turbulence in the solar wind, spectra from voyager 2 data · 2014. 10. 22. · l.l. orionis...

Turbulencein the solar

wind,spectra fromVoyager 2

data

Projectintroduction

Solar windstatisticsfrom V2 data(year 1979,days 1–180)

Spectralanalysis:methodologyandvalidation

Spectralanalysis:syntheticturbulence

Spectralanalysis: V2velocity andmag. fielddata

Rybicki&Presspredictionmethod

Conclusions

Turbulence in the solar wind, spectra fromVoyager 2 data

F. Fraternale1 , L. Gallana1, M. Iovieno1, J.D. Richardson2,D. Tordella1

1Dipartimento di Ingegneria Meccanica e Aerospaziale (DIMEAS)Politecnico di Torino, Italy

2Kavli Institute for Astrophysics and Space ResearchMassachusetts Institute of Technology (MIT), Cambridge, USA

Turbulent Mixing and Beyond WorkshopITCP Trieste, 4–9 August 2014



data

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Conclusions

Table of contents

1. Project introduction

2. Solar wind statistics from V2 data (year 1979, days 1–180)

3. Spectral analysis: methodology and validation

4. Spectral analysis: synthetic turbulence

5. Spectral analysis: V2 velocity and mag. field data

6. Rybicki &Press prediction method

7. Conclusions



data

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Conclusions

Voyager 2 Interstellar Mission

• Voyager 2 is flying now at15.6km/s, 104.7 AU fromEarth, in the Heliosheath,the outermost layer of theheliosphere where the so-lar wind is slowed by thepressure of interstellar gas

• Termination Shock waspassed on Sep 5, 2007

source: M. Opher et al.

source: http://voyager.jpl.nasa.gov

A turbulence hypothesis for themagnetic field in the HeliosheathM. Opher et al, ApJ 734, 2011“Is the magnetic field in the He-liosheath laminar or a turbulentsea of bubbles?”



data

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Conclusions

L.L. Orionis colliding with the Orion Nebula.Image from the Hubble Space Telescope, February 1995

(Credit: NASA, The Hubble Heritage Team (STScI/AURA))



data

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Conclusions

Year 1979: V and B data

-8

-4

0

4

-8

-4

0

4

-8

-4

0

4

8

1 20 40 60 80 100 120 140 160 180 200

time (doy)

(x−

µ)/

σ

VRBR

VTBT

VNBN

8V2 normalized data (1979)

Velocity and magnetic field data from V2, period 1979 (DOY 1–180).RTN heliographic reference frame.



data

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Conclusions

Year 1979: V and B data

-100

0

100

200

-150

0

100

-200

-100

0

100

0 20 40 60 80 100 120 140 160 180

τ (doy)

(v−

µv),

(B

√4πρ−

VA)

(km

/s)

VNBN

VTBT

VRBR

V2 data in Alfvenic units (1979)



data

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Conclusions

Year 1979: V and B moments and PDFs

µ σ2 Sk KuVR 454 1893 0.43 3.41VT 3.21 252.9 -0.99 7.35

VN 0.51 250.3 -0.36 5.80BR -0.04 0.173 0.53 6.71BT 0.06 0.85 -0.72 10.2BN 0.10 0.34 -0.24 7.65

units: km/s, nT

〈ni〉 (cm−3) 0.23

〈T 〉 (K) 27038

βp 0.225VA (km/s) 47.7cs (km/s) 19.3

fci (Hz) 1.49·10−2

fpi (Hz) 101f∗ (Hz) 0.44ri (km) 158λD (m) 5.5

0.001

0.01

0.1

1

-4 -2 0 2 4

V2 velocity data1979, DOY 1-180

RadialNormal

Tangential

PD

F (

m/s

)-1

V - µ /σ( )

σ

0.001

0.01

0.1

1

-4 -2 0 2 4B - µ/σ

σPD

F (

nT

)-1

RadialNormal

Tangential

( )

V2 mag. data1979, DOY 1-180

PDF of V and B standardizedcomponents and comparison witha Normal distribution Evidence ofanisotropy



data

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Conclusions

Year 1979: V and B moments and PDFs

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16 18 20

velocity module mag. field module

χ(k=3)

PD

F

(xi − μi)2 /σ2i

V2 data1979, DOY 1-180

PDF of standardized modules and comparison with a χ2 distribution.

High intermittency?

• Evidence of high Ku(> 3)

• The origin of “intermittency”: advected coherent structures(flux tubes, etc), stochastic Alfvenic fluctuations generated atsolar corona and “frozen” in the wind?

