Turbulence, shocks and particle acceleration

The solar wind as a plasma laboratory• Global morphology• Typical parameters• Measuring the solar wind

Turbulence• What is turbulence?• Hydrodynamic turbulence• Turbulence in collisionless plasmas• Open questions

Shocks– Conservation laws at shocks– Shocks in collisionless plasmas– Open questions

Particle acceleration– Acceleration at shocks– First order Fermi acceleration– Particle transport– Open questions

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Tim Horbury, Imperial College London

Thanks to Chris Chen, Javier Pacheco

The solar wind as a plasma laboratory

• Collisionless, magnetised plasma• Continual, but variable, outflow from Sun’s

corona• Carries magnetic field, waves and turbulence

from corona• A collisionless plasma we can sample

directly

• Supersonic (super-Alfvénic, …)• Hot: >105 K• Rarefied: few per cm3 at Earth• Complex due to solar variability,

solar rotation, and in situ processes• Variable on all measured scales, from sub-second to centuries

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SoHO/LASCO

Measuring the solar wind

• Spacecraft carry instruments to measure key plasma properties

• Bulk plasma parameters– Density, velocity, temperature

• Fields– Magnetic field, electric field

• Particle distribution functions– Essential for kinetic studies

• Energetic particles– Different energy ranges

• These measurements are hard – typically 1% accurate, but coverage in time, cadence, direction, energy, etc. is never as good as we would like

• Every measurement is wrong; the harder you push the data, the more you need to know about how the instruments work– Instrument teams love talking about their instruments; ask them

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This is DSCOVR, launched 2015

Solar wind at 1 AU

• Number density: few per cm3

• Mostly protons and electrons, few percent helium, little bits of other things

• Bulk speed: few hundred km/s (~1keV for a proton)

• Proton thermal speed: ~ 50 km/s– Cold beam

• Plasma beta: ~1• Magnetic field: few nT• Alfven Mach number: a few

• Lots of exciting kinetic effects….

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The real solar wind

Turbulence

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da Vinci 1510

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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 plasmas8

McComb 1995 RPP

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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

• Inertial forces dominate viscosity• Reynolds number is scale dependent• At some scale, viscosity dominates

– Dissipation

Energy input Inertial range

Dissipation

Anisotropic MHD turbulence10

Turbulence in plasmas

Neutral fluids• Motion described by Navier-Stokes

equations Hydrodynamics• Energy transfer by velocity shearPlasmas• 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 plasmas11

Turbulence in the solar wind12

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: transportTurbulence as a universal phenomenon• Comparison with hydrodynamics

D. Vier/SoHO/Hubble/Dimotakis et al

Turbulence13

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

Turbulence14

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

Turbulence15

Alfvén wavesField-parallel Alfvén wave:• B and V variations anti-

correlatedField-anti-parallel Alfvén

wave:• B and V variations

correlated

• See this very clearly in the solar wind

• Most common in high speed wind

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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

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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

Speed Solar wind speed

Alfvén speed

Critical point

Turbulence18

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

MHD turbulence

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Turbulence20

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 turbulence21

Anisotropy of energy transfer

Neutral fluid• No preferred direction

isotropyPlasmas• Magnetic field breaks symmetry

anisotropy

• Shebalin (1983): power tends to move perpendicular to magnetic field in wavevector space

‘hydro-like’ region

Magnetic fieldk||

k

Critical balance• MHD turbulence has two timescales

– Alfven timescale: time for two wavepackets to decorrelate

– Nonlinear timescale: time for eddies to decay and transfer energy

• “Weak” MHD: Alfven time is faster, 3/2 spectrum (Kraichnan, Iroshnikov)

• “Strong” MHD: Nonlinear time is faster, 5/3 isotropic spectrum

• “Critical balance” (Goldreich and Sridhar, 1995): Two timecales balance on every scale

• Predicts 5/3 field-perpendicular spectrum, 2 field-parallel

• Consistent with observations…

Turbulence in space plasmas22

after Wicks et al. 2010 MNRASafter Wicks et al. 2010 MNRAS

k||-2

k⊥-5/3isotropic

anisotropic

inertial range

Anisotropic MHD turbulence23

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

Turbulence beyond MHD

• Collisionless plasmas have no real viscosity

• What happens at the proton gyroscale?

