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 Delaware IGPP, UCLA IAFE, Universidad de Buenos Aires, Argentina EOS, University of New Hampshire Mahidol University,Bangkok, THailand Chulalongkorn 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 turbulence Batchelor and Townsend, 1949 dE/dt ~ -u 3 /L I)Complexity in space + time (intermittency/structures) II)O(1) diffusion/energy decay III)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 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 fluctuation Spectrum, Voyager at 1 AU

Single s/c background: frozen-in flow approx. Space-time correlation assume fluctuation undistorted in fast flow U   measured 1 s/c correlation related To 2-point 1-time correlation by this mixes space- and time- decorrelation, and while useful, needs to be verified (as an approximation) and further studied 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 Correlation function (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 function estimate 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: ≈ 0.32·10 6 to 2.3·10 6 km. Geotail-IMP 8 pair separations (not shown) : ≈ 0.11·10 6 to 0.32·10 6 km. £ 10 6

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 c = 1.3 (±0.003)  10 6 km Cluster/ACE/Wind/Geotail/IMP8 Correlations Separation (10 6 km)

Taylor microscale scale  Determine Taylor scale from Taylor expansion of two point correlation function: Need to extract asymptotic behavior,  need fine resolution  Richardson extrapolation Result is: T = 2400 ± 100 km

Taylor Scale (least Sq. Fit) Taylor Scale (linear 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  ¼  3.4 ion gyroradii Ion gyroradius est. ≈700 km. Ion inertial length est. ≈100 km. TS : 2400 ± 100 km Ion gyrorad Taylor Scale: Least Squares Fit

2-spacecraft two point, single time correlations of SW turbulence correlation (outer, energy-containing) scale c =  £ 10 6 km, ~ 190 R e ~ AU inner (Taylor) scale  Taylor   km ~ 1.6 £ AU another scale: Kolmogoroff or “dissipation scale” d is termination of inertial range Effective Reynolds number of SW turbulence is (L c / l T ) 2 ¼ 230,000

Comparison of correlation functions from 1 s/c (frozen-in) measurements, and 2 s/c (single separation) measurements Two Cluster samples give  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 2 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 T nonlinear (k) < T Alfven (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/B 0 = 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” B0B0

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┴r┴

Magnetic field autocorrelation r┴r┴ <400 km/s> 500 km/s Levels SLOW SW: More 2D-likeFAST 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 structure gives rise to complexity of particle/field line trajectories (non Quasilinear behavior).

2D magnetic turbulence: Rm=4000, t=2, Magnetic field lines [contours of a(x,y)] Electric current density

“Cuts” through 2D turbulence b x (y) Analogous to b N (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

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 AWhite light coronagraph: LASCO C3 Origin of the solar wind