a numerical simulation of magma motion, crustal deformation, and seismic radiation...

May 12, 2003 15:47 Geophysical Journal International gji1936

Geophys. J. Int. (2003) 153, 699–718

A numerical simulation of magma motion, crustal deformation,and seismic radiation associated with volcanic eruptions

Takeshi Nishimura1 and Bernard Chouet21Research Center for Prediction of Earthquakes and Volcanic Eruptions, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan.E-mail: [email protected] Geological Survey 345 Middlefield Road, MS 910, Menlo Park, California 94025

Accepted 2003 January 28. Received 2003 January 27; in original form 2002 April 16

S U M M A R YThe finite difference method is used to calculate the magma dynamics, seismic radiation, andcrustal deformation associated with a volcanic eruption. The model geometry consists of acylindrical reservoir and narrow cylindrical conduit embedded in a homogeneous crust. Weconsider two models of eruption. In the first model, a lid caps the vent and the magma isoverpressurized prior to the eruption. The eruption is triggered by the instantaneous removalof the lid, at which point the exit pressure becomes equal to the atmospheric pressure. Inthe second model, a plug at the reservoir outlet allows pressurization of only the magmaticfluid in the reservoir before the eruption. Magma transfer between the reservoir and conduit istriggered by the instantaneous removal of the plug, and the eruption occurs when the pressureat the conduit orifice exceeds the material strength of the lid capping the vent. In both models,magma dynamics are expressed by the equations of mass and momentum conservation in acompressible fluid, in which fluid expansion associated with depressurization is accounted forby a constitutive law relating pressure and density. Crustal motions are calculated from theequations of elastodynamics. The fluid and solid are dynamically coupled by applying thecontinuity of wall velocities and normal stresses across the conduit and reservoir boundaries.Free slip is allowed at the fluid–solid boundary. Both models predict the gradual depletion ofthe magma reservoir, which causes crustal deformation observed as a long-duration dilatationalsignal. Superimposed on this very-long-period (VLP) signal generated by mass transport arelong-period (LP) oscillations of the magma reservoir and conduit excited by the acousticresonance of the reservoir–conduit system during the eruption. The volume of the reservoir,vent size, and magma properties control the duration of VLP waves and dominant periods ofLP oscillations. The second model predicts that when the magmatic fluid reaches the vent, ahigh-pressure pulse occurs at this location in accordance with the basic theory of compressiblefluid dynamics. This abrupt pressure increase just beneath the vent is consistent with observedseismograms in which pulse-like Rayleigh waves excited by a shallow source are dominant.The strength of the lid plays an important role in the character of the seismograms and indefining the type of eruption observed.

Key words: crustal deformation, fluid dynamics, magma, numerical techniques, seismic-wavepropagation.

I N T RO D U C T I O N

Seismic and geodetic measurements around active volcanoes arevery useful for a quantitative evaluation of volcanic activity. Recentseismic and geodetic data recorded on dense networks of stationshave shed light on the force system associated with volcanic erup-tions. For example, Kanamori et al. (1984) analysed explosion earth-quakes associated with eruptions of Mount St Helens and showedthat these earthquakes were excited by a downward-oriented vertical

single force. They interpreted the source mechanism of the singleforce as a counter force associated with the sudden removal of alid capping a pressurized fluid-filled cavity located at shallow depthbeneath the volcano. Uhira & Takeo (1994) used a moment-tensorinversion technique to analyse seismic signals associated with vul-canian explosions at Sakurajima Volcano, Japan. They showed thatvolcanic explosions at Sakurajima are accompanied by the contrac-tion of a magma chamber at a depth of a few kilometers. Nishimura(1998) considered the deflation process of a magma chamber

C© 2003 RAS 699


700 T. Nishimura and B. Chouet

connected to the surface by a narrow conduit as a seismic source ofexplosion earthquakes, and obtained a mathematical description ofthe observed relation between seismic magnitude and vent radius.These studies strongly suggest that an explosive volcanic eruptionmacroscopically can be represented by the deflation process of areservoir system beneath a volcano.

Seismic observations made close to volcanoes provide fur-ther details of the spatio-temporal properties of these explosivesources. Based on analyses of seismic data and video picturesobtained at Sakurajima Volcano, Japan, Ishihara (1990) demon-strated that an explosive eruption is preceded a few seconds be-fore by an initial break at a depth of about 2 km beneath thesummit crater. Using moment-tensor inversions, Tameguri et al.(2002) suggested that large-amplitude surface waves are generatedfrom the top of the conduit simultaneously with an explosion at thevent.

Long-period (LP) signals in the form of eruption tremor are oftenobserved at many volcanoes during a sustained eruption. Using spec-tral analyses, Nishimura et al. (1995) showed that eruption tremorobserved during the 1989 eruption of Mount Tokachi, Japan, hasa source mechanism similar to that of explosion earthquakes. Mc-Nutt (1994) summarized the characteristics of eruption tremor ob-served around the world and showed that the reduced displacement(Aki & Koyanagi 1981) of eruption tremor, which is an indicationof the power of the tremor source, is proportional to the VolcanicExplosion Index (VEI), which is related to the amount of eruptedmaterials. Although many source models have been proposed to ex-plain the narrow spectral peaks of LP seismicity (Crosson & Bame1985; Chouet 1985, 1986, 1988, 1992, 1996; Chouet et al. 1994;Ferrick et al. 1982; Fujita et al. 1995; Kumagai & Chouet 1999,2000, 2001; Mikada 1992), few source models treat the eruptiontremor.

Geodetic measurements made close to a volcano also yield in-formation on mass transport processes beneath a volcano. Froman analysis of high-sensitivity tilt and strainmeter records obtainedin a vault located 2.7 km from the crater of Sakurajima, Ishi-hara (1990) found that an inflation of the volcano precedes eachexplosive eruption by a few hours. The tilt and strainmeter dataalso show a deflation of the volcano after each eruption. Similardeformation signals associated with volcanic explosions were de-tected by a strainmeter during the 1988–1989 activity of MountTokachi (Miyamachi et al. 1990), and by a tiltmeter located 1.5km from the active vent of Mount Etna, Italy (Bonaccorso & Davis1999). Broad-band seismometers deployed near the active vent ofAso Volcano, Japan, recorded the inflation and deflation phases as-sociated with small phreatic eruptions (Kaneshima et al. 1996).Broad-band recordings illustrating inflation and deflation phasesaccompanying eruptions of Stromboli Volcano, Italy, were obtainedby Chouet et al. (1999, 2003). Similar broad-band measurementswere shown to accompany vulcanian explosions at PopocatepetlVolcano, Mexico (Arciniega-Ceballos et al. 1999). Although themechanisms of inflation phases detected by geodetic and seismicmeasurements may not necessarily originate in the same physicalprocess, these observations show that most volcanic eruptions areassociated with an initial inflation phase, followed by a deflationprocess accompanying the eruption of volcanic materials from thevent.

Several source models have been proposed to explain these char-acteristics of observed seismic and geodetic signals. In most modelsthe magmatic fluid is simplified to an incompressible liquid or per-fect gas, and small pressure fluctuations are assumed to make themathematical treatment of the fluid dynamics and elastodynamics

more tractable. In the case of an eruption, however, it is necessary totake into account the pressure dependence of magma properties asmagma migrates toward the surface because the dominant period ofseismic waves excited by a pressure transient in a volcanic conduitis strongly affected by the depth dependence of magma properties(Neuberg 2000). The discharge of magma is another critical param-eter describing eruptive activity. Several constitutive relationshipsbetween fluid pressure and fluid density accounting for the vesic-ulation process of gases in liquid magma have been proposed toquantitatively evaluate flow motions in a volcanic conduit (Wilsonet al. 1980; Ida 1990; Bower & Woods 1997). These studies gener-ally focus on the dynamics of magma motion only, and do not targetthe seismic radiation and crustal deformation, which represent themost informative data about underground magma dynamics. Thepurpose of the present study is to fill this gap in our knowledge. Weuse numerical simulations to simultaneously estimate the dynamicsof magma flow in the conduit and reservoir, and associated crustaldeformation and seismic radiation during a volcanic eruption. Us-ing the coupled equations of fluid dynamics and elastodynamics, weexamine the basic characteristics of magma motions and associatedelastic displacement field in the surrounding solid.

M AT H E M AT I C A L F O R M U L AT I O NO F A V O L C A N I C E RU P T I O N

We assume the simplified volcanic system illustrated in Fig. 1 toexamine the magma motion and seismic wave radiation associatedwith an eruption. The conduit–reservoir system consists of a magmareservoir connected to the surface by a narrow vertical conduit. Thereservoir has a cylindrical shape with radius r v and length �v , andthe conduit consists of an axially symmetric pipe with radius rc

and length �c. Before an eruption, a lid representing a lava dome orsolidified volcanic plug covers the conduit orifice.

