Nonlinear Processes in Geophysics a Fluid Description For

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Nonlinear Processes in Geophysics (2004) 11: 245–258 SRef-ID: 1607-7946/npg/2004-11-245 Nonlinear Processes in Geophysics © European Geosciences Union 2004

A ﬂuid description for Landau damping of dispersive MHD waves

T. Passot and P. L. Sulem CNRS, Observatoire de la Coteˆ d’Azur, B.P. 4229, 06304 Nice Cedex 4, France Received: 30 June 2003 – Revised: 2 December 2003 – Accepted: 5 January 2004 – Published: 14 April 2004

Part of Special Issue “International Workshops on Nonlinear Waves and Chaos in Space Plasmas”

Abstract. The dynamics of long oblique MHD waves in a and the physical nature of the dissipation processes (Minter collisionless plasma permeated by a uniform magnetic field and Spangler, 1995; Lithwick and Goldreich, 2001). These is addressed using a Landau-fluid model that includes Hall aspects are usually addressed separately, the large-scale dy- effect and electron-pressure gradient in a generalized Ohm’s namics being often described in terms of usual ideal com- law and retains ion finite Larmor radius (FLR) corrections pressible MHD, while the small-scale (collisionless) dissipa- to the gyrotropic pressure (Phys. Plasmas 10, 3906, 2003). tive effects are evaluated using a kinetic approach, assum- This one-fluid model, built to reproduce the weakly nonlin- ing a given turbulence energy spectrum. It would thus be ear dynamics of long dispersive Alfven´ waves propagating desirable to have a fluid description incorporating the domi- along an ambient field, is shown to correctly capture the Lan- nant kinetic effects such as wave-particle resonance and FLR dau damping of oblique magnetosonic waves predicted by corrections. Such a model would provide a self-consistent a kinetic theory based on the Vlasov-Maxwell system. For description of the turbulent cascade and allow a realistic es- oblique and kinetic Alfven´ waves (for which second order timate of its termination scales. It would in particular con- FLR corrections are to be retained), the linear character of tribute to a detailed understanding of collisionless dissipation waves with small but finite amplitudes is established, and the at the origin of heating, and its balance with cooling pro- dispersion relation reproduced in the regime of adiabatic processes. Among the various collisionless dissipation mecha- tons and isothermal electrons, associated with the condition nisms, Landau damping of magnetosonic waves has recently me/mp β Te/Tp, where β is the squared ratio of the been shown to provide a dominant contribution (Lerche and ion-acoustic to the Alfven´ speeds. It is shown that in more Schlickeiser, 2001a). Proper treatment of Landau damping general regimes, the heat fluxes are, to leading order, not gy- in ISM turbulence is also essential for the determination of rotropic and dependent on the Hall effect to leading order. the full fast mode spectrum and thus of the properties of the cosmic ray scattering (Lerche and Schlickeiser, 2001b). Similar problems are encountered in the solar wind where − 1 Introduction magnetic fluctuations show an abrupt transition from a 5/3 power law spectrum at large scale to a steeper slope at the Magnetohydrodynamic (MHD) wave turbulence is ubiqui- scales where proton inertia becomes non-negligible (Li et tous in space plasmas, from the solar wind and planet mag- al., 2001). In the magnetosheath, a realistic description of netospheres to the Interstellar Medium (ISM). In the warm the turbulence reaching the magnetopause could improve the and diffuse ISM for example, weak MHD wave turbulence is understanding of the high level of magnetic field fluctuations believed to be responsible for the observed interstellar scin- that is believed to be at the origin of micro-reconnection and tillation through the density fluctuation they produce (Span- particle penetration in the magnetosphere (Rezeau and Bel- gler, 1991). A remarkable Kolmogorov spectrum for the den- mont, 2001). sity fluctuations is measured over more than twelve decades, In order to capture such phenomena, one needs a descrip- down to scales of the order of 100 km, i.e. the typical ion in- tion going beyond the usual MHD or even the double adia- ertial length (Armstrong et al., 1995). The properties of this batic approximation (Chew etal., 1956) that retains plasma turbulence are still debated. Questions concern the origin anisotropy but neglects heat fluxes. A significant contribu- of this extended inertial range, the nature of the ISM phase tion in the development of such a tool is provided by the from where fluctuations originate (possibly also HII regions) Landau-fluid models of Snyder et al. (1997). A simplified version of this approach was used by Quataert et al. (2002) Correspondence to: T. Passot to describe the magneto-rotational instability in collisionless ([email protected]) 246 T. Passot and P. L. Sulem: Fluid description of Landau damping

1 accretion disks. Such models are however to be generalized ρ (∂t u + u · ∇u) + ∇ · P + b × (∇ × b) = 0, (2) when the fluctuations extend to scales comparable to the ion 4π inertial length, a regime where finite Larmor radius (FLR) where the pressure tensor P is defined relatively to barycen- corrections to the pressure tensor, together with Hall effect tric velocities. In a weakly nonlinear regime (strong ambient and electron pressure tensor gradient in a generalized Ohm’s field), the pressure tensor can be approximated by the sum of law have to be retained. Such an extension developed by Pas- the individual pressures of the various species, even at scales sot and Sulem (2003b), is described in Sect. 2. In Sect. 3, this comparable with the ion Larmor radius. In an expansion in model is shown to accurately reproduce the Landau damp- terms of the ratio of a typical hydrodynamic scale to the ion ing of long magnetosonic waves propagating in an oblique Larmor radius, the pressure tensor P = PG +5 is dominated direction, as predicted by the Vlasov-Maxwell kinetic the- by the gyrotropic contribution ory performed in Appendix A. Section 4 briefly summarizes G X G X the case of parallel Alfven´ waves already discussed by Pas- P = Pr = p⊥r (I − b⊗ b) + pkrb⊗ b, (3) sot and Sulem (2003b) for which the Landau damping does r r not arise as a linear dissipation but enters the derivative non- where the parallel and perpendicular components pkr and linear Schrodinger¨ equation governing the long-wavelength p⊥r for the individual species are governed by dynamics (Rogister, 1971; Mjølhus and Wyller, 1988). Sec- tions 5 and 6 deal with obliquely propagating Alfven´ waves ∂t pkr + ∇ · (u pkr ) + ∇ · (bqkr ) and with the limiting case of kinetic Alfven´ waves respec- +2pkr b· ∇u · b− 2q⊥r ∇ · b = 0, (4) tively. Second order FLR corrections that are relevant in these regimes, are derived in Appendix B. The linear char- ∂t p⊥r + ∇ · (u p⊥r ) + ∇ · (bq⊥r ) + p⊥r ∇ · u acter of the dynamics in the case of waves of small but fi- −p⊥r b· ∇u · b+ q⊥r ∇ · b = 0, (5) nite amplitudes is established, and the computed dispersion and damping are shown to reproduce classical results in the with bdenoting the unit vector along the local magnetic field. me Te The heat flux tensors are furthermore supposed to be domi- regime m β T where β is the squared ratio of the p p nated by the gyrotropic components qk and q⊥ , an assump- ion-acoustic to the Alfven´ speeds. Possible extension of the r r tion whose validity will be discussed in Sect. 4. model to more general conditions are discussed in the Con- The finite Larmor radius corrections to the pressure tensor clusion. are described by the tensor 5 that, when electron inertia is negligible, reduces to the proton contribution 5p. The lat- 2 Generalized MHD with Landau damping ter is to be computed perturbatively, the required accuracy depending on the considered regime. Details are given in As already mentioned, a fluid description going beyond the Appendix B. usual MHD and including a realistic dissipation mechanism With a generalized Ohm’s law that includes the Hall effect is believed to provide a very efficient tool for modeling large- and the contribution of the electron pressure gradient, the in- scale turbulence in a collisionless plasma permeated by a duction equation reads strong ambient magnetic field. − × × = Starting with the Vlasov-Maxwell equations, one easily ∂t b ∇ (u b) derives a hierarchy of equations for the successive moments mpc h 1 1 Gi − ∇ × (∇ × b) × b − ∇ · Pe . (6) of the distribution functions of the various particle species. qp 4πρ ρ When reduced to a one-fluid description, the time evolution To close this system, one needs to specify the evolution of of the density and velocity of the plasma obeys the heat fluxes. In regimes where these quantities are not sen- ∂t ρ + ∇ · (uρ) = 0 (1) sitive to the Hall effect, Passot and Sulem (2003b) suggested

