Multiscale nature of the dissipation range in gyrokinetic simulations of Alfvénic turbulence
D. Told, F. Jenko, J. M. TenBarge, G. G. Howes, G. W. Hammett
MMultiscale nature of the dissipation range in gyrokinetic simulations of Alfvénicturbulence
D. Told ∗ and F. Jenko Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA andMax-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, 85748 Garching, Germany
J.M. TenBarge
IREAP, University of Maryland, College Park, MD 20742, USA
G.G. Howes
Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA
G.W. Hammett
Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA
Nonlinear energy transfer and dissipation in Alfvén wave turbulence are analyzed in the firstgyrokinetic simulation spanning all scales from the tail of the MHD range to the electron gyroradiusscale. For typical solar wind parameters at 1 AU, about 30% of the nonlinear energy transfer closeto the electron gyroradius scale is mediated by modes in the tail of the MHD cascade. Collisionaldissipation occurs across the entire kinetic range k ⊥ ρ i (cid:38) . Both mechanisms thus act on multiplecoupled scales, which have to be retained for a comprehensive picture of the dissipation range inAlfvénic turbulence. Introduction.
Spacecraft measurements find a radialtemperature profile of the solar wind which can only beexplained by the presence of heating throughout the he-liosphere [1]. The key mechanism of heating in the in-ner heliosphere up to ∼
20 AU is thought to be the dis-sipation of turbulent fluctuation energy, and its under-standing and description is one of the outstanding openissues in space physics [2]. Over the past decade, nu-merous studies, both observational [3–7] and theoreti-cal/computational [8–14], have focused on this topic, ex-tracting ever more sophisticated measurements of solarwind fluctuation properties, and accomplishing increas-ingly detailed turbulence simulations.As the solar wind plasma is only weakly collisional,a variety of kinetic effects such as cyclotron damping,Landau and transit time damping, finite Larmor radiuseffects, stochastic heating, or particle acceleration at re-connection sites can contribute to the conversion of fieldenergy to particle energy, and thus determine how col-lisional dissipation will ultimately set in. A kinetic de-scription is crucial in order to judge the relative impor-tance of each of those effects. Due to the complexity ofa nonlinear kinetic system, numerical simulations are es-sential to interpret observations and provide guidance foranalytical theory.In the present Letter, we employ an approach basedon gyrokinetic (GK) theory [15], which is a rigorouslimit of kinetic theory in strongly magnetized plasmas.Due to the assumptions of low frequencies (compared tothe ion cyclotron frequency) and small fluctuation levels,the gyrokinetic model excludes cyclotron resonances andstochastic heating. In absence of these effects, we focuson the energetic properties of kinetic Alfvén wave (KAW) turbulence, which has been demonstrated to be a crucialingredient of solar wind turbulence [16].We address the following key questions: (1) Whichspectral features can be found in a comprehensive simu-lation extending from the magnetohydrodynamic (MHD)range down to the electron gyroradius scale? (2) Whatare the characteristics of nonlinear energy transfer fromlarge to small scales? (3) How is energy dissipated, andhow is the dissipated energy partitioned between ions andelectrons?
Simulation setup.
