Constraints on ion vs. electron heating by plasma turbulence at low beta
aa r X i v : . [ phy s i c s . p l a s m - ph ] A p r Under consideration for publication in J. Plasma Phys. Constraints on ion vs. electron heating byplasma turbulence at low beta
A. A. Schekochihin, , , † Y. Kawazura, ‡ and M. A. Barnes , , Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Clarendon Laboratory,Parks Road, Oxford OX1 3PU, UK Merton College, Oxford OX1 4JD, UK Niels Bohr International Academy, Blegdamsvej 17, 2100 Copenhagen, Denmark University College, Oxford OX1 4BH, UK United Kingdom Atomic Energy Authority, Culham Science Centre,Abington OX14 3DB, UK (compiled on 12 April 2019)
It is shown that in low-beta, weakly collisional plasmas, such as the solar corona,some instances of the solar wind, the aurora, inner regions of accretion discs, theircoronae, and some laboratory plasmas, Alfv´enic fluctuations produce no ion heatingwithin the gyrokinetic approximation, i.e., as long as their amplitudes (at the Larmorscale) are small and their frequencies stay below the ion Larmor frequency (even astheir spatial scales can be above or below the ion Larmor scale). Thus, all low-frequencyion heating in such plasmas is due to compressive fluctuations (“slow modes”): densityperturbations and non-Maxwellian perturbations of the ion distribution function. Becausethese fluctuations energetically decouple from the Alfv´enic ones already in the inertialrange, the above conclusion means that the energy partition between ions and electronsin low-beta plasmas is decided at the outer scale, where turbulence is launched, and canbe determined from magnetohydrodynamic (MHD) models of the relevant astrophysicalsystems. Any additional ion heating must come from non-gyrokinetic mechanisms suchas cyclotron heating or the stochastic heating owing to distortions of ions’ Larmor orbits.An exception to these conclusions occurs in the Hall limit, i.e., when the ratio of the ionto electron temperatures is as low as the ion beta (equivalently, the electron beta is orderunity). In this regime, slow modes couple to Alfv´enic ones well above the Larmor scale(viz., at the ion inertial or ion sound scale), so the Alfv´enic and compressive cascades joinand then separate again into two cascades of fluctuations that linearly resemble kineticAlfv´en and ion cyclotron waves, with the former heating electrons and the latter ions.The two cascades are shown to decouple, scalings for them are derived, and it is arguedphysically that the two species will be heated by them at approximately equal rates.
1. Introduction
One of the most fundamental questions in plasma astrophysics is what determines thetemperatures of different particle species, ions ( T i ) and electrons ( T e ). We know thata system with different T i and T e is not in equilibrium and so must have an intrinsic(although not necessarily overwhelming) tendency to relax to an equi-temperature state.We do not, however, know of any mechanisms other than Coulomb collisions that † Email: [email protected] ‡ New address: Frontier Research Institute for Interdisciplinary Sciences and Department ofGeophysics, Tohoku University, Aramaki aza Aoba 6-3, Aoba-ku, Sendai 980-8578, Japan.
A. A. Schekochihin et al. would equalise the temperatures. There are no instabilities of a spatially homogeneousequilibrium with T i = T e Maxwellian ions and electrons and so there is no obvious wayin which, e.g., turbulence could result and quickly equalise the temperatures. In theabsence of such fast dynamical processes, collisions are all that remains. In a large classof astrophysical and space plasmas where collisions are not very frequent, temperatureequalisation by collisions is extremely slow: the relevant collision frequency is the ion-electron one, ν ie , which is a factor of mass ratio, m e /m i , smaller even than the electroncollision frequency and a factor of ( m e /m i ) / smaller than the ion one. This means that,for most practical purposes, an “incomplete” equilibrium with T i = T e must be assumed(e.g., Braginskii 1965)—and that is indeed what is observed in the solar wind (see, e.g.,Cranmer et al. T i /T e —a question that is also of great interest in the context of extragalacticplasmas, e.g., accretion discs, where only T e is measured, but knowledge of T i is requiredfor the understanding of basic plasma processes and model building (e.g., Quataert 2003;Sharma et al. et al. et al. et al. b , a ;Chandran et al. T i and T e can be changed via heating or cooling processes resultingfrom energy exchange between the mean (equilibrium) particle distributions and fluctu-ations (or waves), which are ubiquitously present in space and astrophysical plasmas—while temperature difference does not drive fluctuations, there are plenty of free-energysources that do (background gradients, large-scale stirring, etc.). The free energy of thesefluctuations is processed through phase space by various nonlinear (e.g., turbulence) andlinear (e.g., phase mixing) mechanisms, brought to suitably small scales and thermalised,giving rise to ion or electron heating (see, e.g., Schekochihin et al. et al. et al. k k ≪ k ⊥ ,implying that the Larmor scales ( ρ i , ρ e ) in the perpendicular direction are reachedbefore the Larmor frequencies ( Ω i , Ω e ) (see Schekochihin et al. § et al. § et al. et al. a ). All of these low-frequency heating routes can be treatedin the so-called gyrokinetic (GK) approximation (Frieman & Chen 1982; Howes et al. et al. b , 2011; TenBarge et al. et al. et al. et al. et al. The goal of such analytical and numerical inquiries is to
3D is the only relevant kind of simulations in this context because only in 3D can boththe dominant nonlinearity and wave propagation be captured simultaneously (see, e.g., Howes2015). It is only in the last year that 3D full-Vlasov-kinetic simulations of the problem have on vs. electron heating by low-beta plasma turbulence T i /T e and plasma beta, the latter defined to be the ratio of the ion thermal and magneticenergies, β i = 8 πn i T i /B (where n i is the ion density and B is the magnetic field).Analytically, determining ion heating in anything like a definitive fashion has so farturned out to be a rather difficult task, except in the linear approximation (Quataert1998; Quataert & Gruzinov 1999). While progress can be made via modeling basedon physically reasonable conjectures (e.g., Breech et al. et al. et al. et al. et al. et al. b , a ), and, finally, laboratory plasmas such asthe LAPD, custom-made for studies of Alfv´en waves (Carter et al. et al.
2. Epitome
When turbulence in a plasma is stirred up by some large-scale mechanism, this amountsto ion and electron distribution functions being perturbed away from equilibrium. Ifthese perturbations are low-frequency ( ω ≪ Ω i ) and large-scale ( kρ i ≪
1, where ρ i is the ion Larmor radius), they will, via nonlinear interactions, generate further,smaller-amplitude, smaller-scale, higher-frequency fluctuations. Just as in ordinary fluidturbulence, this process can be conceptualised as a cascade of energy—in the case ofkinetic (collisionless or weakly collisional) plasma, a cascade of free energy associated withthe perturbed distribution functions and electromagnetic fields (see Schekochihin et al. et al. become possible (Cerri et al. et al. et al. et al. et al. A. A. Schekochihin et al. et al. et al. k k ≪ k ⊥ .In this anisotropic limit, the fluctuations at scales greater than ρ i can be classified intotwo kinds:(i) Alfv´enic , i.e., incompressible perpendicular MHD perturbations of the velocity andmagnetic field, u ⊥ and b ⊥ —these correspond to Maxwellian perturbations of the iondistribution function with flow velocity u ⊥ = c E × B /B , where E is the perturbedelectric field and B the mean magnetic field (see S09– § compressive , i.e., perturbations of plasma density δn e (= δn i n e /n i by quasineutral-ity), field strength δB , and general perturbations of the ion distribution function involvingparallel flow velocity, temperature and higher moments (see S09– § It is then possible to prove (S09– §
5, 6), for any β i and T i /T e and assuming thatthe equilibrium distribution function is either Maxwellian or satisfies certain constraints(Kunz et al. k ⊥ ρ i < k k v A , where v A = B / √ πm i n i is the Alfv´en speed (Goldreich & Sridhar 1995, 1997),whereas the damping rate cannot be much larger than ∼ k k v th i , where v th i = p T i /m i is the ion thermal speed (S09– § v th i = p T i /m i = √ β i v A ≪ v A , so the damping is expected to be negligible compared to the rate of nonlinear transferof the fluctuation energy towards the Larmor scale (cf. Lithwick & Goldreich 2001).This means that (in a weakly collisional plasma) no thermalisation of any of thatenergy can occur until the fluctuations have reached k ⊥ ρ i ∼
1. At this point, the free-energy cascade is, in general, no longer split into Alfv´enic and compressive, the two typesof fluctuations can couple and the free energy can be shown to be cleanly split again intotwo decoupled cascades only at k ⊥ ρ i ≫
1. These two sub-Larmor dissipation channelsare the cascades of kinetic Alfv´en waves (KAW) and of ion entropy (S09– §
7; that thesub-Larmor-range turbulence in the solar wind is indeed predominantly of the KAWkind appears to be settled: see Salem et al. et al. High-frequency modes, such as fast magnetosonic waves (at inertial-range scales) or whistlers(at sub-ion-inertial scales), can be self-consistently ignored in the anisotropic regime that weare considering—and indeed are ordered out in the GK approximation (see H06– § et al. k k , is observed in the solar wind (Wicks et al. et al. et al. et al. et al. et al. et al. Interestingly, it turns out that even at β i ∼
1, Landau damping in the inertial range can beeffectively suppressed by a nonlinear effect, the stochastic echo (Meyrand et al. on vs. electron heating by low-beta plasma turbulence The ion entropycascade is a nonlinear mixing process in phase space, resulting in fine-scale structurein the ion distribution function and eventually thermalised into ions by collisions—aslarge gradients in v ⊥ form alongside large spatial gradients, even very low collisionalityis enough to dissipate non-Maxwellian perturbations at a finite rate (S09– §§ k ⊥ ρ i small, large and order-unity. Gyrokinetics is such a theory,requiring low frequencies ( ω ≪ Ω i ) but not long wavelengths (see H06 for a tutorial).However, in its general form, it does not make the problem of energy partition betweenspecies any more analytically tractable (although it does make numerical simulationsof this process more feasible: Howes et al. b , 2011; TenBarge et al. et al. et al. et al. et al. β i is assumed small. In this limit, v th i /v A = √ β i ≪
1, so ions cannot stream along the field lines fast enough to couple properly to the electro-magnetic fields associated with the Alfv´en waves, MHD or kinetic, and so perturbations ofthe ion distribution function (other than the Maxwellian E × B drift) stay decoupled fromthe Alfv´enic cascade. As we shall demonstrate below, this makes it possible to provethat, in the low-beta limit, compressive perturbations of the ion distribution functionwill cascade from the inertial range, through the ion Larmor scale and turn into theion entropy cascade at sub-Larmor scales and then into ion heat without exchangingany energy with the Alfv´enic fluctuations. All of the energy of the latter turns intoKAW energy, which includes density perturbations at sub-Larmor scales, but cascadesseparately from the ion entropy and is eventually dissipated on electrons. Proving thisanalytically is accomplished by showing that a certain form of the free-energy invariant, Because k k ≪ k ⊥ , the frequency of the turbulent fluctuations at k ⊥ ρ i ∼ Ω i . If the cyclotron frequency is reached by the KAW cascade at sub-Larmor scales,cyclotron heating can result, linearly, but the wave number range in which it occurs is quitenarrow (see Appendix of Howes et al. a ) and it remains to be seen whether it would beeffective at all in a nonlinear situation (see, however, Arzamasskiy et al. It is perhaps worth emphasising that it is the ion beta, β i , that must be low, while β e may ormay not be (the possibility that it is not is covered by the Hall limit; see the end of this sectionand section 5). The regime in which β i ∼ β e ≪
1, i.e., electrons are colder than ions, ZT e /T i ≪
1, is covered by the theory for order-unity or high β i , which we do not attempt here(for a numerical study of what happens there, see Kawazura et al. β e ∼ β i ∼ d e = ρ e / √ β e ≫ ρ e , the electron inertial effectscome in before the KAW cascade reaches the electron Larmor scale. This modifies the structureof the KAW cascade (Chen & Boldyrev 2017; Passot et al. β e is so low that d e & ρ i , even though β i ∼
1. This ispossible if ZT e /T i ∼ β e ∼ m e /m i , perhaps too extreme a limit. We shall not consider it here.Note that the case of β e ∼ m e /m i and β i ≪ et al. et al. A. A. Schekochihin et al. which reduces to the energy of Alfv´enic and KAW perturbations in the long- and short-wavelength limits, respectively, is conserved across the ion-Larmor-scale transition andthus no Alfv´enic energy can leak into ion heat (see section 4). Therefore, only the energyof what started out as compressive cascade in the inertial range will contribute to ionheating, at least to the extent that the GK approximation holds. Any further ion heatingwill have to come from non-GK mechanisms such as cyclotron heating (Gary et al. et al. et al. et al. et al. et al. et al. at low β i , the energy partition between ions and electronsis determined already at the outer scale of the MHD cascade , where the energy flux splitsinto Alfv´enic and compressive. Once this separation occurs, the ratio between ion andelectron heating rates is fixed. Thus, what in principle is a microscale kinetic effect isin fact fully constrained by fluid dynamics. Since all the action is at the outer scale,the ion-to-electron heating ratio may depend on various nonuniversal circumstances,e.g., presence of equilibrium temperature stratification (which will produce temperatureperturbations), shear, rotation, configuration of magnetic field, etc.These conclusions hold provided β e ∼ β i ≪
1. When β i ≪ β e ∼
1, i.e., whenions are much colder than electrons, ZT e /T i ≫ § E), thesituation changes substantially (section 5). The physical difference between the ZT e /T i ∼ ZT e /T i ≫ c s = p ZT e /m i arelarger than v th i ), remain undamped, and join happily with the AW cascade at a certaintransition scale that is larger than ρ i [it is either the ion inertial or ion sound scale; see(5.19)]. At this transition scale, the Alfv´enic and compressive cascades re-couple and,below the transition scale, turn into cascades of higher-frequency KAW (or, as they aresometimes called in the context of Hall MHD, whistlers) and lower-frequency obliqueion cyclotron waves (ICW). In section 5.5, we argue, with some support from numericalsimulations (Meyrand et al. § § This is, of course, only true assuming that the outer scale is collisional, so the Alfv´enic andcompressive cascades split within the MHD approximation and the transition to collisionlessregime occurs within the compressive cascade (see S09– § δB/B ∼ on vs. electron heating by low-beta plasma turbulence
3. Gyrokinetic Primer
This section is an extended recapitulation of the GK formalism that is required forsubsequent developments. In principle, all of this is already available from H06 andS09 (and, in a form generalised to non-Maxwellian equilibria, from Kunz et al.