We examine two models of eruption. Our first model (hereafternamed Model I) is based on the simple model proposed by Kanamoriet al. (1984) and Nishimura (1998). In this model, the reservoir andconduit are both simultaneously pressurized by the addition of newmagma to the system, and the fluid is further gravitationally pressur-ized by its own weight. Accordingly, the volcanic crust surroundingthe reservoir–conduit system is stressed by the pressure exerted bythe magma in the conduit and reservoir, resulting in an inflationof the crust in the vicinity of the conduit and reservoir. An erup-tion is triggered by the instantaneous removal of the lid capping theconduit orifice. Magma migrates upward in response to the pres-sure gradient established between the reservoir and atmosphere, andcrustal deformation and seismic wave radiation are generated by themigration of magmatic fluid and associated deflation of the reser-voir. We calculate the motions of the fluid and surrounding mediumby using the basic equations of compressible fluid dynamics andelastodynamics.

In our second model (hereafter named Model II), we assumea plug at the reservoir outlet. The fluid in the conduit and reser-voir is gravitationally pressurized by its own weight. Before aneruption, new magma is supplied to the reservoir only, leading toadditional pressurization of the reservoir because of the presenceof the plug, which restricts magma transfer from the reservoir tothe conduit. Magma motions in the reservoir–conduit system, andassociated crustal deformation, are triggered by the instantaneousremoval of the plug. Magma migrates upward in response to thepressure gradient between the reservoir and conduit. As the pres-sure wave triggered by the plug removal reaches the conduit orifice,

C© 2003 RAS, GJI, 153, 699–718


A numerical simulation of volcanic eruptions 701

Con

duit

Reservoir

r

S1 S2 S3

l c

rc

lv

r v

Seismic Stations

Observation Points ofFluid Pressure and Velocity

Axial Symmetry

Lid

z

P01

P02

P03

P04

P05

P06

P07

P08

P09

P10

P11

P12

P13

Plug

P14 P16 P17P15

Figure 1. Configuration of the conduit–reservoir system and observationpoints used in the simulation.

a high-pressure pulse is generated at this location. This pressurepulse exceeds the material strength of the lid, causing the instan-taneous rupture of the lid and triggering an eruption. The upwardmigration of magma is then controlled by the pressure gradient es-tablished between the reservoir and atmosphere. Motions of the fluidand surrounding medium are calculated in the same manner as inModel I.

C A L C U L AT I O N O F M A G M A M O T I O N

We apply the equations of momentum and mass conservation, anda constitutive relation between fluid pressure and fluid density, tocalculate the motion of the magmatic fluid in the reservoir–conduitsystem. An axially symmetric flow of inviscid magma is assumed.The equations of momentum conservation in cylindrical coordinatesare expressed as

∂vr∂t + vr

∂vr∂r + vz

∂vr∂z = − 1

ρ f

∂p∂r ,

∂vz∂t + vr

∂vz∂r + vz

∂vz∂z = − 1

ρ f

∂p∂z − g, (1)

and the equation of continuity is

∂ρ f

∂t+ ∂

∂r(ρ f vr ) + ∂

∂z(ρ f vz) + ρ f vr

r+ ρ f

V

∂V

∂t= 0, (2)

where ρ f is the fluid density, vr and vz are the radial and verticalcomponents of flow velocity, respectively, p is the pressure, and gis the gravitational acceleration. The rightmost partial derivative,∂V /∂t , in the lefthand side of eq. (2) expresses the rate of vol-ume change of the conduit and reservoir system associated with thevariation of flow pressure. The term (1/V )∂V /∂t is given by

1

V

∂V

∂t= 2

rc

∂urc∂t + 1

(�c + �v)∂uzb∂t for 0 < r < rc

and 0 < z < �c,

1

V

∂V

∂t= 2

rv

∂urv

∂t + 1(�c + �v)

∂uzb∂t for 0 < r < rc

and �c < z < �c + �v,

1

V

∂V

∂t= 2

rv

∂urv

∂t + 1�v

∂(uzb − uzt )∂t for rc < r < rv

and �c < z < �c + �v,

where uzt and uzb are the displacements of the top and bottom of thereservoir, respectively, and urc and urv are the displacements of thelateral walls of the conduit and reservoir, respectively.

When the magmatic fluid is depressurized due to upward migra-tion, the bulk density of the fluid decreases because of the vesicu-lation process of volatile gases in solution in the liquid magma. Totake this effect into account, a constitutive relation between the fluidpressure and fluid density is necessary. Most previous studies (e.g.Wilson et al. 1980; Ida 1990; Bower & Woods 1997) formulate thisconstitutive relation by applying Henry’s law for volatile gases. Thepresent study uses the constitutive relation proposed by Ida (1990),which is expressed as

ρm

ρ f= 1 − β + β

(p0

p

), (3)

where ρm is the density of the liquid magma, and p0 is the pressure atwhich the volatiles start to vesiculate. We define the depth at whichvesiculation starts as zm = p0/(ρm g). The parameterβ represents themagma property, with β = 0.5 representing a basaltic magma, andβ = 1.0 representing a dacitic magma. This formulation enables usto easily calculate the density and pressure of the fluid reciprocally.

We solve eqs (1)–(3) by the finite difference method of MacCor-mack (e.g. Anderson 1995) to obtain the spatio-temporal variationsof pressure and velocities in the magmatic fluid.

C A L C U L AT I O N O F C RU S TA LD E F O R M AT I O N A N D S E I S M I CWAV E R A D I AT I O N

We assume that the volcanic crust surrounding the conduit–reservoirsystem is perfectly elastic. Accordingly, the displacement field inthe crust is expressed by the equations of motion in cylindricalcoordinates

ρs∂2ur

∂t2 = ∂τrr∂r + ∂τr z

∂z + τrr − τφφ

r ,

ρs∂2uz

∂t2 = ∂τr z∂r + ∂τzz

∂z + τr z

r+ ρs g,

(4)

C© 2003 RAS, GJI, 153, 699–718



and the stress–strain relations for isotropic medium

τr z = µ(

∂ur∂z + ∂uz

∂r

),

τrr = λ(

∂ur∂r + ∂uz

∂z + urr

)+ 2µ

∂ur∂r ,

τzz = λ(

∂ur∂r + ∂uz

∂z + urr

)+ 2µ

∂uz∂z ,

τφφ = λ(

∂ur∂r + ∂uz

∂z + urr

)+ 2µ

urr ,

(5)

where ui is the i-th component of displacement, ρs is the density ofthe solid, τ i j are the components of stress, and λ and µ are the Lamecoefficients.

We evaluate eqs (4) and (5) by extending the second-order stag-gered finite difference scheme of Ohminato & Chouet (1997) tothe axially symmetric configuration of our model. The method ofOhminato & Chouet (1997) has the advantage that the topographicconfiguration and inhomogeneity of the elastic structure can easilybe implemented in the model. For the time being, however, we limitour consideration to a homogeneous half space with flat free surfacein order to examine the basic characteristics of crustal deformationand seismic wave propagation associated with magma motions.

B O U N DA RY A N D I N I T I A LC O N D I T I O N S

The magmatic fluid forces the walls of the conduit and reservoireither inward or outward, and no vacuum is allowed at the boundarybetween the fluid and crust at all times. Hence, the stresses normalto the conduit and reservoir walls are always balanced with the mag-matic pressure, and the radial velocity of the magmatic fluid at thewall is set equal to the radial velocity of the wall. The radial veloc-ity of the elastic wall is obtained by time derivating the calculatedradial displacement of the wall. We assume a free slip condition atthe conduit and reservoir walls in accordance with our assumptionthat the magma is an inviscid fluid.

The initial conditions of the magmatic fluid and elastic solid areobtained by considering the gravitational load due to the weight ofthe fluid in the conduit–reservoir system, and an additional excesspressure, �P. In Model I, �P represents the overburden pressureapplied by the lid obstructing the vent. This pressure is taken asthe orifice pressure. In Model II, �P is applied only to the fluid inthe reservoir, and represents the excess pressure associated with theinflux of new magma originating from deeper regions. The pressurein the fluid at depth z and time t < 0 is given by

p(z; t < 0) =z∫

0

ρ(z′)g dz′ + patm, for 0 < z < zt ,

p(z; t < 0) = p(zt ) + �P +z∫

zt

ρ(z′)g dz′, for z > zt , (6)

in which z′ is the integral parameter representing depth, and patm

represents the atmospheric pressure. The depth zt = 0 in Model I,and zt = �c in Model II, where �c is the depth to the reservoiroutlet. The initial condition of the displacement field in the crustis calculated with eqs (4) to (6), in which we apply the boundaryconditions between the fluid and solid. The explicit forms of thefinite difference equations in the solid and fluid, and grid geometriesused in our calculations are given in Appendix A.

