NSO Summer School Lecture 2: Solar Wind Turbulence

1. NSO Summer SchoolLecture 2:Solar Wind Turbulence Charles W. Smith Space Science Center University of New Hampshire Charles.Smith@unh.edu

2. Good References (Still): • Space Physics – May-Britt Kallenrode, Springer 2001. (~ $70) • Turbulence -- Uriel Frisch, Cambridge University Press 1995. (~ $35 to $55) • Magnetohydrodynamic Turbulence – Dieter Biskamp, Cambridge 2003. (~ $110) • The Solar Wind as a Turbulence Laboratory -- Bruno and Carbone, on line @ Living Reviews in Solar Physics (free)

3. f5/3 Analysis: Day 49 of 1999 • “Classic” upstream wave spectrum for older, well-developed event (as expected). • Broad spectrum of upstream waves. • Not much polarization. No real surprises here. Upstream spectrum from prior to the onset of the SEP. Power law form has no net polarization and there is a spectral break to form the dissipation range. This is a typical undisturbed solar wind interval.

4. Newborn Interstellar Pickup Ions Interstellar neutral enter the heliosphere at small speeds, are ionized either by charged particle collision or photo-ionization, are “picked up” by the solar wind & IMF, and convected back to termination region. In the process their circulation of the IMF provides “free energy” for exciting magnetic waves that can provide energy for heating the thermal plasma. VSW

5. Waves Due to Pickup Ions Traditional plasma theory suggests that wave spectra due to pickup ions will reach LARGE enhancements in the background power levels (left, Lee & Ip, 1987). They don’t! (below, Murphy et al., 1995) Wave growth is slow and turbulent processes overwhelm the plasma kinetics of wave excitation (for once).

6. Why Study Turbulence • Turbulence is just about the most fundamental and most ubiquitous physics on Earth. • Seen in every naturally occurring fluid that is disturbed • Responsible for atmospheric weather • Reason refrigerators work (and heaters) • Reason internal combustion engines work • Has become a code-word for “disturbance”, “complexity”, and “nonlinearity”, but it is much more. • Is the process by which a fluid (or gas) attempts to self-organize its energy. • Where does all that energy go!!!!!

7. What is Turbulence? It is the nonlinear evolution of the fluid away from its initial state and toward a self-determined state.

8. Hydrodynamic Turbulence:Laminar vs. Turbulent Flow Interacting vortices lead to distortion, stretching, and destruction (spawning).

9. Hydrodynamic Eddie Dynamics

10. One Example of Turbulence Contributed by Pablo Dimitric of the Bartol Research Inst.

11. Examples of Gravitational Turbulence Alexei Kritsuk and colleagues

12. So what’s happening in all these cases? Everything! Starting with the fluid undergoing a complex evolution that moves energy between the scales.

13. Fluid Pressure Dissipation of Energy Mass Density Convective derivative (time variability following the flow) Incompressibility (constant mass) Navier-Stokes Equation

14. 0 0 0 Energy Conservation Energy Conserving

15. Dissipation at large wave numbers Wave Vector Dynamics

16. Turbulent Flow Dynamics Vortices of dissimilar size lead to convection rather than destruction. Interacting vortices lead to distortion, stretching, and destruction (spawning).

17. A Closer Look at Vortex Distortion L VL

18. Kolmogorov’s First Hypothesis of Similarity As phrased by Frisch (1995): “At very large, but not infinite Reynolds numbers, all the small-scale statistical properties are uniquely and universally determined by the scale L, the mean energy dissipation rate , and the viscosity .” - Kolmogorov (1941) (Reprinted in Proc. R. Soc. London A, 434, 9, 1991.)

19. Verification of Kolmogorov Prediction Inertial rangespectrum ~ -5/3 Grant, Stewart, and Moilliet, J. Fluid Mech., 12, 241-268, 1962. Spectral steepening with dissipation

20. Ion Inertial Scale Inertial range spectrum ~ 5/3 Inertial rangespectrum ~ -5/3 Spectral steepening with dissipation Grant, Stewart, and Moilliet, J. Fluid Mech., 12, 241-268, 1962. Spectral steepening with dissipation A Universal Spectrum f -1“energy containing range”  = V3/L f -5/3 “inertial range” If the inertial range is a pipeline, the dissipation range consumes the energy at the end of the process. Magnetic Power f -3“dissipation range” This is the essence of hydrodynamic turbulence applied to MHD! Few hours 0.5 Hz

21. MHD Equations The Navier-Stokes equation represents a system where the nonlinear terms move energy without changing the total energy and viscosity dissipates energy at the smallest scales. Can the MHD equations do something comparable?

