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Turbulence and Intermittency in the Solar Wind

Turbulence and Intermittency in the Solar Wind. R. Bruno 1 In collaboration with : 3 Carbone, V., 1 Bavassano, B., 1 D'Amicis, R., 4 Sorriso, L., 5 Pietropaolo, E 1 Istituto di Fisca dello Spazio Interplanetario, INAF, Rome , Italy

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Turbulence and Intermittency in the Solar Wind

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  1. Turbulence and Intermittency in the Solar Wind R. Bruno1 In collaborationwith: 3Carbone, V., 1Bavassano, B., 1D'Amicis, R., 4Sorriso, L., 5Pietropaolo, E 1 Istituto di Fisca dello Spazio Interplanetario, INAF, Rome, Italy 3 Dipartimento di Fisica, Universita` della Calabria, Rende, Italy. 4 Liquid Crystal Laboratory, INFM-CNR, Rende, Italy. 5 Dipartimento di Fisica, Universita` di L'Aquila, Italy. Turbulence and Multifractals in Geophysics and SpaceBelgian Institute for Space Aeronomy Brussels, Belgium 09-11 June 2010 [background pictureadaptedfromChanget al., 2004]

  2. Interplanetary fluctuations show a well developed turbulence spectrum lT lC • Correlative Scale/Integral Scale: • the largest separation distance over which eddies are still correlated. • Taylor scale: • The scale size at which viscous dissipation begins to affect the eddies. • it marks the transition from the inertial range to the dissipation range. • Kolmogorov scale: • The scale size that characterizes the smallest dissipation-scale eddies typical IMF power spectrum in at 1 AU [Low frequency from Bruno et al., 1985, high freq. tail from Leamon et al, 1999] (Batchelor, 1970) Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  3. Interplanetary fluctuations show a well developed turbulence spectrum lT lC • Correlative Scale/Integral Scale: • the largest separation distance over which eddies are still correlated. • Taylor scale: • The scale size at which viscous dissipation begins to affect the eddies. • it marks the transition from the inertial range to the dissipation range. • Kolmogorov scale: • The scale size that characterizes the smallest dissipation-scale eddies typical IMF power spectrum in at 1 AU [Low frequency from Bruno et al., 1985, high freq. tail from Leamon et al, 1999] (Batchelor, 1970) Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  4. Large scales: Origin of K-1 (flicker noise) Ulysses • Observed in a wide varietyofsystems • first interpretationfor the IMF due toMatthaeus and Goldstein, 1986: • withinAlfvénicradius, mergingofmagneticstructures • multiplicativeprocess and characteristiclengthslognormallydistributed • spectrumwould derive from a superposition ofseveralsimilarsamplesrecordedduring long enoughtimeinterval • It can beshown (Montroll and Shlesinger, 1982) that the spectrumS(f)~1/f Matthaeus et al., ApJ, 2007 Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  5. Large scales: Origin of K-1 (flicker noise) • Observed in a wide varietyofsystems • first interpretationfor the IMF due toMatthaeus and Goldstein, 1986: • withinAlfvénicradius, mergingofmagneticstructures • multiplicativeprocess and characteristiclengthslognormallydistributed • spectrumwould derive from a superposition ofseveralsimilarsamplesrecordedduring long enoughtimeinterval • It can beshown (Montroll and Shlesinger, 1982) that the spectrumS(f)~1/f • Numericalmodeling(Dmitruket al., 2002-2004): • Upwardtraveling low frequencywaves at coronal base are capableofself-generating1/f noise in density and B • 1/f notpresent in similarhydrodynamicssimulations (roleofmagneticfield) CompensatedspectrumbyDmitruket al., 2002-2004 Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  6. Large scales: Origin of K-1 (flicker noise) • Observed in a wide varietyofsystems • first interpretationfor the IMF due toMatthaeus and Goldstein, 1986: • withinAlfvénicradius, mergingofmagneticstructures • multiplicativeprocess and characteristiclengthslognormallydistributed • spectrumwould derive from a superposition ofseveralsimilarsamplesrecordedduring long enoughtimeinterval • It can beshown (Montroll and Shlesinger, 1982) that the spectrumS(f)~1/f • Numericalmodeling(Dmitruket al., 2002-2004): • Upwardtraveling low frequencywaves at coronal base are capableofself-generating1/f noise in density and B • 1/f notpresent in similarhydrodynamicssimulations (roleofmagneticfield) • Full disk magnetograms(Nakagawa & Levine, 1974): • the1/k spectral region found in photospheric observations seems to be a clear candidate for involvement in the appearance of the interplanetary 1/f signal. Nakagawa and Levine, 1974 Possible link between the structuredsurfaceof the sun and 1/f scaling in IMF