Notation: Alfv´enic Fields
The electric and magnetic fields are described by the scalar potential φ and vector po-tential A . It is convenient to introduce dimensionless versions of φ and of the componentof A parallel to the equilibrium magnetic field B = B ˆ z : ϕ = ZeφT i = 2 Φρ i v th i , A = A k ρ i B = − Ψρ i v A , (3.1)where − e is the electron charge, Ze the ion charge, T i the ion equilibrium temperature, v th i = p T i /m i the ion thermal speed, m i the ion mass, ρ i = v th i /Ω i the ion Larmorradius, Ω i = ZeB /m i c the ion Larmor frequency, c the speed of light, v A = B / √ πm i n i the Alfv´en speed, and n i the equilibrium ion density. In what follows, we shall drop theion species index everywhere except for some iconic quantities (e.g., ρ i ) or where thereis a possibility of ambiguity (e.g., T i vs. T e ).In the above, we have also introduced the stream function Φ (= cφ/B ) of the E × B flow associated with φ and the flux function Ψ giving (in velocity units) the magnetic-fieldperturbation perpendicular to B : u ⊥ = ˆ z × ∇ ⊥ Φ, b ⊥ = ˆ z × ∇ ⊥ Ψ. (3.2)Physically these perturbations are Alfv´en waves (AW). In the inertial range of magnetisedplasma turbulence, they decouple from all other modes (the fast modes, which are orderedout in the GK approximation, and the slow, or compressive, modes) and satisfy the“Reduced MHD” equations (RMHD, first derived by Kadomtsev & Pogutse 1974 andStrauss 1976; for a GK derivation, see S09– § ∂Ψ∂t = v A ∇ k Φ, dd t ∇ ⊥ Φ = v A ∇ k ∇ ⊥ Ψ. (3.3)Here the nonlinearities are hidden in the convective time derivative and in the spatial A. A. Schekochihin et al. derivative along the perturbed field lines:dd t = ∂∂t + u ⊥ · ∇ ⊥ = ∂∂t + { Φ, . . . } = ∂∂t + ρ i v th i { ϕ, . . . } , (3.4) ∇ k = ∂∂z + b ⊥ v A · ∇ ⊥ = ∂∂z + 1 v A { Ψ, . . . } = ∂∂z − ρ i {A , . . . } , (3.5)where { f, g } = ( ∂ x f )( ∂ y g ) − ( ∂ x g )( ∂ y f ). These derivatives will appear ubiquitously inwhat follows. 3.2. Gyrokinetic Equation
Our starting point is standard, slab, Maxwellian gyrokinetics (see the derivation inH06 or a summary in S09– § f = F + δf, δf = − ϕ ( r ) F + h ( R ) , R = r + ρ , ρ = v ⊥ × ˆ z Ω , (3.6)where R is the GK spatial coordinate (centre of Larmor ring), whereas r is the usualspatial coordinate (position of the particle). Then the GK equation for the evolution of h is ∂h∂t + v k ∂h∂z + ρ i v th {h χ i R , h } = ∂ h χ i R ∂t F + C [ h ] + 2 v k h a ext i R v F . (3.7)Here the GK potential gyroaveraged at constant R (an operation denoted by anglebrackets) is h χ i R = ZeT i (cid:28) φ − v · A c (cid:29) R = ˆ J ϕ − v k ˆ J A + ˆ v ⊥ ˆ J δBB , (3.8)where ˆ v k = v k /v th , ˆ v ⊥ = v ⊥ /v th , δB is the perturbation of the magnetic field along itself(related to A ⊥ ), which is also the perturbation of the field’s strength; the gyroaveragingBessel operators are defined in terms of their Fourier space representations:ˆ J ↔ J ( a ) = 1 − a . . . , ˆ J ↔ J ( a ) a = 1 − a . . . , a = k ⊥ v ⊥ Ω = ˆ v ⊥ k ⊥ ρ i . (3.9)Obviously, a ↔ − ˆ v ⊥ ρ i ∇ ⊥ = − ˆ v ⊥ ˆ ∇ ⊥ , where we denote ˆ ∇ ⊥ = ρ i ∇ ⊥ . We shall use the ˆ J notation (Kunz et al. h . . . i R (or with h . . . i r , the gyroaverageof an R -dependent quantity at constant r ), as proves convenient.The last term in (3.7) represents energy injection by means of an external parallelacceleration a ext . This will be a convenient model of the excitation of compressiveperturbations for further calculations dealing with free-energy budgets. Finally, thecollision operator C [ h ] contains both the ion-ion and ion-electron collisions, but thelatter are negligible in the mass-ratio expansion adopted below.3.3. Isothermal Electron Fluid
We supplement the ion GK equation (3.7) with two fluid equations arising from theisothermal approximation for electrons, which is a result of expansion in the electron-ionmass ratio and holds at k ⊥ ρ e ≪ and with field equations that follow from quasineu-trality and Amp`ere’s law in the same approximation (this system of equations was derived This is as good a place as any to address a certain resentment that a reader with a predilectionfor mathematical rigour (e.g., Eyink 2015, 2018) might experience towards approximateequations valid in restricted scale subranges. Generally speaking, nonlinear solutions of suchapproximate equations will not stay within their own bounds of validity and develop gradients onscales that are smaller than allowed by the assumed ordering. This is, of course, what turbulence on vs. electron heating by low-beta plasma turbulence §
4, implemented numerically by Kawazura & Barnes 2018 and simulated to someuseful effect by Kawazura et al. ∂ A ∂t + v th ∇ k ϕ = v th ∇ k Zτ δnn + η ∇ ⊥ A , (3.10)dd t (cid:18) δnn − δBB (cid:19) + ∇ k u k e = − ρ i v th (cid:26) Zτ δnn , δBB (cid:27) , (3.11) δnn = − ϕ + ˆ J h, (3.12) u k e v th = 1 β i ˆ ∇ ⊥ A + ˆ v k ˆ J h + J ext , (3.13)2 β i δBB = (cid:18) Zτ (cid:19) ϕ − Zτ ˆ J h − ˆ v ⊥ ˆ J h. (3.14)Here τ = T i /T e , δn/n is the relative electron density perturbation (which is the sameas the ion one, by quasineutrality), u k e the parallel electron flow velocity, and overlinesdenote velocity integrals: ( . . . ) = (1 /n i ) R d v ( . . . ); note that the integrals are taken atconstant r and so R -dependent quantities under them must be gyroveraged at constant r , hence the appearance of the ˆ J and ˆ J operators. In the above system, (3.10) is theparallel component of Ohm’s law (electron’s momentum equation), (3.11) is the electroncontinuity equation, (3.12) is the statement of quasineutrality, (3.13) and (3.14) are theparallel and perpendicular components, respectively, of Amp`ere’s law (the perpendicularone is equivalent to the statement of perpendicular pressure balance; see S09– § J ext ≡ j k ext /en e v th (e.g., TenBarge et al. η is the magnetic diffusivity) to representdissipation of energy into electron heat and to allow flux unfreezing at small scales (animportant concern: see Eyink 2015, 2018; Boozer 2018). Formally, this effect is outside themass-ratio ordering that gave us the hybrid equations introduced above and would haveto be brought in alongside electron inertia and electron-collisional effects (see S09– § This is reasonable because the precise details of how the energy is removed does, or, indeed, is: a cascade to smaller scales, in pursuit of dissipation. In such a cascade, thesmallest scales are typically reached in ∼ one turnover time, regardless of how wide the fullrange of available scales is. Therefore, formally, any system of equations restricted to a subrangeof scales is only valid for ∼ one turnover time; “non-ideal” effects associated with dissipation atsmaller scales come in after that (e.g., ideal-MHD solutions do not stay ideal for long, howeversmall is the resistivity or other flux-unfreezing effects; see, e.g., Boozer 2018). This limitedvalidity is, however, sufficient for analysing basic linear and nonlinear interactions that governthe transfer of energy through the scale subrange that is under consideration, as long as thistransfer can be assumed local to this subrange. The approximate equations can also be usefullysimulated numerically as long as some regularisation at small scales is provided and assumingthat the nature of this regularisation is unimportant—i.e., that as long as a free-energy sinkis present at the smallest scale of the considered subrange, its detailed microphysics does notaffect the behaviour of larger scales (this, of course, does not always have to be the case, buttends to be). Equation (3.10) is the electron parallel momentum equation, representing the force balancebetween the parallel electric field (the left-hand side), parallel pressure gradient (the first termon the right-hand side) and the collisional drag force, which is the resistive term. Technically A. A. Schekochihin et al. from the system should not matter, so long as it happens at scales smaller than the ionLarmor scale and does not introduce artificial coupling between ions and electrons. Thesefeatures—forcing and resistivity—will be useful in working out free-energy budgets.3.4.
