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Volcanic Eruptions: Cyclicity during Lava Dome Growth 1
Volcanic Eruptions:
Cyclicity during Lava Dome Growth
OLEG MELNIK
1,2
, R. STEPHEN J. SPARKS
2
, 1
ANTONIO COSTA
2,3
, ALEXEI A. BARMIN
1
2
1
Institute of Mechanics, Moscow State University, 3
Moscow, Russia 4
2
Earth Science Department, University of Bristol, 5
Bristol, UK 6
3
Istituto Nazionale di Geoﬁsica e Vulcanologia, 7
Naples, Italy 8
Article Outline 9
Glossary 10
Deﬁnition of the Subject 11
Introduction 12
Dynamics of Magma Ascent During Extrusive Eruptions 13
ShortTerm Cycles 14
LongTerm Cycles 15
Future Directions 16
Acknowledgments 17
Bibliography 18
Glossary 19
Andesite Magma or volcanic rock is characterized by in 20
termediate SiO
2
concentration. Andesite magmas have 21
rheological properties that are intermediate between 22
basalt and rhyolite magmas. Silica content in andesites 23
ranges from approximately 52 to 66 weight percent. 24
Common minerals in andesite include plagioclase, am 25
phibole and pyroxene. Andesite is typically erupted at 26
temperatures between 800 to 1000°C Andesite is par 27
ticularly common in subduction zones, where tectonic 28
plates converge and water is introduced into the man 29
tle. 30
Basalt Magma or volcanic rock contains not more than 31
about 52% SiO
2
by weight. Basaltic magmas have 32
a low viscosity. Volcanic gases can escape easily with 33
out generating high eruption columns. Basalt is typi 34
cally erupted at temperatures between 1100 to 1250°C. 35
Basalt ﬂows cover about 70% of the Earth’s surface and 36
huge areas of the terrestrial planets and so are the most 37
important of all crustal igneous rocks. 38
Bingham liquid is a ﬂuid that does not ﬂow in response 39
to an applied stress until a critical yield stress is 40
reached. Above the critical yield stress, strain rate is 41
proportional to the applied stress, as in a Newtonian 42
ﬂuid. 43
Bubbly ﬂow A multiphase ﬂow regime, in which the gas 44
phase appears as bubbles suspended in a continuous 45
liquid phase. 46
Conduit A channel, through which magma ﬂows towards 47
the Earth’s surface. Volcanic conduits can commonly 48
be approximately cylindrical and typically a few 10’s 49
meters across or bounded by near parallel sides in 50
a magmaﬁlled fracture. Conduits can be vertical or in 51
clined. 52
Crystallization Conversion, partial or total, of a silicate 53
melt into crystals during solidiﬁcation of magma. 54
Degassing n. (degas v.) The process by which volatiles 55
that are dissolved in silicate melts come out of solu 56
tion in the form of bubbles. Open and closedsystem 57
degassing can be distinguished. In the former, volatiles 58
can be lost or gained by the system. In the latter, the 59
total amount of volatiles in the bubbles and in solution 60
in the magma is conserved. 61
Diﬀerentiation The process of changing the chemical 62
composition of magma by processes of crystallization 63
accompanied by separation melts from crystals. 64
Dome A steepsided, commonly bulbous extrusion of 65
lava or shallow intrusion (cryptodome). Domes are 66
commonly, but not exclusively, composed of SiO
2
 67
rich magmas. In domeforming eruptions the erupted 68
magma is so viscous, or the discharge rate so slow, 69
that lava accumulates very close to the vent region, 70
rather than ﬂowing away. Pyroclastic ﬂows can be gen 71
erated by collapse of lava domes. Recent eruptions pro 72
ducing lava domes include the 1995–2006 eruption of 73
the Soufrière Hills volcano, Montserrat, and the 2004– 74
2006 eruption of Mount St. Helens, USA. 75
Dyke A sheetlike igneous intrusion, commonly vertical 76
or near vertical, that cuts across preexisting, older, ge 77
ological structures. During magmatism, dykes trans 78
port magma toward the surface or laterally in fracture 79
like conduits. In the geologic record, dykes are pre 80
served as sheetlike bodies of igneous rocks. 81
Explosive eruption A volcanic eruption in which gas ex 82
pansion tears the magma into numerous fragments 83
with a wide range of sizes. The mixture of gas and 84
entrained fragments ﬂows upward and outward from 85
volcanic vents at high speed into the atmosphere. De 86
pending on the volume of erupted material, erup 87
tion intensity and sustainability, explosive eruptions 88
are classiﬁed as Strombolian, Vulcanian, subPlinian, 89
Plinian or Mega–Plinian; this order is approximately 90
in the order of increasing intensity. Strombolian and 91
Vulcanian eruptions involve very shortlived explo 92
sions. 93
Please note that the pagination is not ﬁnal; in the print version an entry will in general not start on a new page.
Editor’s or typesetter’s annotations (will be removed before the ﬁnal T
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2 Volcanic Eruptions: Cyclicity during Lava Dome Growth
Extrusive ﬂow or eruption A nonexplosive (nonpyro 94
clastic) magma ﬂow from a volcanic conduit during 95
a lava domebuilding eruption or lava ﬂow. 96
Maﬁc Magma, lava, or tephra with silica concentrations 97
of approximately SiO
2
<55%. 98
Magma Molten rock that consists of up three compo 99
nents: liquid silicate melt, suspended crystalline solids, 100
and gas bubbles. It is the raw material of all vol 101
canic processes. Silicate magmas are the most common 102
magma type and consist of long, polymeric chains and 103
rings of Si–O tetrahedra, between which are located 104
cations (e. g. Ca
2+
, Mg
2+
, Fe
2+
, and Na
+
). Anions (e. g. 105
OH
, F
, Cl
, and S
) can substitute for the oxygen 106
in the silicate framework. The greater the silica (SiO
2
) 107
content of the magma, the more chains and rings of 108
silicate tetrahedra there are to impede each other and 109
hence the viscosity of the magma increases. The pres 110
sure regime and composition of the magma control the 111
minerals that nucleate and crystallize from a magma 112
when it cools or degasses. 113
Magma chamber A subsurface volume within which 114
magma accumulates, diﬀerentiates and crystallizes. Ig 115
neous intrusions can constrain the form and size of 116
some magma chambers, but in general the shape and 117
volume of magma chambers beneath active volcanoes 118
are poorly known. Magma reservoir is an equivalent 119
term. 120
Melt Liquid part of magma. Melts (usually silicate) con 121
tain variable amounts of dissolved volatiles. The pri 122
mary volatiles are usually water and carbon dioxide. 123
Newtonian liquid A liquid for which the strain rate is 124
proportional to the applied stress. The proportionality 125
coeﬃcient is called the viscosity. 126
Microlite Crystal with dimensions less than 100 µm. Usu 127
ally microlites crystallize at shallow levels of magmatic 128
system. 129
Phenocryst Crystal with dimensions larger than 100 µm. 130
Usually phenocrysts growin magmatic reservoirs prior 131
to an eruption and or are entrained by magma in the 132
chamber. 133
Pyroclastic ﬂow or surge A gasparticle ﬂow of pyro 134
clasts suspended in a mixture of hot air, magmatic 135
gas, and ﬁne ash. The ﬂow originates by the gravita 136
tional collapse of a dense, turbulent explosive eruption 137
column at the source vent, or by dome collapse, and 138
moves downslope as a coherent ﬂow. Pyroclastic ﬂows 139
and surges are distinguished by particle concentration 140
in the ﬂow, surges being more dilute. Variations in par 141
ticle concentration result in diﬀerences in the deposits 142
left by ﬂows and surges. 143
Silicic Magma, lava, or tephra with silica concentrations 144
of approximately SiO
2
> 55%. The magmas are com 145
monly rich in Al, Na÷ and K÷ bearing minerals. Sili 146
cic magmas are typically very viscous and can have 147
high volatile contents. Rhyolite is an example of a sili 148
cic magma. 149
Volatile A component in a magmatic melt which can be 150
partitioned in the gas phase in signiﬁcant amounts 151
during some stage of magma history. The most com 152
mon volatile in magmas is water vapor H
2
O, but there 153
are commonly also signiﬁcant quantities of CO
2
, SO
2
154
and halogens. 155
Definitionof the Subject 156
We consider the process of slow extrusion of very viscous 157
magma that forms lava domes. Domebuilding eruptions 158
are commonly associated with hazardous phenomena, in 159
cluding pyroclastic ﬂows generated by dome collapses, 160
explosive eruptions and volcanic blasts. These eruptions 161
commonly display fairly regular alternations between pe 162
riods of high and low or no activity with time scales from 163
hours to years. Usually hazardous phenomena are asso 164
ciated with periods of high magma discharge rate, thus, 165
understanding the causes of pulsatory activity during ex 166
trusive eruptions is an important step towards forecasting 167
volcanic behavior, especially the transition to explosive ac 168
tivity when magma discharge rate increases by a feworders 169
of magnitude. In recent years the risks have increased be 170
cause the population density in the vicinity of many active 171
volcanoes has increased. 172
Introduction 173
Many volcanic eruptions involve the formation of lava 174
domes, which are extrusions of very viscous, degassed 175
magmas. The magma is so viscous that it accumulates 176
close to the vent. Extrusion of lava domes is a slow and 177
longlived process, and can continue for many years or 178
even decades [71,83,85]. Typical horizontal dimensions of 179
lava domes are several hundred meters, heights are of an 180
order of tens to several hundred meters, and volumes sev 181
eral million to hundreds of million cubic meters. Typical 182
magma discharge rates (measured as the increase of dome 183
volume with time in dense rock equivalent (DRE)) can 184
reach up to 20–40 m
3
/s, but are usually below10 m
3
/s [83]. 185
Domebuilding eruptions are commonly associated 186
with hazardous phenomena, including pyroclastic ﬂows 187
and tsunamis generated by dome collapses, explosive 188
eruptions and volcanic blasts. Domebuilding eruptions 189
can also contribute to ediﬁce instability and sector 190
collapse, as occurred on Montserrat on 26 December 191
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 3
1997 [87]. Lava dome activity can sometimes precede 192
or follow major explosive eruptions; the eruption of 193
Pinatubo, Philippines (1991) is an example of the for 194
mer [37], and the eruption of Mount St. Helens, USA 195
(1980–1986) is an example of the latter [89]. 196
Several lava dome eruptions have been documented 197
in detail and show quite complex behaviors. Substantial 198
ﬂuctuations in magma discharge rate have been docu 199
mented. In some cases these ﬂuctuations can be quite 200
regular (nearly periodic), as in the extrusion of lava in 201
1980–1982 on Mount St. Helens [89] and in the 1922–2002 202
activity of the Santiaguito lava dome, Guatemala [35]. In 203
these cases, periods of high magma discharge rate alternate 204
