-8/3 spectrum in kinetic Alfvén wave turbulence: implications for the solar wind
DDraft version July 1, 2019
Typeset using L A TEX twocolumn style in AASTeX62 k − / ⊥ spectrum in kinetic Alfv´en wave turbulence: implications for the solar wind Vincent David
1, 2 and S´ebastien Galtier
1, 2, 3 Laboratoire de Physique des Plasmas, ´Ecole polytechnique, F-91128 Palaiseau Cedex, France Univ. Paris-Sud, Observatoire de Paris, Univ. Paris-Saclay, CNRS, Sorbonne Univ. Institut universitaire de France
ABSTRACTThe nature of solar wind turbulence at large scale is rather well understood in the theoretical frame-work of magnetohydrodynamics. The situation is quite different at sub-proton scales where the mag-netic energy spectrum measured by different spacecrafts does not fit with the classical turbulencepredictions: a power law index close to − / − / − / − / k f ∼ ( t ∗ − t ) − / ,with t < t ∗ . This solution appears when we relax the implicit assumption of stationarity generallymade in turbulence. The agreement between the theory and observations can be interpreted as anevidence of the non-stationarity of solar wind turbulence at sub-proton scales. Keywords: plasma physics – solar wind – turbulence – waves INTRODUCTIONThe solar wind is a collisionless plasma characterizedby fluctuations of its primary fields over a huge rangeof frequencies. One of the most spectacular proper-ties reported from in situ measurements is a spectrumof magnetic fluctuations from frequencies f ∼ − Hzto ∼ f b ∼ f < f b ) from the sub-proton scales( f > f b ) where ions and electrons are decoupled, andwhere signatures of kinetic Alfv´en waves (KAW) canbe found (see e.g. Sahraoui et al. (2010); Salem et al.(2012); Chen et al. (2013)). Note that signatures ofother types of waves are also found (see e.g. Naritaet al. (2011); Roberts et al. (2015)). Despite severalyears of studies, the nature of solar wind turbulence atsub-proton scales remains under debate (in this paperwe restrict our attention to scales greater than the elec- Corresponding author: S´ebastien [email protected] tron gyroscale). A reason is that the magnetic energyspectrum reported is generally close to f − / (Alexan-drova et al. 2012; Podesta 2013; Sahraoui et al. 2013)which is far from the classical predictions of strong and(weak) wave turbulence (Biskamp et al. 1996; Galtier &Bhattacharjee 2003; Galtier 2006a,b; Schekochihin et al.2009; Voitenko & de Keyser 2011; Galtier & Meyrand2015; Cerri et al. 2016; Passot et al. 2018) for which thepower law indices are − / − / f − / is found. A possibility is thatthe latter power law is a spectrum predicted by a classi-cal turbulence theory modified by some kinetic dissipa-tion (see e.g. Passot & Sulem (2015)). Note that sev-eral studies have been devoted to the question of solarwind heating and the evaluation of the energy cascaderate at MHD scales, which can be seen as a proxy formeasuring the heating rate (see e.g. Sorriso-Valvo et al.(2007); Vasquez et al. (2007); MacBride et al. (2008); a r X i v : . [ phy s i c s . s p ace - ph ] J un David and Galtier
Osman et al. (2011); Banerjee et al. (2016); Hadid et al.(2017)).The Letter is organized as follows. In section 2 weintroduce a model of KAW turbulence, first derived byPassot & Sulem (2019), and its phenomenology. In sec-tion 3 we present its non-stationary solution which is aself-similar solution of the second kind. The numericalvalidation of our theory is given in section 4. A discus-sion is given in section 5 about the applications to solarwind turbulence at sub-proton scales. A conclusion isfinally proposed in the last section. MODEL OF KAW TURBULENCENonlinear diffusion models are often used in theanalysis of both strong (Leith 1967; Connaughton &Nazarenko 2004; Matthaeus et al. 2009; Thalabard