The nonlinear GK system of equa-tions is solved using the Eulerian code
GENE [17] tostudy the dynamics of KAW turbulence in three spa-tial dimensions. In order to model the energy injectionat the outer scales of the system, a magnetic antennapotential, whose amplitude is evolved in time accordingto a Langevin equation [18], is externally prescribed atthe largest scales of the simulation domain. The drivenmodes are (0 , , ± and (1 , , ± , where ( i, j, k ) aremultiples of the lowest wave numbers in ( k x , k y , k z ) , re-spectively. The mean antenna frequency is chosen to be ω a = 0 . ω A ( ω A being the frequency of the slowestAlfvén wave in the system), the decorrelation rate is setto γ a = 0 . ω A , and the normalized antenna amplitudeis set to A (cid:107) , = ω A B √ δ/C k ⊥ √ N v A (setting δ = 2 ,N = 4 , C = 1 ), in accordance with the critical balancecondition ω lin ∼ ω nl [18].The physical parameters are chosen to be similar tosolar wind conditions at 1 AU, with β i = 8 πn i T i /B = 1 , T i /T e = 1 . Proton and electron species are includedwith their real mass ratio of m i /m e = 1836 . The elec-tron collisionality is chosen to be ν e = 0 . ω A (with ν i = (cid:112) m e /m i ν e ), a value small enough to not inhibit a r X i v : . [ phy s i c s . p l a s m - ph ] J u l kinetic effects, but large enough to reduce resolution re-quirements in velocity space.In order to maximize the effective dynamic range, thesimulation domain is extended significantly compared topreviously published work, to include scales larger thanthe ion gyroradius, allowing for a free distribution of en-ergy into the KAW or the ion entropy cascade [19] as theion gyroradius scale is passed. The evolution of the gyro-center distribution is tracked on a grid with the resolution (cid:0) n x , n y , n z , n v (cid:107) , n µ , n σ (cid:1) = (512 , , , , , . Theplane perpendicular to the background magnetic field isresolved by fully dealiased grid points, covering aperpendicular wavenumber range . ≤ k ⊥ ρ i ≤ . (or . ≤ k ⊥ ρ e ≤ . ), thus extending into the regimewhere electron finite-Larmor-radius effects become im-portant. Here, ρ σ = √ T σ m σ c/eB with the species index σ . The number of grid points in the perpendicular planeis thus increased by a factor of 36 with respect to thelargest runs of this kind published to date [9]. 96 pointsare used to resolve the dynamics along the backgroundfield (the z direction), and × gridpoints are chosento represent the (cid:0) v (cid:107) , µ (cid:1) domain, where v (cid:107) is the velocityalong the guide field, and µ = mv ⊥ / B is the magneticmoment with respect to the guide field. The domain sizesin velocity space are chosen to extend up to 3 thermal ve-locities v T σ in both parallel and perpendicular velocitiesfor each species σ , where v T σ = (cid:112) T σ /m σ .Our simulations are performed using the same iterativeexpansion scheme as in Ref. [9], where simulations areinitially run with low resolution and are then restartedseveral times with an increasingly fine grid, until the tar-get resolution is reached. The total runtime is chosen tospan several antenna oscillation periods τ A (in this case t end = 7 . τ A ) in order to ensure that a quasi-steadystate has been reached. Diagnostic methods . The key results of this study areobtained using a set of sophisticated energy diagnostics(partially introduced in Refs. [20–23]), which enable stud-ies of energy source, transfer and dissipation spectra sep-arately for each species, and which are applied to KAWturbulence for the first time here. In particular, we an-alyze the time derivative of the spatially averaged freeenergy density, which can be expressed in the case of anantenna-driven electromagnetic system as ∂ t E = (cid:60) (cid:88) σ (cid:88) k (cid:28) πB m σ ˆ d µ d v (cid:107) (cid:18) h σ k T σ F σ + q σ v (cid:107) c C A (cid:107) ant ,σ k (cid:19) ∗ ∂ t g σ k (cid:29) + (cid:60) (cid:88) k (cid:28) k ⊥ π A ∗ (cid:107) tot , k ∂ t A (cid:107) ant , k (cid:29) . (1)Here, the sum over k denotes a summation over allwavenumber pairs ( k x , k y ) , and the angle brackets in-dicate a spatial average along the guide field. f σ k is the perturbed gyrocenter distribution, and h σ k = f σ k + (cid:0) q σ φ σ k + µB (cid:107) σ k (cid:1) F σ /T σ is its nonadiabaticpart. The overbar denotes an average over the gyro-ring,and F σ is a Maxwellian background distribution withbackground density n σ and temperature T σ . The mag-netic potential A (cid:107) tot , k = A (cid:107) k + A (cid:107) ant , k is understoodto contain also the contribution due to the Langevin an-tenna A (cid:107) ant , k , which is necessary for a complete accountof the energy contained in the system. The time deriva-tive ∂ t g σ k = ∂ t ( f σ k + q σ v (cid:107) A (cid:107) k F σ /cT σ ) is the quantityexplicitly evolved in the GK Vlasov equation as imple-mented in GENE, and C = k ⊥ (cid:30)(cid:32) k ⊥ + (cid:88) σ π q σ B m σ c T σ ˆ v (cid:107) J ( λ σ ) F σ d v (cid:107) d µ (cid:33) is a factor arising from the antenna-modified Ampere’slaw, with λ σ = k ⊥ (cid:112) m σ µ/B q σ . By replacing ∂ t g σ k inEq. (1) with any of the various terms contributing to itsevolution, we can assess the impact of that term on theevolution of the free energy density. The nonlinear trans-fer function (i.e. the free energy balance contribution ofthe nonlinear term) thus reads T kpq = πB m σ (cid:60) ˆ d v (cid:107) d µ [ p x q y − p y q x ] (cid:2) χ σ p h σ q − χ σ q h σ p (cid:3) × (cid:20) h σ k T σ F σ + q σ v (cid:107) C A (cid:107) ant ,σ k /c (cid:21) , (2)with k + p + q = 0 . Compared to the definition used inRefs. [22, 23], there is an additional term involving theantenna potential, and the electrostatic approximationhas been dropped by using the full electromagnetic po-tential χ σ = φ σ − v (cid:107) A (cid:107) tot ,σ /c + µB (cid:107) σ /q σ . Note thatthe new antenna potential term does not satisfy the samesymmetry properties as the rest of the transfer function,consistent with the fact that the antenna acts as an en-ergy source through the nonlinear term (but also throughthe parallel advection term). This source can be quanti-fied by measuring the symmetric part of the above trans-fer function. Field energy spectra.
