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Space Physics Seminar, UCLA, March 13, 2009

Statistical description of turbulence in the Earth's plasma sheet & in the solar wind Z. V örös Institute of Astro- and Particle Physics University of Innsbruck Austria. Space Physics Seminar, UCLA, March 13, 2009. The solar wind and the Earth‘s plasma sheet; What is turbulence?

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Space Physics Seminar, UCLA, March 13, 2009

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  1. Statistical description of turbulence in the Earth's plasma sheet & in the solar wind Z. Vörös Institute of Astro- and Particle Physics University of Innsbruck Austria Space Physics Seminar, UCLA, March 13, 2009

  2. The solar wind and the Earth‘s plasma sheet; What is turbulence? Statistical description of turbulence in terms of 2nd order (spectral) and higher order statistics; Structures versus turbulence – non-stationarity; Alfvenic and 2D turbulence - anisotropy Interdependence of statistical moments: non-universal behaviour and cross-scale coupling. THEMIS perspectives OUTLINE UCLA Space Physics Seminar, 2009

  3. The solar wind as a turbulence laboratory The Earth‘s plasma sheet as a turbulence laboratory The physical systems: UCLA Space Physics Seminar, 2009

  4. High latitudes: high speed wind from polar coronal holes Low latitudes: fast and slow streams, strong fluctuations Solar magn. dipole structure Supersonic (super-Alfvénic, …) Hot: >105 K Density: few per cm3 at Earth Complex due to solar variability, solar rotation, and local processes Variable on all scales Collisionless The solar wind as a turbulence laboratory UCLA Space Physics Seminar, 2009 SOLAR MINIMUM: http://solarphysics.livingreviews.org/Articles/lrsp-2005-4/ A review paper by Bruno & Carbone

  5. High latitudes: high speed wind from polar coronal holes Low latitudes: fast and slow streams, strong fluctuations Solar magn. dipole structure Supersonic (super-Alfvénic, …) Hot: >105 K Density: few per cm3 at Earth Complex due to solar variability, solar rotation, and local processes Variable on all scales Collisionless Low beta The solar wind as a turbulence laboratory UCLA Space Physics Seminar, 2009 Burlaga & Vinas, JGR, 2004 Early observations of magnetic PSD indicate that the fluctuations are self-similar Russell (1972).

  6. e.g. Kivelson & Russel, Intro to Space Physics,1995 Magnetospheric fields, currents and plasma regions UCLA Space Physics Seminar, 2009 e.g. Walker et al., Space Sci.Rev., 1999 ~ 30-50 RE Turbulence, where are you?? e.g. Hughes, 1995; in K&R

  7. The fundamental role of magnetic reconnection UCLA Space Physics Seminar, 2009 Dungey global convection Flux tubes • The dayside MSphere is compressed by the SW, the nightside is stretched out into magnetotail; • For southward IMF, magnetospheric plasma circulation is driven by reconnection on the dayside magnetopause; • Magnetotail convection • is affected/driven by nightside reconnection in the tail. • Sources of the plasma in the plasma sheet: • solar wind + ionosphere. DNL Diffusion dominates Rm≤1 In the solar wind: During southward IMF, the merged field lines are transported toward the tail, reconnected in the distant tail and then transported back toward the dayside (e.g. Baumjohann & Treumann, 1996). Unbalanced transport  substorms.

