Waves%20and%20turbulence%20in%20the%20solar%20wind

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

2
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

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

4
What is turbulence?

• Fluid phenomenon
• Nonlinear energy transfer between scales
• Occurs when inertial forces dominate viscous
  forces
• Important in many engineering problems

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

6
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

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

8
The great power law in the sky

• Measure interstellar density fluctuations using
  scintillations
• Consistent with Kolmogorov scaling over many
  orders of magnitude

9
The turbulent solar wind
f -1
turbulence
f -5/3
waves

• Fluctuations on all measured scales

• Power spectrum
• Broadband
• Low frequencies f -1
• High frequencies f -5/3

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

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

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

13
Interpreting spacecraft measurements

• In the solar wind (usually),
• VA 50 km/s, VSW gt300 km/s
• Therefore,
• VSWgtgtVA
• Taylors 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

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
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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

16
Propagation direction of Alfvén waves

• Waves are usually propagating away from the Sun

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 gtgt e-
• Z gtgt Z-

18
Inward and outward spectrum

• Note
• When egtgte-, magnetic field spectrum e
  spectrum
• Define Normalised cross helicity
• ?C (e-e-)/(ee-)
• ?C 1 for pure, outward waves
• ?C 0 for mixed waves

Fast wind
Slow wind
19
Speed dependence of turbulence

• Character of fluctuations varies with wind speed

20
Stream dependence cross helicity

• Wavelets measure time and frequency dependence
  of waves

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
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
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

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

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

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

27
Turbulence and CIRs
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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

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

30
Large scale power dependence

• Ulysses measure large scale trends in power
  levels

31
Latitude dependence?

• Horbury latitude dependence, due to coronal
  overexpansion
• Bavassano non-power law scaling, due to
  nonlinear effects
• Answer is unclear

32
Problems

• Q1 Alistair
• Q2 Chris Alice
• Q3 Ute
• Q4 Daniel Lottie

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
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?

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)ltb(t?)-b(t)m gt
• 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?

37
Intermittency and K41/IK65

• For IK65, expect m/4
• Neither is right whats going on?

38
Structure function scaling

• Intermittency burstiness
• p model (Meneveau and Sreevinasan. 1987)
• Uneven distribution of energy between daughter
  eddies
• p0.5 equality
• Often get p0.7-0.8 in hydrodynamics
• See that here too
• But.. No physics....

• Dots data
• Square p model fit

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.

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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

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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 k0

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?

43
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

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

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
46
The reduced spectrum

• For a given spacecraft frequency f, this
  corresponds to a flow-parallel scale
• ?VSW / f
• and a flow-parallel wavenumber
• k2?f / VSW
• But sensitive to all waves with
• kVSW2?f
• ? reduced spectrum

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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, ?BV

Definition ?BV angle between magnetic field
and flow
48
Anisotropy and the power spectrum
Slab fluctuations
kB
2D fluctuations
k?B
B
49
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

50
Anisotropy

• Large scale magnetic field induces anisotropy

• 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)

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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
• Cant measure variations perpendicular to the
  flow
• How can we measure the 3D structure of the
  turbulence?

Flow
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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
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

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
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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

57
k-filtering

• Narita et al., 2006
• Use k-filtering to identify different components
  of turbulence

58
Anisotropy and k-filtering

• Sahraoui et al., 2006
• Here, in magnetosheath Taylors hypothesis not
  satisfied
• Single spacecraft cant do this analysis
• Anisotropic energy distribution in k space

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

60
Turbulence and energetic particles

• Field-line wandering
• Resonant scattering slab waves
• Apparent power levels slab vs 2D
• Anomalous diffusion (next slide)

61
Superdiffusion

• Anomalous diffusion Zimbardo et al.
• Result of anisotropy in k space

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

63
Steeper spectrum at kinetic scales

• Alexandrova et al., 2007
• Note Taylors hypothesis not necessarily
  satisfied

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

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

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

67
Turbulent reconnection

• Turbulence can bring differently directed
  magnetic fields together
• Trigger reconnection
• Retino et al., Nature, 2007

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...

69
Waves and motion in the chromosphere
70
X-ray bright points ubiquitous reconnection
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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

72
Summary results to date

• Anisotropy
• Perpendicular cascade
• K41-type cascade
• Intermittency
• Similar to hydro
• Large scale dependence
• Distance/latitude/wind speed variability

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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?