JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. B9, PAGES 20,433-20,447,SEPTEMBER 10, 1996
Strombolian explosions
1. A large bubble breaking at the surface of a lava
column
as a source
of sound
S. Vergniolle
1
Institute of Theoretical Geophysics,Universityof Cambridge,Cambridge,England
G. Brandeis 2
Laboratoire de Dynamique des Syst•mes G6ologiques,Institut de Physiquedu Globe de Paris
Abstract. Strombolianactivity consistsof a seriesof explosionscausedby the
breakingof large overpressurized
bubblesat the surfaceof the magmacolumn.
Acousticpressurehas been measuredfor 36 explosionsat Stromboli. We propose
that soundis generatedby the vibration of the bubble beforeit bursts. Oscillations
are driven by an initial overpressureinside the bubble, assumedto be initially
at rest, just below the magma-air interface. Inertia effectscausethe bubble to
overshootits equilibrium radius. Then the bubble becomesunderpressurizedand
contractsbecauseof gascompressibility.
Theseoscillationsare only slightlydamped
by viscouseffects in the magma layer above the bubble. The bubble cannot
completemore than one cycleof vibration becauseof instabilitiesdevelopingon
the magmalayer that lead to its breaking,near the minimumradius. Assuminga
simplegeometry,we modelthis vibrationand constrainthe radiusand lengthof the
bubble and the initial overpressure
by fitting a syntheticwaveformto the measured
acousticpressure.The fit betweensyntheticand observedwaveformsis very good,
both for frequency,m 60 rad s-1, and amplitude.The initial bubbleradiusis m 1
m, and the length varies betweenseveraland a few tens of meters. From the initial
overpressure,
approximately
10• Pa, we calculatethe maximumradialvelocityof
ejecta,m 30 m s-1. The generallygoodagreement
betweendata andpredictions
of
our model leads us to suggestthat acousticmeasurementsare a powerful tool for
the understandingof eruption dynamics.
Introduction
at depth, carries physicalinformation about the dynamics of Strombolian activity, which may lead to a better
Volcanic eruptions present different regimes, which understandingof the system.
can be understood and classified in the framework of
Stromboli volcano has been in a quasi-permanent
a two-phaseflow [Vergniolleand Jaupart, 1986]. In state of eruptive activity for more than 2000 years. Acbasalts,the differential motion of the gas with respect tivity of Stromboli is characterized by a series of exto the liquid produceseither an annular flow in Hawai- plosions,occurringwith a regular intermittency, from
ian fire fountains or a slug flow in Strombolian explo- 10 min to 1 hour. The height reachedby ejecta above
sions[Vergniolleand Jaupart,1986]. The gas,formed the vent varies from a few meters to a hundred meters,
withvelocities
between
50and100m s-• [Chouetet al.,
1974; Weill et al., 1992].Their originlies in the breaking of a metric, overpressurizedbubble at the surfaceof
1Now at Laboratoire de Dynamique des Syst•mes G6ologiques,Institut de Physique du Globe de Paris.
SNow at UMR 5562 CNRS, Groupe de Recherchesde
G6od6sie Spatiale, Observatoire Midi-Pyr6n6es, Toulouse,
France.
the magmacolumn(Figure 1) [Blackburnet al., 1976;
Wilson,1980;Braun and Ripepe,1993]. Suchgaspockets, almost as large as the volcanicconduit, are formed
intermittently at depth in a shallow magma chamber
Copyright 1996 by the American GeophysicalUnion.
by coalescence
of a foamlayer [Jaupartand Vergniolle,
1988, 1989; Vergniolleand Jaupart,1990].
Paper number 96JB01178.
studiedwhen formedby underwaterexplosions[Cole,
0148-0227/ 96/ 96JB-01178509.00
1948; Davies and Taylor, 1950; Taylor, 1963; Taylor and
Large bubbles, over a meter in diameter, have been
20,433
20,434
VERGNIOLLE AND BRANDEIS: STROMBOLIAN EXPLOSIONS, 1
tensivelystudied,either in engineering[Boulton-Stone
and Blake,1993;Leighton,1994]or in the ocean[Spiel,
1992]. In the ocean,small bubbles(below2 mm),
formed by breaking gravity wavesat the surfaceof the
water [Thorpe,1980; Spiel, 1992],producesignificant
levelsof soundin the ocean[Longuet-Higgins,
1990].
