2 nd IWF Turbulence Workshop 30.9 – 4.10. 2013, GRAZ

1. 2nd IWF Turbulence Workshop 30.9 – 4.10. 2013, GRAZ Turbulence in the heliosphere Zoltán Vörös Space Research Institut Graz, Austria Project support: P24740-N27 FP7/2007-2013 - 313038/STORM

2. - Fields and plasma properties in the solar wind show fluctuations over a wide range of temporal & spatial scales ranging from solar cycle/rotation scales down to the electron scales. Fundamental plasma phenomena, such as turbulence or/and magnetic reconnection exhibit multi-scale coupling features. Turbulent interactions are usually supposed to be local in the Fourier/physical/phase space, but can be also nonlocal. Here we advocate that certain aspects of complex plasmas can be understood statistically only. However,... Large number of degrees of freedom PROLOGUE

3. The solar wind as a plasma laboratory; Turbulence: textbook phenomenologies; Are the phenomenologies supported by experiments? New ingredients After all, what is turbulence? OUTLINE

4. Properties of the solar wind near the Earth n is proton density, Vsw is solar wind speed, B is magnetic field strength, A(He) is He++/H+ ratio, Tp is proton temperature, Te is electron temperature, T is alpha particle temperature, Cs is sound speed, CA is Alfven speed (after Gosling). The solar wind is a nearly collisionless, supersonic and super-Alfvenic flow of electrons, protons, alpha particles and other less abundant ions, permeated by the interplanetary magnetic field. There exist large fluctuations around the mean values in all parameters.

5. Fully developed fluid/plasma turbulence There is no general theory based on first principles L PHENOMENOLOGIES Isotropic & homogeneous & incompress. & stationary & no dissipation The turbulent cascade is a superposition of ‚eddies‘ of sizes D D Fluc- tuations [e.g. Frisch, 1995, Salem et al., ApJ09] Energy Eddy interaction or transfer, or ‚turn-over‘ time Models depend on how is defined Energy exchange rate

6. Fully developed (neutral) fluid turbulence Kolmogorov phenomenology (1941) –K41 [after Salem et al., ApJ09] Isotropic & homogeneous & incompress. & stationary & no dissipation  Since Bessel-Parseval theorem and Spectral energy density

7. Fully developed plasma turbulence There is no general theory based on first principles. Iroshnikov (1963) & Kraichnan (1965) phenomenology – IK65 Isotorpic fluctuations; Energy transfer is due to ‚weak‘ interactions between Alfvenic fluctuations moving in opposite directions along B with VA. v and B fluctuations are of the same order of magnitude ‚Weak‘ means that wave packets undergo multiple collisions before they change. One interaction takes an Alfven time Therefore, in comparison with K41 energy transfer time is times longer, that is: and  [e.g. Biskamp, 2003 Salem et al.,ApJ09]

8. Fully developed turbulence - solar wind Observations B f -1 K41 f -5/3 f -2 [Wicks et al. , MNRAS, 2010] IK65 V • Problems: • - competition: other types of fluctuations • local magnetic field introduces anisotropy • field and plasma parameters scale differently • only V shows IK65 scaling • excess of magnetic over kinetic energy • non-stationarity & intermittency • structures in the solar wind f -3/2 [Salem et al., ApJ 2009] Ev/Eb excess of magnetic energy

9. Other types of fluctuations For example: pressure anisotropy driven fluctuations Mirror instability threshold Oblique firehose instability threshold (Bale et al. 2009)

10. Anisotropy B f -1 Goldreich & Sridhar phenomenology (1995) –GS95 Eddies of size are elongated along mean B, the fluctuations develop in the perpendicular components: ‚Critical balance‘ or strong interaction condition: f -5/3 f -2 [Wicks et al. , MNRAS, 2010] Alfven time Eddy turnover time Spacecraft const.  [Horbury et al., 2008 Chen et al., 2010] VSW

11. Anisotropy Observations show that a Kolmogorov cascade develops in perpendicular to B direction confirming GS95. However, numerical simulations show (Mason et al. 2008) that large average magnetic fields can lead to IK65 perpendicular cascade: • Actually,recent investigations (Boldyrev PRL 2006; Chen et al. ApJ 2012) • indicate that Alfvenic fluctuations can be 3-dimensionally anisotropic • in a scale-dependent manner! • direction of the mean B • due to alignment of and near proton gyroscales perp.dir.(1) • perp.dir.(2) perpendicular to perp.dir.(1). • Alignment depends on  non-universality, compressional effects.

