Turbulence and Intermittency in the Solar Wind R. Bruno 1 In collaboration with: 3 Carbone, V., 1...
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Transcript of Turbulence and Intermittency in the Solar Wind R. Bruno 1 In collaboration with: 3 Carbone, V., 1...
Turbulence and Intermittency in the Solar Wind
R. Bruno1
In collaboration with: 3Carbone, V., 1Bavassano, B., 1D'Amicis, R., 4Sorriso, L., 5Pietropaolo, E
1 Istituto di Fisca dello Spazio Interplanetario, INAF, Rome, Italy3 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, Belgium09-11 June 2010[background picture adapted from Chang et al., 2004]
typical IMF power spectrum in at 1 AU[Low frequency from Bruno et al., 1985, high freq. tail from Leamon et al, 1999]
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
Interplanetary fluctuations show a well developed turbulence spectrum
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
lC lT
2
T
Cmeff
R
(Batchelor, 1970)
typical IMF power spectrum in at 1 AU[Low frequency from Bruno et al., 1985, high freq. tail from Leamon et al, 1999]
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
Interplanetary fluctuations show a well developed turbulence spectrum
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
2
T
Cmeff
R
(Batchelor, 1970)
lC lT
Large scales: Origin of K-1 (flicker noise)
Matthaeus et al., ApJ, 2007
Observed in a wide variety of systems
first interpretation for the IMF due to Matthaeus and Goldstein, 1986:• within Alfvénic radius, merging of magnetic structures• multiplicative process and characteristic lengths
lognormally distributed• spectrum would derive from a superposition of several
similar samples recorded during long enough time interval
• It can be shown (Montroll and Shlesinger, 1982) that the spectrum S(f)~1/f
Ulysses
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Compensated spectrum by Dmitruk et al., 2002-2004
Observed in a wide variety of systems
first interpretation for the IMF due to Matthaeus and Goldstein, 1986:• within Alfvénic radius, merging of magnetic structures• multiplicative process and characteristic lengths
lognormally distributed• spectrum would derive from a superposition of several
similar samples recorded during long enough time interval
• It can be shown (Montroll and Shlesinger, 1982) that the spectrum S(f)~1/f
Numerical modeling (Dmitruk et al., 2002-2004):• Upward traveling low frequency waves at coronal base
are capable of self-generating 1/f noise in density and B• 1/f not present in similar hydrodynamics simulations
(role of magnetic field)
Large scales: Origin of K-1 (flicker noise)
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Nakagawa and Levine, 1974
Observed in a wide variety of systems
first interpretation for the IMF due to Matthaeus and Goldstein, 1986:• within Alfvénic radius, merging of magnetic structures• multiplicative process and characteristic lengths
lognormally distributed• spectrum would derive from a superposition of several
similar samples recorded during long enough time interval
• It can be shown (Montroll and Shlesinger, 1982) that the spectrum S(f)~1/f
Numerical modeling (Dmitruk et al., 2002-2004):• Upward traveling low frequency waves at coronal base
are capable of self-generating 1/f noise in density and B • 1/f not present in similar hydrodynamics simulations
(role of magnetic field)
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.
