bubble growth processes in magma surrounded by an elastic...

al Research 155 (2006) 307–322www.elsevier.com/locate/jvolgeores

Journal of Volcanology and Geotherm

Bubble growth processes in magma surrounded by an elastic medium

Youichi Shimomura, Takeshi Nishimura ⁎, Haruo Sato

Department of Geophysics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan

Received 13 April 2005; received in revised form 30 March 2006; accepted 6 April 2006Available online 5 June 2006

Abstract

Gas bubble growth in magma plays an important role in volcanic explosivity, and many previous studies have undertakentheoretical and experimental analyses of gas bubble expansion in melt. These previous studies commonly assume constantambient pressure, but it is more natural to assume that the magma is stressed by the crust or volcanic edifices as the volumeof gas bubbles increases. Here we present a bubble growth model that takes into account the elasticity of the surroundingmedium; we use this model to examine temporal changes in the bubble growth process. Our model consists of a two-dimensional dike filled with compressible viscous melt and numerous tiny gas bubbles. We use the cell model proposed byProussevitch et al. [Proussevitch, A., Sahagian, D. L., Anderson, A. T., 1993. Dynamics of diffusive bubble growth inmagmas: isothermal case. J. Geophys. Res. 98, 22283–22307] to describe the interaction between the numerous gas bubblesand the melt. The bubble growth process is formulated by the diffusion equation of volatiles in the melt, the mass balancebetween bubbles and melt, and the momentum equation of the melt. We also introduce pressure balance equations betweenthe melt and the surrounding elastic medium [Nishimura, T., 2004. Pressure recovery in magma due to bubble growth.Geophys. Res. Lett. 31]. Using the finite difference scheme, we numerically calculate temporal changes in bubble radius andmelt pressure within magma subjected to sudden depressurization. Simulation results show that the elasticity of thesurrounding medium strongly controls the bubble growth process. Under high effective rigidity, the final bubble radius isseveral times smaller than for zero rigidity, and the time required for bubble growth is an order of magnitude quicker.Temporal changes in the melt pressure are particularly dependent on the elasticity of the surrounding medium. Melt pressureeffectively recovers and even exceeds the given pressure drop for conditions of high rigidity and/or small initial bubbleradius. Although bubble growth in magma has previously been investigated mainly from geological samples and theoreticalperspectives, our model can quantitatively evaluate pressure changes in magma that can also be detected by seismic andgeodetic measurements.© 2006 Elsevier B.V. All rights reserved.

Keywords: bubble growth; magma; elastic medium; dike

⁎ Corresponding author. Tel.: +81 22 795 6532; fax: +81 22 7956783.

E-mail address: [email protected](T. Nishimura).

0377-0273/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.jvolgeores.2006.04.003

1. Introduction

Many researchers have investigated bubble growthprocesses in magma over the past few decades, asbubble growth of volatiles such as H2O and CO2

within magma plays an important role in theexplosivity of volcanic eruptions and the migrationof magma. Sparks (1978) conducted a fundamental

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http://dx.doi.org/10.1016/j.jvolgeores.2006.04.003

308 Y. Shimomura et al. / Journal of Volcanology and Geothermal Research 155 (2006) 307–322

study on the relationship between bubble growthprocesses and controlling variables such as magmaascent rate, viscosity of melt, and diffusivity coeffi-cient of volatiles. He numerically examined the bubblegrowth process in an ascending magma on the basis ofa simple physical model. The model assumes a singlegas bubble in an infinite melt, and diffusive bubblegrowth obeys a parabolic growth law as presented byScriven (1959). Proussevitch et al. (1993) proposed acell model that simplifies the interaction betweennumerous gas bubbles and surrounding melt. Theauthors used the cell model to examine temporalchanges in bubble growth by solving the diffusionequation under the condition of a constant ambientpressure for rhyolitic and basaltic magma. Prousse-vitch and Sahagian (1996) further applied the cellmodel to an ascending magma to examine temporalchanges in bubble radius and the amount of saturatedvolatiles in rhyolitic and basaltic melt. They showedthat bubble radius increases rapidly as the magmareaches shallow depths; in this study, magma ascentwithin a vertical conduit was simply expressed byreducing ambient pressure at a constant rate.

To compare these numerical simulation results withexperimental data, Lyakhovsky et al. (1996) performeda bubble growth experiment under high-pressureconditions of 150 MPa and high temperature of780–850 °C. They decompressed rhyolitic melthydrated from 150 MPa to 15–145 MPa and showedthat the measured bubble radius and number density ofbubbles are well explained by the bubble growthmodel of Proussevitch et al. (1993). Lensky et al.(2004) recently obtained analytical solutions for thefirst, second, and third stages of bubble growth, whichare controlled by viscous resistance, diffusion ofvolatiles, and decompression due to magma ascent,respectively. The solutions are in good agreement withtheir experimental data and numerical solutions basedon the bubble growth model of Proussevitch et al.(1993). These previous studies have mainly involvedtheoretical investigations of magma dynamics in avolcanic conduit and interpretations of the character-istic features of geological samples.

Geodetic measurements such as tilt and strainmeters and GPS networks enable the quantitativeevaluation of magma activity beneath volcanoes (e.g.,Okada and Yamamoto, 1991; Voight et al., 1998).Since magma density is lower than the density of thecrust, the buoyancy force is generally considered themost probable driving force of magma intrusions,however recent advances in geodetic and seismicmeasurements reveal pressure increases that appear to

be related to bubble growth within magma. Forexample, Yamamoto et al. (2001) and Fujita et al.(2002) observed step-like changes in tilt of the groundsurface caused by the intermittent opening of cracksover a 2-minute period at depths of about 6–8 kmduring the 2000 eruption at Miyakejima Volcano,Japan. The authors interpreted crack development toresult from vesiculation processes within the magma.Excitation of volcanic low-frequency earthquakes andexplosion earthquakes are also considered to berelated to rapid bubble growth in magma. Forexample, recent seismic observations at SuwanosejimaVolcano, Japan, detected a deflation that triggeredrapid expansion of the magma in the conduit at theinitiation of small Vulcanian eruptions (Tameguri etal., 2004; Iguchi, 2005). It is not easy to explain suchrapid expansion following deflation processes bybuoyancy alone.