  7. Possible link alsobetween the structuredcoronal base and in-situobservationsof density fluctuations (Telloniet al., 2009) Proton density fluctuations* at 2.3 RSUN (UVCS/SOHO)are comparedtoHelios 2measurementsperformed at 64 RSUN Similarcoronalimprintf-2at largescales Telloniet al, ApJ, 706, 238, 2009 (*) the fluctuations of Lyαemission observed in corona can be generally associated with electron density variations ne [Morgan et al., 2004, Bemporad et al., 2008, Telloni et al., 2009]

  8. 1 hour Gaussianityoflarge scale fluctuations 1 day Proton Gyrofrequency flatness • No intermittency at these scales • Fluctuations are already decorrelated (Gaussian) for time delays << stream duration Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  9. dB-dValignment vs scale and heliocentricdistance As the windexpandsturbulenceevolution and compressive effectsdecoupledB-dV in fast wind, startingfrom the largestscales [seeliterature in Tu and Marsch 1995, Bruno and Carbone 2005] Helios 2 observations Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  10. Poweranisotropy at differentscales Largescales are scarselyanisotropic 2 Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  11. Intermediate scales characterized by turbulence scaling K-5/3 Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  12. The presenceof a Kolmogorovscalingimpliesthat Since Z-cannotbeofsolarorigin, the conditionaboverequires generation mechanisms at work: In the ecliptic, velocityshear (Coleman, 1968) and dynamicalignment (Dobrowolnyet al., 1982, Matthaeuset al., 2008) At high latitude, parametricdecayisabletoreproduceUlyssesobservations (Bavassanoet al, 2000, Malaraet al, 2000) Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  13. dB-dValignment vs scale and heliocentricdistance Intermediate scales are the mostAlfvénicwithin fast wind [seeliterature in Tu and Marsch 1995, Bruno and Carbone 2005] Helios 2 observations Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  14. Poweranisotropy at differentscales smallscales are stronglyanisotropic, partly due toAlfvénicfluctuations 20 min 2 Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  15. First anisotropystudy in termsof k and k// in the solarwindbyBieberet al., 1996 Analysisperformed in slow wind Kdominates on k// asweanalyzedirections at largerangleswithmagneticfield Turbulencemadeof 2D+SLAB [Matthaeuset al., 1990] Analysisperformed in slow wind SimilarresultsbyHorburyet al., 2008 2D Thus, the slab turbulence due to Alfvénic fluctuations would be a minor component of interplanetary MHD turbulence SLAB Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  16. MHD turbulence in terms of R and C (scale < 1hr) SLOW WIND FAST WIND 0.9 AU 0.9 AU [adaptedfrom Bruno et al., 2007] Alfvénicpopulation Thisdifferenceaffects the anisotropy Advectedstructureswithmagneticenergyexcess Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  17. Anomalous scaling of IMF is well inside inertial range Inertial range flatness Proton Gyrofrequency Proton Gyrofrequency Intermittency starts within the inertial range at about 2 decades away from the dissipation range. Intermittency at these scales doesn’t seem to be due to dissipative processes