Large scales: Origin of K-1 (flicker noise)
Possible link between the structured surface of the sun and 1/f scaling in IMF
Telloni et al, ApJ, 706, 238, 2009
Proton density fluctuations* at 2.3 RSUN (UVCS/SOHO) are compared to Helios 2 measurements performed at 64 RSUN
Similar coronal imprint f-2 at large scales
Possible link also between the structured coronal base and in-situ observations of density fluctuations (Telloni et al., 2009)
(*) the fluctuations of Lyα emission observed in corona can be generally associated with neutral hydrogen density variations [Bemporad et al., 2008, Telloni et al., 2009] Turbulence and Multifractals in Geophysics
and Space Brussels 9-11 June 2010
• No intermittency at these scales• Fluctuations are already decorrelated (Gaussian) for time delays << stream
duration
Gaussianity of large scale fluctuations
Proton Gyrofrequencyfla
tnes
s
1 day
1 hour
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
102 103 104 105
0
30
60
90102 103 104 105
0
30
60
90
b^
v an
gle
time scale(sec)
F A S T W I N D 0.3 AU 0.7 AU 0.9 AU
b^
v an
gle
S L O W W I N D 0.3 AU 0.7 AU 0.9 AU
As the wind expands turbulence evolution and compressive effects decouple dB-dV in fast wind, starting from the largest scales
[see literature in Tu and Marsch 1995, Bruno and Carbone 2005]
Helios 2 observations
dB-dV alignment vs scale and heliocentric distance
ttBtV
tBtV
)()(
)()(cosˆ 1
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Large scales are scarsely anisotropic
2
Power anisotropy at different scales
Intermediate scales characterized by turbulence scaling K-5/3
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
The presence of a Kolmogorov scaling implies that 0 zz
Since Z- cannot be of solar origin, the condition above requires generation mechanisms at work:
1) In the ecliptic, velocity shear (Coleman, 1968) and dynamic alignment (Dobrowolny et al., 1982, Matthaeus et al., 2008)
2) At high latitude, parametric decay is able to reproduce Ulysses observations (Bavassano et al, 2000, Malara et al, 2000)
102 103 104 105
0
30
60
90102 103 104 105
0
30
60
90
b^
v an
gle
time scale(sec)
F A S T W I N D 0.3 AU 0.7 AU 0.9 AU
b^
v an
gle
S L O W W I N D 0.3 AU 0.7 AU 0.9 AU
Intermediate scales are the most Alfvénic within fast wind
[see literature in Tu and Marsch 1995, Bruno and Carbone 2005]
Helios 2 observations
dB-dV alignment vs scale and heliocentric distance
ttBtV
tBtV
)()(
)()(cosˆ 1
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
small scales are strongly anisotropic,partly due to Alfvénic fluctuations
2
Power anisotropy at different scales
20 min
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
First anisotropy study in terms of k and k// in the solar wind by Bieber et al., 1996
Turbulence made of 2D+SLAB[Matthaeus et al., 1990]
Similar results by Horbury et al., 2008
K dominates on k// as we analyze directions at larger angles with magnetic field
Thus, the slab turbulence due to Alfvénic fluctuations would be a minor component of interplanetary MHD turbulence
Analysis performed in slow wind
Analysis performed in slow wind
2D
SLAB
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
2.000
25.50
49.00
72.50
96.00
119.5
143.0
166.5
190.0
C
R
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.000
6.500
12.00
17.50
23.00
28.50
34.00
39.50
45.00
C
R
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
max
min
min
max
min
max
2.000
7.375
12.75
18.13
23.50
28.88
34.25
39.63
45.00
C
R
0.9 AU
FAST WIND SLOW WIND
0.9 AU
MHD turbulence in terms of R and C (scale < 1hr)
1
2
22
RC
bv
bv
R
bvC
ee
ee
ee
bv
ee
ee
This difference affects the anisotropy
[adapted from Bruno et al., 2007]
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Advected structures with magnetic energy excess
Alfvénic population
Bavassano et al., 1982 found that anisotropy of magnetic field fluctuations increases with heliocentric distance
[Bavassano et al., 1982]
Analysis performed with the same corotating stream observed at different heliodistances
321
12 /
13 /
anisotropy of magnetic field fluctuations increases with heliocentric distance
field compressibility increasese with heliocentric distance
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
It was shown that anisotropy is reduced by removing intermittent events from |B|, i.e. the most compressive magnetic events outside a normal distribution[Bruno et al., 1999]
[adapted from Bruno et al., 1999]
Local Intermittency Measure technique (Farge et al., 1990; Farge, 1992)
based on wavelet decomposition
)(),(22
,
4,2
FFw
wtLIM
tt
ttt
The Flatness Factor of 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 )
Thus, values of FF()>3 allow to localize events which lie outside the Gaussian statistics and cause Intermittency.
