nasa cluster gi/ rssw1au programs - turbulence theory - single spacecraft observations -...
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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