These geophysical studies suggest that bubblegrowth processes in magma can also be investigatedfrom geophysical observations. However, most of thebubble growth models proposed in previous studiespresume a constant ambient pressure of the melt so thatinteractions between the melt and surrounding mediumare neglected; this is necessary to quantitatively relatebubble growth processes to seismic and geodeticobservations. Recently, Nishimura (2004) examinedhow the elasticity of the surrounding medium affectsbubble growth processes in magma. He examined thefinal bubble radius and pressure in melt subjected to apressure drop which may be caused by dike injection,plug opening of a volcanic pipe, and stress changesinduced by nearby earthquakes, and he found that thepressure in magma is recovered when the effectiveelasticity of surrounding medium is sufficiently large.However, his model did not take into account temporalvariations.

In the present study, therefore, we present a newbubble growth model that takes into account theelasticity of the medium surrounding the melt. Bycombining the cell model of Proussevitch et al. (1993)with the interactions of melt and surrounding elasticmedium (Nishimura, 2004), we are able to examinetemporal changes in bubble radius and the melt pressurewithin a dike.

2. Bubble growth model

2.1. Outline of bubble growth model

We consider a magma chamber embedded in avolcanic edifice or crust. Magma is saturated with

Fig. 1. Schematic illustrations of (a) cell model, (b) an elementary cell,and (c) a dike surrounded by an elastic medium. The dike is filled withcompressible viscous melt and numerous tiny spherical gas bubbles.Magma is represented by a combination of many elementary cells. Rand S is the radius of the elementary cell and gas bubble, respectively,and L is the dike length. Note that the directions of x- and y-axes cantake arbitrary directions (e.g., vertical or horizontal) since nogravitational force is assumed.

309Y. Shimomura et al. / Journal of Volcanology and Geothermal Research 155 (2006) 307–322

volatiles such as H2O and CO2 at shallow depths (e.g.,Tait et al., 1989) and consists of melt and numerous gasbubbles. Equilibrium conditions are initially attained notonly between gas bubbles and melt but also between themelt and elastic medium that surrounds the magmachamber. When the magma chamber is suddenlydepressurized, for example, in response to dikeextension or nearby seismicity, melt pressure decreasesrapidly because a pressure wave propagates through themagma chamber at the speed of acoustic velocity. Gasbubbles retain their original pressure longer than thesurrounding melt because bubble growth is restricted bymelt viscosity. Gas bubbles in the magma begin to growdue to the pressure gradient that accumulates betweenthe gas bubbles and ambient melt. Bubble growth isfurther enhanced by the diffusive flow of volatiles thatare dissolved in the melt as gas bubbles. As a result, thevolume of gas in the magma chamber increases. Sincethe magma is surrounded by the volcanic edifice orcrust, expansion of the magma chamber is restricted andthe melt is compressed by the gas bubbles andsurrounding medium.

We propose a magma chamber filled with compress-ible viscous melt and numerous tiny gas bubbles. Themelt and bubbles are expressed by the cell model ofProussevitch et al. (1993), in which multiple sphericalbubbles of a constant radius are uniformly packed (Fig.1(a)). Each bubble is surrounded by a finite volume ofthe melt, represented by an elementary cell. Theelementary cell is spherical, with a single gas bubblelocated at its center (Fig. 1(b)). We assume nointeraction between neighboring elementary cells suchthat all gas bubbles grow in the same manner. Thissimplification enables us to examine bubble growthprocesses in the entire chamber by studying the growthof just a single bubble. The magma chamber is assumedto be surrounded by an elastic medium. The boundarybetween the melt and surrounding medium is not exactlyexpressed by a combination of the elementary cells, asthe outermost cells do not completely follow the smoothinternal surface of the chamber. However, since thenumber of elementary cells that line the boundary issmall compared with the total number of cells in theentire magma chamber, we neglect the effect of theboundary mismatch.

This model is further simplified as follows. Gravi-tational and other body forces are neglected. Gas (H2O)in the bubbles is assumed to be a perfect gas and thenumber of gas bubbles in the magma chamber ispreserved. The system is assumed to be isothermalthroughout bubble growth, because the diffusivitycoefficient of volatiles is several orders of magnitude

smaller than thermal diffusivity (Sparks, 1978; Prous-sevitch et al., 1993).

2.2. Governing equations

We use the governing equations proposed byProussevitch et al. (1993) for interaction between gasbubbles and melt, and the equations of Nishimura(2004) that describe interactions between the melt andelastic medium. Notations are listed at the end of thetext.


2.2.1. Interactions between bubbles and meltBubble growth induced by the diffusive flow of

volatiles within melt into a gas bubble in eachelementary cell is governed by the equation of diffusionof the volatiles dissolved in the melt:

AcAt

¼ D1r2

A

Arr2AcAr

� �; ð1Þ

where c is the concentration fraction of volatiles in themelt and D is the diffusivity coefficient of volatiles. Inthe present study, the volatiles dissolved in the melt areassumed to be water. The initial condition and boundaryconditions at the surface of the gas bubble (r=R) and thesurface of the elementary cell (r=S) are given by

cðr; 0Þ ¼ c0 ðr > R; t ¼ 0Þ; ð2aÞ

cðR; tÞ ¼ ðKhPgÞ1=2 ðr ¼ R; t > 0Þ; ð2bÞ

AcAr

� �r¼S

¼ 0 tz0ð Þ: ð2cÞ

Eq. (2a) expresses the initial uniform distribution ofvolatiles in melt. At the interface between the bubbleand melt, the volatile concentration can be expressedby Henry's law (2b). The outer surface of theelementary cell is not permeable to volatiles, as givenby Eq. (2c).