et al.2015) and weak wave turbulence (Zakharov & Pushkarev1999; Boffetta et al. 2009; Galtier et al. 2019). Thereare mostly built by using phenomenological argumentsbut a rigorous treatment is sometimes possible in theregime of wave turbulence. The known examples arenonlinear optics (Dyachenko et al. 1992) and MHD(Galtier & Buchlin 2010). In this case, the nonlineardiffusion equations are derived by taking the stronglylocal interactions limit of the kinetic equations; the lat-ter equations being themselves derived in a systematicalway. Recently, such a model has been proposed byPassot & Sulem (2019) for KAW turbulence (a modelalso valid for oblique whistler waves as explained inGaltier & Meyrand (2015)) neglecting the coupling toother types of waves. The derivation can be qualified assemi-analytical because the problem is fundamentallyanisotropic and in the final step of the derivation the au-thors neglected the cascade along the uniform magneticfield to find an expression for the nonlinear diffusionequation. However, the parallel cascade is expected tobe relatively weak and its absence cannot be seen asa drawback of the model. Then, KAW turbulence issimulated numerically in presence of magnetic helicityin order to study the regime of imbalanced weak turbu-lence (Passot & Sulem 2019). This type of model givesin general good quantitative information about the pri-mary system because local interactions are in generalthe main driver of the turbulence cascade.Here, we shall use the diffusion equation proposed byPassot & Sulem (2019) for weak KAW turbulence in ab-sence of magnetic helicity. To be self-consistent (and forpedagogical reasons) a new derivation is proposed byusing only phenomenological arguments. This methodhas the advantage of explaining in a simple way the mainphysical ingredients require to derive a nonlinear diffu-sion model for KAW turbulence. Since the leading nonlinear interaction of KAW isthree-wave interaction (Galtier & Meyrand 2015; Pas-sot & Sulem 2019), the model is a second-order diffusionequation of the type ∂E ( k ⊥ ) ∂t = ∂∂k ⊥ (cid:20) D k ⊥ E ( k ⊥ ) ∂ ( E ( k ⊥ ) /k ⊥ ) ∂k ⊥ (cid:21) , (1)where E ( k ⊥ ) is a one-dimensional magnetic energyspectrum, k ⊥ the perpendicular wavenumber and D k ⊥ a diffusion coefficient that eventually depends on thewavenumber k ⊥ . This equation is constructed in sucha way that it preserves the nonlinearity degree withrespect to the spectrum (quadratic in our case) and,its cascade and thermodynamic solutions. We neglectthe cascade along the strong uniform magnetic field B which defines the parallel direction, hence the presenceof only the perpendicular (to B ) wavenumber k ⊥ . Adimensional analysis of expression (1) gives E ( k ⊥ ) τ ∼ D k ⊥ E ( k ⊥ ) k ⊥ , (2)where τ is the cascade time of weak wave turbulence;thus D k ⊥ ∼ k ⊥ τ E ( k ⊥ ) ∼ k ⊥ ( τ KAW /(cid:15) ) E ( k ⊥ ) . (3)The KAW time is given by the relation τ KAW ∼ ω ∼ k (cid:107) k ⊥ ∼ k ⊥ , (4)where (cid:15) ∼ τ KAW /τ NL (cid:28) τ NL ∼ / ( k ⊥ (cid:112) k ⊥ E ( k ⊥ )). We obtain D k ⊥ ∼ k ⊥ , (5)which leads to the following second-order diffusion equa-tion for KAW turbulence (Passot & Sulem 2019) ∂E ( k ⊥ ) ∂t = C ∂∂k ⊥ (cid:20) k ⊥ E ( k ⊥ ) ∂ ( E ( k ⊥ ) /k ⊥ ) ∂k ⊥ (cid:21) , (6)where C is a positive constant.The constant flux solutions can now be found. Wedefine the energy flux Φ E ( k ⊥ ) as follows ∂E ( k ⊥ ) ∂t = − ∂ Φ E ( k ⊥ ) ∂k ⊥ (7)and introduce the magnetic energy spectrum E ( k ⊥ ) = Ak x ⊥ into equation (6) with A a positive constant. Wefind Φ E ( k ⊥ ) = A C (1 − x ) k x ⊥ . (8) / spectrum in the solar wind x = 1, which corre-sponds to the thermodynamic solution (zero flux), and x = − / E ( k ⊥ ) ≡ Φ = (7 / A C which is positive and thus correspondsto a direct cascade. Therefore, we