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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. D. Vier/SoHO/Hubble/Dimotakis et al. Why study waves and turbulence in the solar wind?. - PowerPoint PPT Presentation
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Turbulence
1
Waves and turbulence in the solar wind
• Turbulence: the basics• Turbulence in plasmas: MHD scales• The solar wind context• Key results• Dissipation• Energetic particle propagation
Tim Horbury
PG Lectures
Turbulence in the solar wind
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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
D. Vier/SoHO/Hubble/Dimotakis et al
Anisotropic MHD turbulence
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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
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What is turbulence?
• Fluid phenomenon• Nonlinear energy transfer between
scales• Occurs when inertial forces
dominate viscous forces• Important in many engineering
problems
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
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Anisotropic MHD turbulence
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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
Energy input
Inertial range
Dissipation
Sampling frequency
Tra
ce
po
wer
lev
els
Energy transfer
Anisotropic MHD turbulence
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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
“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
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Turbulence
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The turbulent solar wind
• Fluctuations on all measured scales
f -1
f -5/3waves
turbulence
Power spectrum• Broadband• Low frequencies: f -1
• High frequencies: f -5/3
Turbulence
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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
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
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Turbulence
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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
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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 = 2f / VSW
• Almost always valid in the solar wind• Makes analysis much easier• Not valid in, e.g. magnetosheath, upper corona
Turbulence
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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
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Alfvén wavesField-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
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Average magnetic field sunwardPositive correlationPropagating anti-parallel to fieldPropagating away from Sun in plasma frame
Propagation direction of Alfvén waves
• Waves are usually propagating away from the Sun
Average magnetic field anti-sunwardNegative correlationPropagating parallel to fieldPropagating away from Sun in plasma frame
Turbulence
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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
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Inward and outward spectrum
Fast wind Slow wind
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
Turbulence
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Speed dependence of turbulence
Slow: coronal hole boundarye+ ~ e-
Fast: within coronal holee+ >> e-
• Character of fluctuations varies with wind speed
Turbulence
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Stream dependence: cross helicity
• Wavelets: measure time and frequency dependence of waves
Fast windPositive cross helicity: anti-sunward Alfvén waves
Sharp transitionTo mixed sense waves
Slow windMixed sense, but very variable
Turbulence
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Dominance of outward-propagating waves
• 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!
Distance from Sun
SpeedSolar wind speed
Alfvén speed
Critical point
Turbulence
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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
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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
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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
Structures
Dissipation
Turbulence
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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
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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 and CIRs
Turbulence in space plasmas
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Turbulence
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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
Turbulence
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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
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Large scale power dependence
• Ulysses: measure large scale trends in power levels
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
Turbulence
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Latitude dependence?
• Horbury: latitude dependence, due to coronal overexpansion
• Bavassano: non-power law scaling, due to nonlinear effects
• Answer is unclear
Problems
• Q1 Alistair +• Q2 Chris + Alice• Q3 Ute• Q4 Daniel + Lottie
Turbulence in space plasmas
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Turbulence
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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
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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?
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?
Turbulence
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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?
Intermittency and K41/IK65
• For IK65, expect m/4
• Neither is right – what’s going on?
Turbulence in space plasmas
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Turbulence
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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
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Kolmogorov vs Kraichnan
• 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….
Inertial range
Turbulence
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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
Anisotropic MHD turbulence
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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 k||=0
‘hydro-like’ region
Magnetic fieldk||
k
“2D”
“Slab”
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
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Anisotropic MHD turbulence
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Anisotropic energy transfer
Eddies
• On average, tend to become smaller perpendicular to field
• Results in long, fine structures along the magnetic field
k||
k
Wavevectors
• Energy tends to move perpendicular to magnetic field
Hydrodynamics
MHD
Anisotropic MHD turbulence
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Anisotropy and 3D magnetic field structureSlab• 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
BkB
Turbulence
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Anisotropy and 3D field structure
• 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
100% slab0% 2D
20% slab80% 2D
Matthaeus et al 1995
Anisotropic MHD turbulence
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The reduced spectrum
• 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
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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, θBV
kVkk 3d )( SWfPfP
BV
Definition: BV – angle between magnetic field and flow
Anisotropic MHD turbulence
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Anisotropy and the power spectrum
B
k||B
BkB
“Slab” fluctuations
“2D” fluctuations
Po
wer
Field/flow angle0° 90°
Po
wer
Field/flow angle0° 90°
BV
Anisotropic MHD turbulence
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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
Bieber et al., J. Geophys. Res., 1996
Expected if just slab
Fit with ~20% slab, 80% 2D
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
• Large scale magnetic field induces anisotropy
0 2000 4000 6000 80000
1000
2000
3000
4000
5000
6000
7000
8000
9000
spatial separation - parallel/km
spat
ial s
epar
atio
n -
perp
endi
cula
r/km
r||
r
0 2000 4000 6000 80000
1000
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5000
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9000
spatial separation - parallel/km
spat
ial s
epar
atio
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perp
endi
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r/km
0.1
0.2
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0.4
0.5
0.6
0.7
0.8
r||
r
0 2000 4000 6000 80000
1000
2000
3000
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6000
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spatial separation - parallel/km
spatia
l separa
tion -
perp
endic
ula
r/km
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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r||
r
Correlation function anisotropy
• Dasso et al., 2005• Left: slow wind (2D)• Right: fast wind (slab)
Turbulence in space plasmas
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Turbulence
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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• Can’t measure variations perpendicular to the flow• How can we measure the 3D structure of the turbulence?
Flow
Turbulence
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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
Turbulence
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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
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
Turbulence
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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
k-filtering
• Narita et al., 2006• Use k-filtering to identify different components of turbulence
Turbulence in space plasmas
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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
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Turbulence
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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 and energetic particles
• Field-line wandering
• Resonant scattering: slab waves
• Apparent power levels: slab vs 2D
• Anomalous diffusion (next slide)
Turbulence in space plasmas
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Superdiffusion
• Anomalous diffusion – Zimbardo et al.• Result of anisotropy in k space
Turbulence in space plasmas
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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
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Steeper spectrum at kinetic scales
• Alexandrova et al., 2007• Note: Taylor’s hypothesis not necessarily satisfied
Turbulence in space plasmas
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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
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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
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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
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Turbulent reconnection
• Turbulence can bring differently directed magnetic fields together
• Trigger reconnection• Retino et al., Nature, 2007
Turbulence in space plasmas
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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
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Waves and motion in the chromosphere
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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
Turbulence in space plasmas
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Turbulence
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Summary: results to date
Anisotropy• Perpendicular cascade• K41-type cascade
Intermittency• Similar to hydro
Large scale dependence• Distance/latitude/wind speed variability
Turbulence
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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?
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