In Model I, an eruption is triggered at t = 0 by setting the magmapressure at the conduit orifice equal to the atmospheric pressure. In

Model II, fluid motion is triggered at t = 0 by removing the plug toallow the magma transfer between the reservoir and conduit. In bothmodels, we calculate the spatio-temporal dependences of pressureand velocities in the fluid, and displacements and stresses in the solidas follows: (1) using eqs (1) to (3), fluid motions in the conduit andreservoir are estimated at time t = δt from the fluid motions at t = 0;(2) elastic motions are calculated at t = δt from the elastic motionsat t = 0 and fluid pressure at the conduit and reservoir walls at t = 0;and (3) magma motions are calculated at t = 2δt using the boundaryconditions for velocities at the walls and magma motions in theconduit and reservoir at t = δt . Steps (2)–(3) are repeated throughsuccessive time steps until the end of the simulation. In Model I,the pressure at the conduit orifice remains fixed at the atmosphericpressure during the eruption. In Model II, an eruption starts whenthe lid cannot sustain the fluid pressure in the conduit. This conditionoccurs when the pressure at the top of the conduit exceeds the lidstrength. In our calculations, spurious pressure ripples leading tounrealistically large estimates of pressure, can occur at the top ofthe conduit when a strong shock with step-like onset impinges thisboundary (see Appendix B for details). The peak amplitude of theripples can exceed the mean amplitude of the reflected shock waveby about 50 per cent at the lid so that the lid strength tends to beoverestimated in our calculations. However, this overestimation doesnot affect the solid motions as discussed in more detail in AppendixB. At the start of the eruption, the magma pressure at the conduitorifice is set equal to the atmospheric pressure as in Model I. In bothmodels, we apply a sine function to gradually decrease the pressureto atmospheric pressure over an interval of 0.12 s in order to preventnumerical instabilities associated with an abrupt change of pressureat the conduit orifice.

The time step δt is fixed by the numerical stability condition. Forthe elastic medium, the numerical stability condition is given by(e.g. Virieux 1986)

δt = csδds/(√

2α) (cs < 1), (7)

where α is the P −wave velocity in the solid, δds is the grid sizefor the elastic medium, and cs is a constant used to guarantee robustnumerical stability. We use cs = 0.2 to avoid numerical instabilitiesthat may originate at the corners of the conduit and reservoir. Thestability condition in the fluid is given by the expression

δt = c f δd f /(√

2a) (c f < 1), (8)

where a is the acoustic velocity in the fluid, δdf is the grid size forthe fluid, and cf is a constant used to guarantee robust stability ofthe numerical solution in the fluid. The acoustic velocity of magmaranges over 162–165 m s−1 in the present study. Due to the presenceof gas bubbles in the liquid magma, the acoustic velocity of thefluid is much lower than the P −wave velocity assumed for the crust(3460 m s−1).

We assume a constant δt in both the solid and fluid and selectδdf = δds/nf so that the constant cf = nf csa/α (nf ≥ 1), where nf

represents the ratio of grid sizes. This procedure is similar to thatused by Chouet (1986) in calculations of the response of a fluid-filledcrack to a step in pressure.

The computational domain has dimensions of 5000 m in thez-direction and 2500 m in the r-direction. We set nf = 4 in our cal-culations and use a grid of 12.5 × 12.5 m in the solid, and 3.125 ×3.125 m in the fluid. The grid interval of 12.5 m in the solid issufficiently small compared to the wavelengths of seismic wavestargeted by our model (e.g. the wavelength is 400 m for a shearwave at the frequency of 5 Hz). While a much finer grid may be re-quired in the fluid to accurately portray the sharp pressure transient

C© 2003 RAS, GJI, 153, 699–718



expected to occur at the eruption onset, the 3.125 m grid is smallenough to adequately interpret the fluid motions in our simulations.Our numerical tests indicate that nf = 4 is an optimum value forsimultaneous calculations of the fluid and solid motions, and thatusing nf ≥ 8 induces high-frequency noise in the elastic responseof the solid as a result of abrupt stress variations occurring withinthe space of a few grids in the solid. Using smaller grids in both thefluid and solid would allow the resolution of sharp onsets in pressureand high-frequency elastic motions, albeit at the expense of moreextensive computations.

As the spatial grid interval in the fluid is smaller than that usedin the solid, radial velocities and normal stresses in the solid at thefluid–solid boundary are interpolated with cubic spline functions toobtain the radial velocities and pressure in the fluid at this boundary.

As defined, our problem has axial symmetry (see Fig. 1), andradial velocities in the fluid and radial displacements in the solidalways vanish on the vertical axis of symmetry r = 0. Accordingly,the solution reduces to that of calculating the wavefields in the quar-ter space r ≥ 0 and z ≥ 0. The absorbing boundary conditions ofClayton & Engquist (1977) are used to set up the radiation condi-tions at the edge of the domain in the r −direction, and stress-freeconditions are applied at the free surface z = 0.

The vertical displacement field in the solid (eq. 4) is representedby the sum of two fields

ρs∂2(uz0 + uz1)

∂t2 = ∂(τr z0 + τr z1)∂r + ∂(τzz0 + τzz1)

∂z

+ (τr z0 + τr z1)r + ρs g,

which can be split into

ρs∂2uz0

∂t2= ∂τr z0

∂r+ ∂τzz0

∂z+ τr z0

r+ ρs g

and

ρs∂2uz1

∂t2= ∂τr z1

∂r+ ∂τzz1

∂z+ τr z1

r,

where uz0 represents the static deformation due to the gravitationalforce applied to the elastic body, and uz1 represents the field asso-ciated with the pressurization of the reservoir–conduit system andsubsequent magma motion and eruption.

In our approach, the gravitational body force in eq. (4) is only usedto calculate the static deformation field due to this force, and thisterm is removed from the equation when calculating the dynamicfield associated with the pressurization and eruption phases so as toavoid spurious high-frequency numerical noise associated with thisterm. Thus, we only calculate the displacements uz1 relative to thestatic solution obtained for uz0, and apply radiation conditions at thebottom boundary of the computational domain. Unlike our treatmentof the elastodynamics equations, the gravitational acceleration termin eq. (1) describing the fluid dynamics remains present throughoutthe various phases of calculations. As a result, this term pulls downthe fluid–solid system, thereby causing a mismatch in the total netforce balance. To verify that this mismatch does not cause unac-ceptable errors, we computed solutions in which the gravitationalforce remains present in both the solid and fluid throughout the cal-culations. To prevent the gravitational force from pulling down allof the solid in these calculations, the vertical displacement uz0 wasset to zero at the bottom boundary of the computational domain.The base of the computational domain was set sufficiently far fromthe base of the reservoir to prevent any bias in the calculated defor-mation field associated with this bottom boundary condition. Oncethe crustal deformation due to the gravitational force had converged

to an acceptable level of accuracy, we pressurized the reservoir–conduit system and started the magma motion by removing the lidor plug. Except for the presence of significant components of high-frequency numerical noise, the simulated waveforms were foundto be quite similar to those obtained when the gravitational bodyforce is removed from eq. (4) during the pressurization and erup-tion phases. This consistency between the two types of solutions isexpected because the mass of the fluid is negligibly small comparedto the mass of the solid body. Therefore, the gravitational term ineq. (4) can safely be neglected when computing the dynamic elasticmotions resulting from the pressurization and eruption phases.

S I M U L AT I O N R E S U LT S

We assume the density of the liquid magma (2500 kg m−3), den-sity of the solid (2700 kg m−3), depth at which vesiculation starts(zm = 5000 m), and P- and S-wave velocities in the solid (3460 and2000 m s−1, respectively). We examine eruptions from the sameshallow magma conduit–reservoir system (rc = 50 m, r v = 200 m,�c = 400 m, �v = 200 m) for both Models. We further simulateeruptions for Model II with an extended pipe (rc = 50 m, r v =50 m, �c = 400 m, �v = 200 m) to examine the importance of thelid strength at the conduit orifice on the fluid and elastic motions.

The reservoir assumed in our simulation is quite shallow com-pared to reservoir depths of a few kilometers typically assumed forvolcanoes. The simulation of a deeper system, however, would re-quire extensive computations because of the larger spatial extent ofthe computational domain required for these calculations. Further-more, processes associated with the opening and collapse phasesof the volcanic conduit would need to be accounted for in a deepconduit system as the conduit is subjected to a high overburdenpressure in such a system. To avoid lengthy calculations associatedwith deeper systems, the present study focuses on shallow conduit–reservoir systems.

R E S U LT S F O R M O D E L I

We first investigate the general characteristics of magma motion,crustal deformation, and seismic radiation based on simulation re-sults obtained for Model I. Fig. 2 shows snapshots of the pressurefield in the magma. Prior to the eruption, the magma in the con-duit and reservoir is overburdened by its own weight and by the lidweight, so that the magma pressure increases with depth. When theeruption starts at t = 0, the magma in the reservoir starts to moveupward in response to the pressure gradient established between thereservoir and atmosphere. As shown in the snapshots from t = 0to t = 9 s, a rarefaction wave propagates downward in the con-duit. The propagation speed of the rarefaction wave is about 180 ms−1, which is slightly higher than the fastest acoustic velocity in themagma (165 m s−1).

The pressure in the reservoir gradually decreases with time, asindicated by the change in colour in the reservoir from red, to yellow,to light green. For a more detailed examination of small fluctuationsin magma pressure, we show in Fig. 3 the spatial variations of thetime derivative of pressure, ∂ p/∂t , in the reservoir. As seen in thesnapshots from t =3.7 to t =7 s, a region with high ∂ p/∂t repeatedlyappears at the bottom circular edge and near the center of the magmareservoir, with a dominant period of about 2.5 s. As the reservoirheight and radius are both 200 m, and the acoustic wave velocity ofthe magma at the reservoir depth is about 165 m s−1, this long-periodoscillation is interpreted as an acoustic oscillation of the magmaticfluid in the reservoir.

