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. Why study waves and turbulence in the solar wind?
D. Vier/SoHO/Hubble/Dimotakis et al
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. The inertial range
Energy input
Inertial range
Dissipation
Sampling frequency
Trace power levels
Energy transfer
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
turbulence
waves
f -5/3
Fluctuations on all measured scales
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. Propagation direction of Alfvén waves
Average magnetic field anti-sunward
Negative correlation
Propagating parallel to field
Propagating away from Sun in plasma frame
Average magnetic field sunward
Positive correlation
Propagating anti-parallel to field
Propagating away from Sun in plasma frame
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. Speed dependence of turbulence
Character of fluctuations varies with wind speed
Fast: within coronal hole
e+ >> e-
Slow: coronal hole boundary
e+ ~ e-
Turbulence

20. Stream dependence: cross helicity
Sharp transition
To mixed sense waves
Fast wind
Positive cross helicity: anti-sunward Alfvén waves
Slow wind
Mixed sense, but very variable
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. 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
Alfvén waves
Inertial range
Dissipation
Structures
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. Large scale power dependence
Radial power trend
High frequency Alfvén waves
Faster than r -3 radial scaling: energy transfer
P  f -5/3
Low frequency Alfvén waves
r -3 radial scaling (WKB): non-interacting
P  f -1
Ulysses: measure large scale trends in power levels
Note: latitudinal power scaling!
Lower power at higher latitudes
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
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
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~ 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. Anisotropy of energy transfer
k|| k
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
“2D”
‘hydro-like’ region
“Slab”
Magnetic field
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. Anisotropic energy transfer
Hydrodynamics
k|| k
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. 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 B
k||B B
kB
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. 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 B V
Definition: BV – angle between magnetic field and flow
Anisotropic MHD turbulence

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

49. Evidence for wavevector anisotropy
Bieber et al., J. Geophys. Res., 1996
Expected if just slab
Fit with ~20% slab, 80% 2D
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. Anisotropy Large scale magnetic field induces anisotropy r|| r r|| r
Isotropic
k equal in all directions
Hydrodynamic case
Slab
k aligned with B-field
Alfvén wave propagation 2D
k perpendicular to B-field
Three-wave interactions

51. Correlation function anisotropy
Dasso et al., 2005
Left: slow wind (2D)
Right: fast wind (slab)
Turbulence in space plasmas

52. Limitations of a single spacecraft
Solar wind flows radially away from Sun, over spacecraft
Time series is a one dimensional spatial sample through the plasma
Can’t measure variations perpendicular to the flow
How can we measure the 3D structure of the turbulence?
Flow
Turbulence

53. Combining data from two spacecraft
Compare between spacecraft
Provides information about variations across flow
Varying time lag corresponds to varying scale and direction of separation vector
Limitations on scales and directions
Flow
Turbulence

54. Using multiple spacecraft
Each pair of spacecraft gives a plane on which we can measure the correlation
Four spacecraft give six planes:
We have a large range of angles and scales over which we can measure the turbulence structure
Turbulence

55. Results Axisymmetry around field direction project onto a 2D plane
Bin and average the data
superimpose grid
grid square: (22) 103 km
Fit to an “elliptical model”
Measure of anisotropy: /
related to the ratio of the parallel to perpendicular correlation lengths
 = 0.84,  = 0.49, / = 1.71

56. Anisotropy and Kolmogorov turbulence
Why is the MHD cascade Kolmogorov-like, not Kraichnan-like?
Kraichnan:
Equal populations of oppositely-propagating Alfvén waves
Decorrelation slows cascade
Solar wind:
Dominated by one propagation direction
Anisotropy: wavevectors usually perpendicular to field
Wave speed: V = VAcos()
Perpendicular wavevectors, not propagating: no decorrelation
 Kolmogorov cascade
Turbulence

57. k-filtering Narita et al., 2006
Use k-filtering to identify different components of turbulence
Turbulence in space plasmas

58. Anisotropy and k-filtering
Sahraoui et al., 2006
Here, in magnetosheath: Taylor’s hypothesis not satisfied
Single spacecraft can’t do this analysis
Anisotropic energy distribution in k space
Turbulence in space plasmas

59. Turbulence and energetic particles
Energetic particles follow and scatter from magnetic field
Ulysses: particles at high latitudes
Unexplained “latitudinal transport”
Must be related to 3D structure of magnetic field
This is poorly understood at present
Turbulence

60. Turbulence and energetic particles
Field-line wandering
Resonant scattering: slab waves
Apparent power levels: slab vs 2D
Anomalous diffusion (next slide)
Turbulence in space plasmas

61. Superdiffusion Anomalous diffusion – Zimbardo et al.
Result of anisotropy in k space
Turbulence in space plasmas

62. Kinetic processes What happens when we reach non-MHD scales?
Kinetic processes important
Hydrodynamics: viscosity causes dissipation
Collisionless plasmas: no real viscosity
What causes dissipation?
Waves become dispersive
Turbulence in space plasmas

63. Steeper spectrum at kinetic scales
Alexandrova et al., 2007
Note: Taylor’s hypothesis not necessarily satisfied
Turbulence in space plasmas

64. E and B spectrum in the kinetic regime
MHD: E=-VxB
Not so on kinetic scales
Evidence for kinetic Alfven waves?
Bale, 2005
See also: Galtier, Hall MHD
Turbulence in space plasmas

65. Kinetic instabilities
Evidence for evolution of kinetic distribution limited by instabilities
Instability thresholds for ion cyclotron (solid), the mirror (dotted), the parallel (dashed), and the oblique (dash-dotted) fire hose
Figure from Matteini et al., 2007
Evidence for generation of waves due to this
Turbulence in space plasmas

66. Magnetosheath fluctuations
Magnetosheath is very disturbed: shocked
Very different environment
Turbulence very different too
Alfven “filaments”
Spatially localised Alfven ion cyclotron waves
Alexandrova et al., 2006
Turbulence in space plasmas

67. Turbulent reconnection
Turbulence can bring differently directed magnetic fields together
Trigger reconnection
Retino et al., Nature, 2007
Turbulence in space plasmas

68. Turbulence and coronal heating
How is the corona heated?
Turbulent cascade dissipating photospheric motions as heating?
Nano-scale reconnection events?
Old UVCS evidence for enhanced heating of heavy ions – due to wave-particle interactions?
Recent HINODE evidence for waves, but also reconnection events...
Turbulence in space plasmas

69. Waves and motion in the chromosphere

70. X-ray bright points: ubiquitous reconnection

71. Other stuff Magnetic holes
Common features: short decreases in field magnitude
Mirror modes?
Reconnecting current sheets
Solitons (Rees et al., 2006)
Tens of seconds
Increases in field magnitude, decreases in density
Very rare
Turbulence in space plasmas

72. Summary: results to date
Anisotropy
Perpendicular cascade
K41-type cascade
Intermittency
Similar to hydro
Large scale dependence
Distance/latitude/wind speed variability
Turbulence

73. Summary: unanswered questions
3D structure
What is the 3D form of the turbulence, particularly the magnetic field?
How does this control energetic particle transport?
Intermittency
Is solar wind intermittency “the same” as in hydrodynamics?
Dissipation
Mechanism?
Coronal heating
What can we learn about coronal conditions from the solar wind?
Turbulence