NASA Cluster GI/ RSSW1AU programs

- Turbulence theory

- single spacecraft observations

- Multispacecraft ACE –Wind-Cluster-Geotail, IMP data

Correlation and Anisotropy in solar wind turbulence

W H Matthaeus

Collaborators: J. M. Weygand, S. Dasso, C. W. Smith, M. G. Kivelson, J. W. Bieber, P. Chuychai, D. Ruffolo, P. Tooprakai

Bartol Research Institute and Department of Physics and Astronomy, University of DelawareIGPP, UCLA

IAFE, Universidad de Buenos Aires, ArgentinaEOS, University of New Hampshire

Mahidol University,Bangkok, THailandChulalongkorn University, Bangkok, Thailand

Mean flow and fluctuations

• In turbulence there can be great differences between mean state and fluctuating state

• Example: Flow around sphere at R = 15,000

Mean flow Instantaneous flow

VanDyke, An Album of Fluid Motion

Essential properties of turbulenceBatchelor and Townsend, 1949

dE/dt ~ -u3/L

I) Complexity in space + time (intermittency/structures)II) O(1) diffusion/energy decayIII) wide range of scales, ~self similarity

K41

Large scale features of the Solar Wind: Ulysses

• High latitude– Fast

– Hot

– steady

– Comes from coronal holes

• Low latitude– slow– “cooler” (40,000 K @ 1

AU)

– nonsteady

– Comes from streamer belt

McComas et al, GRL, 1995

MHD scale turbulence in the solar wind

•Powerlaw spectra cascade

•spectrum, correlation function

Magnetic fluctuationSpectrum, Voyager at 1 AU

Single s/c background: frozen-in flow approx.

Space-time correlation

assume fluctuationundistorted in fast flow U

measured 1 s/c correlation relatedTo 2-point 1-time correlation by

this mixes space- and time- decorrelation, and whileuseful, needs to be verified (as an approximation) and furtherstudied to unravel the distinct decorrelation effects

What multi s/c can tell us

• Spatial correlations R(r) fit, or full functional form

• When we have enough samples, R(r ,r)

• examine frozen-in flow approx. (predictability)

• Infer the Eulerian (two time, 1 pt) correlation

Problem: We do not have hundreds or thousands of s/c to use.So, we must average two point correlations at different places and times.

Variability, Similarity and PDFs

R(r) 2 R ( r / )

Similarity variables: turbulence energy, correlation scale

e.g., for Correlationfunction

(per unit mass)

^

•Variance is approx. log-normally distributed•v, b fluctuations are approx. Gaussian• Normalization separates these effects defines ensemble

PDF of component variances

• Variances are approx. log-normal

Suggests independent (scale invariant) distribution of coronal sources

PDF of B components at 1AU

• When normalized to remove variability of mean and variance, component distributions are close to Gaussian

”primitive fields” are ~Gaussian,but derivatives are intermittent

Padhye et al, JGR 2001; Sorriso-Valvo et al, 2001

Mean in interval I

Energy interval I

Structure functionestimate interval I

Correlation function estimate interval I

Data: ACE-Wind Geotail-IMP 8

• 1 min data.

• 12 hr intervals.

• Subtract mean field in interval.

• Normalize correlation estimate by observed variance.

• ACE-Wind pair separations: ≈ 0.32·106 to 2.3·106 km.

• Geotail-IMP 8 pair separations (not shown) : ≈ 0.11·106 to 0.32·106 km.

£

106

£ 106

Data: Cluster Correlations in SW

• 22 samples/sec• 1 hr intervals.• 6 separations/interval (4 s/c) • Mean removed, detrended.

• Normalize correlation estimate by observed variance.

• Black dash: SW intervals.• Blue Dash: plasma sheet

intervals. (Weygand SM24A-3)

Solar Wind: 2 s/c magnetic correlation function estimates

Cluster in the SW

Geotail-IMP 8

ACE-Wind

Correlation scale from CSrRrR /exp0

c = 1.3 (±0.003) 106 km

Cluster/ACE/Wind/Geotail/IMP8 Correlations

Separation (106 km)

Taylor microscale scale

• Determine Taylor scale from Taylor expansion of two point correlation function:

• Need to extract asymptotic behavior,

need fine resolutionRichardson

extrapolation

• Result is:

2

2

21

TSbb

rrR

T = 2400 ± 100 km

Tay

lor

Sca

le (

leas

t S

q. F

it)

Tay

lor

Sca

le (

lin

ear

Fit

)

SW Taylor Scale • Estimate T from quadratic fits to S(r)

with varying max. separation

• Linear fit to trend of these estimates from 600 km to r-max for every r-max.

• Extrapolate each linear fit to r=0 (call this a refined estimate of T)

• Look for stable range of extrapolations

T stable from about 1,000 to 11,000 km.

