Magnetohydrodynamic (MHD) turbulence in the solar wind is observed to show
the spectral behavior of classical Kolmogorov fluid turbulence over an inertial
subrange and departures from this at short wavelengths, where energy should be
dissipated. Here we present the first measurements of the electric field
fluctuation spectrum over the inertial and dissipative wavenumber ranges in a
$\beta \gtrsim 1$ plasma. The $k^{-5/3}$ inertial subrange is observed and
agrees strikingly with the magnetic fluctuation spectrum; the wave phase speed
in this regime is shown to be consistent with the Alfv\'en speed. At smaller
wavelengths $k \rho_i \geq 1$ the electric spectrum is softer and is consistent
with the expected dispersion relation of short-wavelength kinetic Alfv\'en
waves. Kinetic Alfv\'en waves damp on the solar wind ions and electrons and may
act to isotropize them. This effect may explain the fluid-like nature of the
solar wind.Comment: submitted; 4 pages + 3 figure

Bieber

F. S. Mozer

H. Reme

H. J. Beinroth

P. J. Kellogg

P. C. Filbert

S. D. Bale

T. S. Horbury

English

arXiv

10.12.13 KB. Ok to add published version to spiral, APS polic

Bale, SD

Kellogg, PJ

Mozer, FS

Horbury, TS

Reme, H

Spiral - Imperial College Digital Repository

PRL 94, 215002 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending3 JUNE 2005Measurement of the Electric Fluctuation Spectrum of Magnetohydrodynamic TurbulenceS. D. Bale,1 P. J. Kellogg,2 F. S. Mozer,1 T. S. Horbury,3 and H. Reme41Department of Physics and Space Sciences Laboratory, University of California, Berkeley, California 94720, USA2School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA3The Blackett Laboratory, Imperial College, London SW7 2BW, United Kingdom4CESR, 31400 Toulouse, France(Received 23 February 2005; published 2 June 2005)0031-9007=Magnetohydrodynamic (MHD) turbulence in the solar wind is observed to show the spectral behaviorof classical Kolmogorov fluid turbulence over an inertial subrange and departures from this at shortwavelengths, where energy should be dissipated. Here we present the first measurements of the electricfield fluctuation spectrum over the inertial and dissipative wave number ranges in a  * 1 plasma. Thek5=3 inertial subrange is observed and agrees strikingly with the magnetic fluctuation spectrum; the wavephase speed in this regime is shown to be consistent with the Alfvén speed. At smaller wavelengths ki 1 the electric spectrum is enhanced and is consistent with the expected dispersion relation of short-wavelength kinetic Alfvén waves. Kinetic Alfvén waves damp on the solar wind ions and electrons andmay act to isotropize them. This effect may explain the fluidlike nature of the solar wind.DOI: 10.1103/PhysRevLett.94.215002 PACS numbers: 52.35.Ra, 52.35.BjTurbulence is ubiquitous in astrophysical plasmas; tur-bulent processes are thought to a play role in cosmic rayand energetic particle acceleration and scattering [1], ad-vection dominated accretion flows, and perhaps solar/stel-lar wind acceleration. Yet key aspects of the physics ofturbulence in magnetized plasmas are poorly understood,in particular, the physics of dissipation at small scales. Theclassical scenario of magnetohydrodynamic turbulence isthus: fluctuations in the plasma are driven at some large‘‘outer’’ scale and decay by interacting locally in k space.Eddies at some scale k exchange energy with eddies atnearby spatial scales, possibly as a three-wave or higherorder interaction [2–4], with the resulting net flow ofenergy to smaller spatial scales (larger k); this cascade ofenergy occurs over an ‘‘inertial subrange’’ of k space andcan be shown to predict a power spectrum that scales ask5=3. At the smaller scale of the ion thermal gyroradius,ki  1, the ions become demagnetized and the plasmacan no longer behave as a simple fluid; the turbulent energyis then thought to be damped on the thermal plasma byLandau or transit time damping. However, the details ofthis