12. Different scalings V & B B: K41 or IK65 V: IK65  How to estimate if V and B scale differently?? Possibilities: • Estimate the scalings and energy transfer for each quantity • B, V, n, T, E, etc. • 2. Combine B and V to third order moments with no prescribed spectra. Third-order moments in hydrodynamics: Kolmogorov 4/5 law Theoretical result obtained from Navier-Stokes equation, for incompressible, isotropic, stationary, homogeneous turbulence with no dissipation: The third-order moment of speed fluctuations along the bulk flow scales linearly with separation

13. Third-order moments in MHD Elsässer field: The incompressible MHD equations, containing resemble the Navier-Stokes eqs. Nonlinear interactions exist between fluctuations propagating in opposite directions: . The MHD equations for and are evaluated in two points and then vector differences are formed and combined to second-order correlation tensor assuming homogeinity, isotropy, stationarity, vanishing dissipation and projecting to longitudinal direction [Politano and Pouquet]: Yaglom‘s law Kolmogorov 4/5 law Dissipation tensor kinematic viscosity

14. Yaglom‘s law Yaglom‘s law is valid under the assumption of: incompressibility, stationarity homogeinity and isotropy. Solar wind does not fully comply with... Sorriso_Valvo et al. 2007 High-speed polar wind data by Ulysses at distance 3-4 AU heliolatitude 55 – 35 degrees observation length: 20 intervals of 10 days (Sorriso-Valvo et al., 2007) V,B correlations lost Solar wind in ecliptic is more structured (MacBride et al. 2008) Attempt to account for compressible effects (Carbone et al. 2009) Mean energy transfer rate include density Carbone et al. 2009 W+ 102 J/Kg s Y+ Stationarity? to reach convergence estimating 1 year long data intervals are needed (Podesta et al. 2009)  stationarity has to be studied Nonlinear cascade is significantly enhanced by density fluctuations

15. Excess of magnetic over kinetic energy Possible explanations: 1.) local dynamo effect (Grappin et al. 1983) 2.) Compressible fluctuations can change the energy exchange rate 3.) Nonlocal coupling in k-space (Galtier, 2006); nonlocal coupling induces intermittency (Vörös et al., 2006, 2007) 4.) Radial evolution of Alfvenic turbulence (Bruno et al., 2007) Axcess of magnetic energy is associated with magnetic structures  intermittency e.g. Salem et al. 2009 Inertial range

16. Radial evolution of Alfvenic fluctuations in the solar wind 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. Alfven mode propagation inward outward Normalized X-helicity [-1sC+1] -measures correlations between V and B measures predominance of energy associated with Z+ or Z- Normalized residual energy [-1sR+1] measures predominance of kinetic or magnetic energies Useful quantities For an Alfvén mode: |sC|=1; sR=0 16

17. Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr) 0.3 AU Helios-2 fast wind observations Alfvénic population 17

18. Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr) 0.3 AU Helios-2 fast wind observations Alfvénic population 0.7 AU 18

19. Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr) 0.3 AU Helios-2 fast wind observations Alfvénic population 0.7 AU 0.9 AU A new population, characterized by magnetic energy excess , appears (Bruno et al., 2007) 19

20. FAST WIND SLOW WIND The radial evolution of fast wind and slow wind is different. Alfvenic fluctuations are stronger in the fast wind. As the fast wind expands, the Alfvenic component decays due to the nonlinear cascade. The magnetically dominated structures become visible at ~1 AU. In the slow wind the Alfvenic fluctuations are weaker and do not mask the magnetic structures. Magnetic structures introduce/increase intermittency (Bruno et al., 1999 D‘Amicis et al. 2010) However: - only a few events analyzed; - the radial evolution of other turbulent related quantities, e.g. anisotropy, is questionable; 0.3 AU 0.7 AU 0.9 AU 20 (Bruno et al., 2007)