• We observe a spectrum at scales below the ion gyroscale

• It seems that there is an additional cascade to smaller scales

• Seems to be kinetic Alfven waves but this is still an open question

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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 plasmas25

Collisionless plasma turbulence: open questionsMHD scales• How does the magnetic field affect the MHD cascade?• How are compressive fluctuations related to the cascade?• How does the balance of Alfvenic fluctuations affect the cascade?

Kinetic scales• What is the nature of the cascade below the proton gyroscale?• Is energy removed at this scale?• What happens at the electron gyroscale?

How universal is turbulence between neutral fluids and plasmas?

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Shocks

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What is a shock?

• Occurs when velocity change is larger than wave speed

• Acts to decelerate and deflect supersonic flow around an obstacle

• Compresses and heats the fluid

• Fluid phenomenon• But, shock transition is kinetic, not

fluid

Shocks and waves: information speed

• Body moving through fluid• Waves propagate upstream:

send information to deflect flow

• If body moves faster than waves, something else must propagate information – shock

• Fundamentally nonlinear• Related to wave mode

• Mach number: ratio of inflow speed to sound speed

• Plasmas: 3 wave speeds, so 3 Mach numbers

How do shocks form?

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• Consider a sound wave in an Ideal gas (adiabatic)• Compressional wave – perturbation in pressure P

where

• Small P • No significant change in cs within wave• Wave profile does not evolve

• Large P• Wave steepens• Shock forms• Mach number

• shock strength

Pcs vp CC

sflow cVM

What happens at a shock?

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• Consider ideal “thin” shock

• Propagating/standing shock?• Work in rest frame of shock

• Mass IN = Mass OUT• Momentum IN =Momentum OUT• Energy IN = Energy OUT

• Internal distribution of energy given by• Equation of state

• Easy for Ideal gases nkTPV kTP

Shocks: fluid view

• Take a fluid approximation– Mass density – Flow velocity v– Pressure p (isotropic)– Assume static, 1D shock– Assume perfect gas– Assume fluid reacts adiabatically

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Rankine-Hugoniot relations

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Hydrodynamic shocks: kinetic view

• Where does the kinetic energy go?– Heating

• Collisions rapidly thermalise the particles

• Shock thickness around the collisional mean free path

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Shocks Space Physics 2009

Astrophysical shocks• Supernovae• Jets• Bowshocks• Blast waves

Shocks in the solar system

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• Planetary bow shocks

• CME and CIR shocks

• Termination shock

• Heliospheric bow shock (maybe not?)

Plasma shocks

• More complex than hydrodynamic shocks– Multiple wave modes possible

• Alfven mode shock: no compression, “discontinuity”• Fast mode shock: compressive• Slow mode shock: compressive

• Presence of magnetic field• Interactions via charges and fields – not just collisions

• Collisionless shocks possible– Particles can travel far upstream– Must more complex behaviour– Energy partitioning

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MHD Rankine-Hugoniot relations

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0

0

0

tntn

n

n

uBu

B

u

B

0

02

012

0

0

22

00

22

tn

tn

n

nnn

Bu

Bpu

BBupuu

Bu

Bu

Mass conservation

Continuity of Bn

Continuity of Et

Energy conservation

Momentum conservation || n

Momentum conservation n

• R-H relations true for all discontinuities, not just shocks• If inflow velocity → subsonic;; solutions revert to MHD waves: fast, slow, Alfvén

Fast and Slow Mode Shocks

• MHD shocks retain characteristics of linear wave modes• Alfven mode

– not compressive so no shock (in isotropic plasma)– rotational discontinuity

• Fast mode– density and |B| correlated– so B turns away from normal– occurs more commonly