The bursting of gas bubbles at the surface of a liquid has been studied for small bubbles, and these fluid
dynamic studies were recently combined with acoustic
measurements
[Spiel,1992]. Similarly,we havecarried
out acoustic measurements
at Stromboli
volcano in or-
der to understandthe detailedprocesses
occurringwhen
a large bubble, formed in a highly viscousfluid, breaks
at an air-liquid interface.
At Stromboli, visual observationsof the western vent
show that sound and ejecta are produced simultaneously when bubbles break at the surface of the lava
betweenexplosions(P. Allard, personalcommunication, 1993), linking aerial explosionsto the breaking
of large overpressurizedbubbles. Bubbles rising in a
tube slightly larger than the bubble size correspondto
slugflow, which has beenwidely studiedin chemicalen-
gineering[Wallis, 1969;Butterworthand Hewitt, 1977;
Kay and Nedderman,1985]. Most previousstudieshave
been applied to small tubes in which capillary effectsare
important or to fluid liquids, with viscositieslessthan 1
Pa s. We can compareStrombolito a large-scaleexper-
iment in whichthe tube hasa radiusof m 1 m, a length
abovea few hundredof meters[Giberti et al., 1992],
and a viscosityabove 100 Pa s. In theseconditions,the
Reynolds number of the bubble, around 80, indicates
that the viscosityof the magma cannot be neglected,
and the bubble rise speed, independent of the bubble
length,is around1.6In s-1 [Wallis,1969;Butterworth
and Hewitt, 1977]. Furthermore,in chemicalengineering, studieson large bubblesneglectthe influenceof any
air-liquid interface on the bubble behavior.
Previous studiesof acousticson volcanoesare sparse
[Richards,1963; Woulffand McGetchin,1976]. Gas
velocitiesaround100m s-1 in Strombolianactivityled
WoulffandMcGetchin[1976]to suggest,
withoutquantitative acoustic measurements,that the sourceof the
sound generatedby Strombolian explosionsis a dipole.
Figure 1. Photographsof a bubbleburstingon a lava
flow at Etna (Italy) in May 1983 {courtesyof J.-L.
Cheminde).(a) The bubbleprotrudesat the lava surfaceshowinga hemispherical
cap. (b) When the bubble
bursts, the thicknessof the magma abovethe bubble is
smaller than the bubble size and of the order of the size
Recent acousticmeasurements
[Vergniolleand Brandeis,1994]includinglowfrequencies,
between4 Hz and
a few kilohertz, showedthat the sourceis a monopole
and led us to suggestthat the soundis producedby the
vibration of a large bubble due to a suddenoverpressure
at the exit of the vent. In an alternate model, Buck-
inghamand Garc4s [1996] suggestthat the sourceof
of ejecta. (c) After bursting,the lavasurfacebecomes sound at Stromboli is explosiveand embodied deep in
flat again.
the magma column.
Bubble oscillationshave long been recognizedto be
Davies,1963]. The violentformationof the gaspocket a sourceof sound[Batchelor,1967; Leighton,1994].
Ascendingbubbles adjust their size to the decreasing
the surroundingwater,followedby strongoscillations
of pressure.In a magma, viscousforcessignificantlydelay
the newlyformedbubble[Cole,1948;Leighton,1994]. the growth to equilibrium, and overpressurecan build
Small bubbles in an infinite liquid have been also ex- up insidethe bubble [Chouetet al., 1974; Blackburn
under rapid chemical reaction sendsa shockwave into
VERGNIOLLE AND BRANDEIS: STROMBOLIAN EXPLOSIONS,1
et al., 1976].With sucha mechanism,
the velocityof
ejectais relatedto overpressure,
estimated
around600
40
20,435
2•,• •Part3
•
30
Pa at Stromboll. However,it breaksdown for larger
overpressures,
asexisting
at Heimaey
(2.5x 104 Pa),
20
becauseit impliesunreasonably
largevelocityfor the
10
bubblegrowth[Blackburn
et al., 1976].
In a previous
study[Vergniolle
andBrandeis,
1994],
0
basedon small oscillations,the measuredfrequencywas
related to the radius of the bubble, around i m. In
-10
thispaper,wedevelopthe complete
setof equations
for
-20
the generalcaseandexpress
the amplitude
of oscilla-
-30
tions as a function of the bubble overpressure.Then we
show how to estimate the size of the bubble, its initial
-40
overpressure,
and the velocityof ejectafrom the measuredintensityof acousticpressure.We alsodiscuss
the mechanisms
by whicha largeoverpressurized
bub-
13
EXPLOSION
111
•
•
•
•
•
•
13.5
14
14.5
15
15.5
16
16.5
I (a)l
•Part3•
•
17
TIME (s)
ble can break at the surfaceof a lava column. Finally,
• 100
wepresentalternatemodelsandcompare
the results.