C© 2003 RAS, GJI, 153, 699–718



0.00 s

1.02 s

2.04 s

3.06 s

0.10 0.74 1.38

Pressure, MPa

4.08 s

5.10 s

6.12 s

7.14 s

8.16 s

9.18 s

400 m

600

m

Figure 2. Snapshots of the pressure field in the magma conduit and reser-voir in Model I.

Fig. 4 shows the temporal variations of pressure, p(t), and timederivative of pressure, ∂ p/∂t , at several locations in the conduit andreservoir. Fig. 4(a) clearly shows that the magma conduit and reser-voir deflate almost exponentially for up to 30 s or longer. The timeconstant of deflation is fixed by the ratio of the cross-sectional areaof the vent to reservoir volume. A large vent area and small reser-voir volume result in a rapid deflation, while a small vent area andlarge reservoir volume result in a slow deflation. Gradually decay-ing oscillations with dominant periods near 2.5 s and shorter-periodovertones are observed in the plots of ∂ p/∂t (Fig. 4b) as the magmapressure in the reservoir decreases over the 40 s interval. Althoughthe dominant oscillation period is close to 2.5 s, slight variations inperiodicities are noticeable at different locations. These variationsreflect the complex configuration of the conduit–reservoir system,as well as pressure dependence of the acoustic speed in the magmaticfluid. The fluid velocities (not shown here) display similar decay-ing LP oscillations. These results indicate that the magma graduallymigrates upward in response to the pressure difference between themagma reservoir and atmosphere, and that LP oscillations caused byacoustic resonance of the reservoir are superimposed on the pressuredecay associated with mass transport.

Fig. 5 shows snapshots of the radial displacement field radiatedin the solid during the eruption. Deflation of the conduit propagatesdownward from the top of the conduit to the reservoir. Measurementsof the group velocity of the waves indicate that the group velocitydecreases from the P-wave velocity (3460 m s−1) to about 100 m s−1

as the frequency decreases. These dispersion features are the resultof the dynamic coupling between the fluid and solid as theoreticallypredicted by Biot (1952) for a fluid-filled pipe. Similar dispersioncharacteristics were obtained by Chouet (1986) for a fluid-filledcrack.

Deflation motions are identified in the vertical and radial dis-placements of the ground surface recorded in the near field of theconduit–reservoir system (Fig. 6). These slow displacements (VLPcomponents) result from the gradual deflation of the magma reser-voir. Oscillations with a period of about 2.5 s are observed super-imposed on the VLP components of displacements due to deflation.As distances from the vent increase, LP oscillations become moreevident (for example, compare the synthetics obtained at distancesof 100 and 1000 m). The period of 2.5 s is nearly equal to the reso-nance period of the reservoir and the main axes of the particle orbitsat the free surface point toward the reservoir, indicating that theseLP signals result from acoustic resonance in the magma reservoir.

Fig. 7 compares the amplitude spectra of the vertical componentof velocity obtained at an epicentral distance r = 500 m for β = 0.5and β = 1.0. The dominant frequency obtained for a magma char-acterized by β = 0.5 (basalt) is slightly lower than that obtained fora magma characterized by β = 1.0 (dacite). Note also the frequencyshifts and differences in excitations in the subdominant spectralpeaks associated with the two types of magma. Similar changesin the spectral peaks are observed when the depth at which vesic-ulation starts is varied. Since the acoustic velocity of the magmaa = √

∂p/∂ρ f varies with β and zm (see eq. 3), the resonant pe-riod of the reservoir is strongly affected by the properties of themagma. Simulations in which only the initial excess pressure, �P ,is changed, show that the displacements and velocities are linearlyproportional to the initial excess pressure in the reservoir.

Results for Model II

We simulate the fluid and elastic motions for �P = 1 MPa andσ lid = 2 MPa. Fig. 8 shows snapshots of the pressure field in the

C© 2003 RAS, GJI, 153, 699–718



3.67 s

3.98 s

4.29 s

4.59 s

4.90 s

-0.31 -0.14 0.04 MPa s−1

5.21 s

5.51 s

5.82 s

6.12 s

6.43 s

6.74 s

7.04 s

Conduit

200m

Figure 3. Snapshots of the spatial variations of the time derivative of pressure, ∂ p/∂t , in the magma reservoir in Model I.

C© 2003 RAS, GJI, 153, 699–718



0 10 4020 30Time (s)

MPa

0 5 10Time(s)

MPa s−1

(a)

(b)

15

1.00

1.04

1.08

1.13

1.16

1.20

1.24

1.20

1.20

1.20

1.35

1.18

0.71

0.49

0.41

0.23

0.33

0.20

0.18

0.17

P01

P03

P05

P07

P09

P11

P13

P14

P15

P16

P01

P03

P05

P07

P09

P11

P13

P14

P15

P16

Figure 4. Temporal variations of (a) p(t) and (b) ∂ p(t)/∂t at several ob-servation points in the conduit and reservoir system in Model I. See Fig. 1for locations of observation points. Small-amplitude high-frequency oscil-lations observed near the ends of the signals at P03, P05, P07, and P09are numerical artefacts caused by the sharp corner between the conduit andreservoir. The starting times in (a) and (b) are arbitrary.

0.00 s

1.02 s

2.04 s

3.06 s

4.08 s

-1.4e+06 0.0e+00 1.4e+06

Radial Displacement,nm

5.11 s

6.13 s

7.15 s

8.17 s

9.18 s

Magma Reservoir

600m

600m

Figure 5. Snapshots of the radial displacement field (in units of 10−9 m)in the solid in Model I.

C© 2003 RAS, GJI, 153, 699–718



x10 nm6(a) Vertical DisplacementsUp

(b) Radial Displacements Outward x10 nm6

3.76

3.12

2.25

1.52

1.01

0.75

0.61

0.51

0.45

0.40

100

200

300

400

500

600

700

800

900

1000

0 10 20 30 40Time(s)

2.59

2.03

1.86

1.62

1.35

1.11

0.91

0.76

0.63

0.53

100

200

300

400

500

600

700

800

900

1000

0 10 20 30 40Time(s)

Outward

Up(c) Particle Orbits ( 0.3-0.6 Hz)

500mUp

1000m

Figure 6. Vertical (a) and radial (b) displacements at the free surface (inunits of 10−9 m) at various distances from the conduit axis in Model I.Numbers at the left of each trace represent epicentral distances from theconduit axis (in units of m). (c) Particle orbits obtained from bandpass filtered(0.3–0.6 Hz) seismograms at epicentral distances of 500 and 1000 m. Thetime windows within which the particle orbits are obtained have a durationof 4 s and start about 3 s after the removal of the lid.

10-1 10 0 10 110 -2

10 -1

10 0

FREQUENCY (Hz)N

OR

MA

LIZ

ED

AM

PLI

TU

DE

β=0.5

β=1.0

Figure 7. Comparison between amplitude spectra obtained for differentmagma properties expressed by different factors β. The spectra are obtainedat an epicentral distance of 500 m from the conduit axis.

conduit–reservoir system, and Fig. 9 illustrates the spatio-temporalevolution of fluid pressure in the conduit and reservoir during theinitial pressurization and after the reservoir plug is removed. Noticethe compression shock propagating upward and rarefaction shockpropagating downward in the snapshots from t = 0.26 to t = 1.28 sin Fig. 8. The propagation speed of the rarefaction shock (170 ms−1) is slower than that of the upgoing compression shock (280 ms−1). The wave front of the compression shock is a step-like signalwith amplitude of about 0.3 MPa. These characteristics of upgo-ing and downgoing shocks are fully consistent with observations ofunsteady wave motions in shock tube experiments, and with numer-ical simulations of shock waves in a conduit (Morrissey & Chouet1997). Oblique shock waves, generated at the orifice of the reservoir,are reflected several times in the conduit (see snapshots from 1.28to 2.30 s in Fig. 8). Small fluctuations, visible for a few secondsbehind the step-like signals in Fig. 9(b), are caused by the superpo-sition of oblique shock reflections and pressure oscillations in theconduit–reservoir system. When the shock wave reaches the top ofthe conduit, it causes a sudden build-up of pressure at this location.This surge of pressure is the result of the compressibility of the fluidand is discussed in detail in the following section. When the pressureat the bottom surface of the lid exceeds the lid strength, the lid isremoved and the eruption starts, at which point the pressure at theorifice is set equal to the atmospheric pressure (see pressure traceat P01 in Fig. 9(b); see also Appendix B for details concerning theimplementation of this condition). The pressure wave reflected fromthe conduit orifice then propagates downward into the reservoir (seet > 1.8 s in Figs 8 and 9b). While the high-pressure waves travel-ling in the conduit are dominant in this model, small-scale pressurefluctuations associated with the resonance of the reservoir are alsoobserved as in Model I.

