Master student:Alexander Tsupko

Scientific supervisor:associate professor

Liudmyla V. Kozak

Experimental Study of Turbulent Processes in the Solar Wind Plasma

and Earth’s Magnetosphere

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Taras ShevchenkoNational University of Kyiv

(QS World University Rankings: 501-550)

Faculty of PhysicsAstronomy & Space Physics Department

Kyiv, Ukraine19-Jun-2013

I. Goals of the Study

II. Description of the Cluster II Mission

III. Turbulence Theories

IV. Data for Analysis

V. Statistical Methods

a) Analysis of the PDFs (Probability Distribution Functions)• Results

b) Analysis of the Excess Kurtosis (4th Statistical Moment)• Results

c) Self-Similarity Analysis• Results

• Diffusion

VI. Summary

Outline

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I. Goals of the Study

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Goals of the StudyMy primary purpose is to investigatefluctuations of the magnetic field which occur in the boundary regions of the Earth’s magnetosphere.

Pictures show different examples of turbulence.

Data for the study collected by ESA’s Cluster II spacecraft whose aim, among others, is to build a 3-D map of magnetosphere.

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IntermittencyIf you were to measure velocity field in a point of a flow under submarine, you would obtain picture like this:

Phenomenon of alternation of quasilaminarand turbulent forms of motion is called intermittency.

Ratio t1/t2 is called coefficient of intermittency.

Pulsations of velocity cause additional transfer of momentum, energy, etc.

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Vel

oci

ty

Time

II. Description of the Cluster II Mission

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An Illustration of the Four Cluster IISpacecraft Penetrating the Earth’s Bow Shock

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LAUNCH DATES: 16-Jul-2000 and 09-Aug-2000

MISSION END: 31 December 2014

LAUNCH VEHICLE: 2 Russian Soyuz launchers

LAUNCH MASS: 4800 kg

MISSION PHASE: Operational

Cluster Mission Parameters Summary

ORBIT:Elliptical polar (i=90°) orbit, period: 57 hoursPerigee: 19 000 km, apogee: 119 000 km; a = 69 000 km, e = 0.725

ACHIEVEMENTS:The four Cluster spacecraft are providing a detailed three-dimensional map of the magnetosphere, with surprising results

THE MISSION:Cluster is currently investigating the Earth’s magnetic environment and its interaction with the solar wind in three dimensions. Science output from Clustergreatly advances our knowledge of space plasma physics, improves the modeling of the magnetosphere and understanding of physical processes occurring in it.

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Number Acronym Instrument Measurement Purpose

1 CISCluster Ion Spectroscopyexperiment

Ion times-of-flight (TOFs) and energies from 0 to 40 keV

Composition and 3D distribution of ions in plasma

--- --- --- --- ---

6 FGMFluxgate Magnetometer

Magnetic field B magnitude and direction

B vector and event trigger to all instruments except ASPOC

--- --- --- --- ---

11 RAPID

Research with AdaptiveParticle Imaging Detectors

Electron energies from 30 to 1500 keV, ion energies from 20 to 450 keV

3D distributions of high-energy electrons and ions in plasma

Instrumentation on Each Cluster Satellite

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III. Turbulence Theories

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The small scale turbulent motions are statistically isotropic;

is inertial range;

where

is velocity increment.

/ 3 /3( ) ,

n n n

nr C ru

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,

( ) ( ) ( )ru u x r u x

First Hypothesis:(hereafter, for very high

Reynolds numbers)

Second Hypothesis:

Third Hypothesis:

Kolmogorov’s Theory of Turbulence

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Kraichnan Model (MHD Turbulence)

Strong turbulence

Weak turbulence

(total = mean + fluctuations)

Non-linear time scale, Kolmogorov-like

energy spectrum:5 /3

( )E k k

Alfvén time scale, Iroshnikov-Kraichnan

spectrum:3/ 2

( )E k k

0B B b

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IV. Data for AnalysisFor investigating of the turbulent processes in Earth’s magnetosphere there have been used measurements of magnetic field which had been taken with frequency 22.5 Hz.

This means that time step is 0.0445 seconds. Choosing different multiples of this time step I obtain data sets which may then be studied in different ways.

20000 40000 60000 80000

U T

0

15

30

45

60B, nT

04/03/2004

35000 40000 45000 50000 55000 60000 65000 70000

t, сек

0

20

40

60 В, нТл

15/05/2010

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V. Statistical Methods

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a) Analysis of the ProbabilityDistribution Functions (PDFs)

Here you can see the example of non-Gaussian (called Lévy) distributions.

Example of evolution of non-Gaussian distribution with changes of step of observed fluctuations:

P(O) ~ τ-s,

wheres ~ 0.5 – Gaussian distributions > 0.5 – Lévy distribution

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0

1( , ) exp( ) cosp x k kxdk

2

0

2

( )1( ) exp

22

x xp x

PDFs of Magnetic Field Fluctuations

-9 -6 -3 0 3 6 9

dB, nT

0.01

0.10

1.00

10.00 P(dB)0.0445 s

1 s

2 s

5 s

-16 -8 0 8 16

dB, nT

0.01

0.10

1.00

10.00 P(dB)0.0445 s

1 s

2 s

5 s

-4 -2 0 2 4

dB, nT

0.01

0.10

1.00

10.00 P(dB)0.0445 s

1 s

2 s

5 s

Functions of distribution of probability density of magnetic field fluctuations on different shifts of time are shown on these graphs.