•
• Partl •
• Part2•
80
6O
40
Characteristics
of Stromboli
AlthoughStrombollhas two activecraters,we focus
on the easternone, which undergoesone explosionper
hour on the average. Four ventshave been recognized
inside the eastern crater in September1971. The only
observedopeningis circular,roughly0.5 m in diameter
[Chouetet al., 1974].Therearenodatafor the volcanic
conduitdiameterat depth. Gibertiet al. [1992]have
o
-:20
-40
-60
-80
-100
14
14.5
15
15.5
16
16.5
17
17.5
18
estimated the chamber depth to be a few hundredsof
TIME (s)
meters. However, the exact value of these two parameters are not crucial for the reasoning,sinceour study Figure 2. Acousticpressuremeasuredat 250 m from
(a) 111and (b) 112of the
is centeredon the understandingof a shallowsourceof the sourceduringexplosions
sound.
eastern vent in 1992.
In April 1992, we recordedthe acousticpressurein ex-
cellentweatherconditions
(nofog,no rain,no wind)at
cm,smallcompared
to the radius(Figure1) [Vergniolle
a distanceof 250 m for 36 explosionsof the easternvents
andBrandeis,1994].We approximatethe bubbleshape
of Stromboll. Acousticpressurestarts closeto zero and
by a cylindricaltail andan hemispherical
head,getting
increases
(Figure2). The radianfrequencyis low, less
distorted
at
the
air-magma
interface
(Figure
3a). Most
than 60 pads-1 and the amplitudeat a distanceof 250
m is between a few and 50 Pa. After one singlecycle
of the observedventsconsistof an openingsurrounded
by a small coneof ejecta,givingto the total structure
of vibration (i.e. the strongimpulseevent in acous- the shape of a funnel. Therefore no external limitation
tic pressure)the bubblebursts,emittingrather quiet exists on the bubble growth when the bubble reaches
higherfrequencies.However,Vergniolleand Brandeis
the surface.Becausethe magmais closeto the surface
[1994]focusedonly on the frequency
contentandsmall
for
the easternvents[Vergniolleand Brandeis,1994],
oscillationsof such a bubble. In this present paper, we
there is no amplificationof the soundinsidethe upper
broadenour scopeand includeamplitudeof the acoustic
part of the volcanicconduit(seeAppendixA), andthe
pressurein order to bring constraintson suchquantidistortion due to the propagationfrom the sourceto
ties as bubblediameter and length, thicknessof magma
themicrophone
issmall[Vergniolle
andBrandeis,1994].
layer, and overpressure.
We do not considerthe exact vent geometrysinceit is
not known accurately.
Finally, we have to specifythe temperature insidethe
The Model
bubble. Before vibration, the bubble is in thermal equilibrium
with the magma in the chamber. However,the
Assumptions
followingcalculationsare weaklysensitiveto the specific
A typical Strombolian bubble has a radius of • 1 temperaturevalue. It will be chosenin the rangeof valm and its magma thicknessh at equilibrium, assumed uesobtainedfrom experimentalstudies[Francalanci
et
to be closeto the averagediameter of ejecta, is • 2 al., 1989]and considered
equalto 1373K.
20,436
VERGNIOLLE
ANDBRANDEIS:
STROMBOLIAN
EXPLOSIONS,
1
h
Air
Figure 3b. Sketchof a vibratingbubblereachingthe
air-magmainterface.The hemispherical
capabovethe
bubble vibrates in air as a membrane of thickness h.
The bubble has radius R and length L.
and Brandeis,1994]. Bubbleson the surfaceof lava
.....:•½.,,
...........
.•iiii;•;:•;.';.::.•
....
,•.•:.
flowsat Etna have been observedbursting in May 1983
(Figure1). Any bubblein an infiniteliquidoscillates:
inertia causesthe bubble to overshootits equilibrium
radius and the compressibilityof gas is the restoring
force[Batchclot,1967].Here,we assume
for simplicity
'..•;..
that the oscillationsare triggeredby a suddenoverpres-
sureinsidea large bubblereachingthe magma-airinterface. Four possiblesourcesof overpressure
can exist
................
