Why take a systems approach to
the complexity of the
magnetosphere?
Asks questions other approaches usually don’t ...
... so gets answers other approaches won’t ...
Nick Watkins
British Antarctic Survey (NERC)
Cambridge, UK
nww@bas.ac.uk
ISSI Meeting 2012
1
Thank co-authors 1998Gary Abel (BAS)
Tom Chang (MIT)
Sandra Chapman (Warwick)
Gareth Chisham (BAS)
Dan Credgington (BAS/UCL)
Richard Dendy (Culham)
Andy Edwards (BAS/DFO)
Mervyn Freeman (BAS)
Christian Franzke (BAS)
John Greenhough (Warwick/Edinburgh)
Bogdan Hnat (Warwick)
Khurom Kiyani (Warwick)
Dave Riley (BAS/Cambridge)
Sam Rosenberg (BAS/Cambridge)
Raul Sanchez (Oak Ridge)
Sunny Tam (MIT/)
2
Some Reviews:
Vassiliadis, Rev. Geophys., 2006
Dudok de Wit, in Space Plasma Simulation, 2003
Freeman and Watkins, Science, 2002
Watkins, Nonlinear Proc. Geophys., 2002
Chapman and Watkins, Space Sci. Rev., 2001
Watkins et al., JASTP, 2001
Klimas et al., JGR, 1996
Sharma, Rev. Geophys. (Supp.), 1995
3
Why take a (complex) systems
approach ?
... & secondly, because complexity is itself a
frontier of modern science (reaching out
towards a “science of sciences” [Neil
Johnson, “Two’s Company, Three’s
Complexity”])
Not surprising: frontiers of 21st century
physics include the very large, the very small,
and the very complex (where physics meets
geophysics, biology, economics, society ...)
4
Why take a (complex) systems
approach ?
... and exploits frontiers with &
connections into other areas of
physics in a natural unforced
way ...
5
Complex, not “just”
complicated
“Warm”
Adapted from
Rudolf Treumann
Few effective variables
“Cold”
Many
independent
variables
“Hot”
“Complex systems are those with many strongly interdependent variables. This
excludes systems with only a few effective variables, the kind we meet in
elementary dynamics. It also excludes systems with many independent variables;
we learn how to deal with them in elementary statistical mechanics. Complexity
appears where coupling is important, but doesn't freeze out most degrees of
freedom”.
Cosma Shalizi , “Physics Today” [ 2005]
6
Obviously
nonspecialists still
want to know that
these answers are
useful ...
“The Guardian”
[ 2000]
... examples are
likelihood of an
extreme event
[Koons, 2001], return
time to next peak
[Choe et al., 2002],
heating effect of an
auroral index “burst”
[Uritsky et al., 2001]
and value of AE index
...
7
PDF of AE
Watkins, NPG, 2002. 10 years AE (l), 6 months (r).
Bilognormality noted in |AL| Vassiliadis et al, 1996; AE
Consolini & de Michelis 1998.
8
Illustrate only 3 techniques ...
Others at ISSI will cover state of the art
techniques, I have chosen two linear ones that
any physicist will be comfortable with: the
power spectral density (PSD) & autocorrelation
function (ACF) ...
... to preface a less familiar one. Burst (size and
duration) measures introduced into space
physics by Consolini (1997) and Takalo (1994),
& directly inspired by SOC.
At each stage will try to show how the extra info
changes our perspective- & to “join the dots
...“- will advocate more of this process
9
... & 1 complex system ...
Solar wind
Ionosphere
Magnetosphere
Example of AE: 12 magnetometers,
sensing ionospheric currents.
Ultimately energy is turbulent SW...
10
Convection
(DP2)
solar wind
magnetosphere
flow
Sun
• magnetic pole
equator
Courtesy Mervyn Freeman
 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.
Substorms
(DP1)
solar wind
BANG!
magnetosphere
 Irregular, large-scale
releases of energy in
magnetotail
-substorms.
 Intense magnetic fieldaligned currents
accelerate particles to
cause aurora.
Courtesy Mervyn Freeman
How is AE defined ?
Indices estimate auroral dissipation.
via upper (AU) & lower (AL) envelope
of magnetic perturbations from 12 magnetometers
underneath auroral electrojets. Total envelope
(AE)=AU-AL
13
Fourier spectrum S(f)
[Tsurutani]
Note absence of
the peak he was
expecting-an LCR
circuit-like
resonance on
substorm time
scales O(2 hrs)
Tsurutani et al,
GRL,1991
... and existence of
ever increasing LF
power rather than
a flat white noise
... Low pass
14
SW???
S(f) of d/dt AE
A. Pulkkinen et al, JGR, 2006
15
Modelling HF spectrum ...
