Turbulence and dissipation in magnetized space plasmas

Transcription

1 Turbulence and dissipation in magnetized space plasmas Fouad Sahraoui Laboratoire de Physique des Plasmas LPP, CNRS-Ecole Polytechnique-UPMC-Observatoire de Paris, France Collaborators: L. Hadid, S. Huang, S. Banerjee, N. Andrés, S. Galtier, K. Kiyani, G. Belmont, M. Goldstein, L. Rezeau and many others

2 Outline 1. Part I: Turbulence in space plasmas (solar wind and planetary magnetosheaths): Introduction, space instrumentation, data analysis techniques 2. Part II: MHD Turbulence 3. Part III: Kinetic turbulence

3 Part I: Turbulence in space plasmas 1. Plasmas in the Universe 2. Space plasmas: Sun, Solar wind, Planetary magnetospheres 3. Why do we need to study plasma turbulence? 4. In-situ space instrumentation and related measurements 5. Multispacecraft data analysis techniques (e.g., the k- filtering)

4 Turbulence in the Univers Turbulence is ubiquitous in the Univers It covers all scales, from quantum to cosmological ones! Observed heating and particle acceleration (i.e. jets) in astrophysical objects are caused by turbulence dissipation Solar corona heating Turbulence in galaxies & nebulas

5 Accretion disks of black holes [Hawley & Balbus, 2002 (simulations)] Articst s view: NASA/JPL-Caltech Matter spirals into the black hole, converting huge gravitational potential energy into heat: - Magnetorotational Instability (MRI) drives turbulence [Balbus, 1992] - Turbulence cascades nonlinearly to small scales - Kinetic mechanisms damp turbulence and lead to plasma heating Emitted radiations (e.g. X-rays) are function of the plasma heating

6 The Sun Solarcorona T (K) 10 6 Solar wind 10 5 Solar wind visible Very strong heating in the transition region

7 The solar wind [Richardson & Paularena, 1995] The solar wind plasma is generally: Fully ionized (H +, e - ) Non -relativistic (V A <<c), V~ km/s Collisionless

8 Sun-Earth coupling SUN Solar Wind Magnetized planet Magnetic field & plasma particles Magnetosheath Magnetosphere

9 Planetary magnetospheres

10 Turbulence in fusion devices Turbulence is the main obstacle to plasma confinement A better understanding of turbulent transport A better control A longer confinement

11 Any common physics? N~10 6 cm -3 T i ~10 12 K B~10 6 nt N~5 cm -3 T i ~10 K B~10 nt Sun-Earth ~10 11 m

12 Near-Earth space plasmas [Scheckochihin et al., ApJ, 2009] β = Pression thermique Pression magnétique 0.4 NT 2 B [Vaivads et al., Plasma Phys. Contr. Fus., 2009]

13 Remote sensing (distant plasmas) Bernard Lyot, the inventor of coronograph (photo Observatoire de Paris) Bernard Lyot Telescope at Observatoire du Pic du Midi (photo P. Petit)

14 In-situ measurements (space plasmas) = + = = E j E B B E ρ µ t t c ( ) ( ) + = + + = + + = + + = + B u E u u u u B u E u u u u i i i t e e e e e e e t e e e e e t n e p n n m n n e n p n m n n n ).( 0 ).( ).( 0 ).( Plasmas A coupled system of equations = = B E ε ρ ( ) + = + + B u E u u u i i i i i i i t i i e n p n m n ).( e n n e e n n e e i e e i i = = ρ u u j Ideally, a space plasma physicist would like to measure: - B & E: 3 components over a broad range of frequencies [DC, MHz] - N i,e, V i,e, T i,e : in 3D at all energies (ev, MeV) and with high resolutions

15 In-situ measurements THOR spacecraft: 10 instruments (currently under phase A study at ESA) 4m SCM EFI-HFA EFI-HFA 2.5m 50m FGM spin Z GSE X Y EFI-SDP

