Universality of solar wind turbulent spectrum from MHD to electron scales
Olga Alexandrova, Joachim Saur, Catherine Lacombe, Andre Mangeney, Jeremy Mitchell, Steve J. Schwartz, Patrick Robert
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Universality of solar wind turbulent spectrum from MHD to electron scales
O. Alexandrova ∗ and J. Saur Institute of Geophysics and Meteorology, University of Cologne,Albertus-Magnus-Platz 1, 50923, Cologne, Germany.
C. Lacombe and A. Mangeney
LESIA, Observatoire de Paris, CNRS, UPMC, Universit´e Paris Diderot, 5 place J. Janssen, 92190 Meudon, France.
J. Mitchell and S. J. Schwartz
Blackett Laboratory, Imperial College London, London SW7 2AZ, UK.
P. Robert
LPP, 10–12 avenue de l’Europe 78140 Velizy France. (Dated: October 23, 2018)In order to investigate the universality of magnetic turbulence in space plasmas we analyze seventime periods in the free solar wind of different origin, slow or fast, and under different plasmaconditions. The orientation of magnetic field to the flow velocity was always quasi-perpendicular.Unique combination of three instruments on Cluster spacecraft which operate in different frequencyranges give us the possibility to resolve spectra up to 300 Hz. We show that spectra measured underdifferent plasma conditions have a similar shape. Such a quasi-universal spectrum consists of threeparts: two power laws and an exponential domain. At MHD scales, Kolmogorov’s law ∼ k − / is found. At scales smaller than the ion characteristic scales, a k − . law is observed. At scales kρ e ∼ (0 . − ρ e is the electron gyroradius, the magnetic spectrum follows an exponentiallaw exp( − k / ), indicating the onset of dissipation. This is the first observation of an exponentialmagnetic spectrum in space plasmas. We show that among several spatial kinetic plasma scales, theelectron Larmor radius plays the role of a dissipation scale in space plasma turbulence. PACS numbers: 52.35.Ra,94.05.-a,96.60.Vg,95.30.Qd
Space plasmas are usually in a turbulent state, andthe solar wind is one of the closest laboratories of spaceplasma turbulence, where in-situ measurements are pos-sible thanks to a number of space missions [1]. Thesemeasurements obtain time series which provide access tofrequency spectra or to spectra of wave vectors along theflow. It is well established that at MHD scales (below ∼ . ∼ f − / . However, the characteristics of turbulencein the vicinity of the kinetic plasma scales (such as the in-ertial lengths λ i,e = c/ω pi,e , c being the speed of light and ω pi,e the plasma frequencies of ions and electrons, respec-tively, the Larmor radii ρ i,e and the cyclotron frequencies ω ci,e = eB/m i,e ) are not well known experimentally andare a matter of debate. It was shown that at ion scales theturbulent spectrum has a break, and steepens to ∼ f − s ,with a spectral index s that is clearly non-universal, tak-ing on values in the range − − ∼ ∗ Electronic address: [email protected]; Institute of Geo-physics and Meteorology, University of Cologne, Albertus-Magnus-Platz 1, 50923, Cologne, Germany. are difficult and our knowledge is very poor. Denskat[4] using Helios data obtained high resolution magneticspectra at 2 distances from the Sun: up to 50 Hz at1 AU, and up to 470 Hz at 0 . kλ e ≃ kρ e ∼ β (the ratio betweenplasma and magnetic pressures) was ∼ ρ e and λ e .Measurements of solar wind turbulent spectra in thevicinity of ion and electron plasma scales may clarify ourunderstanding of the processes of dissipation (or disper-sion) of turbulent energy in collisionless plasmas. A num-ber of processes may be considered at these scales: cy-clotron damping at f ci and f ce of Alfv´en and whistlerwaves, respectively [7]; scattering of oblique whistlerwaves at f ci < f < f ce [8]; linear dissipation of kineticAlfv´en waves at 1 /ρ i < k < /ρ e [9, 10].In this paper we use the Cluster spacecraft [11] datato analyze the free solar wind of different origin, fastand slow, and under different plasma conditions. WhileSahraoui et al. [6] use the FluxGate Magnetometer(FGM) [12] and STAFF-Search Coil (SC) [13] at theburst mode, which allow them in principle to investigateturbulence spectra up to 180 Hz, we complete these in-struments with the STAFF-Spectra Analyzer (SA), en-abling us to increase considerably the upper frequencylimit, up to 4 kHz. However, as was shown in [6], above100 Hz the instrument noise becomes a significant issuewhich we take into account in our analysis.As suggested in [13], we use measurements in the mag-netospheric lobe (precisely, the data on 5 April 2001,06:00-07:00 UT) as the noise level of the instrument. Thefinal spectra were obtained by substracting the lobe spec-trum from the solar wind spectra. A similar procedurehas been applied by Lin [14] for the Ulysses spacecraftdata. The maximal frequency in our analysis is definedas the highest frequency where the measured spectrum ishigher