Contexte physique : couplage vent solaire-magn

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Measuring and understanding Space Plasmas
Turbulence Fouad SAHRAOUI Post-doc researcher
at CETP, Vélizy, France Now visitor at IRFU
(January 22nd- April 18th 2005)
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Outline

• What is turbulence ?
• How we measure turbulence in space plasmas?
• Magnetosheath ULF turbulence, Cluster data,
  k-filtering technique.
• Theoretical model
• ? General ideas on weak turbulence theory in
  Hall-MHD

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Classical examples
Turbulence is observable from quantum to
cosmological scales! But what is common to
these images?
Slide borrowed from Antonio Celani
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What is turbulence (1)?
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What is turbulence ? (2)

• Essential ingredients
• Many degrees of freedom (different scales)
• All of them in non -linear interaction
  (cross-scale couplings)
• Main characterization
• Shape of the power spectrum
• (But also higher order statistics, pdf,
  structure functions, )

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Role of turbulence in space

• Basically the same consequences as in
  hydrodynamics
• (more efficient diffusion, anomalous
  transports, )
• But still more important because in collisionless
  media
• no normal transport at all ? role of the
  created small scales
• And of different nature because plasma
  turbulence
• Existence of a variety of linear modes of
  propagation
• (? incompressible hydrodynamics)
• Role of a static magnetic field on the
  anisotropies

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Turbulence in the magnetosheath
104km
10 km
Creates the small scales where micro-physical
processes occur ? potential role for driving
reconnection But how ?
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Turbulent spectra and the cascade scenario
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Theory vs measurements (1)

• Turbulence theories predict spatial (i.e.
  stationnary) spectra
• Incompressible fluid turbulence (K-1941) ? k -5/3
• Incompressible isotropic MHD (IK-1965) ? k -3/2
• Incompressible anisotropic MHD (SG-2000) ? k? -2
• Whistler turbulence (DB-1997) ? k 7/3

But measurements provide only temporal spectra,
here B2?sc-7/3
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Theory vs measurements (2)
How to infer the spatial spectrum from the
temporal one measured in the spacecraft frame
B2?sc-7/3 ? B2k ????

• Few contexts (e.g. solar wind) using Taylors
  hypothesis
• v gtgt v? ? ?sc k.v ? B2(?sc) B2(kv)
• Only the k spectrum along the flow is accessible
  (2 dimensions are lost)

• General contexts (e.g. magnetosheath)
• v v? ? Taylors hypothesis is useless
• The only way is to use multi-spacecraft
  measurements and appropriate methods

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Cluster data and the k-filtering method
Provides, by using a NL filter bank approach, an
optimum estimation of the spectral energy density
P(w,k) from simultaneous multipoints measurements

• Had been validated by numerical simulations
  (Pinçon Lefeuvre, JGR, 1991)
• Applied for the first time to real data with
  CLUSTER (Sahraoui et al., JGR, 2003)

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How it works?

• S(?) 12x12 generalized spectral matrix
• S(?)?B(?)BT(?)?
• with BT(?)B1T(?),B2T(?),B3T(?),B4T(?)

• H(k) spatial matrix related to the tetrahedron
• HTId3?e-ik.r1,Id3 e-ik.r2,Id3 e-ik.r3,Id3
  e-ik.r4

? V(?,k) matrix including additional information
on the data (??Bi 0).
? it allows the identification of multiple k for
each wsc More numerous the correlations are,
more trustable is the estimate of the energy
distribution in k space ? it works quite well
with the 3 B components, but will still be
improved by including the 2 E components (That is
why Im at IRFU!)
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limits of validity
Generic to all techniques intending to correlate
fluctuations from a finite number of points.
Two main points to be careful with

• Relative homogeneity /Stationarity
• Spatial Aliasing effect (l gt spacecraft
  separation)

Two satellites cannot distignuish between k1 and
k2 if ?k.r12 2?n
For Cluster ?k ? n1 ?k1 ? n2 ?k2 ? n3 ?k3 with
?k1(r31?r21)2?/V, ?k2(r41?r21)2?/V,
?k3  (r41?r31)2?/V V  r41.(r31?r21)
(Neubaur Glassmeir, 1990)
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What can we do with P(w,k) ?1- modes
identification
For each wsc

1. the spatial energy distribution is calculated
   P(wsc,kx,ky,kz)

• the LF linear theoretical dispersion relations
  are calculated and Doppler shifted
  f(wsc,kx,ky,kz)0
• Ex Alfvén mode wsc-kz VAk.v

1. for each kz plan containing a significant
   maximum, the (kx,ky) isocontours of
   P(wsc,kx,ky,kz) and f(wsc,kx,ky,kz)0 are then
   superimposed

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Application to Cluster magnetic data
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Mirror mode identification
Result The energy of the spectrum is injected by
a mirror instability well described by the linear
kinetic theory (Sahraoui et al., Ann., 2004)
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Studying higher frequencies
Observation of mirror structures over a wide
range of frequencies in the satellite frame, but
all prove to be stationary in the plasma frame.
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What can we do with P(w,k) ?2- calculating
integrated k-spectra

• But how can we interpret the observed small
  scales k? 3.5 ?