! d v qk 1 Tk + th,r H∇ r = v ∇ r (7) q k (0) 3π th,r k (0) dt 8 − 3π vth,r p 1 − T π (1 8 ) kr 8 kr r ! (0) ! ! d π q⊥ T⊥ T |b| − v H∇ r = −v ∇ r + ⊥r − 1 , (8) th,r k (0) th,r k (0) (0) dt 2 B0 vth,r p⊥r T⊥r Tkr

where ∇k denotes the derivative and H the Hilbert transform essentially be viewed as a linear version of Eqs. (32–35) of along the ambient field, and where the temperatures are given Snyder et al. (1997) and was shown to accurately describe the by pkr = nTkr and p⊥r = nT⊥r . The coefficients vth,r refers parallel propagation of weakly nonlinear dispersive Alfven´ to the (equilibrium) thermal velocities of the r species and waves in the long-wavelength limit. Indeed, as discussed in the superscript (0) to the equilibrium state. This closure can Sect. 4, reductive perturbative expansions performed on the T. Passot and P. L. Sulem: Fluid description of Landau damping 247 above system and on the primitive Vlasov-Maxwell equa- damping is small enough to be comparable with dispersion tions lead to asymptotic equations that identify up to the re- and nonlinearity, for example in the case of hot electrons and placement of the plasma response function by suitable Pade´ cold ions, a Korteweg-de Vries equation with Landau damp- approximants. ing is obtained (Janaki et al., 1992). We do not restrict our- Note that, consistently with the weakly nonlinear assump- selves here to this special regime and compute the Landau tion underlying the derivation of the heat flux equations, the damping that arises at the level of the linear dispersion rela- Landau damping is evaluated by integration along the am- tion. bient field lines, which makes its numerical evaluation easy From the induction and the density equations, one gets to when using a Fourier spectral code. Convective derivatives leading order are nevertheless retained in Eqs. (7–8) in order to preserve (1) (1) Galilean invariance. bx ux + cos α = 0 (9) B0 V0 b(1) u(1) 3 Oblique magnetosonic waves z − sin α x = 0 (10) B0 V0 We here demonstrate that the model described in Sect. 2 (1) (1) (1) ux uz ρ correctly reproduces the dissipation rate of oblique mag- sin α + cos α − = 0, (11) V V ρ(0) netosonic waves predicted by kinetic theory in the long- 0 0 wave limit. Assuming a propagation in a direction making that leads to simple expressions in terms of the normal- an angle α with the ambient field taken along the z–axis, ized magnetic perturbation A defined by |b| = 1 + A + · · ·. B0 it is convenient to perform the change of reference frame With the scaling associated with magnetosonic waves, one 0 0 x = x α − z α z = x α + z α b(1) b(1) u(1) cos sin and sin cos . We has z = A and thus writes x = − 1 A, x = 1 A concentrate on perturbations depending only on the coordi- B0 B0 tan α V0 sin α (1) (1) z0 and uz = 1 ρ − A . The linearization of the pressure nate along the direction of propagation, that is stretched V cos α ρ(0) 1/2 0 0 as ξ = (z − V0t). The small parameter measures the equations gives amplitude of the magnetosonic waves that are selected by (1) (1) (1) (1) perturbing the equilibrium state in the form pk u u qk − r + sin α x + 3 cos α z + cos α r = 0 (12) (0) (0) = (0) + (1) + · · · = (0) + (1) + · · · p V0 V0 V p ρ ρ ρ , p⊥r p⊥r p⊥r , k 0 kr (0) (1) (1) (1) (1) (1) pkr = p + p + · · · , p u u q kr kr − ⊥r + 2 sin α x + cos α z + cos α ⊥r = 0, (13) (0) (0) (1) (1) V0 V0 bz = B0 + bz + · · · , bx = bx + · · · , p⊥ V0p⊥r (1) (1) uz = uz + · · · , ux = ux + · · · , that leads to 3/2 (1) 3/2 (1) by = by + · · · , uy = uy + · · · (0) (1) ! V0p p ρ(1) (1) = kr kr − + qkr 3 2A (14) The dispersion and the nonlinearities then act on the slow cos α p(0) ρ(0) time τ = 3/2t. The reductive perturbative expansion is kr ! straightforward and, in the absence of Landau damping, V p(0) p(1) ρ(1) (1) = 0 ⊥r ⊥r − − q⊥r A . (15) leads to the usual Korteweg-de Vries equation (Kakutani et cos α p(0) ρ(0) al., 1968; Kever and Morikawa, 1969; Chakraborty and Das, ⊥r 2000). In the regimes where the strength of the Landau The closure equations provide

  (1) (1) (1) ! v cos α qk v pk ρ −V + th,r H r = th,r cos α r − (16)  0 q  (0) 3π (0) (0) 8 − 3π vth,r p 1 − p ρ π (1 8 ) kr 8 kr r (1) (1) (1) (0) ! π q⊥r p⊥r ρ T⊥r −V − vth,r cos αH = −vth,r cos α − + ( − 1)A . (17) 0 2 (0) (0) ρ(0) (0) vth,r p⊥r p⊥r Tkr

After substitution, one easily expresses the transverse where c = V0 and r vth,r cos α pressure perturbations as

(1) (1) (0) 1 p⊥r ρ T⊥r W (c ) = (19) = + 1 − W2(cr ) A (18) 2 r q (0) ρ(0) (0) 1 − π c H − c2 p⊥r Tkr 2 r r 248 T. Passot and P. L. Sulem: Fluid description of Landau damping can be viewed as the two-pole approximant of the plasma The parallel pressure perturbations involve the four-pole ap- response function proximant of the plasma response function, in the form Z −ζ 2/2 r 1 ζ e π −c2/2 W(cr ) = √ P dζ + cr e r H. (20) 2π ζ − cr 2

√ 1 − 2 − + 2 (8 3π)cr 2πcr H 4 W4(cr ) = √ √ . (21) 1 4 − + 3 + 1 − 2 − + 2 cr (3π 8) 2cr H 2 (16 9π)cr 3 2πcr H 4 One has p(1) (1) kr 2 −1 ρ 2 −1 = c + W (cr ) − c − 1 + W (cr ) A. (22) (0) r 4 ρ(0) r 4 pkr Turning to the momentum equation, one gets

(1) (0) (1) (0) (0) ! (1) 2 (1) (1) ! u p p pk p b B b b −V 2 x + ⊥ sin α ⊥ + − ⊥ cos α x − 0 cos α x − sin α z = 0 (23) 0 V ρ(0) (0) ρ(0) ρ(0) B 4πρ(0) B B 0 p⊥ 0 0 0 (1) (0) (0) ! (1) (1) u pk p b pk −V 2 z − cos α − ⊥ z + cos α = 0. (24) 0 (0) (0) (0) V0 ρ ρ B0 ρ

(1) = P (1) (1) = P (1) After substitution of the total parallel and perpendicular pressures p⊥ r p⊥r and pk r pkr , one obtains