Before focusing on the nonlineartransfer physics, we analyze the spectra of the magneticand electric field energy, which can be directly comparedto spacecraft observations. As is common practice, wecompute 1-D spectra of E E ⊥ , E B (cid:107) , and E B ⊥ vs. k ⊥ ρ i by summing the energy of all ( k x , k y ) modes within agiven k ⊥ shell. Shells are linearly spaced and dividedinto 384 bins; a short-time average over about . τ A is performed, as well as an average in z direction. Theresults are displayed in Fig. 1. Here, the solid vertical linedenotes the boundary to the ’corner modes’, for which theangle integration in ( k x, k y ) ceases to pick up completecircles, causing the artificial spectral break.In the range k ⊥ ρ i (cid:46) , an MHD-type spectrum can beobserved, which exhibits a very small amount of com-pressive fluctuation energy with a flat spectrum, and Perpendicular wavenumber k ⊥ ρ i -7 -6 -5 -4 -3 -2 -1 N o r m . f i e l d f l u c t u a t i o n e n e r g y k − . k − . k − . k ⊥ ρ e = c | E ⊥ | / v Ti E B ⊥ E B || k − . ⊥ Figure 1. Normalized field energy spectra. Power law expo-nents obtained from the B ⊥ energy spectra within the dottedsections are printed into the plot. electric and magnetic field energy spectra decaying ap-proximately with the same power law. The power lawexponent is close to the Goldreich-Sridhar estimate of -5/3 [24], but the confidence level at small wavenumbersis low as there are few modes per shell.As the range of k ⊥ ρ i ∼ is crossed, all spectrasteepen, and the turbulence becomes more compressible(evidenced by the increased ratio (cid:12)(cid:12) B (cid:107) (cid:12)(cid:12) / | B ⊥ | ). For (cid:46) k ⊥ ρ i (cid:46) , all quantities exhibit rather well-definedpower law spectra, until a further steepening of the spec-tra sets in at k ⊥ ρ i ≈ , accompanied by a crossing ofthe parallel and perpendicular magnetic fluctuation en-ergy. These spectral features are consistent with previoussimulations using a fraction of the present dynamic range[12]. As the choice of parameters is (except for the colli-sionality) similar to near-Earth solar wind measurements,in Fig. 1 we plot the power law exponent E B ∝ k − . ⊥ obtained from the measurements of Refs. [4, 25] for com-parison, which agrees within about 15% with our averageexponent of -3.17, measured between < k ⊥ ρ i < . Nonlinear energy transfer.
