GEOPHYSICAL RESEARCH LETTERS, VOL. 27, NO. 9, PAGES 1367-1370, MAY 1, 2000
Analytical determination of power-law index for the
Chapman et al. sandpile (FSOC) analog for
magnetospheric activity- a renormalization-group
analysis
SunnyW. Y. Tam,• Tom Chang,
• S.C. Chapman,
2 andN. W. Watkins3
Abstract.
Recent suggestionand experimental indicationsthat the magnetotail dynamicsexhibit selforganizedcritical behavior have re-motivatedinterest
in sandpile(avalanche)models.Someexamplesof specific interest for geomagneticactivity have the property that internal avalanchesexhibit inversepower law
statistics whereas systemwideavalancheshave a welldefinedmean. Here, we apply the conceptof renormalization group to such a model. We demonstratethat
invariant analysis based on the renormalization-group
theory can explain the powerlaw distributionof energy
releaseby internal avalanchesin the large-scaleregime
of these systems.
The possibilityof FSOC in magnetospheric
dynamics
has re-motivatedinterestin sandpile(avalanche)models. In such models, the probability distributions of
energyreleasedby avalanches,and of avalanchelength,
should display scale-free, inverse power law statistics
that are a characteristic
of FSOC [Bak ½tal., 1987;see
also,Jensen,1998].In contrastto the scale-freebehavior exhibited in the internal dynamicsof the magnetosphere, substorms,the events of energy releaseof the
magnetotail have well-definedcharacteristicscalesin
both intensity and time, a specificclassof sandpilemodels is of particular relevanceto the magnetosphericdynamics.
This class of models exhibits
two scale-free
re-
gionsin the statisticsof its internal avalanches,whereas
1.
systemwideeventshave a well-definedmean [Chapman ½tal., 1998]. Chapman½tal. [1999]focusedon
It has been suggestedthat the Earth's magnetotail the statistics of the internal eventsfor large systems,
Introduction
may be describedby the stochasticbehavior of a nonlin-
and showed that there is a characteristic
broken inverse
ear dynamicalsystemnearforcedand/or self-organized powerlaw signaturein the statisticsfor energyrelease.
criticality (FSOC)[Chang, 1992, 1998, 1999]. There These authors have also established numerically that
is increasingexperimentalevidenceconsistentwith this the two regimesof different powerlaw index correspond
idea. For example, FSOC provides a possibleexplanation for the intermittent turbulence recently observedin
to small and large avalanchesrespectively.
In this study, we utilize the conceptsof renormaliza-
the magnetotail[Lui et al., 1988;Lui, 1998;Angelopou- tion group (RG) to analyticallydeterminethe power
loset al., 1996,1999].In-situ magneticfieldpowerspec- law index for the large internal avalanchesin this class
tra in the magnetotail [Hoshinoet al., 1994]showeda of models.
power law dependence,which is one of the characteristic
of the scale-freeFSOC behavior. Moreover, analysisof a
"burst size" distribution
based on AE data also revealed
2. Sandpile Model
a power law. This may indicate the absenceof a charac-
teristicscalewithin the magnetospheric
system [Consolini,1997],or resultfrom FSOC-likescale-freebehavior in the turbulentsolarwind [Freemanet al., 2000].
Lui et al. [2000]usedPOLAR UVI data to compile
The avalanche("sandpile")numericalalgorithmon
which this study is based is described in more detail
in Chapmanet al. [1999,and references
therein]. We
review it briefly here. The sandpile is representedby a
one-dimensionalgrid of N equally spacedcellsone unit
a probability distribution for the sizes, and integrated apart, eachwith sand at height hj and local gradient
intensities(energyreleased)of patchesof brightnessin zj - hj- hj+l. The sandpileis assumedto havean
the nightsideaurora, ranging from the smallestscales "angleof repose"below which it is alwaysstable. The
to substorm breakups. They found that the substorm
heightshj andthe gradientszj are measured
relativeto
associatedevents had a characteristic mean, whereas all the valuesat the angle of repose. Each cell is assigned
other eventsshowedpower law probability distributions
with index closeto -1. Crucially the slopeof this index
is unchangedby the level of magnetospheric
activity.
