Nick WatkinsBritish Antarctic Survey,

Cambridge, UK

2011, Tuesday, 12th July

Complexity, extreme bursts, and volatility

bunching in solar-terrestrial physics

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].

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

“Standard model” of Solar Terrestrial Physics

Solar wind

Magnetosphere

Ionosphere

• Reconnection-driven plasma convection-”loading”

• Magnetospheric substorms-”unloading”

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

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!

Multiscale magnetosphere ...

Solar wind

Magnetosphere

IonosphereData

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”

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?

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

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

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”.

....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

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

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

Dealing with ambiguityDifficulty 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].

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.

“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

“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.

1 11( ) ( ) ( ) ( )

H H

H HR

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

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

Modelling bursts

Has potential wider application to prediction of “typical” burst size in fractal time series.

Reported scaling of N*()

as to the

Brownian walkers

Brownian “upstarts”

Brownian “survivors”

Surviving activity only

Have repeated

with fBm, LFSM,

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 multifractalmodels between them and Mantegna & Stanley, Nature, 1998.

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 104

-600

-400

-200

0

200

400

600

incre

ments

, 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”

Volatility clustering in AE

100

101

102

103

10-3

10-2

10-1

100

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

The Kesten process

0 2 4 6 8 10 12

x 105

-40

-30

-20

-10

0

10

20

30

40-normalised Kesten process

x(t

)/

(x)

0 2 4 6 8 10 12

x 105

-100

-80

-60

-40

-20

0

20

40Walk 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.

Multifractality

-30 -20 -10 0 10 20 3010

-6

10-5

10-4

10-3

10-2

10-1

100

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

But “mild” volatility clustering ?

100

101

102

103

104

105

106

10-2

100

102

104

FFT of Kesten process x

Raw

PS

D

0 10 20 30 40 50 60 70 80 90 100-5000

0

5000

10000ACF of the Kesten process x itself (NOT y)

AC

F(

)

100

101

102

103

101

102

103

Log log plot of ACF of the square of Kesten process x

AC

F(

)

Decay of ACF of square of X is

slower than exponential, but

finite ranged

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

The Bohr Atom

• “The Bohr model of the atom ... was wrong, yet it turned out to be fruitful.”

– Gene Stanley, Nature 2008

Rydberg formula