Ignoring LF component, NB quite a few ways
to make an f--2 high frequency spectrum, e.g.:
A nonstationary Brownian random walk
- addition of many random variables
- f--2 all the way down
A stationary Ornstein-Uhlenbeck process
-relaxation of a white noise driven system
- LF is white, HF is f--2 , turnover set by
correlation time (aka AR(1))
A “random telegraph” [c.f. Jensen book;
Watkins et al, Fractals, 2001]
-State changes: high to low at Poisson intervals
16
Correlation structure
Example dataset: 4 days of AE by Shan, Hansen,
Goertz & Smith GRL, 1991. Search for correlations
motivated by search for low dimensional chaos-but
good system measure outlives its first application
17
Autocorrelation Function
Shan et al, op. cit.
ACF is essentially
same information as
PSD presented
differently-as
function of lag,
rather than S(f)
18
Does ACF stabilise [Takalo] ?
After Takalo & Timonen, GRL,1994
19
Limit of ACF
ACF stabilises on periods of a few weeks,
as it should because 1/f spectrum is also
repeatable ...
... BUT sum (terms of ACF) blows up as
longer and longer periods studied-classic
indicator of long range dependence(LRD) aka “Joseph effect” [Mandelbrot]
20
Long Range Dependence
ß=1
Illustrate using
fractional
Brownian
motion, here ß
is index of power
spectral slope
ß=2
Courtesy
21
Gary Abel
Models for LRD/Joseph effect:
As before can model LRD using
nonstationary but H-selfsimilar models,
e.g. fractional Brownian motion ...
... Or stationary but not exactly Hselfsimilar ones ...
... So what’s this H-selfsimilarity then ?
22
Self-similarity(affinity) and H
S. d.  of differences
grows with time
separation as return
probability P(0) shrinks
Brownian walk: “the
“normal” model of
natural fluctuations …”
Mandelbrot (1995)
23
Single exponent: H=J=1/2
σ~ J
Return pdf
~ -H
24
Courtesy Bogdan Hnat
LRD alters J (and ß), H no
longer 1/2
ß=1
ß=2
Courtesy
25
Gary Abel
There is another route to
H > ½: heavy-tailed jumps
Consolini &
De Michelis,
1998
26
“Noah”effect- heavy(ish) tails
in ∆AE [Consolini]
E.g.Chapman et al,
NPG [2004]
pdf of
AE at
15 min
Alpha-stable Lévy flight27
Courtesy Andy Edwards
Either/both LRD or heavy tails
will make simple Brownian
H=1/2 random walk unsuitable
Complexity- in time series manifests via
anomalous “burstiness”.
Time series counterpart of anomalous transport ...
....may seem like another buzzword so ...
28
COFFEE
Ballistic ~ t
?
?
bla
h
Anomalous ~ t^H
After a really good poster session, diffusive t^1/2
?!
Blah
Spatiotemporal phenomena
need spatiotemporal diagnostics
We need complementary ways of probing this sort of
data, & (in my view) NOT ONLYsimple toy models for
insight-and calibration of the diagnostics-BUT ALSO
intermediate complexity models (c.f. climate science)
AND physical theories of burstiness.
One probe is the sort of finite range scaling collapses that
Sandra is talking about ...another is burst size/duration –
I’ll talk about some toy models for understanding what
these do ...
30
Multiscaling ???
Pdf of returns
σ
Return pdf and σ scale same way in top 3 plots
(all auroral) but differently in bottom one (solar wind)
Watkins et al, Space Science Reviews [2005]
31
Could an H-selfsimilar model
still capture this ?
[standard Lévy motion, sLm] and fBm, however, are far from
exhausting the anomalies found in nature ... many
phenomena exhibit both the Noah and Joseph effects and
fail to be represented by either sLm or fBm ...
One obvious bridge, fractional Lévy motion, is interesting
mathematically, but has found no concrete use".
– Mandelbrot, 1995.
32
Linear fractional stable motion
X H  ( t )  C
1
H 




R 
H  1
(t  s ) 
 ( s)
H  1 




dL ( s )
LFSM X(t) as given in Stoev and Taqqu, Fractals [2004].
Standard model in stats less well known in physics than fBm
etc. Parameters d (“Joseph”) and α (“Noah”) both contribute
to H, the selfsimilarity exponent:
d  H 1/
Allows subdiffusive H to coexist with superdiffusive α.
We have derived kinetic equation for this in Watkins et al.,
33
(PRE,2009).
Watkins et al
[SSR, 2005]
Modelling with LFSM
34
(q) vs q plot for LFSM: ß (i.e d) varying,  fixed at 1.5
Watkins et al,
[Space Sci.