with longer periods of low magma discharge rate or no ex 205
trusion. In some volcanoes, such as Shiveluch, Kamchatka, 206
the intervals of no extrusion are so long compared with the 207
periods of dome growth that the episodes of dome growth 208
have been described as separate eruptions of the volcano 209
rather than episodes of the same eruption. Other dome 210
building activity can be nearly continuous and relatively 211
steady, as observed at Mount St. Helens in 1983 [89] and at 212
the Soufrière Hills Volcano, Montserrat between Novem 213
ber 1999 and July 2003. In yet other cases the behavior can 214
be more complex with quite sudden changes in magma 215
discharge rate, which cannot be related to any welldeﬁned 216
regularity or pattern (e. g. Lascar volcano, Chile, [57]). 217
Pauses during lava domebuilding eruptions are quite 218
common. For example, at Mount St. Helens there were 219
9 pulses of dome growth with a period of ~74 days, a du 220
ration of 1–7 days and no growth in between [89]. The 221
Soufrière Hills Volcano Montserrat experienced a long 222
(20 months) pause in extrusion after the ﬁrst episode of 223
growth [72]. On Shiveluch volcano in Kamchatka episodes 224
of dome growth occurred in 1980, 1993 and 2000, follow 225
ing a major explosion in 1964 [28]. Each episode of dome 226
growth began with magma discharge rate increasing over 227
the ﬁrst few weeks to a peak of 8–15 m
3
/s, with a grad 228
ual decline in magma discharge rate over the following 229
year. In between the episodes very minimal activity was 230
recorded. 231
Fluctuations in magma discharge rate have been docu 232
mented on a variety of timescales from both qualitative 233
and quantitative observations. Several lava dome erup 234
tions are characterized by extrusion of multiple lobes and 235
ﬂow units [68,94]. In the case of the Soufrière Hills Vol 236
cano, extrusion of shear lobes can be related to spurts 237
in discharge rate and is associated with other geophysi 238
cal changes, such as onset of seismic swarms and marked 239
changes in temporal patterns of ground tilt [90,91,94]. 240
These spurts in discharge rate have been fairly regular 241
for substantial periods, occurring every 6 to 7 weeks over 242
a 7 month period in 1997 [21,87,91]. These spurts are com 243
monly associated with large dome collapses and pyroclas 244
tic ﬂows and, in some cases, with the onset of periods 245
of repetitive Vulcanian explosions [14,26]. Consequently 246
the recognition of this pattern has become signiﬁcant for 247
forecasting activity for hazard assessment purposes. In 248
the Soufrière Hills Volcano and Mount Pinatubo much 249
shorter ﬂuctuations in magma discharge rate have been 250
recognized from cyclic variations in seismicity, ground tilt, 251
gas ﬂuxes and rockfall activity [23,91,93]. This cyclic ac 252
tivity has typical periods in the range of 4 to 36 hours. 253
Cyclic activity has been attributed to cycles of gas pressur 254
ization and depressurization with surges in dome growth 255
related to degassing, rheological stiﬀening and stickslip 256
behavior [23,49,61,91,98]. 257
Dome eruptions can show transitions to explosive ac 258
tivity, which sometimes can be linked to spurts in magma 259
discharge rate. For example, in 1980, periodic episodes 260
of lava dome extrusion on Mount St. Helens were initi 261
ated by explosive eruptions, which partly destroyed the 262
dome that had been extruded in each previous extru 263
sion episode [89]. At Unzen Volcano, Japan a single Vul 264
canian explosive eruption occurred in June 1991 when 265
the magma discharge rate was at its highest [68]. At the 266
Soufrière Hills Volcano, repetitive series of Vulcanian ex 267
plosions have occurred following large dome collapses in 268
periods when magma discharge rates were the highest of 269
the eruption [26,88]. In the case of Lascar Volcano, Chile, 270
an intense Plinian explosive eruption occurred on 18 and 271
19 April, 1993, after nine years of dome extrusion and oc 272
casional shortlived Vulcanian explosions [57]. 273
Lava dome eruptions require magma with special 274
physical properties. In order to produce a lava dome 275
rather than a lava ﬂow, the viscosity of the magma must 276
be extremely high so that the lava cannot ﬂow easily 277
from the vent. High viscosity is a consequence of factors 278
such as relatively low temperature (typically 750–900°C), 279
melt compositions rich in networkforming components 280
(principally Si and Al) eﬃcient gas loss during magma de 281
compression, and crystallization as a response to cooling 282
and degassing. Viscosities of silicarich magmas, such as 283
rhyolites and some andesites, are increased by several or 284
ders of magnitude by the loss of dissolved water during 285
decompression. Many, but not all, domes also have high 286
crystal content (up to 60 to 95 vol%), with crystallization 287
being triggered mostly by degassing [10,86]. In order to 288
avoid fragmentation that leads to an explosive eruption, 289
magma must have lost gas during ascent. Consider, for ex 290
ample, a magma at 150 MPa containing 5wt% of dissolved 291
water decompressed to atmospheric pressure. Without gas 292
loss the volume fraction of bubbles, will be more than 293
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4 Volcanic Eruptions: Cyclicity during Lava Dome Growth
99%. Typical dome rock contains less than 20 vol% of bub 294
bles, although there is evidence that magma at depth can 295
be more bubblerich (e. g., [13,74]). On the other hand, 296
very commonly there is no change in temperature or bulk 297
magma composition in the products of explosive and ex 298
trusive eruptions for a particular volcano. This suggests 299
that the properties of magma that are conducive to the for 300
mation of lava domes are controlled by physicochemical 301
transformations that occurred during magma ascent to the 302
surface. 303
Two other important factors that inﬂuence whether 304
lava domes or ﬂows form are topography and discharge 305
rate. The same magma can form a dome if the discharge 306
rate is low, and a lava ﬂow if the rate is high [29,92]. The 307
discharge rate is controlled by overall conduit resistance 308
that is a function of viscosity, conduit size and shape, and 309
driving pressure (the diﬀerence betweenchamber pressure 310
and atmospheric pressure). Additionally the same magma 311
can form a dome on low slopes, such as a ﬂat crater (e. g. 312
the maﬁc andesite dome of the Soufrière Volcano, St. Vin 313
cent; [40]) and a lava ﬂow on steep slopes. 314
Prior to an eruption, magma is usually stored in a shal 315
low crustal reservoir called a magma chamber. For sev 316
eral volcanoes magma chambers can be detected and 317
characterized by earthquake locations, seismic tomogra 318
phy, petrology or interpretation of ground deformation 319
data [55]. Typical depths of magma chambers range from 320
a few kilometers to tens of kilometers. Volumes range 321
from less than one to several thousand km
3
[55], but are 322
usually less than a hundred km
3
. Magma chambers are 323
connected to the surface by magma pathways called con 324
duits. There is evidence that the conduits that feed lava 325
dome eruptions can be both dykes or cylindrical. Dykes of 326
a few meters width are commonly observed in the interior 327
of eroded andesite volcanoes. Dyke feeders to lava domes 328
have been intersected by drilling at Inyo crater, Califor 329
nia, USA[56] and at Mount Unzen [67]. Geophysical stud 330
ies point to dyke feeders; for example faultplane solutions 331
of shallow volcanotectonic earthquakes indicate pressure 332
ﬂuctuations in dykes [75,76]. Deformation data at Unzen, 333
combined with structural analysis, indicate that the 1991– 334
1995 dome was fed by a dyke [68]. Dykes are also the only 335
viable mechanism of developing a pathway through brit 336
tle crust from a deep magma chamber to the surface in the 337
initial stages of an eruption [50,77]. 338
Cylindrical conduits commonly develop during lava 339
dome eruptions. The early stages of lava dome eruptions 340
frequently involve phreatic and phreatomagmatic explo 341
sions that create near surface craters and cylindrical con 342
duits [12,73,87,89,96,99]. These explosions are usually at 343
tributed to interaction of magma rising along a dyke with 344
ground water. Cylindrical conduits formed by explosions 345
are conﬁned to relatively shallow parts of the crust, proba 346
bly of order hundreds of meters depth and <1 km, as indi 347
cated by mineralogical studies [73]. Examples of such ini 348
tial conduit forming activity include Mount Usu (Japan) 349
Mt. St. Helens, and Soufrière Hills Volcano [12,87,99]. 350
Many lava dome eruptions are also characterized by Vul 351
canian, subPlinian and even Plinian explosive eruptions. 352
Examples include Mount Unzen, Mount St.Helens, San 353
tiaguito and Soufrière Hills Volcano [68,87,89,96]. Here 354
the fragmentation front may reach to depths of several 355
kilometers [54] with the possibility of cylindrical con 356
duit development due to severe underpressurization and 357
mechanical disruption of conduit wallrocks. Subsequently 358
domes can be preferentially fed along the cylindrical con 359
duits created by earlier explosive activity. On the Soufrière 360
Hills Volcano, early dome growth was characterized by ex 361
trusion of spines with nearly cylindrical shape [87]. 362
Observations of magma discharge rate variations on 363
a variety of timescales highlight the need to understand 364
the underlying dynamic controls. Research has increas 365
ingly focused on modeling studies of conduit ﬂow dynam 366
ics during lavadome eruptions. We will restrict our dis 367
cussions here to mechanisms that lead to cyclic and quasi 368
periodic ﬂuctuations in magma discharge rate on various 369
timescales, mainly focusing on longterm cycles. Issues 370
concerning the transition between explosive and extrusive 371
activity are discussed in detail in [45,80,81,97] and in the 372
special volume of Journal of Volcanology and Geother 373
mal Research dedicated to modeling of explosive erup 374
tions [79]. Several papers consider the processes that oc 375
cur at the surface and relate dome morphology and dimen 376
sions with controlling parameters. 377
Combined theoretical, experimental and geological 378
studies identify four main types of dome: spiny, lobate, 379
platy, and axisymmetric [5,29,30]. These types of dome re 380
ﬂect diﬀerent regimes which are controlled by discharge 381
rates, cooling rates and yield strength, and the viscosity 382
of the domeforming material. In recent years mathemati 383
cal modeling has beenused to semiquantitatively describe 384
spreading of lava domes, including models based on the 385
thin layer approximation [1,2] and fully 2D simulations 386
of lava dome growth which account for viscoelastic and 387
plastic rheologies [32,33]. 388