Mass balance of volatiles at the interface between thegas bubble and melt can be expressed as

ddt

43pR3qg

� �¼ 4pR2Dqm

AcAr

� �r¼R

; ð3Þ

where ρg and ρm are the densities of the gas and melt,respectively. The left-hand side of the equation showsthe mass change of volatiles in the gas bubble, while theright-hand side is the diffusive flux of volatiles acrossthe bubble-melt interface.

The gas in the bubble obeys the equation of a perfectgas:

Pg

qg¼ BT

M; ð4Þ

where B is the gas constant, T is the temperature, and Mis the molecular weight of water.

The momentum equation of a Newtonian liquid thatsurrounds a spherical bubble can be written as

Pg−Pm ¼ 2rR

þ 4gdRdt

1R−R2

S3

� �; ð5Þ

where σ and η are the surface tension and viscosity ofthe melt, respectively. Melt pressure in Eq. (5) can beused as the melt pressure for the entire magma chamber.

2.2.2. Interaction between melt and elastic mediumWhen a magma chamber changes in size and volume

and/or gas bubbles expand to compress the melt, thevolume change causes a pressure release or buildupwithin the melt. This relation can be represented by

dPm ¼ −KldVm

Vm; ð6Þ

where dPm and dVm are the pressure change and volumechange of the compressible melt, respectively, and Kl isthe bulk modulus of the melt. The volume of the melt Vm

is written by

Vm ¼ V−4

3pR3ngV0; ð7Þ

where V is the total volume of the magma chamber (i.e.,melt plus gas bubbles), V0 is the initial volume of themagma chamber, and ng is the number density ofbubbles.

For a two-dimensional dike for which a cross-sectionin the x–y plane is elliptical (Fig. 1(c)), the aspect ratio αcharacterizes the shape of the dike and is expressed as

a ¼ Apd L2

; ð8Þ

where A is the cross-sectional area of the dike and L isthe length of the long axis (here defined as the x-axis). Ifwe suppose that the dike opens only in the y-direction,keeping the length of the x-direction constant, thepressure change in the melt dPm is related todeformation of the dike (e.g., Okamura, 1990; Nishi-mura, 2004):

dPm ¼ l̄dVV0

; ð9Þ

where μ̄ is the effective rigidity of the surroundingelastic medium. The effective rigidity is expressed as

l̄ ¼ 2lð1−mÞ a; ð10Þ

where μ and ν are the rigidity and the Poisson's ratio ofthe surrounding elastic medium, respectively. The dikemay also extend its length in x-direction, but we neglectsuch dike extension in this study to avoid a generation ofpressure gradient in the magma which complicates theproblem to be solved; and also, this study is a firstattempt to examine the elasticity on the bubble growth,

Table 1Typical values and variation range in parameters used in the presentsimulation

Property Symbol Typical value Variation range

CommonInitial aspect ratio α 10−2 10−6–10−1

Initial bubble radius R0 10−5 m 10−6–2×10−5 mInitial melt pressure Pm0 25 MPa 25–100 MPaPressure drop ΔP 1 MPa 0.2–10 MPaSurface tension σ 0.2 N m−1 0.1–0.3 N m−1

Rhyolitic magmaThe number density

of bubblesng 1011 m−3 1010–1012 m−3

Viscosity η 106 Pa s 104–106 Pa sDiffusivity D 10−11 m2 s−1 10−13–10−10 m2 s−1

Basaltic magmaThe number density

of bubblesng 109 m−3 109–1011 m−3

Viscosity η 50 Pa s 5–5000 Pa sDiffusivity D 10−8 m2 s−1 10−7–10−10 m2 s−1

Table 2Properties of magma and surrounding elastic medium as used in thepresent simulation

Property Symbol Value

CommonPoisson's ratio ν 0.25Elastic rigidity μ 1.08×1010 PaGas constant B 8.31 J K−1 mol−1

Molecular weightof water

M 0.018 kg mol−1

Rhyolitic magma Basaltic magmaHenry's constant Kh 1.6×10−11 Pa−1 9.0×10−12 Pa−1

Melt density ρ 2200 kg m−3 2600 kg m−3

Bulk modulus of melt Kl 1.375×1010 Pa 1.625×1010 PaTemperature T 1300 K 1500 K


so we assume a thin dike (aspect ratio<10−1) that pre-sumably opens mainly in the y-direction for simplicity.

Eq. (9) is also applicable to magma chambers of othershapes. For example, the effective rigidity of a sphericalmagma chamber is expressed by

l̄ ¼ 43l: ð11Þ

Similar relationships between pressure change of themelt and volume change of the magma are also obtainedfor a cylindrical chamber that can represent, forinstance, a volcanic pipe (conduit):

dPm ¼ 2l

ffiffiffiffiffiVV0

r−1

� �: ð12Þ

We finally suppose that no new bubble is created andthat neither coalescence nor collapse of bubbles occursduring bubble growth. Since the number of gas bubblesis assumed to be preserved, the volume of the magma isrelated to the initial magma volume by

43pS3ngV0 ¼ V : ð13Þ

We numerically solve Eqs. (1)–(10) and (13) by thefinite difference method, mainly modifying the algo-rithm presented by Proussevitch et al. (1993). Weexecute the programs by substituting the initial condi-tions of the magma and cease all calculations when thefollowing condition is satisfied:

Rðt þ DtÞ−RðtÞRðtÞ < 10−5; ð14Þ

where Δt is the time step for each calculation. All of thefinal bubble radii calculated from our model are wellmatched with the solutions (Nishimura, 2004) withinerrors of ±5%, and most are within ±2%.

