Reduction of compressibility and parallel transfer by Landau damping in turbulent magnetized plasmas
aa r X i v : . [ phy s i c s . p l a s m - ph ] S e p Reduction of compressibility and parallel transfer by Landaudamping in turbulent magnetized plasmas
P. Hunana, ∗ D. Laveder, T. Passot, P. L. Sulem, and D. Borgogno Universit´e de Nice Sophia Antipolis,CNRS, Observatoire de la Cˆote d’Azur,BP 4229 06304, Nice Cedex 4, France Dipartimento di Energetica, Politecnico di Torino,corso Duca degli Abruzzi 24, 10138 Torino, Italy (Dated: June 25, 2018)
Abstract
Three-dimensional numerical simulations of decaying turbulence in a magnetized plasma are per-formed using a so-called FLR-Landau fluid model which incorporates linear Landau damping andfinite Larmor radius (FLR) corrections. It is shown that compared to simulations of compressibleHall-MHD, linear Landau damping is responsible for significant damping of magnetosonic waves,which is consistent with the linear kinetic theory. Compressibility of the fluid and parallel energycascade along the ambient magnetic field are also significantly inhibited when the beta parameteris not too small. In contrast with Hall-MHD, the FLR-Landau fluid model can therefore correctlydescribe turbulence in collisionless plasmas such as the solar wind, providing an interpretation forits nearly incompressible behavior. ∗ Electronic address: [email protected] . INTRODUCTION Hydrodynamics and Magnetohydrodynamics (MHD) are the central descriptions used tostudy turbulence in the solar wind and in a wide range of natural systems. Specifically for thesolar wind, MHD description yielded a great success in our understanding of observationaldata (e.g. see reviews by Goldstein et al. [1], Tu and Marsch [2], Bruno and Carbone [3],Horbury et al. [4], Marsch [5], Ofman [6]). Observational studies show that the solar wind istypically found to be only weakly compressible (e.g. see Matthaeus et al. [7], Bavassano andBruno [8] and the reviews cited above) and usual turbulence models which predict the energyspectra are derived in the framework of an incompressible MHD description (Iroshnikov [9],Kraichnan [10], Goldreich and Sridhar [11, 12], Galtier et al. [13, 14], Boldyrev [15, 16],Lithwick et al. [17], Chandran [18], Perez and Boldyrev [19], Podesta and Bhattacharjee [20],see also reviews by Cho et al. [21], Zhou et al. [22], Galtier [23] and Sridhar [24]). Theoreticalmodels which describe the radial evolution of spatially averaged solar wind quantities are alsousually developed in the framework of incompressible MHD formulated in Els¨asser variables(e.g. Zhou and Matthaeus [25–27], Marsch and Tu [28], Zank et al. [29], Smith et al. [30],Matthaeus et al. [31], Breech et al. [32]). It is however well known that the solar windis not completely incompressible and many phenomena which require compressibility areobserved in the solar wind, such as the evolution of density fluctuations (e.g. Spangler andArmstrong [33], Armstrong et al. [34], Coles et al. [35], Grall et al. [36], Woo and Habbal[37], Bellamy et al. [38], Wicks et al. [39], Telloni et al. [40]), magnetic holes, solitonsand mirror mode structures (e.g. Winterhalter et al. [41], Fr¨anz et al. [42], Stasiewicz etal. [43], Stevens and Kasper [44]) or strong temperature anisotropies which trigger, and arelimited by, micro-instabilities such as mirror and fire-hose (e.g. Gary et al. [45], Kasperet al. [46], Hellinger et al. [47], Matteini et al. [48], Bale et al. [49]). The importance ofcompressibility was stressed by Carbone et al. [50], who compared the observational datawith the energy flux scaling laws of Politano and Pouquet [51], which are exact relations ofincompressible MHD. Carbone et al. determined that the scaling relations can better fit thedata if the relations are phenomenologically modified to account for compressibility. Theincorporation of weakly compressional density fluctuations was partially addressed by so-called nearly incompressible models, which expand the compressible equations with respectto small sonic Mach number (Matthaeus and Brown [52], Zank and Matthaeus [53–55])2nd which were recently formulated in the presence of a static large-scale inhomogeneousbackground (Hunana and Zank [56], Hunana et al. [57, 58], see also Bhattacharjee et al.[59]). These models however specifically assume, and cannot explain, why the solar wind isonly weakly compressible. The theoretical compressible MHD models developed in the waveturbulence formalism (e.g. Kuznetsov [60], Chandran [61]) also cannot address this issue.Describing the solar wind with fully compressible MHD or Hall-MHD formalisms yieldsseveral problems. Most importantly, these compressible descriptions introduce sound wavesand slow magnetosonic waves. As elaborated by Howes [62], slow magnetosonic waves arestrongly damped by Landau damping in the kinetic Maxwell-Vlasov description. The pres-ence of fast and slow magnetosonic waves naturally implies higher level of compressibilityand overestimates the parallel energy transfer. Also, numerical simulations of compressibleHall-MHD performed by Servidio et al. [63] in the context of the magnetopause boundarylayer showed that compared to the usual turbulence in compressible MHD simulations, whichconsists of Alfv´en waves, the Hall term is responsible for decoupling of magnetic and velocityfield fluctuations. In compressible Hall-MHD regime, Servidio et al. observed spontaneousgeneration of magnetosonic waves which transform to a regime of quasi perpendicular “mag-netosonic turbulence”. This is in contrast with observational studies which typically showthat turbulence in the solar wind predominantly consists of quasi perpendicular (kinetic)Alfv´en waves (e.g., Bale et al. [64], Sahraoui et al. [65, 66]). Finally, all usual MHD orHall-MHD models cannot address how the energy is actually dissipated at small scales. Itis well known that solar wind plasma is almost collisionless and therefore the classical vonKarman picture of energy being dissipated via viscosity is not