fouad sahraoui
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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.
Physical context : the magnetosheath
• Collisionless plasma • Ideal MHD : magnetopause = impermeable frontier• However, penetration of the solar wind particles Role of the magnetosheath turbulence?
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
k-filtering method Pinçon & Lefeuvre (LPCE, 1991)
CLUSTER
B1
B2
B3
B4
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
P(f =0.37Hz,k)
Sahraoui et al., 2003
kx
ky
k z
Application to Cluster magnetic data
Physical interpretation ?
2nd secondary maximum
principal maximum
1st secondary maximum Magnetosheath (18/02/2002)
Comparison of the maxima to LF linear modes
• Isocontours of P(,k) (f =0.37 Hz fci)
• Theoretical dispersion relations transformed to the satellite frame
Mirror
(~ 0 fci)
Alfvén
(~ 5.9 fci)
Slow(~ 0.3 fci)
“Fast”
(~ 6.1 fci)
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
Necessity to develop a new theory of weak turbulence for the Hall-MHD system
1. Identification of linear modes + small fluctuations (B <<B0) 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
ki
/ci
ideal MHD domaine
Hall term
E + vB = )(vti d
em
1+2 MHD-Hall
Non ideal Ohm’s Law :
2. Scales > ci and compressibility incompressible ideal MHD
Weak turbulence theory in Hall-MHD system
0B4
T2
TT TD
Tv.D-DD .).(. 2
20
3
00
122222Ats
i
txAAst V
ρ
Tρρ
TCδVVC
)(.).()()()()(
)(1).(1)()().()(
03
0
1
vvv.Tv.v.
.vv.vTvv.
4
2
δδδδδδδρδγpδργδpδpδρδpδρT
δδμ
δδμ
δδρδδρδδρδρδT
0tt
00t
bbb
bbbbavec
• 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 ?
Advantage of the Hamiltonian formalism
21
0
22
0
22
0
222
ρCδρ
μδδρa siii
i
bv
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
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 ?
• bi-fluid Hamiltonian formulation :
ltl
BF
ltl
BF
nδφHδ
φnδ
Hδ)(
ltl
BF
ltl
BF
λδμHδ
μδλHδ
AD
DA
tBF
tBF
δHδ
δHδ
HBF is canonical with respect to the variables
)(),,(),,( DA,llll λμφn
eilllll
l
lll
lBF dnUqμ
nλφn
mH
,
2
21 rA
rD.AD dΦΦnnqμε
ei
2
0
2
0 21
21
HBF corresponds to the total energy of the bi-fluid system
HMHDH
ree yx dλμμλn
μλn
qBφn
m eilllll
lll
lli
i
,
2
0
21
21
yx ee e
ee
ei μ
nqB
λn
qB
qμnU 00
2021)(
rdλnμ
μnλ
ee
ee
e
e
2
21
The generalized Clebsch variables (nl,l), (l,l) are suffisiant for a fully description of the MHD-Hall
ltl
HMHD
ltl
HMHD
nδφ
Hδ
φnδ
Hδ)(
ltl
HMHD
ltl
HMHD
λδμ
Hδ
μδλ
HδThe canonical equations of the Hall-MHD:
Sahraoui et al., 2003
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 ) :
kkfPP //
~, kk
kkgS )( //k
Total characterization of the observed
spetra
1 + 2
3. . . .