•,
(a) '•
inside a bubble. The first one is surface tension at the
gas-liquidinterfaces.The secondoneis relatedto the
Figure 3a. Experimentshowing
the strongchangein significant
delayimposed
to the growthof a risingbubgeometry
experienced
by a bubblereaching
the liquid- bledueto the magmaviscosity.The third oneis related
air interhce. The liquid is a viscoussiliconoil (12.5 to the effectof the air-magmainterfaceon the moving
Pa.s).Notetheincredein diameter
of thebubblecap
in
and the small thicknessof the liquid abovethe bubble. bubble,whichinducesa departurefrom steadiness
In this study,we assumethat the magmais a Newtonian liquid. This hypothesis
can be violatedeitherfor
dynamicalreasons,
or because
of the coolingof magma.
A viscous fluid can loose its Newtonian behavior if the
the upwardsvelocity.The fourth oneis relatedto the
energetic
formationof the gaspocketby coalescence
of
a foamlayerat thetopof themagmachamber
[Jaupart
and Vergniolle,1989].All thesesources
of overpressure
will be comparedto our estimatesof the bubbleoverpressurededucedfrom acousticmeasurements.
A bubblevibratingat a magma-airinterfacepresents
time duringwhicha forceis appliedto it is verysmall
[LandauandLifshitz,1987].For Strombolian
magma, three modes of vibration. The first mode is related
this relaxationtime is around4 x 10-8 s [Webband to surfacetension, the secondone is related to gravDingwell,1990],much'lessthan the observed
vibration ity waves,and the third one is only due to volume
andis calledthe volumemode[Lu et al., 1989;
time (0.1 s). Thereforethe departurefrom the New- changes
tonianrheologycan be only due to the occurrence
of Vergniolleand Brandeis,1994].The highfrequencyof
crystals.Strombolianejectacontainsbetween10 and the volumemode comparedto surfacetensionor grav45%of phenocrysts
[Francalanci
et al., 1989].However, ity modes[Vergniolle
andBrandeis,1994]indicates
a
there are no accurate data on the occurrenceof crys-
strongrestoring
force,makingit hardto excite[Lu et
tal contentin lava itself, and T. Trull (personalcom- al., 1989].Overpressure
buildsup insidea Strombolian
munication,1995)suggests
that thisparameteris close bubblerisingin a tubeandinduces
its strongvibration
to 25% for Strombolian lava. In these conditions, the
at themagma-air
interface.Thisleadsto changes
in the
magmarhelogycanbeconsidered
asNewtonian
[Shaw, bubblevolume: energywill go preferentiallyinto the
1969; van der Molen and Paterson, 1979; Ryerson et
al., 1988].
Description of the Source
volumemode, directly drivenby overpressure,
rather
than into the surface tension mode, which is not ener-
geticfor largebubbles,
or intothe gravitymode.Hence
the presentstudywill be focusedonly on the volume
We proposethat the soundrecordedat the vent of mode. A Strombolianbubble reachingthe surfaceis
into the magmaandalmosthalfway
Stromboliis producedby the vibrationof a metricbub- halfwayimmersed
in
air
despite
a
thin
layerof magmaaboveit (Figure
blepriorto burstingandcloseto thesurface[Vergniolle
VERGNIOLLE AND BRANDEIS: STROMBOLIAN EXPLOSIONS, 1
20,437
3b). Becauseof the largedifference
in viscositybetween Equations for the Bubble Vibration
air and magma, the motion of the immersedpart of
In orderto expressthe variationsin acousticpressure,
we
model the bubble motionsin responseto a sudden
Therefore the bubble vibration is mostly concentrated
overpressure.
The bubbleoscillatesin the magmawhich
into its hemispherical
cap (Figure3a). For simplicity,
is
a
newtonian
viscousfluid. In Appendix B we show
we ignore the elasticity of the magma shell.
the bubbleis restricted[VergniolleandBrandeis,1994].
The
vibration
of the interface
between the bubble
that the only significantsourceof dampingon the bubble oscillations
is due to viscousforcesin the magma
and the magmais transmittedradially throughthe thin
layer of magma and reachesthe magma-air interface. layer above the bubble.