applicable to the solar wind.It is evident that new and more realistic models have to be introduced which can overcomesome of these drawbacks.The most realistic approach is of course the fully kinetic Vlasov-Maxwell description. Itis however analytically intractable, and even the biggest kinetic simulations cannot resolvethe large-scale turbulence dynamics. Two leading approaches appear to be promising tosubstitute MHD in describing solar wind turbulence : Gyrokinetics and Landau fluids.Gyrokinetics (e.g. Schekochihin et al. [67], Howes et al. [68]) was originally developedfor simulations of fusion in tokamaks. It is a kinetic-like description, which averages outthe gyro-rotation of particles around a mean magnetic field and therefore makes the kineticdescription more tractable, mostly by eliminating fast time scales. Derived directly from the3inetic theory, gyrokinetics has a crucial advantage of being asymptotically correct. Landaufluid description, on the other hand, is a fluid-like extension of compressible Hall-MHD, inwhich wave dissipation is incorporated kinetically by the modeling of linear Landau damping,thus retaining a realistic sink of energy. Other linear kinetic effects such as finite Larmorradius corrections are also incorporated.The simplest Landau fluid closure was considered by Hammett and Perkins [69] and theassociated dispersion relations were numerically explored by Jayanti, Goldstein and Vi˜nas[70]. The Landau fluid model was further advanced by Snyder, Hammett and Dorland [71],who considered the largest MHD scales, starting from the guiding center kinetic equation.The approach was reconsidered and refined to its present form with incorporation of Hallterm and finite Larmor radius (FLR) corrections by Passot, Sulem, Goswami and Bugnon[72–75]. This Landau fluid approach starts with the Vlasov-Maxwell equations and derivesnonlinear evolution equations for density, velocity and gyrotropic pressures. In the simplestformulation the model is closed at the level of heat fluxes by matching with the linearkinetic theory in the low frequency limit. Kinetic expressions usually contain the plasmadispersion function which is not suitable for fluid-like simulations. Landau fluid closure istherefore performed in a way as to minimize occurrences of this plasma dispersion functionand, where not possible, this function is replaced by a Pad´e approximant. This eliminatesthe time non-locality and also results in the presence of a Hilbert transform with respectto the longitudinal coordinate (in the direction of the ambient magnetic field) which inthe fluid formalism is associated with linear Landau damping. Further details about thedevelopment of Landau fluid models are thoroughly discussed in the papers cited above.The Landau fluid approach should however be contrasted with the more classical gyrofluidmodels (e.g. Dorland and Hammett [76], Brizard [77], Scott [78]), which are derived bytaking fluid moments of the gyrokinetic equation and for which a similar closure scheme isapplied afterwards.The Landau fluid approach has the following advantages. In contrast with Hall-MHD,it contains separate equations for parallel and perpendicular pressures and heat fluxes. Ittherefore allows for the development of temperature anisotropy, which is observed in thesolar wind. Noticeably, in contrast with gyrokinetics or gyrofluid models, the Landau fluidmodel does not average out the fast waves. Compared with these approaches, it also hasan advantage in that the final equations including the FLR corrections are written for the4sual quantities measured in the laboratory frame. Existing spectral MHD and Hall-MHDcodes can therefore be modified to Landau fluid description relatively easily. Also impor-tantly, even though gyrokinetics is a reduced kinetic description, it is still 5-dimensional andtherefore naturally quite difficult to compute. While current largest numerical simulationsof gyrokinetics (Howes et al. [79]) require thousands CPU-cores for a fluid-like 128 × res-olution, the FLR-Landau fluid model requires computational power only slightly larger thanthe usual Hall-MHD simulations. The results presented here, which employ a resolution of N = 128 grid points in all three directions, were calculated using 32 CPU-cores.Landau fluid models can be developed with several levels of complexity. For these firstLandau fluid simulations of three dimensional turbulence, we use a simplified version of themost general Landau fluid model [72], where we constrain ourselves to isothermal electronsand leading order corrections in terms of the ratio of the ion Larmor radius to the consideredscales. A similar model was used by Borgogno et al. [80], who studied the dynamicsof parallel propagating Alfv´en waves in a medium with an inhomogeneous density profile.They numerically showed that the observed Alfv´en wave filamentation and later transitionto the regime of dispersive phase mixing is consistent with particle-in-cell simulations. Inthis paper we concentrate on freely decaying turbulence and compare these to simulationsof compressible Hall-MHD. Numerical integration of the full Landau fluid model in onespace dimension was presented by Borgogno et al. [81], who investigated the dynamics veryclose to the mirror instability threshold and showed the presence of magnetic holes. Resultsconsidering quasi-transverse one-dimensional propagation in the full Landau fluid model arepresented in [82, 83]. II. THE MODEL AND ITS NUMERICAL IMPLEMENTATION