Free-Energy Budget
The δf gyrokinetics conserves (except for explicit sources and sinks) a quadratic normof the perturbations, known as the free energy (see S09– § W = Z d r V X s Z d v T s δf s F s + | δ B | π ! , (3.15)where V is the volume of the plasma. Here the perturbed ion distribution function is givenby (3.6), the perturbed electron distribution function under the mass-ratio expansion is δf e = ( δn/n ) F e (see S09– § W = v Z d r V " h h i r F − ϕ − ϕ δnn + Zτ δn n + 2 β i (cid:18) | ˆ ∇ ⊥ A| + δB B (cid:19) , (3.16)where we have dropped the prefactor of m i n i .Since R d r h h i r /F = R d R h /F , we may derive the evolution equation for W bymultiplying (3.7) by h/F , integrating over the entire GK phase space and using (3.10–3.14) opportunely. The result isd W d t = ε AW + ε compr − Q i − Q e , (3.17)where the sources are the injection rates of the Alfv´enic and compressive perturbations, ε AW = v Z d r V ∂ A ∂t J ext = − Z d r V E · j ext , (3.18) ε compr = Z d r V a ext v k ˆ J h = Z d r V a ext u k i , (3.19) speaking, the latter is proportional to the difference between the electron velocity u k e and the ionvelocity u k i = v k ˆ J h , which is worked out from (3.13). Including normalisations, the resistiveterm is then ν ei ( u k e − u k i ) cm e /eρ i B = η (cid:0) ∇ ⊥ A + β i J ext /ρ i (cid:1) , where ν ei is the electron-ioncollision frequency and η = ν ei d e . However, if η is small, it will only matter when it multiplies ∇ ⊥ , as A develops small-scale structure. Since we assume J ext to be a large-scale quantity, itcan be dropped wherever it multiplies η . A minor nuance is that, in numerical practice, resistivity alone is usually insufficient toterminate a turbulent cascade described by (3.10) and (3.11)—one must have a small-scaleregularisation term in (3.11) as well (Kawazura & Barnes 2018). Formally, such a term wouldrepresent collisionless and/or collisional electron damping at and below the electron Larmorscale. This too is electron heating. For the purposes of analytical energy budgets considered inthis paper, the resistive term is a sufficient representative for it. on vs. electron heating by low-beta plasma turbulence Q i = − v Z d R V hC [ h ] F > , (3.20) Q e = η v β i ρ i Z d r V | ˆ ∇ ⊥ A| = η Z d r V |∇ ⊥ A k | πm i n i > . (3.21)We have restored dimensions these expressions to make their physical meaning moretransparent. Note that in (3.18), the final expression—the work done by the electricfield against the external current—is obtained by noticing that there is a perpendicularcurrent associated with j k ext , which is small in the GK expansion (because, to avoidinjecting charges, ∇ · j ext = 0) and so, to lowest order in k k /k ⊥ , Z d r V c ∂A k ∂t j k ext = Z d r V c ∂ A ∂t · j ext = Z d r V (cid:18) c ∂ A ∂t + ∇ φ (cid:19) · j ext , (3.22)with the expression for − E able to be completed with ∇ φ under the integral because ∇ · j ext = 0.In steady state, (3.17) is the overall free-energy budget, which says that the totalinjection is equal to the total dissipation. The main purpose of this paper is to work outmore restrictive energy budgets that constrain Q i and Q e separately.3.5. Separating Alfv´enic Perturbations
We now rearrange the perturbed distribution function in a way that has the effect ofseparating the Alfv´enic part of the distribution function from its “compressive” part: h = h ϕ i R F + g = ˆ J ϕF + g ⇒ δf = ( h ϕ i R − ϕ ) F + g, g = h δf i R . (3.23)The field equations (3.12–3.14) become δnn = − (1 − ˆ Γ ) ϕ + ˆ J g, (3.24) u k e v th = 1 β i ˆ ∇ ⊥ A + ˆ v k ˆ J g + J ext , (3.25)2 β i δBB = − Zτ δnn + (1 − ˆ Γ ) ϕ − ˆ v ⊥ ˆ J g, (3.26)where two more Bessel operators have arisen:ˆ Γ ↔ J ( a ) F = I ( α ) e − α = 1 − α + . . . , α = k ⊥ ρ i ↔ −
12 ˆ ∇ ⊥ , (3.27)ˆ Γ ↔ ˆ v ⊥ J ( a ) J ( a ) a F = − (cid:2) I ( α ) e − α (cid:3) ′ = 1 − α + . . . . (3.28) If we had retained the η J ext term in (3.10) (dismissed in footnote 8) and theion-electron part of C [ h ], the electron heating term would have turned out to be Q e = ν ei ( Zm e /m i ) R d r ( u k e − u k i ) /V = (4 πη/c ) R d r j k /V , the total Ohmic heating. This isthe same as (3.21) if we drop all terms that are small in the mass-ratio expansion, only retaininginstances of η multiplying the highest spatial derivatives of A . Note that this is a different rearrangement than in S09– § §
5, is different in detail. We shall see that this is a more convenientapproach. A. A. Schekochihin et al.
The GK equation (3.7), rewritten in terms of g , becomes ∂∂t (cid:18) g − ˆ v ⊥ ˆ J δBB F (cid:19) + ρ i v th (cid:18)(cid:26) h ϕ i R , g − ˆ v ⊥ ˆ J δBB F (cid:27) + (cid:26) ˆ v ⊥ ˆ J δBB , g (cid:27)(cid:19) + v k (cid:28) ∇ k (cid:18) g + Zτ δnn F (cid:19) + ρ i {A − hAi R , ϕ − h ϕ i R } F (cid:29) R = C [ g + h ϕ i R F ] + 2 v k h a ext i R v F . (3.29)This has been derived by using (3.10) to express h ∂ A /∂t i R and after some manipulationof gyroaverages. In terms of g , the free energy (3.16) becomes W = v Z d r V " h g i r F + ϕ (1 − ˆ Γ ) ϕ + Zτ (cid:12)(cid:12)(cid:12) (1 − ˆ Γ ) ϕ − ˆ J g (cid:12)(cid:12)(cid:12) + 2 β i (cid:18) | ˆ ∇ ⊥ A| + δB B (cid:19) , (3.30)where, by definition, (1 /V ) R d r ϕ (1 − ˆ Γ ) ϕ = P k (1 − Γ ) | ϕ k | .For some upcoming derivations, it will be useful to have the zeroth moment of (3.29).We integrate (3.29) over velocities at constant r , use (3.24) to express ˆ J g and subtract(3.11) from the resulting equation, using (3.25) for u k e and so far neglecting nothing.The outcome isdd t (cid:20) (1 − ˆ Γ ) ϕ + (1 − ˆ Γ ) δBB (cid:21) − v th ∇ k (cid:18) β i ˆ ∇ ⊥ A + J ext (cid:19) = ρ i (cid:10) {hAi R − A , v k g } (cid:11) r − ρ i v th "(cid:28)(cid:26) h ϕ i R − ϕ, g − ˆ v ⊥ ˆ J δBB F (cid:27) + (cid:26) ˆ v ⊥ ˆ J δBB , g (cid:27)(cid:29) r − (cid:26) Zτ δnn , δBB (cid:27) + h C [ g + h ϕ i R F ] i r . (3.31)Note that (3.10) and (3.31) have the makings of the RMHD system (3.3): this emergesfrom any long-wavelength approximation where one can neglect the δn term in (3.10), aswell as (1 − ˆ Γ ) δB/B and all of the right-hand side of (3.31). This is indeed how RMHDis derived from gyrokinetics in the limit of k ⊥ ρ i ≪ §§
4. Reduced Dynamics and Heating at Low Beta
We shall now show that no ion heating occurs in the low-beta regime, viz., at β i ≪ β i and β e = Zβ i /τ . There are two interestinglimits:(i) β e ∼ β i ≪ Z/τ ∼ β e ∼ β i ≪ Z/τ ∼ β − i ≫
1, cold ions)—section 5. If we are to be consistent, we must retain in (3.29) a forcing term associated with theresistive term in (3.10). As we explained in footnote 8, the full form of this resistive term is ν ei ( u k e − u k i ) cm e /eρ i B = ν ie ( u k e − u k i ) /v th , where ν ie = ( m e n e /m i n i ) ν ei is the ion-electroncollision frequency. The additional term that belongs in the left-hand side of (3.29) is, therefore,2 ν ie v k h u k e − u k i i R F /v , which is minus the ion-electron friction force. But this is cancelled bythe linearised ion-electron collision operator, which, to lowest order in the mass-ratio expansion,is just the ion-electron friction (see, e.g., Helander & Sigmar 2005). Thus, from now on, we maydrop the resistive term in (3.29) as long as the collision operator in this equation is understoodto contain the ion-ion collisions only. on vs. electron heating by low-beta plasma turbulence A , ϕ , δn , δB , u k e and g .4.1. Ordering
Working in the limit β e ∼ β i ≪
1, we let Zτ = β e β i ∼ , k ⊥ ρ i ∼ , (4.1)the latter assumption meaning that we are able to treat the Larmor-scale transitiondirectly.Since we wish to be able to handle Alfv´enic perturbations, and since we wish theirlinear frequency ( k k v A ) and their nonlinear interaction rate ( k ⊥ u ⊥ ) to be able to becomparable, we stipulate δB ⊥ B ∼ u ⊥ v A ∼ k k k ⊥ ∼ ǫ, (4.2)where ǫ is the basic GK expansion parameter, with no further β i -related factors, of whichwe shall now keep a close watch. In view of (4.1), this assumption implies δB ⊥ B ∼ k ⊥ A k B ∼ k ⊥ ρ i A ⇒ A ∼ ǫ, (4.3) u ⊥ v A ∼ ck ⊥ φv A B ∼ k ⊥ ρ i p β i ϕ ⇒ ϕ ∼ ǫ √ β i . (4.4)Examination of (3.24–3.26) then suggests that δnn ∼ gF ∼ ϕ ∼ ǫ √ β i , δBB ∼ ǫ p β i . (4.5)4.2. Equations
With this ordering, the kinetic equation (3.29) becomes, to lowest order in β i , ∂g∂t + ρ i v th {h ϕ i R , g } = C [ g + h ϕ i R F ] + 2 v k h a ext i R v F . (4.6)If we ignore collisions and assume no external forcing ( a ext = 0), then g = 0 is a goodsolution of this equation (these assumptions will be relaxed in section 4.5). The fieldequations (3.24–3.26) turn into simple constitutive relations δnn = − (1 − ˆ Γ ) ϕ, u k e v th = 1 β i ˆ ∇ ⊥ A + J ext , δBB = β i (cid:20) Zτ (1 − ˆ Γ ) + (1 − ˆ Γ ) (cid:21) ϕ. (4.7)Using the first two of these in (3.10–3.11), we find that the latter become, to lowest order, ∂ A ∂t + v th ∇ k (cid:20) Zτ (1 − ˆ Γ ) (cid:21) ϕ = η ∇ ⊥ A , (4.8)dd t (1 − ˆ Γ ) ϕ = v th β i ∇ k ˆ ∇ ⊥ A + v th ∇ k J ext . (4.9)These equations are the same as those derived by Zocco & Schekochihin (2011) in thelimit of ultra-low beta ( β e ∼ m e /m i ), except the electron inertia and the coupling tonon-isothermal electron kinetics have now been lost—the price (painless to pay, in thecontext of present study, because energetics are not affected) for considering somewhathigher β e .4 A. A. Schekochihin et al.