damping process are not known and there are fewreported measurements in this regime of k space.Observations of the magnetic spectrum show break-points near ki  1, above which the magnetic spectrumtypically becomes steeper [5–8]. This has been interpretedvariously as evidence of kinetic Alfvén waves [5], whistlerwave dispersion [9], and ion cyclotron damping of Alvénwaves [10].Here we report the first measured power spectrum ofelectric fluctuations in solar wind turbulence. The inertialsubrange is clearly evident and follows the magnetic fluc-tuation spectrum. At large wave numbers ki  1, theelectric spectrum is enhanced.Data are used from experiments on the Cluster space-craft. Cluster flies four spacecraft, as a controlled tetrahe-05=94(21)=215002(4)$23.00 21500dron, in an inclined orbit with apogee at 19 Earth radii(RE). From December to May each year, the spacecraft exitthe terrestrial magnetosphere on the dayside and makemeasurements in the solar wind. We use approximately195 min of data during the interval 00:07:00–03:21:51 on19 February 2002, when Cluster was at apogee and spentseveral hours in the ambient solar wind; all of our data arefrom Cluster spacecraft four.The electric field is measured by the electric field andwaves experiment (EFW) [11]; EFW is a double-probeexperiment which measures the floating voltage of 8 cmdiameter current-biased spheres extended on 44 m wires inquadrature. These spheres, as well as the spacecraft, areilluminated by the Sun and emit photoelectrons whichcause the surfaces to charge positive with respect to theplasma. The surfaces attract a return current of thermalelectrons which provide the electrical coupling to theplasma. Systematic variations in this coupling, due tochanging illumination or variations in surface propertiesand work function, are a large source of background noisein EFW at the spacecraft spin period (4 sec) and harmonics.This is discussed more below. EFW measures the electricfield on two orthogonal sensor pairs in the spacecraft spinplane at 25 samples= sec . These two components are ro-tated into X and Y components in the GSE (geocentricsolar ecliptic) coordinate system. Since the GSE Y direc-tion represents the orientation with best symmetry for solarillumination, this component of the electric field is gener-ally less noisy; we use the GSE Y electric field Ey for all ofour analysis. However, at any given instance, Ey is com-posed of data from all four electric field probes, each withslightly different photocoupling to the plasma. We there-fore apply a finite impulse response filter to the data tonotch out the primary perturbations at the spin tone andsome harmonics.2-1  2005 The American Physical SocietyPRL 94, 215002 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending3 JUNE 2005The magnetic field is measured by the fluxgate magne-tometer instrument [12]; three-component magnetic fieldvectors are sampled at 22 samples= sec . In our analysis,we use the GSE Z component of the magnetic field Bz forreasons that are explained below. Moments of the solarwind ion distribution (velocity, density, and temperature)are computed from the ion spectrum measured by theCluster ion spectroscopy experiment [13].Figure 1 shows an overview of the data used in thefollowing analysis; panels (a) and (b) are wavelet spectro-grams and will be discussed below. Panel (c) shows the twocomponents of measured electric field Ex and Ey in GSEcoordinates. Panel (d) show the magnetic field data. Panels(e), (f), and (g) show the plasma density, plasma ion i(ratio of plasma to magnetic pressure), and Alfvén Machnumber. The average ion beta is i  5, average Alfvénspeed vA  40 km=s, and the average solar wind velocityis vsw  347; 4:9;32:6 km=s (in GSE coordinates),over the entire interval. During the interval between 00:30and 00:50, the magnetic field is nearly tangent to theEarth’s bow shock (as per a calculation assuming straightBP1BP1BP2electricmagnetic(a)(b)(c)(d)(e)(f)(g)FIG. 1 (color). Wavelet and time series data of solar windturbulence. From the top down, the five panels show (a) thewavelet spectrogram of Ey, as a function of ki, (b) a similarwavelet spectrogram of Bz, (c) the X and Y components of themeasured electric field, (d) the vector magnetic field, (e) plasmaion density, (f) plasma ion , and (g) the Alfvén Mach number.This entire interval was used for the spectral analysis of Ey andBz. The spectral breakpoints are called out.21500field lines [14]); however, Cluster summary plots of elec-tron and plasma wave data show no evidence of connectionto the shock. All of our data are ambient solar wind.To compute power spectra, the electric field data Ey(25 samples= sec ) were subsampled onto the time tags ofthe magnetic field data Bz (22 samples= sec ) by linearinterpolation; a total of exactly 218 points are used. Thepower spectral density (PSD) was computed using bothfast Fourier transform (FFT) and Morlet wavelet [15]schemes. The FFT was computed as follows: the datainterval was divided into 64 contiguous ensembles oflength 4096 (186 sec); this gives an inherent bandwidthof f  1=186 Hz. To minimize spectral leakage, eachensemble was ‘‘whitened’’ by applying a first-order differ-ence algorithm; the PSD was computed by FFT, then thespectrum was postdarkened [1] and divided by the band-width of the FFT. Since the data are prewhitened, nowindow function was applied before the FFT. The electricfield spectra were then ‘‘cleaned’’ by interpolating over thenarrow band spikes resulting from the spin-associatedsignals described above. A final spectrum was computedas the average of the 64 ensembles. Figure 2 shows the FFTpower spectra of Ey and Bz (in black). Wavelet spectrawere computed by first producing the (complex) FFT of Eyand applying the spectral cleaning (interpolation) to thereal and imaginary parts, at positive and negative frequen-FIG. 2 (color). Power spectral density of electric Ey andmagnetic fluctuations Bz as a function of frequency, computedfrom FFT (black) and Morlet wavelet (red) algorithms. The FFTspectrum of the electric field (upper panel) shows the effect ofnotch filters and residual spin-tone data.2-2FIG. 3 (color). The wavelet (upper) and FFT (lower) powerspectra of Ey (green) and Bz (black) binned as a function of wavenumber ki (and offset for clarity) in panel (a). The electricspectra are multiplied by factors to lie atop the magnetic spectra.The spectrum is Kolmogorov k5=3 over the interval ki 20:015; 0:45; a spectral breakpoint occurs for both Ey and Bzat ki  0:45. A second breakpoint occurs for the electricspectrum at ki  2:5 above which the electric spectrum ismore exponential. Panel (b) shows the ratio of the electric tomagnetic spectra in the plasma frame; the average Alfvén speed( vA  40 km=s) is shown as a horizontal line. The red line is afitted dispersion curve, discussed in the text. Panel (c) showsboth the cross coherence of Ey with Bz (as blue dots with errorbars) and the correlation between the electric and magneticpower (as black dots).PRL 94, 215002 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending3 JUNE 2005cies. An inverse FFT restores the cleaned signal and aMorlet wavelet spectrogram was computed from thiscleaned Ey, as well as the original Bz. The wavelet has136 log-spaced frequencies; the final wavelet PSD is com-puted as the square of the spectrum averaged over time.The wavelet PSD is also shown in Fig. 2 (in red). Thewavelet spectrum extends to lower frequencies than theFFT, which is composed of ensembles of smaller dataintervals; however, these very low frequencies lie belowthe ‘‘cone of influence’’ and are unreliable [15]. Here werestrict our interpretation to the region where the FFT andwavelet spectra agree. The FFT electric spectrum in Fig. 2shows clearly the effect of the notch filters and residualspin-harmonic spikes. The wavelet PSD, with its muchlarger bandwidth, mostly averages over these residual fea-tures although a depression near the notched portion of thespectrum can be seen. The FFT and wavelet PSD spectraagree remarkably well for both electric and magneticfields.Of course, our (human) scheme of measuring timemeans little to the solar wind plasma, so there is littlereason to expect the data to be inherently organized by apower spectrum in Hertz. Since the solar wind is super-Alfvénic (Fig. 1), the phase speed vA of the Alfvénicfluctuations is much less than the wind speed itself; hencethe measured frequency spectrum is actually a Doppler-shifted wave number spectrum !  