Considering a neutral bi-fluid consisting of protons (ions) and isothermal electrons, theLandau fluid model consists of evolution equations for proton density ρ = m p n (where m p is the proton mass and n the number density), proton velocity u p , proton paralleland perpendicular pressures p k p , p ⊥ p and heat fluxes q k p , q ⊥ p , together with the inductionequation for magnetic field b . The equations are normalized and density, magnetic fieldand proton velocity are measured in units of equilibrium density ρ , ambient magnetic field B , and Alfv´en speed V A = B / √ πρ , respectively. Pressures are measured in units of5nitial proton parallel pressure p (0) k p and heat fluxes in units of p (0) k p V A . The total pressurein the momentum equation has the form of a tensor. Defining ˆ b = b / | b | as a unit vectorin the direction of local magnetic field, the proton pressure can be cast in the form p p = p ⊥ p n + p k p τ + Π , where τ = ˆ b ⊗ ˆ b , and n = I − ˆ b ⊗ ˆ b , with I being the unit tensor. FiniteLarmor radius corrections to the gyrotropic pressures are represented by Π . Operator ⊗ represents the usual tensor product and in the index notation, for example, τ ij = ˆ b i ˆ b j .Electrons are assumed to be isothermal with the scalar pressure p e = nT (0) e , where T (0) e isthe electron temperature. Parameter R i , whose inverse multiplies the Hall-term and alsothe FLR corrections, is defined as R i = L/d i , where d i is the ion inertial length and L isthe unit length. The proton plasma beta is defined with respect to parallel pressure and β k = 8 πp (0) k p /B . The density, momentum and induction equations of the FLR-Landau fluidmodel can then be expressed as ∂ρ∂t + ∇ · ( ρ u p ) = 0 , (1) ∂ u p ∂t + u p · ∇ u p + β k ρ ∇ · ( p ⊥ p n + p k p τ + Π + p e I ) − ρ ( ∇ × b ) × b = 0 , (2) ∂ b ∂t = ∇ × ( u p × b ) − R i ∇ × (cid:20) ρ ( ∇ × b ) × b (cid:21) . (3)Dropping, for simplicity, indices p for proton velocity u p and proton pressures p ⊥ p , p k p , theevolution equations for perpendicular and parallel pressures reads (neglecting the work doneby the FLR stress forces) ∂p ⊥ ∂t + ∇ · ( p ⊥ u ) + p ⊥ ∇ · u − p ⊥ ˆ b · ( ∇ u ) · ˆ b + ∇ · ( q ⊥ ˆ b ) + q ⊥ ∇ · ˆ b = 0 , (4) ∂p k ∂t + ∇ · ( p k u ) + 2 p k ˆ b · ( ∇ u ) · ˆ b + ∇ · ( q k ˆ b ) − q ⊥ ∇ · ˆ b = 0 . (5)Assuming an ambient magnetic field of amplitude B in the positive z-direction, a semi-linear description of the finite Larmor radius corrections in the pressure tensor neglecting6eat flux contributions can be expressed asΠ xx = − Π yy = − h p ⊥ i R i ( ∂ y u x + ∂ x u y ) , (6)Π xy = Π yx = − h p ⊥ i R i ( ∂ y u y − ∂ x u x ) , (7)Π yz = Π zy = 1 R i (cid:2) h p k i ∂ z u x + h p ⊥ i ( ∂ x u z − ∂ z u x ) (cid:3) , (8)Π xz = Π zx = − R i (cid:2) h p k i ∂ z u y + h p ⊥ i ( ∂ y u z − ∂ z u y ) (cid:3) , (9)Π zz = 0 , (10)where h p ⊥ i and h p k i represents the instantaneous averaged ion pressures over the entiredomain, whose time variation is aimed to take into account the evolution of the global prop-erties of the plasma. Finally, parallel and perpendicular heat fluxes q k , q ⊥ evolve accordingto ddt + p πβ k − π ) H ∂ z ! q k = 11 − π β k ∂ z ( p k − ρ ) , (11) ddt − p πβ k H ∂ z ! q ⊥ = β k T (0) ⊥ p T (0) k p ∂ z " − T (0) ⊥ p T (0) k p ! | b | − T (0) k p T (0) ⊥ p p ⊥ − ρ ! , (12)where T (0) ⊥ p , T (0) k p are the initial perpendicular and parallel proton temperatures and d/dt isthe convective derivative. The operator H , which is defined as H f ( z ) = − π V P Z + ∞−∞ f ( z ′ ) z − z ′ dz ′ , (13)reduces in the Fourier space to a simple multiplication by ik z / | k z | and, is the signature ofthe linear Landau damping.To gain physical insight into a quite complicated model (1)-(12), it is useful to momentar-ily consider just the largest scales by putting 1 /R i →
0, which eliminates the nongyrotropiccontributions to the pressure tensor and which also eliminates the Hall term. The result-ing set of equations still contains the linear Landau damping and with the exception ofisothermal electrons, it is analogous to the model of Snyder, Hammett and Dorland [71]. Ifthis model is further simplified by elimination of eq. (11), (12) and by instead prescribing q k = q ⊥ = 0, the resulting model collapses to the double adiabatic model (Chew et al. [84],7ee also Kulsrud [85]) and the Landau damping disappears. The presence of ion Landaudamping in the system (1)-(12) is therefore a result of closure equations (11), (12) for theheat fluxes, which contain the operator H . At least in the static limit, this closure can beviewed as a modified Fick’s law where the gradient operator that usually relate the heat fluxto the temperature fluctuations is here replaced by a Hilbert transform that is a reminis-cence of the plasma dispersion function arising in the linear kinetic theory, and is a signatureof Landau (zero-frequency) wave-particle resonance. The effect of Landau damping in thesystem (1)-(12) might be better understood by solving the linearized set of equations. Inthe Appendix we consider waves which propagate parallel to the ambient magnetic field.It is shown that except the Alfv´en waves, linear waves have a frequency with a negativeimaginary part, and are therefore damped. This corresponds to linear Landau damping.The model used for the simulations presented here has several limitations. First of all, itcontains electrons which are assumed isothermal, a regime in fact often assumed in hybridsimulations which provide a kinetic description of the ions and a fluid description of theelectrons. In the future, more realistic simulations will be performed with inclusion ofindependent evolution equations for parallel and perpendicular electron pressures and heatfluxes. This will also result in the presence of electron Landau damping, which is absent inthe model presented here and which seems to play an important role in solar wind turbulence.Another main limitation appears to be the form of finite Larmor radius corrections, whichare derived as a large-scale limit of FLR corrections of the full Landau fluid model andwhich are therefore significantly simplified. The FLR corrections (6)-(10) are sufficient forthe simulations of freely-decaying turbulence, which do not lead to significant temperatureanisotropies. However, our preliminary simulations which employ forcing and lead to strongtemperature anisotropies show that the FLR corrections (6)-(10) are overly simplified andmight lead to artificial numerical instabilities. For simulations with strong temperatureanisotropies, a more refined description of FLR corrections is required (see Passot and Sulem[72], Borgogno et al. [81]).To explore the behavior of the FLR-Landau fluid model, we performed simulations offreely decaying turbulence. The code we used is based on a pseudo-spectral discretizationmethod, where spatial derivatives are evaluated in Fourier space. The time stepping isperformed in real space with a 3rd order Runge-Kutta scheme. Spatial resolution is N =128 and the size of the simulation domain is L = 16 × (2 π ) in each direction. The Hall8arameter is R i = 1, implying that the lengths are measured in the units of ion inertiallength d i . Velocity and magnetic field fluctuations of mean square root amplitude h u i / = h b i / = 1 / kd i = m/
