SM51B-0535
Scaling of the solar wind driver and the Akasofu’s ǫ parameter at solar maximum.
1
1
1
1
2
Bogdan Hnat , S. C. Chapman , K. Kiyani , G. Rowlands and N. W. Watkins
1
Centre for Fusion Space and Astrophysics, University of Warwick, Coventry, CV4 7AL UK; Natural Complexity Programme, British Antarctic Survey, High Cross, Madingley Road, Cambridge, UK
9. Models of the probability distributions.
5. Power spectra estimates
Earth magnetosphere is constantly driven by turbulent and intermittent solar
wind. Observations suggest that the multi-scale nature of this coupling is a fundamental aspect of magnetospheric dynamics. We examine the statistical properties of fluctuations in Akasofu’s ǫ, which represents the energy input from
the solar wind into the magnetosphere, and the magnetic field energy density
of the solar wind at solar maximum. Previous studies suggested that, at solar
maximum, these fluctuations are approximately self-similar and their probability distributions have similar functional form. We examine scaling properties of
these quantities in detail, obtain values of their scaling exponents and examine
a fractional Levy walk as a possible model for their statistics.
• Scaling of the power spectrum is estimated using the wavelet based joint estimator of Abry and Veitch.
B2
80
ǫ
log10(σ P)
25
20
75
j
β = 1.65 ± 0.02
70
−5
−1
65
5
10
Octave j
• Driven by intermittent solar wind[2], dissipation via complex current systems
15
5
10
Octave j
• Possible solar signatures identified in the inertial range of the solar wind[3]
5. Systematic conditioning of B 2 and ǫ data
• Self-similar scaling in magnetic field energy density at solar maximum[4]
• Extreme events obscure true scaling of the high order moments
3
0%
0.001%
0.005%
0.05%
0.1%
0.5%
2
ζ(m)
ζ(m)
2
1
1.5
1
0
2
4
6
0
ǫ
2
4
Moment m
6
PL(x) =
• each moment has to converge to an answer independent of the conditioning
threshold
• Solar Max. year 2000: ∼ 4.5×10 samples of B [WIND]
/µ0)l02 sin4(Θ/2)
where
|By |
l0 = 7RE , Θ = arctan( Bz ).
Ps(δxτ
−α
0
0.65
0.55
4
1
2
3
4
Percentage of pnts removed
5
0
1
2
3
4
Percentage of pnts removed
5
), α obtained from data
• Generalised structure functions (GSF) [6]:
R Lp
PN
m
m
Sm(τ ) ≡ h|δx| i ≡ 1/N i |δxi| = −Ln |δx|mP (δx, τ )d(δx) ∝ τ ζ(m)
• Conditioning 1 [8]: fluctuations δx>Lp,n(τ ) are excluded due to poor statistics. For A = 10σ(τ ) we eliminate ≤ 1% of points.
• Conditioning 2 [9]: repeatedly compute ζ(m) whilst successively removing
outlying points
10
Z
dke
−ikx −γ|k|µ
e
10. Conclusions:
• We have identified an approximate self-similar scaling in the fluctuations of
B 2 and ǫ at solar maximum
• The probability distribution of ǫ is well approximated by a Fokker-Planck
equation[8], and differs from The PDF of B 2 for small fluctuations.
Solar Minimum
2
0
δx(τ ≈ 30 min)/σ
• The probability distribution of B 2 fluctuations has power law tails and can be
approximated by the Lévy distribution.
0.6
0
• Fluctuations on timescale τ : δx = x(t + τ ) − x(t), where x = B 2 or ǫ
• PDF rescaling [6,7]: P (δx, τ )=τ
0.7
0.65
4. Statistical Scaling–Methods
−α
ζ(2)
0.75
ζ(2)
• ǫ=v(B
2
0.75
−10
• Unconditioned signals have similar power spectrum, but differ in their scaling
properties when conditioned
0.7
ζ(2)
• Solar Max.(01/00−12/01): ∼ 6×10 samples of ǫ [ACE]
δε
0.75
• Solar Min.(08/95−07/97): ∼ 6×105 samples of ǫ [WIND]
−4
with µ = 1.4 and γ = 0.3.
• Mono-scaling could be concluded from the linear relation ζ(m) = αm after
conditioning, however
2
−3
∂τ P (δx, τ )=∇δx[A(δx)P (δx, τ ) + B(δx)∇δxP (δx, τ )]
5. Single moment change for conditioning
δ B2
−2
. The red line is a Lévy PDF calculated from the characteristic function as
Moment m
• Solar Min. year 1996: ∼ 4.5×105 samples of B 2 [WIND]
−1
1.5
0
0
5
−0.5
0
0.5
1
log10(δ x( τ ≈ 30 min) / σ)
0
• (Right) Probability distributions of B 2 (blue markers) and ǫ (green markers).
Dashed black line corresponds to a self-similar solution of the Fokker-Planck
equation[9]:
0.5
5
slope: 2.37 ± 0.20
δ B2
δε
• (Left) Positive tails of distribution functions for B 2 (blue markers) and ǫ
(green markers). The tail of B 2 distribution is consistent with, but not unique
to, the limiting form of a Lévy PDF, PL(|δB 2| → ∞) ∝ |δB 2|−(1+µ). This is
not as clear in the case of ǫ where only very large events exhibit power law
tail.
15
2.5
B2
3. Data Sets
−3
−4
β=1.63 ± 0.05
10
• Complex dynamics but simple statistics[1]
−2
j
15
y
y
5
Coupling mechanism between solar
wind and the magnetosphere is still
not understood.
1
−1
2. The Earth’s magnetosphere and the solar wind
2-D simulation of MHD turbulence:
magnetic field strength [5].
δ B2
δε
0
log10(σ P(δ x,τ ≈ 30 min))
1. Abstract
• Conditioning improved scaling in higher moments, rate of convergence of a
single moment discriminates between fractal and multi-fractal scaling
• Both B 2 and ǫ are approximately self-similar at solar maximum
• Scaling persists for temporal scales 5 − 100 minutes
Bibliography:
[1] M. I. Sitnov et al., JGR 105 (2000).
[2] S. C. Chapman and N. W. Watkins, Space Sci. Rev. 95 (2001).
[3] K. Kiyani, S. C. Chapman, B. Hnat, N. M. Nicol, Phys. Rev. Lett. 98, 211101 (2007).
[4] B. Hnat et al., Geophys. Res. Lett., doi:10.1029/2007GL029531 (2007).
[5] J. A. Merrifield, T. D. Arber, S. C. Chapman, R. O. Dendy, Phys. Plasmas 13, (2006).
[6] R. N. Mantegna & H. E. Stanley, Nature 376, 46–49 (1995).
[7] S. C. Chapman, B. Hnat, G. Rowlands, N. W. Watkins, NPG 12, 767-774 (2005)
[8] B. Hnat, S. C. Chapman, G. Rowlands, JGR 110 (2005).
[9] K. Kiyani, S. C. Chapman and B. Hnat, Phys. Rev. E 74, 051122 (2006).
Acknowledgements: We thank R. P. Lepping and K. Ogilvie for provision of data
from the NASA WIND spacecraft, the ACE Science Center for providing the ACE data.