In order to study the non-linear energy transfer, it is useful and necessary to reducethe data by subdividing the perpendicular wavenumberplane into shells (see also Ref. [22]), which we define asthe region ≤ k ⊥ ≤ k for the th shell and k ( n − / ≤ k ⊥ ≤ k n/ for the shells numbered ≤ n ≤ N − ,where we set k = 0 . and N = 25 . Thus, the entire k ⊥ range present in the simulations is covered, with goodresolution also for k ⊥ ρ i < , while at the same time en-suring that only the lowest shell < k ⊥ < k containsthe externally driven modes.With this setup, we analyze the net nonlinear shell-to-shell energy transfer, which is obtained by summingover all q wavenumbers in Eq. (2). The resulting matrix(including the symmetric terms due to the antenna, andnormalized for each k ⊥ scale) is displayed for the elec- k ⊥ ρ i F r o m p ⊥ ρ i − − − − − Figure 2. Nonlinear shell-to-shell transfer function for elec-trons, normalized to the maximum absolute value of eachwavenumber scale. tron species in Fig. 2. Numerical inspection shows thatthe antenna source acts almost exclusively on the lowestshell, and diminishes very quickly for higher shell num-bers. Studying the conservative transfer more closely,one can observe that in the range k ⊥ ρ i (cid:46) , while localenergy transfer dominates, there are some nonlocal con-tributions connecting disparate k ⊥ scales. In the range k ⊥ ρ i > , on the other hand, the nonlinear transfer isquite local ( k ⊥ ≈ p ⊥ ) , i.e. dominated by direct energytransfer between neighboring shells. Nonlocal mediation.
Beyond the net energy transfer,we now extend the analysis to differentiate between dif-ferent mediators, i.e. q wavenumbers. To this end, weevaluate the transfer function of Eq. (2) with triply fil-tered inputs, i.e., with fields and distributions condensedinto shells K, P, Q . Even with the limited number ofwavenumber shells used here, this diagnostic is extremelyexpensive (approximately ∝ N , or about 150,000 core-hours here), and is thus only evaluated instantaneouslyfor a single timestep. Its results can be visualized in acompact way, e.g., by means of Kraichnan’s locality func-tions [26]. The so-called infrared (IR) locality function isdefined (following the notation of Ref. [22]) as Π( k p | k c ) = N (cid:88) K = c +1 N (cid:88) P =1 p (cid:88) Q =1 + p (cid:88) P =1 N (cid:88) Q = p +1 T K,P,Q and retains, for a fixed shell k c with a varying ’probe’wavenumber k p , only transfers for which at least one leg p or q is smaller than k p . Thus, starting with k p = k c (retaining all transfers) and then moving the probe k p away from k c , the most local transfers are successivelyremoved. For an extensive description of this setup, werefer the reader to Sec. V of Ref. [23]. − k p ρ i − Π ( k p | k c ) / Π ( k c | k c ) k ⊥ ρ i = k c ρ i = 0 . k c ρ i = 0 . k c ρ i = 2 . k c ρ i = 5 . k c ρ i = 11 . k c ρ i = 22 . k c ρ i = 35 . k c ρ i = 44 . Figure 3. Infrared locality functions for several shells k c , nor-malized to the total nonlinear energy transfer through k c , ver-sus the probe wavenumber k p ρ i . For the curves with k c ρ i (cid:38) ,a change in slope is apparent when the probe k p crosses theion gyroradius scale. For several k c shells, we show the corresponding IR lo-cality functions Π ( k p | k c ) / Π ( k c | k c ) in Fig. 3. By plottingthe curves versus the probe wavenumber k p instead of theconventional ratio k p /k c , Fig. 3 highlights the existenceof a meaningful physical scale length at k ⊥ ρ i ∼ , indi-cating a lack of self-similarity. Indeed, the locality func-tion curves for k c ρ i (cid:38) exhibit a transition in their slopethat occurs close to the ion gyroradius scale, k p ρ i ∼ :for k p ρ i > the nonlinear energy transfer is rather non-local, with a locality exponent between 2/3 and 1/3; for k p ρ i < , a more local exponent of 4/3, as in Navier-Stokes turbulence [27], is found. As a consequence ofthis property, for (cid:46) k c (cid:46) . , nonlocal transfers me-diated by fluctuations in the tail of the MHD range at k p ρ i (cid:46) are responsible for at least 30% of the total en-ergy transfer through these shells. Note that this doesnot contradict the above observation that the net non-linear transfer for large k ⊥ is local. Indeed, the nonlineartriad k + p + q = 0 for such nonlocal interactions is char-acterized by | q | (cid:28) | k | , | p | and thus | k | ≈ | p | , consistentwith a local net transfer between k and p . Finally, wenote that while all of the above statements were illus-trated with results for the electron species, the nonlinearion energy transfer (not shown) exhibits the same char-acteristics, though with an even more pronounced nonlo-cality (exponent ∼ / ), and at least 50% of the transfermediated by modes in the tail of the MHD range. Collisional dissipation.