(0) (1) " (0) p(0) ! (0) (0) !# p⊥ 2 ρ 2 2 p⊥ k 2 2 X p⊥r T⊥r sin α + v − V + − cos α + sin α 1 − W (cr ) A = 0 (25) ρ(0) ρ(0) A 0 ρ(0) ρ(0) ρ(0) (0) 2 r Tkr " (0) # (1) " " (0) (0) ## X pkr − ρ p X pkr − −V 2 + cos2 α c2 + W 1(c ) + V 2 + cos2 α ⊥ − c2 + W 1(c ) A = 0 (26) 0 (0) r 4 r (0) 0 (0) (0) r 4 r r ρ ρ ρ r ρ

B2 where the Alfven´ wave velocity v is deﬁned by v2 = 0 . with, following Ferriere` and Andre´ (2002), the operators A A 4πρ(0) The dispersion relation is then readily obtained. It is con- 2 X 2 2 −1 (0) = + p(0) p Ck vkr cr W4 (cr ) (27) venient to introduce the parameters v2 = ⊥r , v2 = kr , ⊥r ρ(0) kr ρ(0) r 2 = P 2 2 = P 2 2 = 2 − 2 v⊥ r v⊥r , vk r vkr and v1 v⊥ vk together and

(0) ! 4 2 2 X 2 T⊥r 2 X v⊥r C⊥ = v⊥ + v⊥ 1 − W2(cr ) = 2v⊥ − W2(cr ). (28) r (0) v2 r Tkr r kr One has

4 h 2 2 2 2 2 2 2 i 2 V0 − vA + v1 + Ck cos α + vA + C⊥ sin α V0 2 2 2 2 h 2 2 2 4 i 2 2 + (vA + v1)Ck cos α + (vA + C⊥)Ck − v⊥ sin α cos α = 0, (29)

which, up to the replacements of the plasma response func- hot electrons (in an isothermal state). In these limits, cp 1 2 4 6 tion by suitable Pade´ approximants, identifies with the dis- and W(cp) ≈ −1/cp − 3/cp − 15/cp, while ce 1 q persion relation directly derived from the Vlasov-Maxwell and W(c ) ≈ 1 − c2 + π c H. Note that the Pade´ ap- system taken in the long-wave limit (Appendix A), or from a e e 2 e mixed fluid-kinetic formalism (Ferriere` and Andre,´ 2002). proximants of the plasma response function are constructed in a way that reproduces the adiabatic and isothermal lim- A simple expression for the Landau damping is obtained its. As a consequence, the Landau-fluid predicts exact re- under the condition vth,p V0 vth,e, associated with sults in the considered regime. For the sake of simplicity cold protons (following an adiabatic equation of state) and and easy comparison with the literature, we assume isotropic T. Passot and P. L. Sulem: Fluid description of Landau damping 249 temperatures for the equilibrium state. Taking into account the case of a finite angle of propagation, the smallness of the ratio me/mp and also the assumption me Te 1 Te 2 2 r β where β = (that results from the con- Ck C π r me 1 λ mp Tp 2 mp ⊥ vA = = β 1 − √ H , (30) q v2 v2 2 m β cos α dition v V v ), one has c = me λ √1 with A A p th,p 0 th,e e mp cos α β λ = V0 . One then easily gets the following formulas valid in that are to be substituted in the dispersion relation vA

2 2 ! " 2 2 2 (0)2 !# Ck C Ck Ck C⊥ p λ4 − λ2 1 + cos2 α + ⊥ sin2 α + cos2 α + sin2 α − = 0. (31) 2 2 2 4 4 (0)2 vA vA vA vA vAρ One obtains r r r r p π me λ p π me λ λ4 − λ2 1 + β − β H + cos2 α β − β 1 + 2β sin2 α H = 0. (32) 2 mp cos α 2 mp cos α

A solution of the form λ = λR +λI H with λI λR satisﬁes or to leading order p 1 + β ± (1 + β)2 − 4β cos2 α λ2 = , (34) R 2 4 2 2 λR − (1 + β)λR + β cos α = 0 (33) with a dissipative part prescribed by

r r 2 p π me 1 h 2 2 2 i 2(2λR − β − 1)λI = β −λR + cos α(1 + 2β sin α) . (35) 2 mp cos α

It is then convenient to write

2 2 2 2 2 2 2 2 2 −λR + cos α(1 + 2β sin α) = (−λR + β cos α) sin α − (λR − 1 − β sin α) cos α. (36)

Using the equation satisﬁed by λR, one has 4 Parallel Alfven´ waves

β2 sin2 α cos2 α λ2 − 1 − β sin2 α = − . (37) A long-wave perturbative expansion of the Landau-fluid R − 2 + 2 λR β cos α model was performed in the case of parallel Alfven´ waves by Passot and Sulem (2003b). The stretched coordinate along This enables one to write the normalized damping rate λI in 0 the propagation is ξ = (z − V0t) and the perturbation am- the form 1/2 plitudes obeys the scalings bx ∼ by ∼ ux ∼ uy = O( ) (0) (0) rπ r m sin2 α and uz ∼ bz − B0 ∼ ρ − ρ ∼ P − P = O(), with a p e / λI = − β · dynamical time τ = 3 2t. The FLR corrections play a role 8 m cos α p at the level of the wave dispersion coefficient. In the present 2 2 2 2 4 (λ − β cos α) + β cos α regime, they can accurately be captured when using the ap- · R , (38) 2 − − 2 − 2 (2λR β 1)(λR β cos α) proximation given in Yajima (1966) that, in a local frame where the z-axis is taken along the local magnetic field, reads that also rewrites r r 2 p p π me sin α [1] = − [1] = − ⊥p + λI = − β · 5xx 5yy ∂yux ∂xuy (40) 8 mp cos α 2p " # [1] = 2β2 cos4 α 5zz 0 (41) · 1 + . (39) p + − 2 2 − 2 [1] [1] ⊥p (1 β 2β cos α)λR β cos α 5xy = 5yx = − ∂yuy − ∂xux (42) 2p Equation (38) reproduces Eq. (5.5.2.10) of Akhiezer et [1] [1] 1 2 5yz = 5zy = 2pkp∂zux + p⊥p (∂xuz − ∂zux) (43) sin α p al. (1975) up to the factor cos α whose absence seems to be a misprint. This factor is present in Janiki et al. (1992), but is [1] [1] 1 5xz = 5zx = − 2pkp∂zuy + p⊥p ∂yuz − ∂zuy (44). there multiplied by a different expression. p 250 T. Passot and P. L. Sulem: Fluid description of Landau damping

Here, = qpB0 is the proton cyclotron frequency, and the To leading order , the y-components of the induction and p cmp superscript [1] in the FLR tensor components refers to the momentum equations give order of the expansion in terms of 1/p. For parallel Alfven´ (1) (1) uy 1 by waves, the Landau damping does not act as a linear pro- = − , (45) cess but rather affects the nonlinear coupling. A long-wave V0 cos α B0 reductive perturbative expansion performed on the result- and prescribes the propagation velocity of the Alfven´ waves ing Landau-fluid model reproduces the derivative nonlinear as V0 = 30 cos α with Schrodinger¨ equation obtained by an equivalent asymptotics performed on the Vlasov-Maxwell system, up to the replace- 2 (0) (0) B p⊥ pk ment of the response function of the plasma by suitable Pade´ 32 = 0 + − = v2 + v2 , (46) 0 4πρ ρ ρ A 1 approximants in Eqs. (18) and (22) for the pressure pertur- 0 0 0 bations. This agreement also holds in the case of multi- where we used notations defined in Sect. 3. dimensional Alfven´ wave trains where mean fields involving At order 3/2, the magnetic field divergenceless implies an averaging along the direction of propagation are driven by the presence of transverse gradients (Passot and Sulem, (1) 1 (1) bx = − bz (47) 2003a). tan α and the x-component of the induction equation