Data for magnetopause, postshock and solar wind plasma, respectively from left to right. (Events for 2010.)

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Dependance P(O) on the Time Step τ

0.10 1.000.10

1.00

10.00

SW

FSH

M SH

P0

, s

M P

Experimental points are fitted with a straight line P(O) ~ τ-s. Since scales on axes are logarithmical, exponent s is just the tangent of angle of line’s inclination. See the table.

04/03/04 02/05/09 01/05/08 20/02/05 10/03/06 05/04/07 08/08/08 15/05/10

SW 0.52 0 .67 0 .56 0 .6 0 .48 0 .45 0 .55 0 .52

FSH 0.7 0 .72 0 .8 0 .67 0 .68 0 .6 0 .76 0 .75

PSH 0.98 0 .9 1 .01 0 .89 0 .95 0 .92 0 .97 0 .9

M SH 0.95 0 .8 0 .9 0 .82 0 .94 0 .87 0 .91 0 .87

M P 0.65 0 .81 0 .7 0 .66 0 .8 0 .67 0 .73 0 .8

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Excess kurtosis defined by a relation

where

are structure functions of the fourthand the second order respectively, is shown at the left.

It’s clearly that at the small timescales there is intermittency present which causes the statistics to be non-Gaussian.

4

4( ) ( ) ( ) ,S B t B t

b) Analysis of the Excess Kurtosis

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0 1 2 3 4 5 6

, s0

5

10

15

20

25 K

M P

FSH

M SH

SW

MP – Magnetopause SW – Solar Wind

MSH – Magnetosheath FSH – Foreshock

4

2

2

( )( ) ,

( ( ))

SK

S

2

2( ) ( ) ( )S B t B t

Structure function of the order q is defined as follows:

It is the main quantity which successfully describes all the complexity of a turbulent process.

Theory yields that where:

At high orders q, we observe slight discrepancies - the exponent becomes a non-linear function of order q:

ESS-analysis comprises the comparison of different order structure functions:

c) Self-Similarity Analysis (Theory)

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( ) ( ) ( ) .q

qS B t B t

( )( ) ,

qS

( ) / 3q q ( ) / 4q q

( ) / ( )( ) ( ) .

q p

q pS S

3D log-Poisson cascading model

Kolmogorov’s theory

Kraichnan’stheory

3( ) (1 ) [1 ]3 1

qq

q

Self-Similarity Analysis(Experiment)

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-2.0 -1.0 0.0 1.0 2.0Log(S(3))

-2

0

2

4

6

8

Lo

g(S

(q))

q=3

q=4

q=5

q=6

q=7

q=8

q=9

q=10

1 2 3 4 5Log(S(4))

-2

0

2

4

6

8

10

12

14

16

Lo

g(S

(q))

q=0.5

q=1

q=1.5

q=2

q=3

q=4

q=5

q=6

q=7

q=8

q=9

q=10

0 1 2 3 4 5 6 7 8 9

q

0

1

2

3 q

K41

SL

B, SW

B, FSH

B, M SH

B, M P

B, PSH

0 1 2 3 4 5 6 7 8 9

q

0

1

2

q

PP

IK

B, MP

B, PSH

B, MSH

B, FSH

B, SW

Kolmogorov Kraichnan

Diffusion

Anomalous diffusion

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Parameters of diffusion processes

Дата Положение ß Δ (1 1)R 1 / 2s R

04/03/2004 MSH MP

PSH

0.5 0.26

0.47

0.4 0.12

0.8

0.81 0.33

0.9

0.9 0.68

0.95

02/05/2009 MSH MP

0.56 0.25

0.74 0.18

0.58 0.54

0.79 0.77

08/08/2008 FSH

PSH MP

0.62

0.5 0.34

0.51

0.9 0.15

0.31

0.9 0.29

0.66

0.95 0.65

15/05/2010 MSH

FSH PSH

MP

0.52

0.45 0.71

0.17

0.6

0.4 2.22

0.05

0.64

0.48 0.91

0.25

0.82

0.74 0.96

0.63

05/04/2007 MSH FSH

PSH MP

0.51 0.4

0.52 0.48

0.8 0.34

0.89 0.2

0.76 0.52

0.82 0.22

0.88 0.76

0.91 0.61

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VI. Summary Three different approaches yield similar results.

Analyzing data for magnetosheath, foreshock, postshockregions, and magnetopause on timescales less than 1 s I obtainedevidence of the existance of intermittent processes. Oppositely, ontimescales greater than 1 s we observe Gaussian statistics.

Turbulent processes in the magnetosheath obey3D log-Poisson cascading model.

Both Kolmogorov’s and Kraichnan’s theories do NOT hold inhighly inhomogeneous intermittent regions likepostshock, magnetosheath and magnetopause. Vice versa, solarwind region corresponds to Kraichnan’s predictions and foreshocktends to satisfy both theories.

Analyzing generalized diffusion coefficient, it was shown thatturbulent processes have the character of superdiffusion.

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Acknowledgement:• (data for the study kindly granted by) Anthony T.Y.

Lui, Johns Hopkins University APL, Laurel, Maryland, USA

Thanks a lot for attention!

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