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Fouad SAHRAOUI

Fouad SAHRAOUI. PhD Thesis at the university of Versailles , 2003 Magnetic Turbulence in the Terrestrial Magnetosheath : a Possible I nterpretation in the F ramework of the Weak Turbulence Theory of the Hall-MHD System Supervised by Gérard Belmont & Laurence Rezeau

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Fouad SAHRAOUI

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  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 : • 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. • 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 Cluster : STAFF-SC ; 18/02/2002 Power law spectrum ofthe Kolmogorovtype 1941 (k -5/3) Cascade en f -2.3 • Questions : • Importance of the Doppler effect and the shape of the spectrum in the plasma frame ? How to infer the k(spatial) spectrumfrom the(temporal) one ? • Nature of the non linear effects : weakor strong ? Coherent structures ? « linear » modes? How to answer ? New possibilities : Cluster multipoints data and the k-filtering technique

  4. CLUSTER B2 B3 B1 B4 k-filtering methodPinçon & Lefeuvre (LPCE, 1991) From the multipoint measurements of a turbulent field, it provides an estimationof thespectral energy density P(w,k) using a filter bank approach Hypotheses :stationnarity + homogeneity • Has been validated by numerical simulations (Pinçcon et al, 1991) • Applied for the first time to real data (Sahraoui et al., 2003)

  5. P(f=0.37Hz,k) Magnetosheath (18/02/2002) 1st secondary maximum principalmaximum 2ndsecondary maximum kz Sahraoui et al., 2003 ky kx Application to Cluster magnetic data Physical interpretation ?

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

  7. 1+2 MHD-Hall /ci fast mode intermediate mode Non ideal Ohm’s Law : E + vB =  ideal MHD domaine ci Hall term slow mode ki Necessity to develop a new theory of weak turbulence for the Hall-MHD system • Identification of linearmodes+small fluctuations (B <<B0) •  interpretation in the framwork of the weak turbulence theory • weak turbulence theory: developped essentiellement inincompressibleidealMHD (Galtier et al., 2000) • Scales > ciandcompressibility  incompressible idealMHD

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

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

  10. How to build a canonical formulationof 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 How to deal with the bi-fluid system ? by generalizingthe variationnalprinciple : Lagrangian of the compressiblehydrodynamic(Clebsch variables) +electromagnetic Lagrangian + introduction of new Lagrangian invariants

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

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

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

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