  18. LocalIntermittencyMeasuretechniqueallowsto locate intermittentevents (Farge, 1990) Cluster s/c 3 data (Bruno and Carbone, 2005) The waiting times are distributed according to a power law PDF(Δt) ~Δt-β long range correlations Thus, the generating process is not Poissonian. Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  19. LocalIntermittencyMeasuretechniqueallowsto locate intermittentevents (Farge, 1990) Cluster s/c 3 data thisintermittenteventmarks the borderbetweendifferent plasma regions Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  20. LocalIntermittencyMeasuretechniqueallowsto locate intermittentevents (Farge, 1990) Cluster s/c 3 data Vz Vx Vy Bz Twodifferentregions are identified and fluctuationswithineachofthem are Alfvénic Bx By Magneticfield in Alfvénunits Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  21. Severalstudies on eventssimilarto the previousonecontributedto formulate the spaghetti-likemodel [Bruno et al., 2001] A spacecraftcrossingthisstructurewould record a seriesofintermittenteventsembedded in anoceanofGaussian, Alfvénicfluctuations s/c trajectory [adaptedfrom Bruno et al., 2001] the sharp directional changes of magnetic field occurring typically across interwoven flux tubes can lead to slow magnetic reconnection and additional heating of the solar wind. [Vöröset al., 2010] Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  22. Several contribution have been given in this direction in the past years (Thiemeet al., 1988, 1989;Tu et al., 1989, 1997;Tu and Marsch, 1990, 1993;Bieber and Matthaeus, 1996; Crooker et al., 1996; Bruno et al., 2001, 2003, 2004; Chang and Wu, 2002; Chang, 2003; Chang et al., 2004; Tu and Marsch, 1992, Chang et al., 2002, Borovsky, 2006, 2009, Li, 2007, 2008, Vörös et al., 2010) Similar pictures 2D+SLAB [Tu and Marsch, 1992] [Bieber, Wanner and Matthaeus, 1996] 2D+SLAB Structuresproducingintermittencymightbeeitherofsolar nature and advectedby the wind or locallygeneratedbynon-linearturbulenceevolution (Chang et al., 2002)

  23. Thasmallestscalescharacterizedbysteeperscaling Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  24. Power anisotropy study from Podesta, 2009 KAWsdissipation?? [curves arbitrarily shifted vertically] KAWscascade • Inertialrange: • energycascadedirectedprimarilyperpendicularto the meanmagfield [Shebalinet al., 1983; Oughtonet al., 1994; Matthaeuset al., 1996] • poweranisotropyincreaseseswithwavenumber • “Dissipation” range: • The newcascade (KAWs?) starts at ki 1 [i.e. S/C  0.5Hz, in this case] when the fluid-likebehaviorbreaks down • poweranisotropyincreaseseswithwavenumber • The secondpeakmarksbeginningofKAWsdissipation? [q angle between local mean field and radial direction]

  25. Observational evidence for Alfvén waves – KAWs transition in the solar wind [Bale et al., 2005, Cluster data] wavelets electric field and magnetic field k−5/3 inertial sub-range is observed up to ki~1 (Electricspectrumarbitrarily offset tooverlapmagneticspectrum) FFT the wave phase speed in this regime is shown to be consistent with the Alfvén speed The red curve approximates the dispersion relation forKAWs (vk2i2). For whistler waves (vki). itwouldrunmuchlowerfor ki>1 [Bale et al., 2005] Good correlation between the electric and magnetic power (as black dots) and good cross-coherence of Ey with Bz (as blue dots) suggest KAW Moreover, Sahraoui et al., 2009 extended the study to the electron gyroscale where they identified dissipative processes of KAWs

  26. A newcascadingrangebeyondfciconfirmedbybyAlexandrovaet al., 2008, 2009 using Cluster magneticobservations Intermittency increases (Alexandrovaet al., 2008, 2009) (Alexandrovaet al., 2008, 2009) The presenceof a power-lawspectrum (insteadof a roughexponentialcutoff) and the increaseofintermittencysuggest the presenceof a newcascadingrange (Stawickiet al., 2001, Baleet al., 2005, Sahraouiet al., 2006, 2009) The cascadehas a compressiblecharacter. Compressibilityseemstogovern the spectralindexk-7/3+2awherea is the compressibility (Alexandrovaet al., 2008, 2009)

