Turbulence in the Solar Wind – Generation, Evolution and

Dissipation

Eckart Marsch

MPI für Sonnensystemforschung, Katlenburg-Lindau

Talk in the Solar Physics Seminar on Thursday, 15 May 2008

Water jetMHD simulation: current density

Turbulent wake behind obstacle

Turbulence

Helios 2

• Solar corona and origin of the solar wind• Solar wind and heliospheric magnetic field• Fluctuations - scales and parameters• Magnetoacoustic and Alfvénic fluctuations• Alfvén waves in the solar corona• Turbulence spectra and radial evolution• Cross-helicity, anisotropy, compressibility• Spectral properties and cascading• Scaling behaviour and intermittency• Kinetic features and dissipation of turbulence

Overview

Corona of the active sun

EIT - LASCO C1/C2, 1998

Schwenn and Srivastava, 2000

Fe XII 19.5 nm

Fe XIV 530.3 nm

Visible light Thomson scattering

T=25 d

Changing corona and solar wind

McComas et al., GRL, 2000

North Heliolatitude / degree South

SOHO and Ulysses

45 30 15 0 -15 -30

-45

Coronal magnetic field

Banaszkiewicz et al., 1998;

Schwenn et al., 1997

LASCO C1/C2 images (SOHO)

Current sheet is a symmetric disc anchored at high latitudes !

Dipolar, quadrupolar, current sheet contributions

Polar field: B = 12 G

Heliospheric current sheet

Alfvén, 1977

Parker, 1963

Solar wind fast and slow streams

Marsch, 1991

Helios 1976

Alfvén waves and small-scale structures

Dynamic stream interactions

Dynamic processes in inter-planetary space

• Amplitude steepening (n ~ r-2, Br ~ r-2)

• Compression and rarefaction

• Velocity shear

• Nonlinearity by advection (V)V

• Shock formation (co-rotating)

Temporal scales

Phenomenon Frequency Period Speed (s-1) (day) (km/s)

Solar rotation: 4.6 10-7 25 2

Solar wind expansion: 6 - 3 10-6 2 - 4 800 - 300

Alfvén waves: 3 10-4 1/24 50 (1AU) Ion-cyclotron waves: 1 - 0.1 1 (s) (VA) 50

Coulomb collisions: 10-5 - 10-6 1 - 10Turbulent cascade: generation + transport

inertial range kinetic range + dissipation

Fluctuations

Typical day in April 1995 of Ulysses plasma and field observations in the polar (420 north) heliosphere at 1.4 AU

Horbury & Tsurutani, 2001

• Sharp changes in field direction

• Large Component variations

• Weak compressive fluctuations

radial

tangential

normal

Incompressible MHD equations

Momentum

Induction

Actual Alfvén velocityKinematic viscosity

Magnetic diffusivity

Magnetic Reynolds number 1010 in solar wind

Background Alfvén velocity

Ptot is the total pressure (thermal plus magnetic)

Rapid dynamic alignment

Matthaeus et al., PRL, 2008

Two-dimensional incompressible MHD simulation, showing areas where cos < -0.7 (black), I cos I < 0.7 (gray), and cos > 0.7 (white).

Alignment of flow and magnetic field vectors occurs within the turnover time t = l/v of an eddy of scale size l.

Elsässer variables

Alfvén waves

Dissipation Forcing

Advection, nonlinearity

parallel antiparallel

z = v b

Elsässer, 1950

Alfvénic fluctuations (Helios)

Neubauer et al., 1977 V = - VA

Alfvénic fluctuations (Ulysses)

Horbury and Tsurutani, 2001 V = + VA

Radial variation

Bavassano et al., JGR, 105, 15959, 2001

Elsässer ratio: re = e-/e+ based on average values over 0.1 AU wide intervals of hourly variances of z .

Radial variation of energy e(r) = 1/2 (z±)2 , with wave amplitude at 1-hour period.