Proton Gyrofrequency
Intermittency at these scales doesn’t seem to be due to dissipative processes
Inertial range
Anomalous scaling of IMF is well inside inertial range
Proton Gyrofrequency
Intermittency starts within the inertial range at about 2 decades away from the dissipation range.
flatn
ess
20
PDFs of other parameters can be rescaled
The PDF of , dr dB2, drV2, dVB2 can be rescaled under the following change of variable
),(),( xPxP s(Hnat et al., 2002, 2003, 2004)
The height of the peaks rescales
This study showed that:• the distribution is non Gaussian but it is stable and symmetric and can be described by a single parameter monofractal
•The process can be described by a finite range Lévy walk (scales up to 26 hours)•A Fokker-Planck approach can be used to study the dynamics of PDF(db2)
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Local Intermittency Measure technique allows to locate intermittent events(Farge, 1990)
56000 60000 64000 68000 72000
260
280
300
320
340
360
380
spee
d &
res
idua
ls [
km/s
ec]
day 22.02.2001[sec]
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
Local Intermittency Measure technique allows to locate intermittent events(Farge, 1990)
56000 60000 64000 68000 72000
260
280
300
320
340
360
380
spee
d &
res
idua
ls [
km/s
ec]
day 22.02.2001[sec]
Cluster s/c 3 data
180 190 200 210 220
-290
-280
-270
-260
-250
-240
-230
Maximum variance plane
Max
Var
MedVar
this intermittent event marks the border between different plasma regions
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Local Intermittency Measure technique allows to locate intermittent events(Farge, 1990)
56000 60000 64000 68000 72000
260
280
300
320
340
360
380
spee
d &
res
idua
ls [
km/s
ec]
day 22.02.2001[sec]
Cluster s/c 3 data
ByBx
Bz
VyVx
Vz
Two different regions are identified and fluctuations within each of them are Alfvénic
Magnetic field in Alfvén units
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Several studies on events similar to the previous one contributed to formulate the spaghetti-like model [Bruno et al., 2001]
A spacecraft crossing this structure would record a series of intermittent events embedded in an ocean of Gaussian, Alfvénic fluctuationss/c trajectory
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.[Vörös et al., 2010]
[Tu and Marsch, 1992]
[Bieber, Wanner and Matthaeus, 1996]
2D+SLAB
(Chang et al., 2002)
Similar pictures
2D+SLAB
Several contribution have been given in this direction in the past years (Thieme et 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)
Structures producing intermittency might be either of solar nature and advected by the wind or locally generated by non-linear turbulence evolution
Some remarks
Compressive events increase intermittency
Intermittent events can increase the power anisotropy of fluctuations
Then: compressive events might play a role in power anisotropy
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
With this in mind, we look at power anisotropy beyond fC
[Leamon et al., 1998, 1999, Bale et al.,
2005, Hamilton et al., 2008, Podesta,
2009, Sahraoui et al., 2009, etc…]
Some remarks
Compressive events increase intermittency
Intermittent events can increase the power anisotropy of fluctuations
Then: compressive events might play a role in power anisotropy
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Power anisotropy study from Podesta, 2009
[cur
ves
arbi
trar
ily s
hift
ed v
ertic
ally
]
[ q angle between local mean field and radial direction]
KAWs cascade
KAWs dissipation??
Inertial range: • energy cascade directed primarily perpendicular to the mean
mag field [Shebalin et al., 1983; Oughton et al., 1994; Matthaeus et al., 1996]• power anisotropy increaseses with wavenumber
“Dissipation” range: • The new cascade (KAWs?) starts at ki 1 [i.e. S/C 0.5Hz, in
this case] when the fkuid-like behavior breaks down• power anisotropy increaseses with wavenumber• The second peak marks beginning of KAWs dissipation?