  27. A newcascadingrangebeyondfciconfirmedbybyAlexandrovaet al., 2008, 2009 using Cluster magneticobservations Intermittencyincreases (Alexandrovaet al., 2008, 2009) (Alexandrovaet al., 2008, 2009) • Hollweg, 1999: • KAWs become strongly compressive when ki1. • The compression is accompanied by a magnetic field fluctuation δB‖ such that the total pressure perturbation δptot ≈ 0

  28. Intermittency: differentconclusionsin the “dissipation” rangebyKyaniet al., 2009 multifractal monofractal [Kiyaniet al., PRL 2009] Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  29. Anisotropy: Perriet al., 2008, 2009, performed a minimum variancestudy in thisfrequencyrange, focusing on the anisotropyof the fluctuations Cluster mag data resolutionf=22Hz (Perri et al., 2009) (Perri et al., 2009) Power law Time series and probability distribution function for the max eigenvalue at three different scales (46s, 1.5s, 0.2s) Intermittent behavior • The PDFs evolve with the scale, becoming power laws at scales smaller than the ion cyclotron scale. • Similarbehaviourfor the othereigenvalues

  30. Looking at the distributionofangularfluctuationsofBvector on a time scale of 0.045 sec (i.e.22Hz) ~6 minutes Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  31. Looking at the distributionofangularfluctuationsofBvector on a time scale of 0.045 sec (i.e.22Hz) ~6 minutes Distributionobtainedfrom Cluster 1,2,3,4 data at the time scale of 0.045s Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  32. Sametypeofdistribution at largerscales, within the inertialrange whistlers and KAW ? Alfvénic turbulence Coherent structures a [°] (Bruno et al., 2004) Distributionobtainedfrom Cluster 1,2,3,4 data at the time scale of 0.045s DistributionobtainedfromHelios data at the time scale of 6s, high frequencyterminationof the inertialrange Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  33. A Toy Model to reproduce Cluster observations a We allow the tip of a vector to move randomly on the surface of a sphere. The distribution of the angle a between 2 successive orientations of the vector follows the double-lognormal distribution obtained from Cluster observations Distributionobtainedfrom Cluster 1,2,3,4 data at the time scale of 0.045s Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  34. moreover, weaddedcompressions in the fieldintensity • wetriedtoreproduce the same compressive levelfound in real data (similarspectra) • compressionsrevealedto play animportantrole in thistoymodel (seenextslides) Compressibility(f)  S|B|(f)/SC(f) Compressionsadded: s|B|/<|B|>~3% Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  35. Max eigenvaluebehaviorbyPerriet al., 2008, 2009 Cluster mag data resolution f=22Hz (Perri et al., 2009) (Perri et al., 2009) Power law Time series and probability distribution function for the max eigenvalue at three different scales (46s, 1.5s, 0.2s) Intermittent behavior • The PDFs evolve with the scale, becoming power laws at scales smaller than the ion cyclotron scale. • Similarbehaviourfor the othereigenvalues Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  36. Max eigenvaluebehaviorfromToyModel (from Toy-Model) (from Toy-Model) The Toy Model reproduces qualitatively the results obtained in the solar wind Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  37. Max eigenvalue behavior from Toy Model (from Toy-Model) (from Toy-Model) Gaussiancharacter intermittent character Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  38. Time behavior of the angle Qbetween minvar direction and mean field direction at three different scales. (from Toy-Model) (from Perri et al., 2009) Similarprofilesobtainedfromtoymodel Resultsfrom Cluster Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  39. PDFs of the angle  between Minvar direction and local B direction in the Solar Wind (from Perri et al., 2009) (from Toy-Model) Dt=46 sec Dt=1.5 sec Dt=0.2 sec Striking similarity between these PDFs and those found in the SW Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  40. The effect of compressions (from Toy-Model) (from Perri et al., 2009) Dt=46 sec Dt=1.5 sec Dt=0.2 sec Withcompressions W/O compressions Without compressions the toy doesn’t work well