than twice the lobe spectrum, before subtraction.We select seven time intervals of 42 minutes when Clus-ter was at apogee (19 Earth radii) and spent one hour ormore in the free solar wind: the electric field data at theelectron plasma frequency show no evidence of magneticconnection to the bow shock [P. Canu, private commu-nication, 2009]. In Table I, the dates of the intervalsare shown as ymmdd, and their starting times are de-noted by t i . Average plasma parameters for the selectedintervals are given also in this table. Magnetic field mea-surements were obtained from Cluster 1. Ion moments(density N , velocity V and perpendicular temperature T ⊥ i ) are measured by the CIS/HIA experiment [15] onCluster 1. The ion parallel temperatures are not prop-erly determined in the solar wind by the CIS instrument[I. Dandouras, private communication, 2009]. Electronsare measured by the PEACE instrument [16], mostlyon Cluster 2. One can see from Table I that the meanfield/flow angle, Θ BV , is always larger than 60 ◦ . Otherplasma parameters are rather variable: V varies from ∼
360 km/s to 670 km/s, the total perpendicular plasmabeta, β i ⊥ + β e ⊥ = 2 µ nk ( T i ⊥ + T e ⊥ ) /B , varies be-tween 0.7 and 3.3, the Alfv´en speed V a ∈ [30 , V thi,e = p kT ⊥ i,e /m i,e are the ion and electron perpen-dicular thermal speeds, respectively; ρ i,e = V thi,e /ω ci,e are the corresponding Larmor radii. During these sevenintervals we never observe quasi-parallel whistler waves,characterized by a quasi-circular right-hand polarization,which can be captured by STAFF-SA instrument. Thetwo intervals 3 and 5 display the most intense spectra andare observed in the fast solar wind, a few hours down-stream of an interplanetary shock.Figure 1(top) shows the magnetic spectrum P ( f ) forinterval 5. It is calculated using the Morlet wavelettransform, as was done in [17]. One can clearly recog-nize here two power-laws and an exponential ranges: Atlow frequencies, the spectrum is ∼ f − . consistent withKolmogorov’s law. Between f ci and f λ i ≃ f ρ i (where f λ i = V / πλ i and f ρ i = V / πρ i ), the first break appears.At higher frequencies, the spectrum follows an ∼ f − . law. However, at 10 ≤ f ≤
100 Hz, the spectrum is nolonger a power-law, but follows approximatively an expo-nential function exp( − a ( f /f ) . ). At higher frequencies, TABLE I: Solar wind parameters for selected time periods.Nb 1 2 3 4 5 6 7ymmdd 10405 20219 30218 31231 40122 40127 50112 t i (UT) 22:36 01:48 00:18 10:48 05:03 00:36 02:00 B (nT) 7.3 7.0 15.5 10.9 15.5 9.5 13.6 N (cm − ) 3 29 7 22 20 8 33 T ⊥ i (eV) 17 7 40 10 61 10 14 T ⊥ e (eV) 36 7 18 16 28 21 16 V (km/s) 540 365 670 430 635 430 440Θ BV ( ◦ ) 85 65 80 75 85 80 85 β ⊥ i β ⊥ e f ci (Hz) 0.11 0.11 0.24 0.17 0.24 0.15 0.21 f ce (Hz) 205 195 435 305 435 265 380 λ i (km) 130 40 85 50 50 80 40 λ e (km) 3 1 2 1 1 2 1 ρ i (km) 60 40 40 30 50 35 30 ρ e (km) 2 1 0.5 1 1 1 0.5 V a (km/s) 95 30 130 50 75 70 50 V thi (km/s) 40 25 60 30 75 30 35 V the (km/s) 2.5 1.1 1.8 1.7 2.2 1.9 1.7FIG. 1: Top: Magnetic power spectral density for interval5, measured by Cluster/FGM (up to 1 Hz), STAFF–SC (upto 10 Hz) and STAFF–SA ( f ≥ f λ i,e correspond to the Doppler-shifted λ i,e and f ρ i,e to ρ i,e .Power-laws f − . and f − . are shown. Dashed-dotted lineindicates exponential fit ∼ exp( − a ( f/f ) . ), with f = f ρ e and the constant a ≃
9. Bottom: Compensated spectrumby f . at low frequencies, f . for middle part, and by theexponential for high frequency part. FIG. 2: (a) Magnetic spectra for 7 time periods of 42 min-utes; spread of f ci,e for the 7 intervals is shown (b) k -spectranormalized over P ; characteristic wave numbers, k ρ i = 1 /ρ i etc., are shown. f > f λ e , the spectrum is too close to the noise level (seethe black solid line) to draw any firm conclusions.To demonstrate the above scaling laws, Fig-ure 1(bottom) shows compensated energy spectra. Thelow frequency part of the spectrum was compensated by f . (solid line), the middle range – by f . (dashed) andthe high frequency part – by exp( a ( f /f ) . ) (dashed-dotted). The combined compensated spectrum is indeedvery flat up to f λ e .The spectra for the seven intervals are presented inFigure 2(a). Horizontal bars indicate the spread of f ci,e among these seven independent observations. One cansee that the spectra have similar shapes. Their intensityis, however, different. To superpose the spectra, we beginby applying Taylor’s hypothesis, which should be validfor the whole frequency range, as far as quasi-parallelwhistler waves are not observed during selected inter-vals (as mentioned above). Thus, we assume that thefrequency–spectra are indeed Doppler shifted k –spectra P ( k ) = P ( f ) V / π with k = 2 πf /V . Then, we deter-mine a relative intensity of the j -th spectrum, S j with j = 1 , ...,