16 Instruments overview: fields (1) Fluxgate magnetometer: B measurements in [DC, 1Hz] ± B ext =0 -B i B i B ext 0

17 Instruments overview: fields (2) Search-coil magnetometer (SCM): [0.1Hz, ~1MHz] dφ Lenz s law (induced voltage): VC = N = j2π f N S µ eff Bcosθ dt θ µ eff effective permeability of the core B S [m 2 ] L N [turns] e V C ~ LC RC C V core d

18 G V / B Gain [dbv/nt] Resonance f0 = 1 2π LCC Frequency log f A feedback ractionis needed to obtain a flat response function Dual band SCM: LF [1Hz, 4kHz] HF [1kHz, 1MHz] y LF-SC (Y) x LF-SC (X) Solar Orbiter/SCM (LPC2E) z DB-SC (Z) BepiColombo/SCM (LPP- Univ. Kanasawa, JP)

19 Instruments overview: fields (3) Electric field: [DC, 1MHz] 4m EFI-HFA P 2 50m spin P 1 RBSP/EFW (from the THOR proposal) 2.5m FGM n e = C exp( VSC / 2) 1 C E 21 Spacecraft potential V sc electron density n e Hadid et al., 2016b

20 Instruments overview: fields (4) Onboard wave analyzers THR Solar Orbiter/LF analyzer THOR/FWP (Field and Wave Processing Unit Courtesy THOR proposal) TNR/HFR High time resolution measurement of N e and T e

21 Instruments overview: particles (1) (Ion and electron) mass spectrometers MSA BepiColombo ion+ Energy selection 0 -V Angle selection LEF Carbon foil Departure of the Time Of Flight analysis ST LEF START STOP ion + ion - neutral STOP STOP -15 kv +15 kv MCP LEF End of the flight : TOF gives the m/q ratio MCP ST

22 Output measurements : The nature of the particles (m/q) Their direction and energy/velocity Velocity distribution function (VDF) Moments of the VDF : density, velocity, temperature <v> N v th

23 Instruments overview: particles (2) Particle Processing Unit (PPU) Solar Orbiter/PPU (Courtesy of TSD/RTI --from THOR proposal). +Energetic particles + ASPOC + active sounder + THOR/PPU (Courtesy of TSD/RTI and THOR proposal).

24 NS equation: E (k) t Back to turbulence: phenomenology V + F i = V. V P + ν 2 V k -5/3 Inertial range k i k d k Courtesy of A. Celani Hydro: Scale invariance down to the dissipation scale 1/k d Collisionless Plasmas: - Breaking of the scale invariance at ρ i,e d i,e - Absence of the viscous dissipation scale 1/k d

25 Solar wind turbulence Typical power spectrum of magnetic energy at 1 AU Matthaeus & Goldstein, 82 f 5/3 Does the energy cascade or dissipate below the ion scale ρ i? Leamon et al. 98; Goldstein et al. JGR, 94 Richardson & Paularena, GRL, 1995 (Voyager data)

26 How to analyse space turbulence? Turbulence theories generally predict spatial spectra: K41 (k -5/3 ); IK (k -3/2 ), Anisotropic MHD turbulence (k -5/3 ),Whistler turbulence (k 7/3 ),... Example of measured spectra in the SW But measurements provide only temporal spectra (generally with different power laws at differe) How to infer spatial spectra from temporal ones measured in the spacecraft frame? B 2 ~ω sc -α B 2 ~k // -β k -γ?

27 The spatio-temporal ambiguity (1) Spacecraft measurements show highly variable phenomena. With 1 point measurement one cannot distinguish space effects from temporal effects Monochromatic wave Single Observer crossing the wave

28 The spatio-temporal ambiguity (2) A minimum of 4 spacecraft is needed to sample the 3 directions of space (e.g., ESA/Cluster and NASA/MMS missions) Monochromatic wave Multipoint measurements

29 The Taylor frozen-in flow assumption In the solar wind (SW) the Taylor s hypothesis can be valid at MHD scales High SW speeds: V ~600km/s >> V ϕ ~V A ~50km/s ω ω spacecraft = plasma + k.v k.v = k V V Inferring the k-spectrum is possible with one spacecraft But only along one single direction

