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

2. Outline  Cusp  Models of turbulence  Multifractal structure of cusp turbulence  Anisotropy in the cusp  Cusp  Models of turbulence  Multifractal structure of cusp turbulence  Anisotropy in the cusp Uppsala, November, 2005

3. Cusp 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 Uppsala, November, 2005

4. Turbulent magnetic field in the cusp POLAR mission Uppsala, November, 2005 f < 10 -2 Hz f < 10 2 -10 3 Hz

5. Examples of power spectra of the magnetic field fluctuations, measured in the cusp (POLAR satellite) By110497_1 B091096_5 Uppsala, November, 2005

6. Magnetospheric cusp magnetic field (POLAR satellite) cusps spirals ramps The singularity strength : Hölder exponenth(x 0 ) Hölder exponent h(x 0 ) - a measure of the regularity of the function g at the point x 0 - the statistical distribution of the singularity exponents h. Singularity spectrum D(h) - a humped shape (h min - strongest singularity; h max – weakest singularity) Uppsala, November, 2005

7. a maximum in the modulus of the wavelet transform coefficients Singularities Modulus maximum of WT ‘any point ( x 0,a 0 ) of the space-scale half-plane which corresponds to the local maximum of the modulus of considered as a function of x ’ Maxima line the 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 (WT) Wavelet Transform (WT) A tool for detecting the singularities a - scale, b – translation or dilation,  * - conjugated transforming function Uppsala, November, 2005

8. Energy injection Inertial range Dissipation range........ Richardson cascade Kolmogorov phenomenology (1941) Self-similarity in the inertial range Localness in the interaction Uppsala, November, 2005

9. P model (Meneveau and Sreenivasan ‘87) Energy injection Inertial range Dissipation range........ Uppsala, November, 2005

10. Calculation of the scaling properties of turbulence Structure functions Structure functions of a measured fluctuating parameter g(x): !fundamental quantity in classical theory of turbulence! Singularity spectrum Singularity spectrum (Parisi and Frisch,1985)Legendretransform D(h) - statistical distribution of the singularity exponents h Uppsala, November, 2005

11. Scaling law of the partition function along the maxima line: Singularity spectrum D(h) of the WTMM function  (q): Wavelet based partition function (): Wavelet based partition function (Muzy, Bacry, Arneodo, 1991): WTMM L(a) - a set of all the maxima lines l existing at a scale a; b l (a) - the position, at a, of the maximum belonging to the line l l={b l (a), a} is pointing towards a point b l (0) (when a goes to 0 ) which corresponds to a singularity of g Relation between  q and  q Uppsala, November, 2005

12. Extended structure function models (Tu et al. 1996, Marsch and Tu 1997) - scaling exponents for the Kolmogorov-like cascade: - scaling exponents for the Kraichnan-like cascade : P-model Uppsala, November, 2005

13. Method: WTMM partition functions Constructing partition functions – sums of the WT located in the modulus maxima (define the singularity) singularity spectrum Set of locations and strength of the singularities – singularity spectrum MF fluctuations – singular behavior The problem Uppsala, November, 2005

14. Fractional brownian signal WTMM partition functions WTMM partition function exponents Singularity spectrum Muzy, Bacry & Arneodo (1994) Uppsala, November, 2005

15. Devil’s staircase signal WTMM partition functions WTMM partition function exponents Singularity spectrum Muzy, Bacry & Arneodo (1994) Uppsala, November, 2005

16. Comparison 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 Uppsala, November, 2005

17. Probability distribution functions for different time delays Data sampling frequency - 8.333 Hz Uppsala, November, 2005

18. Kolmogorov – like turbulence B z < 0 p - model turbulence B z > 0 Results for 9 Oct 1996 case Uppsala, November, 2005

19. Results for 11 Apr 1997 case B x > 0 B y < 0 B z > 0 Kolmogorov – like turbulence p - model turbulence Uppsala, November, 2005

20. HYDRA / POLAR Uppsala, November, 2005

21. 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

22. Uppsala, November, 2005 B~90 nT  B~10 nT BzBzBzBz B xy SPC north B 56 dusk antisun (B xy, B z, B 56 ) (  B 1,  B 2, B 0 ) Anisotropy features of the magnetic field

23. Uppsala, November, 2005 Power spectra in parallel and perpendicular directions  ~ 1.62  ~ 2.41 f -5/3  ~ 1.21  ~ 1.93  ~ 5

24. Uppsala, November, 2005 Extended Self-Similarity Analysis

25. PDF in parallel and perpendicular directions  = 6,12,24,48,96,192  t

26. 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 2. Conclusions about the anisotropy in the cusp

27. Uppsala, June, 2005 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 Taylor’s hypothesis

28. Structure function  (q) and  (q) Uppsala, June, 2005

29. Power spectra of 11 April 1997 case By110497_1 By110497_2 By110497_3 -2.15 (0.06 – 0.78 Hz)