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 Turbulence

Why study waves and turbulence in the solar wind? 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 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

The inertial range Energy input Inertial range Dissipation Sampling frequency Trace power levels Energy transfer 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  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

“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 f -1 turbulence waves f -5/3 Fluctuations on all measured scales Power spectrum Broadband Low frequencies: f -1 High frequencies: f -5/3 Turbulence

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 Turbulence

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 Intermittency Similar high order statistics to hydrodynamics Field-aligned anisotropy Fundamental difference to hydrodynamics Turbulence

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 = 2f / VSW Almost always valid in the solar wind Makes analysis much easier Not valid in, e.g. magnetosheath, upper corona Turbulence

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

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 Turbulence

Propagation direction of Alfvén waves Average magnetic field anti-sunward Negative correlation Propagating parallel to field Propagating away from Sun in plasma frame Average magnetic field sunward Positive correlation Propagating anti-parallel to field Propagating away from Sun in plasma frame Waves are usually propagating away from the Sun Turbulence

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

Inward and outward spectrum 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

Speed dependence of turbulence Character of fluctuations varies with wind speed Fast: within coronal hole e+ >> e- Slow: coronal hole boundary e+ ~ e- Turbulence

Stream dependence: cross helicity Sharp transition To mixed sense waves Fast wind Positive cross helicity: anti-sunward Alfvén waves Slow wind Mixed sense, but very variable Wavelets: measure time and frequency dependence of waves Turbulence

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 Turbulence

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

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

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 Dissipation Structures Turbulence

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

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

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 Turbulence

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

Large scale power dependence Radial power trend High frequency Alfvén waves Faster than r -3 radial scaling: energy transfer P  f -5/3 Low frequency Alfvén waves r -3 radial scaling (WKB): non-interacting P  f -1 Ulysses: measure large scale trends in power levels Note: latitudinal power scaling! Lower power at higher latitudes Turbulence

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

Problems Q1 Alistair + Q2 Chris + Alice Q3 Ute Q4 Daniel + Lottie Turbulence in space plasmas

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

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? Turbulence

Discontinuities vs 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 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

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 Turbulence

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…. Turbulence

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 Turbulence

Anisotropy of energy transfer k|| k 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 “2D” ‘hydro-like’ region “Slab” Magnetic field 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 Hydrodynamics k|| k 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

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 B k||B B kB 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 Turbulence

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 k·VSW=2f → “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 B V Definition: BV – angle between magnetic field and flow Anisotropic MHD turbulence

Anisotropy and the power spectrum “Slab” fluctuations Power Field/flow angle 0° 90° B B V k||B Power Field/flow angle 0° 90° “2D” fluctuations kB B Anisotropic MHD turbulence

Evidence for wavevector anisotropy Bieber et al., J. Geophys. Res., 1996 Expected if just slab Fit with ~20% slab, 80% 2D 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

Anisotropy Large scale magnetic field induces anisotropy r|| r r|| r 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

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? Flow 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 Flow Turbulence

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 Turbulence

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

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 Turbulence

k-filtering 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

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

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

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? Turbulence