7, as P ( j ) = h S j /S i , where S is a referencespectrum, and h·i indicates a mean over the range of wavevectors 10 − < k < − km − . With this normalization,the rescaled spectra may be nearly superposed as shown FIG. 3: Relative spectral intensity P as a function of (a)magnetic and (b) kinetic energies; (c) P as a function of theion cyclotron period and (d) the electron gyroradius. Linearfits with corresponding slopes are shown by solid lines. in Figure 2(b).One expects that the spectral level, P , depends onthe solar wind kinetic, thermal or magnetic energy. Thescatter plots shown in Figure 3(a) and (b) indicate clearlya power law dependance of P on the magnetic energy,and a less clear dependance on the kinetic energy, andthermal energy (not shown).To understand the meaning of the observed de-pendence on the magnetic energy, one may use aKolmogorov-like phenomenology. Suppose first that thesolar wind magnetic turbulence dissipates through an ef-fective diffusion mechanism of ∼ η ∆ B ( η being a – prob-ably turbulent – magnetic diffusivity), and second, thatthe observed turbulence is quasi stationary. In such acase, there is a balance between the energy input fromnonlinear interactions at large scales and the energy drainfrom the dissipation at small scales. This implies that theenergy transfer rate ǫ depends on the dissipation scale ℓ d as ǫ = η ℓ − d ; thus P ∼ ǫ / ∼ ℓ − / d . The depen-dences observed in Figure 3(c) and (d), P ∼ (1 /f ci ) − . and P ∼ ρ − . e , are very close to the prediction of thisphenomenological model. More statistics are needed toconfirm the observed exponents. We can state, however,that the observed dependences imply that ρ e and/or f ci and/or f ce play an important role in the dissipation pro-cesses in collisionless plasmas. Let us now confirm theseresults.From the balance between the energy input and thedissipation, for the Kolmogorov’s spectrum E ( k ), it fol-lows as well that E ( k ) ℓ d /η is a universal function of kℓ d [18, 19]. Figure 4 tests which of the kinetic scalesis to be used as ℓ d to recover a universal function fromthe observed spectra. We assume for simplicity that η FIG. 4: Universal Kolmogorov function ∝ ℓ d E ( k ) for hypoth-esized dissipation scales ℓ d as a function of (a) kρ i , (b) kλ i ,(c) kρ e and (d) f/f ce . is constant, despite the varying plasma conditions. Onecan see that the ρ i and λ i –normalizations are not effi-cient to collapse the spectra together. Normalization on λ e gives the same result as for λ i . At the same time, thenormalization on ρ e and f ce bring the spectra close toeach other. This confirms that the electron gyroradius ρ e and/or cyclotron periods of the particles are impor-tant in the dissipation.With the present observations it is not possible to dis-tinguish between ρ e and cyclotron periods as far as thereis a clear correlation between ρ e and B . We can ar-gue, however, that if the cyclotron period had been theonly dissipation scale, the turbulent cascade would havestopped by the cyclotron damping of Alfv´en waves at f ci showing an exponential cut-off at this scale [20]. Solar wind observations show the contrary: the turbulent spec-trum continues up to electron scales. Thus, we concludethat ρ e is the dissipation scale of magnetic turbulence inthe solar wind, but we can not exclude that at f ci and f ce there is a partial dissipation by cyclotron damping.In the present letter we analyzed high resolution mag-netic spectra from MHD to electron scales. We show herefor the first time that whatever the plasma conditions andthe solar wind regime, slow or fast, the magnetic spectrahave similar shape. This indicates a certain universal-ity, at least for the quasi-perpendicular configuration be-tween B and V . Such a quasi-universal spectrum consistsof three parts: two power-laws and an exponential do-main. At MHD scales it follows a Kolmogorov’s ∼ k − / spectrum, in agreement with previous observations. Be-tween f ci and Doppler shifted λ i and ρ i a spectral break isobserved. Above the break, it follows a k − . power-law.At scales, of the order of kρ e ≃ (0 . −
1) it follows an ex-ponential exp( − a ( f /f ) / ). This is the first observationof an exponential magnetic spectrum in space plasmas.Such spectra were predicted by the anisotropic dissipa-tion model of Gogoberidze [8]. The author suggests thatsmall scale fluctuations with oblique k are diffused onoblique fluctuations from the inertial range. This diffu-sion is anisotropic and it gives an ∼ exp( − k ∆ α/ ) spec-trum in the dissipation range, where ∆ α = α ⊥ − α k isthe difference between the energy diffusion scaling per-pendicular and parallel to B .It is a long standing problem to distinguish between therole of different kinetic scales in space plasmas. We showfor the first time that the role of dissipation scale in spaceplasma turbulence is played by the electron gyroradius,as assumed by several previous authors [6, 10, 21]. Acknowledgments
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