5 Oblique Alfven´ waves (1) 2 + 2 (1) (1) 30 bz (vA v1e) by ux = − cos α ∂ξ . (48) Turning to the case of obliquely propagating Alfven´ waves, tan α B0 p B0 the reductive perturbative expansion now involves the scal- Furthermore, the density equation gives 1/2 (1) (2) 1/2 (1) (2) ings by = (by +by ···), uy = (uy +uy +· · ·), (1) (1) 2 2 (1) while the same expansions as in Sect. 3 are used for the other ρ b (v + v ) by u(1) = 3 − 3 z + A 1e sin α ∂ . (49) ﬁelds. As previously, we also assume that the disturbances of z 0 (0) 0 ξ ρ B0 p B0 the equilibrium state vary only along the direction of propa- 1/2 0 gation whose coordinate is stretched as ξ = (z − V0t). The x and z-components of the momentum equation ex- We also introduce the slow time τ = 3/2t. press the pressure perturbations as

(1) (1) 2 2 − 2 2 p 1 1 b v⊥p v⊥p 2vkp cos α ⊥ = 3 u(1) + v2 x + sin α ∂ u(1) − ∂ u(1) (0) 0 x 1 ξ y ξ y ρ tan α tan α B0 2p p sin α (1) (1) ! (1)2 cos α b b by + v2 x − z − v2 (50) A α B B A 2 sin 0 0 2B0 (1) (1) 2 − 2 (1)2 pk sin α b v⊥p 2vkp by = 3 u(1) + v2 x − sin α ∂ u(1) − (v2 + 2v2 ) , (51) (0) 0 z 1 α B ξ y A 1 2 ρ cos 0 p 2B0

(1) (1)2 (1) = bz + by (1) that also rewrite as linear functions of ρ , A B 2 and ∂ξ by , in the form 0 2B0 (1) " 2 2 # (1) p 3 cos α v⊥p by ⊥ = −v2 A − 0 v2 + v2 − v2 + v2 + sin α ∂ (52) (0) A A 1e 1p kp ξ ρ p sin α 2 B0

(1) (1) (1) pk ρ 3 by = 32 − v2 + 2v2 A + 0 v2 + v2 + v2 − v2 sin α∂ . (53) (0) 0 (0) A 1 A 1e 1p kp ξ ρ ρ p B0

At order 2, the y-component of the momentum equation required corrections, expressed by the tensor 5[2], are com- quires an accurate description of the FLR corrections. The puted in Appendix B. Together with the induction equation usual approximation (40)-(44) is not sufﬁcient and the re- at the same order, one then obtains

" (2) (2) # " (1) (1) ! (1) 2 uy by (1) 2 bx bz by V0 cos α ∂ξ + ∂ξ = cos α ∂τ uy − cos α ∂ξ v1 sin α − cos α V0 B0 B0 B0 B0 (1) p(1) ! (1) (1)3 # v2 ! p⊥ k by 2 by cos α ⊥p 2 2 2 (1) + cos α − − v cos α + sin α + 2v cos α ∂ξξ u ρ(0) ρ(0) B 1 3 kp x 0 B0 p 2 T. Passot and P. L. Sulem: Fluid description of Landau damping 251

1 1 + v2 cos2 α sin α ∂ u(1) − cos α ∂ u(1) − ∂ sin α5[2] + cos α5[2] (54) ⊥p ξξ z ξξ x (0) ξ xy yz p ρ " (2) (2) # (1) (1) ! (1) ! uy by by (1) by (1) by V0 cos α ∂ξ + ∂ξ = ∂τ + cos α ∂ξ uz + sin α ∂ξ ux V0 B0 B0 B0 B0 2 (1) (1) ! 2 " (1) (1)2 # v b b v b by + A cos α ∂ cos α x − sin α z + 1e ∂ cos2 α − sin2 α x + 2 sin α cos α ξξ B B ξξ B 2 p 0 0 p 0 2B0 (1) (1) 1 pk − p⊥ + cos α sin α ∂ e e . (55) ξξ (0) p ρ

The correction term 5[2] is due to the distortion of the magnetic ﬁeld lines, to the time derivative of 5[1] and also to the contributions of the Hall effect and of the electron pressure gradient. One has

(1)2 ! 1 3 by ∂ sin α5[2] + cos α5[2] = 2 sin α cos α 0 v2 − v2 ∂ (0) ξ xy yz kp ⊥p ξξ 2 ρ p 2B0 " # 1 32 b(1) + −32 v2 − v2 3 α − 0 v2 α 2 α − v2 + v2 v2 3 α ∂ y . 2 0 2 kp ⊥p cos ⊥p cos sin A 1e 1p cos ξξξ (56) p 4 B0 The corresponding solvability condition reads

(1) 2 + 2 " v2 ! # by 30 vA v1e 1 30 ⊥p 2 2 2 2 2 230∂τ − ∂ξξ A + + 2v1e sin α + v⊥p − 2vkp cos α ∂ξξ A B0 tan α p p tan α 2 (1) 2 2 " 2 # (1) 3 ρ v + v v⊥p by − 0 v2 sin α cos α∂ − A 1e cos α sin2 α + v2 − 2v2 cos2 α ∂ ⊥p ξξ (0) 2 ⊥p kp ξξξ p ρ p 2 B0 2 2 (1) 2 " 2 # (1) v + v b 3 v⊥ b − A 1e v2 − v2 2 α∂ y + 0 p 2 α + v2 − v2 2 α ∂ y 2 ⊥p kp cos ξξξ 2 sin 2 kp ⊥p cos ξξξ p B0 p 4 B0 (1) (1) 3 pk − p⊥ + 0 cos α sin α∂ e e = 0. (57) ξξ (0) p ρ

At this point, one needs to express the perturbations tion of the perpendicular and parallel pressures in the form of density, pressure and magnetic field as a function of (1) (1) 2 2 (1) p ρ v + v by the wave amplitude. These relations are a priori provided ⊥r − − A + A 1e sin α ∂ = 0 (58) (0) (0) ξ ρ 30p B0 by the dynamical equations for the pressures and the heat p⊥r fluxes. It however turns out that the scaling associated (1) (1) 2 2 (1) pk ρ v + v by with oblique Alfven´ waves invalidates the assumption that r − 3 + 2A − 2 A 1e sin α ∂ = 0. (59) (0) (0) ξ ρ 30p B0 the non-gyrotropic parts of the heat flux tensor can be ig- pkr nored in the equations for the gyrotropic pressure compo- Assuming also isotropic temperatures for the unperturbed nents. The gyrotropic components of the heat flux are in- plasma, we get for the protons deed of order while the non-gyrotropic contributions in- (1) (0) " (1) 2 (1) # volve the product of the equilibrium pressure with the cur- p⊥p Tp ρ v by = βv2 + A − A sin α ∂ (60) rent which is also of order . This situation contrasts with (0) A (0) (0) ξ ρ T ρ 30p B0 that of oblique magnetosonic and parallel Alfven´ waves for e (1) (0) " (1) 2 (1) # which qkr ∼ q⊥r = O() while the current scales like pkp Tp ρ 2v by = βv2 3 − 2A + A sin α ∂ .(61) 3/2. We are thus led to restrict the present analysis to (0) A (0) (0) ξ ρ Te ρ 30p B0 oblique Alfven´ waves in a plasma with adiabatic protons and isothermal electrons, a regime where the above Landau fluid Concerning the equation of state for isothermal electrons, model remains valid. This regime associated with the condi- the relations obtained in the case of parallel propagation are expected to remain valid since the electrons move very tion vth,p vA vth,e, corresponds to a situation where me β Te . rapidly (relatively to the propagation velocity of the wave) mp Tp along the magnetic field lines in a way that equalizes temper- In the adiabatic regime, all the heat flux components are atures, making the electron pressure insensitive to the prop- zero so that we can use the equations governing the evolu- agation direction of the wave. Taking the isothermal limit 252 T. Passot and P. L. Sulem: Fluid description of Landau damping

r r c = √1 me → 0 for the electron plasma response function p π me e β mp + β HA . (63) in Eqs. (18) and (22) (that can still be used in the present 2 mp regime), we get (1) (1) We observe that A and ρ are linear functions of ∂ξ by . We thus reach the conclusion that the solvability condition gov- (1) erning the evolution of by does not include nonlinear terms (1) " (1) r r # p⊥ ρ p π me and reduces to a linear Airy equation, as in the usual MHD e = v2 β − β HA (62) (0) A (0) ρ ρ 2 mp description (Mio, Ogino, and Takeda, 1976; Gazol, Passot, and Sulem, 1998). (1) ( r (1) pk π r m ρ Substituting the total pressures in Eqs. (52) and (53), we e = v2 β − pβ e H (0) A (0) Tp ρ 2 mp ρ obtain in the considered limit β 1, Te