C© 2003 RAS, GJI, 153, 699–718



0.00 s

1.20MPa

0.26 s

1.20MPa

0.51 s

1.20MPa

0.77 s

1.20MPa

0.10MPa Pmax

Pressure

1.02 s

1.20MPa

1.28 s

1.20MPa

1.53 s

1.23MPa

1.79 s

1.03MPa

2.04 s

1.03MPa

2.30 s

1.05MPa

400m

600m

ReflectedWave

ObliqueShock

Figure 8. Snapshots of the pressure field in the conduit–reservoir systemin Model II. Notice the oblique shock waves visible in the conduit above thereservoir for t > 1.4 s. The pressure in each snapshot is normalized by themaximum pressure, Pmax, (red colour) indicated at the right of the reservoirplug.

0 2 3 4 51Time(s)

(b)

P01

P02

P03

P04

P05

P06

P07

P08

P09

P10

P11

P12

P13

MPa

2.03

0.40

0.37

0.38

0.38

0.41

0.40

0.39

0.96

1.01

1.03

1.04

1.07

0 10 20 30 40Time(s)

(a)

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

1.0

170m s−1

P01

P02

P03

P04

P05

P06

P07

P08

P09

P10

P11

P12

P13

0.5

280m s−1

Build-Up of Pressure MPa

Figure 9. Temporal variations of p(t) at several observation points (P01-P13) in the conduit and reservoir system in Model II during the first 40 s ofthe pressurization and eruption process (a), and during the initial 5 s of thisprocess (b). The black strip at the top of (a) identifies the enlarged sectionof record shown in (b). The dashed lines in (b) mark the propagations of thecompression and rarefaction shocks in the conduit and reservoir.

C© 2003 RAS, GJI, 153, 699–718



Fig. 10 shows snapshots of the vertical and radial displacementfields in the solid. A large VLP deflation motion of the groundsurface starts with the initiation of magma motion and resultingcontraction of the reservoir. Superimposed on this VLP motion areoscillatory motions with periods near 2.5 s due to the acoustic res-onance of the reservoir. A pulse-like signal is also observed in theground response to the eruption in this model. This signal is radi-ated by the sudden build-up of pressure at the vent resulting from theimpact of the high-pressure fluid upon the lid capping the conduitorifice. This pulse is not observed in the seismograms obtained inModel I.

The pulse-like signal excited by the pressure build-up at the ventis conspicuous in eruptions from an extended vertical pipe (rc = 50m, r v = 50 m, �c = 400 m, �v = 200 m, �P = 1 MPa). Fig. 11shows snapshots of the fluid pressure in the pipe and radial velocitiesin the solid during the interval t = 1.07 − 2.09 s. As illustrated inthe snapshots at t = 1.58 s, a pulse-like signal is excited at theconduit orifice by the build-up of pressure in the volcanic pipe.This signal is radiated in the solid. Fig. 12(a) compares verticaland radial displacements at an epicentral distance r = 500 m fordifferent material strengths of the lid, σ lid, ranging between 1 and0.2 MPa. The pulse-like signals are clearly observed for σ lid =1 MPa, however the amplitude of this signal becomes smaller asthe strength of the lid decreases. This means that the seismogramscontain information about the boundary condition at the conduitorifice, and that the material strength of the lid can be estimatedfrom the amplitude of the pulse-like signal. The particle orbits forthe pulse display retrograde elliptical motions characteristic of aRayleigh wave (Fig. 13). Note that the overall characteristics of thewaveforms in Fig. 12 are distinct from those seen in Fig. 10. Theseresults indicate that the seismic waves are markedly affected by thesystem of forces resulting from the conduit–reservoir configurationconsidered.

Fig. 12(b) shows the vertical velocity of the fluid at P01 (seeFig. 1). The initial velocity varies from 270 to 340 m s−1 for differ-ent lid strengths. Note, however, that these velocities are stronglydependent on the boundary condition applied at the vent, which ar-bitrarily sets the fluid pressure equal to the atmospheric pressure atthe start of the eruption in this model. Following this initial phase,the vertical velocities of the fluid become essentially independent ofthe applied boundary condition, indicating that the subsequent fluxof magmatic fluid from the vent is independent of the lid strength.

D I S C U S S I O N

Mechanism of high-pressure build-up at the conduit orifice

Our simulation results for Model II display an abrupt increase ofpressure at the vent, where the pressurized high-velocity fluid mi-grating upward from the reservoir is forced to a sudden stop. Theseresults indicate that a pressure build-up exceeding the incident wavepressure by more than twofold can be achieved at the top of the con-duit. This large pressure amplification is the result of the nonlinear-ity of the compressible fluid dynamics. Fig. 14 provides a simplifiedrepresentation of the fluid motion in the top region of the conduit. Afluid with Mach number Mi and pressure pi is propagating upward,forming a shock front with Mach number Ms propagating ahead ofthe fluid (Fig. 14a). The fluid reaches the lid capping the conduitwhere it is reflected, forming a shock wave propagating downwardwith a Mach number Mr (Fig. 14b). We denote as pr the pressurein the fluid behind the downgoing shock wave.

0 10 20 30 40Time(s)

100

200

300

400

500

600

700

800

900

1000

0 10 20 30 40Time(s)

x10 nm6(a) Vertical Displacements Up

100

200

300

400

500

600

700

800

900

1000

(b) Radial Displacements Outwardx10 nm6

Pulse-like Signal

Pulse-like Signal

4.56

3.79

2.97

2.28

1.81

1.50

1.31

1.22

1.13

1.04

1.75

1.45

1.50

1.35

1.11

9.52

8.53

7.69

7.03

6.46

Figure 10. (a) Vertical and (b) radial displacements (in units of 10−9 m)at the free surface in Model II. Numbers at the left of each trace representepicentral distances from the conduit axis (in units of m). The initial phase isexcited by the removal of the plug at the reservoir outlet, and the followingLP signals result from the acoustic resonance of the reservoir. The pulse-likesignals are Rayleigh waves excited by the high-pressure build-up at the vent.

C© 2003 RAS, GJI, 153, 699–718



-1.8e+6 1.8e+6 nm s−1

Radial Velocity

1.07 s

1.20

1.33 s

1.01

1.58 s

0.86

1.84 s

0.59

0.10

2.09 s

0.48

PmaxMPa

Fluid Pressure

600m

Figure 11. Snapshots of fluid pressure in the volcanic pipe (left) and radialvelocity field (in units of 10−9 m s−1) in the solid (right) during the intervalt = 1.07 − 2.09 s in Model II. The maximum fluid pressure in each snapshotis indicated at the bottom right of the volcanic pipe.

1.25

e+05

nm

2.

11e+

05 n

m

1.09

e+05

nm

1.

57e+

05 n

m

1.03

e+05

nm

1.

56e+

05 n

m

Inflation of Volcanic Pipe Rupture of Plug andStart of Magma Motion

Radial

Vertical

Radial

Vertical

Radial

Vertical

0 5 10s

σlim= 1.0 MPa

σ lim= 0.5 MPa

σlim= 0.2 MPa

Outward

Up

Pulse-Like Signal

σlim= 1.0 MPa

σlim= 0.5 MPa

σlim= 0.2 MPa

(a) Displacements at an epicentral distance of 500m from the conduit axis

(b) Vertical Fluid Velocity at P01

340m

s−1

34

0m s

−1

340m

s−1

0 5 10s

End of Inflation Phase

Figure 12. Effect of lid strength σ lid. (a) Vertical and radial displacements(in units of 10−9 m) at an epicentral distance of 500 m. (b) Vertical fluidvelocity at the vent. An extended pipe with radius of 50 m and length of600 m is assumed in this calculation (rc = 50 m, rv = 50 m, �c = 400 m, �v

= 200 m). The leftmost dashed line in (a) marks the onset of inflation of themagma reservoir (the reservoir is the segment of pipe �c < z < �c + �v), thedashed line immediately to the right marks the end of the inflation phase,and the rightmost dashed line marks the time of plug removal and start ofmagma motion in the pipe. Note that the inflation of the reservoir at the baseof the pipe causes a deflation of the ground surface at an epicentral distanceof 500 m (e.g. Bonaccorso & Davis 1999).

C© 2003 RAS, GJI, 153, 699–718



Outward

Up

0

Displacement seismograms at an epicentral distance of 500 m from the conduit axis

Particle Orbit

1 2 3 4 s

Radial

Vertical

Up

Outward

Down

Inward

1.52

e+05

nm

9.

96e+

04 n

m

Figure 13. Vertical and radial displacements (in units of 10−9 m) at an epicentral distance of 500 m from the volcanic pipe in Model II. The particle orbitshows a retrograde motion typical of a Rayleigh wave. The seismograms and particle orbit are bandpass filtered from 1 to 8 Hz.

The fluid behaviour in this simplified model is expressed by theRankine-Hugoniot equation. The Rankine-Hugoniot equation givesthe pressure ratio pr/pi in the form (e.g. Anderson 1982; Matsuo1994)

pr

pi= 2γ Mr

2 − (γ − 1)

γ + 1, (9)

where γ is the ratio of the specific heat of the fluid at constantpressure versus specific heat of the fluid at constant volume. TheMach numbers Mr, Ms, and Mi are related to each other by

Mr =√

2γ Ms2 − (γ − 1)

(γ − 1)Ms2 + 2

, (10)

and

Mi = (Ms

2 − 1) [(

γ Ms2 − γ − 1

2

) (γ − 1

2Ms

2 + 1

)]− 12

.

