How local stresses control magma-chamber ruptures, dyke injections, and eruptions in composite volcanoes

Transcription

1 How local stsss contol magma-chamb uptus, dyk injctions, and uptions in composit volcanos Agust Gudmundsson Dpatmnt of Stuctual Gology and Godynamics, Goscinc Cnt, Univsity of Göttingn, Gmany Abstact To assss th pobability of a volcanic uption duing an unst piod, w must undstand magma-chamb uptu and dyk popagation to th sufac, as wll as dyk ast at dpth in th volcano. Dyk popagation and ast dpnd stongly on th local stsss in th individual mchanical lays which constitut th volcano. Th local stsss a pimaily dtmind by th loading conditions (tctonic stss, magmatic pssu, o displacmnt) and th mchanical poptis of th lays. In th absnc of stss monitoing of volcanos, th local stsss must b infd fom modls, ith analytical o numical. This pap viws many analytical and numical modls of local stsss aound magma chambs, as wll as analytical modls and numical xampls of dyk-injction and uption fquncis. Most analytical modls of magma chambs igno th mchanical poptis of th individual lays and thi contacts, assum th volcano to bhav as a homognous, isotopic, lastic half spac o a smi-infinit plat, and a of two main typs: nucli of stain and cavitis. Th bst-known nuclus of stain is th point-souc Mogi modl, usd to xplain sufac dfomation as a sult of ith incas o dcas in magma pssu in a chamb whos dpth is also infd fom th sufac data. Th modl xplains stsss and displacmnts fa away fom th chamb, but nith th stss concntation aound th chamb, which dtmins if and wh chamb uptu and dyk injction tak plac, no th shap, siz, and likly tctonic volution of th chamb. In th cavity o (two-dimnsional) hol modl th magma chamb has a finit siz. Thus, th local stsss at, and away fom, th bounday of a chamb can b calculatd. Fo vaious loading conditions, an analytical cavity modl givs a cud indication of th local stsss in a volcano and its sufac dfomation. Howv, vaiation in mchanical poptis, and contacts, btwn lays a ignod. Th analytical cavity modl thus cannot b usd fo dtaild analyss of th local stsss in a composit volcano. Th numical modls psntd h show that th local stsss in a volcano dpnd stongly on th magma-chamb gomty and th mchanical poptis of its lays which a oftn contasting, paticulaly at shallow dpths. Fo xampl, lava flows, wldd pyolastic units, and intusions may b vy stiff (with a high Young s modulus), whas young and non-wldd pyoclastic and sdimntay units may b vy soft (with a low Young s modulus). Consquntly, th local stsss may chang abuptly fom on lay to th nxt; fo xampl, on lay may favou dyk popagation whil an adjacnt lay favous

2 dyk ast. No dyk-fd uption can occu if th is any lay along th potntial path of th dyk to th sufac wh th stss fild is unfavouabl to dyk popagation. If such a lay occus, th dyk nomally bcoms astd and an uption is pvntd. Th psnt sults indicat that duing unst piods composit volcanos commonly dvlop local stsss that ast dyks and pvnt uptions, in agmnt with fild obsvations. Ths sults undlin th nd fo in situ stss monitoing of volcanos to assss th pobability of dyk-fd uptions. Kywods volcanic uption, dyk injction, magma chamb, custal stss, layd cust, volcanotctonic modlling Contnts. Intoduction. Th two basic modls. Nucli of stain. Cavitis. Stss concntation aound chambs. Hols subjct to intnal pssu. Cavitis subjct to intnal pssu. Cavitis subjct to xtnal tnsion 4. Stsss naby chambs analytical modls 5. Stsss naby chambs numical modls 5. Homognous, isotopic host ock 5. Htognous, anisotopic host ock 6. Dyk injction fom a dp-satd svoi 7. Dyk injction fom a doubl magma chamb 8. Application to intusion and uption fquncis 9. Discussion 0. Conclusions. Intoduction In cnt dcads, th has bn considabl pogss in th gnal undstanding of th hazads involvd onc an uption has statd. Th impovd undstanding applis, in paticula, to th dynamics of uptiv columns (Spaks t al., 997) and th fomation and mchanics of tanspot of pyoclastic ocks (Fisch and Schminck, 984; Cas and Wight, 987; Fundt and Rosi, 00; Schminck, 004). But th has also bn considabl advancmnt in th gnal knowldg of th volution of basaltic lava flow filds such as occu woldwid at th sufacs of mantl plums and oth basaltic povincs (Walk, 99; Kilbun and Lops, 99; Slf t al., 996; Rossi, 996; Calvai t al., 00), Most volcanic unst piods, howv, do not sult in an uption (Nwhall and Dzuisin, 988). Evn thos unst piods wh magma-divn factus (dyks o inclind shts) a known to b injctd fom a shallow magma chamb do not nomally sult in an uption (Pollad t al., 98; List and K, 99; Rubin, 995; Bonafd and Rivalta, 999; Pinl and Jaupat, 000; Gudmundsson, 00, 00; Acoclla and Ni, 00; Stwat t al., 00, 005; Rivalta t al., 005). Sinc naly all volcanic uptions a supplid with magma though dyks and inclind shts, it follows that fo an uption to occu a dyk o a sht must b abl to popagat fom a magma chamb to th sufac. Th initiation of a dyk and its vntual popagation to th sufac o, altnativly, ast at som dpth in th volcano, dpnd on th stat of stss in th volcano. This stss stat is contolld, fist, by th mchanical poptis of th

