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Similarity of Magmatism and Volcanism JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104,NO. B7, PAGES 15,425-15,438,JULY 10, 1999 Statistical self-similarity of magmatism and volcanism Jon D. Pelletier Division of Geologicaland Planetary Science,California Institute of Technology,Pasadena Abstract. Magraatismand volcanismexhibit spatial and temporal clusteringon a wide range of scales.Using the spatial pair-correlationfunction, the number of pairs ofmagmatic or volcanicevents whose separation isbetween r- «Ar andr + «Ar per unit area, we quantify the spatial clusteringof raagraatismand volcanismin several data sets. Statistically self-similarclustering characterized by power law spatial pair-correlation functions is observed. The temporal pair-correlation function is usedto identify self-similartemporal clusteringof raagraatismand volcanismin the Radiometric Age Data Bank of 11,986 dated intrusive and extrusive rocks in the North American Cordillera. The clusteringof raagraatismand volcanismin space and time in this data set is found to be statistically self-similarand identical to thoseof distributed seismicity.The frequency-sizedistributions of eruption volume and areal extent of basaltic flows are also found to be self-similarwith power laws analogousto the Gutenburg-Richterdistribution for earthquakes.In an attempt to understandthe origin of statisticalself-similarity in raagraatismand volcanismwe presentone end-membermodel in which the ascentof magma through a disordered crust of variable macroscopicpermeability is modeled with a cellular-automaton model to create a distribution of magma supply regionswhich erupt with equal probability per unit time. The model exhibits statistical self-similarity similar to that observed in the real data sets. 1. Introduction magmatic activity when the whole Cordillerean region is viewed during later Cenozoic time." Spatiotempo- Magmatism and volcanismare phenomenacharacter- ral clusteringof volcanismis recognizedin models in ized by variable intensityon a wide rangeof spatial and which the volcanicintensity of a region dependson the temporal scales. For example, volcanic eruptions are intensityof neighboringregions [Conner and Hill, 1995; episodic: like great earthquakes,they often occur with- Gonditand Conner,1996]. How magmasupplied from out warning after a long period of quiescence. Such a subductingslab upwellsthrough the continentalcrust episodicityoccurs over a broad range of timescalesfrom to erupt episodically at localized points is not well un- historic timescalesto much longer timescalesrecorded derstood. The purposeof this paper is to quantify the in deep-seasediment cores as discussedby Kenneff et aL variability of magmatic and volcanicintensity in space [1977],Kenneff [1981], and Kenneff [1989]. They docu- and time, including the frequency-sizedistribution and mentedepisodes of intensevolcanism separated by long the spatiotemporalclustering of events,using a variety quiescentperiods on the timescalesof millionsof years of techniquesand data sets. In all cases,the measures in PacificOcean sedimentcores. Similarly, volcanism is of magmatismand volcanismare found to exhibit power localizedin space. Magma eruptsin volcaniczones with law relationshipsindicative of statistical self-similarity. up to hundredsof individualvolcanic vents separated by Self-similarbehavior in spaceand time has prevously largedistances with little or no volcanism[Cuffanti and been documentedin hot spot volcanismby Shaw and Weaver,1988; Conner,1990; $irnkin, 1993]. The rate Ghouet[1991]. A modelof fluid invasionin porousme- and spatial concentrationof magmaticand volcanicac- dia has been introduced in the physicsliterature which tivity dependon the scaleof observation.For example, gives rise to statistically self-similar behavior. We pro- with respectto the Cenozoicmagmatic history of the pose this model as one possiblemodel of large-scale NorthAmerican Cordillera, Armstrong and Ward[1991, magma migration through a crust of variable macro- p. 13203]observed that "althoughmagmatism is always scopicpermeability for the formation of magma supply locally episodic,the episodesblur into a continuum of regions. The organization of the text is as follows. First, ev- i•• i• ,,••+• +•o+ +• ...... •+i•,• frequency-size Copyright1999 by the American GeophysicalUnion. distribution of volcanic eruptions and basalt flows is a power law analogousto the Gutenberg-Richterlaw for Paper number 1999JB900109. earthquakes. Since there is evidence that the size of an 0148-0227/99/1999JB900109509.00 eruption scaleswith the size of the magma supply re- 15,425 15,426 PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM gion that producedit, theseresults suggest that magma canic output is observedto be a decreasingpower law supply