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

Outline of the talk

1. Turbulence/Intermittency2. Turbulence in the Cusp 3. CLUSTER data4. Probability Distribution Functions (PDF)5. Flatness6. Correlation coefficients (auto and cross)7. Conclusions & Perspectives

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

Classical turbulence (Kolmogorov 41)

Richardson cascade

• Self-similarity

• Localness of interactions

driving scale

inertial scales

dissipation scale

Two main hypotheses :

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

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)

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)

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.

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

CLUSTER data

Inner magnetosphere Cusp and crossings regions Magnetosheath

Densities from the WHISPER experiment

How can we detect/quantify intermittency ?

• Probability distribution functions (PDFs)• Flatness • Multi-fractal analysis• Continuous Wavelet Transform

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=Bx,By,Bz or B2 )

is the time that separates two observations of a fluctuating component :

= t . 2n

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

PDFs in the inner magnetosphere

Non-scaled PDFs Scaled PDFsB2

PDFs in the cusp region

Non-scaled PDFs Scaled PDFsB2

PDFs in the magnetosheath

Non-scaled PDFs Scaled PDFsB2

FLATNESS

• The flatness F is related to higher moments of the fluctuations :

F =<P(t,)4> / (<P(t,)2>)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

FLATNESS IN THE INNER MAGNETOSPHERE

FLATNESS IN THE CUSP REGION

FLATNESS IN THE MAGNETOSHEATH

CORRELATION COEFFICIENTS

= cross correlation coefficient between

Pi and Pj 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

CORRELATION COEFFICIENTS

• Cluster 1 & 4

• Comp. Bz

• Complete data

Dynamic nature of the

turbulent eddies

COMPARISON MACRO/MICRO-SCALES

CLUSTER 1 & 4

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).