waves and turbulence in the solar wind

Turbulence

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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


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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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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


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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


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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


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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


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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


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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?


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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


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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?


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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


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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


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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”


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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


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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


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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

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Correlation function anisotropy

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


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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


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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


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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)


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Superdiffusion

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


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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


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Steeper spectrum at kinetic scales

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


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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


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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


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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


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Turbulent reconnection

• Turbulence can bring differently directed magnetic fields together

• Trigger reconnection• Retino et al., Nature, 2007


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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...


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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


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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?

waves and turbulence in the solar wind

Documents

solar systemwaves

solar windturbulence

solar cyclethis

space plasmasturbulence

solar cyclecosmic ray

study waves

lesser whirls

energy transfer rate