•Center for Space Research,MassachusettsInstitute of
Technology,Cambridge,MA 02139,USA
a criticalgradientz•; if thelocalgradient
exceeds
this,
the sand is redistributed to neighboring cells and iteration producesan avalanche. The critical gradients
on each of the N nodes are selected randomly from a
top-hatprobability
distribution
P(z•).
The system consideredhere is edge-driven. Sand is
2Departmentof Physics,WarwickUniversity,Coventry addedto cell I at a rate g, and the length and time are
CV4 7AL, UK
normalized to this loading rate such that unit volume
aBritishAntarcticSurvey,Madingley
Road,Cambridgeof
CB3 0ET, UK
Copyright2000 by the AmericanGeophysicalUnion.
sand is added in unit
time.
As soon as the critical
gradientat cell i is exceeded,the sand is redistributed.
The redistribution is instantaneous, and it conservesthe
Papernumber2000GL000030.
total volumeof sand. Sandwill propagateto cell 2, and
if the local gradientexceedsthe critical value there, to
0094-8276/00/2000GL000030505.00
cell 3, and so on. Within an avalanche,the sand is in1367
TAM
1368
ET AL.: ANALYTICAL
DETERMINATION
OF FSOC INDEX
stantaneously "flattened" back to the angle of repose
AXE,but proportional
to (AXE)
-• at largeAXE,with
(the regimeis of uniformgradient).The propagation
of
transition
at avalanches of size L •
64 cells. Both the
powerlaw indicesand the locationof the transitionin
thusoccursif hk- hk+l > z•. This resultsin sandbeing Pz(AXE)wereshownto be robustagainstthe paramedepositedto cell k + 1 from cells 1 through k, suchthat ter values
of thetop-hatdistribution
P(z•). Of importhe new heightshi, h2,.-., hk+• are all equal. If now tance to magnetosphericdynamics,the systemwas also
the critical value of the gradient at cell k q-1 is exceeded, examinedunderfastfuelingg > (z*) andthe powerlaw
the avalanchecontinues to k + 2, and so on. This iter- regimefor large eventswas robust under thesecondian ongoingavalanchefrom one cell, k, to the next, k+ 1,
ative procedure is repeated until the avalanchereaches tions [Chapman
et al., 1999;Watkinset al., 1999].
a cell where the gradient is below critical. The critical
In this study, we have obtained the statisticsof the
gradients within the post-avalancheregion are then re- avalanchesbased on their size, and found the normal-
generated
fromP(z]). Moresandisadded
at cell1until ized probabilityPœ(L) for avalanchesof size L. Figit again becomesunstable,triggeringanother avalanche.
An avalanchemay be entirely an internal arrangement
of sand, or may continue until it spreads acrossall N
cells of the pile in which casethe entire sandpile is emptied and returns to the angle of repose. Note that the
system size N has to be finite under these redistribution rules. If N were infinite, we would have to allow for
avalanchesof infinite length. Such an avalanchewould
take an infinite amount of time to complete the slide,
thereby violating the assumption of an instantaneous
redistribution process.
The total energy in the systemof the sandpileis:
E-Eh j .
(1)
j----1
before
and
A specialcase of the model has been discussedby
Helanderet al. [1999]. These authorsshowedthat
the problemis exactly solvableanalyticallyfor the case
P(z•) = 5(z• - zo). The system
exhibitsonlyone
regime:Pz cr (AXE)
-•. It canalsobe shown
that
AE cr L for L >> 1, although it was not mentioned
in theirpaper.Onecantherefore
inferthat Pœcr L-•
also in their model. In the following, we shall apply an
RG analysis
to demonstrate
that the PœcrL- 1 behavmodels.
3. Renormalization-group
The energy dissipatedby an avalanche,AXE,is just the
in E
-1 at large L.
ior at large L is universalto a certain classof sandpile
N
difference
ure i showsPœ(L)for the samedata in Chapmanet al.
[1999];we seethat Pœexhibitsa powerlaw with index
after
the event.