• Intermittency interests a broad range of scales



data

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Conclusions

Autocorrelations

-0.20

0.20.40.60.8

1

-0.20

0.20.40.60.8

1

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30 35 40 45 50

τ (days)

Rii

/σ2 i

Autocorrelations - V2 data (1979, DOY 1-180)

VR VRBRBR

VT VTBTBT

VNVNBNBN

Rii(τ) =< x(t)x(t+ τ) >



data

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Conclusions

Cross-correlations tensor: off-diagonal terms

-0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30 35 40 45 50

τ (days)

BRVTBTVR

BRVNBNVR

BTVNBNVT

Rij

/σiσ j

Cross-correlations - V2 data (1979, DOY 1-180)

Rii(τ) =< x(t)y(t+ τ) >



data

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Conclusions

Cross-correlations tensor: diagonal terms

-0.4

-0.2

0

0.2

0.4

BRVR

BNVN

BTVT

0 5 10 15 20 25 30 35 40 45 50

τ (days)

Rii

/σ2 i

Summary:

• Averages are computed on 57970 points for V, and 124080points for B, spanning the whole 180 days period

• Evidence of a 25 days periodicity. Minimum of solar activityin 1979

• High cross-correlation VRBR → not in-phase

• High cross-correlation VRBT → not in-phase

• Low Alfvenic one-point correlation (this is often the case in theslow-wind periods)



data

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Conclusions

Data reconstruction techniques

V2 velocity and mag. data are discontinuous and irregularlyspaced. In the whole year 1979 there is 45% of missing velocitydata, 25.4% in the period here considered (DOY 1–180). Aboutmag. data, the percentage is 23.8%. These values are about 97%in 2012.To perform an accurate spectral analysis on these kind of datasets, a reconstruction technique may be mandatory. In thefollowing, the effect of two interpolation/recovery methodologieson averaged turbulent spectra will be discussed.

• Linear interpolation

• Maximum likelihood reconstruction and realizations constrainedby data1

1Rybicki & Press, ApJ 398, 1992



data

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Conclusions

Data reconstruction techniques

To discuss the effects of averaging,interpolating and applyingwindowing techniques, two 1D sequences of synthetic turbulencedata have been generated from imposed spectral properties:

• Synt 1→ E3D(n/n0) = (n/n0)β

(n/n0)α+β

• Synt 2 → E3D(n/n0) = (n/n0)β

(n/n0)α+β ∗[1− exp(n−ntotγ + ε)

]β = 2, α = 5/3, n0 = 11, γ = 104, ε = 10−1

The Synt 1 sequence reproduces the Kolmogorov inertial range ofcanonical turbulence, while Synt 2 reproduces both the inertialand the dissipative part of the spectrum.

• Synthetic data are scaled on a 180 days time grid (∆t = 100 s,ntot = 155520)

• The same gaps of V2 velocity data are projected on thesesequences

• Spectral analysis is performed. Parameters: Lg, Ls



data

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Conclusions

Effect of no/linear interpolation on Synt 1 data

101

102

103

104

105

106

107

108

10-6 10-5 10-4 10-3 10-2

V2

(Km

2/s

2/H

z)

Ns = 84 (Ls = 6 − 8 hrs )Ns = 84 (Ls = 6 − 8 hrs ), HannNs = 320 (Ls = 3 − 4 hrs ) α: -1.76Ns = 320 (Ls = 3 − 4 hrs ), Hannoriginal: slope -1.66

Synthetic turbulence1,effects of averaging withno recovery

-1.66

gaps of V2 data(1979, DOY 1-180)

α: -1.72 101

102

103

104

105

106

107

10-5 10-4 10-3 10-2

V2

(Km

2/s

2/H

z)

Lg = 0.5 hrsLg = 1 hrsLg = 4 hrsoriginal: slope -1.66

Synthetic turbulence1,effects of linear recovery,Hann windowing


-1.66

Ls = 24 hrs

α: -1.79α: -1.90α: -2.05

10 1

10 2

10 3

10 4

10 5

10 6

10 7

10 -5 10 -4 10 -3 10 -2

f (Hz)

V2

(Km

2/s

2/H

z)

Ns = 480 (Ls = 6 hrs )Ns = 200 (Ls = 12 hrs )Ns = 68 (Ls = 24 hrs )original: slope -1.66

Synthetic turbulence1,effects of linear recovery,no windowing


-1.66

Lg = 1 hr

α: -1.77α: -1.78α: -1.68

101

102

103

104

105

106

107

10-5 10-4 10-3 10-2

f (Hz)