(1) r r 2 (1) Tp ρ p π me vA cos α by β 1 + + 1 − β H A = − ∂ (64) (0) ξ Te ρ 2 mp p sin α B0 r r (1) r r (1) p π me ρ p π me vA by β − 1 − β H + 1 + β H A = sin α∂ , (65) (0) ξ 2 mp ρ 2 mp p B0

(1) whose solutions ρ and A are to be reported in the solvabil- that identiﬁes with the equation obtained in the framework of ρ(0) ity condition (57). Hall-MHD for a plasma viewed as a polytropic gas (Mio et Let us ﬁrst neglect the Landau damping and compute al., 1976; Gazol et al., 1998). An explicit form of the dissipation rate due to Landau (1) (1) ρ v 1 by damping is easily obtained in the small β limit. Writing R = − A ∂ (66) (0) ξ (1) = (1) + (1) (1) = (1) + (1) ρ p sin α B0 ρ ρR HρI and A AR HAI , one gets 2 ! (1) to leading order v cos α β by A(1) = − A − ∂ , (67) R ξ r r p sin α sin α B0 (1) p π me (1) A = β A (69) I 2 m R where the subscript R refers to the non dissipative part. Sub- p (1) r r (1) ! stituting in Eq. (57), we simply get ρ (1) p π me ρ (1) − I + A = β R − A . (70) (0) I (0) R ρ 2 mp ρ b(1) v3 cos α b(1) ∂ y + A ( 2 α − β) ∂ y = , τ 2 cos 2 ξξξ 0 (68) B0 2p sin α B0 which leads to the dynamical equation

" # b(1) v3 cos3 α r π m 1 b(1) ∂ y + A + β e 3 α 2 α + H ∂ y = , τ 2 2 cos tan 2 ξξξ 0 (71) B0 2p sin α 2 mp tan α B0 that reproduces the Landau damping dissipation rate given equations where the coordinates refer to the stretched ones) by Akhiezer et al. (1975).

(1) 2 p v⊥p ⊥ − ∂ u(1) + v2 A = 0 (72) (0) x y A ρ 2p 6 Kinetic Alfven´ waves (1) (1) (1) pk b 3 b − v2 ∂−1∂ x − 0 v2 − v2 ∂ y ( 1 z x ⊥p 2 kp x ρ 0) B0 p B0 When describing Alfven´ waves propagating almost perpen- (1)2 by dicularly to the ambient ﬁeld (cos α 1), a different order- + v2 + 2v2 = 0 (73) ∼ = 1/2 = A 1 2 ing is to be used, namely by uy O( ), bz O(), 2B0 3/2 1/2 bx ∼ ux ∼ uz = O( ), together with ∂x = O( ), (0) (0) (1) (1) 2 pk − p⊥ by by ∂t = −30∂z ∼ and ∂τ = O( ). This corresponds to a ∂ u(1) + ∂ − v2 ∂ = 0 (74) t y (0) z A z propagation angle α such that cos α = O(1/2). The long- ρ B0 B0 (2) (2) wave expansion also involves the contributions u and b b(1) y y y − (1) = arising at order 3/2. The expansion leads to the following ∂t ∂zuy 0 (75) B0 T. Passot and P. L. Sulem: Fluid description of Landau damping 253

(1) (0) (1) (1) (1) ∂t ρ + ρ ∂xux = 0 (76) ∂xbx + ∂zbz = 0 (78) (1) 2 + 2 (1) bz vA v1e by ∂t + ∂xux = − ∂xz (77) B0 p B0

! b(2) ρ(1) b(1)b(1) ∂ u(2) − 32∂ y + ∂ u(1) − 3 ∂ u(1) + u(1)∂ u(1) − v2 ∂ x y t y 0 z B τ y 0 (0) z y x x y 1 x 2 0 ρ B0 p(1) − p(1) (1) (1)3 (1) (1) ! k ⊥ by 2 by 2 by bz +∂z + v + v ρ(0) B 1 3 1 2 0 B0 B0 2 (1) (1) (1) (1) 2 (1)2 v⊥p b by b by v⊥p by 3 + ∂ v − v2 x ∂ − v2 z ∂ − 2 3 ∂ + 0 v2 ∂ u(1) = 0 (79) xx x A B x B A B z B 0 xz 2 2 ⊥p zxx y 2 p 0 0 0 0 p 2B0 4p (2) (1) (1) ! (1) (1) ! 2 (1) by (2) by (1) bz by (1) by vA bz ∂t − ∂zuy + ∂τ − ∂z uy + ∂x ux − uy − ∂xz B0 B0 B0 B0 B0 p B0

2 (1) (1)2 ! (1) (1) v b by 1 pk − p⊥ − 1e ∂ x − 2∂ + ∂ e e = 0, (80) xx B xz 2 xz (0) p 0 2B0 p ρ

(1) (1) (1) bz (1) by with ∂z u − ∂x u = 0, since the dynamics is assumed one-dimensional (∂x = sin α∂ξ and ∂z = cos α∂ξ ). y B0 y B0 (1) (1) by Using ∂t = −3 ∂z, this leads to v = −3 , and the solvability condition reads 0 y 0 B0

(1) " (1) (1) ! (1) # (1) ! (1) by pk − p⊥ by by 3 b 23 ∂ − ∂ + 2v2 A + 3 ∂ u − 0 v2 ∂ z 0 τ z (0) 1 0 x x A xz B0 ρ B0 B0 p B0 (1) (1)2 ! (1) (1) 3 b by 3 pk − p⊥ − 0 v2 ∂ x − 2∂ + 0 ∂ e e 1e xx B xz 2 xz (0) p 0 2B0 p ρ 2 (1)2 2 (1) v⊥p 3 by 3 by − ∂ u(1) − 2 0 v2 − v2 ∂ + 0 v2 ∂ = 0. (81) xx x kp ⊥p xz 2 2 ⊥p xxz B 2 p p 2B0 4p 0

(1)2 (1) (1) One then computes 30 by by bz − v2 + v2 ∂ + 32 ∂ . A 1e xz 2 0 z 2 (83) (1) (1) p 2B B pk p 0 0 − ⊥ + 2v2 A = (0) (0) 1 ρ ρ In the equation resulting from the substitution of the above (1) (1) (1) 30 2 2 by 2 bz quantities in Eq. (81), one eliminates bz by means of 3v⊥p − 4vkp ∂x + 30 (82) 2p B0 B0 Eqs. (76) and (78) to get (1) and uses Eq. (78) to express ux . One also uses the assump- (1) (1) 2 2 (0) b ρ v + v by tion of one-dimensionality to write 3 z = 3 − A 1e ∂ , (84) 0 0 (0) xz (1) ! B0 ρ p B0 (1) by 30∂x ux = B0 which leads to