FFT
wavelets
Observational evidence for Alfvén waves – KAWs transition in the solar wind [Bale et al., 2005, Cluster data]
Moreover, Sahraoui et al., 2009 extended the study to the electron gyroscale where they identified dissipative processes of KAWs
electric field and magnetic field k−5/3
inertial sub-range is observed up to ki~1 (Electric spectrum arbitrarily offset to overlap magnetic spectrum)
the wave phase speed in this regime is shown to be consistent with the Alfvén speed
The red curve approximates the dispersion relation for KAWs (vk2i
2). For whistler waves (vki). it would run much lower for 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 bluedots) suggest KAW
eciAei
ecithei
ei
ei
V
V
,,
,,
/
/
,
,
29
(Alexandrova et al., 2008, 2009)
The presence of a power-law spectrum (instead of a rough exponential cutoff) and the increase of intermittency suggest the presence of a new cascading range (Stawicki et al., 2001, Bale et al., 2005, Sahraoui et al., 2006, 2009)
The cascade has a compressible character. Compressibility seems to govern the spectral index k-7/3+2a where a is the compressibility (Alexandrova et al., 2008, 2009)
A new cascading range beyond fci confirmed by by Alexandrova et al., 2008, 2009 using Cluster magnetic observations
Inte
rmitt
ency
incr
ease
s
(Alexandrova et al., 2008, 2009)
(Alexandrova et al., 2008, 2009)
Hollweg, 1999:• KAWs become strongly compressive when ki 1. • The compression is accompanied by a magnetic field fluctuation δB ‖ such that the total pressure perturbation δp tot ≈ 0
Inte
rmitt
ency
incr
ease
s
(Alexandrova et al., 2008, 2009)
A new cascading range beyond fci confirmed by by Alexandrova et al., 2008, 2009 using Cluster magnetic observations
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Different conclusions about intermittency in the “dissipation” range
[Kiyani et al., PRL 2009]
monofractal
multifractal
Time series and probability distribution function for the max eigenvalue at three different scales (46s, 1.5s, 0.2s)
(Perri et al., 2009)(Perri et al., 2009)
Perri et al., 2008, 2009, performed a minimum variance study in this frequency range, focusing on the anisotropy of the fluctuations
• The PDFs evolve with the scale, becoming power laws at scales smaller than the ion cyclotron scale.
• Similar behaviour for the other eigenvalues
What is the possible mechanism to reproduce this behavior?
Cluster mag data resolution f=22Hz
Intermittent behavior
Power law
Looking at the distribution of angular fluctuations of B vector 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
~6 minutes
Looking at the distribution of angular fluctuations of B vector on a time scale of 0.045 sec (i.e.22Hz)
Distribution obtained from 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
Same type of distribution at larger scales, within the inertial range
Distribution obtained from Cluster 1,2,3,4 data at the time scale of 0.045s
[°] a
Alfvénic turbulence Coherent
structures
(Bruno et al., 2004)
Distribution obtained from Helios data at the time scale of 6s, high frequency termination of the inertial range
whistlers and KAW ?
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
Distribution obtained from 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
Compressions added: s|B|/<|B|>~3%
• moreover, we added compressions in the field intensity• we tried to reproduce the same compressive level found in real data (similar spectra)• compressions revealed to play an important role in this toy model (see next slides)
Compressibility(f) S|B|(f)/SC(f)
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Time series and probability distribution function for the max eigenvalue at three different scales (46s, 1.5s, 0.2s)
(Perri et al., 2009)(Perri et al., 2009)
minimum variance study by Perri et al., 2008, 2009
• The PDFs evolve with the scale, becoming power laws at scales smaller than the ion cyclotron scale.