Energy distribution of the identified mirror
structures ?
(v,n) 104 (v,Bo,) 110 (n,Bo) 81
First direct determination of a fully 3-D
k-spectra in space anistropic behaviour is
proven to occur along Bo, n, and v
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Towards a new hydrodynamic-like turbulence theory
for mirror sturctures
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Main conclusions

• Power spectra provide most of the underlying
  physics on turbulence
• First 3-D k-spectrum evidence of strong
  anisotropies (Bo, v, n)
• Evidence of a 1-D direct cascade of mirror
  structures from an injection scale (Lv1800 km)
  up to 150 km with a new law kv-8/3

• Main consequences
• Turbulence theories nothing comparable to the
  existing theories compressibility, anisotropy,
  kineticfluid aspects,
• ? need of a new theory of a fluid type BUT which
  includes the observed kinetic effects (under work
  )
• Reconnection
• - How can the new law be used in reconnection
  models ? open
• - Necessity to explore much smaller scales ?
  MMS (2010?)

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Theory general presentation
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Different approaches

• Many different theoretical approaches of
  turbulence
• Phenomenological
• A priori assumptions on the isotropy
• use of the physical equations through crude,
  but efficient, dimensional arguments
• Ex K41? k -5/3
• IK ? k -3/2
• Statistical weak vs strong turbulence
• Find statistically stationary states by solving
  directly the physical equations
• ? huge calculations requiring numerical
  investigations

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Weak/wave turbulence

• is applicable only when linear solutions exist
  a(k,t)akei?t
• Two basic assumptions
• weak non linear effects ? perturbation theory H
  Ho ?H
• ?H?Ho with ? ltlt1
• Scale separation 1/? lt ?WT ltlt ?NL

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Weak turbulence theory in Hall-MHD
Weak turbulence theory mainly developed in
incompressible ideal MHD (Galtier et al., 2000 ?
k? -2) Few recent developments for EMHD (but
still incompressible)
But observations (e.g. magnetosheath) strongly
suggest the presence of scales ? gt ?ci and
compressibility ? Hall-MHD
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Hall-MHD a step between ideal MHD and bi-fluid
Bi-fluid
Hall-MHD
w kc
w/wci
w/wci
fast
intermediate
Hall-MHD domain
slow
fast
fast
intermediate
ideal MHD domain
kr
slow
kr
6 propagation modes
3 propagation modes
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Weak turbulence theory in Hall-MHD

• Using the physical variables ??, ?v, ?b
  intractable directly

• Problem
• No way to diagonalize the system, i.e. express it
  in terms of only 3 variables, x1, x2, x3, each
  characteristic of one mode. The physical
  variables always remain inextricably tangled in
  the non linear terms

• Solution Hamiltonian formalism of continuous
  media
• Has proved to be efficient in other physical
  fields particle physics, quantum field theory,
  , but is still less known in plasma physics

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Advantage of the Hamiltonian formalism

• It allows to introduce the amplitude of each
  mode

as a canonical variable of the system

• Canonique formulation (to be built)
• Appropriate canonical transformation
• Diagonalisation

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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

How to deal with the bi-fluid system ?
by generalizing the variationnal principle
Lagrangian of the compressible hydrodynamic
(Clebsch variables) electromagnetic Lagrangian
introduction of new Lagrangian invariants
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New Lagrangian invariant
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Bi-fluid canonical description
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Réduction to Hall-MHD

1. Néglecting the displacement current

? Intermediate regime Reduced Bi-Fluid
non-relativistic, quasi-neutral BUT still keep
the electron inertia (? ?ce)
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(Sahraoui et al., Phys. Plas., 2003)
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Future steps for a weak-turbulence theory

• Hall-MHD
• Derive the kinetic equations of waves
• Find the stationary solutions
• Power law spectra of the Kolmogorov-Zakharov type
  ?

• Beyond Hall-MHD
• See how to include mirror mode (anisotropic
  Hall-MHD?) and dissipation.