V2

(Km

2/s

2/H

z)

Ns = 480 (Ls = 6 hrs )Ns = 200 (Ls = 12 hrs )Ns = 68 (Ls = 24 hrs )original: slope -1.66



-1.66

Lg = 1 hr

α: -1.77α: -1.74α: -1.68



data

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Conclusions


101

102

103

104

105

106

107

10-5 10-4 10-3 10-2

f (Hz)

V2

(Km

2/s

2/H

z)

Lg = 0.5 hrs (Ls = 24 hrs)Lg = 1 hr (Ls = 24 hrs)Lg = 4 hrs (Ls = 24 hrs)original spectrum

Synthetic turbulence2,effects of linear recovery,no windowing


-1.75

α: -1.804

α: -1.829

α: -1.841

101

102

103

104

105

106

107

10-5 10-4 10-3 10-2

f (Hz)

V2

(Km

2/s

2/H

z)

Lg = 0.5 hrs (Ls = 24 hrs)Lg = 1 hr (Ls = 24 hrs)Lg = 4 hr (Ls = 24 hrs)original spectrum



-1.75

fit: -1.750fit: -1.798fit: -1.833

• Effect of segmentation: increase in slope of about 5% in theinertial range .

• Effect of linear interpolation: function of Lg (length of “filled”gaps). This interpolation transfers energy to the low frequences,resulting in an increase (about 6%) in the slope, especially inthe high-frequency range (f > 10−3Hz).



data

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Conclusions


• Effect of windowing: the Hann window function allows toeliminate spurious energy due to discontinuities (≈ 1/f) atthe boundary of each segment. The effect is minimal at lowwavenumbers. In the high-frequency range, on the one hand asignificant increase ( up to 23%) of the slope is found to be afunction of Lg, on the other hand any change in slope of thereal spectrum can be followed.Energy correction factor for Hann: 1.632

• Without windowing, the segmentation error doesn’t allow torepresent the correct slope, in the general case (see the analysison Synt 2 data). These cases can be recognized by a flatteningin the high-frequency range of the spectrum. Averaging longsegments helps.



data

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Conclusions

V2 velocity spectra at 5 AU (pre-Jupiter)

102

103

104

105

106

10-5 10-4 10-3 10-2

VR

(Km

2/s

2/H

z)

−no recov, Ls = 3 4 hrsno recov, Ls = 3 − 4 hrs , Hannlin. recov, Lg = 0 .5 hrs, Ls = 12 hrslin. recov,Lg = 0. 5hrs, Ls = 12 hrs, Hann


-1.66

α: -1.56α: -1.49α: -1.60α: -1.51

102

103

104

105

106

10-5 10-4 10-3 10-2

VT

(Km

2/s

2/H

z)



-1.66

α: -1.58α: -1.57α: -1.55α: -1.47

102

103

104

105

106

10-5 10-4 10-3 10-2

f (Hz)

VN

(Km

2/s

2/H

z)



-1.66

α: -1.60α: -1.54α: -1.52α: -1.49

102

103

104

105

106

10-5 10-4 10-3 10-2

f (Hz)

E(K

m2/s

2/H

z)


α: -1.60α: -1.55α: -1.56α: -1.50

V2 kinetic energy1979, DOY 1-180

-1.66



data

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Conclusions

V2 mag. field spectra at 5 AU (pre-Jupiter)

10-2

10-1

100

101

102

103

10-5 10-4 10-3 10-2

B2 R

(nT

2/H

z)


-1.66


α: -1.77α: -1.77α: -1.74α: -1.72

10-2

10-1

100

101

102

103

10-5 10-4 10-3 10-2

B2 T

(nT

2/H

z)


-1.66

α: -1.90α: -1.94α: -1.74α: -1.72−no recov, Ls = 2 3 hrs

no recov, Ls = 2 − 3 hrs , Hannlin. recov, Lg = 0 .5 hrs, Ls = 48 hrslin. recov,Lg = 0. 5hrs, Ls = 48 hrs, Hann

10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

f (Hz)

B2 N

(nT

2/H

z)


-1.66

α: -1.88α: -1.90α: -1.87α: -1.90−no recov, Ls = 2 3 hrs

no recov, Ls = 2 − 3 hrs , Hannlin. recov, Lg = 0 .5 hrs, Ls = 48 hrslin. recov,Lg = 0. 5hrs, Ls = 48 hrs, Hann 10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

f (Hz)

