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

Creates the small scales where micro-physical processes occur potential role for driving reconnection

But how ?

~104km

~10 km

Turbulent spectra and the cascade scenario

Energy injection

FGM data in the magnetosheath 18/02/2002

Energy cascadeTowards

dissipation

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

How to infer the spatial spectrum from the temporal one measured in the spacecraft frame: B2~sc

-7/3 B2~k ????

Theory vs measurements (2)

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

• 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)

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

How it works? CLUSTER

B1

B2

B3

B4

S(): 12x12 generalized spectral matrix

S()=B()BT()

with BT()=[B1T(),B2

T(),B3T(),B4

T()]

H(k): spatial matrix related to the tetrahedron

HT=[Id3e-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).

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 ( > spacecraft separation)

For Cluster: k  n1 k1  n2 k2  n3 k3

with: k1=(r31r21)2/V, k2=(r41r21)2/V,

k3 = (r41r31)2/V

V = r41.(r31r21) (Neubaur & Glassmeir, 1990)

Two satellites cannot distignuish between k1 and k2 if : k.r12= 2n

What can we do with P(,k) ?1- modes identification

kz

kx

ky

kz1

kz2 kz2

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

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

2. the LF linear theoretical dispersion relations are calculated and Doppler shifted: f(sc,kx,ky,kz)=0

Ex: Alfvén mode: sc-kz VA=k.v

For each sc:

Application to Cluster magnetic dataMagnetosheath (FGM-18/02/2002)

Limit imposed by the Cluster minimum separation d~100 km:

max~kmaxv ~ 2 v /min~ 2 v /d

In the magnetosheath: v ~200 km/s

fmax ~ 2Hz !

cpthkv //

cp

k(max)

instability

Mirror mode identification

Mirror : fsat~ 0.3fci ; fplasma~ 0

ko~0.0039 rd/km; (ko,Bo) = 81°

Result:

The energy of the spectrum is injected by a mirror instability well described by the linear kinetic theory (Sahraoui et al., Ann., 2004)

ko~0.3~ k(max)

Linear kinetic theory instability if

11

//TT

measurements: 4;28.01//

T

T

f0 = 0.11Hz

fci=0.33Hz

fci~0.33Hz

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.

Mirror: f1~ fci; fplasma~ 0

k1 ~ 3ko ; (k1,Bo) = 82°

f1=0.37Hzfo=0.11Hz

Mirror : fo= 0.11Hz ; fplasma~ 0

ko~0.3~ k(max); (ko,Bo) = 81°

Mirror: f2~ 4 fci; fplasma~ 0

k2 ~ 10ko ; (k2,Bo) = 86°

f2 = 1.32Hz

What can we do with P(,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

First direct determination of a fully 3-D k-spectra in space: anistropic behaviour is proven to occur along Bo, n, and v

(v,n) ~ 104°(v,Bo,) ~ 110°

(n,Bo) ~ 81°

Towards a new hydrodynamic-like turbulence theory for mirror sturctures

fsc-7/3 temporal signature in the satellite frame of kv

-8/3 spatial cascade

),k,kP(k)P(k,knk

nvv //

//scf

sc,fPP )()( kkA double integration: and

a hydrodynamic-like mirror mode cascade along v: B2~kv-8/3

(Sahraoui et al., submitted to Nature) Li~1800km Ls~150km

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|eit

Two basic assumptions:

1. weak non linear effects perturbation theory: H= Ho +H

H=Ho; with <<1

Scale separation: 1/ < WT << NL

kk1

k2

kk1

k2

k=k1+k2 ; = 1+ 2

: collisions integrale depending only

upon the correlators

and the coupling coefficients

)( 21 kk NNS

11

*

kkkkk δNaa

21kkkV

)()( 2121 kkk kk NNSddNt (Zakharov et al., 1992)

2. random phase hypothesis transition to statistical description in terms of the waves kinetic equations

Weak turbulence theory in Hall-MHD

But observations (e.g. magnetosheath) strongly suggest the presence of

scales > ci and compressibility 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)

mnmnp

tBJvv.v )()()(

E + vB = 0

)(vti d

em

6 propagation modes

Bi-fluid

Hall-MHD domain

Hall-MHD

3 propagation modes

kideal MHD domain

fast

slow

intermediatefast

= kc/ci/ci

fast

inte

rmed

iate

slow

k

Hall-MHD: a step between ideal MHD and bi-fluid

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

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

bbbbwith

Weak turbulence theory in Hall-MHD

• 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 mediaHas proved to be efficient in other physical fields: particle physics, quantum field theory, …, but is still less known in plasma physics

1232

332

3122

222

3212

112

xVxxωx

xVxxωx

xVxxωx

t

t

t

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 ?

Generalized vorticity: )( AvBvΩmq

mq

New Lagrangian invariant

conservation of lAv dmq

μ

C

.

(new )

generalized circulation generalized Clebsch variable

AvvAv

mq

mq

tFrozen-in equation:

For each fluid

dtLS

dtdμμλnnφnUnmeil

llltlllltllllll

,

2 ..21 rvvv

dtdnnqΦnnqμ

Φεeeiieit rvvAAA

.21

22

0

20

Bi-fluid canonical description

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

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

Réduction to Hall-MHD1. Néglecting the displacement current

Φc

μc

tt

2022

211 jA

Intermediate regime «Reduced Bi-Fluid» : non-relativistic, quasi-neutral BUT still keep the electron inertia ( ~ ce)

...21

2

drqμ

nλ

φnm

H eee

eee

eBF A

2. Néglecting the electron inertia ( << ce) MHD-Hall

0 BvE e

BvEv.vv ii

i

ii

iiiit m

qnmP

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 sufficient to describe the whole MHD-Hall

ltl

HMHD

ltl

HMHD

nδφ

Hδ

φnδ

Hδ)(

ltl

HMHD

ltl

HMHD

λδμ

Hδ

μδλ

HδHamiltonian canonical equations of Hall-MHD:

(Sahraoui et al., Phys. Plas., 2003)

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 ?

kkgS )( //k

Beyond Hall-MHD:See how to include mirror mode (anisotropic Hall-MHD?)

and dissipation.