kvsw. This is oftencalled Taylor’s hypothesis and might not be considered tohold at large wave numbers, especially if waves are presentwith phase speeds greater than the solar wind speed (suchas whistler waves).As discussed above, it is considered that the fluidlikebehavior of the wind breaks down at near ki  1, there-fore ki is a natural parameter for organization of thepower spectrum. The top panel of Fig. 3 shows the FFTand wavelet power spectra organized by ki, instead offrequency. For the FFT spectrum, the local values of jvswj,Ti, and jBj are used to compute k  !=vsw and the thermalion gyroradius i  vi=ci averaged over each (186 sec)ensemble; the Ey and Bz power spectra are then interpo-lated onto a linearly spaced set of values ki 2 0:006; 10.Since solar wind parameters vary slightly in each en-semble, this also has the effect of smearing (averaging)over the narrow band interference in the FFT PSD of Ey.The wavelet spectrograms are time averaged onto 4 secintervals and then interpolated onto a set of log-spacedvalues of ki; panels (a) and (b) of Fig. 1 show these scaledspectrograms as a function of time. In Figs. 1(a) and 1(b),the fluctuation power has been divided by k5=3 to high-light fluctuations above the average spectrum of the inertialrange. The electric and magnetic wavelet spectrograms arethen averaged to compute the composite spectra inpanel (a) of Fig. 3.Between ki  0:015 and 0.45, the wavelet and FFTspectra of electric and magnetic fluctuations show power21500law behavior with indices of k1:7, which is consistent withthe Kolmogorov value of 5=3. Both Ey and Bz showbreakpoints at near ki  0:45; the magnetic spectrumbecomes steeper with an index k2:12, while the electricspectrum becomes enhanced. As discussed above, steepmagnetic spectra have been observed previously [6,8].Above ki  0:45, the electric spectrum is a power-lawlike k1:26 to ki  2:5. Above this second breakpoint, anexponential expki=12:5 better fits the spectrum. Atthese higher wave numbers, the electric field data are noisyand show harmonics of the spin tone (as shown above). Totest the validity of these data, we perform two analyses.The black dots of panel (c) in Fig. 3 show the correlationbetween the electric and the magnetic wavelet power as afunction of ki. It can be seen that the fluctuations arestrongly correlated through the inertial range (with coeffi-cient  1), remain well correlated between the two break-2-3PRL 94, 215002 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending3 JUNE 2005points ki 2 0:45; 2, and begin to lose correlationquickly above the second breakpoint. A wavelet cross-spectral analysis (between Ey and Bz) was also com-puted; the blue bars show the cross-spectral coherence,with 1 error bars, also as a function of ki. Again, Eyand Bz are strongly coherent through the inertial rangeand past the first breakpoint. We conclude that the electricand magnetic spectra are physical and well correlated up tothe second spectral breakpoint. Above ki  2:5 it isdifficult to assess the quality of the data. If electrostaticwaves are present, there is no expectation of correlationwith B; however, in this initial study, we cannot eliminatethe possibility of systematic noise at these frequencies.Additionally, the effects of low pass filters on both theEFW and FGM experiments may modify the spectra atthese highest (ki > 3) frequencies.To estimate the phase speed of the