B2

(nT

2/H

z)

no recovery, Ls = 2 − 4 hrs, Hannno recovery, Ls = 2 − 4 hrs

linear rec, Lg = 0.5 hrs ,Ls = 48 hrslinear rec, Lg = 0.5 hrs ,Ls = 48 hrs, Hann

-1.66


α: -1.86α: -1.88α: -1.79α: -1.82



data

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Conclusions

V2 spectra at 5 AU (pre-Jupiter)Velocity:

• The observed frequency range constitute the inertial range

• All computed slopes (10−4 < f < 2 · 10−3 Hz) are flatter thanthe Kolmogorov one:α = −1.53± 0.07

• Computed slopes may be slightly overestimated

• A peak is located at f = 0.0026 Hz for T and N components:is it physical or instrumentation-related? (no relation withfci, fpi, f

∗))

Magnetic field:

• Computed slopes (10−4 < f < 2 · 10−3) are lower than thereference one:α = −1.81± 0.09

• Observed steepening for f > 3·10−3 Hz should not be linked tointerpolation issues: the situation recalls that of Synt 2 case,blue (no recovery) and violet (small gaps filled) give the sameresult.

• Anisotropy is higher with respect to the velocity field



data

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Conclusions

G.B. Rybicki &W.H. Press prediction

• Minimum variance prediction (interpolation):y = s + n irreg. spaced vector data with errors n

s∗ =∑Mi=1 d∗iyi + x∗ s∗ =true value at a particular point

s∗ = ST [S + N]−1y s∗ =min. variance estimate for s∗

Assuming stationary process:

Sij = 〈sisj〉 = f(ti − tj) is the correlation matrix, estimated from data

Nii = 〈n2i 〉 is the errors diagonal matrix ni →∞ in “new” points

The min. variance estimation is not, however, a typical realization of the

underlying process.

• Minimum variance prediction + Gaussian processTo obtain a typical realization, a Gaussian process is added to the min. var.

estimate:

s∗ = u∗ + s∗

If realizations constrained to data are desired:

u = V diag(λ1/21 , ..., λ

1/2M )r where

λi = eig(Q), Q = [S−1+N−1]−1, r = rand(µ = 0, σ2 = 1)



data

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Conclusions

R&P reconstruction

101

102

103

104

105

106

107

10-5 10-4 10-3 10-2

f (Hz)V

2(K

m2/s

2/H

z)

R&P recov, Lg = 4 hrs , Ls = 24 hrs, HannR&P recov, Lg = 4 hrs , Ls = 96 hrs, Hannoriginal: slope -1.66

Synthetic turbulence1,effects of R&P* recovery


lin. recov,Lg = 4hrs , Ls = 24 hrs, Hann

* Rybicki & Press,ApJ, 398, 1992

10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

f (Hz)

B2 R

(nT

2/H

z)

no recov, Ls = 2 − 4 hrs, Hannlin. rec., Lg = 0.5 hrs , Ls= 48 hrs, HannR&P recov*,Lg= 4 hrs ,Ls = 48 hrs, Hann

-1.66

-3.0

α: -3.11α: -2.84α: -2.58



102

103

104

105

106

10-5

10-4

10-3

10-2

f (Hz)

VR

(Km

2/s

2/H

z)


-1.66

no recov, Ls = 3 − 4 hrs ,Hannlin. recov,Lg = 0.5 hrs , Ls = 12 hrs, Hann

Lg = 4 hrs , Ls = 12 hrs, HannR&P recov*,




data

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Conclusions

Final considerations and future development• Kolmogorov (-5/3) or Iroshnikov–Kraichnan (-3/2) cascade?

Debated question. Many works suggested K41 as the moreconsistent for SW turbulence (Goldstein, GRL 1995), but inrecent works values close to IK for velocity and K41 for mag.field are observed (Safranova et al. PRL 2013 at 1AU, Podestaet al. ApJ 2007, 1AU)We provide spectra at 5 AU, supporting these recent observa-tions.

• The high frequency range. Break frequency/ies, dissipation orfurther cascades. Different mechanisms had been proposed toexplain the steepening of collisionless SW spectra in the highfreq. range (foreshock waves, Landau damping of KAW, wavedispersion). Up to what regime will we be able to observe, inthe Heliosheath region?

• Heliosheath data are very sparse. How to get reliable spec-tra when a data loss of 95% is present? We have startedto apply the Compressive Sensing technique to this problem.CS is a very recent paradigm for data acquisition, providingreconstruction for a broad class of sparse signals.

turbulence in the solar wind, spectra from voyager 2 data · 2014. 10. 22. · l.l. orionis...

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