The system of equations (4.8–4.9) turns into RMHD (3.3) when k ⊥ ρ i ≪
1: this isshown by using 1 − ˆ Γ ≈ − ˆ ∇ ⊥ / k ⊥ ρ i ≫
1, using1 − ˆ Γ ≈
1, one obtains the β ≪ § et al. ∂Ψ∂t = v A (cid:18) Zτ (cid:19) ∇ k Φ + η ∇ ⊥ Ψ, ∂Φ∂t = − v A ∇ k ρ i ∇ ⊥ Ψ (4.10)(we have dropped J ext because it occurs at large scales). The relationship between themagnetic field and Ψ is still the same as in (3.2). While Φ is still the stream function for the E × B velocity, this is now the velocity of the electron flow (ions are much slower becauseof gyroaveraging). These equations describe what is sometimes referred to as the turbu-lence of Kinetic Alfv´en Waves (KAW)—although, like the Alfv´enic (RMHD) turbulencein the inertial range, it is expected to be strong and critically balanced and so does notliterally consist of waves (see S09– § et al. Z/τ ≫ β e ≪ Zτ (1 − Γ ) ≈ k ⊥ ρ ∼ , ρ s = r Z τ ρ i = c s Ω , c s = r ZT e m i , (4.11)where c s is the sound speed and ρ s ≫ ρ i is the “sound radius”, setting a transition scale.In this regime, the electron-pressure-gradient term [the right-hand side of (3.11)] is non-negligible and so the AW dynamics become dispersive: using (3.1) and (4.11) in (4.8–4.9),we arrive at a simple modification of RMHD equations (3.3) (cf. Bian & Tsiklauri 2009): ∂Ψ∂t = v A ∇ k (cid:0) − ρ ∇ ⊥ (cid:1) Φ, dd t ∇ ⊥ Φ = v A ∇ k ∇ ⊥ Ψ. (4.12)There is then a second transition in (4.8–4.9) at k ⊥ ρ i ∼
1, to ERMHD (4.10).4.3.
Linear Theory
These transitions become particularly transparent if we consider the linear dispersionrelation for the system (4.8–4.9): ω = k k v k ⊥ ρ i (cid:18) − Γ + Zτ (cid:19) ≈ k k v (1 + k ⊥ ρ ) , k ⊥ ρ i ≪ , Z/τ k k v k ⊥ ρ i , k ⊥ ρ i ≫ . (4.13)The k ⊥ ρ i ≪ k ⊥ ρ i ≫ β ≪ § Z/τ ≫
1, it becomes ω ≈ k k v k ⊥ ρ and so the transition betweenthe long- and short-wavelength frequencies is seamless. Thus, in this limit, the transitionbetween the AW and KAW cascades occurs at k ⊥ ρ s ∼ Free-Energy Budget
The nonlinear system (4.8–4.9) has a conserved energy W = v Z d r V (cid:20) ϕ (1 − ˆ Γ ) ϕ + Zτ (cid:12)(cid:12) (1 − ˆ Γ ) ϕ (cid:12)(cid:12) + 2 β i (cid:12)(cid:12) ˆ ∇ ⊥ A (cid:12)(cid:12) (cid:21) , (4.14) on vs. electron heating by low-beta plasma turbulence g = 0 limit of (3.30). Note that, whereas δn/n doesappear in (4.14) (the second term), δB is energetically (and dynamically) insignificant[see (4.7)].At k ⊥ ρ s ≪
1, (4.14) reduces to the energy of Alfv´en waves W AW = 12 Z d r V (cid:0) | ∇ ⊥ Φ | + | ∇ ⊥ Ψ | (cid:1) = 12 Z d r V (cid:0) | u ⊥ | + | b ⊥ | (cid:1) , (4.15)conserved by RMHD (3.3). When k ⊥ ρ s ∼ k ⊥ ρ i ≪ W = 12 Z d r V (cid:0) | ∇ ⊥ Φ | + ρ |∇ ⊥ Φ | + | ∇ ⊥ Ψ | (cid:1) , (4.16)the energy of the system (4.12). At k ⊥ ρ i ≫ W becomes the energy of low-beta KAWperturbations described by (4.10): W KAW = Z d r V (cid:20)(cid:18) Zτ (cid:19) Φ ρ i + 12 | ∇ ⊥ Ψ | (cid:21) . (4.17)The existence of the invariant (4.14), valid uniformly at small, order-unity and large k ⊥ ρ i , means that no damping of anything and, therefore, no ion heating occurs at anywave number, until resistivity kicks in and causes electron heating: it is easy to ascertainthat d W d t = ε AW − Q e , (4.18)where ε AW is given by (3.18) and Q e by (3.21). In steady state, Q e = ε AW .4.5. Energy Partition in the Presence of Compressive Cascade
In the above, we assumed the g = 0 solution for the kinetic equation (4.6). Thiscorresponds to a situation in which only Alfv´enic perturbations are stirred up at thelargest scales: indeed, the relations (4.7) imply that the compressive fields δn and δB peter out at k ⊥ ρ i ≪
1. Let us now relax this assumption. Mathematically, thiswould correspond, e.g., to restoring the external parallel acceleration term in (4.6). Thevariance of the forced kinetic scalar described by (4.6) with a ext = 0 is conserved by thenonlinearity:dd t v Z d r V h g i r F − v Z d r V (cid:28) gC [ g + h ϕ i R F ] F (cid:29) r = Z d r V a ext h v k g i r = ε compr , (4.19)where ε compr is the energy flux in the compressive cascade [cf. (3.19)]. In steady state(d / d t = 0), we have a balance between this compressive input power and the collisionalterms [cf. (3.17)]: ε compr = Q i + Q x , Q x = v Z d r V hh ϕ i R C [ h ] i r = v Z d r V ϕ h C [ h ] i r , (4.20)where Q i is given by (3.20). Thus, all the compressive energy becomes ion heat, with theexception of the collisional energy exchange Q x with Alfv´enic perturbations, which is, aswe are about to argue, small when collisions are weak. The implication is that all theAlfv´enic energy is destined, via the AW cascade smoothly transitioning into the KAWcascade, to be dissipated into electron heat, ε AW = ε KAW = Q e . (4.21) Note the typo in S09– § Φ in Eq. (246). A. A. Schekochihin et al.
We will confirm this directly in section 4.6.If the collision frequency is small compared to the forcing or nonlinear-advection timescales in (4.6), the only way for the collision terms to balance the finite energy flux is for g to develop small scales in phase space, thus activating large derivatives in C [ h ]. Thisis indeed what happens, as the nonlinear term successfully pushes g towards small scalesin both R and v ⊥ , viz., towards δv ⊥ /v th ∼ ( k ⊥ ρ i ) − ≪
1, via a process known as theentropy cascade (see S09– § β i (cf. Tatsuno et al. et al. et al. (2019) appear toconfirm the presence of such an ion-heating route in low-beta GK turbulence.The ion heating rate Q i [see (3.20)] is positive definite and by this process it will berendered finite, i.e., independent of the ion collision rate, however small the latter is. Letus estimate the size of Q x in comparison to Q i . Clearly, only the parts of ϕ and h thatvary on fine scales in position and velocity space matter in Q x and Q i , the contributionfrom large scales being small because the collision frequency is small. The GK collisionoperator is a diffusion operator both in velocity and position (see, e.g., Abel et al. k ⊥ ρ i ≫
1, when collisions become important, Q i ∼ v ν ii ( k ⊥ ρ i ) h F , (4.22) Q x ∼ v ν ii ( k ⊥ ρ i ) / hF ϕ ∼ v ν ii k ⊥ ρ i h F , (4.23)where ν ii is the ion collision frequency. Thus, Q x ≪ Q i . Here Q x loses out compared to Q i by one factor of ( k ⊥ ρ i ) / because of the gyroaveraging under the velocity integral of C [ h ] and by another factor of ( k ⊥ ρ i ) / because, as will be evident from (4.24–4.25), wemust order ϕ ∼ ˆ J g ∼ ( k ⊥ ρ i ) − / h/F in order for the compressive perturbations to haveany relevance. In fact, (4.23) is probably an overestimate because Q x is not sign-definiteand so there will also be a tendency for the small-scale variation within it to average outunder integration. In any event, it is clear that when collisions are weak, the collisionalenergy exchange can be neglected.4.6. Effect of Compressive Cascade on Alfv´enic Cascade
For completeness, let us ascertain that the notion that non-zero g has no energeticeffect on the AW and KAW cascades is consistent with the dynamical equations for thelatter. We allow g/F ∼ ϕ as per (4.5). In this case, v k ˆ J g is still one-order subdominantin (3.25) and δB/B is still small compared to δn/n , but there is now a contribution from g to δn/n in (3.24). The resulting pair of equations, replacing (4.8–4.9), is ∂ A ∂t + v th ∇ k (cid:26) ϕ + Zτ h (1 − ˆ Γ ) ϕ − ˆ J g i(cid:27) = η ∇ ⊥ A , (4.24)dd t h (1 − ˆ Γ ) ϕ − ˆ J g i − v th β i ∇ k ˆ ∇ ⊥ A = v th ∇ k J ext , (4.25)coupled to (4.6).The quantity in the square brackets in (4.24) and (4.25) is − δn/n , so these equationscan be thought of as evolution equations of A and δn/n , the latter’s relationship to ϕ on vs. electron heating by low-beta plasma turbulence g . Alternatively, (4.25) can be recast asdd t (1 − ˆ Γ ) ϕ − v th β i ∇ k ˆ ∇ ⊥ A = v th ∇ k J ext − ρ i v th h{h ϕ i R − ϕ, g }i r + h C [ g + h ϕ i R F ] i r (4.26)if one uses the evolution equation for ˆ J g derived by integrating (4.6) over the velocityspace [(4.26) can also be obtained by applying the ordering (4.5) to (3.31)]. Thisemphasises the nonlinear FLR coupling of ϕ to g .These equations support a generalised version of the (collisionless) invariant (4.14): f W = v Z d r V (cid:20) ϕ (1 − ˆ Γ ) ϕ + Zτ (cid:12)(cid:12)(cid:12) (1 − ˆ Γ ) ϕ − ˆ J g (cid:12)(cid:12)(cid:12) + 2 β i (cid:12)(cid:12) ˆ ∇ ⊥ A (cid:12)(cid:12) (cid:21) , (4.27)which is the low-beta limit of (3.30), excluding the variance of g , which is still conservedindependently [see (4.19)]. Indeed, using (4.26) to work out the time derivative of thefirst term and (4.24) and (4.25) for the other two terms, we getd f W d t = ε AW − Q e + Q x , (4.28)where ε AW is given by (3.18), Q e by (3.21) and Q x in (4.20). The nonlinear terms havevanished by cancellation and because Z d r V ϕ h{h ϕ i R − ϕ, g }i r = Z d R V h ϕ i R {h ϕ i R , g } = 0 (4.29)(after swapping the order of the v and r integration and changing the integration variablefrom r to R ).Combining (4.28) and (4.19), we recover the overall conservation law (3.17), as indeedwe must, because the free energy is W = f W + v Z d r V h g i r F . (4.30)However, we now have more restrictive and, therefore, more informative energy balances(4.20) and (4.28) (with d f W / d t = 0 in steady state). Since, as we argued in section 4.5, Q x is small, we conclude that Q i = ε compr , Q e = ε AW , (4.31)so compressive energy goes into ions, Alfv´enic into electrons. Thus, while non-zero g doesinsinuate itself into the dynamics of Alfv´enic perturbations, there is no energy exchangebetween the two cascades. 4.7. Ultra-Low Beta
Formally, there is an interesting very-low-beta limit that is outside the validity of ourtheory so far. Namely, if β e ∼ m e /m i , we can no longer use the isothermal-electron-fluidapproximation introduced in section 3.3. The equations in this case are quite similarto (4.6) and (4.24–4.25), except in (4.24) there is now an electron-inertia term and apiece of parallel pressure gradient that contains a non-zero parallel electron temperatureperturbation. The latter has to be calculated from the electron drift-kinetic equation, thusopening up an electron heating route via parallel heat transport and Landau damping. Our choice of forcing in (4.6) has ensured that the contribution of g to density is not affectedand so the compressive driving does not stir up Alfv´enic perturbations. A. A. Schekochihin et al.