Internal
avalanchescan also be characterized by their length,
that is, the number of cellsinvolvedin the event. Chap-
Analysis
The concept of RG involvesthe searchfor invariance
properties under continued coarse-grainingand rescal-
ing of the system [Wilson and Kogut, 1974; Chang
et al., 1992]. When one coarse-grainthe system,and
man et al. [1999]focusedon the statisticsof large(5000
changethe rescalingparameter(e.g. the sizeof the syscell) systems.Their resultsshowedthat for slowfueling tem), transformations
involvingcertainparametersmay
rate (g << (z*)), thenormalized
probabilityPro(AXE)
ex- arise. These transformations may causesome parame-
hibitstwodistinct
regimes'
PBcr(AXE)
-ø'65at small ters to
evolve non-linearly. That would imply certain
fixed points in the parameter space. To study the behavior of theseparametersnear the fixed point, one may
linearize
10ø
the transformations.
That
would
allow
one
to find power law relations between these parameters
and the scalingparameter about the fixed points. Once
these power law indices are obtained, the behavior of the
system near these fixed points can be understood, and
useful information about the system may be extracted,
such as the probability distribution of avalanchesizes.
Here, we shall demonstrate step-by-step how this procedure can be applied to the sandpile system described
10-'
10-2
above.
Let us consider an ongoing avalanche propagating
from cells k- 1 to k in the sandpile system. We shall
define 0k to be the conditional probability that this
avalanche stops at cell k, given that it has already
propagated there. The probability for this avalanche
to reach k + 1 is thus 1- 0k, and that for it to reach
10-3
10-'•10
o
10•
102
L
103
k+2 wouldbe (1-- 0k)(1-- 0k+•). Now we coarsegrain the system by combining every pair of cells, and
renumberthe cells(i.e. cells1 and :2become1', cells3
Figure 1. The probability distribution of avalanche and 4 become2', etc.). This processreducesthe size
lengths for four runs, correspondingto top-hat distri- of the system, and the length of the avalanchesby half
butions
P(zj) withmeanvaluei anddifferent
widths. (L -+ L/2). The rearrangement,in particular,comMarked with symbolsare caseswith widths 0.01 ([2), bines cells k and k + i into l', where the new label l'
0.1 (,) and 1 (o), and markedwith barsis the analyt- numericallyequals(k + 1)/2. The probabilityfor the
ically soluble case where the critical gradient takes a same avalancheto propagate from l' to the next cell
single value.
(whichconsists
of the originalcellk + 2) is then 1 - Qt,
TAM ET AL.: ANALYTICAL
DETERMINATION
by definition. We can seethat if the Q's are to be similarly definedin the rescaledsystem,we must set
OF FSOC INDEX
L'=L/2,
and Q'=2Q-Q2.
1369
(7)
Near Q = 0, the last term in (7) is small. Thus, one
Q•, - Q• + Q•+• - Q•Q•+•.
can show that under the transformation,
BecauseQk and Qk+• are associatedwith adjacentcells,
we expecttheir valuesto be similar. We thus drop their
Q -• 1/L + 0(1/L2).
(8)
subscripts,and make them equal. Equation(2) then
Or QL is an invariant under the RG transformation for
large L and small Q. Note that this result is true only
for avalanchesof large L, where Q is closeto the fixed
n(Q) =
point. In this regime, the probability that an avalanche
which has reacheda cell k will go no further is inversely
which would make the system look the same after a
proportional to the length of the avalanche itself. Of
levelof renormalization.Note that we may renormalize course,if the avalanchestopsat cell k, its !ength will be
the system again. Every subsequentlevel requiresthe
leads to a transformation
rescaling
L -->L/2, andthe transformation
Q -->R(Q),
in order to make the system look the same. The transformation supports two fixed points: Qa - 0, and
Qa = 1. The case Q = 1 means that an avalanche
will alwaysstop propagatingonceit reachesa particular cell; this caseis irrelevantto the sandpilesystem.
The caseQ = 0, on the other hand, correspondsto the
situationwherean avalanchewill alwaysgo on to the
next cell. We shall argue that this case is relevant to
the sandpilesystemby showingthat Q• is small as k
becomeslarge.
For an ongoingavalancheto propagatefrom cellsj- 1
to j, the averageheightof the sandfrom 1 to j- 1 must
exceed
hj + z;. Therefore,
for an avalanche
to reachk,
k. Wecanthusexpress
theresultasQ• crk-• +O(k-2).
