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Measuring and understand ing Space Plasmas Turbulence Fouad SAHRAOUI

Measuring and understand ing Space Plasmas Turbulence Fouad SAHRAOUI Post-doc researcher at CETP, Vélizy, France Now visitor at IRFU (January 22 nd - April 18 th 2005). Outline. What is turbulence ? How we measure turbulence in space plasma s ?

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Measuring and understand ing Space Plasmas Turbulence Fouad SAHRAOUI

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

  2. 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

  3. Classical examples Turbulence is observable from quantum to cosmological scales! Butwhat is common to these images? Slide borrowed from Antonio Celani

  4. What is turbulence (1)?

  5. 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 powerspectrum • (But also higher order statistics, pdf, structure functions, …)

  6. 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 becauseplasma turbulence: Existence of a variety of linear modes of propagation (≠ incompressible hydrodynamics) + Role of a static magnetic field on the anisotropies

  7. Turbulence in the magnetosheath ~104km ~10 km Creates the small scales where micro-physical processes occur  potential role for driving reconnection But how ?

  8. Energy injection Energy cascade Towards dissipation FGM data in the magnetosheath 18/02/2002 Turbulent spectra andthe cascade scenario

  9. 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 Butmeasurements provide only temporal spectra, here B2~sc-7/3

  10. 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 ???? • 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) • General contexts (e.g. magnetosheath) : • v ~ v  Taylor’s hypothesisis useless • The only way is to use multi-spacecraft measurements and appropriate methods

  11. k1 k2 kj k3 Cluster data and the k-filtering method Provides, by usinga 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)

  12. CLUSTER B2 B3 B1 P(,k)=Trace[V(,k) (VT(,k)HT(k)S-1()H(k)V(,k) )–1 VT(,k)] B4 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 HT=[Id3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 I’m at IRFU!)

  13. 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 > 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)

  14. kz2 What can we do with P(w,k) ?1- modes identification For each wsc: • 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 VA=k.v • for eachkzplancontaininga significant maximum,the (kx,ky) isocontours of P(wsc,kx,ky,kz) and f(wsc,kx,ky,kz)=0 are then superimposed

  15. Magnetosheath (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 ! Application to Cluster magnetic data

  16. f0 = 0.11Hz  fci=0.33Hz Mirror :fsat~ 0.3fci ; fplasma~ 0 ko~0.0039 rd/km; (ko,Bo) = 81° Linear kinetic theory  instability if measurements: kor ~0.3~ k(gmax)  instability k(gmax) Mirror mode identification Result: The energy of the spectrum is injected by a mirror instabilitywell described by the linear kinetic theory (Sahraoui et al., Ann., 2004)

  17. fo=0.11Hz f1=0.37Hz f2 = 1.32Hz fci~0.33Hz Mirror:f2~ 4 fci;fplasma~ 0 k2 ~ 10ko ; (k2,Bo) =86° Mirror :fo= 0.11Hz ; fplasma~ 0 kor ~0.3~ k(gmax); (ko,Bo) = 81° Mirror:f1~ fci;fplasma~ 0 k1 ~ 3ko ; (k1,Bo) = 82° 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.

  18. 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

  19. and A double integration: • 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 Towards a new hydrodynamic-like turbulence theory for mirror sturctures

  20. Main conclusions • Power spectra provide most of the underlying physics on turbulence • First 3-Dk-spectrum:evidence of stronganisotropies(Bo, v, n) • Evidence of a 1-Ddirect cascadeof mirror structures from an injection scale(Lv~1800 km) up to 150 km with a new law kv-8/3 Main consequences: 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 …) Reconnection: - How can the new law be used in reconnection models ? open … - Necessity to explore much smaller scales  MMS (2010?)

  21. Theory: general presentation

  22. 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

  23. k=k1+k2 ; = 1+2 • random phase hypothesistransition to statistical description in terms of the waves kinetic equations k1 k k1 (Zakharov et al., 1992) k k2 :collisionsintegrale depending only upon the correlators and the coupling coefficients k2 Weak/wave turbulence • is applicable only whenlinear solutions exist: a(k,t)=|ak|eit • Two basic assumptions: • weaknon linear effects perturbation theory: H= Ho +H • H=Ho; with  <<1 • Scale separation:1/ < WT << NL

  24. E + vB= 0 Weak turbulence theory in Hall-MHD Weak turbulence theory mainly developed inincompressible 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 > ciand compressibility Hall-MHD

  25. Hall-MHD: a step between ideal MHD and bi-fluid Bi-fluid Hall-MHD w/wci w = kc w/wci fast intermediate Hall-MHD domain slow fast fast intermediate ideal MHD domain kr slow kr 3 propagation modes 6 propagation modes

  26. with 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

  27. Advantage of the Hamiltonian formalism • It allows to introduce the amplitude of each mode asa canonical variable of the system Canonique formulation (to be built) + Appropriate canonical transformation = Diagonalisation

  28. How to build a canonical formulationof 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 generalizingthe variationnalprinciple : Lagrangian of the compressiblehydrodynamic(Clebsch variables) +electromagnetic Lagrangian + introduction of new Lagrangian invariants

  29. For each fluid Frozen-in equation:  Generalized vorticity:  conservation of (new m) generalized circulation  generalized Clebsch variable New Lagrangian invariant

  30. HBF corresponds to the total energy of the bi-fluid system HBF is canonical with respect to the variables Bi-fluid canonical description

  31. Néglecting the electron inertia ( << ce)MHD-Hall Réduction to Hall-MHD • Néglecting the displacement current Intermediate regime«Reduced Bi-Fluid» : non-relativistic, quasi-neutral BUT still keep the electron inertia ( ~ ce)

  32. Hamiltonian canonical equations of Hall-MHD: The generalized Clebsch variables (nl,l), (l,l) are sufficient to describe the whole MHD-Hall (Sahraoui et al., Phys. Plas., 2003)

  33. 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.

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