1.
Nick Watkins
British Antarctic Survey,
Cambridge, UK
2011, Tuesday, 12th July
Complexity, extreme bursts, and volatility
bunching in solar-terrestrial physics
2.
1. BAS/Warwick research on multiscale complexity in Earth’s
magnetosphere began ~15 years ago [e.g. Chapman et al, GRL, 1998;
Watkins et al, GRL,1999; Freeman et al, PRE, 2000]. Within this my
personal focus has been on self-similar and multifractal time series
models.
2. For sigma phi audience & this workshop, highlight work in progress on i)
temporal scaling of bursts above threshold in monofractal time series
[Carbone & Stanley, PRE, 2004; Watkins et al, PRE, 2009], and ii) a
multifractal feature, “volatility clustering”. Show that some space physics
time series share this property, well known in some financial ones [see
also Engle Nobel lecture, Mantegna & Stanley book; Rypdal & Rypdal,
JGR, 2011]. Talk about a simple linear stochastic model, the Kesten
process studied intensively by Sornette. Advocate use of this toy model
for framing “null” hypotheses about volatility bunching.
3. Relevance goes beyond solar-terrestrial physics to broader issue of
model choice and diagnostics for complex systems in complex
environments which may be neither weakly coupled or slowly driven
[Freeman & Watkins, Science, 2002].
3.
Thank many colleagues including:
Tim Graves (Cambridge), Dan Credgington (Now UCL) ,
Sam Rosenberg (Now Barclays Capital),
Christian Franzke (BAS), Bogdan Hnat (Warwick),
Sandra Chapman (Warwick), Nicola Longden (BAS),
Mervyn Freeman (BAS),
Bobby Gramacy (Chicago)
Watkins et al, Space Sci. Rev., 121, 271-284 (2005)
Watkins et al, Phys. Rev. E 79, 041124 (2009a)
Watkins et al, Phys. Rev. Lett. , 103, 039501 (2009b)
Watkins et al, submitted to AGU Hyderabad Chapman Conference Proceedings
Franzke et al, submitted Phil. Trans. Roy. Soc., arXiv:1101.5018
4.
“Standard model” of Solar
Terrestrial Physics
Solar wind
Magnetosphere
Ionosphere
• Reconnection-driven plasma convection-”loading”
• Magnetospheric substorms-”unloading”
5.
Convection (DP2)
• Mass, momentum and
energy input from
reconnection at solar
wind - magnetosphere
interface.
• Plasma circulation from
day to night over poles
and from night to day
around flanks.
• magnetic pole
equator
Sun
flow
solar wind
magnetosphere
6.
Substorms (DP1)
• Irregular, large-scale
releases of energy in
magnetotail
-substorms.
• Intense magnetic field-
aligned currents
accelerate particles to
cause aurora.
solar wind
magnetosphere
BANG!
7.
Multiscale magnetosphere ...
Solar wind
Magnetosphere
Ionosphere
Data
Heavy tailed pdf
of size of bursts
above threshold
in AE auroral index
Tsurutani et al (1990) left, and Consolini (1997,98) above:
drew attention to multiscale behaviour in 1D auroral time series
Used a “burst” diagnostic derived from SOC.
Reviews incl. Freeman & Watkins, 2002; Watkins et al, 2001.
Averaged spectrogram
of AE-”1/f” at low freq.
“Burst”
8.
Often forgotten (or not realised) that Bak et
al’s aim was to unify heavy tails in amplitude
with “1/f” noise in time, via a physics-inspired
model.
The physical inspiration for SOC just
happened to be from condensed matter, not
from solar terrestrial physics ... & v 1.0 of the
model didn’t produce 1/f noise in output ...
Why an SOC approach?
9.
So a question, 1997-98, was …
22 April 2014 9
Does SOC apply to magnetospheric energy release events ?
[Consolini 1997; Chapman et al, 1998; Uritsky & Pudovkin, 1998 ] ?
Lui et al, GRL, 2000
10.
The joy of fractals ...
• "It makes me so happy. To be at the beginning
again, knowing almost nothing...a door like
this has cracked open five or six times since
we got up on our hind legs. Its the best
possible time to be alive”
– Tom Stoppard, Arcadia
11.
Scenario: “… the internal relaxations of the magnetosphere
statistically follow power laws that have the same index independent
of the overall level of activity, and that both the internal and global
events are consistent with the behaviour of a finite size avalanche
model. ...
… The onset of local avalanches in the sandpile model can be
physically related to the merging of coherent structures around
Alfvenic resonances [Chang, 1998, 1999] or current disruption by
kinetic instabilities [Lui, 1996] in the magnetotail”.
