Waves and turbulence in the solar wind

1. Waves and turbulence in the solar wind Tim Horbury PG Lectures • Turbulence: the basics • Turbulence in plasmas: MHD scales • The solar wind context • Key results • Dissipation • Energetic particle propagation Turbulence

2. D. Vier/SoHO/Hubble/Dimotakis et al Why study waves and turbulence in the solar wind? Effect on the Earth • Can trigger reconnection, substorms, aurorae, … Understanding solar processes • Signature of coronal heating, etc. Application to other plasmas • Astrophysics: particle propagation • Dense plasmas: transport Turbulence as a universal phenomenon • Comparison with hydrodynamics Turbulence in the solar wind

3. Cosmic rays and the solar cycle • Cosmic ray flux at the Earth is modulated by the solar cycle • This is due to variations in the magnetic barrier in the solar system • Waves and turbulence in the solar wind form a key part of this barrier Anisotropic MHD turbulence

4. What is turbulence? • Fluid phenomenon • Nonlinear energy transfer between scales • Occurs when inertial forces dominate viscous forces • Important in many engineering problems Anisotropic MHD turbulence

5. The Richardson cascade Bigger whirls have little whirls, That feed on their velocity; And little whirls have lesser whirls, And so on to viscosity. Lewis Fry Richardson, 1920 Turbulence in space plasmas

6. Energy input Inertial range Trace power levels Energy transfer Dissipation Sampling frequency The inertial range • If energy input is steady, and far from dissipation scale, have a steady state Inertial range • K41: k-5/3 spectrum • We observe this in hydrodynamic fluids • Note: energy transfer rate is analytic in hydrodynamics Anisotropic MHD turbulence

7. Turbulence in plasmas Neutral fluids • Motion described by Navier-Stokes equations  Hydrodynamics • Energy transfer by velocity shear Plasmas • On sufficiently large scales, can treat plasma as a fluid  Magnetohydrodynamics • Multiple, finite amplitude waves can be stable • Presence of a magnetic field • Breaks isotropy • Key difference to neutral fluids Anisotropic MHD turbulence

8. “The great power law in the sky” • Measure interstellar density fluctuations using scintillations • Consistent with Kolmogorov scaling over many orders of magnitude Turbulence in space plasmas

9. The turbulent solar wind f-1 • Fluctuations on all measured scales turbulence f-5/3 waves • Power spectrum • Broadband • Low frequencies: f -1 • High frequencies: f -5/3 Turbulence

10. Solar wind as a turbulence laboratory • Characteristics • Collisionless plasma • Variety of parameters in different locations • Contains turbulence, waves, energetic particles • Measurements • In situ spacecraft data • Magnetic and electric fields • Bulk plasma: density, velocity, temperature, … • Full distribution functions • Energetic particles • The only collisionless plasma we can sample directly Turbulence

11. Importance of the magnetic field • Magnetic field is often used for turbulence analysis • Precise measurement • High time resolution • Low noise • For MHD scales, this is often sufficient • (but more about velocity later...) • For kinetic scales, have to be more careful Turbulence in space plasmas

12. Fundamental observations of waves and turbulence Alfvén waves • Waves of solar origin Active turbulent cascade • Not just remnant fluctuations from corona Intermittency • Similar high order statistics to hydrodynamics Field-aligned anisotropy • Fundamental difference to hydrodynamics Turbulence

13. Interpreting spacecraft measurements • In the solar wind (usually), VA ~50 km/s, VSW >~300 km/s • Therefore, VSW>>VA • Taylor’s hypothesis: time series can be considered a spatial sample • We can convert spacecraft frequency f into a plasma frame wavenumber k: k = 2f / VSW • Almost always valid in the solar wind • Makes analysis much easier • Not valid in, e.g. magnetosheath, upper corona Turbulence

14. Interpreting spacecraft measurements • Solar wind flows radially away from Sun, over spacecraft • Time series is a one dimensional spatial sample through the plasma • Measure variations along one flow line Flow Turbulence

15. Alfvén waves Field-parallel Alfvén wave: • B and V variations anti-correlated Field-anti-parallel Alfvén wave: • B and V variations correlated • See this very clearly in the solar wind • Most common in high speed wind Turbulence

16. Average magnetic field sunward Positive correlation Propagating anti-parallel to field Propagating away from Sun in plasma frame Average magnetic field anti-sunward Negative correlation Propagating parallel to field Propagating away from Sun in plasma frame Propagation direction of Alfvén waves • Waves are usually propagating away from the Sun Turbulence