recover the well-known solutions of the problem (Galtier & Bhattachar-jee 2003; Galtier & Meyrand 2015; Passot & Sulem2019). NON-STATIONARY REGIMETime-dependent solutions of the KAW turbulenceequation (6) will be studied further analytically and nu-merically. We will demonstrate the existence of a non-trivial solution (called sometimes anomalous scaling) inthe sense that it cannot be derived with the usual turbu-lence phenomenology or theory. This property is relatedto the finite capacity of the system which is linked to theconvergence of the integral (cid:90) + ∞ k i E ( k ⊥ ) dk ⊥ , (9)where k i is the scale of magnetic energy injection. Thisproperty is satisfied when x < −
1, a situation found inKAW turbulence.The non-stationary spectrum can be modeled as a self-similar solution of the second kind (see e.g. Falkovich &Shafarenko (1991); Thalabard et al. (2015)) taking theform E ( k ⊥ ) = 1 τ a E (cid:18) k ⊥ τ b (cid:19) , (10)where τ = t ∗ − t , and t ∗ is a finite time at which themagnetic energy spectrum reached the largest availablewavenumber. By introducing the above expression into(6) we find the condition a = 4 b + 1 . (11)A second condition can be found by assuming that E ( ξ ) ∼ ξ m far behind the front. Then, the station-arity condition gives the following relation a + mb = 0 . (12)Finally, the combination of both relations gives m = − ab = − − b . (13)The latter expression means that we have a direct rela-tion between the power law index m of the spectrum andthe law of the front propagation which follows k f ∼ τ b .For example, if we assume that the stationary solution – k E ( k ) k Figure 1.
Time evolution (every 1000 dt ) of the magneticenergy spectrum E ( k ⊥ ) from t = 0 (blue) to t ∗ (dark red).A k − / ⊥ spectrum emerges over three decades. the Kolmogorov-Zakharov spectrum – is established im-mediately during the front propagation, then m = − / b = − / a = − / k f ∼ ( t ∗ − t ) − / . (14) NUMERICAL SIMULATIONWe now study numerically the time evolution of themagnetic energy spectrum described by the KAW tur-bulence equation (6) with C = 1. A linear hyper-viscousterm of the form − ηk ⊥ E ( k ⊥ ) is added to equation (6)in order to introduce a sink at small scale to avoid thedevelopment of numerical instabilities at the final timeof the simulation ( t > t ∗ ) when the stationary state es-tablishes; we take η = 10 − . A logarithmic subdivisionof the k ⊥ -axis is used with k ⊥ i = 2 i/ and i an integervarying between 0 and 160. Note that this resolutionis far too large to model the sub-proton scales whereelectron inertia is neglected. This choice is however nec-essary to reach a clear conclusion about the values of thepower law indices (see below). The Crank-Nicholson andAdams-Bashforth numerical schemes are implementedfor the nonlinear and dissipative terms respectively. Theinitial condition ( t = 0) corresponds to a spectrum lo-calized at large scale with E ( k ⊥ ) ∼ k ⊥ exp( − ( k ⊥ /k ) )and k = 5. No forcing is added at t >
0. The time-stepis dt = 2 × − .In Figure 1 we show the time evolution of the mag-netic energy spectrum from t = 0 to t ∗ . During thisnon-stationary phase a clear power law spectrum in k − / ⊥ is formed behind the front. To check if this spec-trum corresponds to the self-similar solution of the sec- David and Galtier
Figure 2.
Temporal evolution of the spectral front k f for t ≤ t ∗ in linear-logarithmic coordinates (blue). A sharp in-crease of k f is observed from which we can define preciselythe singular time t ∗ = 6 . × − . Inset: The temporalevolution of k f as a function of t ∗ − t (orange) in doublelogarithmic coordinates. The black dashed line correspondsto ( t ∗ − t ) − . . For comparison two other values of t ∗ aretaken (green and blue). k E ( k ) k Figure 3.