(11)

Fig. 15 illustrates the relation between Mi and the pressure ratio,pr/pi, for γ = 1.01 − 1.4. These results show that the pressureincreases by about 70 per cent when a fluid with Mi = 0.5 impactsthe lid. For supersonic waves with Mi = 2 impacting the lid, thepressure increases by more than 600 per cent.

The basic characteristics of the simulation results obtained withModel II are in agreement with the theoretical prediction, althoughsignificant differences are noted between model and theory. For ex-ample, the downgoing high-pressure wave in our model (see Figs 9and 11) is small and does not display a step-like onset as in theschematic illustration in Fig. 14(b). Rather, it shows marked fluc-tuations of amplitude at P02 to P04. These differences originate inthe triggering condition used for the eruption, and geometry of thereservoir. In our simulation the lid is removed and the pressure at theorifice of the conduit is set to gradually decrease to the atmospheric

pressure over an interval of 0.12 s when the pressure exceeds thelid strength. Pressure fluctuations induced by acoustic resonance inthe reservoir then affect the fluid motions in the conduit. To obtain areflected shock wave similar to the theoretical prediction and bettercompare our numerical results with the theory, we simulate the fluidmotion in an extended pipe (rc = 50 m, r v = 50 m, �c = 400 m,�v = 400 m) with infinite lid strength (σ lid = ∞). Fig. 16 showsthe temporal variation of pressure beneath the conduit orifice. Theinitial excess pressure �P is 1.0 MPa. The downward propagatingshock wave reflected at the lid displays a clear step-like front inagreement with the schematic illustration in Fig. 14. The large over-shoot of pressure at P01 does not fit the theoretical prediction butis the result of numerical artefacts in our finite-difference scheme(see Appendix B). We further simulate the fluid motions for �P= 0.1, 0.2, 0.5, 1.5 and 2.0 MPa, and estimate the ratios of pr/pi

and incident Mach number Mi at P02 (50 m below the lid) for each�P. The results are shown by solid circles in Fig. 15. The numericalresults coincide with the theoretical values for γ = 1.01. Therefore,we conclude that a large build-up of pressure can occur just beneaththe vent in a volcanic eruption. It is worth noting here that this pres-sure pulse is not detectable by geodetic measurements because ofits short duration.

Comparison between simulations and field observations

We have examined the magma motions, crustal deformation, andseismic wave radiation associated with two models of a volcaniceruption. Many seismograms associated with various types of erup-tions have been observed at different volcanoes, which allow a com-parison between our simulation results and field observations.

Both Models I and II predict the LP seismic waves that are of-ten detected in seismic measurements performed in the vicinity oferupting volcanoes. These waves become more evident in the farfield. Changes in the dominant period of eruption tremor have been

C© 2003 RAS, GJI, 153, 699–718



shock front

Ms

Mi

shock front

Mr

Mi

lid

lid

p

p i

p

p i

pr

(a)

(b)

Figure 14. Schematic illustration of the fluid motion at the conduit orifice.The conduit orifice is sealed by a lid.

observed at several volcanoes (e.g. Chouet et al. 1997; Hagerty et al.1997). Slight changes in the dominant period of the signal are pre-dicted by our model, due to a slight change in the acoustic velocityof magma with depth. However, large changes in the dominant pe-riod can not be accounted for by our model. Using a more realisticconstitutive relation between fluid pressure and density, in whichthe dynamics of the vesiculation process is fully accounted for, mayimprove our model.

In both Models I and II, VLP motions originate in the deflationof the reservoir–conduit system. VLP signals have been observed atMount St Helens (Kanamori et al. 1984), and Mount Tokachi andMount Asama (Nishimura & Hamaguchi 1993), and have been inter-preted as the result of a reaction force associated with an eruption(Kanamori et al. 1984). Recent broad-band seismic observationshave shed additional light on the spatio-temporal properties of thesources of seismic waves associated with explosions. For example,Tameguri et al. (2002) demonstrated that the dominant signals of

Mach Number of Incident Fluid (M )

Pre

ssur

e R

atio

(p

/p )

r i

γ=1.01

γ=1.2

γ=1.4

1 2 3 0

1

2

3

4

5

6

7

8

9

10

0

i

Figure 15. Relation between the Mach number of the incident fluid andpressure in the shock generated by reflection at the conduit orifice. The ver-tical scale is normalized by the pressure of the incident wave. The simulatedresults are shown by solid circles (see text for explanations).

explosion earthquakes at Sakurajima Volcano are Rayleigh wavesoriginating from a source located immediately beneath the vent.This Rayleigh wave source is activated about 1 s after an initial ex-plosion occurs at a depth of a few kilometers beneath the vent. AVLP inflation preceding an eruption by a few tens of second has alsobeen observed to accompany vulcanian explosions at PopocatepetlVolcano, Mexico (Arciniega-Ceballos et al. 1999). The basic char-acteristics of Model II, which show an initial motion of the fluid andseismic wave radiation at depth prior to an eruption, and subsequentexcitation of a Rayleigh wave at the vent at the eruption onset, areconsistent with these observations.

Although Vulcanian-type eruptions often accompany explosionearthquakes, many eruptions are dominated by eruption tremor. Forexample, at Mount St Helens the first catastrophic blast at 0830on 1980 May 18 was preceded by strong explosion earthquakes(e.g. Kanamori & Given 1983), while the eruption starting threehours later was not explosive but characterized by a sustained fluxof fragmented materials causing eruption tremor. Sakurajima Vol-cano is famous for its Vulcanian-type eruptions, but it sometimesshows Strombolian-type eruptions accompanying BL-type earth-quakes. Based on analyses of explosion and BL-type earthquakes,Iguchi (1995) suggested that the boundary conditions at the ventmay control the type of eruption at Sakurajima. Explosion earth-quakes (Vulcanian-type eruptions) mark the destruction of lavadomes obstructing the vent, and BL-type earthquakes (Strombolian-type eruptions) occur when the conduit is filled with fluid magmaand the vent is unobstructed. These observations suggest that eitherModel I, or Model II with a small σ lid, may both represent ‘weak’eruptions.

Other considerations

Our models should only be viewed as seminal models because ofthe simplifying assumptions we have used to describe a volcanic

C© 2003 RAS, GJI, 153, 699–718



P01

P02

P03

P04

P05

0 1 2 3 4 5

P06

Inci

dent

Sho

ck W

ave

pi

pr

Reflected S

hock Wave

Time(s)

2.0M

Pa

Figure 16. Temporal variations of p(t) at locations P01 through P06 alongthe conduit axis. The large overshoot of pressure at P01 is the result ofnumerical artefacts.

eruption. In these models, the pressure at the vent is arbitrarily setequal to the atmospheric pressure during an eruption. This assump-tion does not take into account the overburden pressure due to thevolcanic plume, which is expected to fluctuate as the eruption pro-ceeds. The accompanying fall-back of volcanic materials from theplume into the conduit provides an additional feedback mechanismthat will induce pressure fluctuations at the vent. The flow behaviourin the atmosphere will affect the underground flow of magma andcontrol the long-period characteristics of the observed eruption sig-nals. To incorporate such effects in future eruption models, the flowin the atmosphere will need to be calculated simultaneously with thefluid dynamics in the conduit and reservoir, and elastodynamics inthe solid. Furthermore, the configuration of the magma conduit andreservoir we have used may be too simplistic. An irregular shapefor the reservoir system, in which the cross-sectional area of theconduit and reservoir are varied, may be more realistic as suggestedby the complex distributions of dyke widths observed in volcanicfields. As pointed out by Wilson et al. (1980), the conduit may beeroded by the flow of magma. Erosion may be especially importantin the shallowest part of the conduit. Conduit and reservoir collapsesmay also occur in response to mass withdrawal from the system. Assuch collapses occur mostly during the late stages of eruptions, oursimulation results are still useful for an understanding of the overall

characteristics of an eruption, and are especially adequate to de-scribe the process leading up to and acting during the initial phaseof an eruption.

We considered an inviscid fluid under the assumption that strongvesiculation of volatile gases will significantly reduce the viscosityof the magma (e.g. Wilson et al. 1980). Other types of magma flow,such as the slug flow discussed by Chouet et al. (1999); Chouet etal. (2003), and fragmentation process of magma in the conduit, arenot described by eq. (3) and are not a target of our simulations. Al-though our calculations are strictly limited to the simplified aspectsof an eruption, we believe that the overall characteristics of magmamotions in the shallow reservoir, and seismic wave radiation excitedby the eruption are well reproduced.

C O N C L U S I O N S

We have used the finite difference method to calculate the magmamotions, seismic radiation, and crustal deformation associated witha volcanic eruption. Our main results are summarized as follows.