3. Simulation results

We examine temporal changes in bubble radiuswithin a two-dimensional dike for two major groups ofmagma: rhyolitic and basaltic magma. As the bubblegrowth process is affected by the surrounding elasticmedium, the initial values of gas bubbles (bubbleradius and number density of bubbles), physicalparameters of the melt (melt pressure, viscosity of themelt, and the diffusivity of water), and the givenpressure drop of the melt, we calculate the bubbleradius and melt pressure by changing one parameterand fixing the others to investigate the way in whicheach parameter controls the bubble growth process.Typical values and variation ranges for each parameterare listed in Table 1, while the physical properties ofrhyolitic and basaltic magma and the surroundingelastic medium are listed in Table 2. In the followingtext we describe temporal changes in bubble radius andpressure recovery that is defined as the increase in meltpressure after a given pressure drop ΔP.

3.1. Effects of the elasticity of the surrounding medium

Since the aspect ratio of the dike α controls theeffective rigidity, we calculate temporal changes inbubble radius and pressure recovery for different aspectratios while fixing the values of other physicalparameters. The aspect ratios of dikes observed in thefield are in the order of 10−4 for basaltic magma(Dvorak and Dzurisin, 1997), with smaller aspect ratiosexpected for rhyolitic magma. In the present study, we


therefore examine initial aspect ratios that range from10−6 to 10−1. The small aspect ratio of 10−6 represents alow effective rigidity of the surrounding elastic mediumwhile the large aspect ratio of 10−1 represents a higheffective rigidity (see Eq. (10)). In our calculations, theaspect ratio changes with time as the volume of the dikeincreases due to bubble growth.

Fig. 2(a) shows temporal changes in bubble radiusand pressure recovery for rhyolitic magma after asudden pressure drop of 1 MPa. The bubble radiusgradually increases over a logarithmic scale at thebeginning of bubble growth. All growth curves initiallyfollow the same trend, suggesting that the effect of thesurrounding medium is negligibly small during the earlystages of bubble growth. Then, the bubble radiusgradually increases with time, but the growth rate isdependent on aspect ratio. The rate of increase in bubbleradius is about 1/3 for large aspect ratios of 10−4 and10−6, which is slightly smaller than the value of 1/2expected from parabolic growth associated with diffu-sive processes under constant ambient pressure. How-ever, the rate of increase for aspect ratios of 10−2 and10−1 decreases to about 1/4 or 1/16. These resultsindicate that the surrounding elastic medium plays a role

Fig. 2. Temporal changes in bubble radius and pressure recovery in (a) rhyolit10−6 to 10−1. A small aspect ratio represents low effective rigidity, while a lgrowth curves and pressure recovery curves for small aspect ratios (10−4 an

in restricting bubble growth and decreasing the growthrate. The rate of increasing bubble radius for each aspectratio gradually decreases and the bubble radius finallyconverges on a constant value to attain an equilibriumcondition. Final bubble radius and the lapse timerequired for attaining the final equilibrium conditionare greatly affected by aspect ratio: the final bubbleradius is smaller and lapse time required for bubblegrowth shorter for larger aspect ratio.

One of the most important aspects derived from ourmodel is the pressure recovery in melt (right-handpanel in Fig. 2(a)). A large aspect ratio can build uppressure in melt. For aspect ratios of 10−2 and 10−1,melt pressure eventually recovers to 0.87 and0.99 MPa, respectively, given a pressure drop of1 MPa. This reflects the effect of the surroundingmedium within which melt pressure is built up whengas bubbles expand to increase the melt volume. Incontrast, only very small pressure recoveries of lessthan 0.07 MPa are observed for the small aspect ratiosof 10−4 and 10−6, which are approximately the sameas the case of constant ambient pressure. Thedifferences in the increase rate of bubble radiusdescribed above are explained by this pressure buildup.

ic magma and (b) basaltic magma for various aspect ratios ranging fromarge aspect ratio indicates high effective rigidity. Note that the bubbled 10−6) overlap.

Table 3Results of simulations for rhyolitic magma

Parameter Rf

(×10−5 m)Tgrowth(s)

Pre

(MPa)

α10−1 1.3 136 0.9910−2 2.1 1005 0.8710−4 3.7 3367 0.05610−6 3.8 3486 0.0006

R0 (m)2×10−5 2.5 460 0.8510−5 2.1 1005 0.872×10−6 2.1 1181 1.0010−6 2.2 1128 1.17

ng (m−3)

1010 4.3 5461 0.881011 2.1 1005 0.871012 1.2 82 0.85

Pm0 (MPa)25 2.1 1005 0.8750 1.9 2250 0.6775 1.8 3253 0.51100 1.7 3956 0.41

η (Pa s)104 2.1 1005 0.87105 2.1 1005 0.87106 2.1 1005 0.87

D (m2 s−1)10−10 2.1 104 0.8710−11 2.1 1005 0.8710−12 2.1 10054 0.8710−13 2.1 1×105 0.87

σ (N m−2)0.1 2.1 1005 0.860.2 2.1 1005 0.870.3 2.1 1005 0.98

ΔP (MPa)0.2 1.4 1168 0.181 2.1 1005 0.872 2.6 817 1.7210 4.3 374 8.58


The gradual pressure buildup is considered to makedecreasing rate of gas density in bubbles smallercompared with the case of no pressure recovery so thatthe inflow rate of volatiles into gas bubbles decreasesdue to an increasing saturation concentration.