To which might add multi site reconnection, made more explicit by
Klimas et al, 2000
The SOC paradigm, led Lui, Chapman et al [GRL, 2000]; to
study spatial “blobs” defined by thresholding in UVI images “…
using the global auroral energy deposition as measure of the
energy output of themagnetosphere”.
12.
....spatial signals
Integrated power in Polar UVI
blobs exceeding a brightness
threshold----subdivided into
substorm, quiet time and
pseudobreakup.
Lui, NPG, 2002
Prediction in Chapman
et al, GRL, 1998; Watkins
et al GRL, 1999.
Test in Lui, Chapman et al,
GRL, 2000
13.
22 April 2014 13
Lui, Chapman et al,
GRL, 2000
Continuing Question: What would
magnetotail exhibiting multisite
reconnection [e.g. Daughton et al,
2011] look like in ionosphere ?
Uritsky et al, JGR, 2002
(& Freeman & Watkins
Science commentary),
and their recent papers
14.
log s
log
P(T)
log
P()
logT
log
Poynting flux in solar wind plasma from NASA Wind
Spacecraft at Earth-Sun L1 point Freeman, Watkins & Riley
[PRE, 2000]. Dialogue on this topic is one of several directions
research area has proceeded along post 2002 (c.f. Rypdal &
Rypdal, JGR, 2010b).
log
P(s) size
length
waiting time
Ambiguity: magnetosphere non-
autonomous, what about driver ?
Bursts seen in solar
wind Poynting flux
15.
Dealing with ambiguity
Difficulty of attributing complex astrophysical phenomena
uniquely to SOC has led me to back up one stage, and to get
interested in the known models for non-Gaussian, temporally
correlated stochastic processes. Partly to try and see what
physics was embodied in any given choice, partly for
“calibration” of the measurement tools. [e.g. Watkins NPG,
2002; Watkins et al, SSR, 2005; PRL, 2009]
In similar sense to Eliazar & Klafter’s work the models go
beyond the CLT. They do not embody general “laws”, but map
out a range of widely observed “tendencies”. We have become
particularly interested in Mandelbrot’s models and their close
relatives [e.g. Watkins et al, PRE, 2009].
16.
4 “giant leaps” made by Mandelbrot between
1963 and 1974---”well known” but history is informative
1. BBM remarks heavy tailed fluctuations in 1963 in cotton
prices---applies alpha-stable model & self-similarity idea
2. BBM hears about River Nile and “Hurst effect”. Initially (see
his Selecta) believes this will also be explained by heavy tails.
But when sees that fluctuations are ~ Gaussian
applies self-similarity [Comptes Rendus,1965] in form of a
long range dependent (lrd) model, the roots of fBm. BBM’s classic series of papers
with Van Ness and Wallis (68-69) on fBm in maths & hydrology literatures.
3. BBM demonstrates a new self-similar model, fractional hyperbolic motion, in
1969 paper with Wallis on “robustness” [sic] of R/S. Combines 1 & 2 (heavy tails & lrd).
4. BBM becomes dissatisfied with purely self-similar models, develops multifractal
cascade, initially in context of turbulence [JFM, 1974]. Later applications
of multifractal models include finance.
17.
“Noah effect”- e.g. Lévy
flights where < 2
increases tail fatness
=1
e.g. Hnat et al, NPG [2004]
=2
“Levy flight”: applied to magnetometer data by Consolini
Black line is AE differenced at ~ 15 minutes
1. BBM observes heavy tailed fluctuations in 1963 in
cotton prices--- alpha-stable model , self-similarity
idea
18.
“Joseph effect”-e.g. fractional
Brownian (fBm) walk: steepness
of log(psd) with log(f) increases
with memory parameter d
d=-1/2
d=0
S(f) ~ f-2(1+d)
Fractional Brownian motion
model: applied to AE by Takalo
and Timonen, 1994 et seq.
2. BBM hears about River Nile and “Hurst
effect”. Initially (see his Selecta) believes
this will be explained by heavy tails,
but when he sees that fluctuations are ~
Gaussian applies self-similarity [Comptes
Rendus1965] in the form of a
long range dependent (lrd)
model, roots of fractional Brownian
motion.
BBM’s classic series of papers on
fBm in mathematical &
hydrological literature with
Van Ness and Wallis in 1968-1969.
19.
1 1
1
( ) ( ) ( ) ( )
H H
H H R
X t C t s s dL s
H = d+1/α: allows
H “subdiffusive” (i.e. < ½) while
α “superdiffusive” (i.e. <2).
Memory kernel:
Joseph effect
α-stable jump:
Noah effect
LFSM of today is a stable successor to
Mandelbrot’s model
3. BBM demonstrates a new self-similar model, fractional hyperbolic
motion, in 1969 paper with Wallis on robustness of R/S. Combines
effects 1 & 2 (heavy tails & lrd).
