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Waves and turbulence in the solar wind

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Tim Horbury

PG Lectures

- Turbulence: the basics
- Turbulence in plasmas: MHD scales
- The solar wind context
- Key results
- Dissipation
- Energetic particle propagation

Turbulence

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

- 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

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

Anisotropic MHD turbulence

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

Energy input

Inertial range

Trace power levels

Energy transfer

Dissipation

Sampling frequency

- 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

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

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

Turbulence in space plasmas

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

- 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

- 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

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

- 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 = 2f / VSW

- Almost always valid in the solar wind
- Makes analysis much easier
- Not valid in, e.g. magnetosheath, upper corona

Turbulence

- 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

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

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

- Waves are usually propagating away from the Sun

Turbulence

- 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

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

Slow: coronal hole boundary

e+ ~ e-

Fast: within coronal hole

e+ >> e-

- Character of fluctuations varies with wind speed

Turbulence

Fast wind

Positive cross helicity: anti-sunward Alfvén waves

Sharp transition

To mixed sense waves

Slow wind

Mixed sense, but very variable

- Wavelets: measure time and frequency dependence of waves

Turbulence

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

- 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

- 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

Alfvén waves

Inertial range

Dissipation

Structures

- 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

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

Turbulence

- 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

Turbulence in space plasmas

- 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

- 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

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

- Ulysses: measure large scale trends in power levels

Turbulence

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

Turbulence

- Q1 Alistair +
- Q2 Chris + Alice
- Q3 Ute
- Q4 Daniel + Lottie

Turbulence in space plasmas

- 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

- 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

- 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?

- 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

- For IK65, expect m/4
- Neither is right – what’s going on?

Turbulence in space plasmas

- 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

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

- 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

k

‘hydro-like’ region

“2D”

“Slab”

k||

Magnetic field

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

- 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

k

k||

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

B

k||B

kB

B

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

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

- For a given spacecraft frequency f, this corresponds to a flow-parallel scale
=VSW / f

- …and a flow-parallel wavenumber
k||=2f / VSW

- But sensitive to all waves with
k·VSW=2f

→“reduced spectrum”

Anisotropic MHD turbulence

V

B

- 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

Power

B

0°

90°

Field/flow angle

V

B

Power

0°

90°

Field/flow angle

“Slab” fluctuations

k||B

“2D” fluctuations

kB

B

Anisotropic MHD turbulence

Fit with ~20% slab, 80% 2D

Expected if just slab

Bieber et al., J. Geophys. Res., 1996

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

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

- Dasso et al., 2005
- Left: slow wind (2D)
- Right: fast wind (slab)

Turbulence in space plasmas

- 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

- 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

- 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

Results

- Axisymmetry around field direction
- project onto a 2D plane

- Bin and average the data
- superimpose grid
- grid square: (22) 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

- 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

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

Turbulence in space plasmas

- 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

- 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

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

Turbulence in space plasmas

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

Turbulence in space plasmas

- 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

- Alexandrova et al., 2007
- Note: Taylor’s hypothesis not necessarily satisfied

Turbulence in space plasmas

- MHD: E=-VxB
- Not so on kinetic scales
- Evidence for kinetic Alfven waves?
- Bale, 2005
- See also: Galtier, Hall MHD

Turbulence in space plasmas

- 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

- 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

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

Turbulence in space plasmas

- 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

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

Anisotropy

- Perpendicular cascade
- K41-type cascade
Intermittency

- Similar to hydro
Large scale dependence

- Distance/latitude/wind speed variability

Turbulence

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