Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Mutual information as a measure of spatial
correlation properties of the turbulent solar wind as
seen by WIND and ACE.
R. T. Wicks, S. C. Chapman, R. O. Dendy
Centre for Fusion, Space and Astrophysics
University of Warwick
Coventry
UK
r.wicks@warwick.ac.uk
Thanks to the WIND and ACE magnetometer and SWE teams
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
1
2
3
4
5
6
Motivation
Mutual Information
Theory
Histograms
Spacecraft
ACE and WIND
Solar Wind: CMEs
Data
MI of Lagged Timeseries
Solar Wind: Quiet
Turbulent Data
Mean Correlation
Detailed Correlation
Anisotropy
Conclusions
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Motivation
Multiple spacecraft measurements are becoming more available,
giving us a new insight into the spatial structure of Solar wind
turbulence.
Spacecraft give sparse measurements in a continuous field.
Understanding the spatio-temporal structure of turbulence in the
solar wind is difficult from this.
Particularly as the study of turbulence requires the correlation
structure of these strongly non-linear time series.
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Theory
Histograms
Theory
Developed by Shannon in 1949
Mutual information (MI) is a measure of shared entropy between
two signals (X and Y )
I (X ; Y ) = H(X ) + H(Y ) − H(X , Y )
H(X ) = −
P
i
P(Xi )log2 (P(Xi ))
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Theory
Histograms
Histograms
50
0.04
40
0.03
30
P(θ)
Raw
data
X
Probabilities P(X ), P(Θ) and P(X , Θ) estimated by frequency of
occurrence
20
0.02
0.01
10
0
−3 −2 −1
0
θ
1
2
0
−3 −2 −1
3
0.06
0
θ
1
2
3
50
40
0.04
X
P(X)
X
position
histogram
Θ value
histogram
30
20
0.02
10
0
0
10
20
X
30
40
50
−3 −2 −1
0
θ
1
2
3
Joint
probability
distribution
P(X , Θ)
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
ACE and WIND
ACE and WIND
150
100
Y (Re)
50
0
−50
−100
−150
0
50
100
150
200
250
300
X (Re)
30
20
In 2005 and 2006 both ACE and WIND
are in halo orbits around L1
Z (Re)
10
0
−10
−20
−30
0
50
100
150
200
250
300
X (Re)
This gives a good range of scales and
orientations with a large separation
(from under 10 Re to over 100 Re)
30
20
Z (Re)
10
0
−10
−20
−30
−150
−100
−50
0
Y (Re)
50
100
150
GSE coordinates
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Data
MI of Lagged Timeseries
CME Data
|B| (nT)
20
CME as measured by
ACE and WIND
ACE
WIND
15
10
5
0
130
131
132
133
134
135
136
|b| (nT)
3
|b| = |B|t −|B|1hour
1.5
0
−1.5
−3
130
131
132
133
Days of 2001
134
135
136
Note large excursion in
|B| and accompanying
wave train.
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Data
MI of Lagged Timeseries
MI of Lagged Timeseries
Mutual information of a
CME as measured
between ACE and WIND
I (|B|ACE ; |B|WIND )
4 minute cadence data,
400 points used for MI
≈ 1.11 days.
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Turbulent Data
Mean Correlation
Detailed Correlation
Anisotropy
Turbulent Data
ACE and WIND data
MI calculated with lag
Min MI 0.0001
Max MI 1.42
MI gives indication of
typical coherence length
within solar wind
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Turbulent Data
Mean Correlation
Detailed Correlation
Anisotropy
Mean Correlation Coefficient
Mean Correlation
0.8
0.6
0.4
0.2
0
−0.2
−180
0.8
−120
−60
0
60
120
180
|B|
Bx
By
Mean MI
Bz
0.6
0.4
0.2
−180
|B| peak smaller
than components
→ Alfvénic
turbulence?
−120
−60
0
60
Lag (minutes)
MI peaks sharper
than linear
correlation.
120
180
MI and linear
correlation agree
on time lag and
ordering of the
data.
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Turbulent Data
Mean Correlation
Detailed Correlation
Anisotropy
Detailed Correlation
Linear Correlation
identifies the convecting
structures.
Distinct signature in MI.
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Turbulent Data
Mean Correlation
Detailed Correlation
Anisotropy
Anisotropy
All components show
anisotropy. Anisotropic
B
0
0
25
0
0
50
By
50
S
6
Bx
50
25
25
r⊥ (104 km)
peak in MI most clearly
seen in By and Bz .
|B|
50
25
50
Bz
50
25
rk
0
0
r⊥
25
25
0
0
50
4
r|| (10 km)
25
50
Motivation
Mutual Information
Spacecraft
Solar Wind: CMEs
Solar Wind: Quiet
Conclusions
Data from two spacecraft, ACE and WIND, has been analysed
using mutual information for the first time.
Mutual information has been demonstrated on large scale
structures (CMEs) that convect past both spacecraft.
On average both MI and linear correlation find large scale
convecting structures, although MI is more precise.
When viewed in more detail, MI reveals distinct fine scale
structure, reflecting nonlinear structure/dynamics.
Preliminary results on vector information suggest anisotropy
(anisotropic SW turbulence?).