Dynamics of Magma Ascent 389
During Extrusive Eruptions 390
In order to understand the causes of cyclic behavior dur 391
ing extrusive eruptions, ﬁrst we need to consider the un 392
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 5
derling dynamics of volcanic systems, and discuss physical 393
and chemical transitions during magma ascent. 394
The physical framework for the model of a volcanic 395
system is shown in Fig. 1. Magma is stored in a chamber 396
at depth L, with a chamber pressure P
ch
that is higher than 397
hydrostatic pressure of magma column and drives magma 398
ascent. Magma contains silicate melt, crystals and dis 399
solved and possibly exsolved volatiles. During ascent of the 400
magma up the conduit the pressure decreases and volatiles 401
exsolve forming bubbles. As the bubble concentration be 402
comes substantial, bubble coalescence take place and per 403
meability develops [27,45], allowing gas to escape from 404
ascending magma, both in vertical (through the magma) 405
and horizontal (to conduit wallrocks) directions. If magma 406
ascends slowly, gas escape results in a signiﬁcant reduc 407
tion of the volume fraction of bubbles; such a process is 408
termed open system degassing. In comparison closed sys 409
tem degassing is characterized by a negligible gas escape. 410
Reduction in bubble content, combined with relatively low 411
gas pressures and eﬃcient decoupling of the gas and melt 412
phases, prevents magma fragmentation and, thus, devel 413
opment of explosive eruption [59,60,82]. 414
Due to typical low ascent velocities (from millime 415
ters to a few centimeters per second) magma ascent 416
times to the Earth’s surface from the magma chamber 417
range from a several hours to many weeks. These ascent 418
times are often comparable with the times that are re 419
quired for crystals to grow signiﬁcantly and for heat ex 420
change between magma and wallrocks. The main driving 421
force for crystallization is related to exsolution of volatiles 422
from the magma, leading to increase in the liquidus tem 423
perature T
L
, and development of magma undercooling 424
^T = T
L
– T [10]. Crystallization leads to release of latent 425
heat, and magma temperature can increase with respect to 426
the initial temperature [6]. As a consequence of increasing 427
crystal content, magma viscosity increases by several or 428
ders of magnitude [16,20,21] and magma becomes a non 429
Newtonian ﬂuid [78]. As will be shown later, crystalliza 430
tion induced by degassing can become a key process in 431
causing variable ﬂow rates. 432
Due to the long duration of extrusive eruptions, 433
the magma chamber can be replenished with signiﬁcant 434
amounts of magma from underlying sources [41,65]. Re 435
plenishment can lead to pressure buildup in the magma 436
chamber, volatile and heat exchange betweenhost and new 437
magmas. The composition of the magma can also change 438
over time, due to diﬀerentiation, crustal rock assimilation, 439
or magma mixing. Thus, any model that explains magma 440
ascent dynamics needs to deal with many complexities. 441
Of course there is no single model that can take into ac 442
count all physical processes in a volcanic system. Addi 443
tional complications arise from the fact that the physical 444
properties of magma at high crystal contents, such as rhe 445
ology or crystal growth kinetics and geometry of volcanic 446
systems, are typically poorly constrained. Several issues re 447
garding the dynamics of multiphase systems have not been 448
resolved theoretically, especially for cases where the vol 449
ume fractions of components of the multiphase system are 450
comparable. 451
Below we will present a review of existing models that 452
treat cyclic behavior during extrusive eruptions on diﬀer 453
ent timescales. 454
ShortTermCycles 455
Cyclic patterns of seismicity, ground deformation and vol 456
canic activity (Fig. 2 from [23]) have been documented at 457
Mount Pinatubo, Philippines, in 1991 [37] and Soufrière 458
Hills volcano, Montserrat, British West Indies, in 1996– 459
1997 [90,91]. At Soufrière Hills, periodicity in seismicity 460
and tilt ranged from ~4 to 30 h, and the oscillations in 461
both records continued for weeks. Cyclic behavior was ﬁrst 462
observed in the seismicity (RSAM) records beginning in 463
July 1996, when the record of dome growth constrained 464
the average supply rate to between 2 and 3 m
3
/s [88]. The 465
oscillations in the RSAM records initially had low ampli 466
tudes, and no tiltmeasurement station was close enough 467
to the vent to detect any pressure oscillations in the con 468
duit. By August 1996, RSAM records showed strong os 469
cillatory seismicity at domegrowth rates between 3 and 470
4 m
3
/s. Tilt data, taken close enough to the vent (i. e., 471
Chances Peak [90]) to be sensitive to conduit pressure os 472
cillations, are only available for February 1997 and May– 473
August 1997. In the latter period dome growth rate in 474
crease from ~5 m
3
/s in May to between 6 and 10 m
3
/s in 475
August. Both nearvent tilt and RSAM displayed oscilla 476
tory behavior during this period and were strongly corre 477
lated in time. Similar RSAM oscillations having periods of 478
7 to 10 h were observed at Mount Pinatubo following the 479
climactic eruption in 1991 [37]. At both volcanoes, oscilla 480
tion periods were observed that do not ﬁt any multiple of 481
Earth or ocean tides. 482
The cyclic activity at both Pinatubo and Soufrière Hills 483
Volcano are strongly correlated with eruptive behavior 484
and other geophysical phenomena. In the Pinatubo case 485
and on Soufrière Hills Volcano in August, September and 486
October 1997, the cycles were linked to shortlived vol 487
canic explosions. In the case of the Soufrière Hills Volcano, 488
explosions in August 1997 occurred at the peak in the tilt. 489
The peak in tilt also marked the onset of episodes of in 490
creased rock falls [7,8] and [91] is attributed to increased 491
magma discharge rates. SO
2
ﬂux data show that the cy 492
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6 Volcanic Eruptions: Cyclicity during Lava Dome Growth
Volcanic Eruptions: Cyclicity during Lava Dome Growth, Figure 1
Schematic view of the volcanic system. In the upper part the conduit is cylindrical with a radius R. A transition from the cylinder to
a dyke occurs at depth L
T
. The length scale for the transition from cylinder to dyke is w
T
. The dyke has an elliptical crosssection
with semiaxis lengths a
0
and b
0
. The chamber is located a depth L. In the text we used also the following auxiliary variables: D D 2R
for conduit diameter, L
d
D L L
T
for the dyke vertical length, W
d
D 2a
0
for the dyke width, and H
d
D 2b
0
for the dyke thickness.
After [20]
cles are linked to surges in gas release, which reach a peak 493
about an hour after the tilt peak [93]. Green et al. [31] have 494
shown that several families of near identical long period 495
earthquakes occur during the tilt cycle, starting at the in 496
ﬂexion point on the upcycle and ﬁnishing before the in 497
ﬂexion point on the downcycle. 498
Several models [23,49,70,98,98] have been proposed to 499
explain the observed cyclicity. In these models [23,98] the 500
conduit is divided into two parts. Magma is assumed to be 501
forced into the lower part of volcanic conduit at a con 502
stant rate. In Denlinger and Hoblitt [23] magma in the 503
lower part of the conduit is assumed to be compressible. 504
In Wylie et al. [98] the magma is incompressible but the 505
cylindrical conduit is allowed to expand elastically. In both 506
models the lower part of the conduit, therefore, acts like 507
a capacitor that allows magma to be stored temporally in 508
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 7
Volcanic Eruptions: Cyclicity duringLava Dome Growth, Figure 2
Examples of cyclic behavior in a lava dome eruption. RSAM
records from Mt. Pinatubo, Philippines following its climatic
eruption in June 1991 (a, after [23]). Radial tilt, triggered earth
quakes, and hybrid earthquakes for 17 May to 6 August 1997, at
Soufrière Hills Volcano (b, after [21,90]). Parts of three 5–7 week
cycles are shown, with each cycle showing high amplitude tilt
and seismicity pulsations with short time scale that last for sev
eral weeks after the start of the cycle. The onset of a cycle is rapid,
as detailed in c for the cycle initiating on 22 June 1997
order to release it during the intense phase of the eruption. 509
In the upper part of the conduit friction is dependent on 510
magma discharge rate, with a decrease in friction result 511
ing in an increase in discharge rate, over a certain range 512
of discharge rates. In Denlinger and Hoblitt [23], when 513
magma discharge rate reaches a critical value, magma de 514
taches from the conduit walls and a stickslip transition 515
occurs. Rapid motion of magma leads to depressurization 516
of the conduit and a consequent decrease in discharge rate, 517
until at another critical value the magma again sticks to the 518
walls. Pressure starts to increase again due to inﬂux of new 519
magma into the conduit. On a pressuredischarge diagram 520
the path of eruption is represented by a hysteresis loop. In 521
Wylie et al. [98] the friction is controlled by volatilede 522
pendent viscosity. Volatile exsolution delay is controlled 523
by diﬀusion. When magma ascends rapidly, volatiles have 524
no time to exsolve and viscosity remains low. Depressur 525
ization of the upper part of the conduit leads to a decrease 526
in magma discharge rate and an increase in viscosity due 527
to more intense volatile exsolution. 528
In Neuberg et al. [70] a steady 2D conduit ﬂow model 529
was developed. The full set of Navier–Stokes equations for 530
a compressible ﬂuid with variable viscosity was solved by 531
means of a ﬁnite element code. Below some critical depth 532
the ﬂowwas considered to be viscous and Newtonian, with 533
a noslip boundary condition at the wall. Above this depth 534
a plug develops, with a wall boundary condition of fric 535
tional slip. The slip criterion was based on the assumption 536
that the shear stress inside the magma overcomes some 537
critical value. Simulations reveal that slip occurs in the 538
shallow part of the conduit, in good agreement with loca 539
tions of longperiod volcanic earthquakes for the Soufrière 540
Hills Volcano, [31]. However, some parameters usedin the 541
simulations (low crystallinity, less than 30% and high dis 542
charge rate, more than 100 m
3
/s) are inconsistent with ob 543
servations. 544
Lensky et al. [49] developed the stickslip model by 545
incorporating degassing from supersaturated magma to 546
gether with a sticking plug. Gas diﬀuses into the magma, 547
which cannot expand due to the presence of a sticking 548
plug, resulting in a build up of pressure. Eventually the 549