17. Spectral analysis of Alfvén waves • For an Alfvén wave: b = v, • Where b = B /(0)1/2 • Calculate Elsässer variables: e = b v • Convention: e+ corresponds to anti-sunward (so flip e+ and e- if B away from Sun) • Also power spectrum Z (f) = PSD (e) • If pure, outward Alfvén waves, e+ >> e- Z+ >> Z- Turbulence

18. Inward and outward spectrum Note: • When e+>>e-, magnetic field spectrum ~ e+ spectrum Define: Normalised cross helicity C = (e+-e-)/(e++e-) • C = 1 for pure, outward waves • C = 0 for mixed waves Fast wind Slow wind Turbulence

19. Slow: coronal hole boundary e+ ~ e- Fast: within coronal hole e+ >> e- Speed dependence of turbulence • Character of fluctuations varies with wind speed Turbulence

20. Fast wind Positive cross helicity: anti-sunward Alfvén waves Sharp transition To mixed sense waves Slow wind Mixed sense, but very variable Stream dependence: cross helicity • Wavelets: measure time and frequency dependence of waves Turbulence

21. Dominance of outward-propagating waves Speed Solar wind speed • Solar wind accelerates as it leaves the corona • Alfvén speed decreases as field magnitude drops • Alfvén critical point: equal speed (~10-20 solar radii) • Above critical point, all waves carried outward Therefore, • Outward-propagating low frequency waves generated in corona! Critical point Alfvén speed Distance from Sun Turbulence

22. Active turbulent cascade in fast wind • Bavassano et al (1982) • Fast wind: “knee” in spectrum • Spectrum steepens further from the Sun • Evidence of energy transfer between scales: turbulent cascade after Bavassano et al 1982 Turbulence

23. Interpretation • Initial broadband 1/f spectrum close to Sun • High frequencies decay, transfer energy • Spectrum steepens • Progressively lower frequencies decay with time (distance) • Breakpoint in spectrum moves to lower frequencies • Breakpoint is the highest frequency unevolved Alfvén wave Turbulence

24. Alfvén waves Inertial range Dissipation Structures Summary: spectral index in fast wind • Ulysses polar measurements • Magnetic field component • Inertial range • Development of cascade • 1/f Alfvén waves at low frequencies Not shown or considered here: • Dissipation at higher frequencies • Structures at lower frequencies Turbulence

25. High and low latitudes: power levels • Low latitudes: fast and slow streams • Stream-stream interactions • Big variations in power levels • Cross helicity often low, highly variable Turbulence

26. High and low latitudes: power levels • High latitudes at solar minimum • Dominated by high speed wind form coronal hole • Power levels very steady • Cross helicity steady and high Therefore, • Ulysses polar data are ideal for detailed analysis of turbulence and waves Turbulence

27. Turbulence and CIRs Turbulence in space plasmas

28. Large scale variations in power levels • Power in high speed wind, low and high latitudes • Ulysses agrees well with Helios • Data taken 25+ years apart • Increasing scatter in Helios reflects stream-stream interactions Turbulence

29. Power dependence on distance • WKB (Wentzel-Kramer-Brillouin) • Assume propagation of waves through a slowly changing medium • Solar wind density scales as r -2 •  power scales with distance as r -3 • We expect this for low frequency Alfvén waves: • Non-interacting • No driver or dissipation, especially in high latitude polar flows Turbulence

30. Radial power trend Low frequency Alfvén waves r -3 radial scaling (WKB): non-interacting P f-1 High frequency Alfvén waves Faster than r -3 radial scaling: energy transfer P f-5/3 Note: latitudinal power scaling! Lower power at higher latitudes Large scale power dependence • Ulysses: measure large scale trends in power levels Turbulence

31. Latitude dependence? • Horbury: latitude dependence, due to coronal overexpansion • Bavassano: non-power law scaling, due to nonlinear effects • Answer is unclear Turbulence

32. Problems • Q1 Alistair + • Q2 Chris + Alice • Q3 Ute • Q4 Daniel + Lottie Turbulence in space plasmas

33. Intermittency • Distributions of increments are not Gaussian • Well known in hydrodynamics, also present in solar wind MHD • More ‘big jumps’ than expected • Is this a signature of the turbulence, or solar wind structure? Sorriso-Valvo et al., 2001 Turbulence

34. Identifying intermittent events • What causes these large jumps? • Identify individual events, study in detail • Discontinuities? • Are large jumps part of the turbulent cascade, or are they structures? Turbulence

35. Discontinuities vs turbulence • Turbulence • Field-perpendicular cascade generates short scales across the field • Tube-like structures • Not topological boundaries • Flux tubes • Sourced from Sun (Borovsky) • Topological boundary? • How to decide? • Composition changes?