30 1. At MHD scales, even if the Taylor assumption is valid, inferring 3D k-spectra from an ω-spectrum is impossible 2. At sub-ion and electron scales scales Vϕ can be larger than V sw The Taylor s hypothesis is invalid MHD scales Sub-ion scales 1 & 2 Need to use multi-spacecraft measurements and appropriate methods to infer 3D k-spectra

31 The k-filtering technique (1) Goal: estimation of the spectral energy density P(ω,k) from the multipoint measurements of a turbulent field Method: it uses a filter bank approach: the filter bank is constructed to absorb all signals, except those corresponding to plane waves with a specified frequency and wave vector, which pass unaffected. By going through all frequencies and wave vectors, one gets an estimate of the wave-field energy distribution P(ω,k) [Pinçon & Lefeuvre, 1991; Sahraoui et al., 2003, 2004, 2006; 2010; Narita et al., 2010; Grison et al., 2005; Tjulin et al., 2005; Roberts et al., 2012]

32 The k-filtering technique (2) k 1 k 2 k 3 k j

33 The k-filtering technique (3) k. B = 0 ωb = k E

34 The k-filtering technique (4)

35 The k-filtering technique (5) [Tjulin et al., 2005]

36 The k-filtering technique (6) Spatial Aliasing effect: Two satellites cannot distinguish between k 1 and k 2 if : k.r 12 = 2πn (n=1,2, ) For Cluster: k = n 1 k 1 + n 2 k 2 + n 3 k 3 with: k 1 =(r 31 r 21 )2π/V, k 2 =(r 41 r 21 )2π/V, k 3 = (r 41 r 31 )2π/V V = r 41.(r 31 r 21 ) [Neubaur & Glassmeir, 1990]

37 END of Part I

38 PART II: MHD scale turbulence 1. Theoretical models: HD, incompressible and compressible MHD 2. Turbulence at MHD scales in the solar wind Energy cascade rate: incompressible vs compressible models Spatial anisotropy and scaling properties 3. Comparisons with magnetosheath turbulence

39 Incompressible HD turbulence u(x) l x u(x) u(x+l) S3( l) =< δul >=< [ u( x + l) u( x)] >= εl 5 ε is the energy cascade (dissipation) rate E(k)= u k2 ~ k -5/3

40 Incompressible MHD turbulence: equations and phenomenolgy (1). B 2 v B B. B Incompressible + v. v = p + + t 8π 4π MHD equations. v = 0 B = ( v B) tt Elsässer variables: = 0 ± B z = v ± = v ± 4πρ 0 v A

41 Incompressible MHD turbulence: third order law and energy casade rate (2) ± ± z m v. z + z. z = p t A Linear term: k v A z + Nonlinear term: k v z + m ± Ratio of nonlinear to linear terms χ = k k v // v A χ~1 Critically Balanced turbulence Third order law [Politano & Pouquet, PRE, 1998] ± ε ( δz ± ) 4 δz ± l = ε l 3 2 m is the cascade rate of the pseudo-energies E ± z = v ± 1 ± = z 2 ± B 4πρ 0 z m

42 Compressible isothermal MHD turbulence: Equations Compressible MHD equations ρ v t + v. v = p + 2 B + 8π B. B 4π B t = ( ) v B. B = 0 P = C S 2 ρ Isothermal closure C s sound speed (constant)

43 Compressible MHD turbulence: linear solutions ω/ω ci fast mode mode rapide intermediate mode mode intermédiaire 1 mode d Alfvén mode lent slow mode kρ i

44 Compressible isothermal MHD turbulence: 3 rd order law and energy casade rate (3)

45 Additional assumptions: Neglect the source terms Isotropy Uniform β

46 The solar wind The solar wind plasma is generally: Fully ionized (H +, e - ) Non -relativistic (V A <<c), V~ km/s Collisionless [Richardson & Paularena, 1995]