Value is

TS = 2400 ± 100 km

¼3.4 ion gyroradii

• Ion gyroradius est. ≈700 km.

• Ion inertial length est. ≈100 km.

TS: 2400 ± 100 km

2.9 5.7 8.6 11.4 14.2 17.1

Ion gyrorad.

2.9 5.7 8.6 11.4 14.2 17.1

Taylor Scale: Least Squares Fit

2-spacecraft two point, single time correlations of SW turbulence

• correlation (outer, energy-containing) scale

c = £ 106 km, ~ 190 Re ~ 0.008 AU

• inner (Taylor) scale Taylorkm ~ 1.6 £ 10-5 AU

• another scale: Kolmogoroff or “dissipation scale” d is termination of inertial range

Effective Reynoldsnumber of SW turbulenceis

(Lc/)2 ¼ 230,000

Comparison of correlation functions from 1 s/c (frozen-in) measurements, and

2 s/c (single separation) measurements

Two Cluster samplesgive two 1 s/c estimates of R(r)for a range of r one 2 s/c estimate of R(r)R= s/c separation

1 s/c

1 s/c

2 s/c

2 s/c

1 s/c

1 s/c

Deviation from frozen-in flow is a measure of temporal decorrelation, i.e., connection to Eulerian single point two time correlation fn in progress)

Spectral Anisotropy

Anisotropy in MHD associated with a large scale or DC magnetic field

Shebalin, Matthaeus and Montgomery, JPP, 1983

Preferred modes of nearly incompressible cascade

• Low frequency quasi-2D cascade: – Dominant nonlinear activity involves k’s such that

Tnonlinear (k) < TAlfven (k)– Transfer in perp direction, mainly

– k perp >> k par

• Resonant transfer: Shebalin et al, 1983

– High frequency Z+ wave interacts with ~zero frequency Z- wave to pump higher k? high frequency wave of same frequency

• Weak turbulence: Galtier et al

See: two time scale

derivation of Reduced MHD (Montgomery, 1982)

All produce essentially perpendicular cascade!

Cross sections B/B0 = 1/10

Jz and Bz in an x-z plane Jz and Bx, By in an x-y plane

Solar Wind Quasi-Perpendicular cascade…..plus “waves”

B0

Maltese cross• Several thousand samples of ISEE-3 data• Make use of variability of ~1-10 hours mean magnetic

field relative to radial (flow) direction

Quasi-2D

Quasi-slab

r ‖

r┴

Magnetic field autocorrelation

r┴

<400 km/s > 500 km/s

Levels 1000 1200 1400 1600 1800 2000

SLOW SW: More 2D-like FAST SW: More slab-like

r ‖

Correlations in fast and slow wind, as a function of angle between observation direction and mean magnetic field

Spatial structure and complexity

Models that are 2D or quasi-2D transverse structuregives rise to complexity of particle/field line trajectories (non Quasilinear behavior).

2D magnetic turbulence: Rm=4000, t=2, 10242

Magnetic field lines [contours of a(x,y)]

Electric current density

“Cuts” through 2D turbulence bx(y)

Analogous to bN(R) in SW magnetic field data. Compare with ~5 day Interval at 1 AU

Magnetic field lines/magnetic flux surfaces for model solar wind turbulence

A mixture of

2D and slab

fluctuations

in the “right”

proportion

Magnetic structure is

spatially complex

“halo” of low SEP density over wide lateral region

“core” of SEP with dropouts

IMF with transverse structure and topological “trapping”

Piyanate Chuychai, PhD thesis 2005

Ruffolo et al.2004

Orbit of a selected field lines in xy-plane

Radial coordinate (r) vs. z

Particle trapping, escape and delayed diffusive transport

Tooprakai et al, 2007

Dissipation and Taylor scales: some clues about plasma dissipation

processes

Solar Wind Dissipation

• steepening near 1 Hz (at 1 AU) -- breakpoint scales best with ion inertial scale

• Helicity signature proton gyroresonant contributions ~50%

• Appears inconsistent with solely parallel resonances

• kpar and kperp are both involved

• Consistent with dissipation in oblique current sheets

Leamon et al, 1998, 1999, 2000

k

Dissipation scale and Taylor scales (ACE at 1 AU)

T > d cases are like hydro T < d cannot occur in hydro, it is a plasma effect.

Further study of the relationship between these curves may provide clues about plasma dissipation

clouds: red

(C. Smith et al)

Summary

• Correlation functions– 2 pt 1 time, 1 pt 2 time, predictability

• Anisotropy– Incompressible: dominant perp cascade– Low freq quasi 2D + waves

• Structure and complexity– Diffusion and topology

• Dissipation and Taylor scales– What limits mean square gradients in a plasma?

Activity in the solar chromosphere and corona: SOHO spacecraft

UV spectrograph: EIT 340 A White light coronagraph: LASCO C3

Origin of the solar wind