3 ocks that constitut th volcano and th associatd custal sgmnt and, scond, by th shap, dpth, and loading conditions of th souc magma chamb o chambs. In solid mchanics, load is a wod that nomally mans th focs, stsss, o pssus applid to a body and xtnal to it matial (Bnham t al., 996). Accodingly, in this pap loading conditions f to th stsss and magmatic pssus applid to th magma chambs in th analytical and numical modls. To undstand and assss th hazad duing an unst piod, w must know th stat of stss in th volcano and, in paticula, th stss concntation aound th magma chamb o chambs that supply magma to its uptions. Mchanically, patly o compltly solidifid magma chambs (plutons) that hav poptis diffnt fom thos of th host ock a analogous to inclusions in an lastic body; compltly moltn chambs a analogous to cavitis. In two-dimnsional modls, cavitis a fd to as hols. All cavitis and inclusions in an lastic body distub th stss fild of that body and giv is to stss concntations (Fig. ). Stss concntations aound magma chambs a sponsibl fo thi uptus and dyk o sht injctions duing piods of unst. As a sult of stss concntation, a local stss fild dvlops aound th chamb and in its vicinity. This local fild dtmins whth an injctd sht intusions bcoms a sill, an inclind sht, o a subvtical dyk (Fig. ). In this pap th wod dyk is usd mostly as a gnic tm, coving both pop dyks and inclind shts. Whn ncssay, howv, a distinction is mad btwn subvtical dyks and inclind shts. Also, whn discussing sults applicabl to composit volcanos it is implid that th sam sults may apply to composit ift zons. To undstand th mchanics of a composit volcano (cntal volcano, statovolcano) on must know th stss filds associatd with its souc magma chamb. Som authos hav modlld th host ock of th chamb as viscolastic (Bonafd t al., 986; Folch t al., 000). H, howv, th focus is on host-ock bhaviou that can b dscibd to a fist appoximation as lastic. Whn th ock hosting th chamb is modlld as homognous (with th sam poptis vywh) and isotopic (with poptis indpndnt of diction of masumnt), analytical solutions xist fo th local stss fild (.g., Gudmundsson, 988; Pinl and Jaupat, 00). Som of ths solutions a viwd blow. Gnally, howv, th ock hosting th chamb is htognous and anisotopic. Th xist som analytical solutions fo magma chambs (hols, cavitis) in anisotopic bodis (Savin, 96; Lkhnitskii, 968; Tan, 994), as wll as fo vy simpl aspcts of micomchanical (micopola) thois (Sadd, 005). Howv, bcaus of th complxitiy of th ovall poptis of htognous, lastic matials (Nmat-Nass and Hoi, 999), simpl, closdfom analytical stss solutions such as can b applid to magma chambs in htognous and anisotopic custal sgmnts a gnally not availabl. It follows that th stss filds aound magma chambs in anisotopic and htognous custal sgmnts must nomally b obtaind using numical modls. Many numical modls of stss filds aound magma chambs a givn blow. It should b mphasisd that although most of th fild and numical xampls a fom Icland, th sults psntd h a applicabl to any composit volcano. This follows bcaus th condition fo magma-chamb uptu and dyk injction (Eq. ) is univsal and dos not dpnd on th tctonic gim within which th volcano is locatd. Also, in th numical modls th magma-chamb shaps o th mchanical poptis of th lays, of cous, do not dpnd on th associatd tctonic gim. Futhmo, vn if th loading conditions usd in th modls may b achd in diffnt ways in diffnt tctonic gims, th calcualatd stsss dpnd only on th loading conditions usd and not on how thy w achd. Fo xampl, if th conditions fo dyk injctd (Eq. ) a achd though duction in th minimum compssiv pincipal stss, it dos not matt if that duction is achd though divgnt plat movmnts (at an ocan idg), doming (in a continntal

4 4 gion), o a a sult of xtnsion in a tantnsion gion associatd with a majo stik-slip fault. Most of th fild xampls a fom composit volcanos in Icland; ths a taditionally fd to as cntal volcanos (Walk, 96; Samundsson, 978; Gudmundsson, 000). Som a statovolcanos that is high abov thi suoundings, oths a caldas (Fig. ). Insid th ift zon, th volcanos tnd to b locatd na th cnt of th associatd volcanic systm, and upt much mo fquntly than th st of th systm, hnc th nam cnal volcano. Bcaus of this tadition, in this pap th tms cntal volcano and volcano a commonly usd fo th composit volcanos and caldas of Icland. Th gnal mphasis, howv, is on th tms composit volcano and composit ift zon fo two main asons. Fist, all volcanos and ift zons a composit in that thy consist of lays with (oftn widly) diffnt mchanical poptis (Fig. 4). Also, som lays may b agadd as locatd in a matix o units of vy diffnt mchanical poptis. Scond, th mchanical bhaviou and local stsss of composit volcanos a fomally simila to thos of gnal composit matials (Danil and Ishai, 994; Tan, 994; Kaw, 997; Hy, 998). By mphasising this similaity, local stsss in volcanos can b compad with, and undstood in tms of, gnal sults on stss vaiations in composit matials. Many, phaps most, composit volcanos a supplid with magma fom shallow magma chambs which, in tun, civ thi magmas fom dp svois. Such a pai is fd to as a doubl magma chamb (Fig. 5). Commonly, th dp-satd chamb o svoi is much lag than th shallow chamb and locatd in th low cust o at th cust-mantl bounday (Gudmundsson, 000). A singl magma flow fom th dp svoi may thn tigg many dyk injctions, and vntually uptions, fom th shallow chamb. It follows that th dp svoi has gat ffcts on th uptu fquncy of th shallow chamb and, indictly, on th uption fquncy of th associatd volcano. This pap has th main aims. Th fist aim is to viw xisting analytical and numical modls on th local stsss that dvlop in majo composit volcanos and ift zons supplid with magma fom shallow chambs and dp svois. To mak th tatmnt compltly up-to-dat, I also discuss th sults of sval nw numical modls. Th basic modls a considd. Fist, singl and doubl magma chambs in a homognous, isotopic custal sgmnt. Fo this cas both analytical and numical modls a psntd. Scond, numical modls of th stss filds aound singl magma chambs in a layd custal sgmnt. And, thid, numical modls of th stss filds aound doubl magma chambs in a layd custal sgmnt. Th scond aim is to viw xisting, and psnt som nw, sults on th conditions of magma-chamb uptu and dyk injction. Paticula attntion is givn to th ffcts that abupt changs in local stss filds hav on th popagation and ast of dyks. Also, simpl stss analysis is usd to xplain th typical injction fquncis of dyks associatd with singl and doubl magma chambs. Th thid aim is to viw and xtnd th cunt knowldg of th conditions fo dyks aching th sufac in composit volcanos and ift zons, that is, th conditions fo dyk-fd volcanic uptions. Th main attntion is givn to th ffcts of mchanical laying in composit volcanos and ift zons on th local stss filds, and how ths lagly dtmin whth an injctd dyk is abl to ach th sufac and supply magma to an uption. In Sction 8 of th pap, som of th main thotical sults a usd to quantify th stss conditions fo magma-chamb uptu and th fquncy of dyk injctions fo ctain spcifid bounday conditions.. Th two basic modls

