GEOPHYSICAL RESEARCH LETTERS, VOL. 26, NO. 16, PAGES 2617-2620, AUGUST 15, 1999
Robustness of collective behaviour in strongly driven
avalanche models' magnetospheric implications
N. W. Watkins•'2, S.C. Chapman
2, R. 0. Dendy3'2,G. Rowlands
2
Abstract. The hypothesisof self-organisedcritical- 1987;Lu, 1995]. Observationalmotivationis provided
ity (SOC) predictsthat certainopen dissipativesys- by power law features of magnetosphericindex data,
tems evolve to a critical state where all energy release notably AE which is an indicator of energy dissipated
statistics display power law distributions for event oc- by the magnetosphere
into the ionosphere.A significant
currence, size and duration. This has motivated "sand- result [Chapmanet al., 1998]is that a classof sandpile
pile" simulationsof magnetospheric
energyconfinement
andreleaseevents("avalanches"),
previousexamples
of
which have taken the limit whereenergyinflow ("fuelling") is slow relative to dissipation,and either uniform or random. However the magnetosphericsystem
has both slow and fast periods mixed together in observations,and naturally modulated fuelling. We have
models [Dendyand Helander,1998],seealso [Pinho
and Andfade,1998],yieldsystemwide
avalanches
where
the statisticshavea well definedmean(intrinsicscale)
whereas the internal avalanchestatistics are scale free;
analogousto the differing characteristicsof internal and
external magnetosphericenergyreleaseevents.
A limitation of standard sandpile models is that their
developedan avalanchemodel with variable, modulated fuelling, which would correspondin principle to magnefuelling rate. The power law form for the distribution tosphericloading, is assumedto be slow. This raisesa
of energy releaseevents is the least ambiguouscurrent central question. Given that in reality magnetospheric
indicator of SOC; we showthat this is preservedfor the loading can be strong, and fluctuate substantially, to
large avalanchesin such a system under both constant what extent doesstandard sandpilephenomenologyperand varying loading and sosuchsystemsare remarkably sist? Does the sandpile(or magnetosphere)
act as a
efficient at eliminating small scale information about nonlinear filter, with the observedavalanchesretaining
their fuelling.
magnitude and frequency information contained in the
fuelling(or loading)process?
Tsurutaniet al. [1990]describeda brokenpowerlaw
Introduction
AE spectrum; this has been claimed as evidenceof SOC
Most investigationsof the magnetosphericsubstorm [Consolini,1997; Uritsky and Pudovkin,1998], but a
problem have focusedon single substorm episodesor power law spectrum for AE cannot be uniquely interevent groups. An alternative is to use long-term statis- preted as the output of an SOC system becauseother
tics of magnetosphericenergyreleaseto infer constraints types of physical system can also produce scale free
on the underlying physical processes. This has usu- spectra. Furthermore AE is a compound index in that
ally required the definition of substorm "events" via a it mixes driven and unloading effects, a fact reflected
thresholdingprocess. There is, however,growing in- both by its powerspectrum [Freemanet al., 1998]and
terest in relating the characteristicsof global energy structurefunction [Takaloand Timonen,1998]. One
transport in the solar wind-magnetosphere-ionosphere may alsodefineindividual energyreleaseevents,and exsystemto "sandpilemodels" [Consolini,1997;Dendy amine the statistics of their distribution; the signature
and Helander, 1998; Chapman et al., 1998; Uritsky of an SOC system would be a "power law" probabil-
and Purlorkin,1998],whichdissipateenergyby means
o•f avalanches. This followsthe suggestion [Chang,
1992; Chang,1998a;Chang,1998b]that the magnetosphereis in a self organisedcritical (SOC) state. In
ity densityfor this quantity. Consolini[personalcommunication,1998] usedAE data taken over a 10 year
period to constructthe probabilitydensityD(s) of a
burst measure s; the density obtained was fitted by an
cutoffpowerlawD(s) ~ (s-•)e-•//• with
sandpile models, the distribution of avalanchemagni- exponentially
tude and frequencymay displayscalefree, inversepower c• = 1.23q-0.01 and/• = (1.17q-0.01)x 106,extendlaw statisticsthat are characteristicof SOC [Bak et al., ing the result obtained for the year 1978 by Consolini,
[1997].A genericfeatureof thistypeof statisticalexper2Spaceand Astrophysics,University of Warwick, UK
imental evidenceis that long runs of data are required,
and in the magnetosphericsystemit is unavoidablethat
3EURATOM/UKAEA FusionAssociation,Culham,UK
both the instantaneous
XBritish Antarctic Survey, Cambridge, UK
value and the recent mean of the
loading rate will have strong variation. Another class
of evidence is derived from in situ data. Here, long
runs of data are not available, but a given region in
the magnetospherecan be observedrepeatedly under
Copyright1999by the AmericanGeophysical
Union.