Next, we study the spectralproperties of the collisional dissipation rate by measuringthe contribution of the collision term to the free energybalance. The resulting graphs are presented in Fig. 4for both electron and ion species, as well as their sum.About 70% of the total dissipation is found to arise fromelectron collisions, which exhibit a broad peak around k ⊥ ρ i ∼ − . Qualitatively, this peak is consistent withelectron Landau damping acting on the magnetic energy k ⊥ ρ i k ⊥ ( d E / d t ) c o ll totalElectrons Ions Figure 4. Normalized, short-time averaged collisional dissi-pation for electrons, ions, and its total value. Curves aremultiplied by k ⊥ so the area under the curve is proportionalto the energy dissipation rate. spectrum shown in Figure 1. Despite peaking at theserelatively small k ⊥ wavenumbers, electron dissipation re-mains strong throughout the spectrum, and begins to in-tensify somewhat at k ⊥ ρ i (cid:38) . At k ⊥ ρ i ∼ , where iontransit-time damping is expected to transfer field energyto ion particle energy, there is in fact little ion heating. Atthese scales the ion free energy (not shown) is comparableto the magnetic fluctuation energy, but it is cascaded tosmaller scales in both position and velocity space, and isdissipated close to the electron gyroradius scale (around k ⊥ ρ i ∼ ). This observation is consistent with an ionentropy cascade and the fact that ν i (cid:28) ν e [9, 19, 28].Taking into account both species’ contributions, we findan essentially flat dissipation spectrum throughout thekinetic wavenumber range, contrasting with some inter-pretations of solar wind data [4, 5] which suggested thatthe electron gyroradius scale acts as the dominant dissi-pation scale. Conclusions.
In the present study, the first gyrokineticsimulation of kinetic Alfvén wave turbulence coupling allscales from the tail of the MHD range to the electrongyroradius scale was performed, with the goal of analyz-ing fundamental properties of nonlinear energy transferand collisional dissipation for parameters relevant to thesolar wind. It was found that nonlinear energy transferin the kinetic range, particularly for k ⊥ ρ i (cid:38) , is con-siderably more nonlocal than hydrodynamic turbulence,as suggested by previous theoretical considerations [29],and is to a significant percentage ( > k ⊥ ρ i ∼ , whilethe net energy transfer occurs mainly between nearest-neighbor shells. For T e /T i = 1 and β i = 1 , similar tothe near-Earth solar wind, 70% of the injected energyis dissipated through the electron species, whose dissi-pation spectrum peaks around k ⊥ ρ i ∼ − , consistentwith electron Landau damping. The ion free energy, onthe other hand, is cascaded to small scales and dissipatedaround k ⊥ ρ i ∼ . These findings underscore the pres-ence of strong dissipation throughout the kinetic range k ⊥ ρ i (cid:38) , justifying the common notion of a ’dissipa-tion range’, and demonstrating a coupling across multiplescales of both transfer and dissipation. Acknowledgments . The authors acknowledge fruit-ful discussions with F. Muller, A. Bañón Navarro andM.J. Pueschel. The research leading to these results hasreceived funding from the European Research Councilunder the European Union’s Seventh Framework Pro-gramme (FP7/2007-2013)/ERC Grant Agreement No.277870, NSF CAREER Award AGS-1054061, and U.S.DOE Award No. DEFG0293ER54197. Furthermore, thiswork was facilitated by the Max-Planck/Princeton Cen-ter for Plasma Physics. The Rechenzentrum Garching(RZG) is gratefully acknowledged for providing compu-tational resources used for this study. Parts of this re-search also profited from resources of the National EnergyResearch Scientific Computing Center, a DOE Office ofScience User Facility supported by the Office of Scienceof the U.S. Department of Energy under Contract No.DE-AC02-05CH11231. ∗ [email protected][1] J. D. Richardson and C. W. Smith, Geophys. Res. Lett. , 1206 (2003).