The soundspeedin magmais above2500rn s-1 [Rivers
and Carmichael,1987;Kressand Carmichael,1991]for
the shoshonite
composition
of Stromboli[Francalanci
et
al., 1989].The magmais largelyincompressible
for the
radial motion consideredhere. Thereforeno energyis
lost by transmissionthrough the thin layer of magma
[Pierce,1981]:the magma-airinterfacevibratesas the
bubble
The bubble vibrates as a thin membrane of thickness
h. Its headgrowsbut remainssphericalwith a radiusR.
The part of the bubblewhichremainsin the cylindrical
tube hasa lengthL (Figure3a). Hencethe bubblehas
a volumeVgequalto
2•R 3
does.
Radiation
Pattern
of the
where Ro is the initial radius. Note that in the follow-
Source
ing, indexeso, g, and eq referto initial conditions,gas,
The sourceis a thin layer of magma, pushedby a and equilibriumvalues,respectively.The highviscosity
variation of pressureinside the bubble. We have shown of magmaimpedesany significantdrainageof magma
that the sourceis a monopoleas its amplitudedecreases abovethe bubble,duringthe shorttime allowedfor the
inversely proportional to the distance between the mi- bubbleto vibrate. The volumeof magmaabovethe
crophoneand the vent. Therefore the propagationof
bubbleis conserved
and the liquidstreches,
following
pressure
wavesis radial[Vergniolle
andBrandeis,1994],
the variations in bubble radius
2
and in that case, the source is a simple one. The ex/•2h-/•eqheq.
(4)
cesspressurecan be describedfrom the linear theory of
sound. In that case,the basicvariable is the rate of mass Becauseheat transfer insidelarge bubblesis adiabatic
outflowfromthe source,•, oftencalledthe strengthof [Plesset
andProsperetti,
1977],the pressure
pginside
the source[Lighthill,1978].Acousticpressure
Pa½
emit- thebubblefollows
thevariations
of its volumeVg
ted at the sourceat time t will reachthe microphone
Pg• -- PgeqV•eq,
(5)
at a time t + r/c, wherer is the distancefrom the vent
and c is the soundspeedin air (340 rn s-• at 900 rn where7 is the ratio of specificheats,equalto 1.1 for hot
abovesealevel[Lighthill,1978]). Herethe magma-air gases[Lighthill,1978]. For a thin layerof magma,the
interfaceis the radiatingbodywhichis half a sphereof contributionof its weight to the equilibriumpressure
radius equal to the bubble radius R. It emits sound in insidethe bubble is small, and the equilibriumpresair in half a sphereof radiusr, the distancebetweenthe surepgeqcanbe considered
asequalto the atmospheric
vent and the microphone.For sucha monopolesource, pressure
Pair,105Pa. The temperature
T is homogethe excesspressurePa½-Pair at time t is
neousinsidethe gasduringvibration[Prosperertl,
1986]
and obeysthe law of perfectgases
•(t -- •lc)
Pac -- Pair
4•rr
pgVg
6
2rrr
d2 4rrR3(t--r/c)]
Pair(1)
= dt•
T- TopgoVg
ø.
(6)
Therefore it will passivelyfollow variations in volume
wherePairand flairare respectively
atmospheric
pres- and pressure.Supposethat the bubble,initially at rest
sureand air density(1.1 kg m-• at 900 m abovesea at the magma-airinterface,is suddenlyoverpressurized
level[Batchelor,1967]). Finally,we obtainthe excess by an amountAP. The bubblestarts to grow and vipressure in air
brate in response
to that pressurechange.Pressureand
volumefollow the adiabatic law; hencewe can calculate
Pac
--Pair
-- [2/•2(t
--r/c)+•(t--r/c)l•(t
--r/c)] their variations.The equilibriumradiusReqchanges
x
PairR(t- r/c)
(2)
where/•/is
theradialvelocity
andJ•istheradialaccel-
from Ro accordingto the adiabatic law
Req-•
•__O+L
1+
Pair
-L
(7)
eration of the hemisphericalcap. This equationrelates
the variationsin acousticpressureto the bubble vibra- where L is the bubble length. The bubble radius R
tion.
variesaroundits equilibriumradiusReqby
20,438
VERGNIOLLE AND BRANDEIS- STROMBOLIAN EXPLOSIONS,1
R- Req(1+ e),
where e is a dimensionless
bubble radius.