fluctuations, we useFaraday’s law and compute the ratio of the electric andmagnetic spectra. Since the electric field measurements aremade in the spacecraft (unprimed) frame, we need toLorentz transform to the plasma (primed) frame by ~E ~E0  ~vsw  ~B. Panel (b) of Fig. 3 shows the phase speedvki E0ykiBzkiEykiBzki vx  vzBxBz; (1)where vx, vz, Bx, and Bz are the average x and z compo-nents of the solar wind velocity and magnetic field. The iondata are sampled at 0:25 samples= sec ; this necessitatesusing averages of vsw in the Lorentz transformation. Theblack dots in panel (b) are computed from the waveletspectrum, while the blue line is computed from the FFTspectrum. The average Alfvén speed vA  40 km= sec isshown as a horizontal bar. Over, and even below, theinertial range ki 2 0:015; 0:4, the phase speed is con-sistent with the local Alfvén speed; this is strong evidenceof the Alfvénic nature of the cascade. The red curve inpanel (b) is a fit of the function v01 k22i  to the FFTcurve, where v0 is a free parameter which finds a best fit atv0  55 km=s. This function approximates the dispersionrelation of kinetic Alfvén waves [16,17]. The cold-plasmawhistler wave phase speed goes as v  ki1=2vAabove !>ci, i.e., linear with ki, and would form amuch shallower dispersion above ki  1 than is observedin panel (b) of Fig. 3. This leads us to believe that the21500Alfvén waves in the inertial subrange eventually disperseas ‘‘kinetic’’ Alfvén waves above ki  1, becoming moreelectrostatic and eventually damping on the thermalplasma. Plasma heating by linear dissipation of kineticAlfvén waves at   1 has been studied in the context ofaccretion flows [16]. There it was found that Landau andtransit time damping contribute to both proton and electronheating at short wavelengths, which is enhanced for higher. Kellogg [18] computed the level of electric field fluc-tuations required to stochastically thermalize protons to1 AU in the solar wind; he found that a spectral density ofE2  1011 V=m2 Hz1 was sufficient. This is 1 order ofmagnitude less than our observed levels (Fig. 2). It is,therefore, plausible to conclude that the observed electricspectrum is responsible for isotropizing the solar windprotons and may be the mechanism by which the solarwind maintains its fluidlike characteristics.Cluster data analysis at UC Berkeley is supported byNASA Grant Nos. NAG5-11944 and NNG04GG08G.2-4[1] Bieber et al., J. Geophys. Res. 98, 3585 (1993).[2] F. Waleffe, Phys. Fluids 4, 350 (1992).[3] P. Goldreich and S. Sridhar, Astrophys. J. 485, 680 (1997).[4] C. S. Ng and A. Bhattacharjee, Astrophys. J. 465, 845(1996).[5] R. J. Leamon et al., J. Geophys. Res. 104, 22 331 (1999).[6] R. J. Leamon et al., J. Geophys. Res. 103, 4775 (1998).[7] M. L. Goldstein et al., J. Geophys. Res. 99, 11 519 (1994).[8] H. J. Beinroth and F. M. Neubauer, J. Geophys. Res. 86,7755 (1981).[9] O. Stawicki, S. P. Gary, and H. Li, J. Geophys. Res. 106,8273 (2001).[10] S. P. Gary, J. Geophys. Res. 104, 6759 (1999).[11] G. Gustafsson et al., Space Sci. Rev. 79, 137 (1997).[12] A. Balogh et al., Space Sci. Rev. 79, 65 (1997).[13] H. Reme et al., Space Sci. Rev. 79, 303 (1997).[14] P. C. Filbert and P. J. Kellogg, J. Geophys. Res. 84, 1369(1979).[15] C. Torrence and G. P. Compo, Bull. Am. Meteorol. Soc.79, 61 (1998).[16] E. Quataert, Astrophys. J. 500, 978 (1998).[17] S. R. Cranmer and A. A. van Ballegooijen, Astrophys. J594, 573 (2003).[18] P. J. Kellogg, Astrophys. J 528, 480 (2000).

Physical Review Letters

Measurement of the electric fluctuation spectrum of magnetohydrodynamic turbulence

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Measurement of the Electric Fluctuation Spectrum of Magnetohydrodynamic Turbulence

Measurement of the electric fluctuation spectrum of magnetohydrodynamic
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Measurement of the electric fluctuation spectrum of magnetohydrodynamic turbulence

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