16, where m ∈ [1 , p ⊥ = p k = 1, density ρ = 1 and heat fluxes q ⊥ = q k = 0 are initializedin the entire domain. The temperature of the electrons is T (0) e = 1 and is therefore equalto proton temperatures, which can be defined as T ⊥ = h p ⊥ /ρ i and T k = h p k /ρ i . In thesimulations presented here, both T ⊥ and T k stay rather close to their initial value. Thecompressible Hall-MHD model consists of eq. (1)-(3), where the divergence of the pressuretensor in eq. (2) is substituted with the usual gradient of scalar pressure and, assumingthe adiabatic law p = ρ γ , in the normalized units it is equal to β /γ ∇ ρ γ , where β is theusual plasma beta defined as β = c s /V A and the sound speed c s = γp/ρ . Adiabatic index γ = 1 .
66 is used.As shown for example by Hirose et al. [86] and Howes [62], it is not straightforward tocompare Hall-MHD and Vlasov-Maxwell kinetic theory because the associated dispersionrelations are quite different if in the kinetic description the proton (ion) temperature T p is not negligible with respect to the electron temperature T e . We choose to follow Howes[62], who, in order to compare Hall-MHD with the kinetic theory (which is representedhere by the FLR-Landau fluid model), defined the necessary relation between β and β k as β = β k (1 + T p /T e ) / (2 T p /T e ). For equal proton and electron temperatures T p = T e thisrelation yields β = β k . Landau damping alone is not sufficient to run the code and somekind of artificial dissipation is needed to terminate the cascade. We here resorted to use afiltering in the form of { − tanh[( m − . N/ / } , where m is the mode index, appliedeach time step on all fields for both Hall-MHD and Landau fluid regimes. Because of thefiltering, it is crucial to have identical time steps dt = 0 .
128 in both models. In most ofthe simulations we used β = β k = 0 .
8. In normalized units, the Alfv´en speed is equal tounity, the usual MHD sound speed c s = √ β = 0 .
894 and the turbulent sonic Mach number M s = h u i / /c s = 0 .
14. In section IV., we also consider simulations with β = β k = 0 . M s = 0 .
25) and β = β k = 0 . M s = 0 . -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) HMHDb x , θ =0 ° -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) Landaub x , θ =0 ° FIG. 1: Left and right polarized Alfv´en waves for the propagation angle of 0 ◦ for Hall-MHD(left) and Landau fluid (right), as revealed by frequency analysis of b x modes with wavenumbers k x = 0 , k y = 0 , k z d i = m/
16, where m = 1 (red), m = 2 (green), m = 4 (blue), m = 8 (black).Theoretical predictions for peaks calculated from dispersion relations for given k are shown on thetop axis. III. IDENTIFICATION OF THE MHD MODES
A wave analysis procedure was implemented in the code, which consists in choosing fewmodes in spatial Fourier space for each field and recording their value every 20 time steps.After the run, the time Fourier transform is performed and frequency-power spectra areobtained for each mode. This procedure makes it possible to identify at which frequencythere is maximum power for each mode, and by comparing these with frequencies obtainedfrom theoretical dispersion relations it allows to uniquely identify which waves are presentin the system. This method was previously used for incompressible MHD by Dmitruk andMatthaeus [87], therefore detecting Alfv´en waves. Considering the dynamics associatedwith waves propagating in the direction parallel to the ambient magnetic field (propagationangle θ = 0 ◦ ), Fig. 1 shows frequency-power spectra of four modes with wavenumbers k x = 0 , k y = 0 , k z d i = m/
16, with m = 1 , , ,
8, recorded from the component b x for Hall-MHD (left) and for FLR-Landau fluid (right). For Hall-MHD, these correspond to polarizedAlfv´en waves which obey the dispersion relation ω = ± k / (2 R i ) + k p k/ R i ) . Thetheoretical frequency values which are expected from this dispersion relation for given k are10lotted on the top axis of the figure as black vertical lines. Figure 1 shows that the resolutionof 128 is sufficient to clearly distinguish between left and right polarized Alfv´en waves. Incontrast with Hall-MHD, for which it is possible to write the general dispersion relationfor frequency ω as a relatively simple polynomial of 6th order, for FLR-Landau fluids thegeneral dispersion relation would be uneconomically large to write down. In general, it isnecessary to numerically solve the determinant obtained from linearized equations (1)-(12)for a given wavenumber k after assuming linear waves. FLR-Landau fluid (1)-(12) consistsof 11 evolution equations in 11 variables and, together with the divergence free constraintfor the magnetic field, therefore yields general dispersion relation for frequency ω in theform of a complex polynomial of 10th order. This represents 5 forward and 5 backwardpropagating waves, with some solutions having highly negative imaginary part and whichare therefore strongly damped. A similar situation is encountered in the Vlasov-Maxwellkinetic theory which essentially yields an infinite number of strongly damped solutions. Forthe propagation angle θ = 0 ◦ and additional constraint T ⊥ = T k = 1, it is however possibleto obtain an analytic solution for the circularly polarized Alfv´en waves as ω = ± k R i (cid:18) β k (cid:19) + k s (cid:18) k R i (cid:19) (cid:18) − β k (cid:19) , (14)with two other solutions obtained by substituting ω with − ω . The dispersion relation isquite similar to that of Hall-MHD with additional terms proportional to β k and resultingfrom the finite Larmor radius corrections. Expected theoretical frequencies obtained fromthis analytic solution are plotted on the top axis of Fig. 1 for the Landau fluid regime(right). They again match quite precisely. Note also the moderately strong damping ofparallel Alfv´en waves in Landau fluid regime, which can be seen for the last mode m = 8.Landau damping does not act directly on linear constant amplitude Alfv´en waves obeyingrelation (14), which are exact solutions of linearized FLR-Landau fluid model. However,nonlinear parallel Alfv´en waves, which are of course present in the full model, cause produc-tion of density (sound) fluctuations. Sound waves in the FLR-Landau fluid model, as wellas in the kinetic theory, are heavily damped by Landau damping as shown below and thisprocess therefore also results in damping of Alfv´en waves. Mjølhus and Wyller [88] stud-ied the kinetic derivative nonlinear Schr¨odinger equation (KDLNS) for parallel propagatinglong-wavelength Alfv´en waves where they refer to this effect as nonlinear Landau damping,because it is acting on Alfv´en waves in a nonlinear way.11 -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) HMHDu z , θ =0 ° S A -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) SA Landauu z , θ =0 ° FIG. 2: Sound waves for the propagation angle of 0 ◦ for Hall-MHD (left) and Landau fluid (right),from frequency analysis of u z modes with the same wavenumbers as in Fig. 1. Sound waves (S)are heavily damped for the Landau fluid, which is consistent with the kinetic theory. The spectraalso show weak presence of Alfv´en waves (A), which are visible for the first two modes. Further exploring propagation angle of θ = 0 ◦ , Fig. 2 shows frequency power spectraobtained from component u z and which should therefore predominantly display sound waves(almost identical spectra can be obtained from component ρ ). The same modes with k z d i = m/
16, where m = 1 , , , ω = kc s , where the sound speed c s = √ β . Sound waves (S)are clearly presented in Fig. 2 for Hall-MHD (left) with quite sharp peaks which matchthe theoretical dispersion values shown on the top axis. Weak presence of polarized Alfv´enwaves (A) for modes m = 1 , ω = k q (3 + T (0) e ) β k / T i = T e and β = β k = 1, noted that the kinetic solution corresponding to the slow wave has a higherphase speed than the kinetic solution corresponding to the Alfv´en wave.Considering the propagation angle of θ = 45 ◦ , Fig. 3 shows frequency-power spectra incomponent b x for wavenumbers k y = 0 , k x d i = k z d i = m/
16, where m = 1 , , , b y andcorresponding spectra are shown in Fig. 4.Finally, considering purely perpendicular propagation with θ = 90 ◦ , Fig. 5 showsfrequency-power spectra obtained from the density field ρ and wavenumbers k y = 0 , k z =0 , k x d i = m/
16, where m = 1 , , ,
8. For Hall-MHD regime (Fig. 5 left) the spectral peakscorrespond to magnetosonic waves with the usual dispersion relation ω = k √ β . Inthe Landau fluid regime, linearized set of equations (1)-(12) with the additional constraint T ⊥ = T k = 1 can be shown to yield the dispersion relation for perpendicular magnetosonicwaves in the form ω = k vuut β k T (0) e ! + (cid:18) β k k R i (cid:19) . (15)The dispersion relation clearly shows the effect of inclusion of isothermal electrons (forsimulations presented here T (0) e = 1) and also the effect of finite Larmor radius corrections13 -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) HMHDb x , θ =45 ° S A F -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) Landaub x , θ =45 ° SA F
FIG. 3: Slow (S) and fast (F) magnetosonic waves for the propagation angle of 45 ◦ for Hall-MHD (left) and Landau fluid (right), from frequency analysis of b x modes with wavenumbers k y = 0 , k x d i = k z d i = m/
16, where m = 1 (red), m = 2 (green), m = 4 (blue), m = 6 (black). Thepresence of Alfv´en waves (A) is also visible. Theoretical predictions for slow and fast waves areshown on the top axis. which are represented by the last quadratic term. Theoretical predictions from Hall-MHDand FLR-Landau fluid dispersion relations are again shown on the top axis. To clearly showthe shift of the peaks between the two regimes, we also added the theoretical Hall-MHDfrequencies to the top axis of FLR-Landau fluid regime (Fig. 5 right) and represent themwith the small magenta lines. IV. FLOW COMPRESSIBILITY
Compressibility of the flow can be evaluated by decomposing the velocity field into itssolenoidal and non-solenoidal components and by calculating the associated energies accord-ing to X k | u k | = X k | k × u k | | k | + X k | k · u k | | k | , (16)where the left-hand side corresponds to the total energy E U in velocity field, the first term inthe right-hand side corresponds to the energy E in in the solenoidal component and the secondterm E c originates from the compressible one. Relation (16) can be therefore expressed as E U = E in + E c and the compressibility of the flow can be evaluated as a ratio of compressible14 -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) HMHDb y , θ =45 ° -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) Landaub y , θ =45 ° FIG. 4: Alfv´en waves for the propagation angle of 45 ◦ for Hall-MHD (left) and Landau fluid(right), from frequency analysis of b y modes with the same wavenumbers as in Fig. 3. Theoreticalpredictions for the Alfv´en waves frequencies are shown on the top axis. and total energy E c /E U . Time evolution of E c /E U for Hall-MHD and FLR-Landau fluidregime with β = β k = 0 . E c /E U which represents compressibility is significantly lower in Landau fluid regime and is thereforea result of presence of Landau damping. The question then arises of the influence of thesonic Mach number in the compressibility evolution. The time evolution of E c /E U for β = β k = 0 .