5 5 Th most commonly usd modls of a magma chamb a of two basic typs. On typ gads th magma chamb simply as a pssu souc in an lastic custal sgmnt, commonly a point souc, without any finit-siz gomty. This modl is pimaily usd to xplain sufac dfomation as obtaind fom godtic masumnts duing piods of unst. Th oth basic typ gads th magma chamb as a finit-siz body in an lastic custal sgmnt: a hol fo two-dimnsional modls, and a cavity fo th-dimnsional modls. Both basic modls can b psntd though analytical solutions, in which cas th lastic custal sgmnt is nomally gadd as homognous and isotopic. Fo alistic modlling of stss filds and sufac dfomation associatd with a composit volcano, howv, th custal sgmnt hosting th chamb must b gadd as htognous and, in paticula, as layd. Fo such host ocks, numical modls a nomally usd.. Nuclii of stain Duing piods of unst in a volcano its sufac dfomation is taditionally xplaind in tms of a pssu chang in th associatd magma chamb modlld as a nuclus of stain. This is th so-calld Mogi modl, a widly usd modl in volcanology (Mogi, 958). In this modl, th chamb is gadd as a concntatd (point) foc of an infinitsimal volum (Fig. 6). Th stsss and displacmnt poducd by a nuclus of stain locatd at a ctain distanc blow th sufac of a smi-infinit lastic body o an lastic half spac can b obtaind though analytical solutions. Ths solutions w initially divd by Mlan (9) and Mindlin (96) and usd in gology by Andson (96). Using nucli of stain, Andson (96) was abl to xplain, in fomal tms, th tnds of dyks and inclind shts injctd fom a magma chamb locatd in a homognous, isotopic lastic half spac. Sinc dyks and shts a mostly xtnsion factus thy follow th tnds (tajctois) of th maximum pincipal compssiv stss,, and a ppndicula to th minimum compssiv, maximum tnsil, pincipal stss, (Gudmundsson, 00). Fom his nuclus-of-stain modl, Andson (96) calculatd th ointations of th pincipal stsss and, thby, th tnds of idal dyks and inclind shts fa away fom th magma chamb (th stain nuclus) itslf (Fig. 7). Using a nuclus of stain fd to as cnt of compssion (Lov, 97) fo a magma chamb, Andson (96) also psntd a modl of collaps-calda fomation as bing th sult of magmachamb undpssu, that is, compssion o contaction of an associatd magma chamb. Fo half a cntuy, nucli of stain hav bn outinly usd to account fo sufac dfomation in volcanos (Mogi, 958; Bonafd t al., 986; Davis, 986; McTigu, 987; Dlany and McTigu, 994; Battaglia t al., 00). Most of ths modls assum that th stsss and displacmnts gnatd by th nuclus cospond to b thos poducd by an xcss magmatic pssu o, fo collaps caldas in Andson s (96) modl, magmatic undpssu in a sphical chamb associatd with th volcano (Figs. 6, 7). Anoth common modl assumption is that th magma chamb is locatd at compaativly gat custal dpths. This implis that th chamb dpth blow th sufac of th composit volcano is lag in compaison with th chamb diamt. Displacmnts gnatd by nucli of stain can oftn b fittd to th obsvd sufac dfomation duing a piod of volcanic unst (Fig. 6). Invsion of th sufac data may thn giv a cud indication of th dpth to th pssu chang associatd with that paticula unst piod. It is nomally assumd that th point pssu o pssu chang occus somwh na th top of an associatd magma chamb. Somtims th pssu chang dos indd occu na th top of a chamb, but duing many unst piods that assumption is not waantd. Fo xampl, th appant pssu chang may b pimaily associatd with stss concntations and, vntually, sht injctions fom th magins of th chamb. Also, faulting and vaious oth tctonic and thmal pocsss may caus sufac dfomation duing unst piods. Intpting ths as pssu changs na th top of a magma chamb

6 6 may sult in a mistakn pictu of th physical pocsss sponsibl fo th sufac dfomation. Fo ths asons, th physical maning of modls which us nucli of stain to intpt sufac dfomation of volcanos is commonly obscu. Th a also oth asons why nucli of stain must b gadd as unsatisfactoy modls of magma chambs. On is that a nuclus-of-stain modl dos not mak it possibl to daw any futh conclusions gading th infastuctu o tctonic volution of th composit volcano that is undgoing unst. A paticula nuclus-of-stain modl can nomally b fittd to th sufac dfomation of a volcano only duing a ctain shot unst piod. Such a modl consids nith th ffcts of a al, finit-siz magma chamb no th tctonic volution of th chamb and th associatd volcano. It follows that a nuclus-ofstain modl tlls us nxt to nothing about th al volcanotctonic pocsss associatd with th unst. Th a fundamntal qustions that ais duing ach unst piod, such as: Is th volcano likly to upt, and if so wh? Altnativly, is th volcano likly to dvlop a collaps calda o gnat a landslid? A nuclus-of-stain modl of th magma chamb dos not hlp answing ths and latd qustions. A scond ason is that th nuclus-of-stain modl can in pincipl only account fo th displacmnt and stsss fa away fom th chamb, but not thos that occu in th vicinity o at th magin of th chamb. This limitation is bcaus th nuclus-of-stain modl substituts actual magma chambs with vanishingly small point soucs. Thus, th magnitud and location of th maximum stss concntation aound th chamb itslf - factos that dtmin if and wh chamb uptu and magma injction taks plac duing an unst piod - cannot b dtmind whn th chamb is modlld as a nuclus of stain. Many xtinct magma chambs cuntly xposd as plutons hav volums of -0 km although som a much lag (Sibbtt, 988; Mash, 989). Similaly, activ magma chambs a widly considd to hav volums anging fom lss than 5 km to about 500 km (Chst, 99). A cosponding, activ sphical chamb would hav a adius fom lss than km to about 5 km. Magma chambs associatd with activ composit volcanos a thus likly to b of considabl volums. Many shallow chambs a also lag in compaison with thi dpths blow th sufacs of th volcanos. Magma chambs, paticulaly shallow ons, a thus likly to dvlop local stss and displacmnt filds fo which nuclus-of-stain modls a not appopiat.. Cavitis Th considations abov indicat that all shallow magma chambs a of finit, and oftn considabl, sizs. Most chambs, paticulaly shallow ons, a locatd in custal sgmnts that, duing most unst piods, bhav as lastic. It follows that an appopiat mchanical modl of a typical magma-filld chamb is a finit-siz cavity o, fo a twodimnsional modl, a hol in an lastic plat o half spac. Many magma chambs, howv, a psumably patially moltn, that is, composd patly of magma and patly of a cystal mush. In that cas, a chamb may b gadd as an inclusion (Goodi, 9; Eshlby, 957; Savin, 96) th stiffnss (Young s modulus) of which incass as th magma faction dcass. Whn using a cavitiy modl, th magma chamb can hav any siz o dpth blow th sufac of its volcano. Th chamb may also b subjct to any typ of loading, such as intnal xcss magmatic pssu (Bonafd t al., 986; Folch and Mati, 998; Folch t al., 000; Gudmundsson, 998), xtnal stsss, o xtnal displacmnts (Savin, 96; Gudmundsson, 988, 00). Fo a cavity modl, it is possibl to calculat th stss concntation aound th chamb, and thus to idntify th potntial gions of uptu and dyk injction. Futhmo, th tajctois of th pincipal stsss in th vicinity of th cavity chamb dtmin th popagation dictions of idal dyks injctd fom th chamb.