(1) Magma pressure gradually decreases with time as the magmamigrates upward in response to the pressure difference between themagma reservoir and atmosphere. LP oscillations of the magmareservoir and conduit are excited by acoustic resonance of theconduit–reservoir system during the eruption. The dominant pe-riod of oscillation depends both on the size of the conduit–reservoirsystem and magma properties. These VLP and LP oscillations aredetected in the displacement field measured at the free surface.

(2) The magmatic fluid is highly pressurized just beneath thevent when the fluid abruptly migrates upward and is instantaneouslystopped by a lid blocking the conduit orifice. This build-up of pres-sure is detected as a single pulse-like Rayleigh wave in the seismo-grams, a feature similar to those observed in actual seismograms ofexplosion earthquakes.

(3) The material strength of the lid at the conduit orifice plays animportant role in the generation of seismic waves. A strong lid leadsto a large-amplitude single-pulse explosion earthquake. A weak lidleads to seismograms dominated by LP oscillations and graduallychanging crustal deformations.

A C K N O W L E D G M E N T S

We are indebted to Phil Dawson and Javier Almendros for criticalreviews. Careful and thoughtful reviews by Takao Ohminato andMasato Iguchi were very helpful for improving the manuscript. Thisstudy was motivated when T.N. visited the USGS with a support ofa Grant for Scientific Research and the Programme for the OverseasResearcher of MEXT, Japan, and is partially supported by ACT-JSTand a Grant for Scientific Research from MEXT (No.14080202).

R E F E R E N C E S

Aki, K. & Koyanagi, R., 1981. Deep volcanic tremor and magma ascentmechanism under Kilauea, Hawaii, J. geophys. Res., 86, 7095–7109.

Anderson, J.D., 1982. Modern Compressible Flow With Historical Perspec-tive, 1st edn, McGraw-Hill, New York.

Anderson, J.D., 1995. Computational Fluid Dynamics, The Basics with Ap-plications, 1st edn, MacGraw-Hill, New York.

Arciniega-Ceballos, A., Chouet, B.A. & Dawson, P., 1999. Very long-periodsignals associated with vulcanian explosions at Popocatepetl Volcano,Mexico, Geophys. Res. Lett., 26, 3013–3016.

Biot, M.A., 1952. Propagation of elastic waves in a cylindrical bore contain-ing a fluid, J. Applied Phys., 23, 997–1005.

C© 2003 RAS, GJI, 153, 699–718



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A P P E N D I X A : E X P L I C I T F O R M S O FT H E F I N I T E D I F F E R E N C E E Q UAT I O N SI N T H E F L U I D A N D S O L I D

Fig. A1 illustrates the grid architectures used in the fluid and solid.The intersections of the thin lines represent the grid nodes at whichthe fluid densities, pressures, and velocities are calculated. Plainsymbols represent grid nodes where elastic stresses and the twoelastic moduli are defined, and open symbols represent grid nodeswhere elastic displacements, body forces and density are calculated.The grid nodes located at the boundary between the fluid and thesolid are

i = ic and j 0 ≤ j ≤ j t, l = lc and m0 ≤ m ≤ mt − 1

for the conduit wall,

i = iv and j t ≤ j ≤ jb, l = lv and mt ≤ m ≤ mb − 1

for the reservoir side wall,

ic ≤ i ≤ iv and j = j t, lc ≤ l ≤ lv − 1 and m = mt

for the reservoir roof,

i 0 ≤ i ≤ iv and j = jb, l0 ≤ l ≤ lv − 1 and m = mb

for the reservoir bottom.

C© 2003 RAS, GJI, 153, 699–718



τrz τiiu , ρs, f ; z u , ρs , f ;r , λ, µ ; , µGrid for Fluid (i,j)

icj0

jt

Cell for Solid (l,m)

i0

jb

iv

lc

m0

lv

mt

r

z

mb

v , v , ρr z f

Cell for Solid (l,m)

12.5m

Grid for Fluid (i,j)

z r

Figure A1. Grid architectures of the fluid and solid used in our finite-difference simulations.

C© 2003 RAS, GJI, 153, 699–718



The algorithm used to calculate the fluid motions is that of Mac-Cormack (Anderson 1995). Consider the variables, ρ f , vr, and vz atgridpoint (i, j). Assuming the flow-field at each gridpoint is knownat time t, we proceed to calculate the flow-field variables at the samegridpoints at time t + δt . This is obtained from

ρ t+δtf,i, j = ρ t

f,i, j +(

∂ρ f

∂t

)av

δt,

vt+δtr,i, j = vt

r,i, j +(

∂vr∂t

)av

δt,

vt+δtz,i, j = vt

z,i, j +(

∂vz∂t

)av

δt, (A1)

where (∂ρ f /δt)av , (∂vr/δt)av , and (∂vz/δt)av are representativemean value of (∂ρ f /δt), (∂vr/δt), and (∂vz/δt), respectively, be-tween times t and t + δt . These mean values are expressed as(

∂ρ f

∂t

)av

= 12

[(∂ρ f

∂t

)t

i, j

+(

∂ρ f

∂t

)t+δt

i, j

],

(∂vr∂t

)av

= 12

[(∂vr∂t

)t

i, j+

(∂vr∂t

)t+δt

i, j

],

(∂vz∂t

)av

= 12

[(∂vz∂t

)t

i, j+

(∂vz∂t

)t+δt

i, j

].

(A2)

The first and second terms in the right-hand sides represent a ‘pre-dictor step’ and ‘corrector step’, respectively. From eqs (1) and (2)the predictor steps for each parameter are expressed by(

∂ρ f

∂t

)t

i, j

= −ρ tf,i+1, jv

tr,i+1, j − ρ t

f,i, jvtr,i, j

δr f

− ρ tf,i, j+1v

tz,i, j+1 − ρ t

f,i, jvtz,i, j

δz f− ρ t

f,i, jvtr,i, j

(i − i0)δr

− ρ tf,i, j

V

(∂V

∂t

)t

i, j

,

(∂vr∂t

)t

i, j= −vt

r,i, j

vtr,i+1, j − vt

r,i, j

δr f− vt

z,i, j

vtr,i, j+1 − vt

r,i, j

δz f

− pti+1, j − pt

i, j

ρ tf,i, jδr f ,(

∂vz∂t

)t

i, j= −vt

r,i, j

vtz,i+1, j − vt

z,i, j

δr f− vt

z,i, j

vtz,i, j+1 − vt

z,i, j

δz f

− pti, j+1 − pt

i, j

ρ tf,i, jδz f

− g, (A3)

where

1

V

(∂Vδt

)t

i, j= 2

rc

utrc,ic, j − ut−δt

rc,ic, j

δt + 1�c + �v

utzb,i, jb − ut−δt

zb,i, jb

δt ,

for i0 ≤ i ≤ ic and j0 ≤ j ≤ j t,

1

V

(∂Vδt

)t

i, j= 2

rv

utrv,iv, j − ut−δt

rv,iv, j

δt + 1�c + �v


zb,i, jb

δt ,

for i0 ≤ i ≤ ic and j t ≤ j ≤ jb,

1

V

(∂Vδt

)t

i, j= 2

rv

utrv,iv, j − ut−δt

rv,iv, j

δt

+ 1�v


zb,i, jb − utzt,i, j t + ut−δt

zt,i, j t

δt ,

for ic ≤ i ≤ iv and j t ≤ j ≤ jb, (A4)

where urc,ic, j and urv,iv, j represent the radial displacements at theconduit and reservoir walls, respectively, uzt,i, j t and uzb,i, jb are the

vertical displacements at the reservoir roof and reservoir bottom,respectively, and δr f and δzf represent the grid intervals in the r andz directions, respectively (in our calculation δr f = δzf = δd/4). Thepressure at each grid node, pt

i, j , is calculated from ρ tf,i, j using eq.