25 (which corresponds to M s = 0 .
25) is displayed in Fig. 7 (left), whereasthe time evolution of E c /E U for β = β k = 0 . M s = 0 .
40) is shownin Fig. 7 (right). The simulations were performed with the same time step dt = 0 . β = β k = 0 . M s = 0 . β = β k = 0 .
25, this reduction is almost insignificantfor simulations with β = β k = 0 .
1. This is an expected effect, as the strength of the Landaudamping is proportional to β k . We note that the turbulent sonic Mach number in thesolar wind is typically small and, for example, analysis of observational data performed byBavassano and Bruno [8] (from 0.3-1.0 AU) showed that the most probable value is between M s = 0 . − . E tot can be evaluated as the sum of the15 -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) HMHD ρ , θ =90 ° -16 -14 -12 -10 -8 -6 -2 -1 po w e r frequency ( ω ) Landau ρ , θ =90 ° FIG. 5: Magnetosonic waves for the propagation angle of 90 ◦ for Hall-MHD (left) and Landau fluid(right), from frequency analysis of density modes with wavenumbers k y = 0 , k z = 0 , k x d i = m/ m = 1 (red), m = 2 (green), m = 4 (blue), m = 8 (black). Theoretical predictions fromdispersion relations are shown on the top axis (long black lines). For comparison, on the rightpanel we also included the frequency predictions from Hall-MHD (small magenta lines). c o m p r e ss i b ili t y time HMHDLandau β = β || =0.8 FIG. 6: Compressibility for Hall-MHD (red line) and FLR-Landau fluid (blue line) evaluated as( P k | k · u k | / | k | ) / P k | u k | for β = β k = 0 .
8. Both regimes start with the identical initialcondition where the velocity field is divergence free. The figure shows that the compressibility isclearly inhibited in the Landau fluid simulation. c o m p r e ss i b ili t y time HMHDLandau β = β || =0.25 c o m p r e ss i b ili t y time HMHDLandau β = β || =0.1 FIG. 7: Compressibility for Hall-MHD (red line) and FLR-Landau fluid (blue line) evaluated as( P k | k · u k | / | k | ) / P k | u k | with β = β k = 0 .
25 (left) and with β = β k = 0 . β = β k = 0 .
25, whereas for simulations with β = β k = 0 . kinetic energy E kin , the magnetic energy E mag and the internal energy E int . In Hall-MHDand FLR-Landau fluid model, the definitions of kinetic and magnetic energy are identicaland equal to E kin = 12 Z ρ | u | dx , E mag = 12 Z | b | dx . (17)However, the definition of internal energy is naturally different in each model. In the Hall-MHD model, the internal energy is defined asHMHD: E int = β o γ ( γ − Z ρ γ dx , (18)whereas in the FLR-Landau fluid model with isothermal electrons, the internal energy isgiven by Landau: E int = β k Z (cid:16) p ⊥ + p k T (0) e ρ ln ρ (cid:17) dx . (19)It is emphasized that because of the filtering, the total energy E tot is not exactly preservedin the Hall-MHD and Landau fluid simulations. During the simulations with β = β k = 0 . E k i n + E m ag time HMHDLandau normalized energy E kin + E mag E i n t time HMHDLandaunormalized internal energy E int
FIG. 8: Normalized mechanical fluctuations energy E kin + E mag (left) and normalized internalenergy E int (right) for Hall-MHD (red line) and FLR-Landau fluid (blue line), in the case β = β k = 0 .
8. Landau damping transfers energy from E kin + E mag into E int . Hall-MHD simulations do not have heating at all and the internal energy could increaseonly through development of density fluctuations. Time evolution of the sum of kinetic andmagnetic energies (normalized to their initial values) is displayed in Fig. 8 (left), where theenergy contained in the ambient magnetic field was subtracted. This figure shows that thissum decays faster for FLR-Landau fluid simulation. Time evolution of internal energy E int is shown in Fig. 8 (right), where energies were again normalized to their initial value. Theinitial jump observed in both simulations reflects a rapid adjustment from initial conditionsthat are not close to an equilibrium state. Later on, the internal energy of Hall-MHDsimulation decreases, whereas the internal energy of FLR-Landau fluid simulation more orless smoothly increases until around time t = 1500. During this time (which correspondsto over 10 time steps) the Landau damping acts strongly and, by mainly damping slowwaves, converts the mechanical energy E kin + E mag into the internal energy E int , whichrepresents heating of the plasma. The question also arises what fraction of mechanical energyis dissipated directly by Landau damping and what fraction is dissipated by the filteringprocess. Unfortunately, we are unaware of any technique how to address this question.We note that for simulations of freely decaying turbulence the heating is quite weak, im-plying that driving the system is necessary to produce significant temperature anisotropies.However, the absence of forcing, which yields only a small amount of heating, makes it easier18o precisely identify various waves in the system as was presented in Sec. III. V. ANISOTROPY OF THE ENERGY TRANSFER
The presence of Landau damping can also be seen in the usual wavenumber velocity andmagnetic field spectra. Considering first simulations with β = β k = 0 .