Papernumber1999GL900586.
0094-8276/99/1999GL900586505.00
2617
2618
WATKINS
ET AL.'
ROBUST
COLLECTIVE
BEHAVIOUR
IN STRONGLY
DRIVEN
SANDPILES
different conditions. Of particular interest as evidence by O•j_l, the next nearestneighhouris multipliedby
for avalanche-typeeventsare bursty bulk flows [Angelopoulos
et al., 1996]in the geomagnetic
tail.
c•j_2c•j_•,and soon. Sincec•xc•2...c•k
_• c•xc•...c•k_l•_
c•c•...c•_• and similarly for higher powers in c•i, it
follows that the relative importance of • g • in depower law statistics still emergefrom sandpilemodels termining the size of the avalanchediminishesas the
that have substantially varying loading, they remain scalej of the avalancheincreases.We consideronly the
useful as a characteristicsignatureto test for in energy class of sandpile models which have scale free power
releaseevent time series. The appearanceof power law law avalanche statistics in the limit of small but nonThis returns
us to the central
issue of robustness.
If
(scalefree) statisticsin suchdata wouldthen support zero • g • ("quiet time"). To give suchstatistics
the hypothesisthat the underlying physicsremained II•__-•c•i
• g • mustdecline
sufficiently
rapidly
with
governed by critical gradients leading to local redis- j for small oscillations
in g (with • g • small) to be
tribution generatingavalancheson all scales. A lim- effectivelydamped by the sandpile.
iting caseof the sandpilealgorithm used in this paper
[DendyandHelandev,1998]hasrecentlybeenshownto
displayself-similarproperties [Helanderet al., 1999]. Sandpile with strong driving
In general, the conditions necessaryfor SOC to oc-
Let us considerthe effectsof rapid fuellingon a physi-
cur constrainthe inflowrate (see [Lu, 1995]for some
proposedcriteria); to emphasise
this Jensen,[1998] cal systemin whichenergytransport and dissipationare
has proposeda category of slowly-driven,interaction- causedonly by avalanches.If the systemdisplaysSOC
over some dynamic range, the event statistics within
dominated,thresholded(SDIDT) systems.The slowly
driven condition is of particular interest in the magne- that range will be of power law form. There are three
characteristic timescalesimplicit in any sandpile algotospheric context becausemost SOC simulationshave
rithm for such a system' the relaxation time rr over
been conducted in the limit where the rate of inflow is
"slow" relative to dissipation. In this paper we pose which an avalanchetakes place; the average time rethequestion:
howrobustaremagnetospherically
rele- quired, followingan avalanche,for instability to recur
vant sandpilemodelsto changesin the inflow ?
at the first cell (we assumecentralfuellingfor simplicity), ru; and the time for oneiterationof the algorithm,
Scale free behaviour under slow, finite
loading
At. The amount of sand added at cell 1 per timestep is
again g, which may be constant, modulatedover time,
or drawn from some random distribution; its ensemble averagedvalue is < g >. Followingthe physically
In a typical sandpilealgorithm,redistributionof sand
at cell j is triggeredif the local gradientzj exceedsa
critical gradient zc. Redistributionmay in turn cause
the neighbouringcell to becomeunstable, triggering
further redistribution and, by iteration, an avalanche.
Avalanchescan be classifiedas internal, meaning rearrangementsof sandwithin the pile, or systemwidewhere
the entire sandpilereturns to its angleof reposeand excesssand actually leavesthe system. In the model con-
sideredin [Chapmanet al., 1998]the valuesfor the critical gradientsat eachcell weredrawnfrom a probability
motivatedargumentsof Dendyand Helander[1998]
and
Chapmanet a/.[1998]we also assumethat the critical
gradient Zcrit for any grid cell and time step is drawn
from some probability distributionP(zc) with mean
< zc >. In the limit whereP(zc) is a delta function,
< Zc > becomesthe "standard" uniform sandpile-wide
Zcrit. It is clear that ru -•< Zc > / < g > and hence,
for instantaneously
relaxingsandpilemodels( •-• (( At
) we identifyslowand fast loadingregimes:
(3)
distribution. The sand added to trigger an avalancheis
spread by the avalancheover j cells, at each of which
fast ß•-• (( • • At
(4)
the averageheightof sandhas increased.Let hj,o and respectively. In the latter case, instability is likely to
hj,x denotethe heightsat cellj immediatelybeforeand be triggered at eachtimestep. We reiterate, that in the
after the avalanche respectively, and < g > the mean
literature,the orderingof time scalesgivenby (3) has
(ensembleaverage)fuellingrate. The amountof sand often been cited as a condition for SOC rather than one
then transferred
to the second cell can be written
c•l(hl,0- h2,0-]-• g •)
(1)
of severalregimesin which aspectsof SOC behaviour
may be observed.