  8. Multi-scale plasma sheet dynamics during substorms UCLA Space Physics Seminar, 2009 • Enhanced dayside merging; • Storage of magnetic field • energy in the tail; • The tail current sheet • becomes thin (100s of km); • Current sheets can store • energy; • An instability of current • sheet (e.g. reconnection) can • explosively release • the energy during the • expansion phase (not always); Baumjohann & Treumann, 1996 (20-30 RE) DNL- Distant Neutral Line; NENL- Near Earth Neutral Line Variety of signatures: particle acceleration, plasma flows, enhanced field-aligned currents, auroral precipitation, enhanced auroral electrojets, etc.  multi-scale physics including turbulence

  9. Bursty bulk flow (BBF) associated turbulence UCLA Space Physics Seminar, 2009 BBF and non-BBF statistics Angelopoulos et al., Phys. Plasmas, 1999 [Consolini, 2004] B=B(t+)-B(t) Remnant flow = cross-tail drift at midnight + small Earthward flow near flanks According to observations, the magnetotail is in a bi-modal state: nearly stagnant, except when driven turbulent by transport-efficient fast flows. Walker et al., ISSI vol, 1999, Angelopoulos, 1993

  10. Turbulence is an irregular motion which, in general, makes ist appearance in fluids when they flow past solid surfaces or even when neighbouring streams of the same fluid flow over one another (Taylor & von Karman, 1937) Turbulence is a phenomenon of instability at high Reynolds numbers…a complete theory of general solutions of the Navier-Stokes equations are called for …it is tied to 3-D (von Neumann, 1949) What is turbulence?? UCLA Space Physics Seminar, 2009

  11. Turbulence is a highly excited state of a system with many degrees of freedom (in most cases a continuous medium) to be described statistically. This excited state is far away from thermodynamic equilibrium and is accompanied by intensive energy dissipation. Such states can be found in fluids, plasmas, magnets, dielectrics, etc.  the problem of turbulence goes far beyond the limits of hydrodynamics and the Navier-Stokes equation (Zakharov, L‘vov, Falkovich, 1992). A modern definition UCLA Space Physics Seminar, 2009

  12. The central issue is to understand the mechanisms by which the energy contained in large-scale turbulent motions is transferred to smaller scales or higher frequencies where it is eventually deposited as thermal energy in the plasma Turbulent cascades UCLA Space Physics Seminar, 2009

  13. Intensive energy dissipation UCLA Space Physics Seminar, 2009 E E Wavefield Turbulence A fluid is heated Power Spectral Density Differences: In turbulence, there is a cascade of turbulence energy from largest to smallest scales. Large scale enery input = energy dissipated by the smallest scales. After injection of E, the multi-scale turbulence channels energy to the smallest scales, the turbulent motion ceases rapidly, the fluid is warmer. Turbulence is highly dissipative. A turbulent motion ceases more rapidly than a residual wave motion, after an injection of the same amount of energy, E.

  14. Heating of the solar wind UCLA Space Physics Seminar, 2009 Adiabatic expansion (no heat is transferred to or from the fluid) • Solar wind plasma is cooling down while it is blown away from the sun • more slowly than it is expected from an adiabatic spherical expansion • Heating is needed to explain the observed temperature radial profile. (Leamon, 1999)

  15. Turbulent heating of the solar wind UCLA Space Physics Seminar, 2009 Leamon, 1999. A recent result: the contribution of turbulent cascades can explain ~80 % of solar wind heating (V. Carbone, personal communication) However, it is not clear yet how the dissipation is happening (e.g. Landau damping, Hall physics, local reconnection, etc.

  16. Statistical description of turbulence UCLA Space Physics Seminar, 2009 • Second order statistics: • - correlation functions; • - power spectral density. • Higher order statistics: • - Probability density function • (statistical moments) • - Structure functions PDFs are fully determined if all the relevant moments are known

  17. Statistical description of turbulence UCLA Space Physics Seminar, 2009 • Second order statistics: • - correlation functions; • - power spectral density. • In the solar wind • - spectral scaling • - fluctuations are Alfvenic • - hydrodynamic and MHD cascades • - spectral anisotropy • - non-stationarity, localized turbulence • 2. In the Earth‘s magnetotail

  18. Spectral scaling in the solar wind UCLA Space Physics Seminar, 2009 Large-scale structures Interaction Turbulence The interaction between large-scale structures and turbulence is largely unexplored JUMPS Burlaga & Vinas, JGR, 2004