With g = 0, the appropriate equations were worked out by Zocco & Schekochihin (2011)and proved to be a useful model for numerical experimentation (Loureiro et al. et al. g = 0 in exactly the same way as thesystem (4.8–4.9) was generalised in sections 4.5 and 4.6. There is no change in the energypartition: by the same arguments as above, the energy of compressive perturbations goesinto ions, the energy of Alfv´enic ones into electrons.
5. Reduced Dynamics and Heating in the Hall Limit
Let us now consider the case of β e ∼ β i ≪
1. This is the so-called Hall limitand the derivation in sections 5.1–5.4 is a reworking (in a slightly different order) of the“Hall RMHD” (S09– § E), which we will need for what follows and which turns out tohave some interesting consequences for the energy partition, detailed in sections 5.5 and5.7. 5.1.
Ordering
In this limit, since β e = Zβ i /τ , the ions are cold and, as we anticipate based onsection 4, the AW physics will become dispersive at k ⊥ ρ s ∼ Zτ ∼ β i ≫ , k ⊥ ρ s ∼ ⇒ k ⊥ ρ i ∼ r τZ ∼ p β i ≪ ⇒ k ⊥ d i ∼ , (5.1)where d i = ρ i / √ β i = ρ s p /β e is the ion inertial scale, which is of the same order as ρ s in this limit.We must adjust all expansions and equations accordingly. Instead of (4.3) and (4.4),we have A ∼ ǫk ⊥ ρ i ∼ ǫ √ β i , ϕ ∼ ǫk ⊥ ρ i √ β i ∼ ǫβ i . (5.2)Since the sound speed and the Alfv´en speed are of the same order in this limit, viz., c s = r ZT e m i = v th r Z τ = v A r β e ∼ v A , (5.3)the AW and the compressive modes (slow waves) have similar frequencies. This allowsus to handle both cascades simultaneously. To avoid prejudice, we order the compressiveperturbations to have similar amplitudes to the Alfv´enic ones: δnn ∼ u k i v A ∼ δBB ∼ δB ⊥ B ∼ ǫ, (5.4)where u k i = v k ˆ J g is the parallel ion flow velocity. The requirement that (3.24–3.26) beconsistent with (5.4) implies that we ought to order gF ∼ ǫ √ β i , ˆ J g ∼ ǫ, (5.5)i.e., to lowest order, the distribution function should have no density moment.5.2. Equations
With these orderings, (3.24–3.26) become, to lowest order in β i (and τ ), g = δnn −
12 ˆ ∇ ⊥ ϕ, u k e = u k i + v th (cid:18) β i ˆ ∇ ⊥ A + J ext (cid:19) , δnn = − β e δBB . (5.6) on vs. electron heating by low-beta plasma turbulence ∂ A ∂t + v th ∇ k (cid:18) ϕ + 2 β i δBB (cid:19) = η ∇ ⊥ A , (5.7) (cid:18) β e (cid:19) dd t δBB = ∇ k (cid:20) u k i + v th (cid:18) β i ˆ ∇ ⊥ A + J ext (cid:19)(cid:21) . (5.8)So all four fields A , ϕ , δB and u k i (the latter representing g ) are coupled and we needtwo more equations to close the system.One of these is (3.31), where applying the ordering of section 5.1 leads to the disappear-ance of the entire right-hand side, as well as of the δB term under the time derivative. Tolowest order, therefore, we are left with a rather familiar equation [cf. the second RMHDequation in (3.3)]: dd t
12 ˆ ∇ ⊥ ϕ + v th β i ∇ k ˆ ∇ ⊥ A = − v th ∇ k J ext . (5.9)The last required equation is the lowest-order version of the kinetic equation (3.29):d g d t = v k ∇ k β i δBB F + C [ g ] + 2 v k a ext v F , (5.10)where we again used the last equation in (5.6). This is consistent with g = 0 to lowestorder, as anticipated in (5.5). If we split off the velocity moment from g , viz., g = 2 u k i v k v F + G , G = 0 , v k G = 0 , (5.11)then (5.10) becomes d u k i d t = v ∇ k δBB + a ext , (5.12)d G d t = C [ G ] . (5.13)The first of these is the final equation that we needed to close the system comprisingalready (5.7), (5.8) and (5.9). The second equation, (5.13), describes a passively advectedkinetic field, which, however, is not coupled to anything and so can be safely put tozero —it is the kinetic version of the MHD entropy mode, whereas the rest of ourequations describe linearly and nonlinearly coupled AW and slow waves (SW). Notefinally that (5.9) is needed because it is not possible to calculate ϕ from the first and lastof the field equations (5.6) and the kinetic equation (5.10). This is because, as assumed in(5.5), the density moment g comes from the next-order part of g not captured in (5.10).It is instructive to rewrite the Hall RMHD equations (5.7–5.9) and (5.12) in “fluid” The resistive term in (5.7) can, in fact, be legitimately retained only if resistivity becomesimportant before the Larmor scale is reached. This is possible formally, but unlikely in reality. Unless it is explicitly forced. The forcing that we have chosen for compressive perturbationshas ended up only driving parallel ion flows. To model energy injection into G , we would needto inject, e.g., temperature perturbations—physically this can happen if there is an equilibriumtemperature gradient (see, e.g., Schekochihin et al. A. A. Schekochihin et al. notation, dropping the forcing terms and resistivity (cf. G´omez et al. ∂Ψ∂t = v A ∇ k (cid:0) Φ + v A ρ H B (cid:1) , (5.14)d B d t = ∇ k (cid:0) v s U − ρ H ∇ ⊥ Ψ (cid:1) , (5.15)dd t ∇ ⊥ Φ = v A ∇ k ∇ ⊥ Ψ, (5.16)d U d t = v s ∇ k B , (5.17)where Φ and Ψ are defined by (3.1), we have denoted B = δBB r β e , U = u k i v A , (5.18)and introduced the Hall transition scale ρ H = d i p /β e = ρ s p β e / ρ i s Z/τ β e (5.19)and the SW phase speed v s = v A p /β e = c s p β e / . (5.20)At k ⊥ ρ H ≪
1, the Alfv´enic and the SW-like perturbations decouple from each otherand revert to standard RMHD equations (see S09– § B d t = v s ∇ k U , d U d t = v s ∇ k B (5.21)(passively advected by the AW via d / d t and ∇ k , without energy exchange). Thus,our new system of equations (5.14–5.17) captures the RMHD regime and describes itstransformation, at the Hall transition scale ρ H , into one in which all four fields Φ , Ψ , B and U are coupled.The system (5.14–5.17) also contains the low- β e limit (4.12). This corresponds to takingthe limit v s →
0. Combining (5.15) and (5.16), we getdd t (cid:18) B + ρ H v A ∇ ⊥ Φ (cid:19) = v s ∇ k U → ⇒ B = − ρ H v A ∇ ⊥ Φ. (5.22)Using this in (5.14) and setting ρ H = ρ s , we get the first equation in (4.12). Thesecond is the same as (5.16). The parallel velocity in this limit decouples and cascadesindependently: d U d t = 0 , (5.23)just like G does in (5.13) and like g did in (4.6). on vs. electron heating by low-beta plasma turbulence Free Energy and Heating
The conserved free energy for (5.14–5.17) [equivalently, for (5.7–5.9) and (5.12)] is f W = 12 Z d r V h(cid:12)(cid:12) ∇ ⊥ Φ (cid:12)(cid:12) + (cid:12)(cid:12) ∇ ⊥ Ψ (cid:12)(cid:12) + v (cid:0) U + B (cid:1)i = Z d r V " v (cid:18) (cid:12)(cid:12) ˆ ∇ ⊥ ϕ (cid:12)(cid:12) + 2 β i (cid:12)(cid:12) ˆ ∇ ⊥ A (cid:12)(cid:12) (cid:19) + u k i δB πm i n i (cid:18) β e (cid:19) . (5.24)The free energy has no access to G , whose variance is individually conserved, as is obviousfrom (5.13). If we forced G (without breaking the ordering of section 5.1), the free energyinjected in this way would remain decoupled and travel all through the Hall range of scalesunconcerned with the wave dynamics, eventually arriving at k ⊥ ρ i ∼ G might beforced is by the presence of an ion temperature gradient. However, there cannot be netheating of the plasma by turbulence produced by temperature gradients: any energythus “borrowed” from the ion thermal bath may only be redistributed between species(Abel et al. W SW = v Z d r V (cid:0) U + B (cid:1) , (5.25)conserved by (5.21). When β e ≪
1, the substitution of (5.22) turns (5.24) into (4.16),with the U part of the free energy splitting off, destined for ion heating. In contrast,at β e ∼
1, the decoupling between the Alfv´enic and compressive cascades is broken at k ⊥ ρ H ∼
1, so we can no longer conclude that the former must heat electrons and the latterions. In order to work out what happens (see section 5.5.4 for a preview of the answer),we must shift our focus to k ⊥ ρ i ∼ k ⊥ ρ H ≫ Linear Theory
The dispersion relation is( ω − ω )( ω − ω ) = ω ω , (5.26)where ω AW = k k v A is the AW frequency, ω SW = k k v s the SW frequency and ω KAW = k k v A k ⊥ ρ H the KAW frequency in the Z/τ ≫ § k ⊥ ρ H ≪ ω KAW ≪ ω AW , ω SW and we recover from (5.26) four low-frequencyMHD waves ω = ± ω AW , ω = ± ω SW . (5.27)At k ⊥ ρ H ≫
1, if ω ≫ ω AW , ω SW , the linear response assumes its KAW form: ω = ± ω KAW = ± k k v A k ⊥ ρ H . (5.28) As it did in the β e ≪ β e ≪ ω SW ≪ ω AW and the Alfv´enic branch in (5.26) obeys the k ⊥ ρ i ≪ A. A. Schekochihin et al.
AWSW KAWICW0.01 0.1 1 10 k ⊥ ρ H ω k ∥ v A Figure 1.
