Waves and turbulence in the solar wind
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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?.

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Waves and turbulence in the solar wind

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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?

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 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

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

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

Energy input

Inertial range

Trace power levels

Energy transfer


Sampling frequency

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

Anisotropic MHD turbulence

Turbulence in plasmas

Neutral fluids

  • Motion described by Navier-Stokes equations


  • Energy transfer by velocity shear


  • 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

“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

The turbulent solar wind


  • Fluctuations on all measured scales




  • Power spectrum

  • Broadband

  • Low frequencies: f -1

  • High frequencies: f -5/3


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

Fundamental observations of waves and turbulence

Alfvén waves

  • Waves of solar origin

    Active turbulent cascade

  • Not just remnant fluctuations from corona


  • Similar high order statistics to hydrodynamics

    Field-aligned anisotropy

  • Fundamental difference to hydrodynamics


Interpreting spacecraft measurements

  • In the solar wind (usually),

    VA ~50 km/s, VSW >~300 km/s

  • Therefore,


  • 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


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



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


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

Propagation direction of Alfvén waves

  • Waves are usually propagating away from the Sun


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-


Inward and outward spectrum


  • 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


Slow: coronal hole boundary

e+ ~ e-

Fast: within coronal hole

e+ >> e-

Speed dependence of turbulence

  • Character of fluctuations varies with wind speed


Fast wind

Positive cross helicity: anti-sunward Alfvén waves

Sharp transition

To mixed sense waves

Slow wind

Mixed sense, but very variable

Stream dependence: cross helicity

  • Wavelets: measure time and frequency dependence of waves


Dominance of outward-propagating waves


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


  • Outward-propagating low frequency waves generated in corona!

Critical point

Alfvén speed

Distance from Sun


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



  • 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


Alfvén waves

Inertial range



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


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


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


  • Ulysses polar data are ideal for detailed analysis of turbulence and waves


Turbulence and CIRs

Turbulence in space plasmas

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


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


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

Large scale power dependence

  • Ulysses: measure large scale trends in power levels


Latitude dependence?

  • Horbury: latitude dependence, due to coronal overexpansion

  • Bavassano: non-power law scaling, due to nonlinear effects

  • Answer is unclear



  • 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


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?

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

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


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….


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



‘hydro-like’ region




Magnetic field

Anisotropy of energy transfer

Neutral fluid

  • No preferred direction

     isotropy


  • Magnetic field breaks symmetry


  • 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

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



Anisotropic energy transfer


  • Wavevectors

  • Energy tends to move perpendicular to magnetic field


  • Eddies

  • On average, tend to become smaller perpendicular to field

  • Results in long, fine structures along the magnetic field

Anisotropic MHD turbulence





Anisotropy and 3D magnetic field structure


  • 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

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


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


    →“reduced spectrum”

Anisotropic MHD turbulence



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

Definition: BV – angle between magnetic field and flow

Anisotropic MHD turbulence




Field/flow angle





Field/flow angle

Anisotropy and the power spectrum

“Slab” fluctuations


“2D” fluctuations



Anisotropic MHD turbulence

Fit with ~20% slab, 80% 2D

Expected if just slab

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

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

Anisotropic MHD turbulence








  • 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

Correlation function anisotropy

  • Dasso et al., 2005

  • Left: slow wind (2D)

  • Right: fast wind (slab)

Turbulence in space plasmas

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?



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



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



  • 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

Anisotropy and Kolmogorov turbulence

  • Why is the MHD cascade Kolmogorov-like, not Kraichnan-like?


  • 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



  • Narita et al., 2006

  • Use k-filtering to identify different components of turbulence

Turbulence in space plasmas

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

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


  • Anomalous diffusion – Zimbardo et al.

  • Result of anisotropy in k space

Turbulence in space plasmas

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

Steeper spectrum at kinetic scales

  • Alexandrova et al., 2007

  • Note: Taylor’s hypothesis not necessarily satisfied

Turbulence in space plasmas

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

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

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

Turbulent reconnection

  • Turbulence can bring differently directed magnetic fields together

  • Trigger reconnection

  • Retino et al., Nature, 2007

Turbulence in space plasmas

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

Waves and motion in the chromosphere

X-ray bright points: ubiquitous reconnection

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

Summary: results to date


  • Perpendicular cascade

  • K41-type cascade


  • Similar to hydro

    Large scale dependence

  • Distance/latitude/wind speed variability


Summary: unanswered questions

3D structure

  • What is the 3D form of the turbulence, particularly the magnetic field?

  • How does this control energetic particle transport?


  • Is solar wind intermittency “the same” as in hydrodynamics?


  • Mechanism?

    Coronal heating

  • What can we learn about coronal conditions from the solar wind?


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