  19. Fluctuations in the SW are often Alfvenic UCLA Space Physics Seminar, 2009 A strong correlation exists between magnetic field and velocity fluctuations (Belcher and Davis, 1971) The sign of the correlation is given by the (wave vector normal and the background magnetic field vector). For an outward directed mean field B0, a negative correlation would indicate an outward directed wave vector k and vice-versa

  20. Alfvén waves Inertial range Dissipation Fluctuations in the SW are often Alfvenic UCLA Space Physics Seminar, 2009 MHD, incompr. case (e.g. Biskamp, 2003) Elsässier variables Large-scale structures After neglecting the nonlinear, viscous and forcing terms, we obtain: waves propagating along Bo waves propagating opposite to Bo

  21. What’s the problem Formal similarity between hydrodynamic and MHD equations UCLA Space Physics Seminar, 2009 Turbulence is the result of nonlinear dynamics Incompressible Navier-Stokes equation u  velocity field P  pressure n  kinematic viscosity Nonlinear Dissipative MHD flows: the same “structure” of NS equations z- Elsasser variables z+ Nonlinear interactions happens only between fluctuations propagating in opposite direction with respect to the magnetic field. Interactions are local.

  22. Difference between hydrodynamic and MHD cascades UCLA Space Physics Seminar, 2009 Within the inertial range, at scale the velocity is the kinetic energy is . In MHD opposite travelling wave packets of size l collide and lose energy to smaller scales The number of collissions needed for a cascade is and In hydrodynamics energy is transferred to the next scale within eddy turnover time and the energy transfer rate  , (Frisch, 1995 Biskamp, 2003)

  23. Scaling and correlations in the SW UCLA Space Physics Seminar, 2009 The local mean magnetic field introduces anisotropy. In units of 1010 cm (Bruno, Carbone, 2005) Contour plot of the 2D correlation function of interplanetary magnetic field fluctuations as a function of parallel and perpendicular distance with respect to the mean magnetic field. (Matthaeus et al., 1990) Inertial range scaling, however, shows scaling index typical for isotropic hydrodynamic turbulence. WHY?

  24. Alfvenic fluctuations in the solar wind UCLA Space Physics Seminar, 2009 Substantial part of the turbulence research in the solar wind is addressing the question about alfvenicity of fluctuations Review papers: Tu, C.-Y., Marsch, E., “MHD structures, waves and turbulence in the solar wind: Observations and theories”, Space Sci. Rev., 1995. Bruno R. and V. Carbone, “The Solar Wind as a Turbulence Laboratory”, Living Rev. Solar Phys., 2005. Normalized X-helicity measures the predominance of the energy associated to one of the two possible Alfven modes [-1sC+1] Normalized residual energy [-1sR+1] Useful quantities For an Alfvén mode: |sC|=1; sR=0 24

  25. Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr) 0.3 AU Alfvénic population 25

  26. Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr) 0.3 AU Alfvénic population 0.7 AU 26

  27. Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr) 0.3 AU Alfvénic population predominance of outward fluctuations (positive values of C) 0.7 AU 0.9 AU A new population, characterized by magnetic energy excess , appears (Bruno et al., 2007) 27

  28. An excess of magnetic energy dominate regions whose local magnetic field is oriented at large angles with respect to the large scale magnetic field (Bruno et al., 2007) background field B  X (Each pixel > 3 elements) At 1 AU more than 25% of the 1hr intervals within fast wind are magnetically dominated 28

  29. Difference between the fast wind and slow wind The high velocity interval shows a remarkable anti-correlation which, since the mean magnetic field B0 is oriented away from the Sun, suggests a clear presence of outward oriented Alfv´enic fluctuations given that the sign of the correlation is the sign[−k ·B0] (Bruno et al., 2007) Bruno and Carbone, LRSP, 2005 29