(1) (1) by 3 3 ρ 23 ∂ + 0 v2 − 2 v2 − v2 ∂ A − 0 v2 ∂ 0 τ ⊥p A 1e xz ⊥p xz (0) B0 2p p ρ 2 (1) (1) (1) v⊥p by 3 pk − p⊥ + 32 − 2 v2 + v2 ∂ + 0 ∂ e e = 0. (85) 2 0 A 1e xxz xz (0) 4p B0 p ρ 254 T. Passot and P. L. Sulem: Fluid description of Landau damping

b(1) 2 As in the previous section, we shall make use of the pressure u(1) = −3 y is the same as Eq. (53) up to the cos α term y 0 B0 sin α equations in order to express ρ(1)/ρ(0) and A. In Eq. (73) that is subdominant in the kinetic Alfven´ wave asymptotics. (1)2 2 for the parallel pressure, we express by /B0 in terms of Assuming as previously adiabatic protons and isothermal (1) A and bz /B0 that is given by Eq. (84). This reproduces electrons together with the hypothesis of isotropic temper- Eq. (52) where sin α∂ξ is here denoted ∂x. Equation (72) atures for the equilibrium state, we get for the perpendicular pressure, together with the relation

(0) ! (1) (0) ! r r (0) (1) Tp ρ Tp p π me vA Tp by β 1 + + 1 + β A − β HA = β ∂ (86) (0) (0) (0) (0) x Te ρ Te 2 mp 2p Te B0 (0) ! (1) (0) ! r r (1) ! (0) ! (1) Tp ρ Tp p π me ρ vA Tp by −1 + β + 3β + 1 − 2β A + β H A − = 1 − 3β ∂ . (87) (0) (0) (0) (0) (0) x Te ρ Te 2 mp ρ p Te B0

When neglecting the Landau damping, this system is solved parallel electric ﬁeld is easily obtained from the generalized as Ohm’s law and, to leading order, read

(1) (0) ! (1) mpc (1) ρ v 3 Tp by Ek = − ∂ n T . (91) = − A 1 − β ∂ (88) (0) z e (0) (0) x qpρ ρ p 2 Te B0 (0) ! (1) ≈ (1) v 3 Tp by Using that E⊥ vAby one recovers the usual relation A = A β 1 + ∂ . (89) (0) x (Lysak 1990) p 2 Te B0 2 Ek = ρs ∂z∇⊥ · E⊥ (92) In the small β limit, the Landau damping in Eq. (85) is due to the anisotropy of the electron pressure fluctuations, where 2 (0) 2 with ρs = Te /(mpp). As mentioned by Hollweg (1999), the contribution of the density fluctuations is dominant. This and seen on Eq. (88), a kinetic Alfven´ wave becomes com- leads to the equation governing the dynamics of long kinetic pressive as soon as the transverse wavenumber k⊥ satisfies Alfven´ waves the condition k⊥vA/p ' 1, i.e. for scales of the order of the ion inertial length. For low β plasmas, this occurs even (1) 3 " (0) ! by v 3 Tp ∂ + A cos α −β 1 + though the wave dispersion is negligible. τ 2 (0) B0 2p 4 Te It is remarkable that kinetic Alfven´ waves with frequen- r r (1) cies ω p, can be described by a single fluid generalized p π me by + β H ∂ξξξ = 0, (90) MHD formalism. This point was noted by Marchenko, Den- 2 mp B0 ton and Hudson (1996) who derived the linear dispersion relation for kinetic Alfven´ waves ignoring Landau damping. where we made the substitution ∂xxz = cos α∂ξξξ , the vari- The crucial ingredients are a generalized Ohm’s law with able ξ denoting the coordinate along the direction of prop- Hall term and electron pressure gradient, FLR corrections agation. Note that this equation is linear. It is valid for up to order 1/2 and, when retaining wave dissipation, an 2 p propagation angles α such that cos α β 1. We re- appropriate form for the electron Landau damping. cover in particular the effect on the dispersion of the proton- electron temperature ratio predicted by Hasegawa and Chen (1976) using a fully kinetic theory. The same dispersion co- 7 Conclusions efficient was also obtained by Belmont and Rezeau (1987) using a bi-fluid approach. We here demonstrate that a one- We have studied the dynamics of dispersive MHD waves fluid approach can reproduce the correct dispersion. As al- propagating in a collisionless plasma permeated by a strong ready mentioned, the extension of the present approach to magnetic field, using a Landau fluid model that retains fi- hotter plasmas and the derivation of the dispersion and dis- nite Larmor radius corrections to the gyrotropic pressures to- sipation of kinetic Alfven´ waves in this regime (Lysak and gether with the Hall effect and the electron pressure gradient Lotko, 1996) requires a specific modeling of the influence of in a generalized Ohm’s law. The predicted Landau damping the Hall effect on the heat fluxes and will be addressed in a of magnetosonic waves with a wavelength large compared forthcoming paper. to the ion inertial length accurately reproduces that derived In a kinetic Alfven´ wave, protons move perpendicularly by a long-wave asymptotics from the Vlasov-Maxwell sys- to the average magnetic field while electrons move mainly tem. It is also demonstrated that the dynamics of small- along the magnetic field lines. As a consequence a parallel amplitude oblique Alfven´ waves is linear, even in the case electric field Ek develops which accelerates electrons. This of quasi-transverse propagation (kinetic Alfven´ waves). The T. Passot and P. L. Sulem: Fluid description of Landau damping 255 dispersion and damping rates, computed in the regime of adi- As in Sect. 3, where α is the angle between the ambi- abatic ions and isothermal electrons often considered in the ent magnetic field B0bz (where bz is the unit vector point- literature, also agrees with the results of the kinetic theory ing along the z-axis) and the direction of propagation of the (Akhiezer et al., 1975). Addressing more general regimes wave, it is then convenient to perform the change of frame would require an extension of the Landau fluid closure de- x0 = x cos α − z sin α, z0 = x sin α + z cos α, the dynamics scribed in Sect. 2 by retaining the influence of the Hall effect being assumed independent of the y variable. We then intro- 1/2 0 on the heat fluxes. Another development concerns the de- duce the stretched variable ξ = (z −V0t) where V0 c scription of FLR corrections. In instances, such as kinetic is the wave velocity in the z direction, together with the slow Alfven´ waves, a description going beyond Yajima’s (1966) time τ = 3/2t. It follows that the spatial gradient rewrites 1/2 1/2 formulas is required. It is obtained in the present paper by a ∇ = ( sin α∂ξ , 0 , cos α∂ξ ). perturbative expansion that correctly retains the effect of the In order to select oblique magnetosonic waves, (1) 3/2 (1) magnetic curvature, a prerequisite to get the exact cancella- we write bx = bx + · · ·, by = by + · · ·, tion of nonlinear terms whose presence would have made the (1) bz = B0 + bz + · · · and thus, from Eq. (A.2), resulting long-wave equation mathematically ill-posed. (1) (1) (1) e = 3/2e + · · ·, e = e + · · ·, e = 3/2e + · · ·, As mentioned in the Introduction, a one-fluid description x x y y z z V0 b(1) = − αe(1) V0 b(1) = αe(1) − αe(1) appears to be an efficient tool to investigate both analyti- with c x cos y , c y cos x sin z , V0 (1) = (1) cally and by means of numerical simulations, the regime of c bz sin αey . magnetohydrodynamic-wave turbulence, as it occurs both in We also expand the distribution function in the form space and laboratory plasmas. Among the questions to be (0) (0) 1/2 (1) addressed are the anisotropy of the cascade, the typical scale fr = Fr + fr + fr + · · · (A.5) at which the cascade is arrested and the compared heating (0) of the electron and proton populations. The role played by where Fr denotes the equilibrium velocity distribution dispersion is of specific interest and motivates the develop- function, assumed rotationally symmetric around the direc- ment of a weak turbulence theory for Hall-MHD (Sahraoui, tion of the ambient field and symmetric relatively to forward Belmont and Rezeau, 2003). and backward velocities along this direction, thus excluding the presence of equilibrium drifts. It is also convenient to express the velocity v in a cylin- Appendix A: Kinetic theory of magnetosonic waves drical coordinate system by defining the velocity space angle φ = tan−1(v /v ). One writes We write the Vlasov-Maxwell equations in the form z y = = = = qr 1 v vx v⊥ cos φ , vy v⊥ sin φ , vz vk (A.6) ∂t fr + v · ∇fr + (e + v × b) · ∇vfr = 0 (A.1) mr c and 1 ∂t b = −∇ × e (A.2) sin φ c ∇v = cos φ ∂v⊥ − ∂φ , 4π X Z 1 v⊥ ∇ × b = q n vf d3v + ∂ e (A.3) c r r r c t cos φ r sin φ ∂v⊥ + ∂φ , ∂vk (A.7) Z v⊥ X 3 ∇ · e = 4π qr nr fr d v, (A.4) Furthermore, qr (v × B z) · ∇ = − ∂ , where r cmr 0b v r φ = qr B0 is the cyclotron frequency of the particles of where fr and nr are the distribution functions and the aver- r mr c age number densities of the particles of species r with charge species r. 1 Expanding to the successive orders, one gets from qr and mass mr . The displacement current c ∂t e turns out to be negligible in the present analysis. Eq. (A.1),