Temporal changes in bubble radius and pressurerecovery for basaltic magma also show similarcharacteristics (see Fig. 2(b)). When the aspect ratiois small (10−4 or 10−6), the increase rate of bubbleradius is equal to 1/2, which is consistent with the rateexpected from parabolic growth. As the aspect ratioincreases, the increase rate decreases slightly. But thedifference is not significant as recognized in rhyoliticmagma because we assumed small number density of109 m−3 (see Section 3.2.2). Significant differences forrhyolitic magma in terms of the time scale of bubblegrowth. The time required for bubble growth inrhyolitic magma is one or two orders of magnitudelonger than growth in basaltic magma. This is mainlybecause of differences in the diffusivity coefficient ofwater in each of the magmas. As the diffusivitycoefficient assumed for basaltic magma (10−8 m2 s−1)is three orders of magnitude larger than that forrhyolitic magma (10−11 m2 s−1), basaltic magmapromotes the diffusive flow of volatiles into bubbles,and bubbles grow faster.

To quantitatively evaluate the characteristics ofbubble radius and pressure recovery, we introducethree parameters: final bubble radius Rf, growth timeTgrowth, and final pressure recovery of the melt Pre. Thefinal bubble radius Rf and the pressure recovery Pre aredefined as the radius and the pressure recovery,respectively, when the convergence condition of Eq.(14) is satisfied. The growth time Tgrowth is defined asthe lapse time from t=0 to the time when the bubbleradius reaches 95% of its final value. In the case ofrhyolitic magma, the final bubble radius varies from3.8×10−5 to 1.3×10−5 m for aspect ratios in the range10−6 to 10−1. The growth time for a large aspect ratio of10−1 is 136 s, and 3486 s for a small aspect ratio of10−6, which is an order of magnitude larger than theformer value. The final pressure recovery varies from0.99 to 0.0006 MPa for the various aspect ratios. Forbasaltic magma, the final bubble radius varies from4.7×10−5 to 17.6×10−5 m for aspect ratios in the range10−1 to 10−6. The growth time for a large aspect ratio is7 s, which is about one-tenth of that for a small aspectratio (76 s). The melt pressure recovers to a value aslarge as 1.01 MPa for an aspect ratio of 10−1, whilepressure recovery is barely observed (0.0006 MPa) foran aspect ratio of 10−6. These results are summarized inTables 3 and 4.

3.2. Effects of the initial conditions of gas bubbles

3.2.1. Initial bubble radiusFig. 3(a) shows temporal changes in bubble radius

and pressure recovery for initial bubble radii rangingfrom 10−6 m to 2×10−5 m within rhyolitic magma.Results are shown for a large dike aspect ratio of 10−2

because sufficient pressure recovery is observed for thiscase; results for other aspect ratios indicate similarcharacteristics except for the magnitude of the finalbubble radius and the pressure recovery. The initialbubble radius influences the final pressure recovery,which is larger for smaller initial bubble radius. Forexample, pressure recovery is estimated to be 0.85 MPafor a large initial bubble radius of 2×10−5 m. In contrast,

Table 4Results of simulations for basaltic magma

Parameter Rf

(10−5 m)Tgrowth(s)

Pre

(MPa)

α10−6 17.6 76 0.000610−4 17.2 74 0.05710−2 9.3 26 0.8910−1 4.7 6.9 1.01

R0 (m)2×10−5 9.3 25 0.8710−5 9.3 26 0.892×10−6 9.8 25 1.0210−6 10.3 24 1.18

ng (m−3)

109 9.3 25 0.891010 4.3 5.3 0.881011 2.1 1.0 0.87

Pm0 (MPa)25 9.3 25 0.8950 8.6 60 0.6975 7.9 90 0.53100 7.3 118 0.42

η (Pa s)5000 9.3 169 0.89500 9.3 169 0.8950 9.3 169 0.895 9.3 169 0.89

D (m2 s−1)10−7 9.3 2.5 0.8910−8 9.3 26 0.8910−9 9.3 255 0.8910−10 9.3 2555 0.89

σ (N m−2)0.1 9.3 25 0.870.2 9.3 25 0.890.3 9.4 25 0.90

ΔP (MPa)0.2 5.7 43 0.201 9.3 26 0.892 11.7 19 1.7410 19.9 8 8.62


for a small initial bubble radius of 10−6 m, pressurerecovery reaches 1.17 MPa, which exceeds the initialpressure drop of 1 MPa. Such over-pressurization isconsidered to mainly result from the release of high gaspressure that originates from the surface tension of gasbubbles (see the first term in the right-hand side of Eq.(5)). The bubble growth time increases slightly withdecreasing initial bubble radius, which is due to theeffects of high viscosity term (Sparks, 1994; Sahagian etal., 1994). But, the variation is not significant andbubbles reach final equilibrium conditions within about1000 s.

Temporal changes in bubble radius and pressurerecovery for basaltic magma are similar to those forrhyolitic magma (Fig. 4(a)). Final bubble radius and

growth time are not strongly dependent on initial bubbleradius, whereas final pressure recovery is as large as1.19 MPa for a small initial bubble radius of 10−6 m.

3.2.2. Number density of bubblesFig. 3(b) shows temporal changes in bubble radius

and pressure recovery of melt for different numberdensities of bubbles ranging from 1010 to 1012 m−3 forrhyolitic magma. For a small number density of1010 m−3, the bubble radius reaches a final value thatis about four times larger than that for a large numberdensity of 1012 m−3. The final bubble radius varies from4.33×10−5 to 1.21×10−5 m for number densities thatrange from 1010 to 1012 m−3. A small number densityresults in such a large bubble separation distance that alarge volume of volatiles is able to flow into eachbubble; bubbles therefore grow larger. Irrespective ofnumber density, all curves of bubble radius follow asimilar trend during early stages of bubble growth;however, the curves begin to diverge in the middlestages and the increase rate decreases as the numberdensity increases. Bubble growth finishes in a shortertime with higher number density of bubbles. This trendis an effect of the surrounding elastic medium, with asimilar process to that explained in Section 3.1. Duringthe early stages of bubble growth, the distance betweenbubbles is large and no significant interaction exists;bubbles therefore grow in a similar manner even fordifferent number densities of bubbles. However, as thebubbles grow and total volume of bubbles expands, theincreasing rate of bubble radius decreases with increas-ing number density. Since the increase rate is notdependent on the number density for a small aspect ratioof 10−6 (see Fig. 3(c)), such restriction of bubble growthis considered to be a result from the increase of pressurein the melt.