7 7 Th-dimnsional cavity modls and two-dimnsional hol modls of magma chambs can also asily b analysd using numical pogams. In such modls th magma chamb may hav any shap, and htognitis can b addd to th custal sgmnt hosting th chamb: fo xampl, lays with contasting mchanical poptis, such as soft pyoclastic lays altnating with stiff lava flows, as a common in many composit volcanos (Gudmundsson, 00; Gudmundsson and Bnn, 004a,b, 005; Tasatti t al., 005). Numical pogams mak it possibl to calculat th stss tajctois fo any kind of loading, volcano gomty, magma-chamb gomty, host-ock laying, anisotopy, and dyk-dyk o dyk-fault intactions (DNatal and Pingu, 99; Gudmundsson and Bnn, 004a,b, 005; Lungaini t al., 005; Tasatti t al., 005). It follows that numical cavity modls can b usd to xplain, in fomal tms, th attitud of all shts and dyks injctd fom magma chambs. Many na-sufac solid ocks a at low tmpatu and pssu and bhav as lastic up to stains of about % (Fam, 98). Expansion of a cavity-lik magma-chamb pio to uptu and dyk injction would aly gnat host-ock stains xcding this limit. At th dpths of most activ magma chambs, howv, ocks a subjct to pssu and tmpatu much high than na th sufac. Incasing tmpatu gnally lows th stiffnss of th ock hosting th chamb (Hudson and Haison, 997; Schön, 004). By contast, th stiffnss tnds to incas with custal dpth patly as a sult of haling and saling of contacts and filling of pos with sconday minals. Th tmpatu and dpth ffcts on host-ock stiffnsss can b takn into account by alting th ffctiv Young s modulus of th ock. Rasonably alistic cavity modls of magma chambs can nomally b povidd on th assumption that th ock bhavs as lina lastic. That assumption is usd in th following sctions.. Stss concntations aound chambs Th simplst finit-siz modl of a magma chamb is a cicula hol. In physical tms, this is a two-dimnsional modl which is suitabl whn on of th magma-chamb dimnsions is vy lag in compaison with th oth two. Fo xampl, th chamb may b longat paalll with th axis of ift zon within which it is locatd. This is so fo som xtinct shallow magma chambs in Icland, fo xampl th Slaufudalu pluton (Figs. 8, 9). Elliptical caldas indicat that longat shallow magma chambs may b quit common (Acoclla t al., 00; Holohan t al., 005). Th a also indications that many dp-satd magma chambs undlying lag pats of th ift-zon volcanic systms a longat paalll to th volcanic systms (Fig. 0). Fo a chamb wh on hoizontal dimnsion of is much lag than th oth two dimnsions, th lag dimnsion may oftn b assumd infinit, in which cas th stss-concntation poblm is ducd to two dimnsions. Many chambs, howv, may b appoximatly sphical, in which cas a th-dimnsional modl is mo appopiat. Stss concntation aound a chamb dcids whth it uptus. A magma chamb uptus and initiats a dyk whn th following quation is satisfid (Gudmundsson, 990): p l p () T 0 H, p l is th lithostatic stss o ovbudn pssu at th uptu sit; p Pt pl is th diffnc btwn th total magma pssu P t in th chamb at th tim of its uptu and th lithostatic stss, and is fd to as xcss pssu - also fd to as ovpssu, (Bonafd t al., 986; Folch and Mati, 998; Folch t al., 000; Pinl and Jaupat, 00); is th minimum compssiv o maximum tnsil pincipal stss, and T 0 th local in situ tnsil stngth, at th uptu sit. Following th tadition in gology, compssiv stss is

8 8 considd positiv. Thus, fo an absolut tnsion to occu, must b ngativ. By contast, th maximum compssiv pincipal stss is always positiv. In subsqunt sctions of this pap, whn has no sign, it is undstood to b th absolut valu of th maximum tnsil pincipal stss, that is,.. Hols subjct to intnal pssu Consid a two-dimnsional magma chamb modlld as a hol with a cicula vtical coss sction of adius R and dpth to cnt d (Fig. ). Th chamb is subjct to xcss magmatic pssu p as th only loading (Eq. ). Th stss x at th sufac of th associatd composit volcano o a ift zon is thn givn by (Jffy, 9; Savin, 96): R ( x d R x 4 p () ( x d R ) Th maximum tnsil stss t at th sufac of th volcano occus at point A (with x = 0; Fig. ), and its magnitud is: 4 p R t () d R Th stss t may gnat tnsion factus at th sufac of th associatd composit volcano duing magma-chamb inflation, that is, incasd magmatic pssu. At th points x = ( d R ), howv, t bcoms compssiv. Its lagst compssiv valu, occus at points x = ( d R ) and has a magnitud of: p R c (4) ( d R ) Th absolut valu of c is qual to /8 of that of t. Th uptu and dyk injction nomally occu wh, at th bounday of th chamb, th tnsil stss concntation duing a paticula unst piod achs a maximum. Th tangntial o cicumfntial stss at th bounday of th chamb is: p ( tan ) (5) wh th angl is dfind in Fig.. Nomally, it is th upp pat of th chamb that uptus, so I focus on that pat h. Fom Eq. 5 it follows that th pak valu of, dnotd by b, occus wh th angl is maximum, namly at th points a and a wh th lin AQ is tangnt to th bounday of th magma chamb (Fig. ). At ths points, th maximum tnsil stss achs th valu: p ( d R ) b (6) d R c,

9 9 Fom Eqs. 5 and 6, w can conclud as follows: If d >.7 R, th maximum tnsil stss associatd with th chamb occus at its bounday, at points a and a (Fig. ), and is givn by Eq. 6. This stss fild favous dyk injction. If d <.7 R, th maximum tnsil stss associatd with th chamb occus not at its bounday but ath at th sufac of th associatd composit volcano, at point A (Fig. ), and is givn by Eq.. This stss fild favous th fomation of tnsion factus at th sufac and is unlikly to tigg dyk injction. If d =.7 R, th maximum tnsil stsss at th sufac of th composit volcano is qual to that at th bounday of th chamb, with a magnitud of p. b t At point B (Fig. ) th tnsil stss is qual to p, whas at points A, a and a th tnsil stss always xcds p. At points A, a and a th stss dpnds on th diffnc btwn th dpth to th cnt of th chamb d and th chamb adius R. Sinc Eqs. and 6 both hav d R in th dnominato, thn whn R d, that is, whn th dpth to th top (point B) of th chamb dcass, b and t can, thotically, bcom many tims gat than p. In natu, howv, th tnsil stss at th bounday of th chamb is limitd by th local tnsil stngth of th host ock, nomally MPa (Haimson and Rumml, 98; Schultz, 995). Whn th condition of Eq. is achd at points a and a, th will b sht o dyk injction that immdiatly laxs th tnsil stss at th bounday of th chamb. Sinc most chambs a at considabl dpths in compaison with thi adii, d would aly b vy simila to R. Also, tnsil stss concntation aound al thdimnsional chambs (cavitis) would nomally b lss than that indicatd by ths twodimnsional (hol) sults.. Cavitis subjct to intnal pssu An idal th-dimnsional magma chamb is an llipsoidal cavity (Gudmundsson, 988) with th shap of a sph, an oblat llipsoid, o a polat llipsoidal (Sadowsky and Stnbg, 947, 949; Tsuchida and Nakahaa, 970; Soutas-Littl, 97). Fo a sphical magma chamb compaativly clos to th sufac of th associatd composit volcano, that is, fo a cavity in an lastic half spac (with a f top sufac), analytical closd-fom solutions hav bn obtaind by K t al. (998). Similaly, Tsuchida and Nakahaa (970) povid analytical solutions fo a cavity in a smi-infinit plat (with f sufacs at its top and bottom) and lastic half spacs. Ths solutions a too complx to b givn h. Fo magma chambs at dpths that a gat in lation to thi sizs th xist simpl analytical solutions fo th stss filds. If th chamb adius R is much small than th distanc d to its cnt fom th f sufac of th composit volcano, thn th stsss at th chamb bounday and byond can b calculatd fom th wll-known quations fo a hollow sph und intnal pssu. In this modl, th chamb is subjct to a total magmatic pssu Pt p pl as th only loading (Eq. ). Th chamb adius is R and th magin of th lastic cust hosting th chamb is also assumd to b a sph with a adius R. Whn compad with R thn R is ffctivly infinit, so that R >> R, and th is lithostatic stss p l at R. Fo this analysis, w us sphical pola coodinats (,, ), wh is th adius vcto (distanc), θ is th angl btwn th adius vcto and a fixd axis z, and is th angl masud aound this axis (Fig. ). Th adial stss away fom th chamb and du to