WKB ~ r -1

Heliocentric distance

Latitudinal variation

Forsyth et al., 19961-hour scale

Magnetohydrodynamic waves

• Magnetosonic waves

compressible

- parallel slow and fast

- perpendicular fast

Cms = (Cs2+CA

2)1/2

• Alfvén wave

incompressible

parallel and oblique

CA = B/(4)1/2

Broad band in k and random phases

Two-component model

• Alfvén waves parallel to the mean field

• 2-dimensional turbulence perpendicular

• Convected structures (discontinuities) and shocks

pressure balanced

Flux tube angular scale: 2° - 4°; supergranule: 20-30 Mm

Compressive fluctuations

Bavassano et al., Ann. Geophysicae, 2004

Colour coding:

Positive correlations between total plasma pressure pt and density n, and kinetic (thermal) pk and magnetic pm pressure, indicating magnetoacoustic fluctuations of the slow mode type.

Left scale:

Time, radial distance, and heliographic latitude of Ulysses.

Compressive fluctuation spectra

Marsch and Tu, JGR, 95, 8211, 1990

Kolmogorov turbulence of passive scalars

Alfvèn wave spectra

Tu et al., GRL, 1990

Coronal Alfvén waves

Tomczyk et al. Science, 2007

The FeXIII 1074.7nm coronal emission line seen with the Coronal Multi-Channel Polarimeter (CoMP) instrument at the National Solar Observatory, New Mexico

Travel time analysis

DePontieu et al. Science, 2007

1

2

Hinode SOT CaII H 396.8 nm

Spicular transverse motion Numerical simulation

Chromospheric Alfvén waves

FW = 100 W/m2

T = 150 – 350 s

v = 20 km/s VA = 200 km/s

Flows and waves in prominences

Okamoto et al., Science, 2007

T = 20000 K

Ca II 396.8 nm

Hinode

Turbulence in the heliosphere

• Nature and origin of the fluctuations

• Spatial evolution with heliocentric distance

• Distribution and spectral transfer of turbulent energy

• Intermittency and microphysics of dissipation

Scaling, non-linear couplings and cascading

Alfvénic correlations: Alfvénicity (cross helicity)

c = (e+ - e-)/(e+ + e-) = 2< V•VA>/< (V)2 + (VA)2 >

Magnetic versus kinetic energy: Alfvén ratio

rA = eV/eB = < (V)2 >/< (VA)2 >

Evolution of cross helicity

c = 2<V • VA> /(V2 + VA

2)

= (e+ - e-)/(e+ + e-)

Roberts et al., J. Geophys. Res. 92, 12023 , 1987

Alfvénic correlations decay radially!

eA(k) = 1/2 d3k e-i k•r<A(0)•A(r)>

Alfvén ratio

rA(k)= eV(k)/eB(k)

Marsch and Tu, J. Geophys. Res., 95, 8211, 1990

slow fast

Alfvén ratio increases radially!

Anisotropy and correlation

„Maltese cross“

Correlations:Alfvén waves and 2-D turbulence

Matthaeus et al., J. Geophys. Res., 95, 20673, 1990

Autocorrelation function

Radial variation of turbulence

• Turbulence intensity declines with solar distance

• Wave amplitudes are consistent between Helios and Ulysses in fast streams from coronal holes

• Variation of spectral breakpoint (decreases) as measured by various S/C

• Slower radial evolution of spectra over the poles

Horbury & Tsurutani, 2001

Length scales in the solar wind

Macrostructure - fluid scales

• Heliocentric distance: r 150 Gm (1AU = 214 Rs)

• Solar radius: Rs 696000 km

• Alfvén waves: 30 - 100 Mm

Microstructure - kinetic scales

• Coulomb free path: lC ~ 0.1 - 10 AU

• Ion inertial length: VA/p (c/p) ~ 100 km

• Ion gyroradius: rL ~ 50 km

• Debye length: D ~ 10 m

• Spacecraft size: d ~ 3 m Microscales vary with radial distance

Cascading and spectral transfer

Energy spectrum change = transfer – dissipation + forcing

Kolmogorov phenomenology for isotropic homogeneous turbulence

Energy cascade: Turbulent energy (per unit mass density), el (Vl)

2,

at scale l is transported by a hierarchy of turbulent eddies of ever decreasing sizes to the dissipation range at scale lD.

l Vl/l (Vl)2 Ek

3/2 k5/2

energy transfer rate: l (Vl)2/

turnover time: l/Vl

wavenumber: k 1/l

energy spectrum: Ekk (Vl)2

Scale invariance: l = (dissipation rate) --> Ek 2/3 k-5/3

Simulation of 3-D MHD turbulence

Ek 2/3 k-5/3 Kolmogorov, 1941

Ek (vA)1/2 k-3/2 Kraichnan, 1965

Müller and Biskamp, Phys. Rev. Lett., 84, 475, 2000

Direct numerical simulation with a spectral code with 5123 modes

Compensated normalized spectrum shows Kolmogorov scaling and sheet-like dissipative structures