Solutions (5.33) of the Hall dispersion relation (5.26) with β e = 1. This is not particularly surprising: the KAW response is the Alfv´enic response with(nearly) immobile ions—and the ion-flow terms in the two magnetic-field equations (5.14)and (5.15) do indeed become subdominant at k ⊥ ρ H ≫
1. Linearly, the KAW are thendescribed by ∂Ψ∂t = v ρ H ∂ B ∂z , ∂ B ∂t = − ρ H ∂∂z ∇ ⊥ Ψ. (5.29)In the Hall limit, there is nothing particularly kinetic about kinetic Alfv´en waves, sothey should probably be called Hall Alfv´en waves (but are sometimes called whistlers);we shall keep the KAW moniker to avoid multiplying entities beyond necessity.There is more to the story at k ⊥ ρ H ≫
1. In this limit, besides the two KAW, (5.26)has two other, low-frequency, solutions: ω = ± ω AW ω SW ω KAW = ± k k v s k ⊥ ρ H = ± Ω k k k ⊥ ≡ ± ω ICW . (5.30)These are oblique ion cyclotron waves (ICW; cf. Sahraoui et al. B = − Φv A ρ H , ∇ ⊥ Ψ = v s ρ H U = Ω U , (5.31)and, consequently, the linearised versions of (5.16) and (5.17) turn into ∂∂t ∇ ⊥ Φ = Ω ∂u k i ∂z , ∂u k i ∂t = − Ω ∂Φ∂z . (5.32)It is more transparent here to go back from U [defined in (5.18)] to u k i as the Alfv´enicnormalisation is no longer physically relevant. These equations, and the correspond-ing dispersion relation (5.30), are mathematically the same as the equations and thedispersion relation for inertial waves in rigidly rotating (with angular velocity Ω/ ω = k k v (cid:26) σ + k ⊥ ρ ± q(cid:2) k ⊥ ρ + (1 + σ ) (cid:3)(cid:2) k ⊥ ρ + (1 − σ ) (cid:3)(cid:27) , (5.33)where σ = v s v A = 1 p /β e (5.34)(the only parameter in the problem). Note that there is no mode conversion, the AW on vs. electron heating by low-beta plasma turbulence β e , tending towards the limit described by (4.12) and (5.22).5.5. Hall Turbulence at Short Wavelengths
The nature of Hall turbulence at k ⊥ ρ H ≫ Ψ = Ψ + e Ψ , Φ = Φ + e Φ, B = B + e B , U = U + e U . (5.35)In everything that follows, overbar will mean KAW-time-scale averaging and overtilde willdesignate KAW-time-scale quantities, which average to zero (with apologies to the reader,who should now forget what overbars and overtildes have been used for previously). Theslow quantities will represent the ICW turbulence and the fast ones the KAW turbulence.We shall venture an a priori guess that the two cascades will decouple completelyat k ⊥ ρ H ≫
1, work out the scalings of all the fields on that basis and then confirm aposteriori that those are consistent with such a decoupling. Namely, we anticipate thatthe nonlinear version of the KAW equations (5.29) will be ∂ e Ψ∂t = v ρ H g ∇ k e B , ∂ e B ∂t = − ρ H ^ ∇ k ∇ ⊥ e Ψ , ∇ k = ∂∂z + 1 v A { e Ψ, . . . } , (5.36)and the nonlinear version of the ICW equations (5.32)dd t ∇ ⊥ Φ = Ωv A ∂ U ∂z , d U d t = − Ωv A ∂Φ∂z , dd t = ∂∂t + { Φ, . . . } . (5.37)In each case, the other two fields play a subordinate role: for KAW turbulence, from(5.16) and (5.17), ∂∂t ∇ ⊥ e Φ = v A ^ ∇ k ∇ ⊥ e Ψ , ∂ e U ∂t = v s g ∇ k e B ; (5.38)for ICW turbulence, (5.31) hold nonlinearly, viz., B = − Φv A ρ H , ∇ ⊥ Ψ = Ω U . (5.39)The physics of these “constitutive relations” will be made transparent in (5.86). Notethat the first equation in (5.38) combined with the second equation in (5.36) also turnsinto a “constitutive relation” between e B and e Φ [cf. (5.22)]: ∇ ⊥ e Φ = − v A ρ H e B . (5.40)The pieces of the free energy (5.24) individually conserved by the systems (5.36) and(5.37) are, respectively, W KAW = 12 Z d r V h(cid:12)(cid:12) ∇ ⊥ e Ψ (cid:12)(cid:12) + v e B i , (5.41) W ICW = 12 Z d r V h(cid:12)(cid:12) ∇ ⊥ Φ (cid:12)(cid:12) + v U i . (5.42)4 A. A. Schekochihin et al.
Here W ICW is just the kinetic energy of the ion motion, perpendicular plus paral-lel, whereas W KAW is the total magnetic energy plus the free energy of the electrondistribution—the latter is the δn /n term in (3.16), now absorbed into e B by way ofthe last equation in (5.6).We are now going to work out all the relevant scalings for KAW (section 5.5.1) and ICW(section 5.5.2) turbulence, then use these scalings to confirm that (5.36–5.39) are correct(section 5.5.3), and finally propose what the energy partition in these circumstancesshould be (section 5.5.4).5.5.1. KAW Scalings
The scalings for a critically balanced cascade of KAW-like fluctuations are a standardproposition (see S09– § The magnetic energy has aconstant flux ε KAW , with the cascade time scale set by the magnetic nonlinearity inside ∇ k in, e.g., the first equation in (5.36):( k ⊥ e Ψ ) τ − ∼ ε KAW , τ − ∼ v A ρ H k ⊥ e B . (5.43)The relationship between e B and e Ψ , and hence the scaling of field amplitudes, is thenfixed by the second equation in (5.36): e B ∼ ω − ρ H k k KAW k ⊥ e Ψ ∼ k ⊥ e Ψv A ∼ (cid:18) ε KAW ρ H v (cid:19) / k − / ⊥ , (5.44)the last relation following from (5.43). Finally, the relationship between the wave fre-quency ω KAW (and, therefore, k k KAW ) and the nonlinear decorrelation rate τ − (and,therefore, k ⊥ ) is set by the critical-balance conjecture: ω KAW = k k KAW v A k ⊥ ρ H ∼ τ − ⇒ k k KAW ∼ (cid:18) ε KAW ρ H v (cid:19) / k / ⊥ . (5.45)The two subordinate fields are found from (5.38): e Φ ∼ ω − k k KAW v A e Ψ = e Ψk ⊥ ρ H , e U ∼ ω − k k v s e B = v s v A e B k ⊥ ρ H . (5.46)It follows from all this that the magnetic and velocity spectra are E e B ∝ k − / ⊥ , E e u ∝ k − / ⊥ (5.47)(cf. Galtier & Buchlin 2007; Meyrand & Galtier 2012).5.5.2. ICW Scalings
The scalings for a critically balanced ICW cascade are perhaps less well established,but also known, in the guise of the scalings for rotating hydrodynamic turbulence(Nazarenko & Schekochihin 2011). Assuming constant energy flux ε ICW and using thefirst equation in (5.37), we find the Kolmogorov scaling (which is no surprise, the Various theoretical considerations (Boldyrev & Perez 2012; Boldyrev et al. et al. et al. on vs. electron heating by low-beta plasma turbulence k ⊥ Φ ) τ − ∼ ε ICW , τ − ∼ k ⊥ Φ ⇒ k ⊥ Φ ∼ ε / k − / ⊥ . (5.48)From either equation in (5.37), U ∼ k ⊥ Φv A . (5.49)The critical-balance conjecture implies ω ICW = Ω k k ICW k ⊥ ∼ τ − ⇒ k k ICW ∼ ε / Ω k / ⊥ . (5.50)Interestingly, it follows from (5.50) that ICW turbulence becomes less anisotropic atsmaller scales. Finally, the subordinate fields (5.39) are
B ∼ Φv A ρ H ∼ U k ⊥ ρ H , k ⊥ Ψv A ∼ v s v A U k ⊥ ρ H . (5.51)The velocity and magnetic energy spectra are, therefore, E u ∝ k − / ⊥ , E B ∝ k − / ⊥ (5.52)(cf. Krishan & Mahajan 2004; Galtier & Buchlin 2007; Meyrand & Galtier 2012).5.5.3. Decoupling of Cascades
The above scalings appear to be consistent with the numerical evidence recentlyreported by Meyrand et al. (2018), who solved the traditional Hall-MHD equations thateffectively describe the β e ≫ v s = v A and ρ H = d i ). They didsee E e u ≪ E B ≪ E e B ≪ E u ∝ k − / ⊥ ; the k − / ⊥ and k − / ⊥ spectra of the magneticperturbations associated with the two different wave modes [see (5.47) and (5.52)] hadpreviously been extracted numerically from Hall MHD by Meyrand & Galtier (2012) (andfrom a shell model by Galtier & Buchlin 2007). Unlike us, Meyrand et al. (2018) thinkthat the KAW turbulence is weak, rather than critically balanced, but we consider theevidence that they present in fact consistent with the possibility of a critically balancedKAW cascade: in particular, both their KAW fluctuations and their ICW fluctuationshave broad frequency spectra and are spatially anisotropic in a scale-dependent way, theformer becoming more anisotropic and the latter less, as k ⊥ increases—in agreement with(5.45) and (5.50). They also see striking evidence that k k KAW ≪ k k ICW , which is indeedwhat (5.45) and (5.50) imply. Finally, and crucially, they show quite unambiguously thatenergy exchange between velocity and magnetic fields (and, therefore, between ICWand KAW fluctuations) peters out at k ⊥ d i ≫
1, i.e., the two cascades are energeticallydecoupled.As promised above, we now confirm that the scalings of sections 5.5.1 and 5.5.2, ifadopted as orderings, do indeed allow the two cascades to decouple (an impatient readerwilling to trust us may skip to section 5.5.4). Let ǫ = ( ε ICW ρ H ) / v A ∼ ( ε KAW ρ H ) / v A , δ = 1( k ⊥ ρ H ) / . (5.53)Here ǫ is just the GK expansion parameter that must enter all field amplitudes. The only Isotropy is achieved at k ⊥ ∼ Ω / ε − / , known as the Zeman (1994) scale in the context ofinertial waves. This scale is, however, outside the GK ordering and so is formally smaller thanany scale present in our considerations. A. A. Schekochihin et al. nontrivial choice about ǫ is that ε KAW ∼ ε ICW , i.e., that the KAW and ICW fluctuationsreceive a priori comparable amounts of energy—equivalently, we assume that the KAWand ICW amplitudes are similar at the Hall transition scale (at k ⊥ ρ H ∼ δ as a subsidiary ordering parameter for the Hall-MHD equations (5.14–5.17): k ⊥ Φv A ∼ U ∼ ǫδ, k ⊥ Ψv A ∼ ǫσδ , B ∼ ǫδ , (5.54) k k ICW ρ H ∼ ǫσ − δ − , ω ICW Ω ∼ ǫσ − δ − , (5.55) k ⊥ e Ψv A ∼ e B ∼ ǫδ , k ⊥ e Φv A ∼ ǫδ , e U ∼ ǫσδ , (5.56) k k KAW ρ H ∼ ǫδ − , ω KAW Ω ∼ ǫσ − δ − , (5.57)where σ is defined in (5.34).Applying the decomposition (5.35) and the above ordering to (5.14), we get, keepingtwo lowest orders, ∂ e Ψ∂t = v A ∂∂z ( Φ + v A ρ H B ) + v ρ H ∇ k e B , ∇ k = ∂∂z + 1 v A { e Ψ, . . . } . (5.58)Averaging this equation over the KAW time scale gives us v A ∂∂z ( Φ + v A ρ H B ) + v A ρ H { e Ψ, e B} = 0 . (5.59)Subtracting this from (5.58), we end up with the first KAW equation in (5.36). Retainingthe lowest order only in (5.59) results in the first ICW constitutive relation in (5.39),assuming that we can ignore any additive corrections to this that are constant along themagnetic field.From (5.17), again keeping only two lowest orders, we getd U d t + ∂ e U ∂t = v s (cid:18) ∂ B ∂z + ∇ k e B (cid:19) , dd t = ∂∂t + { Φ, . . . } . (5.60)Averaging and using (5.59) gives usd U d t = v s (cid:18) ∂ B ∂z + 1 v A { e Ψ, e B} (cid:19) = − v s v A ρ H ∂Φ∂z , (5.61)which is the second ICW equation in (5.37). Subtracting (5.61) from (5.60) leaves us withthe second equation in (5.38), describing small parallel ion flows associated with KAW.Continuing in the same vein, we find that (5.15) becomes, to two lowest orders, ∂ e B ∂t = ∂∂z ( v s U − ρ H ∇ ⊥ Ψ ) − ρ H ∇ k ∇ ⊥ e Ψ . (5.62)The average of this is ∂∂z ( v s U − ρ H ∇ ⊥ Ψ ) − ρ H v A { e Ψ, ∇ ⊥ e Ψ } = 0 , (5.63)which, to lowest order, becomes the second ICW constitutive relation in (5.39) (againignoring any contributions that do not vary along the magnetic field). Subtracting (5.63)from (5.62) gets us the second KAW equation in (5.36). on vs. electron heating by low-beta plasma turbulence t ∇ ⊥ Φ + ∂∂t ∇ ⊥ e Φ = v A (cid:18) ∂∂z ∇ ⊥ Ψ + ∇ k ∇ ⊥ e Ψ (cid:19) . (5.64)Its average is, via (5.63),dd t ∇ ⊥ Φ = v A ∂∂z ∇ ⊥ Ψ + { e Ψ, ∇ ⊥ e Ψ } = v s v A ρ H ∂ U ∂z , (5.65)which is the first ICW equation in (5.37). Subtracting (5.65) from (5.64), we get the firstequation in (5.38) for the small perpendicular flows present in KAW.Thus, the equations for decoupled KAW and ICW cascades, (5.36–5.39), which wereour basis for developing the scalings in sections 5.5.1 and 5.5.2, can indeed be extractedfrom Hall equations (5.14–5.17) if those scalings are assumed. Consistency is the least—and the most—that we can ask for in this approach.5.5.4.