• Similar behaviour for the other eigenvalues
Cluster mag data resolution f=22Hz
Intermittent behavior
Power law
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
10000 15000 20000 250000.00
0.05
0.10
0.1510000 15000 20000 25000
0.0
0.1
0.2
10000 15000 20000 250000.0
0.2
0.4
0.6
time in arbitrary units
1
scale: 5 pts
1
scale: 38 pts
1
scale: 1150 pts
0.1 1101
102
103
104
0.01 0.1100
101
102
103
104
0.01 0.1100
101
102
103
104
PD
F( 1)
scale 1150
PD
F( 1)
scale 38
PD
F( 1)
1
scale 5
The Toy Model reproduces qualitatively the results obtained in the solar wind
(from Toy-Model)(from Toy-Model)
Max eigenvalue behavior from Toy Model
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
10000 15000 20000 250000.00
0.05
0.10
0.1510000 15000 20000 25000
0.0
0.1
0.2
10000 15000 20000 250000.0
0.2
0.4
0.6
time in arbitrary units
1
scale: 5 pts
1
scale: 38 pts
1
scale: 1150 pts
0.1 1101
102
103
104
0.01 0.1100
101
102
103
104
0.01 0.1100
101
102
103
104
PD
F( 1)
scale 1150
PD
F( 1)
scale 38
PD
F( 1)
1
scale 5
(from Toy-Model)(from Toy-Model)
Max eigenvalue behavior from Toy Model
intermittent character
Gau
ssia
n ch
arac
ter
-15 -10 -5 0 5 10 15
1E-4
1E-3
0.01
0.1
1
PD
F
1 /
compression~6% scale 5 scale 1150
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
15500 16000 16500 170000
20
40
60
80
14000 16000 18000 200000
20
40
16400 16500 16600 16700 16800 169000
20
40
60
80
scale 38 pts
scale 1150 pts
time in arbitrary units
scale 5 pts
(from Perri et al., 2009) (from Toy-Model)
Time behavior of the angle Q between minvar direction and mean field direction at three different scales.
Similar profiles obtained from toy model
Results from Cluster
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
(from Perri et al., 2009)
Striking similarity between these PDFs and those found in the SW
PDFs in the Solar Wind
Dt=46 secDt=1.5 secDt=0.2 sec
(from Toy-Model)
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
(from Perri et al., 2009) (from Toy-Model)
The effect of compressions
Dt=46 secDt=1.5 secDt=0.2 sec
With compressions
W/O compressions
Without compressions the toy doesn’t work wellTurbulence and Multifractals in Geophysics
and Space Brussels 9-11 June 2010
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
summary
• k-1 frequency range: possible link with the structured surface of the sunfluctuations are Gaussianpower anisotropy (P/P// >1) increaseses with wavenumber
• k-5/3 inertial range: fluctuations mixed with advected structuresintermittency more than 2 freq. decades before the “dissipation range”, intermittency linked to compressive structures and current sheets. possible flux tube structurepower anisotropy (P/P// >1) increaseses with wavenumber
• Beyond the proton cyclotron freq.: new cascade (KAWs or/and whistlers) intermittency increases again, important role played by compressions confirmed by anisotropy studies
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
(Feng et al., 2007)
These flux tubes should look like interplanetary flux ropes
Force-free structure for which
BJB
(Lundquist, 1950)
WIND data, 20-09-1995
Some helicity mesure should be able to unravel the presence of these structures
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
11231 Im2 )/k(kS)(kH rrm
48
In practice, if collinear measurements are made along the X direction, the reduced magnetic helicity spectrum is given by:
1
1
Im2
k
ZY)(kH
*rm
where Y and Z are the Fourier transforms of By and Bz components, respectively
The tensor S23 is the Fourier transform of the correlation function R23
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
(Goldstein et al., 1991)(Matthaeus and Goldstein, 1982)
)(
)()(
kE
kHkk
b
mm
Adopting the wavelet transform to study Hm in the solar wind
k
(k)](k)B[B
k
(k)][S(k)H z
*y
rrm
Im2Im2 23 k
(k,t)](k,t)W[W(k,t)H zyr
m
*Im2
(k,t)E
(k,t)k H(k,t)σ
r
rm
m
a wavelet is a localized function which can be translated and re-scaled
τ
t'tψ
τ(t)ψt',τ
1
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
Handed-Left 0σ
Handed-Right 0σ
m
m
)(
)(
k
k
re
re
The normalized magnetic helicity (sm) helps to easily estimate
the "handedness“ of magnetic field fluctuations
)()(
)()(
)(
)()(
kEkE
kEkE
kE
kHkk
RL
RL
b
mm
sm
Helios 2
Wavelet analysis on magnetic helicity within 1 day of slow wind at 0.3 AU
BY
B Z
BY
B Z
rectangles indicate limits in scale and time
negative helicity(right-handed)
mixed helicity
sm
Wavelet analysis on magnetic helicity within 1 day of slow wind at 0.3 AU
from 11.6 to 12.8
from 12.8 to 14.2
(negative helicity, right-handed)
sm
Flux rope spotted by Feng et al., 2007
Wavelet analysis on magnetic helicity within 1 day of WIND data containing a flux-rope
Band pass filtered between 15 min and 2 hrs
WIND data, 20-09-1995
Preliminary magnetic helicity analysis shows some similarities during ACE and ULYSSES alignment in 2007
1AU 1.4AU
Sca
le[h
r]
Sca
le[h
r]
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 itMore quantitative analysis is needed.
