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Cusp turbulence as revealed by POLAR magnetic field data E. Yordanova. Uppsala, November, 2005. Outline. Cusp Models of turbulence Multifractal structure of cusp turbulence Anisotropy in the cusp. Uppsala, November, 2005. depressed and irregular magnetic magnetic field

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Cusp turbulence as revealed by POLAR magnetic field data

E. Yordanova

Uppsala, November, 2005


Outline

  • Cusp

  • Models of turbulence

  • Multifractal structure of cusp turbulence

  • Anisotropy in the cusp

Uppsala, November, 2005


Cusp

  • the direction of IMF

  • the tilt of the magnetic dipole

  • the solar wind dynamic pressure

Uppsala, November, 2005


Turbulent magnetic field in the cusp

POLARmission

f < 10-2 Hz

f < 102-103 Hz

Uppsala, November, 2005


Examples of power spectra

of the magnetic field fluctuations, measured in the cusp (POLAR satellite)

By110497_1

B091096_5

Uppsala, November, 2005


Magnetospheric cusp magnetic field (POLAR satellite)

cusps

ramps

spirals

The singularity strength:

Singularity spectrum D(h)

  • the statistical distribution of the singularity exponents h.

Hölder exponenth(x0) - a measure of the regularity of the function g at the point x0

- a humped shape (hmin- strongest singularity; hmax –weakest singularity)

Uppsala, November, 2005


Wavelet Transform (WT)A tool for detecting the singularities

a - scale, b – translation or dilation,

* - conjugated transforming function

Wavelet Transform Modulus Maxima Method (WTMM)

Mallat and Zong(1992)

a maximum in the modulus of the wavelet transform coefficients

Singularities

‘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’

Modulus maximum of WT

the curve, connecting the modulus maxima

Maxima line

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

Singularity exponents

Uppsala, November, 2005


Kolmogorov phenomenology (1941)

Energy injection

Richardson cascade

Inertial range

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

Dissipation range

Self-similarityin the inertial range

Localness in the interaction

Uppsala, November, 2005


P model (Meneveau and Sreenivasan ‘87)

Energy injection

Inertial range

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

Dissipation range

Uppsala, November, 2005


Structure functionsof a measured fluctuating parameter g(x):

!fundamental quantity in classical theory of turbulence!

Singularity spectrum(Parisi and Frisch,1985)

Legendre

transform

D(h) - statistical distribution of the singularity exponentsh

Calculation of the scaling properties of turbulence

Uppsala, November, 2005


WTMM

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

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

Scaling law of the partition function along the maxima line:

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

Relation between q and q

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

Uppsala, November, 2005


- scaling exponents for the Kolmogorov-like cascade:

P-model

- scaling exponents for the Kraichnan-like cascade:

Extended structure function models

(Tu et al. 1996, Marsch and Tu 1997)

Uppsala, November, 2005


The problem

MF fluctuations – singular behavior

Set of locations and strength of the singularities – singularity spectrum

Method: WTMM

Constructing partition functions – sums of the WT located in the modulus maxima (define the singularity)

Uppsala, November, 2005


WTMM partition functions

WTMM partition function exponents

Fractional brownian signal

Singularity spectrum

Muzy, Bacry & Arneodo(1994)

Uppsala, November, 2005


WTMM partition functions

WTMM partition function exponents

Devil’s staircase signal

Singularity spectrum

Muzy, Bacry & Arneodo(1994)

Uppsala, November, 2005


partition function exponents

(power law fit of wavelet coefficients along maxima line)

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

Non-linear behavior

Mean square deviation between numerical and theoretical spectra

Least-square fit of models of turbulence

Comparison with models of turbulence

Uppsala, November, 2005


Data sampling frequency - 8.333 Hz

Probability distribution functions for different time delays

Uppsala, November, 2005


Results for 9 Oct 1996 case

p- model turbulence

Bz > 0

Kolmogorov – like turbulence

Bz < 0

Uppsala, November, 2005


Results for 11 Apr 1997 case

Kolmogorov – like turbulence

p - model turbulence

Bx > 0

By < 0

Bz > 0

Uppsala, November, 2005


HYDRA / POLAR

Uppsala, November, 2005


1. Conclusions about the magnetic field intensity

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

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

Uppsala, June, 2005


SPC

Bz

north

antisun

Bxy

dusk

B56

Anisotropy features of the magnetic field

B~90 nT

B~10 nT

(Bxy, Bz, B56)

(B1, B2, B0)

Uppsala, November, 2005


~ 1.62

 ~ 2.41

Power spectra in parallel and perpendicular directions

f -5/3

 ~ 1.21

 ~ 5

 ~ 1.93

Uppsala, November, 2005


Extended Self-Similarity Analysis

Uppsala, November, 2005



2. Conclusions about the anisotropy in the cusp

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’

Uppsala, November, 2005


Taylor’s hypothesis

V total

For 9 Oct 1996 case – V~100 km/s

POLAR speed is 2 km/s

For 11 Apr 1997 case – V~40 km/s

Uppsala, June, 2005


Structure function

 (q) and (q)

Uppsala, June, 2005


Power spectra of 11 April 1997 case

  • 2.15

  • (0.06 – 0.78 Hz)

By110497_1

By110497_2

By110497_3

Uppsala, June, 2005


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