Energy Partition
We anticipate, and will prove in section 5.7, that the sub-Hall-scale KAW cascade allgoes into the sub-Larmor-scale KAW cascade and thence to electron heating, whereasthe ICW cascade is destined for ion-entropy cascade and thence to ion heating. Thus, in the Hall regime, the energy partition is decided at the Hall scale ρ H . While we do notknow how to determine this energy partition rigorously, a plausible conjecture can bemade.The only parameter in the problem is the ratio σ = v s /v A [equivalently, β e : see (5.34)].As explained at the end of section 5.2, (5.14–5.17) reduce to (4.12) in the limit of σ ≪ β e ). This happens because, sufficiently far into that limit, the finite- k ⊥ ρ H contribution to B from the Alfv´enic fluctuations overwhelms the SW part of B [see (5.22)],while what remains of the SW cascades independently according to (5.23), unbotheredby the Hall-scale transition. The result is again (4.31): the Alfv´enic energy goes intoelectrons, the compressive one into ions.In contrast, when σ ∼
1, there are no small parameters left in the problem and alltime scales and all parts of the free energy (5.24) are of the same order at k ⊥ ρ H ∼
1. At k ⊥ ρ H ≪ Φ and Ψ fluctuations carry ε AW , while U and B fluctuations carry ε compr . Onthe other side of the transition, at k ⊥ ρ H ≫
1, the Ψ and B fluctuations (the magneticenergy) are picked up by the KAW cascade ( ε KAW ) and the Φ and U fluctuations (thekinetic energy) by the ICW cascade ( ε ICW ). It is then natural to conjecture an equalsplit of the power of the original MHD cascade between ε KAW and ε ICW , and, therefore,between electron and ion heating—independently of the relative size of ε AW and ε compr : Q e ∼ ε KAW ∼ ε AW + ε compr ∼ ε ICW ∼ Q i . (5.66)Numerical simulations of the full system (5.14–5.17) with external driving are needed(and will be done) to test this reasoning. A parameter scan in σ should reveal a gradualtransition from [ Q i /Q e ] σ → → ε compr /ε AW to [ Q i /Q e ] σ → → Turbulence-theory literati might appreciate an amusing mathematical similarity between thesituation that has emerged here and the rigidly rotating MHD turbulence at large scales,which also features two co-existing cascades—of inertial and magnetostrophic waves—withdispersion relations and, therefore, scalings similar to ICW and KAW, respectively (Galtier2014; Bell & Nazarenko 2019). A. A. Schekochihin et al.
Helicities
Before, as promised, moving on to the Larmor-scale dynamics, let us, for the sake ofcompleteness and for the benefit of those readers who might be interested in Hall RMHDturbulence per se , offer some discussion of other invariants of the system (5.14–5.17).Famously, Hall MHD equations conserve two helicities, magnetic and “hybrid” (Turner1986; Mahajan & Yoshida 1998). However, in Hall RMHD, owing to the presence of astrong background magnetic field, magnetic helicity is not conserved, except in 2D: H = Z d r V Ψ B , d H d t = Z d r V (cid:18) v s Ψ ∂ U ∂z + v A B ∂Φ∂z (cid:19) (5.67)(see S09– § F.4 and references therein for a discussion of helicity non-conservation ina system with a mean field). What is conserved, however, is the sum of three other“helicities” present in the system, viz., the Alfv´enic cross-helicity, the compressive cross-helicity and the kinetic helicity (note that ∇ ⊥ Φ is the z component of the vorticity ofthe plasma motions): X = Z d r V (cid:20) ( ∇ ⊥ Φ ) · ( ∇ ⊥ Ψ ) + v v s U (cid:18) B + ρ H v A ∇ ⊥ Φ (cid:19)(cid:21) . (5.68)Note that the Hall MHD “hybrid” helicity referred to above is then just H − ( ρ H /v A ) X (not conserved because H is not conserved).In the RMHD limit ( k ⊥ ρ H ≪ X loses its last term (the kinetic helicity) andturns into the standard RMHD cross-helicity, whose conservation reflects the energeticdecoupling of the cascades of the four Elsasser fields Φ ± Ψ and U ± B (see S09– § k ⊥ ρ H ≫
1, when Hall RMHD splits into the KAW equations (5.36)and the ICW equations (5.37), each of these systems conserves its own piece of X : X KAW = Z d r V e Ψ e B , X ICW = v Ω Z d r V U ∇ ⊥ Φ. (5.69)The first of these is the helicity of the KAW turbulence, which is in fact the cross-helicity(5.68) by way of (5.40) and integration by parts (cf. S09– § F.3); the second is the kinetichelicity of the ICW turbulence—the last term in (5.68), dominant when k ⊥ ρ H ≫ X KAW or X ICW being non-zero would indicate an imbalance between coun-terpropagating KAW-like or ICW-like perturbations, respectively, there is no corollarythat such counterpropagating perturbations have energetically decoupled cascades in theway that Elsasser fields do in RMHD. This is because while the conserved quantity X is the difference between the “energies” of the generalised Elsasser fields Φ ± Ψ and U ± ( B + ρ H ∇ ⊥ Φ/v A ), the sum of these “energies” is not the free energy (5.24) and isnot conserved, and neither, therefore, are these “energies” conserved individually. Notealso that these fields are not the eigenfunctions associated with the counterpropagatingmodes: in the k ⊥ ρ H ≫ v A e B ∓ k ⊥ e Ψ for ω = ± ω KAW and v A U ± k ⊥ Φ for ω = ± ω ICW . The energies of these fields are not individually conserved either.The presence of extra invariants does open the possibility of dual or even triple cascadesin Hall RMHD turbulence: in particular, X KAW will cascade to larger scales if it isinjected at small scales (see S09– § F.6; Cho & Kim 2016 and references therein), whereas X ICW is expected to cascade forward, together with the free energy (Chen et al. on vs. electron heating by low-beta plasma turbulence
Larmor-Scale Transition
We are now going to prove that, once the KAW and ICW cascades hit the Larmor scale,the former will be channelled into electron heating (via a sub-Larmor KAW cascade) andthe latter into ion heating (via an electrostatic ion entropy cascade). What follows isformally necessary, as due diligence, but a reader who is not a particular GK aficionadaneed not read it if she trusts our algebra. Qualitative physics discussion resumes insection 6.5.7.1.