  41. Also the typeofDQdistributionplays a role, the samelevelofcompressionisnotlongersufficient extractedfromGaussiandistribution extractedfromuniformdistribution

  42. summary • k-1frequencyrange: • possible link with the structured surface of the sun • fluctuationsare Gaussian • poweranisotropy (P/P// >1) increaseseswithwavenumber • k-5/3inertialrange: • fluctuationsmixedwithadvectedstructures • intermittencymore than 2 freq. decadesbefore the “dissipationrange”, • intermittencylinkedto compressive structures and currentsheets. • possibleflux tube structure • poweranisotropy (P/P// >1) increaseseswithwavenumber • Beyond the protoncyclotronfreq.: • newcascade(KAWs or/and whistlers) • intermittencyincreasesagain, • importantroleplayedbycompressionsconfirmedbyanisotropystudies Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  43. Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  44. Bavassanoet al., 1982 foundthatanisotropyofmagneticfieldfluctuationsincreaseswithheliocentricdistance Analysisperformedwith the samecorotatingstreamobserved at differentheliodistances [Bavassanoet al., 1982] fieldcompressibilityincreasesewithheliocentricdistance anisotropyofmagneticfieldfluctuationsincreaseswithheliocentricdistance Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  45. Itwasshownthatanisotropyisreducedbyremovingintermittenteventsfrom |B|, i.e. the most compressive magneticeventsoutside a normaldistribution [Bruno et al., 1999] Local Intermittency Measure technique (Farge et al., 1990; Farge, 1992) based on waveletdecomposition The Flatness Factorof the wavelet coefficients at a given scale  , i.e. LIM2, is equivalent to the Flatness Factor FF of data at the same scale (Meneveau, 1991 ) [adaptedfrom Bruno et al., 1999] Thus, values of FF()>3 allow to localize events which lie outside the Gaussian statistics and cause Intermittency.

  46. PDFs of other parameters can be rescaled The PDF of dr, dB2, drV2, dVB2can be rescaled under the following change of variable (Hnat et al., 2002, 2003, 2004) The heightof the peaksrescales • Thisstudyshowedthat: • the distributionis non Gaussianbutitisstable and symmetric and can bedescribedby a single parameter monofractal • The process can bedescribedby a finite rangeLévywalk (scales up to 26 hours) • A Fokker-Planckapproach can beusedtostudy the dynamicsof PDF(db2)

  47. Some remarks Compressive eventsincreaseintermittency Intermittentevents can increase the poweranisotropyoffluctuations Then: compressive eventsmight play a role in poweranisotropy Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  48. Some remarks Compressive eventsincreaseintermittency Intermittentevents can increase the poweranisotropyoffluctuations Then: compressive eventsmight play a role in poweranisotropy Withthis in mind, we look at poweranisotropybeyondfC [Leamon et al., 1998, 1999, Bale et al., 2005, Hamilton et al., 2008, Podesta, 2009, Sahraoui et al., 2009, etc…] Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010

  49. These flux tubes should look like interplanetary flux ropes WIND data, 20-09-1995 Force-free structure for which (Feng et al., 2007) (Lundquist, 1950) Some helicitymesureshouldbeabletounravel the presenceofthesestructures

  50. Measurements in the solar wind, onboard a spacecraft, are essentially collinear along the radial direction, so a full Fourier decomposition is not possible and we have to reduce to one-dimensional spectra (reduced spectra) Matthaeus and Goldstein (1982) provided the following expression for the reduced spectrum of Hm The tensor S23is the Fourier transform of the correlation function R23 In practice, if collinear measurements are made along the X direction, the reduced magnetic helicity spectrum is given by: where Y and Z are the Fourier transforms of By and Bz components, respectively 50

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