1
2
3
4
5
1
2
3
4
5
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
(Ruf
folo
et a
l., 2
003,
Chu
ycha
i et a
l., 2
007)
2D dominated
slab dominated
In the 2D turbulence field lines follow contours of constant “a” remaining trapped near some (x,y) point
Some similarity is found between IMF coherent structures and Ruffolo et al (2003) and Chuychai et al. (2007) model
(Bruno et al., in preparation)
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)
sm
mean field
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
(Servidio et al., 2008)
(Ruffolo et al., 2003)
(Chuychai et al., 2007)
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
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
Cluster Toy Model
distribution vs compressibility at different scales
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
0 20 40 60 80
0.0
5.0x102
1.0x103
1.5x103
(from Toy-Model)
PD
F
between minvar and <B>
|B|
/<|B|>~60%window:
1150 38 5
PDFs in the magnetosheath
It is sufficient to increase the compressive level to obtain distributions similar to those recorded in the magnetosheath
(from Perri et al., 2009) (from Toy-Model)
Dt=46 secDt=1.5 secDt=0.2 sec
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
extracted from Gaussian distribution
extracted from uniform distribution
Also the type of DQ distribution plays a role, the same level of compression is not longer sufficient
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)
6 3 0-3-6
-6-3036
-6-3036
6 3 0-3-6
-6-3036
-6-3036
6 3 0-3-6
-6-3036
-6-3036
6 3 0-3-6
-6-3036
-6-3036
CB
from: 49.612to: 49.646
Helios 2, 6s avgs
E
D
from: 49.654to: 49.690
D C
from: 49.628to: 49.674
BA
from: 49.598to: 49.623
Time sequence 2hr 13 min
6 4 2 0 -2 -4 -6 -8
-8
-6
-4
-2
0
2
4
6
-8-6
-4-2
0246
Helios 2, 6sec, 49.628-49.674 (~66min)
Bz
By
Bx
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
[°] a
Alfvénic turbulence Coherent
structuresFlux tubes
[Bruno et al., 2004]
Similar results found by Borovsky (2008)
Alfvénic turbulence
Coherent structuresFlux tubes
Largest jumps due to convected structures like discontinuities
Directional fluctuations follow a double log-normal
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
[Bruno et al., 2004]
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 a double-lognormal distribution
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010
Distribution of magnetic field intensity fluctuations at the smallest scale directly taken from real Cluster S/C1 data
A Toy Model to reproduce Cluster observations
0.1100
101
102
103
0.1
101
102
103
0.1
101
102
103
0.1 1100
101
102
103
0.1 1100
101
102
103
0.1 1100
101
102
103
PD
F(
3/ 1)
scale 1150
PD
F(
3/ 1)
3/
1
scale 5
PD
F(
3/ 1)
scale 38
PD
F(
2/ 1)
scale 1150
PD
F(
2/ 1)
2/
1
scale 5
PD
F(
2/ 1)
scale 38
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
Interplanetary fluctuations show self-similarity properties
30 days
16 hours
72 minutes
)μ(r)v(r)v(
The solution of this relation is a power law:
hC )v(
Turbulence and Multifractals in Geophysics and Space Brussels 9-11 June 2010