1. Turbulence in the magnetosphere studied with CLUSTER data : evidence of intermittency Lamy H. 1, Echim M. 1,2, Darrouzet F. 1, Lemaire J. 3, Décréau P. 4, Dunlop M. 5 1 Belgian Institute of Space Aeronomy, Brussels, Belgium 2 Institute of Space Sciences, Bucharest, Romania 3 Center for Space Radiations, Louvain-La-Neuve, Belgium 4 LPCE/CNRS, University of Orléans, France 5 Rutherford Appleton Laboratory, United Kingdom

2. Outline of the talk 1.Turbulence/Intermittency 2.Turbulence in the Cusp 3.CLUSTER data 4.Probability Distribution Functions (PDF) 5.Flatness 6.Correlation coefficients (auto and cross) 7.Conclusions & Perspectives

3. What is turbulence ? A non-linear phenomenom resulting from the interaction between waves and eddies of many different scales. In a turbulent regime, fluid and plasma parameters vary randomly in time and space Statistical approach

4. Classical turbulence (Kolmogorov 41) Richardson cascade Self-similarity Localness of interactions driving scale inertial scales dissipation scale Two main hypotheses :

5. Self-similarity/Intermittency Self-similar fluctuations : if we magnify an arbitrary part, the statistical properties will be identical Intermittent fluctuations : alternance of intervals with high activity with quiet intervals Brownian motion is self-similar The Devil’s staircase is intermittent

6. Several models of intermittency Smaller eddies are less and less space-filling (ex : the  model, Frisch 1995) The energy transfer rate is scale-dependent (ex : the p- model, Meneveau & Sreenivasan 1987)

7. Turbulence in the magnetosphere Energy transfer from large scales to kinetic scales ? Mass and momentum transfer from one region of the magnetosphere to another (Goldstein 2005)

8. Turbulence in the cusp region Cluster spacecraft allow to distinguish between temporal and spatial fluctuations Nykyri et al. (2004) : using magnetometer data from Cluster, they find evidence that the cusp contains magnetic turbulence. Sundkvist et al. (2005) : discovery of short-scale vortices in the cusp region  another channel to transport plasma particles and energy through the magnetospheric boundary layers.

9. CLUSTER data Outbound pass on February 26, 2001 [3:30:00 – 7:00:00 UT] High resolution Magnetic Field (MF) data from the FGM magnetometer : 8 samples/sec for [3:30:00 – 5:30:00 UT] and 3 samples/sec for [5:30:00 – 7:00:00 UT] A background MF (IGRF + external Tsyganenko 2001) has been subtracted from the data before analyzing the fluctuations. Three distinct regions along the spacecraft trajectory are considered

10. CLUSTER data Inner magnetosphereCusp and crossings regionsMagnetosheath Densities from the WHISPER experiment

11. How can we detect/quantify intermittency ? Probability distribution functions (PDFs) Flatness Multi-fractal analysis Continuous Wavelet Transform

12. Probability density functions (PDFs) PDF = histogram of the fluctuating field P(t) P(t,) = P(t+) – P(t) for a given value  of the temporal scale. ( P=B x,B y,B z or B 2 )  is the time that separates two observations of a fluctuating component :  = t. 2 n where t is the time resolution of the data. Intermittency is associated with increasing departure of PDFs from gaussianity when the scale  decreases. Number of points << than in SW data  statistics is good only up to ~  5

13. PDFs in the inner magnetosphere Non-scaled PDFsScaled PDFs B2B2

14. PDFs in the cusp region Non-scaled PDFsScaled PDFs B2B2

15. PDFs in the magnetosheath Non-scaled PDFsScaled PDFs B2B2

16. FLATNESS The flatness F is related to higher moments of the fluctuations : F = / ( ) 2 = mean on all data considered A fluctuating parameter is intermittent if F increases when considering smaller scales If F remains more or less constant whatever the scale, the fluctuations are self-similar F = 3 for Gaussian fluctuations

17. FLATNESS IN THE INNER MAGNETOSPHERE

18. FLATNESS IN THE CUSP REGION

19. FLATNESS IN THE MAGNETOSHEATH

20. CORRELATION COEFFICIENTS = cross correlation coefficient between P i and P j for the time-lag  Auto-correlation when i = j The Magnetic Field will be correlated with itself within a turbulent eddy and uncorrelated outside the eddy. The value of  for which the auto-correlation coefficient = 1/e gives the temporal scale size of the eddy. The length of the eddy can then be deduced from the flow speed of the plasma

21. CORRELATION COEFFICIENTS Cluster 1 & 4 Comp. B z Complete data Dynamic nature of the turbulent eddies

22. COMPARISON MACRO/MICRO-SCALES CLUSTER 1 & 4

23. CONCLUSIONS & PERSPECTIVES PDFs : gaussian in the inner magnetosphere, non- gaussian in the cusp and magnetosheath Flatness : F takes values close to 3 in the magnetosphere and strongly increases with decreasing scale in the cusp and magnetosheath region These results suggest the presence of intermittent turbulence in the cusp and magnetosheath Correlation analysis : existence of structures with scales comparable to the satellite separation distance. Structures with smaller scales exist as well, suggesting non self-similarity. To test this hypotheses more quantitavely  non- gaussian rescaling of the PDFs (Hnat et al. 2002) + multi-fractal analysis (investigations in progress).