22. Coleman, Phys. Rev. Lett., 17, 207 (1966) Fluctuations in the wind and IMF are presumed to be waves, most likely Alfven waves, that are remnant signatures propagating out from the solar corona. Coleman, Astrophys. J., 153, 371 (1968) Fluctuations arise in situ as result of large-scale interplanetary sources such as wind shear and evolve non-linearly in a manner analogous to traditional hydro-dynamic turbulence. A New View???

23. Dasso et al., ApJL, 635, L181, (2005). Unprocessed features of solar origin… Slow wind is 2D Fast wind is 1D Hamilton et al. (2008) find no evidence of speed-associated geometry at small scales. …and in the dissipation range, the geometry goes to 1D!  Large-scale energy source…  …well-studied waves or turbulence… Matthaeus et al., 1990, Maltese Cross shows significant  component. Bieber et al., 1996 places  component at ~80% of total energy.  …feeds an energy- conserving cascade… Dasso et al. (2005) examine fast and slow winds separately. They find fast winds mostly k || B0 and slow winds mostly  B0. …plasma kinetic physics. …until fluid approx. breaks down. Matthaeus et al., J. Geophys. Res., 95, 20,673 (1990). Inertial Range Wave Vector Anisotropy Magnetic + Velocity Power 1 / (Few hours) 0.2 Hz

24. Timescales

25. How to Measure the Spectral Cascade in Hydrodynamics? …and if isotropic

26. In MHD it looks like: The N-S / MHD Versions Politano and Pouquet, Phys. Rev. E, 57, R21, 1998a. Politano and Pouquet, GRL, 25, 274, 1998b.

27. Slow Wind Hybrid Geometry In an MHD extension of the Kolmogorov 4/5 law in hydrodynamics: Politano & Pouquet, Phys. Rev. E, 57, R21—R24 (1995). Politano & Pouquet, Geophys. Res. Lett., 25(3), 273—276 (1998). MacBride et al., Astrophys. J., 679, 1644--1660 (2008).

28. Building a Heating Model • Dominant component is 2-D. Spectrum evolves quickly. • Construct a turbulence-based model using the 2-D component. • Use concepts familiar from Navier-Stokes turbulence theory. • Wind shear and shock heating are strongest inside ~10 AU. •  Set wind shear term according to fluctuation levels at 1 AU. • Pickup ions only uniform energy source in outer heliosphere. • They drive waves that must couple to turbulence? • On what time scale? And what waves? Propagating how? •  Allow for expansion/cooling.

29. Summary of Interplanetary Turbulence Large scales are dictated by sun. Geometry velocity-ordered… Large-scale energy source… …evolving toward 2D geometry… with compression …feeds an energy- conserving cascade… Magnetic + Velocity Power …plasma kinetic physics. …until fluid approx. breaks down. 1 / (Few hours) 0.2 Hz

30. Extra Slides

31. Why Turbulence in the Solar Wind?

32. Pickup Ions Heat the Solar Wind

33. Roberts et al., J. Geophys. Res., 92(10), 11021, 1987. Decay of HC (Cross-Correlation) HCVB / V2+B2 Strong correlations evident at 1 AU disappear by ~ 4 AU.

34. Milano et al., Phys. Rev. Lett., 93(15), 155005 (2004). Multi-D HC = VB If there is a V and a B so that there is a VB, there must be sufficient energy to support VB. Define C  VB / EV+EB so that 1  C  +1

35. Observations of IMF Spectral Index Magnetic fluctuations at high frequencies show 5/3 power law steepening to 3. Not much evidence of 2!

36. Velocity Spectra Podesta et al., J. Geophys. Res., 111, A10109, 2006.

37. Why do we always see the same spectral form - EVERYWHERE?Why/How does the spectrum change over scales?Why/How does Hc evolve if fluctuations are waves?How do we tap the wave energy to create heat?

