Cusp turbulence as revealed by POLAR magnetic field data E. Yordanova Uppsala, November, 2005.

Cusp turbulence as revealed by POLAR magnetic Cusp turbulence as revealed by POLAR magnetic field datafield data

E. YordanovaE. Yordanova

Uppsala, November, Uppsala, November, 20052005

OutlineOutline

Cusp Cusp

Models of turbulenceModels of turbulence

Multifractal structure of cusp turbulenceMultifractal structure of cusp turbulence

Anisotropy in the cuspAnisotropy in the cusp

Cusp Cusp

Models of turbulenceModels of turbulence

Multifractal structure of cusp turbulenceMultifractal structure of cusp turbulence

Anisotropy in the cuspAnisotropy in the cusp


CusCuspp

• depressed and irregular magnetic magnetic field

• magnetosheath plasma /high density and low energy/

• plasma of ionospheric origin

• the direction of IMF

• the tilt of the magnetic dipole

• the solar wind dynamic pressure


Turbulent magnetic field in the Turbulent magnetic field in the cuspcusp

POLARPOLAR missionmission


f < 10-2 Hzf < 102-103 Hz

Examples of power spectra Examples of power spectra of the magnetic field fluctuations, of the magnetic field fluctuations,

measured in the cusp (POLAR satellite)measured in the cusp (POLAR satellite)

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Magnetospheric cusp magnetic field (POLAR satellite)Magnetospheric cusp magnetic field (POLAR satellite)

cuspscuspsspiralsspirals

rampsramps

0 0

h

ng x P x x C x x

The singularity strength:

Hölder exponentHölder exponent hh((xx00)) - a measure of the

regularity of the function g at the point x0

- the statistical distribution of the singularity exponents h.

SSingularity spectrum Dingularity spectrum D((hh)) - a humped shape (hmin - strongest singularity;

hmax – weakest singularity)


a maximum in the modulus of the wavelet transform

coefficients

Singularities

Modulus maximum of WT

‘any point (x0,a0) of the space-scale

half-plane which corresponds to the local maximum of the modulus of considered as a function of x’

Maxima linethe curve, connecting the modulus maxima

Singularity exponents a power law fit of the wavelet coefficients along the maxima line

Mallat and Zong (1992)Wavelet Transform Modulus Maxima Method (WTMM)Wavelet Transform Modulus Maxima Method (WTMM)

Wavelet Transform (WT)Wavelet Transform (WT) A tool for detecting the singularities

a - scale, b – translation or dilation, * - conjugated transforming function

dxa

bxxg

aabg

*1,


Energy injection

Inertial range

Dissipation range

. . . . . . . . . . . . . . . .

Ric

hard

son

casc

ade

Ric

hard

son

casc

ade

Kolmogorov phenomenology (1941)Kolmogorov phenomenology (1941)

Self-similaritySelf-similarity in the inertial rangein the inertial range

LocalnessLocalness in the interaction in the interaction


P model (Meneveau and Sreenivasan ‘87)P model (Meneveau and Sreenivasan ‘87)

Energy injection

Inertial range

Dissipation range

. . . . . . . . . . . . . . . .


Calculation of the scaling properties of turbulenceCalculation of the scaling properties of turbulence

Structure functionsStructure functions of a measured fluctuating parameter g(x):

0

1~

Lq q

qS l g x l g x dx lL

!!fundamental quantity in classical theory of turbulence!fundamental quantity in classical theory of turbulence!

Singularity spectrumSingularity spectrum (Parisi and Frisch,1985)

q

qS l lLegendre Legendre transformtransform inf 1q qD h qh

D(h) D(h) - statistical distribution of - statistical distribution of the singularity exponentsthe singularity exponents hh


Scaling law of the partition function along the maxima line:Scaling law of the partition function along the maxima line:

, ~ qZ q a a

Singularity spectrum D(h) of the WTMM function Singularity spectrum D(h) of the WTMM function (q):(q): infqD h qh q

Wavelet based partition function Wavelet based partition function ((Muzy, Bacry, Arneodo, 1991))::

'

, sup ' , 'q

la al L a

Z q a T g b a a

WTMM

L(a) - a set of all the maxima lines l existing at a scale a; bl(a) - the position, at a, of the maximum belonging to the line l

l={bl(a), a} is pointing towards a point bl(0) (when a goes to 0) which corresponds to a singularity of g