  30. Several studies suggested that besides Alfvénic fluctuations, coherent structures advected by the wind play an important role in interplanetary turbulence (Tu and Marsch, 1992) In the plane perpendicular to Bo fluctuations are fluid like  Hydrodynamic scaling Spaghetti-like structure: (Bieber et al., 1996) Average magnetic field Local field (Bruno et al., 2001) Similar studies: McCracken and Ness, 1966; Mariani et al., 1973; Thieme et al., 1988, 1989; Tu et al., 1989, 1997; Tu and Marsch, 1990, 1993; Crooker et al., 1996; Bruno et al., 2003, 2004; Chang, 2003; Chang et al., 2004; Borovsky, 2006. 30

  31. Turbulence is local in the solar windfluctuations are non-stationary UCLA Space Physics Seminar, 2009 ‚Bavassano-Villante streams‘ (e.g. Bruno & Carbone, Living Rev. Solar Phys., 2005) Labels show the same corotating streams at different heliocentric distances observed by Helios2 Burlaga & Vinas, JGR, 2004 Spectra computed from the whole year of data

  32. Statistical description of turbulence UCLA Space Physics Seminar, 2009 • Second order statistics: • In the solar wind • - spectral scaling • - fluctuations are Alfvenic • - hydrodynamic and MHD cascades • - spectral anisotropy • - non-stationarity, localized turbulence • - change with heliocentric distance • 2. In the Earth‘s magnetotail • - what drives turbulence? • - are the fluctuations Alfvenic? • - are the fluctuations anisotropic? • - are the fluctuations non-stationary? • - is turbulence localized? • - how fluctuation statistics is changing with RE?

  33. Plasma flows are the drivers of wave power in the Earth’s plasma sheet UCLA Space Physics Seminar, 2009 Compressional comp. in a field-aligned coord. system 9s 65s Volwerk et al., AG, 2004; The data are transformed to a mean field-aligned coordinate system which is calculated from low-pass filtered data; the two transverse components in the mean-field-aligned system are combined to a left- and a right-hand polarized component. • Compressional component dominates • over larger time scales; • Fluctuations are non-stationary; • Spectral indices are different from 5/3; • Spectral power is bulk speed dependent. (Bauer et al., JGR, 95; Borovsky &Funsten, JGR, 2003 Volwerk et al., AG, 2004; Vörös et al., JGR, 2004; Weygand et al., JGR, 2005)

  34. Plasma flows are the drivers of wave power in the Earth’s plasma sheet UCLA Space Physics Seminar, 2009 If the plasma flows drive turbulence in the plasma sheet, we can estimate the Reynolds number from the ratio of flow channel size (L) and the smallest scale (S) in turbulence See e.g. Warhaft, PNAS, 2002 a paper read in the Journal Club L / S ~ (Re)3/4 The large-scale of the flows was determined by Nakamura et al., GRL 2004, using multi-point Cluster measurements: L(north-south)=1.5 – 2 RE L(dawn-dusk)=2-3 RE

  35. Turbulence might be not fully developed in the plasma sheet UCLA Space Physics Seminar, 2009 The small-scale of the flows was determined through statistical analysis (Vörös et al., 2006) L/S=6000 km / 200 km = 30 Re ~ 93 L/S = 12600 km / 100 km ~ 130 Re ~ 630 L/S = 20000 km /50 km Re ~ 3000 Spectral widening with V

  36. Turbulence might be not fully developed in the plasma sheet UCLA Space Physics Seminar, 2009 Another estimation of the effective Reynolds number based on Taylor microscale is by Weygand et al., JGR, 2007: Which is a length associated with the mean square spatial derivatives of b. ~ 2000 km ~ 7 -- 110 In the solar wind ~ 105

  37. Estimation of the scaling parameters in sliding windows UCLA Space Physics Seminar, 2009 (Vörös et al., JGR, 2004) A non-flow period