8, Fig. 9 shows thevelocity spectra with respect to perpendicular and parallel wavenumbers k ⊥ , k k which aredefined as E U = R E u ( k ⊥ ) dk ⊥ = R E u ( k k ) dk k . With respect to k ⊥ , the spectra for Hall-MHD and FLR-Landau fluid are almost identical (Fig. 9 left) whereas with respect to k k (Fig. 9 right), the spectra of Landau fluid are much steeper. Landau damping thereforesignificantly inhibits the parallel transfer. Even though low resolution does not allow toprecisely identify the slopes of the spectra, three straight lines were added to figures andcorrespond to power law solutions k s , where s = − / , − / − /
3. For E u ( k ⊥ ), theclosest spectral index value appears to be − /
3, the spectral range being however quitelimited. The same conclusion with the inhibition of parallel transfer is also obtained forthe magnetic field spectra, which are almost identical to the velocity spectra and are shownin Fig. 10. In contrast, for simulations with β = β k = 0 . E ( k k )displays a similar behavior for Hall-MHD and Landau fluids. The velocity spectra are shownin Fig. 11 and the magnetic field spectra are shown in Fig. 12. This is consistent with resultspresented in the previous section where it was shown that the Landau damping is responsiblefor significant reduction of compressibility for simulations with β = β k = 0 .
8, whereas forsimulations with β = β k = 0 .
1, the Landau damping was much weaker and the reductionof compressibility almost negligible.It is useful to compare our compressible simulations to incompressible simulations. Nat-urally, our compressible Hall-MHD code cannot be run in an incompressible regime, forwhich the turbulent sonic Mach number M s → c s → ∞ . Neverthe-less, incompressible MHD simulations of decaying turbulence were performed, for example,by Bigot et al. [89]. These simulations showed that the combined velocity and magneticfield spectra (they used Els¨asser variable z + ) are much steeper with respect to k k than withrespect to k ⊥ , if the ambient magnetic field is sufficiently strong. Our compressible sim-ulations presented here have initially h b i / /B = 1 / -8 -7 -6 -5 -4 -3 -1 E u ( k ⊥ ) k ⊥ d i E u (k ⊥ ) β = β || =0.8 -8 -7 -6 -5 -4 -3 -1 E u ( k || ) k || d i E u (k || ) β = β || =0.8 FIG. 9: Velocity spectra for Hall-MHD (red) and Landau fluid (blue) with respect to perpendicularwavenumber E u ( k ⊥ ) (left) and with respect to parallel wavenumber E u ( k k ) (right), for β = β k =0 .
8. Spectra were taken at time t = 5248. Straight lines correspond to k − / , k − / and k − / . Thefigure shows that spectra with respect to k k are much steeper in Landau fluid simulation, which isa result of Landau damping. is 1 / /
15, respectively. Considering compressible Hall-MHD, our simulations showedthat spectra with respect to k k (e.g. Fig. 9 right, red line) are steeper than spectra withrespect to k ⊥ (Fig. 9 left, red line). However, these parallel spectra are nowhere near assteep as the parallel spectra of Bigot et al. [89]. Interestingly, their parallel spectra moreresemble our parallel spectra for Landau fluid model (Fig. 9 right, blue line), where the Lan-dau damping was strong ( β k = 0 . VI. CONCLUSION
We have presented the first three-dimensional fluid simulations of decaying turbulencein a collisionless plasma in conditions close to the solar wind. For this purpose, we usedthe FLR-Landau fluid model that extends compressible Hall-MHD by incorporating low-frequency kinetic effects such as Landau damping and finite Larmor radius corrections. It20 -8 -7 -6 -5 -4 -3 -1 E b ( k ⊥ ) k ⊥ d i E b (k ⊥ ) β = β || =0.8 -8 -7 -6 -5 -4 -3 -1 E b ( k || ) k || d i E b (k || ) β = β || =0.8 FIG. 10: Magnetic field spectra for Hall-MHD (red) and Landau fluid (blue), for E b ( k ⊥ ) (left) and E b ( k k ) (right) when β = β k = 0 .
8. Spectra were taken at time t = 5248. -8 -7 -6 -5 -4 -3 -1 E u ( k ⊥ ) k ⊥ d i E u (k ⊥ ) β = β || =0.1 -8 -7 -6 -5 -4 -3 -1 E u ( k || ) k || d i E u (k || ) β = β || =0.1 FIG. 11: Velocity spectra for Hall-MHD (red) and Landau fluid (blue) with respect to perpendicularwavenumber E u ( k ⊥ ) (left) and with respect to parallel wavenumber E u ( k k ) (right) for β = β k =0 .
1. Spectra were taken at time t = 5248. Straight lines correspond to k − / , k − / and k − / . was shown that in spite of the turbulent regime, it is possible to precisely identify linear wavespresent in the system. Comparisons between compressible Hall-MHD and FLR-Landaufluid model showed that when beta is not too small, linear Landau damping yields strongdamping of slow magnetosonic waves in Landau fluid simulations. These waves are indeeddamped in kinetic theory described by the Vlasov-Maxwell equations but not in compressible21 -8 -7 -6 -5 -4 -3 -1 E b ( k ⊥ ) k ⊥ d i E b (k ⊥ ) β = β || =0.1 -8 -7 -6 -5 -4 -3 -1 E b ( k || ) k || d i E b (k || ) β = β || =0.1 FIG. 12: Magnetic field spectra for Hall-MHD (red) and Landau fluid (blue), for E b ( k ⊥ ) (left) and E b ( k k ) (right) when β = β k = 0 .