Now let us considernot only thoseavalanchesthat reach
k, but avalanchesof all lengths. An arbitrary avalanche
that stops at k > K must have propagated through
K, K+i,...,
k-1. Hence, for k > K, we can normalize
PL by the factor (1- Q•)... (1- QK-•), and write:
PL(L = k) cr (1 - QK)(1 - QK+•) . . . (1 - Q•_•)Qk
+ O(Q
ock-• + O(k-2),
(9)
as all the Q's in Eq. (9) are muchsmallerthan unity.
This result is equivalent to
the following conditionsmust be satisfied:
PL(L)crL-• + O(L-2).
(10)
n
•-•jzj > nz•,n- 1,...,k- 1.
(4)
j=l
Therefore,for smallQ• (or largeL, seeFig. 1), our RG
analysishas reproducedto the leading order the scaling law obtained by large-scaleavalanchemodel calculations.
For this avalancheto stop at k, it requires:
k
_<zZ.
(5)
j=l
The analysis above, however,has yet to explain the
power law behavior for PE. Here we look for a relationship between AE and L for large-scaleavalanches,
and then determine the power law between PE and AE
basedon the results above. We recognizethat the en-
ergyof the system,givenby Eq. (1), scalesasthe system
Condition(5) can be combinedwith the n = k- 1 case size N. When we renormalize,the length of the system
in (4) to determinethe rangeof z• that wouldsatisfy is halved. One can thus write the renormalizedenergy
the inequalities:
as E' = El2. To find AE, we simplytake the difference
,
1•'-•
,
k-1
z•_<
zk-•Ejzj <z•- •z•_,.
k
in total energy before and after the avalanche. Thus,
(6)
j=l
AE' = (AE)/2.
We recognizethat AE and L are re-
duced by the same factor under renormalization. Thus,
we establish that AE cr L, or equivalently, the factor
(AE)/L is an invariantin the regimeof large L. With
Notethat the middleor the right-handsideof (6) may Eq. (10), we can concludethat there is a regimefor
benegative,
depending
ontherandomchoices
of z• and large AE where
z•_•. In that case,condition(6) can neverbe satisfied
(becausez• _>0). Otherwise,we can take the ensemP•(AS) cr(AE)-• + O((AE)-2).
(11)
ble averageof the right-handsideof (6) to estimatean
upper bound for the range of z• that satisfiesthe con-
We note that the analysis above is quite robust. Sup-
dition. This estimatedupperboundis {z*l/k , where posewe use a slightly different coarse-grainingprocess:
(z*) = (z•) = (z•_•). Thus,the rangeof z• that con- Instead of combiningevery 2 cellsin the system,we may
tributes to the probability Qk decreasesas k becomes perform the procedure with every 3, 4,...or
M cells.
larger.We thereforeconclude
that Q• is smallfor large Provided M is not too large, such that the Q's are es-
k. Basedon this argument,we expect that if N is sufficiently large, there exists a cell K < N such that for
k > K, Q• is in the regimegovernedby the fixed point
Now we renormalizethe system, and study the behavior near Q = 0. For eachstep of transformation,L
and Q evolve as:
sentially the same, the renormalization transformation
for Q is then:
RM(Q)- i - (1- Q)M.
(12)
There are M fixed points for the transformation RM:
Q a = 0 and Q a = I are still fixed points, while the other
1370
TAM ET AL.: ANALYTICAL
M-
DETERMINATION
2 do not lie within the real line between 0 and
1. These other fixed points are thus irrelevant. Near
Q = 0, the linear behavior discussedabove is retained.
One can showthat now Q• = MQ and L• = L/M.
Therefore, Q L is still an invariant under the RG trans-
formationsin the regimeof largeL and smallQ. Sim-
ilarly,E • = ElM underthe renormalization,
implying
(AE)/L to be an invariantin this regime.Therefore,
OF FSOC INDEX
Monograph, p. 193, WashingtonD.C. American Geophysical Union, 1998.
Chang, T., Self-organizedcriticality, multi-fractal spectra,
sporadic localized reconnections and intermittent turbulence in the magnetotail, Phys. Plasmas, 6, 4137, 1999.
Chang, T. S., D. D. Vvedensky,and J. F. Nicoll, Differential
renormalization-group generatorsfor static and dynamic
critical phenomena, Phys. Reports, 217, 279, 1992.
