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Cusp turbulence as revealed by POLAR magnetic field data E. Yordanova

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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
- magnetosheath plasma /high density and low energy/
- plasma ofionospheric origin

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)

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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

PDF in parallel and perpendicular directions

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

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)

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