(0) r ∂φFr = 0 (A.8) " (1) (1) # (0) qr (1) vkbx v⊥bx (0) r ∂φfr = (ey + ) sin φ∂v⊥ − sin φ∂vk Fr (A.9) mr c c " (1) (1) # (1) qr (1) vkby (1) v⊥by (0) r ∂φfr = (ex − ) cos φ∂v⊥ + (ez + cos φ)∂vk Fr mr c c (0) + sin αv⊥ cos φ + cos αvk − V0 ∂ξ fr . (A.10) 256 T. Passot and P. L. Sulem: Fluid description of Landau damping

(0) Equation (A.8) indicates that Fr is independent of the angle Assuming that the plasma contains only electrons and pro- φ. Equation (A.9) leads to tons with bi-maxwellian equilibrium distribution functions

(1) 3/2 " 2 2 # bx (0) 1 m mr vk mr v f (0) = DF (0) cos φ + f (A.11) F (0) = r exp − − ⊥ (A.22) r r r r 3/2 ( ) (0)1/2 (0) (0) B0 (2π) 0 T⊥r Tkr 2Tkr 2T⊥r with D = (3 − v )∂ + v ∂ and 3 = V / cos α. One 0 k v⊥ ⊥ vk 0 0 and using the quasi-neutrality condition of the equilibrium (0) (0) (0) denotes f = hf i ≡ 1 R f dφ. (0) (0) (0) r r 2π r state np = ne = n , one obtains The solvability condition of Eq. (A.10) together with the divergenceless condition for the magnetic field reads X (0) 2 1 L = −n qr Wr (A.23) T (0) (0) v⊥ (0) r kr vk − 30 f = − vk − 30 ∂v F A r 2 ⊥ r (0) X (0) T⊥r v2 q M = −n qr Wr (A.24) + ⊥ (0) + r (0) (0) ∂vk Fr A ∂vk Fr ϕ (A.12) r Tkr 2 mr (0)2 X T⊥ b(1) N = −2n(0) r W , (A.25) where A = z and the electric potential ϕ is such that (0) r B0 r Tk (1) r ez = −∂zϕ = cos α∂ξ ϕ. Finally, the y-component of Eq. (A.3) gives where we define Wr = W(cr ) with the notations of Sect. 3. Using Eq. (81) of Passot and Sulem (2003b) and intro- Z 1 4π X (1) 3 ducing the notation C⊥ defined in Eq. (28), one recovers − ∂ξ A = qr nr v⊥ cos φ∂φfer d v. (A.13) sin α cB0 r the dispersion relation of the magnetosonic waves given by Eq. (29). Using Eq. (A.10), one obtains

1 1 cos2 α − ∂ A = − 32 − v2 ∂ A B: Second order FLR corrections to the pressure tensor ξ 2 0 1 ξ sin α vA sin α 4π sin α X Z (0) In a regime where the electron inertia is negligible, the FLR + ∂ m n v2 f d3v. 2 ξ r r ⊥ r (A.14) corrections are relevant only for the proton pressure tensor B0 2 r Pp. They obey (Kulsrud, 1983) From Eq. (A.12), one has b b 1 dPp × 5p − 5p × = − + (∇ · up)Pp X Z (0) m n v2 f d3v = 4p(0)A + 2N A + Mϕ (A.15) B0 B0 r dt r r ⊥ r ⊥ r tr +Pp · ∇up + (Pp · ∇up) (B.1) and from Eq. (A.4), where, as usual, the heat transfer contribution to the FLR MA + Lϕ = 0, (A.16) have been neglected. To leading order, the local magnetic where field b is approximated by the ambient field in the left-hand- side and the ion pressure tensor P in the right-hand-side is 2 Z ∞ 2 p X qr nr v⊥ replaced by the gyrotropic contribution PG. This leads to L = 2π d( ) Gr (A.17) p r mr 0 2 the components of the leading order FLR correction tensor Z ∞ 2 2 given by Eqs. (40)–(44). This approximation obtained in a X v⊥ v⊥ M = 2π qr nr d( ) Gr (A.18) local frame where the z-axis points along the local magnetic 2 2 r 0 field is often oversimplified by using the same formula in a Z ∞ v2 v4 fixed frame with the z-axis along the ambient field (Khanna = X ⊥ ⊥ N 2π mr nr d( ) Gr , (A.19) and Rajaram, 1982; Chakraborty and Das, 2000). These two r 0 2 4 references also involve a sign error in the definition of the [1] [1] [1] and components 5xx = −5yy and 5xy . This approximation (0) (0) turns out to be insufficient in some instances, for example in Z 1 ∂Fr ∂Fr Gr = P dvk + π H. (A.20) the case of oblique Alfven´ waves for which a higher order ex- − = vk 30 ∂vk ∂vk vk 30 pansion is needed. For the scaling used in the description of One finally gets oblique Alfven´ waves for example, these corrections involve both terms of order and 3/2. However they do not contain 1 3/2 V 2 − v2 + v2 cos2 α − v2 + 2v2 + N all the contributions arising at order . In particular, one 0 A 1 A ⊥ ρ(0) should also retain the effect of the magnetic field distortion, [1] 2 1 2 2 −1 the fast-time derivative ∂t 5 , and also the Hall-term and · sin α + sin αM L = 0. (A.21) d G(1) ρ(0) electron pressure contribution to dt Pr . T. Passot and P. L. Sulem: Fluid description of Landau damping 257

Denoting by L (respectively R) the corrections of the left Acknowledgements. This work beneﬁted of support from CNRS hand side (respectively right hand side) of Eq. (B.1) involv- programs “Soleil-Terre” and “Physique et Chimie du Milieu Inter- ing magnetic ﬁeld distortion, one obtains stellaire” and from INTAS contract 00-292. 1 [2] [1] Edited by: A. S. Sharma 25xy + Lxx = Rxx + ∂t 5xx + Hxx (B.2) p Reviewed by: R. L. Lysak and another referee [2] (2) 1 [1] 5yy − 5xx + Lxy = Rxy + ∂t 5xy + Hxy (B.3) p