10 0 th chamb xcss pssu is thn (Saada, 98; Gudmundsson, 00): R R Pt pl (7) Sphical symmty implis that th two oth pincipal stsss, and, a qual; thy a givn by: P t R pl R (8) If w us th xcss magma pssu and 8 may b wittn as: p (Eq. ) ath than th total pssu P t, Eqs. 7 R p (9) p R (0) Fo a vy small chamb, with R << d and R 0, but a finit p R, th intnsity of th point xcss pssu S of th chamb is givn by S = p R, th units bing Nm o wok (ngy). A point pssu of this kind is th basis of th Mogi modl (Mogi, 958), which was discussd abov. Fom Eqs. 9 and 0 it follows that th intnsity of th stss fild associatd with a sphical magma chamb subjct to magmatic xcss pssu p as th only loading falls of invsly as th cub of th distanc (adial vcto) fom th chamb. Whn w substitut R in Eqs. 9 and 0, th compssiv stss at th sufac of th magma chamb bcoms p whas th tnsil stss is 0.5 p (Fig. ).. Cavitis subjct to xtnal tnsion Whn a sphical magma chamb is subjct to xtnal tnsil stss, as is common in ift zons, th sulting stsss a vy diffnt fom thos gnatd by intnal magmatic pssu. Consid a sphical magma chamb locatd at a considabl dpth in a ift zon subjct to xtnal tnsil stss. In th coodinat systm dfind in Fig. 4, th unixial tnsil loading is paalll with th z-axis (Soutas-Littl, 97). Fo a ift-zon magma chamb, th tnsil loading diction nomally coincids with that of th spading vcto, and is thus hoizontal. Fo application to ift-zon chambs w thfo imagin that th sph is otatd by 90 so that th z-axis bcoms hoizontal and paalll with th spading vcto. Th chamb is initially in a lithostatic quilibium so that th vtical stss latd to ovbudn pssu is balancd by th magma pssu and may, thfo, b ignod in th analysis. It follows that th only loading that nds to b considd is th tnsil stss. Using th coodinats dfind in Fig. 4, th stsss at th sufac of th sph bcom (Goodi, 9; Timoshnko and Goodi, 970; Soutas-Littl, 97):

11 (7 5 ) 5 cos () (7 5 ) (7 5 ) (5 ) 5 cos () (7 5 ) (7 5 ) Th scond tms in Eqs. and bcom zo and th tnsil stsss thus maximum whn = 90º. Thus, fo = 90º, at th quatoial plan of th sph, th maximum tnsil stsss a: (7 5 ) () (7 5 ) (5 ) (4) (7 5 ) At th top and bottom of th sph, howv, th xtnal tnsil stss compssiv stss of a magnitud: gnats (5 ) (5) (7 5 ) Whn applid to a sphical chamb, it follows fom th considations abov that th top and bottom of th sph a along th hoizontal, quatoial plan of th al magma chamb; and, similaly, that th top and bottom of th chamb li along th quatoial plan of th sph. Consquntly, th compssiv stss obtaind fom Eq. 5 is gnatd at th quatoial plan of th ift-zon chamb whas th tnsil stsss obtaind fom Eqs. and 4 a gnatd at th top and bottom of th ift-zon chamb. Simpl, closd-fom analytical solutions do not xist fo fo calclulating th stss concntation aound a magma chamb of a gnal llipsoidal fom (Fig. 5). Fom quations divd by Sadowsky and Stnbg (947, 949) fo th-dimnsional llipsoidal cavitis, howv, som stimats of th stss concntations aound dp-satd llipsoidal magma chambs can b mad (Gudmundsson, 988). If a two-axial, polat llipsoidal magma chamb has hoizontal width c, hight (vtical o dip dimnsion) b, and lngth (stik dimnsion) a, th atio c/b is fd to as th shap atio of th magma chamb. Fo a two-axial chamb, th dimnsion a is qual to ith c o b. Th focus is on th uppmost pat of th chamb wh, as bfo, uptu and dyk injction is most likly to tak plac. H th tnsil stss concntation at th top of th polat llipsoidal magma chamb is calculatd fo host ocks with diffnt Poisson s atios: 0.5 and 0.0 (Fig. 6). Fo th spcial cas whn c/b = and a =, th chamb is sphical and th tnsil stss concntation can b calculatd fom Eqs. and 4. Whn th chamb has th shap of a tiaxial llipsoid, so that a b c (Fig. 5), th stss concntation facto k at point A on th llipsoidal magma chamb can b calculatd (Fig. 7). Fo a tiaxial magma chamb that is longat paalll with a divgnt plat bounday o a ift zon, two-dimnsional (hol) modls may oftn b usd to calculat th stss concntation. Fo xampl, if th lngth of th chamb paalll with th axis of th ift zon, that is, th stik dimnsion of th chamb, is much gat than its hight, that is, its dip dimnsion (so that a >> b), th tnsil stss at point A at th top of th chamb can b