regionsmay also exhibit a power law cumulative function of the timescale of observation. This indicates frequency-sizedistribution. that as the timescale is increased,more long periods of The spatial clusteringof volcanicvents and basaltic quiescenceare included, reducingthe rates of magma- flows is then considered. We utilize the spatial pair- tism and volcanism. This relationship is another way to correlation function, the number of pairs of volcanic characterizethe episodicityand intermittency of mag- eventswhose separation isbetween r- «Ar andr + •1A r matism and volcanism. per unit area. We find that power laws are applicableto Last, we introducea modelfor the upwellingof magma the spatial pair-correlationfunction for many data sets, throughthe continentalcrust that producesself-affine indicating that the clusteringis statisticallyself-similar. variations in temperature and a self-similarfrequency- As a model for the size distributionand clustering size distribution and spatiotemporal clustering. Our of volcanic events and magma supply regions,we uti- model considersthe penetration of magma in a crust lize the observationthat temperature variations in the of variable macroscopicpermeability. The variable per- uppermost mantle are self-affine. Self-affine functions meability tends to roughen the isothermscharacteriz- are those which have a power spectrum with a power ing the ascent, allowing magma to be advected more law dependenceon wavenumber, k: $(k) c• k-z, easily in regionsof high permeability. Self-affinevaria- where/• is a characteristic exponent. Self-affinetem- tions in isothermic depths are produced by this model. perature variations are inferred from the observation This is analogousto experiments in which fluids pen- of self-affineregional variations in seismicwave speed etrating porous media developinterfaces that are self- at depth within Earth's mantle [Passierand $neider, affine[Buldyrev et al., 1992].The magmasupply regions 1995]. In our model,magma supply regions are those formed in this model of upwellingmagma are allowedto where the isothermic depth is larger than a threshold erupt with a random probability. As a result of the pen- value required for the melt fraction to be greater than etration of magma into new regions, magmatism and the percolation threshold. Synthetic magma supply re- volcanism migrate over time in a realistic way in this gions and distributions of volcanic rocksare constructed model. Spatiotemporal data from a model of magmatic and their statistics compared to those from the western and volcanic history are produced and the statistics are Great Basin. Close agreementis obtained betweenthe comparedto those of the Radiometric Age Data Bank. statistics of real and model magmatic and volcanic re- Identical clustering statistics are obtained, suggesting gions. that this model is one possible end-member model for Spatiotemporalclustering of magmatismin the North magmatic and volcanic processes. American Cordillera is considered with the Radiomet- ric Age Data Bank of 11,986 radiometrically-datedin- trusive and extrusive, basaltic and nonbasalticigneous 2. Frequency-Size Distribution rocks. The large number of rocks in the database en- of Volcanic Eruptions ables us to examine the clusteringin time as a function of the distance separating rocks and, conversely,the The frequency-sizedistribution of volcanicevents from clustering in space as a function of time. It is found a variety of geologicand marine environmentsis ob- that power law statistics are applicable to the tem- servedto be a power law analogousto the Gutenberg- poral pair-correlation function but that the exponent Richter law for earthquakes. The Gutenberg-Richter of the temporal pair-correlationfunction characterizing law states that the number of earthquakes with a seis- the strength of the clustering depends on the spatial mic moment greater than Mo is a power law function separation such that locations close together in space with an exponentclose to -• 2. N(• Mo) • M• -B, have highly correlatedsequences of eventsin time. Sim- whereB • •2 [Kanamoriand Anderson, 1975]. Sirakin ilarly, the clusteringin spaceis strongerfor rockssepa- [1993]has compileddata on Holoceneeruptions from rated by a small interval of time. The pair-correlation small, monthly events to globally catastrophic events functions we compute with the Radiometric Age Data such as the 1815 eruption of Tambora and the eruption Bank are identical to those computed for distributed of the basalt flowsof Yellowstone2 Ma to obtain an ap- seismicityby Kagan and Knopoff[1980] and Kagan and proximate frequency-sizedistribution of volcanic erup- Jackson[1991], indicating that distributedvolcanism tions. The volcanicexplosivity index (VEI), definedto and distributed seismicity
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