Bubble growth time varies by two orders ofmagnitude (from 83 s to 5461 s) with comparablechanges in the number density, while all the final valuesof pressure recovery converge to 0.88 MPa. A largenumber density such as 1015 m−3 may be a moreappropriate value for rhyolitic magma (e.g., Toramaru,1989), but we are not able to simulate such a value dueto instabilities in the numerical simulation; however, ifwe extrapolate our results to such a large number densityof bubbles, the growth time is estimated to be as low as10 s, which is a time scale that can be detected as seismicwaves.

The same features as described above are recognizedfor bubble growth processes in basaltic magma, exceptfor the time scale (Fig. 4(b)). The growth time is muchquicker and varies from 0.07 s to 5.3 s for number

Fig. 3. Temporal changes in bubble radius and pressure recovery in rhyolitic magma for (a) initial bubble radius ranging from 10−6 to 2×10−5 m and(b) number density of bubbles ranging from 1010 to 1012 m−3, for an aspect ratio of 0.01. (c) Bubble radius for number densities of 1010 to 1012 m−3

and an aspect ratio of 10− 6.


densities ranging from 1011 m−3 to 109 m−3. Thisgrowth time is very fast, but such a very short time scale(<1 s) may be out of the range of our model, as quasi-static pressure change is assumed in the melt.

3.3. Effects of the physical parameters of the melt

3.3.1. Initial melt pressureIn this section we examine temporal changes in

bubble growth and pressure recovery for initial melt

pressures ranging from 25 to 100 MPa, whichcorresponds to lithostatic pressure at depths of about1 and 4 km, respectively. The initial concentration ofvolatiles in the melt, which depends on initial meltpressure, varies from 0.7 to 1.4 wt.% for rhyoliticmagma and from 1.2 to 2.4 wt.% for basaltic magma.Fig. 5(a) shows temporal changes in bubble radius andpressure recovery for rhyolitic magma. The finalbubble radius for high initial melt pressure is lessthan that for low initial pressure. The bubble reaches

Fig. 4. Temporal changes in bubble radius and pressure recovery in basaltic magma for (a) initial bubble radius ranging from 10−6 to 2×10−5 m and(b) number density of bubbles ranging from 109 to 1011 m−3.


the final condition within a short time when the initialmelt pressure is low: growth time varies from 1005 to3956 s for initial melt pressures that range from 25 to100 MPa. Since the rate of increase of bubble radiusduring the middle stages of bubble growth (about 102

to 104 s) is almost the same for the four casesconsidered here, delay time is mainly generated duringthe early stages of the bubble growth. This delay isconsidered to originate from an increase in gas densitywithin the bubble (Proussevitch et al., 1993). Pressurerecovery is sensitive to differences in the initial meltpressure and varies from 0.87 to 0.41 MPa for initialmelt pressures of 25–100 MPa.

For basaltic magma, the growth time for a low initialmelt pressure of 25 MPa (25 s) is about four times lessthan that for a high initial melt pressure of 100 MPa(118 s) (Fig. 6 (a)). Pressure recovery varies from 0.42 to0.88 MPa for initial melt pressures in the range 100–25 MPa.

3.3.2. DiffusivityThe diffusivity coefficient of water varies by several

orders of magnitude depending on the temperature of

the melt, and the concentration fraction of volatiles inthe melt, and melt viscosity (e.g., Karsten et al., 1982;Zhang et al., 1991; Zhang and Behrens, 2000). Forexample, diffusivity in rhyolitic magma varies from10−12 to 10−10 m2 s−1 at temperatures of 1000 K–1200 K and concentration fractions of 1–3 wt.%. Weexamined the bubble growth process for diffusivitiesranging from 10−13 to 10−10 m2 s−1. Temporal changesin bubble radius and pressure recovery for rhyoliticmagma are shown in Fig. 5(b). The growth time ofbubbles varies from 104 to 105 s for diffusivities of10−10 to 10−13 m2 s−1. An order of magnitude increasein diffusivity reduces growth time by an order ofmagnitude. The final bubble radius (2.0×10−5 m) andfinal pressure recovery (0.87 MPa) are independent ofdiffusivity.

For basaltic magma, temporal changes in bubbleradius and pressure recovery are shown in Fig. 6(b).Following Proussevitch et al. (1993), we varied thediffusivity from 10−10 to 10−7 m2 s−1. The results aresimilar to those for rhyolitic magma, although growthtimes varied from 2.5 to 2555 s for diffusivities withinthe range 10−7 to 10−10 m2 s−1; this is several orders of

Fig. 5. Temporal changes in bubble radius and pressure recovery in rhyolitic magma for (a) initial melt pressures ranging from 25 to 100 MPa, (b)diffusivity from 10−10 to 10−13 m2 s−1, (c) viscosity from 104 to 108 Pa s, and (d) surface tension from 0.1 to 0.3 N/m. Note that for the pressure dropdepicted in (c), viscosity is set at 5 MPa.