• Slow mode– density and |B| anti-correlated– so B turns towards normal– occurs less commonly

Collisionless shocks: magnetic field angle

Quasi-perpendicular• Ions gyrate back into

shock• Sharp transition

Quasi-parallel• Ions travel upstream• Unstable: generate

waves• Extended, messy shock

Shock

Magnetic field

Ions gyrate into shock

θBN largequasi-perpendicular

Shock

Magnetic field

Ions escape

θBN smallquasi-parallel

Quasi-perpendicular shocks (Bn~90°)

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VSWVMS

BMS

BSW

ESW

ESh

dcs

dsh df~0.68 Vsw/ci

jsh

je jf

|B|

t

Overshoot Ramp Foot

Reflected ionsGyrating ions Figure after Baumjohann and Treumann. 1997(not to scale)

Waves

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Different types of Q shock: the importance of Mach number

• Low Mach number: 1- 3 e.g. Interplanetary shocks: – Sharp transition; little overshoot– Sometimes see clear dispersive

whistler leading shock ramp

• Moderate to High Mach number:. 3-10 e.g.Terrestrial bow shock:– Foot; Ramp; Overshoot;

Downstream wave decay– Ion reflection plays essential

role

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Shock variability• Simulations show non-stationary

structure– Important for particle heating?– Reformation of the ramp– Rippling of the shock front

Lembege and Savioni, 1992

Lembege et al, 2004

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The Earth’s bow shock• Curved geometry

– Shock orientation (Bn) varies• Electrons and ions reflected/escape to

form foreshock– Morphology depends on particle

energy, cross-field drift speed– Not steady state but time varying

• Upstream particles generate waves– ULF wave foreshock

• ULF waves convect back towards the shock – Potential to modify shock

geometry– Seen in modulation of reflected ion

populations• Rankine-Hugoniot only true if include

foreshock From Treumann and Scholer, 2001

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Generation of ion beams Kucharek et al, Ann. Geophys., 2004

• Ion beams originate in the shock ramp – they do not leak from the magnetosheath• The density of the ion beam is higher for high Mach number shocks• Gyrating ions and ion beams come from same population

– Specular reflection not sufficient for ions to escape • scattering/multiple shock interactions?

– Predicted by simulations [Burgess, Ann. Geophys., 1987]

The termination shock

• Very large radius of curvature• Most similar to astrophysical

shocks?• How different is this to

planetary bowshocks?• The Voyager spacecraft have

now sampled this shock

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Termination shock

• ~MeV ions rise by a factor of ~10 for months before shock crossing

• Solar wind slows down• Mass loading by energetic particles

– A “cosmic ray modulated shock”?• Probably more similar to many

astrophysical shocks

Florinski et al., 2009

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Energy partitioning

• Where does the energy go?• Proton heating

– Temperature anisotropy?• Electron heating

– Temperature anisotropy?• Supra-thermal particles• Heavy species: helium, …

• Can depend on θBN, Mach number, …

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Particle acceleration

What are energetic particles?

• Particles with kinetic energies significantly above that of the bulk plasma• They must somehow be accelerated out of the thermal population

Sources• Galaxy

– Supernova shocks, etc.• Sun

– Solar flares• Interplanetary shocks• Pick-up ions

– Interstellar neutrals

• Note: we only detect energetic particles because the interplanetary medium and interstellar medium are collisionless

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Galactic cosmic rays

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http://www.physics.utah.edu/~whanlon/spectrum.html

• We measure fast ions and electrons up to relativistic energies

• These are “cosmic rays”• Flux is modulated by solar cycle

Energetic particles in the heliosphere

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• Also see energetic “tails” on bulk plasma distributions

• Particles are accelerated within the solar system– “Solar energetic particles”

• Highest energy events (rare) reach ~GeVand are seen as cosmic rays– “ground level events”

• Where does this particle acceleration happen?