[2] R. Bruno and V. Carbone, Living Rev. Sol. Phys. , 2(2013).[3] S. D. Bale, P. J. Kellogg, F. S. Mozer, T. S. Hor-bury, and H. Reme, Phys. Rev. Lett. , 215002 (2005),physics/0503103.[4] O. Alexandrova, J. Saur, C. Lacombe, A. Man-geney, J. Mitchell, S. J. Schwartz, and P. Robert,Phys. Rev. Lett. , 165003 (2009), arXiv:0906.3236[physics.plasm-ph].[5] F. Sahraoui, M. L. Goldstein, P. Robert, and Y. V.Khotyaintsev, Phys. Rev. Lett. , 231102 (2009).[6] F. Sahraoui, M. L. Goldstein, G. Belmont, P. Canu, andL. Rezeau, Phys. Rev. Lett. , 131101 (2010).[7] C. H. K. Chen, S. Boldyrev, Q. Xia, and J. C. Perez,Phys. Rev. Lett. , 225002 (2013), arXiv:1305.2950[physics.space-ph].[8] G. G. Howes, W. Dorland, S. C. Cowley, G. W. Ham-mett, E. Quataert, A. A. Schekochihin, and T. Tatsuno,Phys. Rev. Lett. , 065004 (2008), arXiv:0711.4355.[9] G. G. Howes, J. M. TenBarge, W. Dorland, E. Quataert,A. A. Schekochihin, R. Numata, and T. Tatsuno, Phys. Rev. Lett. , 035004 (2011), arXiv:1104.0877[astro-ph.SR].[10] M. Wan, W. H. Matthaeus, H. Karimabadi, V. Royter-shteyn, M. Shay, P. Wu, W. Daughton, B. Loring, andS. C. Chapman, Phys. Rev. Lett. , 195001 (2012).[11] C. S. Salem, G. G. Howes, D. Sundkvist, S. D. Bale, C. C.Chaston, C. H. K. Chen, and F. S. Mozer, Astrophys. J.Lett. , L9 (2012).[12] J. M. TenBarge, G. G. Howes, and W. Dorland, Astro-phys. J. , 139 (2013).[13] J. M. TenBarge and G. G. Howes, Astrophys. J. Lett. , L27 (2013), arXiv:1304.2958 [physics.plasm-ph].[14] K. T. Osman, W. H. Matthaeus, J. T. Gosling, A. Greco,S. Servidio, B. Hnat, S. C. Chapman, and T. D. Phan,Phys. Rev. Lett. , 215002 (2014), arXiv:1403.4590[physics.space-ph].[15] A. J. Brizard and T. S. Hahm, Rev. Mod. Phys. , 421(2007).[16] J. J. Podesta, Sol. Phys. , 529 (2013).[17] F. Jenko, W. Dorland, M. Kotschenreuther, and B. N.Rogers, Phys. Plasmas. , 1904 (2000).[18] J. M. TenBarge, G. G. Howes, W. Dorland, and G. W.Hammett, Comput. Phys. Commun. , 578 (2014),arXiv:1305.2212 [physics.plasm-ph].[19] A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W.Hammett, G. G. Howes, E. Quataert, and T. Tat-suno, Astrophys. J. Suppl. Ser. , 310 (2009),arXiv:0704.0044.[20] A. Bañón Navarro, P. Morel, M. Albrecht-Marc,D. Carati, F. Merz, T. Görler, and F. Jenko,Phys. Rev. Lett. , 055001 (2011), arXiv:1008.3974[physics.plasm-ph].[21] A. Bañón Navarro, P. Morel, M. Albrecht-Marc,D. Carati, F. Merz, T. Görler, and F. Jenko, Phys. Plas-mas. , 092303 (2011).[22] B. Teaca, A. B. Navarro, F. Jenko, S. Brunner, andL. Villard, Phys. Rev. Lett. , 235003 (2012).[23] B. Teaca, A. B. Navarro, and F. Jenko, Phys. Plasmas. , 072308 (2014), arXiv:1404.2080 [physics.plasm-ph].[24] P. Goldreich and S. Sridhar, Astrophys. J. , 763(1995).[25] F. Sahraoui, S. Y. Huang, G. Belmont, M. L. Goldstein,A. Rétino, P. Robert, and J. D. Patoul, Astrophys. J. , 15 (2013).[26] R. H. Kraichnan, J. Fluid Mech. , 497 (1959).[27] R. H. Kraichnan, Phys. Fluids , 1728 (1966).[28] T. Tatsuno, W. Dorland, A. A. Schekochihin, G. G.Plunk, M. Barnes, S. C. Cowley, and G. G. Howes,Phys. Rev. Lett. , 015003 (2009), arXiv:0811.2538[physics.plasm-ph].[29] G. G. Howes, J. M. TenBarge, and W. Dorland,Phys. Plasmas.18