36. Intermittency and structure functions • Structure functions: • S(,m)=<|b(t+)-b(t)|m > • Moments of the distribution of differences at different lags • We are interested in how these scale with time lag: • S(,m)=  (m) • How do the wings of the distribution change with scale? Turbulence

37. Intermittency and K41/IK65 • For IK65, expect m/4 • Neither is right – what’s going on? Turbulence in space plasmas

38. Structure function scaling • Intermittency: burstiness • p model (Meneveau and Sreevinasan. 1987) • Uneven distribution of energy between daughter eddies • p=0.5 = equality • Often get p~0.7-0.8 in hydrodynamics • See that here too • But.. No physics.... • Dots: data • Square: ‘p model’ fit Turbulence

39. Kolmogorov vs Kraichnan Inertial range • Carbone (1992): g(4)==1 for Kraichnan • Use g(3) and g(4) to distinguish Kraichnan from Kolmogorov • Answer: Kolmogorov • Why is it not a Kraichnan cascade? • Answer lies with anisotropy…. Turbulence

40. Field-aligned anisotropy • Power levels tend to be perpendicular to local magnetic field direction • anisotropy • Dots: local minimum variance direction • Track large scale changes in field direction • Small scale turbulence “rides” on the back of large scale waves Turbulence

41. k ‘hydro-like’ region “2D” “Slab” k|| Magnetic field Anisotropy of energy transfer Neutral fluid • No preferred direction  isotropy Plasmas • Magnetic field breaks symmetry anisotropy • Shebalin (1983): power tends to move perpendicular to magnetic field in wavevector space • Goldreich and Sridhar (1995): “critical balance” region close to k||=0 Anisotropic MHD turbulence

42. Critical balance • Goldreich and Sridhar, 1995 • Balance of Alfven and nonlinear timescales • Distinguish hydro-like and MHD-like regimes • What is nature of cascade around this regime? Turbulence in space plasmas

43. k k|| Anisotropic energy transfer Hydrodynamics • Wavevectors • Energy tends to move perpendicular to magnetic field MHD • Eddies • On average, tend to become smaller perpendicular to field • Results in long, fine structures along the magnetic field Anisotropic MHD turbulence

44. B k||B kB B Anisotropy and 3D magnetic field structure Slab • Plane waves • Infinite correlation length perpendicular to magnetic field • Flux tubes stay together 2D (+slab) • Finite perpendicular correlation length • Flux tubes “shred” → Field-perpendicular transport Anisotropic MHD turbulence

45. Anisotropy and 3D field structure 100% slab 0% 2D • Wavevectors parallel to the field: long correlation lengths perpendicular to field (“slab”) • Wavevectors perpendicular to the field: short correlation lengths perpendicular to field (“2D”) • Mixture of slab and 2D results in shredded flux tubes • Consequences for field structure and energetic particle propagation 20% slab 80% 2D Matthaeus et al 1995 Turbulence

46. The reduced spectrum • For a given spacecraft frequency f, this corresponds to a flow-parallel scale =VSW / f • …and a flow-parallel wavenumber k||=2f / VSW • But sensitive to all waves with k·VSW=2f → “reduced spectrum” Anisotropic MHD turbulence

47. V B The reduced spectrum – wavevector space • Spacecraft measure “reduced” spectrum, integrated over wavenumbers perpendicular to flow: • Contribution by slab and 2D varies with field/flow angle, θBV Definition: BV – angle between magnetic field and flow Anisotropic MHD turbulence

48. Power B 0° 90° Field/flow angle V B Power 0° 90° Field/flow angle Anisotropy and the power spectrum “Slab” fluctuations k||B “2D” fluctuations kB B Anisotropic MHD turbulence

49. Fit with ~20% slab, 80% 2D Expected if just slab Bieber et al., J. Geophys. Res., 1996 Evidence for wavevector anisotropy Bieber et al., 1996 • Power levels vary with field/flow angle • Not consistent with just slab • Best fit: 20% slab, 80% 2D Limitations of analysis • Long intervals of data • Magnetic field direction not constant • Adds noise and error to analysis Anisotropic MHD turbulence

50. r r r r|| r|| r|| Anisotropy • Large scale magnetic field induces anisotropy • 2D • k perpendicular to B-field • Three-wave interactions • Isotropic • k equal in all directions • Hydrodynamic case • Slab • k aligned with B-field • Alfvén wave propagation