Rev., 2005]
(m) for  = 1.5 and various 
2







1.5
1
(m)
(q)
= 2.5
= 2.3
= 2.0
= 1.8
= 1.5
“Structure function”
(generalised
variogram)
= 1.3
= 1.1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
1.5
m
2
2.5
3
3.5
4
if monofractal then:
q
Demonstration of known multi-affine, “pseudo
multifractal” behaviour-extends effect in ordinary Levy motion (=2)35
Replot SVD/SSA mode
heirarchy
Sitnov et al, PRE, 2001
36
A very bold postulate …[Bak,
Tang & Wiesenfeld, ‘87-’88]
“In fact, there is one area of physics where the relation between
spatial and temporal power-law behaviour is well-established. At the
critical point for continuous phase transitions, the correlation
function for the order parameter decays spatially ... and temporally
as [power laws]
But in order to arrive at the critical point, one has to fine-tune an
external control parameter such as the temperature or pressure, in
contrast to the phenomena above [1/f noise, spatial fractals] which
occur universally without any fine-tuning.
The explanation is that open, extended, dissipative dynamical
systems may go automatically to the critical state as long as they are
driven slowly: the critical state is self-organised. We see fractals as
snapshots of systems operating at the self-organised critical (SOC)
37
state” - Bak and Chen, Physica D [1989] and “Fractals in Physics”
Avalanches as
bursts ...
BTW, PRE 1988
38
... motivated “burst” diagnostic
Courtesy Mervyn Freeman
Amplitude
Burst with integrated
“size” s taken over
blue area
Threshold
Duration T
Waiting
Time 
time
Measure designed for activity series in sandpile models
Burst size [Consolini, 1997-98]
AE burst size for 10 years (main), 1 year (inset).
“Fractals-what you see is what you wait for ...”
40
Probability density
Burst duration
T (min)
Method introduced by Takalo, 1993 (inset). Main plot
41
Riley, unpublished 2000 for AE 1978-85 (cf Consolini, ‘99)
Durations: AU, AL & SW
1978-88
1995-98
Freeman et al, GRL, 2000
42
Bump in AE family not SW
Method Consolini 1999 preprint; Freeman et. al., op. cit.
43
Poynting flux bursts @ L1
Freeman et al [PRE, 2000]
P(s)
P()
P()
P(T)
44
Bursts in random walk models
Showing work in progress: Reported
partly in Watkins et al ( PRE, 2009),
See second talk at ISSI for details
Bursts show spatiotemporal connectionsneed such a process to model them-show
LFSM-will also study multifractals e.g. pmodel (Watkins et al, PRL, 2009).
45
Adapt recent work of
Kearney and Majumdar
[2005] : simple
scaling even for
Brownian bursts
s ~ T ’,
P(s) ~ s 
with ’=3/2 and =-4/3
s

Show example “calibration”
simulations in Brownian
(=2, =2, H=0.5) case, here
s v  (top)] –steps toward
interesting size-duration scaling
seen in AE by Uritsky et al, 2001
(right)
s
It’s not all noise – new
generation of dynamical
models:
Fokker Planck: Hnat et al, GRl,
JGR, 2002-3
Langevin: Pulkkinen et al, JGR,
2006.
SDE: Anh et al, JGR, 2008
SDE: Rypdal and Rypdal, PRE,
2008
Integrate & fire : FreemanMorley, GRL.
47
Conclusions
Complexity nothing to be scared of-on contrary need
for it can be seen even in investigations of early 1990s
using very familiar methods-PSD and ACF.
Results show evidence of spatiotemporal bursts and
scaling-motivate the use of newer diagnostics.
We are using simple models of burstiness to unpick what
has already been seen and help join the dots-showed you
some preliminary results.
Will feed into dynamical models.
Complements other methods you have seen/will see.
48
AL conditional PDF
49
Ukhorskiy et al
Scaling in AE: Activity-lifetime
Uritsky et al., [GRL]
have measured the dependence of
N( )= <AE(t+  )> -L
a time averaged activity measure
And P() = n ( )/ m
A survival probability (with the n
the number of events with
duration T >  )
on averaging time 
Repeated for solar wind,
(different) scaling relations
found
What does AE measure ?
DP2
DP1
Kamide and Baumjohann, MagnetosphereIonosphere Coupling, p.156 [Springer, 1991]
LFSM: where’s the physics ?
52
LFSM: where’s the physics ?
Extremal models map onto to LFSM !
53
H from low dimensionality ?
Evidence from SVD
Sharma et al, GRL, 1993
54
Problems-LD disappears when
phases randomised ???
Prichard & Price, GRL, 1993
55