47 Typical magnetic power spectrum at 1AU ε Kiyani et al., 2015

48 Estimation of the energy cascade rate : compressible vs incompressible MHD model (1) ± z = v ± B 4πρ 0 ( δz ± ) δ 4 = ε 3 2 m z ± l l

49 Estimation of the energy cascade rate : compressible vs incompressible MHD model (2) Hadid et al., ApJ, 2016

50 Estimation of the energy cascade rate : cross helicity and turbulent Mach number

51 Spatial anisotropy In MHD turbulence, the presence of a mean lagnetic field makes the turbulence anisotropic at small scales [Montgomery et al., 1983] l /L 0 ~1 l /L 0 ~1/10 l /L 0 ~1/100 3D Electron -MHD simulations [Meyrand & Galtier, 2013]

52 Anisotropy and the critical balance conjecture The critical balance conjecture [Goldreich & Sridhar, 1995]: Linear (Alfvén) time ~ nonlinear (turnover) time ω~k // V A ~ k u k // ~ k 2/3 See also [Boldyrev, ApJ, 2005] and [Galtier et al., Phys. Plasmas, 2005] k // k [Chen et al., ApJ, 2010]

53 Single satellite analysis use of the Taylor assumption: ω sc ~k.v sw ~k v V sw V//Bk v =k // V Bk v =k Assumes axisymmetry around B Θ BV 0 B 2 ~ k // -2 Partial evidence of the critical balance [Horbury et al., PRL, 2008]

54 Results confirmed by Podesta, ApJ, 2009 See also Chen et al., PRL, 2010

55 The ESA/Cluster mission The first multispacecraft mission: 4 identical satellites Objetives: 3D exploration of the Earth magnetosphere boundaries (magnetopause, bow shock, magnetotail) & SW Mesurements of 3D quantities: J= xb, Fundamental physics: turbulence, reconnection, particle acceleration, Different orbits and separations (10 2 to 10 4 km) depending on the scientific goal

56 The 4 satellites before launch

57 The k-filtering technique Interferometric method: it provides, by using a NL filter bank approach, an optimum estimation of the 4D spectral energy density P(ω,k) from simultaneous multipoints measurements [Pinçon & Lefeuvre; Sahraoui et al., 03, 04, 06, 10; Narita et al., 03, 06,09] k 1 k 2 k j k 3 We use P(ω,k) to calculate 1. 3D ω-k spectra plasma mode identification e.g. Alfvén, whistler 2. 3D k-spectra (anisotropies, scaling, )

58 Measurable spatial scales Given a spacecraft separation d only one decade of scales 2d < λ < 30d can be correctly determined λ min 2d, otherwise spatial aliasing occurs. d~4000km d~100km λ max 30d, because larger scales are subject to higher uncertainties MHD scales Sub-ion scales ω sat ~kv f max ~k max V/λ min (V~500km/s) d~10 4 km MHD scales d~10 2 km Sub-ion scales d~1 km Electron scales (but not accessible with Cluster: d>100)

59 1- MHD scale solar wind turbulence Position of the Quartet on March 19, 2006

60 FGM data (CAA, ESA) Data overview Ion plasma data from CIS (AMDA, CESR) T T

61 f 1 =0.23Hz~2f ci f 2 =0.9Hz~6f ci To compute reduced spectra we integrate over ~ 1. all frequencies f sc : P( k) = P(f sc, k ) ky,kz 2. all k ~ ~ i,j : P(k ) = P(k,k,k ) x ky,kz x y z

62 Anisotropy of MHD turbulence along B o and V sw Turbulence is not axisymmetric (around B) [see also Sahraoui, PRL, 2006] [Narita et al., PRL, 2010] V sw The anisotropy ( B) is along V sw SW expansion effect?[saur & Bieber, JGR, 1999]

63 Mirror mode turbulence L i ~1800k m L i =1800km L s ~150km B o B o n v v n (v,n) ~ 104 (v,b o ) ~ 110 (n,b o ) ~ 81 Compressible, anisotropic and non-axisymmetric turbulence (along B o, the magnetopause normal n, and the flow v) [Sahraoui+, PRL,2006]