  38. Intermittency vs. Stationarity (Vörös et al. 2007) • Multiple flows: • (Intervals A, B) • V ~ (100-1000) km/s; •  ~ (0.5 – 3); • cf ~ (0 – 150); • frequency ↛wavenumber. • Individual flows: • (e.g. interval C) • V ~ 750+- 150 km/s; •  ~ 2.5 +- 0.3; • cf >> 0 ; • frequency  wavenumber. Multi-scale quasi stationarity is required to detect turbulence! (Large-scale mean flow + small-scale power + steady )

  39. Individual vs. Multiple Flows Stationary intervals Non-stationary intervals (Vörös et al., NPG, 2007) 6-min long intervals 7 hour long interval • Hall phys. • visc./resist. • ? 1/f scaling multiple sources Spectral steepening Inertial range Inertial range smeared

  40. The inertial range of turbulence in the plasma sheet might be hidden due to multiple flow smearing; and as a consequence, wide range of values for the scaling indices were observed by different authors.

  41. Are the fluctuations Alfvenic in the plasma sheet? UCLA Space Physics Seminar, 2009 TWO-POINT Spatial fluctuations between Cluster 1,4: ONE-POINT Time-delayed fluctuations: • Fluctuations are • frequently Alfvenic • Taylor frozen-in • hypothesis valid • during fast flows. (Vörös et al. 2006)

  42. Are the fluctuations anisotropic in the plasma sheet? UCLA Space Physics Seminar, 2009 Scale-dependent variance anisotropy Vörös et al., JGR,2004

  43. Spectral anisotropy Real space structure of turbulent eddies Large eddies are almost isotropic; Smaller eddies are anisotropic and they are elongated along the local mean magnetic field and not along the global meanB; The local mean field is not the same for the large eddies and the small eddies  scale-dependent anisotropy Cho et al., ApJ, 2002 We do not know the structure of the eddies in the plasma sheet (if they exist at all…); Speculation: Magnetic fluctuations at a given scale “feel” a local mean field at a scale which is of an order larger;

  44. Scale-dependent anistropy UCLA Space Physics Seminar, 2009 K41 does not contain anisotropy. The magnetic field, however, naturally introduces an anisotropy. Goldreich & Shridhar (1995) theory: -motions perpendicular to the mean magnetic field are eddies, while those parallel to magnetic field are waves. -motions perpendicular to the magnetic field lines occur over hydrodynamic time scales Perpendicular and wave-like motions parallel to magnetic field are balanced: As previously: 

  45. How to measure anisotropy in turbulence? UCLA Space Physics Seminar, 2009 The degree of spectral anisotropy can be measured through the anisotropy angle (Shebalin et al. 1983): where k is the Fourier wavevector with components parallel and perpendicular to Bo Oughton et al. (1998) have shown: 90 o for fully perpendicular; 0 o for fully parallel; 54 o for isotropic fluctuations.

  46. Case studies UCLA Space Physics Seminar, 2009 G10 C4 C1 C3 Tc-2 C2 Sergeev et al., GRL 2007 JGR, 2008 Vörös et al., JGR, 2008 September 26, 2005

  47. Anisotropy comparisons- Cluster/TC2 UCLA Space Physics Seminar, 2009 Anisotropy is not scale dependent Anisotropy is scale dependent - Expected for MHD cascade

  48. Anisotropy angle in the magnetotail UCLA Space Physics Seminar, 2009 Earthward flows Tailward flows Magnetic reconnection outflow associated turbulence (observed by Cluster) shows scale-dependent anisotropy, TC-2 – ??? (Vörös et al., JGR, 2008) Do the asymmetry appear because of tailward – Earthward flows?

  49. Earthward flow at Cluster (C3) pos. UCLA Space Physics Seminar, 2009 X ~ -19Re

  50. Earthward flow at Cluster (C1) pos. UCLA Space Physics Seminar, 2009 X ~ -15Re A A B B

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