Chapman, S.C., N. W. Watkins, R. O. Dendy, P. Helander,
and G. Rowlands, A simple avalanche model as an analoguefor magnetosphericactivity, Geophys.Res. Lett., 2,5,
the power law behaviors that we found earlier can be
obtainedfrom a more generalanalysisbasedon the RG
theory.
2397, 1998.
We shouldemphasizethat the RG analysisaboveis Chapman, S.C., R. O. Dendy, and G. Rowlands,A sandpile
quite general.I[ appliesnot only to oneparticularsandmodel with dual scaling regimesfor laboratory, spaceand
pile model.In fact, Eq. (10) is true for anyfinite model
astrophysicalplasmas, Phys. Plasmas, 6, 4169, 1999.
whereQ << 1 near the edge.Hence,the analysisis also Consolini, G., Sandpile cellular automata and magnetoapplicableto modelsof higherdimensions,
as long as
spheric dynamics, in Proc. "Cosmic Physics in the Year
the property involvingQ is satisfied.
2000", edited by S. Aiello, N. Iucci, G. Sironi, A. Treves,
4.
Conclusion
We have demonstratedthat the conceptof RG can
and U. Villante, vol. 58, p. 123, Bologna, Italy. Societa
Iraliana de Fisica, 1997.
Freeman, M.P., N. W. Watkins, and D. J. Riley, Evidence
for a solar wind origin of the power law burst lifetime
distribution of the AE indices, Geophys. Res. Lett., in
be appliedto a classof sandpile(avalanche)models.
press, 2000.
The FSOC behaviordemonstratedby this classof systems is of particularinterestin the studyof the mag- Helander, P., S.C. Chapman, R. O. Dendy, G. Rowlands,
and N. W. Watkins, Exactly solvablesandpile with fractal
avalanching, Phys. Rev. E, ,59, 6356, 1999.
in the modelsexhibitstwo scale-freeregions,whereas Hoshino, M., A. Nishida, T. Yamamoto, and S. Kokubun,
that of systemwideevents has a well-definedmean. RG
Turbulence magnetic field in the distant magnetotail:
analysishas allowedus to explain the power-lawrelabottom-up processof plasmoid formation?, Geophys.Res.
tions that characterizethe large-scaleregimeof interLett., 21, 2935, 1994.
nal avalanches
in the sandpilesystems.The analysisis Jensen,H. J., Self-OrganizedCriticality: Emergent Complex
quite general,and is applicableto other sandpilemodBehaviourin Physicaland BiologicalSystems.Cambridge
els, or perhapseven more complicatedavalanchesysUniversity Press, UK, 1998.
tems that exhibit FSOC. In moregeneralapplications, Lui, A. T. Y., R. E. Lopez, S. M. Krimigis, R. W. McEntire,
recursionequationssuchas (3) and (12)may include
L. J. Zanetti, and T. A. Potemra, A casestudy of magmore than one parameterand more complexnonlinearnetotail current sheet disruption and diversion, Geophys.
ity, leading to power-lawrelationsother than -1.
Res. Lett., 1,5,721, 1988.
Lui, A. T. Y., Multiscale and intermittent nature of current
disruption in the magnetotail, in Physicsof SpacePlasmas
Acknowledgments.
The authors would like to thank
(1998), editedby T. Chang,and J. R. Jasperse,no. 15, p.
Henrik Jensen and George Rowlands for valuable discus-
netosphericdynamics; statistics of internal avalanches
sions. SCC was funded by PPARC. NWW is grateful to
MIT/CSR for its hospitality.The workat MIT/CSR is partially supported by grants from AFOSR, NSF, and NASA.
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S. W. Y. Tam and T. Chang, Center for SpaceResearch,
MassachusettsInstitute of Technology, Cambridge, MA
02139, USA.
(e-mail:wyt@space.mit.edu;
tsc@space.mit.edu)
S. C. Chapman, Department of Physics, Warwick
University, Coventry CV4 7AL, UK.
N. W. Watkins, British Antarctic Survey, Madingley
Road, Cambridge CB3 0ET, UK.
(ReceivedJanuary26, 2000;acceptedMarch 16, 2000.)