(8)
The bubble
motion is possiblethrough an exchangebetween the
kinetic energy Ek of its hemisphericalcap and the po-
tential energyEp insidethe gas,whichare
Small Oscillations Without
Damping
Equation (12) has an analyticalsolutionwhen the
variationin bubbleradius• is small (<< 1). Without
damping, the solutionof • is a sine function, whose
amplitude A, radian frequency w, and phase delay •b
are
2
'2
Zk -- Wpl/r•eqheq/r•
(9a)
A = 3•'Pair/•e2q
2
v
Ep-- - fVo
(pg
- Pair)dV.
(9b)
PlReqheq2 q-3L/Req
We can expressthe rate of change of kinetic energy
dEk/dt and potential energydEp/dt as a functionof
• =
the variations in velocity • and accelerationg
dEk
dt
2
(15a)
(sb)
(15c)
where Pext is close to the atmospheric pressurePair,
= 2•rplRe4qheq•g
(10a) 105Pa, for a bubblecloseto the air-magmainterface.
The phase delay, •b,in the sine function correspondsto
dEp
eq
)'•]'-•'
dVg
dt = [p
air- Pgeq(Vg
Vg
(lOb) a bubble initially at its minimum radius. The results
show that the amplitude of the vibration is directly related to the overpressurein the bubble. Similarly, the
Their sum is equal to the rate of energy dissipation,
frequency
of the vibration dependson the size of the
which is assumedto be entirely due to viscousforces
source, here both on the bubble volume and on the
insidethefilm of m.agma
(seeAppendix
B). Thetotal
rate of dissipationFv in a hemisphericalfilm of thick- thicknessof the magma layer: the larger the source,
nessh and viscosity/• can be calculatedin the same
way asfor a sphere[Batchelor,1967]and is equalto
the lower the frequency of vibration is.
Such a sourcewill produce an excessof acousticpres-
surepa½- Pairat a distancer and at a time t + r/c
• - •R
12•rR4I•22•rr2dr24•ryk2heq.
(11)
•6
'•+h
Ac02
•
sin(wt+ •).
Pac-- Pair-- --Pair/r•e3q
(16)
We obtainthe generalequationfor the bubblevibration The radial velocity is maximum for equilibrium, where
acceleration is zero, and zero close to the minimum
or maximum radius, where acceleration is maximum.
Acoustic pressure,which dependsmainly on acceleration, starts at its highestpositivevalue before decreas-
(•,plReUq
12/•
)•+Pair[l-(v-m•'
k'+
%
]](l+e)
2-0
(12)
PlReqheq
whereVgis a functionof e (equation
(8)) andfi• is the ing towardzero (Figure4a). It is clearfrom the comparisonbetweenmodeland data (Figure2) that either
viscousdamping coefficient
hypotheseson small oscillationsor no dampingor both
fiv- 6y
plRe2q
'
(13) are not appropriate. Therefore we need to take into ac-
The first initial condition to be specifiedis the initial
value of the dimensionlessradius eo. The second initial
condition is the initial radial accelerationgo, which dependson the initial forceappliedto the layer of magma.
countthe nonlinearityof equation(12) and the viscous
damping.
The
General
Case
In this section,we solvethe generalcasefor the bubAssumingthat the bubble,at rest at the magma-airin- ble vibration(equation(12)). Equation(12)is an ordinary differential equation of order 2. Its behavior can
terface, is suddenlyoverpressurizedby an amount AP,
this force is directly related to the bubbleoverpressure. be understoodby lookingat the stability of equilibrium
Therefore the initial conditions are
points and their local behavior when assumingsmall
oscillations[Drazin, 1992]. The equilibriumpointsare
APR2o
go =
(14a) found by setting at zero the accelerationin equation
pl/r•e3q
heq
eo =
Ro
(14b) only one physicallypossiblesolution,which is a simple
oscillator.Its phasediagramwill be discussed
later.
Radial accelerationis maximum when the strong viIf the vibrationof a bubbleis not small,equation(12)
/•eq
1.
(12). Ignoringviscousdamping(sincewe showit is
small)andassuming
smalloscillations,
ourequationhas
bration starts, and therefore the initial radial velocity
is equal to zero. These initial conditionscorrespondto
a bubble close to its minimum
radius.
has no analytical solution. We solvedit by numerical
integration(a second-andthird-orderRungeandKutta
method). In orderto find the behaviorof the solution,
VERGNIOLLE
AND BRANDEIS'
STROMBOLIAN
•-
40
EXPLOSIONS,
i
i
i
1
i
20,439
i
i
i
i
i
Model
.5
Contraction
Expansion
.0
--.
