x u e + ∇ x u Te − D v ( ∇ x · u e ) I d (cid:1) , (65a) q e = − D v + 22 ν ee ( n e T e ) ∇ x T e . (65b)23 eferences [1] M. Acheritogaray, P. Degond, A. Frouvelle, and J.-G. Liu. Kinetic formulation andglobal existence for the hall-magneto-hydrodynamics system. Kinetic and Related Mod-els , 4(4):901–918, Nov. 2011.[2] M. Bennoune, M. Lemou, and L. Mieussens. Uniformly stable numerical schemes for theBoltzmann equation preserving the compressible Navier-Stokes asymptotics.
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