1. Fouad SAHRAOUI PhD Thesis at the university of Versailles, 2003 Magnetic Turbulence in the Terrestrial Magnetosheath : a Possible Interpretation in the Framework of the Weak Turbulence Theory of the Hall-MHD System Supervised by Gérard Belmont & Laurence Rezeau Now : Post-doctoral position at CETP (CNES followship) Centre d’étude des Environnements Terrestre et Planétaire, Vélizy, France Main publications : 1.F. Sahraoui, G. Belmont, and L. Rezeau, From Bi-Fluid to Hall-MHD Weak Turbulence : Hamiltonian Canonical Formulations, Physics of Plasmas, 10, 1325-1337, 2003. 2. F. Sahraoui et al., ULF wave identification in the magnetosheath : k-filtering technique applied to Cluster II data, J. Geophys. Res., 108 (A9), 1335, 2003. 3. L. Rezeau, F. Sahraoui & Cluster turbulence team, A case study of low-frequency waves at the magnetopause, Annales Geophysicae,19, 1463-1470, 2001.

2. Physical context : the magnetosheath Collisionless plasma Ideal MHD : magnetopause = impermeable frontier However, penetration of the solar wind particles  Role of the magnetosheath turbulence?

3. The ULF magnetic turbulence in the magnetosheath Questions : 1.Importance of the Doppler effect and the shape of the spectrum in the plasma frame ? How to infer the k (spatial) spectrum from the  (temporal) one ? 2.Nature of the non linear effects : weak or strong ? Coherent structures ? « linear » modes? Power law spectrum of the Kolmogorov type 1941 (k -5/3 ) Cascade en f -2.3 Cluster : STAFF-SC ; 18/02/2002 How to answer ?  New possibilities : Cluster multipoints data and the k-filtering technique

4. k-filtering method Pinçon & Lefeuvre (LPCE, 1991) CLUSTER B1B1 B2B2 B3B3 B4B4 From the multipoint measurements of a turbulent field, it provides an estimation of the spectral energy density P( ,k) using a filter bank approach Has been validated by numerical simulations (Pinçcon et al, 1991) Applied for the first time to real data (Sahraoui et al., 2003) Hypotheses : stationnarity + homogeneity

5. P(f =0.37Hz,k) Sahraoui et al., 2003 kxkx kyky kzkz Application to Cluster magnetic data Physical interpretation ? 2 nd secondary maximum principal maximum 1 st secondary maximum Magnetosheath (18/02/2002)

6. Comparison of the maxima to LF linear modes Isocontours of P( ,k) (f =0.37 Hz  f ci ) Theoretical dispersion relations transformed to the satellite frame Mirror (~ 0 f ci ) Alfvén (~ 5.9 f ci ) Slow (~ 0.3 f ci ) “Fast” (~ 6.1 f ci ) Main results : The observed spectrum in the satellite frame  a mixture of modes in the plasma frame Identification of LF linear modes from a turbulent spectrum  validity of a weak turbulence approach

7. Necessity to develop a new theory of weak turbulence for the Hall-MHD system 1.Identification of linear modes + small fluctuations (  B <<B 0 )  interpretation in the framwork of the weak turbulence theory weak turbulence theory : developped essentiellement in incompressible ideal MHD (Galtier et al., 2000) fast mode intermediate mode slow mode  ci kiki  /  ci  ideal MHD domaine Hall term E + v  B = 1+2  MHD-Hall Non ideal Ohm’s Law : 2.Scales  >  ci and compressibility  incompressible ideal MHD

8. Weak turbulence theory in Hall-MHD system avec Equations of motion in terms of the physical variables ,  v,  b Problème : absence of appropriate variables allowing diagonalisation (mixture of the physical variables in the N.L termes) Solution : Hamiltonian formalism ?

9. Advantage of the Hamiltonian formalism Canonique formulation (to be built) + Appropriate canonical transformation = Diagonalisation It allows to introduce the amplitude of each mode as a canonical variable of the system

10. How to build a canonical formulation of the MHD-Hall system ? Bi-fluide  MHD-Hall First we construct a canonical formulation of the bi-fluid system, then we reduce to the one of the Hall-MHD by generalizing the variationnal principle : Lagrangian of the compressible hydrodynamic (Clebsch variables) + electromagnetic Lagrangian + introduction of new Lagrangian invariants How to deal with the bi-fluid system ?

11. bi-fluid Hamiltonian formulation : H BF is canonical with respect to the variables H BF corresponds to the total energy of the bi-fluid system

12. The generalized Clebsch variables (n l,  l ), ( l,  l ) are suffisiant for a fully description of the MHD-Hall The canonical equations of the Hall-MHD: Sahraoui et al., 2003

13. The future steps 1.Derive the kinetic equations of waves for the Hall-MHD weak turbulence  Power law spetra of the Kolmogorov type: 2.Deduce the k spectrum (integrated in  ) : Total characterization of the observed spetra 1 + 2  3....