1q q

Relation between q and q


Extended structure function modelsExtended structure function models (Tu et al. 1996, Marsch and Tu 1997)(Tu et al. 1996, Marsch and Tu 1997)

- scaling exponents for the Kolmogorov-like cascade:- scaling exponents for the Kolmogorov-like cascade:

/ 3/32 1 1

2/32/32 1 1

5 3log 1

2 2 3

1log 1

3

qqqq P P

P P

- scaling exponents for the Kraichnan-like cascade- scaling exponents for the Kraichnan-like cascade::

/ 4/ 42 1 1

1/ 21/ 22 1 1

3 2 log 14

1log 1

2

qqqq P P

P P

P-model


Method: WTMM Constructing partition partition functionsfunctions – sums of the WT located in the modulus maxima (define the singularity)

Set of locations and strength of the singularities – singularity spectrumsingularity spectrum

MF fluctuations – singular behavior

The problemThe problem


Fractional Fractional brownian signalbrownian signal

WTMM partition WTMM partition functionsfunctions

WTMM partition WTMM partition function exponentsfunction exponents

Singularity spectrum Singularity spectrum

Muzy, Bacry & Arneodo (1994)


Devil’s staircase Devil’s staircase signalsignal

WTMM partition WTMM partition functionsfunctions

WTMM partition WTMM partition function exponentsfunction exponents

Singularity spectrum Singularity spectrum

Muzy, Bacry & Arneodo (1994)


Comparison with models of turbulenceComparison with models of turbulence

partition function exponents (power law fit of wavelet coefficients

along maxima line)Non-linear behavior

Least-square fit of models of turbulence

Through numerical differentiation of the exponents curve singularity spectrum is derived (parabolic shape, typical for the non-linear systems)

Mean square deviation between numerical and theoretical spectra


Probability distribution functions for different time delaysProbability distribution functions for different time delays

Data Data sampling sampling

frequency - frequency - 8.333 Hz8.333 Hz


Kolmogorov – like turbulence

Bz < 0

p - model turbulence

Bz > 0

Results for 9 Oct 1996 caseResults for 9 Oct 1996 case


Results for 11 Apr 1997 caseResults for 11 Apr 1997 case

Bx > 0

By < 0

Bz > 0

Kolmogorov – like Kolmogorov – like turbulenceturbulence p p - model - model

turbulenceturbulence


HYDRA / POLAR


1. Conclusions about the magnetic 1. Conclusions about the magnetic field intensityfield intensity

IMF Bz > 0 – p – model (fluid, fully developed)

IMF Bz < 0 - Kolmogorov- like (fluid, non fully developed)

Uppsala, June, 2005Uppsala, June, 2005


B~90 nT

B~10 nT

BBzz

BBxyxy

SPCSPC

northnorth

BB5656

duskdusk

antisunantisun

(Bxy, Bz, B56)

(B1, B2, B0)

Anisotropy features of the magnetic field


Power spectra in parallel and Power spectra in parallel and perpendicular directionsperpendicular directions

~ 1.62 ~ 2.41

f -5/3

~ 1.21

~ 1.93

~ 5


Extended Self-Similarity AnalysisExtended Self-Similarity Analysis

PDF in parallel and perpendicular directionsPDF in parallel and perpendicular directions

= 6,12,24,48,96,192t

0

2 2 2

2 2 20 0

B B t B t

B B t B t

PSD - different scaling in parallel and perpendicular directions

ESS analysis – parallel fluctuations are characterized by monofractal nature; perpendicular - by a strong intermittent (multifractal) character

PDF – more intermittent character of the fluctuations in perpendicular direction then in parallel

Acknowledgements: E. Yordanova acknowledges the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2001-00314, ‘Turbulent Boundary Layers’


2. Conclusions about the anisotropy 2. Conclusions about the anisotropy in the cuspin the cusp


V total

For 9 Oct 1996 case – V~100 km/sPOLAR speed is 2 km/s

For 11 Apr 1997 case – V~40 km/s

Taylor’s hypothesis

Structure functionStructure function (q) and (q) and (q)(q)



Power spectra of Power spectra of 11 April 1997 case11 April 1997 case

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-2.152.15(0.06 – 0.78 Hz)(0.06 – 0.78 Hz)

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Cusp turbulence as revealed by POLAR magnetic field data E. Yordanova Uppsala, November, 2005.

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