Temporal evolution of the energy flux Φ E ( k ⊥ )in double logarithmic coordinates for the same times as inFigure 1 (same conventions). The flux follows a power law ∼ k − / ⊥ . ond kind introduced above we show in Figure 2 thefront propagation k f ( t ). This front is defined by taking E ( k ⊥ ) = 10 − from Figure 1: we then follow the pointof intersection between this threshold and the spectraltail. From Figure 2 we can define the singular time t ∗ at which the front can reach in principle k ⊥ = + ∞ .Note that a similar situation where the small scalesare reached in a finite time is also observed e.g. in Figure 4.
Temporal evolution (every 1000 dt ) for t > t ∗ ofthe energy spectrum compensated by k / ⊥ (in double loga-rithmic coordinates). Inset: Temporal evolution of the en-ergy flux Φ E ( k ⊥ ) for the same times (in linear-logarithmiccoordinates). Alfv´en wave turbulence (Galtier et al. 2000). The value t ∗ = 6 . × − is chosen. In Figure 2 (inset) we show k f as a function of t ∗ − t : a clear power law is observedover three decades with a power law index of − . a = − , b = − / − / , (15)which therefore demonstrates the self-similar nature ofthe non-stationary solution.As displayed in Figure 3, the non-stationary phaseis characterized by a non-constant energy flux Φ E ( k ⊥ )(computed from the nonlinear terms): we start with aflux localized at small wavenumbers which then developstowards smaller scales without reaching a plateau. Thesolution does not correspond to the constant flux solu-tion derived analytically, but it is fully compatible withthe power law solution ∼ k − / ⊥ when we take x = − / t > t ∗ of the energy spectrum and energy flux (inset),respectively. The classical (stationary) wave turbulencepredictions are finally obtained with an energy spectrumin k − / ⊥ and a constant positive energy flux, as expectedfor a direct cascade. This behavior is specific to a viscoussimulation made in a finite box where the cascade cannotcontinue to smaller scales: the energy accumulates atsmall scale until the viscous term (proportional to the / spectrum in the solar wind − / SOLAR WIND TURBULENCE AT SUB-PROTONSCALESSolar wind turbulence at sub-proton scales (for fre-quencies f > − / − /
3) or weak waveturbulence ( − / k − / ⊥ which coincideswith in situ observations. In this non-stationary phasethe viscous dissipation is negligible. While the absenceof viscous dissipation should be considered as the rightway to tackle the problem of solar wind turbulence atsub-proton scales, since the solar wind is a collisionlessplasma and thus cannot behave like a viscous fluid, wemust nevertheless clarify the meaning and the conse-quences of such assumption. The first clear idea is thatthere is no reason to believe that dissipation at kineticscales should behave like that found in hydrodynamics;Landau damping is a good example. According to ourinterpretation the results obtained here are in favor ofa kinetic dissipation that does not produce a feedback on the inertial range of KAW turbulence. This prop-erty is at odds with fluid turbulence. We might alsoconclude that the kinetic dissipation is simply negligi-ble, however, the presence of kinetic dissipation as asource of plasma heating seems to be necessary to ex-plain the slow (ion) temperature variation with the he-liocentric distance (Richardson et al. 1995). Accordingto our study, we can also think that the observation of aspectral index close to − / CONCLUSIONOur study reveals that the classical hypothesis of sta-tionarity to obtain any turbulence predictions may notbe the best way to understand solar wind turbulenceat sub-proton scales. Instead, the relaxation of this as-sumption opens a new type of solution that is under-stood as a self-similar solution of the second kind. Onthe basis of numerical simulations of a nonlinear dif-fusion model of weak KAW turbulence we show thatthe main scaling behavior observed with spacecrafts – apower law index close to − / Alexandrova, O., Lacombe, C., Mangeney, A., Grappin, R.,& Maksimovic, M. 2012, Astrophys. J., 760, 121 Banerjee, S., Hadid, L. Z., Sahraoui, F., & Galtier, S. 2016,Astrophys. J., 829, L27