I
.L•,_%
I •,2
••
0
A = 0.32
• lO
---if----
:
0.5
Data
• 20
i
-
g
-0.5
_
o
• -10
-1.0
-1.5
-2.0
0
0.04
œmax
I Iœmax
I I I I
0.2
0.4
0.6
0.8
1.0
EXPLOSION
1.2
DIMENSIONLESS
• -30
(a)
•
I()/
-40
I
I
1.4
1.6
1.8
2.0
I
-0.2
111
I
I
0
I
I
0.2
TIME
I
0.4
I
I
0.6
(a)
0.8
TIME (s)
2OO
4 - Expansion
Contraction
i
!
i
A =0.41
Data
I
•
i
Model
150
I
2
I
i 100
I
1
50
o
-1
-2
-50 -
-3
F œmax
-4 A=0.2
I
I
I
I
I
I
I
I
I
t
(b)
-50 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
DIMENSIONLESS
15
I
I
-150 -
EXPLOSION
-200
-0.2
TIME
I
-100-
I
I
RayleighTaylorinstability -
112
I
I
-0.1
0
Ripples
waves
I
I
0.1
0.2
I
0.3
(b)
0.4
TIME (s)
I
Figure 5. Comparisonbetweenthe syntheticand mea-
pansio•,C•ontractio•
suredwaveformsfor explosions
(a) 111 and (b) 112 aC
eastern vent of SCromboliin 1992. For explosion 111,
Ro = 0.8 m, L = 22 m, and AP = 4.7 104 Pa. For
explosion11•, Ro: 1 m, L: 3 m, andAP = 2.5 x 10•
5
Pa. Higher frequencies,attributed Cobubble bursting,
appear closeCo•he minimum radius. The model ceases
0
•o be valid
-5
when •he bubble
bursts.
-10
A=0.4
I.
-15
0
0.2
œmax
I
0.4
I
0.6
I
0.8
I
1.0
DIMENSIONLESS
wt
(c)
1.2
1.4
1.6
1.8
td
TIME
Figure 4. Acousticpressurein time with dimensionless
units. Initial radius Ro is I m, length L is 10 m, and
equilibriumthickness
of magmaabovethe bubbleheqis
2 cm. Maximum emax,minimum eminand equilibrium
dimensionless
radiieeqareshown.(a) AP = 10a Pa
andA = 0.04. (b) AP = 104Pa andA = 0.2. (c)
AP=5x
103PaandA=0.4.
= 2-;
= •
(17a)
(17b)
•d
= Aw
(17c)
•d
= Aw
2
(17d)
2.0
•d
(Pac-- Pair)d
_--(Pac
--Pair)r
(17e)
Pair
ReaqAw2
ß
we expressthe variablesin a dimensionless
form with Figure 4 showsthe stronginfluenceof amplitudeA. As
respectto their valuesfor the smalloscillationcase.The A increases,the differencebetweenpositiveand negaamplitude of the oscillationsA can be usedas a length tive peaksin acousticpressuregetsstronger,and initial
scale and the radian frequencyw as a timescale. The acousticpressuretends to zero. Comparisonwith data
dimensionless
variables
are
(Figure 2) showsclearlythat A needsto be at least
20,440
VERGNIOLLE AND BRANDEIS: STROMBOLIAN EXPLOSIONS, I
4O
Expansion Contraction
2,5 --
ß
20
I
I
2
I
•o
I
Rmin
I
I
o
1.5-
1-
gmin
•,/
-lO
T
-20
I
Expansion Contraction•
0.5-
-30
A = 0.33
0
-0.2
(a)
I
!
I
I
I
•
-0.1
0
0.1
0.2
0.3
0.4
A = 0.33
-40
0.5
-0.2
•
I
I
I
-0.1
0.1
0.2
0.3
TIME (s)
1500 '
Z
¸
I
I (b)
0.4
0.5
TIME (s)
' Expansion
' Contraction
'
'
'
/ • 30
•Re•
1000 -
• 500r•
Rmin
•
o
•
• 0
i -10
• -500•1-1000
•-1500
•a
• -20
-
-0.2
-0.1
i
i
•
i
i
•
0
0.1
0.2
0.3
0.4
0.5
-30
A=0.33
0.5
Req',
1
TIME (s)
1.5
2
(d)
2.5
3
BUBBLE RADIUS (m)
Figure 6. Variationswith timeof (a) bubbleradius,(b) velocity,and(c) acceleration
for a bubble
with Ro = 0.8 m, L = 22 m, and AP = 4.7 x 104Pa. Equilibriumthickness
of magmaabove
the bubble,heq,is 2 crn and amplitudeof smalloscillations
A is equalto 0.33. Corresponding
positionsof minimumRmin,equilibriumReq,and maximumRmaxradii are shown.(d) Phase
trajectory showingthat damping is small.
equal to 0.3 in order to reproducecorrectlythe observed sizesof ejecta [Chouetet al., 1974]. Then we fit the
acousticpressure(Figure4).
calculated
Applications to Strombolian Explosions
determination of the three other parameters,bubble radius, length and overpressure.