64 PART III: Kinetic (sub-ion scale) turbulence

65 Kinetic turbulence 1. Kinetic scales in the SW: Some hotly debated question vs Cluster observations The nature of the cascade or dissipation below ρ i : : KAW? whistler? Others? The nature of the dissipation: wave-particle interactions? Current sheets/reconnection? 2. Conclusions & perspectives (turbulence & the future space missions)

66 I- Theoretical predictions on small scale turbulence E + V B = 1 J en B Pe en 1. Fluid models (Hall-MHD) +... Whistler turbulence (E-MHD): (Biskamp et al., 99, Galtier, 08) Weak Turbulence of Hall-MHD (Galtier, 06; Sahraoui et al., 07) B²~k -7/3-5/2 whistler Alfvén B²~k -5/2 B²~k 2. Gyrokinetic theory: k // <<k and ω<<ω ci (Schekochihin et al. 06; Howes et al., 11)

67 Other numerical predictions on electron scale turbulence 2D PIC simulations gave evidence of a power law dissipation range at kρ e >1 [Camporeales & Burgess, ApJ, 2011]

68 3D PIC simulations of whistler turbulence : k -4.3 at kd e >1 Chang & Gary, GRL 2011

69 First evidence of a cascade from MHD to electron scale in the SW 1. Two breakpoints corresponding to ρ i and ρ e are observed. 2. A clear evidence of a new inertial range ~ f -2.5 below ρ i 3. First evidence of a dissipation range ~ f -4 near the electron scale ρ e B // ² (FGM) B ² (FGM) B // ² (STAFF) B ² (STAFF) Sahraoui et al., PRL, 2009 STAFF-SC sensitivity floor

70 Whistler or KAW turbulence? 1. Large (MHD) scales (L>ρ i ): strong correlation of E y and B z in agreement with E=-VxB FGM, STAFF-SC and EFW data 2. Small scales (L<ρ i ): steepening of B² and enhancement of E² (however, strong noise in E y for f>5hz) Good agreement with GK theory of Kinetic Alfvén Wave turbulence Howes et al. PRL, 11 See also Bale et al., PRL, 2005

71 Theoretical interpretation : KAW turbulence Linear Maxwell-Vlasov solutions: Θ kb ~ 90, β i ~2.5, T i /T e ~4 The Kinetic Alfvén Wave solution extends down to kρ e ~1 with ω r <ω ci [See also Podesta, ApJ, 2010] k // V A /ω cp ω = k V k ρ / β + 2 /(1 + T r / T // A i i i e )

72 E/B : KAW theory vs observations ω r = k V k ρ / β + 2 /(1 + T / T // A i i i e ) Lorentz transform: E sat =E plas +VxB Taylor hypothesis to transform the spectra from f (Hz) to kρ i E/B observations E/B Vlasov Θ kb ~ 90, 1. Large scale (kρ i <1): δe/δb~v A 2. Small scale (kρ i >1): δe/δb ~k 1.1 in agreement with GK theory of KAW turbulence δe²~k -1/3 & δb²~k -7/3 δe/δb~k 3. The departure from linear scaling (kρ i 10) is due to noise in Ey data Sahraoui et al., PRL, 2009

73 Magnetic compressibility Fast magnetosonic Additional evidence of KAW at kρ i >1 Cluster/STAFF-SC data KAW KAW, Θ kb =89.9 [Sahraoui+, ApJ, 2012] [Kiyani+, ApJ, 2012; Podesta+, 2012]

74 3D k-spectra at sub-proton scales of Conditions required: 1. Quiet SW: NO electron foreshock effects 2. Shorter Cluster separations (~100km) to analyze subproton scales 3. Regular tetrahedron to infer actual 3D k-spectra [Sahraoui et al., JGR, 2010] 4. High SNR of the STAFF data to analyse HF (>10Hz) SW turbulence. SW turbulence , 06h05-06h55