12 calculatd appoximatly using th two-dimnsional fomula: b c (6) H, is th maximum pincipal tnsil stss, which occus at point A, and is th mot tnsil stss latd to plat pull. Fo xampl, fo c/b = 0. and a vy longat chamb (b/a 0.0), Eq. (6) givs, in agmnt with Fig. (7). Eq. (6) is simila to th on usd to calculat tnsil stsss at lliptical hols subjct to fluid ovpssu P 0 (thn P 0 is substitutd fo and th tm + in Eq. bcoms -), and is widly applid in studis of factus (Maugis, 000; Sanfod, 00). 4. Stsss naby chambs analytical modls Stss concntations associatd with a magma chamb a nomally confind to th host ocks at and naby th magin of th chamb. At distancs simila to o gat than th diamt of th chamb, th stsss bcom simila to that of th host ock without th chamb (Fig. ). Ths sults follow fom Eqs. 7-0 which show how th stss intnsity falls off with distanc fom a sphical magma chamb. Som composit volcanos may hav chambs with th shap of a vtical cylind. In paticula, chambs of this shap a likly to b patly sponsibl fo th commonly infd (pimialy) latal dyk popgation in som volcanic difics and gional dyk swams (Rubin, 995; Fialko and Rubin, 999; Enst t al., 00; Mg and Kom, 004; Pinl and Jaupat, 004; Klugl t al., 005; Acoclla t al., 006a,b). Extinct chambs, plutons, of this shap a fd to as plugs o ncks whn th diamt is fom mts to a fw thousand mts and as stocks fo lag diamts; th lagst diamts a 0- km. In Icland, sval plugs a known fom odd composit volcanos (.g., Walk, 96). Th stss fild aound a plug-lik magma chamb follows, appoximatly, fom th modl of a cylind (Fig. 8) of an inn adius R and out adius R. Sinc th adius of th composit volcano with th plug-lik chamb is vy much lag than th adius of th plug, w hav R >> R and may assum that R is ffctivly infinit. W us pola coodinats wh th adius vcto is dnotd by and th pola angl by. At its magin, R, th chamb is und total magma pssu P t (Eq. ) whas th compost volcano is subjct to hoizontal compssiv stss H at R. Fo isotopic stat of stss, th maximum ( H ) and th minimum ( ) hoizontal compssiv stsss a qual. Thn th adial compssiv h stss is (Saada, 98; Gudmundsson, 00): R R Pt H (7) Similaly, th tangntial o cicumfntial tnsil stss is: R R Pt H (8) Eqs. 7 and 8 can b simplifid whn th xcss magmatic pssu p is usd ath

13 than th total magmatic pssu P t (Fig. 8). In this cas, th xcss pssu is th diffnc btwn th total magmatic pssu in th cylindical chamb and th gional hoizontal compssiv stss H. Eq. indicats that th chamb will uptu and injct dyks whn p T 0, wh T 0 is th tnsil stngth of th host ock at th chamb magin. If H is qual to th lithostatic stss in th cust hosting th chamb, thn H = h = p l. Substituting p fo P t and putting H = 0 in Eqs. 7 and 8, th adial compssiv stss bcoms: R p (9) Similaly, th cicumfntial tnsil stss bcoms: R p (0) Claly, Eqs. 9 and 0 indicat that th intnsity of th stss fild gnatd by an xcss pssu p in a vtical, cylindical magma chamb falls off as th squa of th distanc fom th magin of th chamb (Fig. 8). Ths sults a simila to thos fo a sphical magma chamb (Fig. ). Thy show that duing unst piods with inflation of a magma chamb, th condition of dyk injction and popagation a commonly mt at, and in th vicinity of, th magma chamb whil th condition of dyk ast a mt at a ctain distanc fom th chamb. 5. Stsss naby chambs numical modls 5. Homognous, isotopic host ock Th analytical modls psntd abov assum th cust holding th chamb to b homognous and isotopic. Numical modls of magma chambs basd on such assumptions indicat th gnal stss concntations and th stss tajctois aound th chambs. All th numical modls in this pap w mad using th finit-lmnt pogam ANSYS and th bounday-lmnt pogam BEASY. A dsciption of th bounday-lmnt mthod in gnal, and that of th BEASY pogam in paticula, is givn by Bbbia & Dominguz (99) and th BEASY hompag ( Similaly, a gnal dsciption of h finit-lmnt mthod is povidd by Zinkiwicz (977) and that of th ANSYS pogam by Logan (00) and th ANSYS hompag ( Jing and Hudson (00) discuss th vaious numical mthods, including th bounday lmnt and th finit lmnt, in contxt of solving ock-mchanics poblms. Th ock stiffnsss usd in th modls a obtaind ith dictly fom laboatoy tsts (Camichal, 989; Afouz, 99; Bll, 000; Myvang, 00), o modifid using infomation on in-situ poptis (Fam, 98; Pist, 99; Schön, 004). Th bounday conditions usd in th modls, as spcifid blow, a obtaind fom th appopiat gological and gophysical fild studis. Th most basic of ths modls is th on of a magma chamb in a custal sgmnt that is vy lag in compaison with th siz of th chamb itslf. A two-dimnsional vsion of this modl shows th stss concntation (Fig. 9) and th potntial paths of dyks injctd fom th magma chamb, that is, th tajctois of th maximum compssiv pincipal stss (Fig. 0). In this and all subsqunt numical modls in this pap, th modls a fastnd at th cons, using th conditions of no displacmnt, so as to avoid igid-body otation and tanslation. Numical modls with small chambs, that is, modls with lag