ACE/SWICS

Solar energetic particles

• Energy in SEPs can reach 10% of the total energy of a coronal magnetic ejection

• Cant be ignored as part of the event

• SEPs flood the heliosphere

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Mewaldt et al., 2007

Impulsive vs gradual events

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•Protons •Protons

Parent solar activity PROTONS

Impulsive event

Gradual event

shock

Acceleration at shocks in the solar wind

• Interplanetary shocks can accelerate particles locally

• In some CMEs, around 10% of the total kinetic energy is in suprathermal particles

• Particles can also be accelerated nearer the Sun, and in flares

• Bothmer et al., GRL, 1995

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Acceleration at shocks: cartoon

• Particles bounce of scattering centres either side of the shock• These scattering centres are converging: particles accelerate• Scattering centres are “turbulence”

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Marc Pulupa

Fermi acceleration

First order Fermi (“diffusive shock acceleration”)• Converging scattering centres around shock• Elastic scatters result in energy gain• Steady state spectrum is a power law function of shock compression ratio• Spectral index of 2 for a strong shock (close to observed)

Second order Fermi• Particles scatter off randomly moving objects (e.g.”gas clouds”)• More likely to hit one coming towards you, so on average gain energy• Statistical• Spectrum quite similar to first order Fermi

• Widely regarded as broadly correct mechanism• Injection problem: how do particles get out of the thermal population?

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Hot ions upstream of the bow shock

First order Fermi-acceleration:

• Particle history:1. Crosses shock2. Scatters off density irregularities

(waves, turbulence) in sheath3. Moves upstream4. Scatters again… from pulsations…?5. Sees upstream and downstream

irregularities are converging at uu-ud

6. Gains energy

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Away from shock

shock

uu ud

ion

Shock normal angle: perpendicular is faster

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Giacalone

Particle transport

• Particles travel from acceleration site to spacecraft

• See dispersion signatures from events occurring near the Sun

• Scattering can occur along the route

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Klecker and Kallenbach, 2003

Particel dropouts: hints about transport

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Mazur et al., 2000, APJ Ruffolo et al., 2003, APJ Lett.

Particle acceleration: open questions

Injection problem• How do we get particles out of the thermal population?

Where are SEPs accelerated?• Flares? Near-Sun shocks? Interplanetary shocks?• Need to go close to the Sun within one scattering mean free path

How representative are planetary bowshock to astrophysical shocks?• Curvature, Mach number, …

Is the termination shock our first astrophysical shock?

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Summary

Turbulence• Importance of the magnetic field• The kinetic cascade

Shocks• Partitioning of energy between species• Importance of curvature

Particle acceleration• Injection problem• Relating solar system to astrophysical shocks

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References: turbulence

• Books– Landau & Lifshitz (1959) “Fluid Mechanics”– Frisch (1995) “Turbulence”– Biskamp (2003) “Magnetohydrodynamic Turbulence”

• Review Papers– Schekochihin & Cowley (2007) “Turbulence and Magnetic Fields in

Astrophysical Plasmas”– Alexandrova et al. (2013) Space Sci. Rev. “Solar Wind Turbulence and the

Role of Ion Instabilities”– Bruno & Carbone (2013) Living Rev. Sol. Phys. “The Solar Wind as a

Turbulence Laboratory”

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References: shoks

• Books– Landau & Lifshitz (1959) “Fluid Mechanics”– Burgess (1995) “Collisionless Shocks” in Kivelson & Russell “Introduction

to Space Physics”

• Review Articles– Treumann (2009) Astron. Astrophys. Rev. “Fundamentals of collisionless

shocks for astrophysical application, 1. Non-relativistic shocks” – Krasnoselskikh et al. (2013) Space Sci. Rev. “The dynamic

Quasiperpendicular Shock: Cluster Discoveries”– Burgess & Scholer (2013) Space Sci. Rev. “Microphysics of Quasi-parallel

Shocks in Collisionless Plasmas”

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