Normalising At to unity, that is one timestep per in
unit time, the fast loading condition becomes• g ••
Zc•. For a givenloadingrate, there will be an effective
o•2{h2,o
- h3,o-]--o•1(hl,O
- h2,o--{-< # >))
(2) minimum avalanchelength neededto dissipatethe energy associatedwith the sand added in eachAt. In the
is transferred to the third cell, where c•2 < 1, and so slow limit this correspondsto lessthan one cell, in the
on. For the final transfer to the jth cell, it followsthat fast limit to many. If the mean fuelling rate • g • is inj-1
< g > is multipliedby IIi= xc•i, whereasthe nearest- creasedtowards and beyond • Zc •, the smallest scale
neighboutheightdifferencehj-l,o- hj,o is multiplied avalancheswill be increasinglyeliminated. This will be
where c•x< i by construction. Similarly, an amount
WATKINS
ET AL.'
ROBUST
COLLECTIVE
BEHAVIOUR
IN STRONGLY
DRIVEN
SANDPILES
2619
reflectedin the lower bound of any rangeof "powerlaw"
event statistics for energy releasein the system.
lO-1
Sandpile with variable driving
o
We now examine how far information concerning
the modulation or fluctuation of g is transmitted in
avalanches,and whether its consequencesmay become
visiblein the eventstatistics("the sandpileas filter").
Insofar as fluctuations in g take the sandpile intermittently into the fast driving regime, the commentsin the
precedingsection apply. In any event, there will exist
a maximumavalanche
scale(numberof cellsn/) above
o
• 10'2
o
00
e
0
0
•
O0
lO-3
ß
0o
0o
o oø
10-4
10'
i
102
i
103
i
104
i
106
**
106
,•E
which the effect of fluctuations in g becomesvanishingly small in the avalanchestatistics. We then have
Figure 1. Normalisedhistogramsof probabilityof energy releaseof sizeAE for all (o) and systemwideonly
two possiblecases. The first is that n! is sufficiently (*) avalanches
for a 5000 cell sandpilewith constant
large for there to be a detectablesignaturein the inter-
fuelling g - 10.
nal statistics(disruptionof the powerlaw), but is still
muchsmallerthan the systemsize. The secondhasn! tem, with g: 0.01. Here we presentresultsfor a larger
of order the system size, in which casethe event statis- grid of 5000 cells,with P(z•) a "top hat" distribution
ticsof bothinternalandexternal(or systemwide)
events covering[0.5,1.5] (seealso [Chapmanet al., 1999]).
reveal a signaturereflectingboth the mean level and the The normalisedhistogramsof probability of energy refluctuation spectrum of g. In the latter casewe expect leaseevents of size AE in the plots shownhere are coma sandpilemodel to yield goodcorrelationbetweenfluc- prised of over 150,000 internal avalanchesand 20,000
tuationsin the inflowand in the external(systemwide) systemwide avalanches. In Figure i we show results
timeseriesin the extremely strongly driven limit.
for constant g: 10. The systemwide event statistics
remain unaffected by the fast loading, illustrating the
Results
and discussion
fact that large avalanchesare "well insulated" against
the fuelling process;howeverthe internal avalanchedisAs a concreteexample of the preceedingarguments tributiondisplaysa dropoutin the numberof the smallwe modified the algorithm usedby Dendy and Helander est events. In this constant loading rate case, there is a
[1998]and Chapmanet al. [1998]to incorporatevari- sharp cutoff at AE -• 100 and eventsbelow this size are
able fuelling. We take a one dimensional,unit spaced of low probability. Above this, the emergent power law
grid of N cells.Sandheightis givenby hi, sothe local form of the energy releasedistribution is preservedup
gradientbecomeszj - hj - hj+l. The pile is assumed to AE of 105. This behaviourmay be comparedwith
stableat the angleof repose.Both hj, zj arethenmea- Figure 2 in which the input fuelling rate has been varsured from this stable state. Each jth cell has a critical
ied. The mean < g > is still 10 but each value of g is
gradientzcj. In commonwith all otheravalanchemod- drawn from a uniform randomdistribution, in the range
els the model of Dendy and Helander[1998]has a fu- [5, 15]. We seethat the effectis to "soften"the peak
elling rule, an instability rule and a redistribution rule, at AE of 100, and to leave some avalanches down to
which together form a repeating cycle. Fuelling occurs
by adding sand at the first cell. We normaliseL and T
to the averageloadingrate g so that unit volumesand
is addedin unit time, (i.e. after 1/g iterations).The
conservative
redistributionrules [Dendyand Helander,
1998;Chapmanet al., 1998]aresuchthat "avalanches"
i.e. redistribution events,occur on all scalesup to the
size of the pile.
lO-1
ooOO
0øø
..
o
o
o
o
I•10-2
o
o
Each avalanchereleasesenergy(but not necessarily
sand)from the sandpile
o
o
o
oo
lO-3
N
O'
N
o
o
o
o
j--1
j-1
OoO
lO-4
lO'
i
i
lOt
lO3
,
i
i
lO4
lO5
lO6
The extent to which the loading rate g is "fast" or
"slow" can be measuredby comparingg to the mean Figure 2.