14 4 computational domains, yild stss sults that a simila to thos obtaind in th modls in th psnt pap. Th stss fild aound magma chambs in a homognous, isotopic cust dpnds on th chamb location and shap, and its tctonic nvionmnt. Fo xampl, whn th chamb is na th sufac of its volcano, th sits of maximum tnsil stss concntation at th magin of th chamb shift towads th upp pat of th chamb (Fig. ). Th ason fo th shift in location of stsss is th vicinity of th ath s f sufac. Fo a chamb clos to th sufac, th sistanc to lastic dfomation of th chamb magin is lss in th diction of th sufac than in any oth diction. Consquntly, th dfomation and stss shift towads th upp magin of th chamb. Th tnds of th sulting dyks, howv, a not much affctd (Gudmundsson, 00). Th loading conditions can hav stong ffcts not only on th location of th sits of maximum stss concntations aound th chamb, but also on th tnds of th injctd dyks. Whn, fo instanc, a shallow magma chamb is pimaily loadd by xtnal tnsion, as is common at divgnt plat boundais and in ift zons, th main tnsil stss concntation occus at th top of th chamb (Fig. ). Also, th potntial paths of th dyks injctd fom th chamb a mostly vtical (Fig. ). By contast, th potntial paths of shts injctd fom a chamb subjct to magmatic xcss pssu as th pimaily loading a mostly inclind (Fig. 0). It should b mphasisd, howv, that th loading conditions, ath than th tctonic pocssd giving is to thm that is of impotanc fo magma chamb uptu and dyk injction. Fo xampl, fo a magma chamb subjct to xtnsion, th tnsil stss concntation aound th chamb, and th vntual uptu of th chamb, dos not dpnd much on whth th xtnsional gim is associatd with tanstnsion along a majo stik-slip fault o with ifting at a divgnt plat bounday. Th shap of a magma chamb also affcts th location of th main stss concntation aound it, as wll as th potntial paths of th injctd dyks and shts (Fig. 4). Thus, fo a sill-lik (oblat llipsoidal) and compaativly shallow chamb subjct to xcss magmatic pssu as th only loading, sht injction tnds to occu at th sits of maximum tnsil stss concntation, namly at th upp latal nds of th chamb (Fig. 4), and th th sht dip is shallow. A homognous, isotopic cust, howv, dos not ally xist. Modls wh th cust is assumd homognous and isotopic giv ctain idas as to stss filds aound magma chambs and associatd mplacmnt of dyks. But such modls a not suitabl fo th dtaild analysis of dyk paths. Futhmo, in a homognous, isotopic cust most buoyant magma-divn factus should ach th sufac to supply magma to uptions. By contast, fild obsvations show that most dyks bcom astd; thy nv ach th sufac to fd uptions (Hais t al., 000; Gudmundsson, 00, 00; Stwat t al., 00, 005; Gudmundsson and Bnn, 005; Gudmundsson and Philipp, 006). 5. Htognous, anisotopic host ock To undstand how th local stss filds aound magma chambs may ncouag dyk ast, w must us numical mthods. This follows pimaily bcaus not only a th stss filds complx, but th mchanical poptis of th ock lays that host th magma chambs a nomally highly vaiabl (Camichal, 989; Bll, 000; Myvang, 00; Gudmundsson, 00, 00; Schön, 004). Whn subjct to loading soft pyoclastic ocks, fo xampl, bhav vy diffntly fom stiff basaltic lava flows o sills, and both typs a common host ocks of magma chambs in composit volcanos. Th a thus abupt changs in mchanical poptis of th lays that constitut composit volcanos, and though which dyks must popagat on thi paths to th sufac. Ths abupt changs in poptis a th main asons fo th complx stss filds of composit volcanos and fo dyk ast. To xplo th ffcts of ths stss filds on dyk mplacmnt, w consid

15 5 sval numical modls of local stsss aound singl and doubl magma chambs of diffnt shaps and locatd in composit volcanos o custal sgmnts whos lays hav contasting stiffnsss. Th stiffnsss usd in th numical modls a basd on th following considations. Laboatoy masumnts indicat that lays of basalt, such as lava flows, and gabbo, such as lag intusions fo xampl, solidifid pats of mafic magma chambs (Fig. 5) ach stiffnsss of 0-0 GPa; volcanic tuffs, by contast, hav stiffnsss as low as GPa (Afouz, 99; Bll, 000). Mo xtm valus a known, howv. Fo xampl, som ocks ach stiffnsss of up to GPa (Myvang, 00). By contast, unconsolidatd ocks may hav static stiffnsss as low as 0.08 GPa fo sand and gavl and 0.00 GPa fo clay (Schön, 004). Poisson s atios of th ocks that commonly constitut composit volcanos, howv, hav a much naow ang. Poisson s atio of many basaltic lava flows is, fo instanc, th sam as that of many volcanic tuffs, o about 0.5 (Bll, 000). In Icland, most activ composit (cntal) volcanos contain high popotion of hyaloclastit (mobg), a pyoclastic ock mad of basaltic bccias and tuffs fomd in subglacial and submain uptions. Static laboatoy stiffnsss of young hyaloclastits a mostly btwn 0.5 GPa and 8 GPa (Jonsson, 97; Oddson, 984; Egilsson t al., 989). Sdimntay ocks, mainly tillits of lat Plistocn ag which a common outsid and insid th composit volcanos, hav simila stiffnsss whas, as indicatd abov, unconsolidatd sdimnts, common in som composit volcanos and ift zons, hav much low stiffnsss (Schön, 004). Na-sufac Holocn and Plistocn basaltic lava flows hav static stiffnsss mostly btwn 0 GPa and 5 GPa (Oddson, 984; Egilsson t al., 989). Th stiffnsss of ths ocks gnally incas with ag and dpth of buial. Th fist two layd modls (Figs. 6, 7) show th ffcts of magma-chamb gomty on th local stsss in a composit volcano. Th magma chamb itslf is locatd in a ock unit of a stiffnss typical fo th cust at th dpth of a fw kilomts, 40 GPa, whas th lays abov that unit fom a pat of th composit volcano and altnat btwn bing vy soft, GPa, and vy stiff, 00 GPa. This is a vy lag vaiation in stiffnss, but still wll within th limits obtaind in laboatoy masumnts, as indicatd abov. Th vy soft lays may psnt tuffs, sdimnts, scoia and soils btwn lava flows whas th vy stiff lays psnt basaltic and intmdiat lava flows and intusions. In th fist layd modl, a magma chamb of a cicula coss sction is subjct to intnal xcss magmatic pssu of 5 MPa as th only loading (Fig. 6). This xcss pssu is chosn so as to b simila to th in situ tnsil stngth of a typical cystallin host ock, nomally MPa (Schulz, 995; Amadi and Stphansson, 997). Th siz of th chamb in lation to th custal sgmnt and lay thicknss is as indicatd in Fig. 6. Th chamb is locatd in a custal sgmnt consisting of svn lays of vy diffnt stiffnsss; GPa (th thin lays), 40 GPa (th lay hosting th chamb), and 00 GPa (th thick lays). Th sam lay stiffnsss and magmatic pssu a usd in th modl in Fig. 7, th only diffnc bing th gomty of th chamb. Compaison of th two modls (Figs. 6, 7) shows th ffct of magma-chamb gomty on th local stss fild, as psntd by th tajctois of th maximum pincipal compssiv stss. Sinc idal dyks tnd to follow th - tajctois, thos injctd fom th cicula chamb in Fig. 6 would b stply inclind o vtical up to th upp contact btwn th cntal thin soft lay and th scond thick lay. Th th - tajctois abuptly bcom hoizontal, in which cas th popagating dyks might chang into sills. This follows sinc most dyks a xtnsion factus so that if a dyk is to continu its popagation in a pat of a custal sgmnt wh is hoizontal, it would hav to popagat in that pat as a hoizontal sill. Mo likly, howv, on mting a pat of a lay o a contact wh wh is hoizontal, a dyk would bcom astd. In th maginal upp