Ordering
We continue to assume the Hall ordering of the temperature ratio vs. plasma beta[see (5.1)], but focus on scales that are of the order of the Larmor radius, a regime thatdoes not appear to have been studied before: Zτ ∼ β i ≫ , k ⊥ ρ i ∼ ⇒ k ⊥ ρ H ∼ k ⊥ ρ i r Zτ ∼ √ β i ≫ . (5.70)The ordering of the time scales and amplitudes must now be adjusted. How to do thiscan be deduced a priori from the k ⊥ ρ H ≫ k ⊥ ρ H ∼ / √ β i , or δ ∼ β / i . Having obtained the orderings, we willthen backtrack to the hybrid ion-electron equations of sections 3.3 and 3.5 and derive anew set of equations valid under our new ordering.Reverting to our old notation, we convert the δ orderings (5.54–5.57) into β i orderingsusing δ ∼ β / i , σ ∼ k k ρ H ∼ k k ρ i √ β i , and k ⊥ Φv A ∼ k ⊥ ρ i v th v A ϕ ∼ ϕ p β i , U ∼ u k i v th p β i , k ⊥ Ψv A ∼ k ⊥ ρ i A ∼ A , B ∼ δBB (5.71)[recall (3.1) and (5.18)]. The resulting ordering is ϕ ∼ u k i v th ∼ gF ∼ ǫβ − / i , A ∼ δBB ∼ δnn ∼ ǫβ / i , ω ICW Ω ∼ k k ICW v th Ω ∼ ǫβ − / i , (5.72) e ϕ ∼ e u k i v th ∼ e gF ∼ e A ∼ δ e BB ∼ δ e nn ∼ ǫβ / i , ω KAW Ω ∼ ǫβ − / i , k k KAW v th Ω ∼ ǫβ / i , (5.73)where bars and tildes continue to mean averaged and fluctuating quantities over theKAW time scale. Note that for ICW, the wave frequency and the ion streaming ratehave turned out to be the same size, whereas for KAW, the former is much larger thanthe latter. This is the basic physical reason why ICW will couple into ion kinetics and,eventually, ion heating, while KAW will not.5.7.2. Field Equations
The field equations (3.24–3.26) are linear, so can be split cleanly into slow- and fast-varying parts. To lowest order (in all cases, ∼ β − / i for the slow fields and ∼ β − / i for This said, (5.86) and (5.89) are perhaps of some technical interest, showing the electrostaticnature of the ICW cascade. A. A. Schekochihin et al. the fast ones), they are (1 − ˆ Γ ) e ϕ = − δ e nn + 1 n i Z d v ˆ J e g, (5.74) e u k e v th = 1 β i ˆ ∇ ⊥ e A + e J ext , (5.75) Zτ δ e nn = − β i δ e BB , (5.76)(1 − ˆ Γ ) ϕ = 1 n i Z d v ˆ J g, (5.77) u k e v th = 1 β i ˆ ∇ ⊥ A + u k i v th + J ext , (5.78) Zτ δnn = − β i δBB + (1 − ˆ Γ ) ϕ − n i Z d v ˆ v ⊥ ˆ J g. (5.79)The external energy-injecting currents are, in fact, supposed to represent energy arrivingfrom much larger scales. It is then a logical choice to set J ext = 0 and treat e J ext asrepresenting the incoming KAW energy [see (5.84)]. In a similar vein, we shall, in (5.91),let e a ext = 0 and treat a ext as representing the incoming ICW energy.5.7.3. Electron Equations
The treatment of the electron equations (3.10) and (3.11) is completely analogous tothe treatment of their counterparts (5.14) and (5.15) in section 5.5.3. We retain terms totwo leading orders, β − / i and β − / i : ∂ e A ∂t + v th ∂∂z (cid:18) ϕ − Zτ δnn (cid:19) = v th ∇ k Zτ δ e nn + η ∇ ⊥ e A , (5.80) ∂∂t δ e nn − δ e BB ! + ∂u k e ∂z = −∇ k e u k e , where ∇ k = ∂∂z − ρ i { e A , . . . } . (5.81)If these are averaged over the KAW time scale and then its average is subtracted fromeach equation, we obtain, using also (5.75) and (5.76), ∂ e A ∂t = − v th β i ^ ∇ k δ e BB + η ∇ ⊥ e A , (5.82) ∂∂t (cid:18) β e (cid:19) δ e BB = v th ^ ∇ k (cid:18) β i ˆ ∇ ⊥ e A + e J ext (cid:19) . (5.83)These are just the KAW equations (5.36) in different notation, but now they are validat k ⊥ ρ i ∼
1, i.e., both above and below the Larmor scale. They are entirely decoupledfrom ion dynamics and so the KAW energy will cascade right through the Larmor scaleand eventually dissipate into electron heat.To restate the last point in terms of a free-energy budget, the system (5.82–5.83) obeysd W KAW d t + Q e = v Z d r V ∂ e A ∂t e J ext = ε KAW , (5.84) Note that since we are now using overbars to denote time averages, we have suspended theoverbar notation for ion velocity integrals and reverted to writing them explicitly. on vs. electron heating by low-beta plasma turbulence W KAW is given by (5.41) [it is the same as the last two terms of (3.30), after using(5.76)], Q e is given by (3.21) (with A → e A ) and e J ext now represents the KAW cascadefrom k ⊥ ρ i ≪
1. In steady state, Q e = ε KAW . (5.85)Returning to the averaged versions of (5.80) and (5.81) and retaining only the lowestorder, we find ∂∂z (cid:18) ϕ − Zτ δnn (cid:19) = 0 , ∂u k e ∂z = 0 . (5.86)With the aid of (5.78) and (5.79), these are readily seen to be the k ⊥ ρ i ∼ Ion Equations
Finally, we treat the ion GK equation (3.29) in the same manner as we did the electronequations in section 5.7.3. To two lowest orders, it is ∂g∂t + ρ i v th {h ϕ i R , g } + ∂∂t e g − ˆ v ⊥ ˆ J δ e BB F ! + v k (cid:28) ∂∂z (cid:18) g + Zτ δnn F (cid:19) + ∇ k Zτ δ e nn F (cid:29) R = C [ g + e g + h ϕ + e ϕ i R F ] + 2 v k h a ext + e a ext i R v F . (5.87)Taking the KAW-time-scale average of (5.87) and then subtracting it from the equation,we get ∂∂t e g − ˆ v ⊥ ˆ J δ e BB F ! = v k * ^ ∇ k β i δ e BB + R F + C [ e g + h e ϕ i R F ] , (5.88)where have used also (5.76) and set e a ext = 0 as promised at the end of section 5.7.2.This is an (irrelevant) imprint of the KAW turbulence on the ion distribution function—the FLR version of (5.38). Note that there is no phase mixing here, either parallel orperpendicular, so, in a weakly collisional plasma, these small perturbations of the iondistribution function have no means of accessing the collision operator and thermalising.Returning to the KAW-time-scale average of (5.87), retaining the lowest-order termsand using the first equation in (5.86), we get ∂g∂t + ρ i v th {h ϕ i R , g } + v k ∂∂z ( g + h ϕ i R F ) = C [ g + h ϕ i R F ] + 2 v k h a ext i R v F . (5.89)Together with (5.77), this is a closed system—the standard electrostatic GK equationsupporting ion hydrodynamics (5.37) at long scales ( k ⊥ ρ i ≪ and the ion entropycascade at sub-Larmor scales (see S09– § et al. k ⊥ ρ i ≪ The first and second equations of (5.38) are recovered by taking the density andparallel-velocity moments, respectively, of (5.88), using, in the case of the density moment,(5.83), and going to the k ⊥ ρ i ≪ This is again derived by taking the density and parallel-velocity moments of (5.89). A. A. Schekochihin et al. the ion entropy cascade at k ⊥ ρ i ≫
1, to become, upon reaching collisional phase-spacescales, ion heat.For the reference of a meticulous reader, the dispersion relation that follows from (5.89)and (5.77) is 1 + ζ Z ( ζ ) = 1 − Γ , ζ = ω | k k | v th , (5.90)where Z ( ζ ) is the plasma dispersion function (Fried & Conte 1961). When k ⊥ ρ i ≪ ζ ≫
1, the right-hand side of (5.90) is ≈ − k ⊥ ρ i / ≈ − / ζ (the “fluid” limit). The result is the ICW dispersion relation (5.30). When k ⊥ ρ i ∼
1, we must have ζ ∼ t v Z d r V (cid:20) n i Z d v h g i r F + ϕ (1 − ˆ Γ ) ϕ (cid:21) + Q i = Z d r V n i Z d v a ext h v k g i r = ε ICW , (5.91)where Q i is given by (3.20) (with h → h ) and, as promised in section 5.7.2, a ext nowrepresents the energy flux into the ICW cascade. In the long-wavelength limit k ⊥ ρ i ≪ G [see (5.13)]. Thedifference between (5.91) and the analogous low- β e equation (4.19) is that the “kinetic-energy” term ϕ (1 − ˆ Γ ) ϕ has now migrated into the ion-heating balance [cf. (4.27) and(3.30)] (removing also the technical complications associated with Q x ). In steady state,(5.91) tells us that Q i = ε ICW , (5.92)restating again that all the ICW energy goes into ion heating.
6. Discussion
The physics of turbulent heating of low-beta GK plasmas was already summarisedand discussed at length in section 2, so we need not repeat that discussion. The headlineresult is the clean separation between the Alfv´enic cascade heating the electrons and thecompressive cascade the ions, at low β i and low β e (section 4). One practical implicationis that it becomes an interesting question, not just in itself, but also for large-scalemodelling of, e.g., detectable emission from astrophysical objects (e.g., Ressler et al. et al. b , a ), how any particular type of MHD turbulence present in theseobjects splits itself into these two cascades—the answer to this question for, e.g., MRIturbulence, is not known, although it can, in principle, be obtained via standard fluidsimulations. In the solar wind, the answer is known observationally, if not necessarilyunderstood theoretically: the compressive cascade carries about 10% of the energy(Howes et al. et al. et al. et al. et al. on vs. electron heating by low-beta plasma turbulence et al. Stochastic Heating
The fraction of the Alfv´enic energy flux arriving to the ion Larmor scale that getsconverted into ion heat via the stochastic heating mechanism is (Chandran et al. Q (stoch) i ∼ ε AW e − /δ , δ ∼ u ⊥ ρ i v th i ∼ √ β i δB ⊥ ρ i B ∼ √ β i δB ⊥ L B (cid:16) ρ i L (cid:17) / , (6.1)where u ⊥ ,ρ i and δB ⊥ ρ i are the typical velocity and magnetic perturbations at the Larmorscale. The last estimate comes from assuming a k − / ⊥ spectrum of the Alfv´enic cascade(replace the exponent 1 / / k − / ⊥ ), to refer δ to the magnetic-field variation δB ⊥ L at the outer scale L . Given L and δB ⊥ L , which are independent,system-specific properties, setting δ ∼ δ ∼ ǫ/ √ β i ,so δ ∼ δB ⊥ L /B ∼ ρ i /L ∼ − , so, if we were to err on the side of caution, we would start disbelieving theGK predictions for β i . − , although it is not hard to play with numbers and lower thisby another factor of 10 in specific circumstances. More careful estimates of the validityof the GK approximation can be found in Howes et al. (2008 a ) and of the importance ofstochastic heating in Chandran et al. (2010) and Chandran (2010). Our purpose here isto emphasise that the constraints that we have placed on the ion heating are pessimistic(from the ions’ viewpoint) and may become unreliable when β i is too low. An interesting corollary is that there might be an intrinsic mechanism that wouldprevent β i from being much lower than the stochastic-heating threshold: indeed, if β i diddrop lower, stochastic heating would become significant and channel turbulent energyinto ions, which would increase β i and shut down stochastic heating. One could imaginesome equilibrium hovering around that threshold in a system where ions, starved ofheating in the GK approximation, were able to cool down and thus lower β i untilstochastic heating turned on.It is perhaps useful to mention two other plausible self-regulation mechanisms impliedby our considerations above.6.2. Energy Redistribution in the Hall Regime
At the price of the rather long derivation in section 5, we learned that the clean energypartition that holds at low β e breaks down at β e &
1. If, in a given low- β i system, electronsare heated preferentially and if that preferential heating leads to electron temperatureincreasing, then the system will be nudged towards the Hall limit. Once T i /T e ∼ β i (equivalently, β e ∼ T i /T e cannot decreasefurther and/or that β i will be pushed up. Thus, low- β i plasma is intrinsically averse toelectrons getting too hot. Note the recent observational analysis by Vech et al. (2017) and theoretical arguments byMallet et al. (2018) and Hoppock et al. (2018), which suggest that stochastic heating may,quantitatively, be more important, at higher values of β i , than previously believed. If T e does not change, changing T i alone cannot, obviously, alter the relative size of τ = T i /T e and β i because both parameters are ∝ T i . A. A. Schekochihin et al.
Collisional Heating
In all of the above (and, in particular, in section 4.5), we have assumed that ioncollisions are sufficiently infrequent for the collision operator to become important onlyat sub-Larmor scales. If, however, ions are starved of heating and are, as a result, cooledby some competing mechanism, their collision frequency will increase. The typical rate atwhich collisional heating happens is [cf. (4.22)] τ − ν ∼ ν ii ( k ⊥ ρ i ) . This is to be comparedwith the turbulent-cascade rate: for Alfv´enic turbulence, τ − ∼ k ⊥ u ⊥ ∼ ε / k / ⊥ .Balancing the two rates gives us a “Kolmogorov scale” τ − ν ∼ τ − ⇒ k ⊥ ν ρ i ∼ ε / ρ / i ν / ii ∝ ε / n − / i B / T / i . (6.2)If T i is so low that k ⊥ ν ρ i .
1, the cascade will be dissipated by ion (perpendicular)viscosity and ion heating will result. Again, one can imagine an equilibrium hoveringaround the transition between the two regimes—collisional and collisionless.While it is not our purpose here to propose macroscopic thermodynamic models ofany specific object, we hope that we have given a more object-oriented reader somefood for thought and perhaps even some useful information, while a fellow kinetic-theoryenthusiast might have enjoyed the ride. Some of the ideas, loose ends and opportunitiesfor numerical verification identified above will be picked up in our own future work.We are indebted to W. Dorland, T. Adkins, S. Balbus, S. Cowley, N. Loureiro, F. Parra,and E. Quataert for many important conversations, to B. Chandran, G. Howes andM. Kunz for detailed comments on the draft manuscript, and to R. Meyrand for a tutorialon Hall MHD The work of YK was supported in full and of MAB and AAS in part bythe UK STFC Consolidated Grant ST/N000919/1; AAS was also supported in part bythe UK EPSRC Grant EP/M022331/1 and, at NBIA, by the Simons Foundation. Theauthors are grateful to the Wolfgang Pauli Institute, Vienna, for its hospitality and thescientific interactions that it enabled, on multiple occasions.
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