Measuring and understanding Space
Plasmas Turbulence
Fouad SAHRAOUI
Post-doc researcher at CETP, Vélizy, France
Now visitor at IRFU (January 22nd- April 18th 2005)
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
Classical examples
Turbulence is observable
from quantum to
cosmological scales!
But what is common to
these images?
Slide borrowed from Antonio Celani
What is turbulence (1)?
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, …)
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
Turbulence in the magnetosheath
v
~104km
~10 km
Creates the small scales where micro-physical processes occur 
potential role for driving reconnection
But how ?
Turbulent spectra and the cascade scenario
Energy
injection
Towards
dissipation
FGM data in the magnetosheath
18/02/2002
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
Theory vs measurements (2)
How to infer the spatial spectrum from the temporal one
measured in the spacecraft frame: B2~sc-7/3  B2~k ????
1. Few contexts (e.g. solar wind): using Taylor’s hypothesis
v >> v  sc =k.v  B2(sc) ~ B2(kv)
Only the k spectrum along the flow is accessible (2 dimensions are lost)
2. General contexts (e.g. magnetosheath) :
v ~ v  Taylor’s hypothesis is useless
The only way is to use multi-spacecraft measurements and appropriate
methods
Cluster data and the k-filtering method
Provides, by using a NL filter bank
approach, an optimum estimation of
the spectral energy density P(,k)
from
simultaneous
multipoints
measurements
k1
k2
k3
kj
• 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)
How it works?
 S(): 12x12 generalized spectral matrix
CLUSTER
B2
S()=B()BT()
B3
with BT()=[B1T(),B2T(),B3T(),B4T()]
B1
 H(k): spatial matrix related to the tetrahedron
HT=[Id3 e-ik.r1,Id3 e-ik.r2,Id3 e-ik.r3,Id3 e-ik.r4]
B4
 V(,k): matrix including additional information on the data (Bi = 0).
P(,k)=Trace[V(,k) (VT(,k) HT(k) S-1() H(k) V(,k) )–1 VT(,k)]
 it allows the identification of multiple k for each sc
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 I’m at IRFU!)
limits of validity
Generic to all techniques intending to correlate fluctuations from a finite
number of points.
Two main points to be careful with:
1. Relative homogeneity /Stationarity
2. Spatial Aliasing effect (l > 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)
What can we do with P(,k) ?
1- modes identification
For each sc:
1. the spatial energy distribution
calculated: P(sc,kx,ky,kz)
2. the LF linear theoretical
dispersion
relations
are
calculated and Doppler shifted:
f(sc,kx,ky,kz)=0
is
kz2
Ex: Alfvén mode: sc-kz VA=k.v
3. for each kz plan containing a significant maximum, the (kx,ky) isocontours of
P(sc,kx,ky,kz) and f(sc,kx,ky,kz)=0 are then superimposed
Application to Cluster magnetic data
Magnetosheath (FGM-18/02/2002)

Limit imposed by the Cluster
minimum separation d~100 km:
max~kmaxv ~ 2 v /lmin~ 2 v /d
In the magnetosheath: v ~200 km/s
 fmax ~ 2Hz !
Mirror mode identification
f0 = 0.11Hz

fci=0.33Hz
Linear kinetic theory  instability if T  1  1
T

measurements: T  1  0.28 ;   4
T

//

//

Mirror : fsat~ 0.3fci ; fplasma~ 0
ko~0.0039 rd/km; (ko,Bo) = 81°

 instability
Result:
The energy of the spectrum is injected by
a mirror instability well described by the
linear kinetic theory (Sahraoui et al., Ann., 2004)
kor ~0.3~ k(max)
 cp
k(max)
kvth // cp
Studying higher frequencies
fo=0.11Hz
f1=0.37Hz
f2 = 1.32Hz
fci~0.33Hz
Mirror : fo= 0.11Hz ; fplasma~ 0
Mirror: f1~ fci; fplasma~ 0
Mirror: f2~ 4 fci; fplasma~ 0
kor ~0.3~ k(max); (ko,Bo) = 81°
k1 ~ 3ko ; (k1,Bo) = 82°
k2 ~ 10ko ; (k2,Bo) = 86°
Observation of mirror structures over a wide range of frequencies in the
satellite frame, but all prove to be stationary in the plasma frame.
What can we do with P(,k) ?
2- calculating integrated k-spectra
But how can we interpret the observed small scales kr ~ 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
Towards a new hydrodynamic-like
turbulence theory for mirror sturctures
A double integration: P(k)   P(f sc ,k) and P(kv )   P(kv ,kn,k// )
k ,k
f sc
n //
 a hydrodynamic-like mirror mode cascade along v: B2~kv-8/3
(Sahraoui et al., submitted to Nature)
Li~1800km
Ls~150km
fsc-7/3 temporal signature in the satellite frame of kv-8/3 spatial cascade
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 (Lv~1800 km) up to 150 km with a new law kv-8/3
Main consequences:
1. Turbulence theories: nothing comparable to the existing
theories: compressibility, anisotropy, kinetic+fluid aspects, …
 need of a new theory of a fluid type BUT which includes the
observed kinetic effects (under work …)
2. Reconnection:
- How can the new law be used in reconnection models ? open …
- Necessity to explore much smaller scales  MMS (2010?)
Theory: general presentation
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
Weak/wave turbulence
is applicable only when linear solutions exist: a(k,t)=|ak|eit
Two basic assumptions:
1. weak non linear effects  perturbation theory: H= Ho +H
H=Ho; with  <<1
 Scale separation: 1/ < WT << NL
2. random phase hypothesis  transition to statistical description in
terms of the waves kinetic equations