1. Spectra were taken at time t = 5248. Spectra for Hall-MHDand Landau fluid model are again almost identical. MHD and Hall-MHD descriptions, which overestimate compressibility and parallel transferin modeling weakly collisional plasmas. The FLR-Landau fluid model can therefore be usefulfor simulating the solar wind, which is typically found to be only weakly compressible.
Acknowledgements
The support of INSU-CNRS “Programme Soleil-Terre” is acknowledged. Computationswere performed on the Mesocentre SIGAMM machine hosted by the Observatoire de laCˆote d’Azur (OCA) and on the JADE cluster of the CINES computational facilities. PHwas supported by an OCA Poincar´e fellowship. The work of DB was supported by theEuropean Community under the contract of Association between EURATOM and ENEA.The views and opinions expressed herein do not necessarily reflect those of the EuropeanCommission.
APPENDIX
To clearly understand how the Landau damping acts in the present system (1)-(12), it isuseful to solve dispersion relations for linear waves propagating in parallel direction to the22mbient magnetic field. A detailed analysis of linear waves for various propagation angleswas elaborated by Passot and Sulem [90]. To simplify the analytic expressions, we define theproton temperature anisotropy as T (0) ⊥ p /T (0) k p ≡ a p , and the normalized electron temperatureas T (0) e /T (0) k p ≡ τ . It can be shown that for parallel propagation angle, the Landau fluidmodel contains four dispersive Alfv´en waves. Two waves obey the dispersion relation whichcan be expressed as ω = ± k R i h β k (cid:16) − a p (cid:17)i + k s β k a p −
1) + (cid:18) k R i (cid:19) h − β k (cid:16) − a p (cid:17)i , (20)with another two solutions obtained by substituting ω with − ω . Obviously, these Alfv´enwaves are independent of the electron temperature τ , which is a consequence of electronsbeing modeled as isothermal. For a more general Landau fluid model which contains evo-lution equation for electron pressures and heat fluxes, the electron temperature τ entersthe dispersion relation for Alfv´en waves. The solutions (20) can become imaginary, if theexpression under the square root becomes negative. At large scales (when 1 /R i →
0) thecondition 1 + β k ( a p − / < a p = 1), the solution (20) naturally col-lapses to the solution (14). The four Alfv´en waves (20) can be eliminated from the generaldispersion relation and this yields a complex polynomial of 6th order in frequency ω . Solu-tions of this polynomial represent 3 forward and 3 backward propagating waves which havea negative imaginary part and are therefore damped. Importantly, it is possible to eliminatethe dependence on β k and wavenumber k and, after applying a substitution Ω = ω/ ( k p β k ),the polynomial of 6th order can be simplified toΩ + Ω i √ π − π −
16 + 6 π + Ω − π (14 + 3 τ ) + 24 + 8 τ −
16 + 6 π + Ω i √ π τ − π (9 + 3 τ ) / − π +Ω π (15 + 5 τ ) / − − π + Ω i √ π (5 + 3 τ ) / − π − τ − π = 0 . (21)Because this polynomial in Ω does not depend on β k or k , the substitution implies thatall 6 waves are linear with k and p β k . The polynomial (21) has to be solved numeri-cally for a given value of τ . The simulations presented here use τ = 1 and numericallysolving polynomial (21) yields Ω = ± . − i . ± . − i . ± . − i . ω = k q β k ( ± . − i . , (22) ω = k q β k ( ± . − i . , (23) ω = k q β k ( ± . − i . . (24)The least damped solution (22) represents the sound wave. The solutions of Landau fluidmodel (1)-(12) for parallel propagation angle are therefore 4 Alfv´en waves (20), 2 soundwaves (22) and 4 waves (23), (24), which are highly damped. These 4 waves (23), (24)do not have an analogy in Hall-MHD description and correspond to solutions of kineticMaxwell-Vlasov description, which contains an infinite number of highly damped solutions.Interestingly, the last solution (24) is not dependent on the value of τ and it can be expressedanalytically as Ω = ±√ − π/ − i √ π/
4. After eliminating these waves from eq. (21), thepolynomial which contains the sound waves (22) and solutions (23) is now of 4th order in Ωand expressed asΩ + i √ π − π Ω −
12 9 π −
16 + ( − π ) τ − π Ω − i √ π (3 + τ ) − π Ω + 2(1 + τ ) − π = 0 . (25)It is of course possible to use Ferrari-Cardano’s relations to solve this polynomial analytically,the final result is however too complicated and it is still more convenient to solve (25)numerically for a given value of τ .If this wave analysis is repeated with the model (1)-(10) with heat flux equations q k = 0, q ⊥ = 0, the same dispersion relation (20) for Alfv´en waves is obtained. However, the onlyother solutions present in the parallel direction are ω = ± k r β k τ ) . (26)These waves have frequencies which are purely real and correspond to undamped soundwaves of double adiabatic model with isothermal electrons.For completeness, considering perpendicular propagation, it can be shown that the heatfluxes vanish and the solutions of the Landau fluid model with isothermal electrons areundamped magnetosonic waves with the dispersion relation expressed as ω = ± k s β k (cid:16) a p + τ (cid:17) + (cid:18) a p β k k R i (cid:19) . (27)24
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