Fig. 6. Temporal changes in bubble radius and pressure recovery in basaltic magma for (a) initial melt pressures ranging from 25 to 100 MPa, (b)diffusivity ranging from 10−7 to 10−10 m2 s−1, (c) viscosity ranging from 5 to 5000 Pa s, and (d) surface tension from 0.1 to 0.3 N/m. Note that for thepressure drop depicted in (c), viscosity is set at 5 MPa.



magnitude quicker than the growth time for rhyoliticmagma. The final bubble radius and final pressurerecovery are estimated to be about 9×10−5 m and0.9 MPa, respectively.

3.3.3. ViscosityVariations in the temperature of the melt and the

concentration fraction of volatiles in the melt alter meltviscosity by several orders of magnitude (e.g., Shaw,1963; Persikov et al., 1990). For example, at atemperature of 1300 K, the viscosity of rhyoliticmagma varies from 104 to 108 Pa s for a concentrationfraction of water that varies from 0.2 to 4 wt.%.Viscosity varies from 105 to 108 Pa s for a concentrationfraction of 2 wt.% at temperatures in the range 1000–1300 K.

Here we examine bubble growth processes forviscosities that range from 104 to 108 Pa s asrepresentative values (Fig. 5(c)). Viscosity restrictsthe motion of gas bubbles (see Eq. (5)), and highviscosity of the melt acts to increase the growth timeof gas bubbles at the beginning of bubble growth. Thedelay time is proportional to the ratio of pressure dropto viscosity; however, if we assume a 1 MPa pressuredrop, the difference in bubble radius is too small todemonstrate, so we present results in Fig. 5(c) thatassume a 5 MPa pressure drop. The trend of bubbleradius growth for a viscosity of 108 Pa s is delayed byabout 10 s from those of other viscosities. Theviscosity of basaltic magma (5–5000 Pa s) is severalorders of magnitude lower than that of rhyoliticmagma; hence, the effects of viscosity are negligiblysmall (Fig. 6(c)) in terms of affecting bubble growthprocesses.

3.3.4. Surface tensionThe nature of surface tension is dependent on

pressure (e.g., Khitarov et al., 1979) and dissolvedwater (e.g., Mangan and Sisson, 2005). In the presentstudy, therefore, we examine three cases of surfacetension that range from 0.1 to 0.3 N m−1. Surfacetension affects the final pressure recovery: as surfacetension increases, the final pressure recovery alsoincreases (Tables 2 and 3). Large surface tensioncauses high gas pressure within the initial bubblesuch that pressure recovery increases. This charac-teristic is similar to the case of initial bubble radius.As shown in Figs. 5(d) and 6(d), temporal changesin bubble radius follow almost the same trend fromthe early stages of bubble growth; slight differencesare only recognized at the cessation of bubblegrowth.

3.4. Pressure drop of the melt

Sudden pressure drops within melt are considered toresult from dike extension, volcanic eruption, or nearbyearthquakes. The magnitude of a sudden pressure dropin the crust or a volcanic edifice can be estimated from,for example, seismic observations. Pressure releasewithin a shallow magma chamber is estimated to be0.1–10 MPa from analyses of explosion earthquakesobserved at several volcanoes around the world(Nishimura and Hamaguchi, 1993; Nishimura, 1998).Stress drops associated with tectonic earthquakes are inthe range 1–10 MPa (e.g., Aki and Richards, 2002).These observations suggest that pressure drops of up to10 MPa are plausible.

Fig. 7(a) shows temporal changes in bubble radiusand pressure recovery for pressure drops ranging from0.2 to 10 MPa within rhyolitic magma. Bubble radiusincreases and growth time decreases with increasingpressure drop. The rate of increase in bubble radius isreduced with smaller pressure drop; this trend isexplained as follows. A large pressure drop lowersambient pressure, and in our model, the initial volatileconcentration is controlled by the initial melt pressure.Therefore, a large pressure drop generates a largeamount of excess volatiles that cannot be dissolved inthe melt. Since a large initial volatile content acts toincrease the rate of bubble growth due to a steepconcentration gradient at the bubble interface (Prousse-vitch et al., 1993), the rate of increase of bubble radius isreduced for a small pressure drop.

Bubble growth time for a large pressure drop of10 MPa is 373 s, which is three times quicker than the1168 s growth time for a small pressure drop of 0.2 MPa.Final bubble radius varies from 1.4 × 10− 5 to4.3×10−5 m for pressure drops ranging from 0.2 to10 MPa.

In the right-hand panel of Fig. 7(a), the pressurerecovery is normalized by each given pressure drop.Pressure recoveries for the given pressure drops, thatrange from 0.2 to 10 MPa, vary from 85% to 90% of thegiven pressure drop. That is, the ratio of pressurerecovery to the given pressure drop is not stronglydependent on the given pressure drop. This trend isillustrated in Fig. 2 of Nishimura (2004), in which thepressure recovery curve tends to become linear as thepressure drop increases over a logarithmic scale.However, when an initial bubble radius is sufficientlysmall, a large pressure recovery or excess pressure isbuilt up, as discussed in Section 3.2.1.

Fig. 7(b) shows temporal changes in bubble radiusand pressure recovery in basaltic magma for pressure

Fig. 7. Temporal changes in bubble radius and pressure recovery for pressure drops ranging from 0.5 to 10 MPa. (a) Rhyolitic magma. (b) Basalticmagma. Each pressure recovery is normalized to the given pressure drop.