pressure exceeds the strength of the plug, which fails in 550
stickslip motion and the pressure is relieved. The magma 551
sticks again when the pressure falls below the dynamic 552
friction value. In this model the time scale of the cycles is 553
controlled by gas diﬀusion. The inﬂuence of permeable gas 554
loss, crystallization and elastic expansion of the conduit on 555
the period of pulsations was studied. 556
A shorter timescale of order of minutes was investi 557
gated in Iverson et al. [43] in relation to repetitive seismic 558
events during the 2004–2006 eruption of Mount St. He 559
lens. The ﬂow dynamics is controlled by the presence of 560
a solid plug that is pushed by a Newtonian liquid, with the 561
possibility of a stickslip transition. Inertia of the plug be 562
comes important on such short timescales. 563
Models to explain the occurrence of Vulcanian ex 564
plosions have also been developed by Connor et al. [15], 565
Jaquet et al. [44] and Clarke et al. [13]. A statistical 566
model of repose periods between explosions by Connor 567
et al. [15] shows that data ﬁt a loglogistic distribution, 568
consistent with the interaction of two competing processes 569
that decrease and increase gas pressure respectively. Ja 570
quet et al. [44] show the explosion repose period data 571
have a memory. The petrological observations of Clarke 572
et al.. [13] on clasts from Vulcanian explosions associated 573
with shortterm cycles support a model where pressure 574
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8 Volcanic Eruptions: Cyclicity during Lava Dome Growth
builds up beneath a plug by gas diﬀusion, but is opposed by 575
gas leakage through a permeable magma foam. Although 576
these models include some of the key processes and have 577
promising explanatory power, they do not consider the de 578
velopment of the magma plug explicitly. This process is 579
considered in [24], where a model of magma ascent with 580
gas escape is proposed. 581
LongTermCycles 582
Figure 3 shows views of three lava domes (Mount St. He 583
lens, USA, Santiaguito, Guatemala and Shiveluch, Rus 584
sia). Measurements of magma discharge rate variations 585
with time are presented below [25,35,89]. Behavior at the 586
ﬁrst two volcanoes is rather regular, whereas at Shiveluch, 587
long repose periods are followed by an initial rapid in 588
crease in eruptive activity with subsequent decrease and 589
complete stop of the eruption. Growth of the lava dome 590
at Unzen volcano, Japan, 1991–1995 was similar to Shiv 591
eluch [25,68]. 592
There are three types of conceptual models that at 593
tempt to explain long term variation in magma discharge 594
rate. Maeda [53], after [42], considers a simple system 595
that contains a spherical magma chamber located in elas 596
tic rocks with a cylindrical conduit located in viscoelas 597
tic rocks. Magma viscosity is assumed to be constant. The 598
magma chamber is replenished with a time dependent in 599
ﬂux rate. The model reproduces discharge rate variation 600
at Unzen volcano by assuming a bellshaped form of in 601
ﬂux rate dependence on time. There are two controver 602
sial assumptions in the model. First, the assumption that 603
the conduit wallrocks are viscoelastic, while the magma 604
chamber wallrocks are purely elastic cannot be justiﬁed 605
because near the chamber rock temperature is the high 606
est and it is more reasonable to expect viscous properties 607
for chamber wallrocks rather than for the conduit. The 608
equation that links conduit diameter with magmatic over 609
pressure assumes viscous rock properties up to inﬁnity. If 610
the chamber is located in viscoelastic rocks, oscillations in 611
discharge rate are not possible. Second, in order to obtain 612
reasonable timescales, the rock viscosity must be rather 613
small, of order of 10
13
Pa s, which is only slightly higher 614
than the typical viscosities of the magma. 615
Another set of models attribute cyclic behavior to heat 616
exchange between ascending magma and wallrocks, which 617
accounts for temperature dependent viscosity [17,95]. The 618
idea of both models is that magma cools down as it as 619
cends, and heat ﬂux is proportional to the diﬀerence be 620
tween the average temperature of the magma and the 621
temperature of the wallrocks. If magma ascends quickly 622
than heat loss is small in comparison with heat advection. 623
Magma viscosity remains low as a consequence and allows 624
high magma discharge rates. In contrast, when magma 625
ascends slowly it can cool substantially and viscosity in 626
creases signiﬁcantly. Both models suggest that, for a ﬁxed 627
chamber pressure, there can be up to three steady state 628
solutions with markedly diﬀerent discharge rates. Tran 629
sition between these steadystate solutions leads to cyclic 630
variations in discharge rate. Whitehead and Helfrich [95] 631
demonstrated the existence of cyclic regimes in exper 632
iments using corn syrup. In application to magma as 633
cent in a volcanic conduit, these models have strong lim 634
itations, because a constant wallrock temperature is as 635
sumed. However, as an eruption progresses the wallrocks 636
heat up and heat ﬂux decreases, a condition that makes 637
periodic behavior impossible for longlived eruptions. For 638
such a longlived eruption like Santiaguito (started in 639
1922) wallrocks are expected to be nearly equilibrated in 640
temperature with the magma, and heat losses from magma 641
became small. It is possible that this decrease in heat ﬂux 642
contributes to a slow progressive increase in temperature 643
that is observed on timescales longer than the period of 644
pulsations. For example, magma at Santiaguito becomes 645
progressively less viscous, resulting in a transition from 646
mainly lava dome to lava ﬂow activity. 647
Models, developed by authors of this manuscript, con 648
sider that degassinginduced crystallization is a major con 649
trolling process for the longterm cyclicity during lava 650
dome building eruptions. There is increasing evidence 651
that there is a good correlation between magma discharge 652
rate and crystallinity of the magma [10,68]. An increase 653
in crystal content leads to an increase in magma viscos 654
ity [16,20,21] and, thus, inﬂuences magma ascent dynam 655
ics. First, we consider a simpliﬁed model of magma ascent 656
in a volcanic conduit that accounts for crystallization and 657
rheological stiﬀening. 658
A Simpliﬁed Model 659
In Barmin et al.. [3] the following simplifying assumptions 660
have been made in order to develop a semianalytical ap 661
proach to magma ascent dynamics. 662
1. Magma is incompressible. The density change due to 663
bubble formation and melt crystallization is neglected. 664
2. Magma is a viscous Newtonian ﬂuid. Viscosity is a step 665
function of crystal content. When the concentration of 666
crystals ˇ reaches a critical value ˇ
the viscosity of 667
magma increases from value µ
1
to a higher value µ
2
. 668
Later on we will consider magma rheology in more de 669
tail (see Sect.“Rheology of CrystalBearing Magma and 670
Conduit Resistance”), but a sharp increase in viscosity 671
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 9
Volcanic Eruptions: Cyclicity during Lava Dome Growth, Figure 3
Observed discharge rate versus time for (a) Mount St.Helens dome growth and (b) Santiaguito volcano and Shiveluch (c). Photos:
Mount St.Helens by Lyn Topinka (1985), Santiaguito by Gregg Bluth (2002), Shiveluch by Pavel Plechov (2001)
over a narrow range of crystal content has been con 672
ﬁrmed experimentally (e. g., [9,48]). 673
3. Crystal growth rate is constant and no nucleation oc 674
curs in the conduit. The model neglects the fact that 675
magma is a complicated multicomponent system and 676
its crystallization is controlled by the degree of under 677
cooling (deﬁned as the diﬀerence between actual tem 678
perature of the magma and its liquidus temperature). 679
Later in the paper a more elaborate model for magma 680
crystallization will be considered. 681
4. The conduit is a vertical cylindrical pipe. Elastic defor 682
mation of the wallrocks is not included in the model. 683
This assumption is valid for a cylindrical shape of the 684
conduit at typical magmatic overpressures, but is vio 685
lated when the conduit has a fracture shape. Real ge 686
ometries of volcanic conduits and their inclination can 687
vary signiﬁcantly with depth. 688
5. The magma chamber is located in elastic rocks and is 689
fed from below, with a constant inﬂux rate. For some 690
volcanoes, like Santiaguito or Mount St.Helens, average 691
magma discharge rate remained approximately con 692
stant during several periods of pulsation. Thus the as 693
sumption of constant inﬂux rate is valid. For volcanoes 694
like Mount Unzen or Shiveluch, there is an evidence of 695
pulselike magma recharge [25,53]. 696
With above simpliﬁcation the system of equations for un 697
steady 1D ﬂow is as follows: 698
d
dt
j ÷
d
dx
ju = 0 ;
d
dt
n ÷
d
dx
nu = 0 (1a) 699
dp
dx
= ÷jg ÷
32ju
ı
2
; j =
_
j
1
. ˇ < ˇ
j
2
. ˇ > ˇ
(1b) 700
d
dt
ˇ ÷u
dˇ
dx
= 4¬nr
2
¡ = (36¬n)
1
3
ˇ
2
3
¡ (1c) 701
Here j is the density of magma, u is the vertical crosssec 702
tion averaged ascent velocity, n is the number density of 703
crystals per unit volume, p is the pressure, g is the accel 704
eration due to gravity, ı is the conduit diameter, ˇ is the 705
volume concentration of crystals, ˇ
is a critical concen 706
tration of crystals above which the viscosity changes from 707
µ
1
to µ
2
, r is the crystal radii, ¡ is the linear crystal growth 708
rate, and x is the vertical coordinate. The ﬁrst two Eqs. (1a) 709
represent the conservation of mass and the number den 710
sity of crystals, the second (1b) is the momentum equa 711
tion with negligible inertia, and the third (1c) is the crystal 712
growth equations with ¡ = constant. We assume the fol 713
lowing boundary conditions for the system (1): 714
x = 0 :
dp
ch
dt
=
;
V
ch
(Q
in
÷ Q
out
) ; ˇ = ˇ
ch
; n = n
ch
x = l : p = 0
715
Here ; is the rigidity of the wallrock of the magma cham 716
ber, V
ch
is the chamber volume, ˇ
ch
and n
ch
are the crystal 717
concentration and number density of crystals per unit vol 718
ume in the chamber, p
ch
is the pressure in the chamber, l 719
is the conduit length, Q
in
is the ﬂux into the chamber and 720
Q
out
= ¬ı
2
u/4 is the ﬂux out of the chamber into the con 721
duit. Both ˇ
ch
and n
ch
are assumed constant. We neglect 722
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10 Volcanic Eruptions: Cyclicity during Lava Dome Growth
the inﬂuence of variations of the height of the lava dome 723
on the pressure at the top of the conduit and assume that 724
the pressure there is constant. As the magma is assumed 725
to be incompressible, the pressure at the top of the con 726
duit can be set to zero because the atmospheric pressure is 727