[2] 1 [1] 5yz + Lxz = Rxz + ∂t 5xz + Hxz (B.4) p

[2] 1 [1] −25xy + Lyy = Ryy + ∂t 5yy + Hyy (B.5) References p [2] 1 [1] Armstrong, J. W., Rickett, B. J., and Spangler, S. R.: Electron den- −5xz + Lyz = Ryz + ∂t 5yz + Hyz (B.6) p sity power spectrum in the local interstellar medium, Astrophys. J., 443, 209–221, 1995. 1 [1] Lzz = Rzz + ∂t 5zz + Hzz , (B.7) Akhiezer, A. I., Akhiezer, I. A., Polovin, R. V., Sitenko, A. G., p and Stepanov, K. N.: Plasma electrodynamics, Vol.1, Pergamon where the tensor H , associated with the Hall term and elec- Press, 1975. tron pressure contributions to the time variation of the direc- Belmont, G. and Rezeau, L.: Finite Larmor radius effects: The two- tion of the magnetic field, is defined as fluid approach, Ann. Geophysicae, 5A, 59–70, 1987. Chakraborty, D. and Das, K. P.: Evolution of nonlinear magne- pkr − p⊥r b ⊗ b = h ⊗ b + b ⊗ h − b · h tosonic waves propagating obliquely to an external magnetic H 2 r r 2 2 r (B.8) |b| |b| field in a collisionless plasma, J. Plasma Phys., 64, 211–226, cmr 1 1 G 2000. with hr = ∇ × (b × (∇ × b) + ∇ · P . qr 4πρ ρ e Chew, G. F., Goldberger, M. L., and Low, F. E.: The Boltzmann For the specific case of oblique Alfven´ waves for which equation and the one-fluid hydromagnetic equations in the ab- the magnetic perturbation is dominated by by (and under the sence of particle collisions, Proc. Roy. Soc. (London), 236, 112– assumption of no dependency in the y-coordinate), the non- 118, 1956. zero contributions of the symmetric tensors L, R and H are Ferriere,` K. M. and Andre,´ N.: A mixed magnetohydrodynamic- given by kinetic theory of low-frequency waves and instabilities in homo- geneous, gyrotropic plasmas, J. Geophys. Res., 107, SMP 7–1, b [1] y 7–17, CiteID 1349, 2002. Lxx = −25xz (B.9) B0 Gazol, A., Passot, T., and Sulem, P. L.: Nonlinear dynamics of [1] by obliquely propagating Alfven´ waves, J. Plasma Phys., 60, 95– Lxy = −5yz (B.10) B0 109, 1998. Hasegawa, A. and Chen, L.: Kinetic processes in plasma heating [1] [1] by Lxz = (5xx − 5zz ) (B.11) by resonant mode conversion of Alfven´ wave, Phys. Fluids, 19, B0 1924–1934, 1976. Lyy = 0 (B.12) Hollweg, J. V.: Kinetic Alfven´ wave revisited, J. Geophys. Res., b 104, 14 811–14 819, 1999. = [1] y Lyz 5xy (B.13) Janiki, M. S., Dusguspa, B., Gupta, M. R., and Som, B. K.: Solitary B0 magnetosonic waves with Landau damping, Physica Scripta, 45, [1] by Lzz = 25xz , (B.14) 368–372, 1992. B0 Kakutani, T., Ono, H., Taniuti, T., and Wei, C. C.: Reductive per- and turbative method in nonlinear wave propagation II, Application by to hydromagnetic waves in cold plasma, J. Phys. Soc. Japan, 24, R = p(0)∂ u (B.15) xx ⊥r z y B 1059–1166, 1968. 0 Kever, H. and Morikawa, G. K.: Korteweg-de Vries equation for b = (0) − (0) y nonlinear hydromagnetic waves in a warm collision-free plasma, Ryy 4pkr 3p⊥r ∂zuy (B.16) B0 Phys. Fluids, 12, 2090–2093, 1969. b Khanna, M. and Rajaram, R.: Evolution of nonlinear Alfven´ waves = (0) − (0) y Ryz p⊥r ∂xux 2pkr ∂zuz (B.17) propagating along the magnetic field in a collisionless plasma, J. B0 b Plasma Phys., 28, 459–468, 1982. = − (0) − (0) y Rzz 4pkr 2p⊥r ∂zuy (B.18) Kulsrud, R. M.: MHD description of plasma, in: Handbook of B0 Plasma Physics, edited by Rosenbluth, M. N. and Sagdeev, R. Z., together with Vol. 1 Basic Plasma Physics edited by Galeev, A. A. and Su- dan, R. N., North Holland, 115–145, 1983. v2 + v2 b = A 1e (0) − (0) y Lerche, I. and Schickeiser, R.: Linear Landau damping and wave Hxz pkp p⊥p ∂zz . (B.19) r B0 energy dissipation in the interstellar medium, Astro. Astrophys., 366, 1008–1015, 2001a. 258 T. Passot and P. L. Sulem: Fluid description of Landau damping

Lerche, I. and Schickeiser, R.: Cosmic ray transport in anisotropic 10, 3887–3905, 2003a. magnetohydrodynamic turbulence, Astro. Astrophys., 378, 279– Passot, T. and Sulem, P. L.: A long-wave model for Alfven´ wave 294, 2001b. trains in a collisionless plasma: II, A Landau-fluid approach, Li, H., Gary, S. P., and Stawicki, O.: On the dissipation of magnetic Phys. Plasmas, 10, 3906–3913, 2003b. fluctuations in the solar wind, Geophys. Res. Lett., 28, 1347– Quataert, E., Dorland, W., and Hammett, G. W.: The magnetoro- 1350, 2001. tational instability in a collisionless plasma, Astrophys. J., 577, Lithwick, Y. and Goldreich, P.: Compressible magnetohydrodyn- 524–533, 2002. qmic turbulence in interstellar plasmas, Astrophys. J., 562, 279– Rezeau, L. and Belmont, G.: Magnetic turbulence at the 296, 2001. magnetopause, a key problem for understanding the solar Lysak, R. L.: Electrodynamic coupling of the magnetosphere and wind/magnetosphere exchanges, Space, Sci. Rev., 95, 427–441, ionosphere, Space Sc. Rev., 52, 33–87, 1990. 2001. Lysak, R. L. and Lotko, W.: On the kinetic dispersion relation for Rogister, A.: Parallel propagation of nonlinear low-frequency shear Alfven´ waves, J. Gophys. Res., 101, 5085–5094, 1996. waves in high-β plasma, Phys. Fluids, 12, 2733–2739, 1971. Marchenko, V. A., Denton, R. E., and Hudson, M. K.: A magne- Sahraoui, F., Belmont, G., and Rezeau, L: Hamiltonian canonical tohydrodynamic model of kinetic Alfven´ waves with finite ion formulation of Hall-magnetohydrodynamics: Toward an appli- gyroradius, Phys. Plasmas, 3, 3861–3863, 1996. cation to weak turbulence theory, Physics of Plasmas, 10, 1325– Minter, A. H. and Spangler, S. R.: Heating of the interstellar diffuse 1337, 2003. ionized gas via the dissipation of turbulence, Astropys. J., 485, Snyder, P. B., Hammett, G. W., and Dorland, W.: Landau fluid mod- 182–194, 1997. els of collisionless magnetohydrodynamics, Phys. Plasmas, 4, Mio, K., Ogino, T., and Takeda, S.: A perturbation method and its 3974–3985, 1997. application to obliquely propagating nonlinear Alfen´ waves, J. Spangler, S. R.: The dissipation of magnetohydrodynamic turbu- Phys. Soc. Japan, 41, 2114–2120, 1976. lence responsible for interstellar scintillation and the heating of Mjølhus, E. and Wyller, J.: Nonlinear Alfven´ waves in a finite-betas the interstellar medium, Astrophys. J., 376, 540–555, 1991. plasma, J. Plasma Phys., 40, 299–318, 1988. Yajima, N.: The effect of finite ion Larmor radius on the propa- Passot, T. and Sulem, P. L.: A long-wave model for Alfven´ wave gation of magnetoacoustic waves, Prog. Theor. Phys., 36, 1–16, trains in a collisionless plasma: I. Kinetic theory, Phys. Plasmas, 1966.