t(Nk)  dk1dk 2 S(Nk1Nk2)
(Zakharov et al., 1992)
S(Nk1Nk2) : collisions integrale depending only
upon the correlators akak*   Nkδk k
1
1
and the coupling coefficients Vkk k
1 2
k
k1
k2
k1
k
k2
k=k1+k2 ; = 1+ 2
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  > ci and compressibility  Hall-MHD
mi
E + vB = 0e dt(v)
(p) J  B
t(v)  v.(v)  

mn
mn
Hall-MHD: a step between ideal MHD
and bi-fluid
Bi-fluid
Hall-MHD
/ci
/ci
Hall-MHD
domain
ideal MHD domain
kr
kr
3 propagation modes
6 propagation modes
Weak turbulence theory in Hall-MHD
• Using the physical variables r, v, b: intractable directly
  2  C 2  V 2DT.D - V 2DT. t .(δv)  C 2 T1     T2    T3   V 2DT. T4 
x
B 
s
A
A
s ρ 
t ρ 
A
2

 t


 0
 0
 0
i
2
 ρ0 
 2

with
 t x1   ω1 x1  Vx2 x3
 
 
2
2)
(δρ
)

δρ

.(
δ
v
T1  δv.
 t x2  ω2 x2  Vx1 x3
T2  δρt(δ2v)  ρ0δv.2(δv)  1 (δb).δb  1 δb.(δb)
 t x3  ω3 x3  
μ0Vx2 x1
μ0

T3  δρt(δp)  ρ0δv.(δp)  γδptδρ   γp0δv.(δρ)
T4  δv.(δb)  δb.(δv)  δb.(δv)
• 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
Advantage of the Hamiltonian
formalism
• It allows to introduce the amplitude of each mode
12
 ρ0δvi
δbi
δρi Cs 
ai  



2
2
μ
2
ρ
0
0


as a canonical variable of the system
2
2
2
2
Canonique formulation (to be built)
+
Appropriate canonical transformation =
Diagonalisation
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
New Lagrangian invariant
For each fluid
Frozen-in equation:
q
   v   q    A  
t    v      A     
v

 


 
m
m
 

 Generalized vorticity:
Ω    v  
 conservation of
q 
 
μ    v  A .dl
m 
 
q
q
B    (v  A)
m
m
(new m)
C
generalized circulation  generalized Clebsch variable







S   L dt

 1

2














m
n
v

U
n

φ

n


.
n
v

λ

μ

v
.

μ

l l l
l
l
l
t
l
l l
l
t
l
l
l  dr dt
l i ,e  2

 ε0

2
2
1
  A   qni  ne Φ  qA.ni v i  ne v e  dr dt
 Φ    t A  
2
2
μ
0


Bi-fluid canonical description

H BF  




2




λl
1
nl φl    μl   ql A   U l nl  dr

l i ,e  2ml
nl




 1 2

2
1
  A   qni  ne Φ  D.Φ  dr
D 

2 μ0
 2ε 0

HBF corresponds to the total energy of the bi-fluid system
 δ H BF 
 δ (nl )   t φl 
 δ H 
BF

  t nl 
 δφl
 δ H BF 
 δλl   t  μl 
 δ H 
BF
  t λl 

 δμl
 δ H BF 
 δA   t D 
 δ H 
BF

  t A 
 δD
HBF is canonical with respect to the variables (nl, φl), (μl, λl ), (A, D)
Réduction to Hall-MHD
1. Néglecting the displacement current
1 2
2
1  Φ 





A

μ
j

t
0
2 t
 2

c
c

 Intermediate regime «Reduced Bi-Fluid» : non-relativistic, quasineutral BUT still keep the electron inertia ( ~ ce)
2. Néglecting the electron inertia ( << ce)  MHD-Hall
t vi   vi. vi   
 Pi  qi
 E  vi  B 
mini mi
E  ve  B  0
H BF 

2
1 n φ   λe  μ   q A  dr  ...
e
e
e

2me e 
ne
H HMHD 




2



qB0
 1
1
λl e x  μl e y   λl μl   μl λl 
ni  φl  

nl
2 nl

 2mi l i ,e 

  U (ni )  1 2

2 μ0 q

  qB0

 qB0

λe   e x  
μe   e y

  ne

 ne



 dr

2

  λe 
 
 μe 
1
       μe      λe   dr
2   ne 
 ne 
 

Hamiltonian canonical  δ H HMHD    φ 
t
l

equations of Hall-MHD:  δ (nl )
 δ H
HMHD 

  t nl 

δφl
(Sahraoui et al., Phys.
Plas., 2003)
 δ H HMHD 
  t  μl 
 δλl
 δ H
HMHD 
  t λl 


δμl
The generalized Clebsch variables (nl,l), (ll,ml) are sufficient to
describe the whole MHD-Hall
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 ?
S k   g ( k//) k


 Beyond Hall-MHD:
See how to include mirror mode (anisotropic Hall-MHD?) and
dissipation.