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Turbulence and Intermittency in the Solar Wind

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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]

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

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

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

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

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

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]

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

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

Poweranisotropy at differentscales

Largescales are scarselyanisotropic

2

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

Intermediate scales characterized by turbulence scaling K-5/3

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

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

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

Poweranisotropy at differentscales

smallscales are stronglyanisotropic,

partly due toAlfvénicfluctuations

20 min

2

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

First anisotropystudy in termsof k and k// in the solarwindbyBieberet al., 1996

Analysisperformed in slow wind

Kdominates 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

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

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

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

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

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

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

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)

Thasmallestscalescharacterizedbysteeperscaling

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

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 ki 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]

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 ki~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 (vk2i2). For whistler waves (vki). itwouldrunmuchlowerfor ki>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

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)

A newcascadingrangebeyondfciconfirmedbybyAlexandrovaet al., 2008, 2009 using Cluster magneticobservations

Intermittencyincreases

(Alexandrovaet al., 2008, 2009)

(Alexandrovaet al., 2008, 2009)

- Hollweg, 1999:
- KAWs become strongly compressive when ki1.
- The compression is accompanied by a magnetic field fluctuation δB‖ such that the total pressure perturbation δptot ≈ 0

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

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

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

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

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

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

- 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

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

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

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

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

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

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

Also the typeofDQdistributionplays a role, the samelevelofcompressionisnotlongersufficient

extractedfromGaussiandistribution

extractedfromuniformdistribution

- 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

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

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

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.

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)

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

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

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

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

Results in the solar wind do not show a definite sign

Within the MHD regime, Hm has no clear dominant sign in the inertial range, especially at high frequencies

(Matthaeus and Goldstein, 1982)

(Goldstein et al., 1991)

Adopting the wavelet transform to study Hm in the solar wind

a wavelet is a localized function which can be translated and re-scaled

unlike the Fourier transform, the wavelet transform allows to unfold a signal both in time and frequency (or space and scale).

Wavelet analysis is particularly useful to study non-stationary processes

Wy(k,t) and Wz(k,t) wavelet transform of By(t) and Bz(t)

Normalized magnetic helicity

The normalized magnetic helicity(sm) helps to easily estimate the "handedness“ of magnetic field fluctuations

Wavelet analysis on magnetic helicity within 1 day of slow wind at 0.3 AU

sm

Helios 2

Wavelet analysis on magnetic helicity within 1 day of slow wind at 0.3 AU

sm

rectangles indicate limits in scale and time

BZ

BY

mixed helicity

negative helicity

(right-handed)

BY

BZ

sm

Wavelet analysis on magnetic helicity within 1 day of WIND data containing a flux-rope

from 11.6 to 12.8

Band pass filtered between 15 min and 2 hrs

(negative helicity, right-handed)

from 12.8 to 14.2

WIND data, 20-09-1995

Flux rope spotted by Feng et al., 2007

Preliminarymagnetichelicityanalysisshows some similaritiesduringACE and ULYSSES alignment in 2007

1AU

1.4AU

1

2

3

4

5

1

2

3

4

5

Scale[hr]

Scale[hr]

Some of the helicity structures seen by ACE at certain scales reach Ulysses, 0.4 AU away in spite of the fact that the stream has strong velocity gradients in it

More quantitative analysis is needed.

Coherent, helical structures observed in the solar wind recall magnetic field lines trapped in filamentary structures dominated by 2D turbulence as in the model by Ruffolo et al., 2003, and Chuychai et al., 2007

slab dominated

2D dominated

(Ruffolo et al., 2003, Chuychai et al., 2007)

(Bruno et al., in preparation)

Some similarity is found between IMF coherent structures and Ruffolo et al (2003) and Chuychai et al. (2007) model

In the 2D turbulence field lines follow contours of constant “a” remaining trapped near some (x,y) point

Recent 2.5D compressible MHD simulations (Servidio et al., 2008) also showed the formation of islands of well defined magnetic helicity

sm

(Servidio et al., 2008)

local field

- In the presence of a mean field along z the islands are the footprints of elongated flux-tubes
- Current sheets are formed locally between the islands

mean field

sm

(Servidio et al., 2008)

Filamentary structures, connected to the sun, would cause the observed (Mazur et al., 2000) SEP “dropouts” from impulsive solar flares (Ruffolo et al., 2003, Chuychai et al., 2007)

Energy range: 20keV/nucleon to 2MeV/nucleon

(Chuychai et al., 2007)

(Ruffolo et al., 2003)

wavelet analysis might help to prove this hypothesis identifying flux tubes

(Bruno et al., in preparation)

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

distribution vs compressibility at differentscales

ToyModel

Cluster

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

PDFs in the magnetosheath

(from Toy-Model)

(from Perri et al., 2009)

Dt=46 sec

Dt=1.5 sec

Dt=0.2 sec

It is sufficient to increase the compressive level to obtain distributions similar to those recorded in the magnetosheath

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

Also the typeofDQdistributionplays a role, the samelevelofcompressionisnotlongersufficient

extractedfromGaussiandistribution

extractedfromuniformdistribution

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

Anomalous jumps of the field vector identified as source of intermittency (Bruno et al., 2001, 2004, Borovsky, 2008, Li, 2008)

Time sequence 2hr 13 min

these directional jumps are generally associated to discontinuities in field magnitude and/or density (Bruno et al., 2001)

Directional fluctuations follow a double log-normal

Alfvénic turbulence

Coherent structures

Flux tubes

a [°]

[Bruno et al., 2004]

Directional fluctuations follow a double log-normal

Similar results found by Borovsky (2008)

Alfvénic turbulence

Coherent structures

Flux tubes

[Bruno et al., 2004]

Largest jumps due to convected structures like discontinuities

These results corroborated by a recent analysis by Li (2008) who studied the behavior of the cumulative probability of having large .

Li identified 2 populations: turbulent fluctuations and current-sheet-like structures

A Toy Model to reproduce Cluster observations

a

Distribution of magnetic field intensity fluctuations at the smallest scale directly taken from real Cluster S/C1 data

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 a double-lognormal distribution

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

Ratio of the eigenvalues

(from Perri et al., 2009)

(from Toy-Model)

The Toy-Model reproduces the behaviour of the ratio of the eigenvalues.

As already noticed by Perri et al., 2009, at smaller scales values of l2/l1<0.1 have a large probability to occur so that fluctuations are quasi-one-dimensional.

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

30 days

Interplanetary fluctuations show self-similarity properties

16 hours

The solution of this relation is a power law:

72 minutes

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