much smaller than magma chamber pressure. 728
From the mass conservation equation for the case of 729
constant magma density, u = u(t) and n = n
ch
every 730
where. Equations (1) can be integrated and transformed 731
from partial diﬀerential equations to a set of ordinary dif 732
ferential equations with statedependent delay represent 733
ing a “memory” eﬀect on crystal concentration (see [3] for 734
details). 735
Results and Applications 736
The general steadystate solution for magma ascent veloc 737
ity variations with chamber pressure is shown in Fig. 4a. 738
Solutions at high magma ascent velocities, when the criti 739
cal concentration of crystals is not reached inside the con 740
duit, result in a straight line (CB), which is the same as for 741
the classical Poiseuille solution for a ﬂuid with constant 742
viscosity. At low ascent velocities there is a quadratic re 743
lationship (OAC) between chamber pressure and ascent 744
velocity (see [3] for derivation of the equation). A key fea 745
ture of the steadystate solution is that, for a ﬁxed cham 746
ber pressure, it is possible to have three diﬀerent magma 747
ascent velocities. We note that for j
2
/j
1
= 1 only the 748
branch CB exists and for 1 < j
2
/j
1
6 2 there is a smooth 749
transition between the lower branch (ODA) and the up 750
per branch (CB) and multiple steadystate regimes do not 751
exist. 752
We ﬁrst consider the case where chamber pressure 753
changes quasistatically and the value of Q
in
is between 754
Q
A
and Q
C
. Starting at point O the chamber pressure in 755
creases, because the inﬂux into the chamber is higher than 756
the outﬂux. At point A, a further increase in pressure is not 757
possible along the same branch of the steadystate solution 758
and the system must change to point B, where the outﬂux 759
of magma is larger than the inﬂux. The chamber pressure 760
and ascent velocity decrease along BC until the point C is 761
reached and the system must change to point D. The cycle 762
DABC then repeats itself. Provided the chamber continues 763
to be supplied at the same constant rate repetition of this 764
cycle results in periodic behavior. The transitions AB and 765
CD in a cycle must involve unsteady ﬂow. 766
Oscillations in magma discharge rate involve large 767
variations in magma crystal content. This relation is ob 768
served on many volcanoes. For example, pumice and sam 769
ples of the Soufrière Hills dome that were erupted during 770
periods of high discharge [88] have high glass contents (25 771
to 35%) and few microlites [65], whereas samples derived 772
from parts of the dome that were extruded more slowly 773
(days to weeks typically) have much lower glass contents 774
(5 to 15%) and high contents of groundmass microlites. 775
These and other observations [34,68] suggest that micro 776
lite crystallization can take place on similar time scales to 777
the ascent time of the magma. 778
Of more general interest is to consider unsteady ﬂow 779
behaviour. We assume that the initial distribution of pa 780
rameters in the conduit corresponds to the steadystate so 781
lution of system (1) with the initial magma discharge rate, 782
Q
0
, being in the lowest regime. The behaviour of an erup 783
tion with time depends strongly on the value of Q
in
. If 784
Q
in
corresponds to the upper or the lower branch of the 785
steady state solution the eruption stabilizes with time with 786
Q = Q
in
and dp
ch
/ dt = 0. However, if Q
in
corresponds 787
to the intermediate branch of the steadystate solution, 788
with Q
in
between Q
A
and Q
C
, periodic behaviour is pos 789
sible. Figure 4b shows three eruption scenarios for diﬀer 790
ent values of the magma chamber volume V
ch
. When V
ch
791
is small the eruption stabilizes with time. In contrast, un 792
damped periodic oscillations occur for values of V
ch
larger 793
than some critical value. For very large magma chamber 794
volumes the transient solution almost exactly follows the 795
steadystate solution, with unsteady transitions between 796
the regimes. The time that the system spends in unsteady 797
transitions in this case is much shorter than the period of 798
pulsations. The maximum discharge rate during the cycle 799
is close to Q
B
, the minimum is close to Q
D
and the average 800
is equal to Q
in
. The period of pulsations increases as the 801
volume of magma chamber increases. 802
Now we apply the model to two welldocumented 803
eruptions: the growth of lava domes on Mount St. He 804
lens (1980–1986) and on Santiaguito (1922present). Our 805
objective here is to establish that the model can repro 806
duce the periodic behaviors observed at these two volca 807
noes. Estimates can be obtained for most of the system pa 808
rameters. However, magma chamber size is not wellcon 809
strained and so the model can be used to make qualitative 810
inferences on relative chamber size. Given the uncertain 811
ties in the parameter values and the simpliﬁcations in the 812
model development, the approach can be characterized as 813
mimicry. Adjustments in some parameters were made to 814
achieve best ﬁts with observations, but the particular best 815
ﬁts are not unique. 816
For Mount St. Helens our model is based on data 817
presented by [89]. Three periods of activity can be dis 818
tinguished during the period of dome growth. The ﬁrst 819
period consists of 9 pulses of activity with average peak 820
magma discharge rates ~15 m
3
/s during 1981–1982. Each 821
pulse lasted from 2 to 7 days (with a mean value of 4 days) 822
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 11
Volcanic Eruptions: Cyclicity during Lava Dome Growth, Figure 4
The general steadystate solution and possible quasistatic evolution of an eruption. After [3]. Different curves correspond to differ
ent values of dimensionless parameter D (ı
2
)/(4V
ch
g) (0.12 – thin line, 0.05 – dotted line, and 0.005 – bold line)
with the average period between the pulses being 74 days 823
and the average discharge rate during the period being 824
Q
1
~0.67 m
3
/s. The second period is represented by con 825
tinuous dome growth and lasted more than a year (368 826
days) with a mean magma discharge rate of Q
2
~ 0.48 827
m
3
/s. During the last period there were 5 episodes of dome 828
growth with peak magma discharge rates up to ~15 m
3
/s, 829
an average period of pulsation of ~230 days and a mean 830
discharge rate of Q
3
~ 0.23 m
3
/s. We assume that the in 831
tensity of inﬂux into the magma chamber, Q
in
, is equal to 832
the average magma discharge rate over the corresponding 833
periods. There might have been a progressive decrease in 834
the intensity of inﬂux during the eruption, but, due to lim 835
itations of the model, we assume that Q
in
changes as a step 836
function between the periods. 837
The bestﬁt model for the eruption is presented in 838
Fig. 5a, with the parameters used for the simulation sum 839
marized in Table 2 in [3]. During the ﬁrst period of the 840
eruption Q
in
= Q
1
. This corresponds to the intermediate 841
branch of the steadystate solution and cyclic behaviour 842
occurs. In the second period Q
in
= Q
2
the system moves 843
to the lower regime and the eruption stabilizes with time. 844
For the third period of the eruption the parameters of the 845
system have been changed, so that Q
in
= Q
3
< Q
2
, corre 846
sponding to the intermediate regime once again and peri 847
odic behaviour occurs. This condition can be satisﬁed by 848
a decrease in the diameter of the conduit, or a decrease 849
in crystal growth rate, or the number density of crystals. 850
All these mechanisms are possible: decrease in the diam 851
eter could be a consequence of magma crystallization on 852
the conduit walls, while a decrease in either crystal growth 853
rate or number density of crystals could be reﬂect ob 854
served changes in magma composition [89]. The inﬂuence 855
of conduit diameter is the strongest because ascent veloc 856
ity, for the same discharge rate, depends on the square 857
of the diameter. The required change in diameter is from 858
18 to 12 m, but this change can be smaller if we assume 859
a simultaneous decrease in crystal growth rate. 860
Since 1922, lava extrusion at Santiaguito has been 861
cyclic [35]. Each cycle begins with a 3–6 year long high 862
(0.5–2.1 m
3
/s) magma discharge rate phase, followed by 863
a longer (3–11 years) low (~0.2 m
3
/s) discharge rate 864
phase. The timeaveraged magma discharge rate was al 865
most constant at ~0.44 m
3
/s between 1922 and 2000. The 866
ﬁrst peak in discharge rate had a value >2 m
3
/s, whereas 867
the second peak had a much smaller value. The value for 868
the second peak is underestimated as it is calculated based 869
on the dome volume only, but does not include the volume 870
of dome collapse pyroclastic ﬂows. Later peaks showan in 871
crease in magma discharge rate until 1960 (Fig. 5b, dashed 872
line). Post1960, the duration of the low discharge rate 873
phase increased, the peak discharge and the timeaveraged 874
discharge rates for each cycle decreased, and the diﬀerence 875
between discharge rates during the high and low discharge 876
rate phases of each cycle decreased. Our bestﬁt model is 877
shown in Fig. 5b and the parameters estimates are listed 878
in Table 2 in [3]. The model reproduces the main features 879
of the eruption, including the period of pulsations, the ra 880
tio between low and high magma discharge rates, and the 881
range of observed discharge rates. We cannot, however, re 882
produce the decrease in the amplitude of pulsations within 883
the framework of the model using ﬁxed parameter values. 884
U
n
c
o
r
r
e
c
t
e
d
P
r
o
o
f
2
0
0
8

0
7

0
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12 Volcanic Eruptions: Cyclicity during Lava Dome Growth
Volcanic Eruptions: Cyclicity during Lava Dome Growth, Figure 5
Discharge rate versus time for a Mount St.Helens dome growth and b Santiaguito volcano. Doted lines represent are the observed
values of discharge rates and the solid lines are the best fit simulations. After [3]
The theory provides a potential method to estimate 885
magma chamber volumes. For Mount St.Helens our esti 886
mate of the chamber size (~0.6 km
3
) is comparable with 887
the total erupted volume in the entire 1980–1986 eruption 888
and is consistent with the fact that geophysical imaging 889
did not identify a large magma body. Santiaguito volcano 890
erupted more than 10 km
3
in the 1902 explosive erup 891
tion [96] and more than 1 km
3
of lava domes since 1922. 892
The bestﬁt model estimate of a large (64 km
3
) chamber is 893
consistent with much larger eruption volumes, long peri 894
ods, and longevity of the eruption in comparison to Mount 895
St. Helens. One limitation of the model is that the supply 896
of deep magma from depth to the chamber is assumed to 897
be constant. 898
Model Development 899