38. Magnetohydrodynamics Hydrodynamics A Suggestive Association Grant, Stewart, and Moilliet, J. Fluid Mech., 12, 241, 1962. Inertial rangespectrum ~ -5/3 Spectral steepening with dissipation

39. Weak Turbulence Theory • One view, popular in plasma theory, is that turbulence is just the 2nd-order interaction of linear waves. • Oppositely propagating low-frequency waves interact to produce a “daughter” wave propagating obliquely in a 3rd direction. • Requires that transfer rates be slow compared with wave periods. • Traditionally give suspect predictions (in my humble opinion)

40. A New View (With Complications) • Complications: • MHD is not hydrodynamics! • …but it contains hydrodynamics! • There are multiple time scales. • There are wave dynamics. • The mean field provides a direction of special importance! • The spectra are not isotropic. • Can we build a theory of MHD turbulence that brings ideas from hydrodynamics into the problem? • Dissipation is NOT provided by the fluid equation! • Dissipation marks the breakdown of the fluid approximation. • Going to need kinetic physics for dissipation!?!

41. N-S Vorticity Equation

42. Homework Problem: • Show that the incompressible MHD equations conserve total energy V2+B2 • In the volume-integrated sense. • Except for viscosity and resistivity terms. • Write down the MHD equations, work the nonlinear terms into something familiar, and apply reasonable boundary conditions as before.

43. What is Obstacle to MHD Theory? • MHD contains 2 timescales: • Fluid timescale of overturning eddies NL ~ v3/L • Wave timescale of propagating fluctuations A • Simple similarity theories fail (potentially) if there are 2 competing timescales. • How do we proceed?

44. What is Kolmogorov Saying? Large-scale fluctuations (eddy’s, waves, shears, ejecta, shocks, whatever) contain a lot of energy, but direct dissipation of that energy is slow (except maybe shocks). The turbulent inertial range cascade converts energy of the large-scale objects into smaller scales until dissipation becomes important. In this manner, the large-scale “structure” of the flow can heat the thermal particles of the fluid. This occurs within the fluid description! Does this apply to the solar wind and other space plasmas? It appears that dissipation in the solar wind occurs outside the fluid description, which complicates and changes the problem.

45. Kraichnan Theory

46. What Spectral Predictions Exist? • Kolmogorov (1941a) Pk ~ 2/3k5/3 (isotropic hydro) • Kraichnan (1965) Pk ~ (VA)1/2k3/2 (MHD) • ---- experiments with liquid metals… • Fyfe et al. (1977) Pk ~ 2/3k5/3 (2D-MHD) • ---- reduced MHD dynamics…Rosenbluth et al. (1976), Strauss (1976, 1977), Montgomery & Turner (1981), Montgomery (1982): (claims kz dominated by wave dynamics while k is dominated by turbulence dynamics.) • Higdon (1984) adopts Fyfe result for 2D (argues a kz = k2/3 separatrix) and gets Pk ~ kz3! • Goldreich and Sridhar (1995) rename Higdon and rMHD results “critical balance”, apply EDQNM theory and gets Pk ~ kz2 . • Boldyrev (2005, 2006) claims Pk ~ k3/2 !

47. 10 Years of ACE Observations Tessein et al., ApJ, submitted (2008).

48. High-Latitude BV Result Horbury et al., unpublished.

49. Evolution of EB k t = 0.5 t = 8 In a 2D MHD simulation: with DC magnetic field, energy is placed in a few k, background noise, energy moves to larger wave vectors and moves away from the mean field direction. Initial input of energy at scales incapable of dissipation evolves toward scales where dissipation can occur. t = 0 t = 1 B0 Ghosh et al., J. Geophys. Res., 103, 23,691, 1998. J. Geophys. Res., 103, 23,705, 1998.

50. Fluid Meets Kinetic Physics • What causes the spectrum to steepen at approx. ion inertial scale? (cyclotron freq.) • Cyclotron damping, Landau damping, current sheet formation, eMHD, …. • We know that dominance of k  k||. • Fluctuations are less transverse. • Breakdown of single-fluid MHD!