-5/3

Power spectrum evolution

Horbury et al., JGR 101, 405, 1996

Turbulence spectrum:

eB(f) = 1/2 (B)2 (f/f0)-

5/3

1

Evidence for dissipation

• Steepening by cascading

• Ion heating by wave sweeping

• Dissipation by wave absorption

Tu and Marsch, J. Geophys. Res. , 100, 12323 ,1995

0.29 AU

0.87 AU

• Self-similarity: Turbulent eddies look alike on any scale in the inertial and dissipation domain -> scale-invariance.

• In the absence of intrinsic scale, the fluctuations exhibit power-law behaviour with exponent h:

• However, MHD simulations and solar wind observations show that smaller eddies become increasingly sparse -> intermittency.

• Spectral transfer (inertial range) rate and dissipation rate become scale dependent and vary rapidly in space and time.

Intermittency

Fluctuation/eddy amplitude:

Energy dissipation rate becomes scale dependent and intermittent (fractal or not space-filling). Structure function has scaling exponent p, depending on geometry.

Scaling of velocity fluctuations

Dissipative structures: 2-D current and vorticity sheets.

Structure function scaling

Burlaga, JGR, 96, 5847, 1991

Sp()=<|V()-V(0)|p>=s(p)

Scaling exponent s(p) of speed increments

s(p) = 1- ln[Pp/3 + (1-P)p/3] P-model of fractal cascade; P=1/2 no intermittency

Voyager 2 near 8.5 AU s=p/3

Variation of intermittency

Bruno et al., J. Geophys. Res., 108, 1130, 2003

Helios, fast solar wind: Bx radial component of magnetic field, By, Bz.

Flatness (Gaussian, 3):

fast

slow

Time scale

Slow wind more intermittent!

Probability distribution functions

Probability distributions

Spatial intermittency

Marsch and Tu, Annales Geophys., 12, 1127, 1994

Helios: fast SW, Vx radial component of flow velocity

exp(-x')

exp(-x'2)

• Non-Gaussian statistics at small scales

• Increased probability of large fluctuations

22.5 h 13.5 min

2.25 h 81 s

Kinetic processes in solar wind

Problems:

• Transport properties of a plasma involving multiple scales......?

• Thermodynamics of solar wind plasma far from equilibrium.....?

• Plasma is multi-component and non-uniform

Complexity beyond MHD paradigm

Kinetic physics required for dissipation

• Plasma is dilute and micro-turbulent

Free energy for micro-instabilities

Resonant wave-particle interactions

Weak and varying collisions

Kinetic dissipation by plasma waves

Marsch, Liv. Rev. Solar Phys. 2004

Ion acoustic wave

Proton beam

Magneto-sonic wave

Ion differential streaming

Ion cyclotron wave

Proton anisotropy

Kinetic wave mode

Free energy source

Parametric decay of Alfvén wave

• Linear Vlasov theory and 1-D hybrid simulations to study the parametric instabilities of a circularly polarized parallel-propagating Alfvén wave.

• Linear and weakly nonlinear instabilities of an Alfvén wave drive ion acoustic-like and cyclotron waves, leading to beam and anisotropic core.

Araneda, Marsch, and Viñas, JGR, 2007; and PRL 2008

Pitch-angle scattering and energy diffusion in micro-turbulent wave field

Summary• Solar wind is a weakly anisotropic turbulent magnetofluid• Alfvénic fluctuations dominate, with an admixture of weak

compressive (slow magnetoacoustic) fluctuations• Alfvén waves mostly come from the corona, compressive

waves from pressure imbalances and stream interactions • Alfvén ratio, cross-helicity, anisotropy evolve radially, as

do the average spectra and their slopes• Turbulence develops towards Kolmogorov spectra, but

intermittency prevails at small scales• Structure functions and probability distributions clearly

reveal non-gaussian statistics• Intermittency is partly due to small convected structures,

such as rotational and tangential discontinuities• Dissipation takes place in the dispersion domain via

damping of plasma waves (ion-cyclotron, ion-acoustic)