In this section we further develop models to examine new 900
eﬀects and relax some of the simpliﬁcations of earlier 901
models. We investigate a number of eﬀects that were not 902
fully explained or considered in previous studies [3,61,62]. 903
The new model incorporates a more advanced treatment 904
of crystallization kinetics based on the theoretical con 905
cepts developedin [38,46], and is calibrated by experimen 906
tal studies in andesitic systems [22,34]. In particular, we 907
distinguish growth of phenocrysts formed in the magma 908
chamber from crystallization of microlites during magma 909
ascent. Previous models have assumed that magma is al 910
ways Newtonian, so we study models of conduit ﬂow as 911
suming nonNewtonian rheology, with rheological prop 912
erties being related to crystal content. Latent heat is re 913
leased during the crystallization of ascending magma due 914
to degassing and we show that this can have an important 915
inﬂuence on the dynamics. Elastic deformation of conduit 916
walls leads to coupling between magma ascend and vol 917
cano deformations. 918
System of Equations We model magma ascent in 919
a dykeshaped conduit with elliptical crosssection using 920
a set of 1D transient equations written for horizontally av 921
eraged variables [20,21]: 922
1
S
d
dt
(Sj
m
) ÷
1
S
d
dx
(Sj
m
V) = ÷G
mc
÷ G
ph
(2) 923
1
S
d
dt
(Sj
mc
) ÷
1
S
d
dx
(Sj
mc
V) = G
mc
(3a) 924
1
S
d
dt
(Sj
ph
) ÷
1
S
d
dx
(Sj
ph
V) = G
ph
(3b) 925
1
S
d
dt
(Sj
d
) ÷
1
S
d
dx
(Sj
d
V) = ÷J (4a) 926
1
S
d
dt
(Sj
g
) ÷
1
S
d
dx
(Sj
g
V
g
) = J (4b) 927
Here t denotes time, x the vertical coordinate, j
m
, j
ph
, j
mc
, 928
j
d
and j
g
are the densities of melt, phenocrysts, microlites, 929
dissolved gas and exsolved gas respectively, and V and V
g
930
are the velocities of magma and gas, respectively. G
ph
, G
mc
931
represent the mass transfer rate due to crystallization of 932
phenocrysts and microlites, respectively, and J the mass 933
transfer rate due to gas exsolution, S is the crosssection 934
U
n
c
o
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 13
area of the conduit. Equation (2) represents the mass con 935
servation for the melt phase, Eqs. (3a) and (3b) are the con 936
servation equations for microlites and phenocrysts respec 937
tively, Eqs. (4a) and (4b) represent the conservation of the 938
dissolved gas and of the exsolved gas respectively. 939
dp
dx
= ÷jg ÷ F
c
(5) 940
V
g
÷ V = ÷
k
j
g
dp
dx
(6) 941
Here p is the pressure, j the bulk density of magma, g the 942
acceleration due to gravity, j is the magma viscosity, k is 943
the magma permeability and j
g
is the gas viscosity. Eq. (5) 944
represents the equation of momentum for the mixture as 945
a whole, in which the pressure drops due to gravity and 946
conduit resistance are calculated for laminar ﬂow in an el 947
liptic pipe. Equation (6) is the Darcy law for the exsolved 948
gas ﬂux through the magma. 949
950
1
S
d
dt
(SjC
m
T) ÷
1
S
d
dx
(SjC
m
VT) 951
= L
(G
mc
÷G
ph
) ÷ C
m
TJ ÷ Q
cl
÷Q
vh
(7) 952
953
Here C
m
is the bulk speciﬁc heat of magma, T is the bulk 954
ﬂowaveraged temperature, L
is latent heat of crystal 955
lization, Q
cl
denotes the total heat loss by conduction to 956
the conduit walls, and Q
vh
denotes the total heat genera 957
tion due to viscous dissipation. Here we consider the case 958
of the latent heat release. This assumption is valid when 959
both Q
cl
~ 0 and Q
vh
~ 0 or when Q
cl
÷Q
vh
~ 0. The 960
study of the eﬀects of both heat loss and viscous heating, 961
which are intrinsically twodimensional [18,19], and their 962
parametrization is the subject of ongoing research. 963
j
m
= j
0
m
(1 ÷˛)(1 ÷ˇ)(1 ÷c) ; j
c
= j
0
c
(1 ÷˛)ˇ (8a) 964
j
d
= j
0
m
(1 ÷ ˛)(1 ÷ ˇ)c ; j
g
= j
0
g
˛ (8b) 965
j = j
m
÷j
c
÷j
d
÷j
g
(8c) 966
˛ =
4
3
¬r
3
b
n;
d
dt
(Sn)÷
d
dx
(SnV) = 0 ; p = j
0
g
RT (9) 967
Here ˛ is the volume concentration of bubble, ˇ is the 968
volume concentration of crystals in the condensed phase 969
(melt plus crystals), and c is mass concentration of dis 970
solved gas (equal to volume concentration as we assume 971
that the density of dissolved volatiles is the same as the 972
density of the melt), j
m
0
denotes the mean density of the 973
pure melt phase, j
c
0
is density of the pure crystal phase 974
(with j
c
= j
ph
÷j
mc
. ˇ = ˇ
ph
÷ˇ
mc
), r
b
is the bubble 975
radius, and n the number density of bubble per unit vol 976
ume. Concerning the parametrization of mass transfer rate 977
functions, we use: 978
J = 4¬r
b
nDj
0
m
_
c ÷ C
f
_
p
_
(10) 979
G
mc
= 4¬j
0
c
(1 ÷ ˇ) (1 ÷˛)
× U(t)
_
t
0
I(o)
__
t
!
U(j) dj
_
2
do
(11a) 980
G
ph
= 3;
s
_
4¬N
ph
ˇ
2
ph
3
_1
3
j
0
c
(1 ÷ˇ) (1 ÷ ˛) U(t) (11b) 981
Here J is parametrized using the analytical solution de 982
scribed in [69], U is the linear crystal growth rate (ms
1
), 983
I is the nucleation rate (m
3
s
1
), which deﬁnes the num 984
ber of newly nucleated crystal per cubic meter, and ;
s
985
is a shape factor of the order of unity, D and C
f
are 986
the diﬀusion and the solubility coeﬃcients, respectively. 987
Concerning the mass transfer due to crystallization G
mc
, 988
we adapt a model similar to that described in [38]. As 989
suming spherical crystals, the Avrami–JohnsonMehl– 990
Kolmogorov equation in the form adopted by [46], for the 991
crystal volume increase rate, is: 992
dˇ
dt
= 4¬Y
t
U(t)
_
t
0
I(o)
__
t
!
U(j) dj
_
2
do 993
where Y
t
= (1 ÷ ˇ)(1 ÷˛) is the volume fraction of melt 994
remaining uncrystallized at the time t. Therefore, we have 995
G
mc
= j
mc
dˇ/ dt. For the phenocryst growth rate () we 996
assume that it is proportional to the phenocryst volume 997
increase rate dˇ
ph
/ dt = 4¬R
2
ph
N
ph
U(t) times the crystal 998
density j
0
c
times the volume fraction of melt remaining 999
uncrystallized at the time t. A detailed description of the 1000
parametrization used for the diﬀerent terms is reported 1001
in [63]. 1002
For parametrizations of magma permeability k and 1003
magma viscosity j we use: 1004
k = k (˛) = k
0
˛
j
(12) 1005
j = j
m
(c. T) 0 (ˇ) j (˛. Ca) (13) 1006
where k is assumed to depend only on bubble volume 1007
fraction ˛. Magma viscosity j depends on water content, 1008
temperature, crystal content, bubble fraction and capillary 1009
number as described in detail in the next section. 1010
Regarding equations for semiaxes, a and b, we assume 1011
that the elliptical shape is maintained and that pressure 1012
change gradually in respect with vertical coordinate and 1013
U
n
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o
r
r
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0
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0
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14 Volcanic Eruptions: Cyclicity during Lava Dome Growth
time so that the plain strain analytical solution for an el 1014
lipse subjected to a constant internal overpressure [64,66], 1015
remains valid: 1016
a = a
0
÷
^P
2G
_
÷(1 ÷2v)a
0
÷2(1 ÷v)b
0
_
(14a) 1017
b = b
0
÷
^P
2G
_
2(1 ÷v)a
0
÷ (1 ÷ 2v)b
0
_
(14b) 1018
where ^P is the overpressure, i. e. the diﬀerence between 1019
conduit pressure and far ﬁeld pressure (here assumed to be 1020
lithostatic for a sake of simplicity), a
0
and b
0
are the initial 1021
values of the semiaxes, v is the host rock Poisson ratio, 1022
and G is the host rock rigidity. 1023
Equations (2)–(14) are solved between the top of 1024
the magma chamber and the bottom of the lava dome 1025
that provides some constant load by using the numeri 1026
cal method described in [63]. The eﬀects of dome height 1027
and morphology changes are not considered in this pa 1028
per. We consider three diﬀerent kinds of boundary con 1029
ditions at the inlet of the dyke: constant pressure, constant 1030
inﬂux rate and the presence of a magma chamber located 1031
in elastic rocks. The case of constant pressure is applica 1032
ble when a dyke starts from either a large magma cham 1033
ber or unspeciﬁed source, so that pressure variations in the 1034
source region remain small. An estimate of the volume of 1035
magma stored in the source region that allows pressure to 1036
be approximated as constant depends on wallrock elastic 1037
ity, magma compressibility (volatile content), and the total 1038
volume of the erupted material. If the magma ﬂowat depth 1039
is controlled by regional tectonics, the case of constant in 1040
ﬂux rate into the dyke may be applicable if total variations 1041
in supply rate are relatively small on the timescale of the 1042
eruption. 1043
For the case where magma is stored in a shallow 1044
magma chamber prior to eruption, and signiﬁcant cham 1045
ber replenishment occurs, the ﬂowinside the conduit must 1046
be coupled with the model for the magma chamber. In this 1047
case, as explained in detail in [63], we assume that the re 1048
lationship between the pressure at the top of the magma 1049
chamber p
ch
and the intensity of inﬂux Q
in
and outﬂux 1050
Q
out
of magma to and from the chamber is given by: 1051
dp
ch
dt
=
4G (K)
(j) V
ch
(3 (K) ÷4G)
(Q
in
÷ Q
out
) (15) 1052
where V
ch
is the magma chamber volume, (j) and (K) are 1053
the average magma density and magma bulk modulus, re 1054
spectively, and G is the rigidity of rocks surrounding the 1055
chamber. 1056
Cases of constant inﬂux and of constant source pres 1057
sure are the limit cases of Eq. (15) in the case of inﬁnitely 1058
small and inﬁnitely large magma chamber volume. We as 1059
sume that the volume concentration of bubbles and phe 1060
nocrysts are determined by equilibrium conditions and 1061
that the temperature of the magma is constant. The eﬀect 1062
of temperature change on eruption dynamics, due to in 1063
teraction between silicic and basaltic magma, was studied 1064
in [63]. 1065
We use a steadystate distribution of parameters along 1066
the conduit as an initial condition for the transient simula 1067
tion. The values are calculated for a low magma discharge 1068
rate, but the particular value of this parameter is not im 1069
portant because the systemdeviates frominitial conditions 1070
to a cyclic or stabilized state, which does not depends on 1071
the initial conditions. 1072
Rheology of CrystalBearing Magma andConduit Resis 1073
tance Magma viscosity is modeled as a product of melt 1074
viscosity j
m
(c. T), the relative viscosity due to crystal con 1075
tent 0 (ˇ) = O(ˇ) ç (ˇ), and the relative viscosity due to 1076
the presence of bubbles j(˛. Ca). The viscosity of the pure 1077
melt j
m
(c. T) is calculated according to [36]. Viscosity 1078
increase due to the presence of the crystals is described 1079
through the function O(ˇ) [16,20,21]. As crystallization 1080
proceeds, the remaining melt becomes enriched in silica 1081
and melt viscosity increases. The parametrization of this 1082
eﬀect is described by the function ç(ˇ) in [21,25]. Eﬀects 1083
of the solid fraction are parametrized as described in [21]. 1084
Eﬀects due to the presence of bubbles are accounted 1085