Nowadays would use linear fractional
stable motion---LFSM, applied in space
plasmas by Watkins et al, 2005:
NB H here is self-similarity exponent
not identical to “Hurst” exponent
except in Gaussian alpha=2 case
20.
1D spreading exponent
• Burst diagnostics previously proposed include
1D version of “spreading” exponent used by
Uritsky et al, GRL, 2001 [c.f. book by Marro &
Dickman].
• Took ensemble time average as function of
time of activity AE of a curve after it has
crossed a threshold at time t.
N*() = <AE(t+ )> - L
21.
Modelling bursts
Has potential wider application to prediction of
“typical” burst size in fractal time series.
Reported scaling of N*()
as to the
25.
Surviving activity only
Have repeated
with fBm, LFSM,
26.
Noah
Meneveau & Srinivasan p-model
4. BBM becomes dissatisfied with purely monofractal models, develops
multifractal cascade, initially in context of turbulence, JFM,1974.
Applied to fluctuations of AE index by
Consolini et al, 1996.
Later multifractal applications studied by
Mandelbrot included finance in late 1990s.
See also Ghashgaie et al, Nature, 1996 who
used multifractal Castaing pdf, and interesting
debate about alpha-stable versus multifractal
models between them and Mantegna &
Stanley, Nature, 1998.
27.
Noah
Natural examples include ionospheric AE index (above), & ice cores (e.g.
Davidsen and Griffin, PRE , 2009), . Rypdal & Rypdal, 2010-11 noted
that effect not seen in monofractal models like LFSM
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10
4
-600
-400
-200
0
200
400
600
increments,r
First differences of AE index January-June 1979
-100 -80 -60 -40 -20 0 20 40 60 80 100
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
lagacf
AE data: acf of returns
-100 -80 -60 -40 -20 0 20 40 60 80 100
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
lag
acf
AE data: acf of squared returns
First differenced AE data
ACF of diff. AE
ACF of (diff. Ae squared)
Why did BBM become dissatisfied ? Partly his eyes told him to:
One effect multifractals capture is “volatility clustering”
28.
Volatility clustering in AE
10
0
10
1
10
2
10
3
10
-3
10
-2
10
-1
10
0
lag
acf
AE data: acf of squared returns
ACF of (diff. Ae squared) for 20 000 minutes after
1979 1st Jan
Time lags up to 1000 minutes
29.
The Kesten process
0 2 4 6 8 10 12
x 10
5
-40
-30
-20
-10
0
10
20
30
40
-normalised Kesten process
x(t)/(x)
0 2 4 6 8 10 12
x 10
5
-100
-80
-60
-40
-20
0
20
40
Walk y made from summing Kesten process x
y(t)=(x)
X(n+1) = A X(n) +B
Where A and B both iid
Normal, <A>,<B>=0,
s d A =0.8, s d B =0.05
Parameter and
distribution choices give
wide range of behaviour.
X
Y= cum. sum of X
Generalises X(n+1)
= λ X(n) +ξ, the AR(1)
Process, to case where
correlation time varies.
30.
Multifractality
-30 -20 -10 0 10 20 30
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Rescaling y() by ()
( y - < y>)/
()P(y(t,))
=1
=10
=100
=1000
-1 0 1 2 3 4 5 6
-1
-0.5
0
0.5
1
1.5
2
2.5
3
(m)
m
Difference pdfs of walk y do not collapse,
instead change shape.
Curvature in function zeta (m),
exponents of mth order structure
functions versus m, indicates
multifractality
31.
But “mild” volatility clustering ?
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
-2
10
0
10
2
10
4
FFT of Kesten process x
RawPSD
0 10 20 30 40 50 60 70 80 90 100
-5000
0
5000
10000
ACF of the Kesten process x itself (NOT y)
ACF()
10
0
10
1
10
2
10
3
10
1
10
2
10
3
Log log plot of ACF of the square of Kesten process x
ACF()
Decay of ACF of square of X is
slower than exponential, but
finite ranged
32.
Conclusion
• In 1D spreading exponents governed by H [Watkins et
al., Chapman Conference Proceedings submitted,
2011]. Further generalisation to multifractals underway.
• Volatility bunching, in sense of correlation of absolute
values of time series, seen in auroral energy dissipation
data [Watkins et al., op cit; Rypdal & Rypdal, JGR,
2011].
• Linear Kesten process shows “weak” volatility bunching
33.
The Bohr Atom
• “The Bohr model of the atom ... was wrong,
yet it turned out to be fruitful.”
– Gene Stanley, Nature 2008
Rydberg formula
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