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. . . .