for by adopting a generalization of [51] for an elliptical 1086
conduit [20,21]. 1087
In the case of Newtonian magma rheology, the fric 1088
tion force in an elliptical conduit can be obtained from 1089
a classical Poiseuille solution for low Reynolds number 1090
ﬂow F
c
= 4j(a
2
÷b
2
)/(a
2
b
2
)V [47]. High crystal or bub 1091
ble content magmas may show nonNewtonian rheology. 1092
One possible nonNewtonian rheology is that of a Bing 1093
ham material characterized by a yield strength t
b
[4]. The 1094
stressstrain relation for this material is given by: 1095
t
i j
=
_
j ÷
t
b
;
_
;
i j
=t > t
b
;
i j
= 0 =t 6 t
b
(16) 1096
Here t
i j
and ;
i j
are the stress and strain rate tensors, t and 1097
; are second invariants of corresponding tensors. Accord 1098
ing to this rheological law, the material behaves linearly 1099
when the applied stress is higher than a yield strength. No 1100
motion occurs if the stress is lower than a yield strength. 1101
In the case of a cylindrical conduit the average velocity 1102
can be calculated in terms of the stress on the conduit wall 1103
U
n
c
o
r
r
e
c
t
e
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2
0
0
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 15
t
w
[52]: 1104
V =
1
12
r
t
3
w
j
_
t
4
b
÷3t
4
w
÷ 4t
b
t
3
w
_
(17) 1105
Here r is the conduit radii. This form of equation gives 1106
an implicit relation between ascent velocity and pressure 1107
drop, and is not convenient to use. By introducing dimen 1108
sionless variables ˘ = jV/t
b
r and O = t
w
/t
b
> 1 rela 1109
tion (17) can be transformed into: 1110
O
4
÷
1
6
(8 ÷3˘) O
3
÷
1
3
= 0 (18) 1111
Following [63] a semianalytical solution can be used for 1112
(18) and the conduit friction force can be expressed ﬁnally 1113
as: 1114
F
c
=
2t
w
r
=
2t
b
O(˘)
r
. 1115
We note that a ﬁnite pressure gradient is necessary to ini 1116
tiate the ﬂow in the case of Bingham liquid, in contrast to 1117
a Newtonian liquid. 1118
Results and Applications 1119
Inﬂuence of nonNewtonian Properties on Eruption Be 1120
haviour Now we compare the dynamics of magma ex 1121
trusion in the cases of Newtonian and Bingham rheology. 1122
We will assume that yield strength is reached when the 1123
concentration of crystals reaches a critical value: 1124
t =
_
t
b
for ˇ > ˇ
cr
0 for ˇ 6 ˇ
cr
(19) 1125
Figure 6a shows a set of steadystate solutions for diﬀer 1126
ent values of t
b
. Values of t
b
and ˇ
cr
depend on crystal 1127
shape, crystal size distribution, magma temperature and 1128
other properties, but here are assumed to be constant. To 1129
illustrate the inﬂuence of Bingham rheology, the value of 1130
ˇ
cr
= 0.65 was chosen so that, for discharge rate larger 1131
then ~5 m
3
/s, the magma has Newtonian rheology (see 1132
Fig. 6a). A more detailed study would require measure 1133
ments of the rheological properties of magma for a wide 1134
range of crystal content and crystal size distributions. As 1135
the value of t
b
the chamber pressure that is necessary to 1136
start the eruption increases. 1137
Figure 6b shows the inﬂuence of these two rheologi 1138
cal models on the dynamics of magma extrusion. In the 1139
case of Bingham rheology, magma discharge rate between 1140
the two pulses is zero until a critical chamber overpres 1141
sure is reached. Then the magma discharge rate increases 1142
rapidly with decrease in crystal content, leading to a sig 1143
niﬁcant reduction of both magma viscosity and the length 1144
of the part of the conduit that is occupied by the Bingham 1145
liquid where ˇ
c
> ˇ
cr
. There is a transition in the system 1146
to the uppermost ﬂow regime and the pressure then de 1147
creases quickly. Because the pressure at the onset of the 1148
pulse was signiﬁcantly larger than in the case of a Newto 1149
nian liquid, the resulting discharge rate in the case of Bing 1150
ham rheology is also signiﬁcantly higher. 1151
Modeling of Conduit Flow during Dome Extrusion on Shiv 1152
eluch Volcano The maximum intensity of extrusion was 1153
reached at an early stage in all three eruptions (Fig. 3). 1154
We therefore suggest that dome extrusion was initiated 1155
by high overpressure in the magma chamber with re 1156
spect to the lithostatic pressure. Depressurization of the 1157
magma chamber occurred as a result of extrusion. With 1158
out magma chamber replenishment, depressurization re 1159
sults in a decrease in magma discharge rate. In open sys 1160
tem chambers replenishment of the chamber during erup 1161
tion can lead to pulsatory behaviour [3]. The following ac 1162
count is derived from[25]. For the 1980–1981 eruption the 1163
monotonic decrease in discharge rate indicates that there 1164
was little or no replenishment of the magma chamber. 1165
During the 1993–1995 and 2001–2004 episodes, however, 1166
the magma discharge rate ﬂuctuated markedly, suggest 1167
ing that replenishment was occurring. The inﬂux of new 1168
magma causes an increase in magma chamber pressure, 1169
and a subsequent increase in magma discharge rate. Dur 1170
ing the 2001–2004 eruption there were at least three peaks 1171
in discharge rate. Replenishment of the magma chamber 1172
with new hot magma can explain the transition from lava 1173
dome extrusion to viscous lava ﬂowthat occurred on Shiv 1174
eluch after 10 May 2004, and which continues at the time 1175
of writing (2007). 1176
We simulated dome growth during the 2001–2002, be 1177
cause this dataset is the most complete and is supported 1178
by petrological investigations [39]. We assume the shape 1179
of the inﬂux curve: 1180
Q
in
=
_
ˆ
_
ˆ
_
0. t < t
s
Q
0
. t
s
6 t 6 t
f
0. t > t
f
(20) 1181
Inﬂux occurs with constant intensity Q
0
between times t
s
1182
and t
f
. We have examined many combinations of values of 1183
these parameters within the constraints provided by obser 1184
vations. The best simulation results use the following val 1185
ues of parameters: Q
0
= 3.8 m
3
/s, t
s
= 77 and t
f
= 240 days. 1186
A more continuous inﬂux, dependent on time, is plausi 1187
ble, but there is no geophysical evidence that allows us to 1188
U
n
c
o
r
r
e
c
t
e
d
P
r
o
o
f
2
0
0
8

0
7

0
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16 Volcanic Eruptions: Cyclicity during Lava Dome Growth
Volcanic Eruptions: Cyclicity during Lava Dome Growth, Figure 6
a Steadystate solutions and dependence of discharge rate on time for Newtonian and Bingham rheology of the magma. Yield
strength is a parameter markedon the curves (values in MPa). For Binghamrheology discharge rate remains zero between the pulses
of activity. Bingham rheology results in much higher chamber pressures prior to the onset of activity and, therefore, much higher
discharge rates in comparison with Newtonian rheology. b Comparison of the period of pulsation in discharge rate for Newtonian
and Binghamrheologies. After [63]
constrain the intensity of the inﬂux, because ground defor 1189
mation data are absent for Shiveluch volcano. The output 1190
of the model gives a magma chamber volume of 12 km
3
, 1191
assuming a spherical chamber. Figure 7a shows the time 1192
dependence of magma discharge rate, and Fig. 7b shows 1193
the increase in the volume of erupted material with time 1194
after 6th June 2001. The timing of magma inﬂux is in good 1195
agreement with the residence time of basaltic magma in 1196
the system, as calculated from the olivine reaction rims. 1197
For further details see [25]. 1198
5 to 7 Weeks Cycles on the Soufriere Hills Volcano: Evidence 1199
for a Dyke? An approximately 5 to 7 week cyclic pattern 1200
of activity was recognized at the Soufrière Hills Volcano 1201
(SHV) [87,91] between April 1997 and March 1998 from 1202
peaks in the intensity of eruptive activity and geophysical 1203
data, including tilt and seismicity (Fig. 2). 1204
In models discussed above, the timescale of pulsations 1205
depends principally on the volume of the magma cham 1206
ber, magma rheology and the crosssectional area of the 1207
conduit. These models might provide an explanation for 1208
the 2–3 year cycles of dome extrusion observed at SHV, 1209
where deformation data indicate that the magma chamber 1210
regulates the cycles. However, the models cannot simulta 1211
neously explain the 5–7 week cycles. Thus another mech 1212
anism is needed. 1213
The evidence for a dyke feeder at SHV includes 1214
GPS data [58], distribution of active vents, and seismic 1215
data [76]. We have assumed that, at dept, the conduit has 1216
an elliptical shape that transforms to a cylinder at shallow 1217
level. In order to get a smooth transition from the dyke at 1218
depth to a cylindrical conduit (see Fig. 1) the value of a
0
in 1219
Eqs. (14) is parametrized as: 1220
a
0
(x) = A
1
arctan
_
x ÷ L
T
w
T
_
÷A
2
(21) 1221
Here L
T
and w
T
are the position and the vertical extent 1222
of the transition zone between the ellipse and the cylinder 1223
and constants A
1
and A
2
are calculated to satisfy condi 1224
tions a
0
(L) = R and a
0
(0) = a
0
, where R is the radius of 1225
the cylindrical part of the conduit and a
0
is the length of 1226
semimajor axis at the inlet of the dyke. The value of b
0
1227
is calculated in order to conserve the crosssection area of 1228
the unpressurized dyke, although it can also be speciﬁed 1229
independently. 1230
In order to decouple the inﬂuence of the dyke geome 1231
try from the oscillations caused by magma chamber pres 1232
sure variations, we have assumed a ﬁxed chamber pressure 1233
as a boundary condition for the entrance to the conduit. 1234
This assumption is valid because the timescale of chamber 1235
pressure variations are much longer than the period of the 1236
cycle (2–3 years in comparison with 5–7 weeks). 1237
Results presented in Figure 8 show that, even with 1238
a ﬁxed chamber pressure, there are magma discharge rate 1239
oscillations. At the beginning of a cycle the magma dis 1240
charge rate is at a minimum, while the overpressure (here 1241
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 17
Volcanic Eruptions: Cyclicity during Lava Dome Growth, Figure 7
a Comparison of calculated and measured discharge rates (a) and volumes of the dome (b) for the episode of the dome growth in
2001–2002. Influx into the magma chamber is shown by a dashed line in a. Time in days begins on 6th June, 2001. After [25]
Volcanic Eruptions: Cyclicity during Lava Dome Growth, Figure 8
a Dependence of magma discharge rate (solid line) and magmatic overpressure at depth of 1 km(dashed line) on time, for a D 240m
and b D 2:25m at the inlet of the dyke. The period of cycle is 46 days, average discharge rate is 6.2m
3
/s, with peak rate about
12 m
3
/s. b Profiles of crosssection areas of the conduit during one cycle. Curve A corresponds to the beginning of the cycle, B – to
a point on the curve of ascending discharge rate, C – to maximum discharge, and D to the middle of descending discharge curve.