Normalisedhistogramsof probability of
valueof zcfrom (3) and(4). Wehavepreviously
studied energyreleaseof sizeAE for all (o) andsystemwide
only
[Chapmanet al., 1998]avalanche
statisticsunderslow (*) avalanches
for a 5000 cell sandpilewith randomly
(g << < zc >) constantloadingrate, for a 500cellsys- fluctuatingfuellingin the range[5,15] and < g >= 10.
2620
WATKINS ET AL.: ROBUST COLLECTIVE
BEHAVIOUR
IN STRONGLY DRIVEN SANDPILES
AE = 10. We interpret this as being due to the range Chang T. S., Sporadic localized reconnectionsand mulriscale intermittent turbulence in the magnetotail, in
of possiblefuelling rates, which means that the smallGeospaceMass and Energy Flow: Results[rom the Inest avalanchesare no longer automatically ruled out in
ternational Solar Terrestrial Physics Program, ed. J. L.
a giveniteration (i.e. for someiterations# is relatively
small).
Conclusions
Horwitz, D. L. Gallagherand W. K. Peterson,Geophysical Monograph104, AmericanGeophysical
Union, Washington, D.C., 193-199, 1998b.
Chapman S.C., N. W. Watkins, R. O. Dendy, P. Helander
and G. Rowlands, A simple avalanche model as an ana-
loguefor magnetospheric
activity, Geophys.Res. Lett., 25,
Overlongperiods,the magnetospheric
driver(that is,
2397-2400, 1998.
the solarwind) is characterised
by both strongvariabil- Chapman S.C., R. O. Dendy, and G. Rowlands,A sandpile
model with dual scalingregimesfor laboratory,spaceand
ity about the mean, and a large dynamic range of mean
astrophysicalplasmas, Phys. Plasmas,in press,1999.
energy input. Any statistical analysisof energyrelease
Consolini
G., Sandpile cellular automata and magnetoevents will therefore require long uninterrupted runs of
spheric dynamics, in Proc. vol. 58, "CosmicPhysics in
data. In order to use such data to test the hypothesis
the Year 2000", S. Aiello, N. Iucci, G. Sironi, A. Treyes
that the magnetospheremay be in a state of SOC, it
and U. Villante (eds.),SocietaIralianadi Fisica,Bologna,
is necessaryto examine how resilient the main observaItaly, 123-126, 1997.
tional signaturesof SOC (notablyscaleinvariantpower Dendy R. O. and P. Helander, Sandpiles, silos and tokamak phenomenology:a brief review, Plasma Physicsand
law statistics)will be under circumstances
of varying
Controlled Fusion, 39, 1947-1961, 1997.
Dendy, R. O. and P. Helander, On the appearance and nonappearanceof self-organisedcriticality in sandpiles,Phys.
Rev. E., 57, 3641-3644, 1998.
Freeman, M.P., Rhodes, C. and Watkins, N. W., On the
two-component nature of the auroral electrojets, AGU
1998 Fall Meeting, EOS Trans., 79 suppl., F793, 1998.
Helander, P., Chapman S.C., R. O. Dendy, G. Rowlands,
and N. W. Watkins, Exactly solvable sandpile with fractal
avalanching, Phys. Rev. E, in press, 1999.
JensenH. J., Self-OrganisedCriticality: Emergent Complex
Behaviour in Physical and Biological Systems,Cambridge
University Press, 1998.
Lu E., Avalanches in continuum dissipative systems, Phys.
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Rev. Lett., 74{,2511-2514, 1995.
Pinho S. T. R, and R. F. S. Andrade, An Abelian model for
Acknowledgments.
The authorswish to acknowledge
rainfall, Physica A, 255, 483-495, 1998.
valuable discussionswith T. S. Chang, G. Consolini, M.P.
Takalo, J., and Timonen, J., Comparison of the dynamics
Freeman, P. Helander, H. J. Jensen,A. Klimas, V. M. Uritof the AU and PC indices, Geophys.Res. Lett., 25, 2101sky, and Z. Voros. SCC would like to acknowledgea P PARC
2104, 1998.
lecturer fellowship.
Tsurutani B., M. Sugiura, T. Iyemori, B. E. Goldstein, W.
D. Gonzalez, S.-I. Akasofu and E. J. Smith, The nonlinear
response of AE to the IMF B• driver: a spectral break at
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(ReceivedJune6, 1999;acceptedJuly 7, 1999.)