(3). Predicted values for ρ f , vr, and vz are obtained from the firsttwo terms of a Taylor series:

ρ t+δtf,i, j = ρ t

f,i, j +(

∂ρ f

∂t

)t

i, j

δt,

vt+δtr,i, j = vt

r,i, j +(

∂vr∂t

)t

i, jδt,

vt+δtz,i, j = vt

z,i, j +(

∂vz∂t

)t

i, jδt, (A5)

By substituting the predicted values ρ f , vr , and vz into eqs (1) and(2) and replacing the spatial derivatives with backward differences,we obtain the corrector steps:

(∂ρ f

∂t

)t+δt

i, j

= −ρ t+δtf,i, jv

t+δtr,i, j − ρ t+δt

f,i−1, jvt+δtr,i−1, j

δr f

− ρ t+δtf,i, jv

t+δtz,i, j − ρ t+δt

f,i, j−1vt+δtz,i, j−1

δz f

− ρ t+δtf,i, jv

t+δtr,i, j

(i − i0)δr − ρ t+δtf,i, j

V

(∂V

∂t

)t+δt

i, j(∂vr

∂t

)t+δt

i, j

= −vt+δtr,i, j

vt+δtr,i, j − vt+δt

r,i−1, j

δr f− vt+δt

z,i, j

vt+δtr,i, j − vt+δt

r,i, j−1

δz f

− pt+δti, j − pt+δt

i−1, j

ρ t+δtf,i, jδr f ,(

∂vz

∂t

)t+δt

i, j

= −vt+δtr,i, j

vt+δtz,i, j − vt+δt

z,i−1, j

δr f− vt+δt

z,i, j

vt+δtz,i, j − vt+δt

z,i, j−1

δz f

− pt+δti, j − pt+δt

i, j−1

ρ t+δtf,i, jδz f

− g, (A6)

where

1

V

(∂V

δt

)t+δt

i, j

= 2rc

ut+δtrc,ic, j − ut

rc,ic, j

δt + 1�c + �v

ut+δtzb,i, jb − ut

zb,i, jb

δt ,

for i0 ≤ i ≤ ic and j0 ≤ j ≤ j t,

1

V

(∂V

δt

)t+δt

i, j

= 2rv

ut+δtrv,iv, j − ut

rv,iv, j

δt + 1�c + �v


zb,i, jb

δt ,

for i0 ≤ i ≤ ic and j t ≤ j ≤ jb,

1

V

(∂V

δt

)t+δt

i, j

= 2rv

ut+δtrv,iv, j − ut

rv,iv, j

δt

+ 1�v


zb,i, jb − ut+δtzt,i, j t + ut

zt,i, j t

δt ,

for ic ≤ i ≤ iv and j t ≤ j ≤ jb. (A7)

The fluid variables at the conduit and reservoir walls are linearlyextrapolated from the flow field values calculated at internal gridnodes. By representing all the fluid variables (i.e. =ρ f , vr, or vz) as

C© 2003 RAS, GJI, 153, 699–718



f , the fluid variables at the boundaries are obtained as

fic, j = 2 fic−1, j − fic−2, j, for j0 ≤ j ≤ j t,

fiv, j = 2 fiv−1, j − fiv−2, j, for j t ≤ j ≤ jb,

fi, j t = 2 fi, j t+1 − fi, j t+2, for ic ≤ i ≤ iv,

fi, jb = 2 fi, jb−1 − fi, jb−2, for i0 ≤ i ≤ iv. (A8)

At the conduit orifice the pressure is set equal to the atmosphericpressure and the other fluid variables are extrapolated as

fi, j0 = 2 fi, j0+1 − fi, j0+2, for i0 ≤ i ≤ ic. (A9)

The explicit forms of the finite-difference equations representingthe equations of motion in the solid (eq. 4) are given by

ρs,l,mut+δt

r,l,m − 2utr,l,m + ut−δt

r,l,m

δt2 = τ trr,l,m − τ t

rr,l−1,mδzs

+ τ tr z,l,m+1 − τ t

r z,l,mδrs

+ τ trr,l,m + τ t

rr,l−1,m − τ tφφ,l,m − τ t

φφ,l−1,m

2(l − 1)δrs,

ρs,l,mut+δt

z,l,m − 2utz,l,m + ut−∂t

r,l,m

δt2 = τ tr z,l+1,m − τ t

r z,l,mδrs

+ τ tzz,l,m − τ t

zz,l,m−1δzs

+ τ tr z,l+1,m + τ t

r z,l,m

2(l − 1/2)δrs+ ρs,l,m g, (A10)

where the subscripts l and m represent the indices of unit cells, andδr and δz represent the unit material cell sizes in the r and z directions,respectively (in our calculation δr s = δzs = δd). The stress-strainrelations (eq. 5) become

τ tr z,l,m = µl,m

(ut

r,l,m − utr,l,m−1

δzs+ ut

z,l,m − utz,l−1,m

δrs

),

τ trr,l,m = λl,m�l,m + 2µl,m

utr,l+1,m − ut

r,l,mδrs,

τ tzz,l,m = λl,m�l,m + 2µl,m

utz,l,m+1 − ut

z,l,mδzs,

τ tφφ,l,m = λl,m�l,m + 2µl,m

utr,l+1,m + ut

r,l,m

2(l − 1/2)δrs,(A11)

where

�l,m = utr,l+1,m − ut

r,l,m

δrs+ ut

z,l,m+1 − utz,l,m

δzs+ ut

r,l+1,m + utr,l,m

2(l − 1/2)δrs.

The fluid and elastic variables are dynamically coupled at the fluid–solid boundary by applying the continuity of velocities in the fluidand solid, and continuity of fluid pressure and stress in the solid, atthe conduit and reservoir boundaries. As the grid size in the fluidis one quarter of the grid size used for the solid in our model, weinterpolate the elastic displacements at the fluid–solid boundary withcubic spline functions and derive the velocities in the fluid from

vtr,ic, j = ut

rc,ic, j − ut−δtrc,ic, j

δt, for j0 ≤ j ≤ j t,

vtr,iv, j = ut

rv,iv, j − ut−δtrv,iv, j

δt ,for j t ≤ j ≤ jb,

vtz,i, j t = ut

zt,i, j t − ut−δtzt,i, j t

δt ,for ic ≤ i ≤ iv,

vtz,i, jb = ut

zb,i, jb − ut−δtzb,i, jb

δt ,for i0 ≤ i ≤ iv, (A12)

where urc,ic, j and urv,iv, j represent radial displacements at the con-duit and reservoir walls interpolated from ur,lc,m and ur,lv,m , respec-tively, and uzt,i, j t and uzb,i, jb represent vertical displacements atthe roof and bottom of the reservoir interpolated from uz,l,mt andur,l,mb, respectively. The fluid pressure and stress field in the solid are

coupled via the relations

τrr,lc,m = pic, jm,

τzz,lc,m = 2τzz,lc+1,m − τzz,lc+2,m,

τφφ,lc,m = 2τφφ,lc+1,m − τφφ,lc+2,m,

τr z,lc,m = 0, for m0 ≤ m ≤ mt − 1,

τrr,lv,m = piv, jm,

τzz,lv,m = 2τzz,lv+1,m − τzz,lv+2,m,

τφφ,lv,m = 2τφφ,lv+1,m − τφφ,lv+2,m,

τr z,lv,m = 0, for mt ≤ m ≤ mb − 1,

τrr,l,mt−1 = 2τrr,l,mt−2 − τrr,l,mt−3,

τzz,l,mt−1 = pil, j t,

τφφ,l,mt−1 = 2τφφ,l,mt−2 − τφφ,l,mt−3,

τr z,l,mt = 0, for lc ≤ l ≤ lv − 1,

τrr,l,mb = 2τrr,l,mb+1 − τrr,l,mb+2,

τzz,l,mb = pil, jb,

τφφ,l,mb = 2τφφ,l,mb+1 − τφφ,l,mb+2,

τr z,l,mb = 0, for l0 ≤ l ≤ lv − 1,

τr z,lv,mb = 0,(A13)

0 1 2

Time(s)

0 m

3.125 m

6.250 m

9.375 m

12.500 m

15.625 m

18.750 m

21.875 m

Ripples

2.0M

Pa

Figure B1. Temporal variation of p(t) at grid nodes along the conduit wallclose to the conduit orifice. The vertical distances from the orifice are indi-cated at the left of each trace.

C© 2003 RAS, GJI, 153, 699–718



where il, jm denote the grid nodes in the fluid that match the gridnodes in the solid at the fluid–solid boundary.

A P P E N D I X B : F L U I D M O T I O N AT T H EC O N D U I T O R I F I C E

As indicated in eq. (A8), the pressure at the conduit orifice is ex-trapolated from the pressure at the nearby two gridpoints in ourfinite-difference scheme. This extrapolation algorithm works wellespecially when the fluctuations of pressure between nearby grid-points are small. However, unrealistic waves may be produced whenan abrupt change of pressure occurs at the conduit orifice. In ModelII, the shock with step-like onset impinging the lid induces artificialripples near the wave front of the reflected shock. Fig. B1 showsplots of the fluid pressure at grid nodes along the conduit wall closeto the lid in an extended pipe (rc = 50 m, r v = 50 m, �c = 400 m, �v

= 400 m) with �P = 1 MPa. We observe ripples with a peak am-

plitude exceeding the mean amplitude of the reflected shock waveby about 70 per cent at the lid. In this case, therefore, the strength ofthe lid is overestimated by about 70 per cent in absolute value. Thepeak amplitude of such ripples decreases to 20 per cent of the meanamplitude of the reflected shock wave for the case of �P = 0.1MPa. These ripples lead to an error in our estimation of the pressurecondition leading to the removal of the lid and start of the eruptionin Model II, but do not affect the solid motions because the rippleamplitudes decrease rapidly with increasing distance from the lid.For example, the peak ripple amplitude exceeds the mean amplitudeof the reflected shock by 20 per cent at a depth of 6.25 m in Fig. B1.The stress σ rr in the solid near the conduit orifice is coupled withthe fluid pressure p at grids located 6.25 m and 18.75 m from thelid. Ripples are relatively weak at these locations and their effectin the elastodynamics is trivial. The ripples can also be reduced viathe expedient of using a smaller grid interval δdf , however, at theexpense of more extensive computations.

C© 2003 RAS, GJI, 153, 699–718

a numerical simulation of magma motion, crustal deformation, and seismic radiation...

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