At the beginning of the cycle, due to large viscosity of magma (at low discharge rate crystal content is high) large magmatic over
pressure develops, reaching a maximum near the transition between the dyke and cylindrical conduit; the dyke inflates providing
temporary magma storage. Minimumdyke volume corresponds to maximumdischarge rate (curve C). After [21]
presented for 1 km depth by a dashed line) and dyke width 1242
are at a maximum. At point A in Fig. 8a the crystal content 1243
and viscosity have reached their maximumvalues. Beyond 1244
this threshold condition, an increase in magma discharge 1245
rate results in decreasing pressure and dyke width. How 1246
ever, crystal content and viscosity also decrease and this 1247
eﬀect decreases friction, resulting in ﬂow rate increase and 1248
pressure decrease. At C the system reaches minimum vis 1249
cosity and crystal content, which cannot decline further. 1250
Thereafter the magma discharge rate decreases, while the 1251
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18 Volcanic Eruptions: Cyclicity during Lava Dome Growth
pressure and dyke width increase. The dyke acts like a ca 1252
pacitor, storing volume during this part of the cycle. 1253
The period of oscillation depends on several parame 1254
ters such as inﬂux rate and dyke aspect ratio a/R. Typi 1255
cally the period decreases with increasing aspect ratio. The 1256
range of calculated periods varies between 38 and 51 days 1257
for semi majoraxis lengths, a, from 175 to 250 m and semi 1258
minoraxes, b, from2 to 4 m. These results match observed 1259
cyclicity at SHV. The start of a cycle is quite sharp (Fig. 2), 1260
with the onset of shallow hybridtype (impulsive, lowfre 1261
quency coda) earthquakes. The change to shorter period 1262
and higher amplitude tilt pulsations indicate a marked in 1263
crease in average magma discharge rate [91,98]. The model 1264
cycles also have rapid onsets. The highamplitude tilt pul 1265
sations lasted for several weeks [91], consistent with the 1266
duration of higher magma discharge rates early in each 1267
5–7 week cycle. Tilt data (Fig. 2) are consistent with the 1268
model in that the episode of high magma discharge is as 1269
sociated with a marked deﬂation that lasts several weeks 1270
(see dashed curve at Fig. 8, representing magmatic over 1271
pressure at 1 km depth). The magma pressure builds up in 1272
the swelling dyke and then reaches a threshold, whereupon 1273
a surge of partly crystallized magma occurs, accompanied 1274
by elevated seismicity. 1275
The models presented above have certain general fea 1276
tures that are necessary to show cyclic behaviour. First of 1277
all, the resistance of the conduit must depend on magma 1278
discharge rate in a way that resistance decreases when 1279
discharge rate increases in some range of discharge rate. 1280
This dependence is reproduced by a sigmoidal curve. Re 1281
sistance is a product of viscosity and velocity, and is 1282
linearly proportional to discharge rate. This means that 1283
magma viscosity must decrease as discharge rate strongly 1284
increases. There may be many reasons for this behaviour, 1285
including crystallization, temperature variation or gas dif 1286
fusion. The second condition is that there must be some 1287
capacitor in the system that can store magma in a period 1288
of low discharge rate and release it in a period of high 1289
discharge rate. The role of this capacitor can be played 1290
by a magma chamber or dykeshaped conduit located in 1291
elastic rocks, or by compressibility of the magma itself. 1292
The volumes of these capacitors are diﬀerent and thus will 1293
cause pulsations with diﬀerent periods. Currently there is 1294
no single model that can account for pulsations with mul 1295
tiple timescales. 1296
Future Directions 1297
Our models indicate that magmatic systems in lava dome 1298
eruptions can be very sensitive to small changes in param 1299
eters. This sensitivity is most marked when the system is 1300
close to the cusps of steadystate solutions. If magma dis 1301
charge rate becomes so high that gas cannot escape eﬃ 1302
ciently during magma ascent, then conditions for magma 1303
fragmentation and explosive eruption can arise. Empiri 1304
cal evidence suggests that conditions for explosive erup 1305
tion arise when magma discharge rates reach approxi 1306
mately 10 m
3
/s or more in dome eruptions [45,83]. Cal 1307
culations show the possibility of such high discharge rates 1308
for the system parameters typical of lava domebuilding 1309
eruptions. 1310
We have illustrated model sensitivity of results by 1311
varying only one parameter at a time on plots of cham 1312
ber pressure and discharge rate. However, magmatic sys 1313
tems have many controlling parameters that may vary 1314
simultaneously. Furthermore, some controlling parame 1315
ters are likely to be interdependent (such as temperature, 1316
volatile content and phenocryst content, for example) and 1317
others may be independent (such as magma temperature 1318
and conduit dimensions). An eruption can be expected to 1319
move through nparameter space, making simulation and 1320
its parameter depiction diﬃcult. Our results are simpli 1321
ﬁed, so system sensitivity and behaviour in the real world 1322
may be yet more complex. A volcanic system may be quite 1323
predictable when it is within a stable regime, but may be 1324
come inherently unpredictable [84,85] when variations in 1325
the parameters move the system towards transition points 1326
and ﬂow regime boundaries. 1327
As in all complex systems there are many controlling 1328
parameters. Our models capture some of the key dynam 1329
ics, but are still simpliﬁed in many respects, so do not fully 1330
capture the real variations. Our models do not, for ex 1331
ample, consider variations in dome height, gas escape to 1332
surrounding rocks, strainrate dependent rheological ef 1333
fects or time dependent changes in conduit diameter. The 1334
model for porosity is based on interpretation of measure 1335
ments of porosity of erupted magma. The role of post 1336
eruptive alterations of pore structure, for example, for 1337
mation of cooling cracks, cannot be easily estimated. The 1338
model of bubble coalescence and permeability formation 1339
is important for understanding gas escape mechanisms 1340
and will provide constraints on transitions between extru 1341
sive and explosive activity. Because the model remains 1D, 1342
lateral distribution of parameters cannot be studied. These 1343
includes: lateral pressure gradients, magma crystallization 1344
on the conduit walls, wallrocks melting or erosion, for 1345
mation of shear zones and shear heating, heat ﬂux to sur 1346
rounding rocks. The models also make the simplifying as 1347
sumption that inﬂux into the chamber from a deep source 1348
is a constant, or given as a function of time. The dynamics 1349
of the magma chamber itself are oversimpliﬁed in all ex 1350
isting conduit ﬂow models. Changes in magma properties 1351
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Volcanic Eruptions: Cyclicity during Lava Dome Growth 19
in magma chamber can aﬀect the longterm evolution of 1352
eruptions. We have considered water as the only volatile 1353
and the addition of other gas species (e. g. CO
2
and SO
2
) 1354
would add further variability. 1355
There are large uncertainties in some parameters, 1356
which are likely to be very strong controls, such as the rhe 1357
ological properties of high crystalline magmas and crys 1358
tal growth kinetic parameters, notably at low pressures 1359
(< 30 MPa) where experiments are very diﬃcult to do 1360
(e. g. [22]). More experiments are necessary to under 1361
stand the rheology of multiphase systems containing melt, 1362
crystals and bubbles. The eﬀects of crystal shape, crystal 1363
size distribution and strain rate remain largely unclear. 1364
Some parameters, such as conduit geometry variation with 1365
depth, are highly uncertain. With so many parameters, 1366
good ﬁts can be achieved by selecting plausible values for 1367
real systems. Barmin et al. [3], for example, were able to re 1368
produce the patterns of discharge rate at Mount St.Helens 1369
and Santiaguito quite accurately. However, such models 1370
are not unique, partly because the actual values of some 1371
parameters may be quite diﬀerent to the assumed values 1372
and partly because of the model simpliﬁcations. 1373
Results obtained from waveform inversions of very 1374
longperiod seismic data over the past few years point to 1375
the predominance of a cracklike geometry for volcanic 1376
conduits [11], and it is becoming increasingly evident that 1377
the details of this geometry, such as conduit inclination, 1378
a sudden change in conduit direction, a conduit bifurca 1379
tion, or a sudden increase in cross section, are all criti 1380
cally important in controlling magma ﬂow dynamics. Fu 1381
ture modeling attempts will need to be closely tied to in 1382
formation on conduit geometry derived from seismology 1383
to provide a more realistic view of volcanic processes. 1384
The full simulation of any particular volcanic erup 1385
tion in such a nonlinear and sensitive system may appear 1386
a hopeless task. However, some reduction in uncertainties 1387
will certainly help to make the models more realistic. Fur 1388
ther experimental studies of crystallization kinetics and 1389
the rheological properties of magma at high crystallini 1390
ties are among the most obvious topics for future research. 1391
Advances in understanding the controls on magma input 1392
into an opensystem chamber would be beneﬁcial, since 1393
the delicate balance between input and output is a prime 1394
control on periodic behaviour. 1395
Further model development includes 2Deﬀects, elastic 1396
deformation eﬀects in dykefed domes and coupling be 1397
tween magma chamber and conduit ﬂow dynamics. Even 1398
with such improvements, large parameter uncertainties 1399
and modeling diﬃculties will remain. In such circum 1400
stances the logical approach is to start quantifying the un 1401
certainties and sampling from themto produce probabilis 1402
tic outputs based on ensemble models where numerical 1403
models of the kind discussed here can be run many times. 1404
A future challenge for numerical models will also be to 1405
produce simulated outputs which compare in detail with 1406
observations, in particular time series of magma discharge 1407
rates. 1408
Acknowledgments 1409
This work was supported by NERC research grant refer 1410
ence NE/C509958/1. OM and AB acknowledges Russian 1411
Foundation for Basic Research (050100228) and Presi 1412
dent of Russian Federation program (NCH4710.2006.1). 1413
RSJS acknowledges a Royal Society Wolfson Merit Award. 1414
The Royal Scoiety exchange grants and NERC grants had 1415
supported the Bristol/Moscow work over the last 10 years. 1416
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