Table 5Relationships between physical parameters and major results of thesimulations

Physicalparameter

Final bubbleradius

Growth time Pressurerecovery

α Large Small Short LargeSmall Large Long Small

R0 Large Small change Small change SmallSmall Small change Small change Large

ng Large Small Short Small changeSmall Large Long Small change

Pm0 Large Small change Long SmallSmall Small change Short Large

η Large No change No change No changeSmall No change No change No change

D Large No change Short No changeSmall No change Long No change

ΔP Large Large Short No changeSmall Small Long No change


drops ranging from 0.2 to 10 MPa. The main features ofthe data are the same as the case for rhyolitic magma,except for the time scale of bubble growth. Bubblegrowth time varies from 43 s to 8 s for pressure dropsranging from 0.2 to 10 MPa. The final bubble radiusvaries from 5.6×10−5 m to 19.9×10−5 m for pressuredrops over the range 0.2 to 10 MPa. Pressure recovery isabout 85–100% of the initial pressure drop (see right-hand panel of Fig. 7(b)).

4. Discussion and conclusions

In this study we examined the effect of the elasticityof the surrounding medium (i.e., effective rigidity oraspect ratio in the dike model) and various physicalparameters on bubble growth processes in magmasubjected to a sudden pressure drop. Our findings aresummarized in Table 5. The most important role ofelasticity in the surrounding medium is the generation ofpressure recovery in the magma. High effective rigiditycan produce a large pressure recovery in magma due tothe volume expansion of gas bubbles. Our numericalsimulation also shows that the elasticity decreases thefinal bubble radius by an order of magnitude and

shortens the duration of bubble growth several times.Bubble radius and pressure recovery follow a similartrend in the early stages of bubble growth, and then, theincrease rate decreases in the middle stages as theelasticity becomes high (i.e., aspect ratio increases).This temporal feature that originates from gradual


pressure increases is recognized in the cases for thenumber density of bubbles, but not for the otherparameters. The final bubble radius is strongly affectedby the number density of bubbles, and the pressure dropand bubble growth time are strongly dependent on thediffusivity of water and the number density of bubbles,but our results indicate significant effects of elasticity onbubble growth processes.

As the elasticity of the surrounding medium can aidpressure recovery in a magma chamber subjected to asudden pressure drop, it may be possible to detectpressure changes within a magma chamber or dike bygeodetic or seismic observations. This may enable us toconstrain physical parameters of the bubble growthprocess. For example, during the 2000 eruption atMiyakejima Volcano, Japan, unusual tilt motions repeat-ed every few minutes can be explained by the opening ofcracks at a depth of 7 km (Yamamoto et al., 2001; Fujitaet al., 2002). The cyclic nature of the cracking indicatesthat the magma chamber must recover from the pressuredrop associated crack opening within a few minutes. Inthis case, we can explain the observed tilting and pressuredrops by modelling a magma with R0 =10

− 7 m,ΔP=1 MPa, ng≥109 m−3, and D≥10−8 m2 s−1, andwith a surrounding medium with ≥92 MPa.

During small explosions at Suwanosejima Volcano,Tameguri et al. (2004) used seismic analyses to detect acompressive motion 0.2–0.3 s following a dilationarymotion at very shallow depths (less than 0.5 km)beneath the crater. Such pressure recovery can beexplained if the magma has R0=10

−5 m, ΔP=1 MPa,ng≥1010 m−3, and D≥10−8 m2 s−1, although ourmodel may be inadequate to model the rapid expansionthat accompanies magma fragmentation. Tilt observa-tions during the 1996–1997 eruption activity atSoufriere Hills Volcano on Montserrat detected avolume expansion of ascending magma at a shallowdepth of less than 1 km (Voight et al., 1998); this may berelated to an expansion of magma due to bubble growthat shallow depths.

In conclusion, we have developed a new bubblegrowth model for magma surrounded by an elasticmedium. For a two-dimensional dike filled withcompressible viscous melt and numerous tiny gasbubbles, we numerically calculated temporal changesin bubble radius and melt pressure via the finitedifference method. The main results of our study aresummarized as follows:

(1) The elasticity of the surrounding medium controlsthe bubble growth process. For high effectiverigidity, the final bubble radius and bubble growth

time are several times smaller and an order ofmagnitude shorter, respectively, than under condi-tions of constant ambient pressure (i.e., no rigidity).

(2) Melt pressure can recover or even exceed thegiven pressure drop when the effective rigidity ofthe surrounding elastic medium is large and/or theinitial bubble radius is small.

(3) For a sufficiently low effective rigidity, the rate ofincrease in bubble radius is approximately pro-portional to the root square of time, as expectedfrom parabolic growth due to diffusion under con-stant ambient pressure. However, for high effec-tive rigidity, the increased rate of bubble radiusdecreases and does not follow parabolic growth.

(4) Pressure drop and recovery are measurable byseismic and geodetic observation; such observa-tions can potentially provide constraints on thecontrolling factors of bubble growth processesthat until now have been mainly discussed interms of analyses of geological samples.

NotationA cross-sectional area of two-dimensional dike

(m2)B universal gas constant (J K−1 mol−1)c concentration fraction of volatiles in meltD diffusivity coefficient of water in melt (m2 s−1)Kh Henry's constant (Pa−1)Kl bulk modulus of melt (Pa)L length of the long axis of two-dimensional dike

(m)M molecular weight of volatile gas (kg mol−1)ng number density of gas bubbles (m−3)P pressure (Pa)R radius of gas bubble (m)r radial coordinate of spherical coordinate

systemS radius of elementary cell (m)T temperature of melt (K)t time (s)V volume (m3)α aspect ratio of two-dimensional dikeη viscosity of melt (Pa s)μ rigidity of surrounding elastic medium (Pa)ν Poisson's ratio of surrounding elastic mediumρ density (kg m−3)σ surface tension of melt (N m−1)

Subscripts0 initial conditiong gasm melt


Acknowledgments

The manuscript was significantly improved byhelpful comments and suggestions by Margaret Man-gan, Eisuke Fujita, and two anonymous reviewers. Thisstudy was partly supported by a Grant for ScientificResearch from MEXT (No.14080202) and ACT-JST.

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