16 6 pats of th modl, howv, th - tajctois main inclind though all th lays, making it possibl fo som inclind shts to ach th uppmost lay and, phaps, th sufac. Fo th sill-lik chamb (Fig. 7) th abupt chang fom vtical to hoizontal - tajctois occus alady in th fist soft lay abov th magma chamb. Thus, fo th givn laying and loading conditions, th diffnc in gomty btwn th chambs in Fig. 6 and Fig. 7 sults in vy diffnt local stss filds. And ths diffnt local stss filds, in tun, ncouag diffnt dyk-injction paths duing unst piods. In paticula, fo th sill-lik gomty (Fig. 7) most dyks injctd fom th upp cntal pat of th chamb would bcom astd just abov its magin, whas som inclind shts injctd fom na its latal nds might popagat to shallow dpths in th cust, possibly aching th sufac to fd uptions. Many shallow magma chambs in Icland and lswh a at dpths of only.5- km blow th sufac of th associatd volcano. Fo a chamb with a top at.5 km dpth, th stiff lays in th modls in Figs. 6 and 7 would b about 80 m thick ach and th thin, soft lays about 90 m. In Icland, hyaloclastit lays foming pats of mountains and th thickst basaltic (pahoho) lava flows may ach thicknsss of sval hundd mts. Th thicknsss of most lays constituting composit volcanos and ift zons, howv, a of th od of tns of mts o lss ath than hundds of mts. To study th ffcts of thinn lays, a 0-lay modl was mad (Fig. 8). Fo a shallow cicula chamb with a top at.5 km dpth, th thicknss of ach of th 0 lays would b 50 m, simila to that of many pyoclastic and sdimntay lays and thick lava flows. As in th modls abov, th lays hav altnating stiffnsss of GPa and 00 GPa whas th chamb xcss pssu is h 0 MPa. Th sults (Fig. 8) show that abupt stss changs occu at many contacts btwn soft and stiff lays, paticulaly in th cntal pat of th modl abov th chamb. Thus, fo ths loading conditions and ock-mchanical poptis many dyks injctd duing unst piods would tnd to bcom astd and uptions pvntd. Th sults also suppot th gnal conclusion basd on fild studis of dyks in odd composit volcanos (Gudmundsson, 00, 00; Gudmundsson and Bnn, 005; Gudmundsson and Philipp, 006) and thotical studis of composit matials (Danil and Ishai, 994; Tan, 994; Kaw, 997; Hy, 998) that contasts in mchanical poptis btwn lays ath than absolut lay thicknss contol th local stsss at lay contacts and, thby, factu ast. Although na-sufac lays may b hoizontal in composit volcanos, and paticulaly in ift zons, mo commonly th lays a inclind (Figs. 4, 5). In composit volcanos that fom topogaphic highs, th uppmost lays may b outwad dipping, but bcom inwad dipping at ctain dpths (Fig. 5). Th sam applis to ift zons. Th inwad dip is pimaily du to th additional load gnatd by th xtusiv and intusiv ock which lads to gadual down-bnding of th associatd custal sgmnt. Fo xampl, in th palaoift zons in Icland th dips of th lava flows incas, on avag, about fo vy 50 m of custal dpth (Fig. 4). At dpth th lays thus dip towads th cnt of th composit volcano o th axis of th ift zon. In many dply odd composit volcanos th cnts towads which th lays dip a occupid by plutons that psnt th uppmost pats of th xtinct, shallow chambs of th volcanos (Figs. 8, 9, 5). Th ffcts of th gadual tilting of th lays with dpth in composit volcanos and ift zons is considd in th modl in Fig. 9. Th stiffnsss a th sam as in th pvious modls, GPa fo th soft lays and 00 GPa fo th stiff lays. Th magma chamb is.5 units in diamt, o.5 km in a 0-km-thick cust, and th only loading is a magmatic xcss pssu of 5 MPa. It should b notd that th gomtis of th stsss a not much dpndnt on whth th magmatic xcss pssu is 5 MPa o 0 MPa.

17 7 In th cntal pat of th custal sgmnt abov th top of th magma chamb th - tajctois clos to and at th contacts btwn th stiff and th soft lays a subpaalll to th contacts. On nting ths local stss filds, popagating dyks would thus tnd to chang into sills (Fig. ) o stop altogth, that is, bcom astd. Also, th dyks that nvthlss mak it to th sufac would not ach it abov th top of th chamb but in an aa shiftd in th dip-diction of th lava pil (Fig. 0). Thus, tilting of th lava pil may b on ason fo th common shifting of th volcanic systms in th down-dip diction of th lava pil as is wll known fom Icland (Hlgason, 985). Many and phaps most composit volcanos a supplid with magma though doubl magma chambs. A shallow chamb is thn fd by a dp-satd chamb, locatd in th low cust o at th cust-mantl bounday (Fig. 5). H I psnt two numical modls of a doubl magma chamb locatd in a layd cust (Figs., ). Th modls a idntical xcpt fo th loading conditions. Th custal sgmnt hosting th shallow sill-lik chamb is 6-7 km thick, and thus appopiat fo fast-spading idgs such as th East Pacific Ris. Th shallow chamb is supplid with magma fom a dp, flat svoi, locatd at th bottom of th custal sgmnt. Th stiffnsss of th lays gadually incas with dpth, fom 0 GPa fo th sufac lay to 50 GPa fo th bottom lay, that is, th oof of th dp svoi. Also, in accodanc with obsvations, all th lays dip towads th cnt of th composit ift zon and th dips gadually incas with dpth. Th local stsss, as indicatd by th - tajctois, a vy diffnt in th two modls. In th modl in Fig. th only loading is intnal magmatic xcss pssu of 5 MPa, both in th dp-satd svois as wll as in th shallow chamb. Th - tajctois tnd to b somwhat inclind abov th maginal pat of th dp-satd svoi, but much mo so in th vicinity of th shallow chamb. By contast, whn th only loading is xtnal tnsion of 5 MPa, th - tajctois main mostly vtical and lagly unaffctd by th shallow chamb (Fig. ). Ths sults a in agmnt with thos obtaind in ali modls of shallow magma chambs (Figs. 0,, 4). Claly, whn th pimay loading is intnal magmatic xcss pssu, th shap of th shallow chamb has gat ffcts on th - tajctois and, thfo, th popagation dictions of th dyks. By contast, whn th pincipal loading is xtnal tnsion, as is common duing many ifting pisods at divgnt plat boundais, th - tajctois tnd to b oughly ppndicula to th diction of th spading vcto, that is, th local tnd of, and thfo subvtical. Thus, loading though xcss magmatic pssu tnds to favou inclind shts whas loading though xtnal tnsion tnds to favou subvtical dyks. 6. Dyk injction fom a dp-satd svoi Consid fist longat magma svois of th typ that psumably undli lag pats of th ift-zon volcanic systms such as at divgnt plat boundais in gnal and in th ift zon of Icland in paticula (Figs. 5, 0, ). Th conditions fo svoi uptu and dyk injction, givn by Eq., a psumably commonly achd though duction of in th oof of th svoi. This duction is dictly latd to plat movmnts which gnat lativ o absolut tnsil stss concntations in th svoi oof as a consqunc of its shap and plat pull (Eq. 6). Th on-dimnsional Hook s law in lasticity lats tnsil stss and stain though th quation: E ()