75 3D k-spectra at sub-proton scales We use the k-filtering technique to estimate the 4D spectral energy density P(ω,k) (d~200km) B // 2 B 2 k 1 k 2 k 3 k j We use P(ω,k) to calculate 1. 3D ω-k spectra 2. 3D k-spectra (anisotropies, scaling, )

76 Comparison with the Vlasov theory Turbulence cascades following the Kinetic Alfvén mode (KAW) as proposed in Sahraoui et al., PRL, 2009 Rules out the cyclotron heating Heating by p- Landau and e-landau resonances [Sahraoui et al., PRL, 2010] β i ~ 2 Τ i /Τ e =3 85 <Θ kb <89 Limitation due to the Cluster separation (d~200km)

77 3D k-spectra at sub-ion scales 1. First direct evidence of the breakpoint near the proton gyroscale in k-space (no additional assumption, e.g. Taylor hypothesis, is used) 2. Strong steepening of the spectra below ρ i A Transition Range to dispersive/electron cascade

78 Journey of the energy cascade through Injection scales Dissipation via e-landau damping B² 1 st cascade k -5/3 k nd KAW cascade k -7/3 1. Turbulence 2. e-acceleration & Heating 3. Reconnection Transition Range: k -4.5 Partial dissipation via p-landau damping kρ e ~ kρ i k -4 Dissipation range Another interpretation in Meyrand & Galtier, 2010

79 Dissipation through reconnection/current sheets Large scale laminar current sheet: reconnection can occur and the can be heated or accelerated (e..g. jets) [Zhong+, Nature Physics, 2010]

80 [Lazarian & Vishniac,1999] Turbulent current sheets 2D Hall-MHD simulation of turbulence: evidence of a large number of reconnecting regions ~ d [e.g., Retinò+, Nature Physics, 2007]

81 Dissipation by wave-particle interaction or via reconnection? Good correlation between enhanced T p and threshold of linear kinetic instabilities Good correlation between enhanced high shear B angles and the threshold of linear instabilities!! Osman et al., PRLs, 2012a,b

82 Higher order statistics and intermittency (1) Self-similar signal intermittent signal [Frisch, 1995]

83 Higher order statistics and intermittency (2)

84 2. Monfractality vs multifractality in the dispersive range: ne ~ 4 cm -3 β i ~ 2 VA ~ 50 km s -1 T i ~ 103 ev B ~4 nt [Kiyani et al., PRL, 2009]

85 Sub-proton scales MHD scales S m Stuctures functions: (τ ) = Scaling: t B ( t + τ ) B( t) m Evidence of monofractality (selfsimilarity) at sub-proton scales, while MHD-scales are multifractal (intermittent) [See also Alexandrova et al., ApJ, 2008]

86 Conclusions The Cluster data helps understanding crucial problems of astrophysical turbulence: Its nature and anisotropies in k-space at MHD and sub-ion scales Its cascade and dissipation down to the electron gyroscale ρ e electron heating and/or acceleration by turbulence Strong evidences of KAW turbulence (ω<< ω ci, k // <<k ) Heating by e-p-landau dampings (no cyclotron heating) Importance of kinetic physics in SW turbulence Turbulence & dissipation are at the heart of the future space missions: ESA/Solar Orbiter (2018), NASA/Solar Probe Plus (2018), THOR (2026?)

87 Turbulence and the future space missions 4 NASA satellites, launch 2015 Higher resolution instrumentations Small separations (~10km) Equatorial orbites

88 Need of multiscale measurements with appropriate spacecraft separations d~1000km d~100km d~10km Narita et al. PRL, 2010 MMS 2015 Sahraoui et al. PRL, 2010

89 Solar Orbiter Exploring the Sun-Heliosphere Connection Launch 2017 Distance : 0.28 AU In-situ measurements & remote sensing

90 Launch 2019 Distance : ~0.03 AU In-situ measurements & remote sensing Solar Probe Plus

91 THOR Turbulent Heating ObserveR Turbulent energy dissipation and particle energization SNSB (2012) ESA (S1 Call, 2012) CNES (TOR/TWINS, call for ideas, 2013) ESA (M4 call, 2015): under Phase A study. Final decision 2017