21. High-order statistics: intermittency & more Turbulent fluctuations are non-Gaussian. We have already mentioned: - Energy transfer rate (4/5 law, Yaglom law): third-order structure function - Intermittency: scaling fourth-order structure function connection to structures. Possible strategies: PDF 1. Sturcture functions if is nonlinear the process is multifractal andd there is a disitribution of structures which are singular or very irregular [eg. Burlaga ,1993, Horbury et al., 1996, Vörös, 1998] 2. Wavelet transform coefficients can be related to structure functions [e.g.Salem et al 2009, Farge et al. 2006] which allow to find singularities or magnetic structures and remove them from the data. 3. Calculation of moments 3rd order: 4. Non-Gaussian PDF models For example, log-normal, Castaing, kappa, log-kappa, etc. (Leubner&Vörös, 2005; Burlaga & Vinas, 2005; Leitner et al. 2009). 4th order: ... and higher moments

22. What is the shape of PDFs in the solar wind? after Burlaga & Vinas 2004 close to Gaussian increasing peakedness skewed PDFs

23. Intermittency: convected structures or turbulence? Numerical simulations show that turbulence produces its own highly localized coherent structures, eg. current sheets, discontinuities (eg. Servidio et al., 2009, 2012). Other structures, such as discontinuities, shocks, flux tubes originate on the Sun. One can localize coherent structures by wavelet thresholding and remove them from time series, making the PDFs gaussian or the structure functions to follow K41 or IK65 scalings (Salem et al., 2009). after wavelet filtering However, if the structures are produced by turbulence, wavelets remove part of the turbulence as well.

24. Intermittency: convected structures or turbulence? Intermittency associated with structures can be nonlocal, why local wavelet thresholding is removing localized structures. An interdependence of Skew and Kurt indicates that large-scale structures influencing small-scale statistics are present resulting in non-locality and non-universal scalings (Warhaft, 2000). Warhaft [2000, 2002] has shown that small-scale PDFs of a passive scalar field in turbulence are skewed if large-scale scalar gradients exists in turbulent flows. Holzer and Siggia (1994) have shown that large-scale gradients not only skew PDFs, but also increase intermittency. In MHD flows the magnitude of B fluctuations resembles dynamical properties of passive scalars (Bershaskii & Sreenivasan, 2004). We have to check how the statistical moments associated with B are related to each other. In hydrodynamics passive scalars in shear flows obey Kurt = a . (Skew)2 + b

25. Moment interdependence near shocks Skewness (B) Kurtosis (B) Multiple shock interval in 2000 (Vörös et al. 2006) Interval with no discontinuities

26. Comparisons Kurtosis Passive scalar statistics in a fluid flow (Chatwin, Robinson, 1997) Skewness Passive scalar statistics near interplanetary shocks (Vörös et al., 2006) Passive scalar statistics in the Earth’s plasma sheet Evidence for nonlocal turbulence interactions in space plasmas (Vörös et al., 2007)

27. The importance of structures 2. Turbulence associated with unstable flux tubes Flux tubes are basic magnetic structures in the solar wind – represent sources of intermittency. The total pressure profile inside flux tubes depends on magnetic field twist. Tubes twisted with the angle of 70o or more are unstable against kink instability which can lead to reconnection (Zaqarashvili et al., in prep.) Twisted flux tubes in an external twisted magnetic field can be unstable to K-H instability even in cases of sub-Alfvenic shears (Zaqarashvili et al., in prep.) and significantly contribute to turbulence in the solar wind.

28. Conclusions The concept of turbulent cascades and nonlinear interactions is important to understand energy redistribution over multiple scales in the solar wind. More effort is needed, however, to understand, for example, intermittency, the role of coherent structures, nonlocality, or to identify the drivers of fluctuations, their evolution, etc. Intermittent structures can be found e.g. by wavelets, but it is not known if they are produced by turbulence or not. Case studies are needed to identify the structures embedded in turbulent plasma and to establish the interrelationships between structures and non-Gaussian turbulence.

29. Epilogue 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) OR 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 (or MHD) and the Navier-Stokes equation (Zakharov, L‘vov, Falkovich, 1992).