In this section,we comparethe acousticpressurepredicted by the model to the measurements. Acoustic
pressure produced by the bubble vibration is entirely
determined by the initial conditions,suchas the bubble
geometry and its pressure,and by the characteristicsof
the magma, such as the viscosityand density,assumed
here to be 400 Pa s, from physical and chemical mod-
Qualitative
waveform
to the recorded
one.
This
allows
Results
The fit between predicted and measured acoustic
pressureis goodfor about onecycle,until high frequenciesappearafter the first cycleof vibration (Figures2
and 5), probablygenerated
by the burstingof the bub-
ble after which our model obviouslyceasesto be valid.
Three different features of the acousticpressureare very
eling [Shaw,1972]and magmacomposition
at Strom- well reproduced:the shapeof acousticpressureduring
boll [Capaldiet al., 1978],and 2700kg m-•, respec- rising time, the relative intensities,and the durations
tively. The four parameters(radius,length,overpres- of positive and negativepeaks.
sure,thicknessof magmalayer) cannotbe determined The behavior of radius, velocity, acceleration, and,
acousticpressureis summarizedon Figindependentlyand simultaneouslyunlessthe regimeis consequently,
highly nonlinear,A > 0.5. We have allowedone param- ures 6 and 7. As a consequence
of the variation in its
eter, the magma layer thicknessh, to vary in a reason- radius(Figure6a), the bubbletakesa distortedshape
able range, from 0.5 to 10 cm, accordingto observed with a head twice as large as its bottom, as observedin
VERGNIOLLE
AND BRANDEIS'
STROMBOLIAN
0.16
EXPLOSIONS,
1500
I
1
20,441
I
I
I
I
I
Expansion Contraction
Rmin
0.14
•1450
0.12
1400
0.1
0.08
I
Rmi
1350
0.06
1300
0.04
gmax
•1250
Re•
0.02
A-0.3
A = 0.33
o
-0.2
1200
0
0.2
0.4
0.6
0.8
1
-0.2
I
I
I
•
I
-0.1
0
0.1
0.2
0.3
TIME (s)
TIME
1.8
I (b)
0.4
0.5
(s)
40
I
Expansion Contraction
I•
1.6
•-• •
•,
•Rmin
&
30
Expansion_
_Contraction
i
1.4
:o
lO
1.2
o
1
•
-10
•
-20
Rmin
0.8
0.6
A = 0.33
0.4
-0.2
•
-0.1
I.
0
I
0.1
Rmax
I
I
i
0.2
0.3
0.4
-30 -
A = 0.33
(c)
-40
0.5
Rmax
•
-0.2
-0.1
I
0
0.1
TIME (s)
0.2
0.3
0.4
(d)
0.5
TIME (s)
Figure 7. Variationsof (a) liquidthickness,
(b) temperature,(c) bubbleinternalpressure,
and
(d) acousticpressurewith time for sameset of parametersand initial conditionsasin Figure6.
laboratoryexperiments(Figure3a). The phasetrajectory,fairly differentfroma circle(Figure6d), exhibitsa
spiralnode [Drazin, 1992],confirmingthat the bubble
is a simplenonlinearoscillatorwith a small damping
coefficient.The dampingis strongercloseto the minimum radius, where the film of magma is the thickest
than closeto the maximumradius(Figure6d). The
magmathicknessh (Figure7a) is smallcomparedto
the bubbleradiusR (Figure6a), in agreement
with the
assumptions.The variationin the gastemperature,althoughherealmost100K (Figure7b), doesnot allowa
significantcoolingof the magmaduringthe shorttime
of vibration, less than 0.1 s.
Source
Characteristics
Numericalvaluesof bubble radius, bubble length,
overpressure
and amplitude A are displayedfor different valuesof the magmalayer thicknessin Table 1. The
skinthicknessh is the mostcriticalparameterof ourcalculations. It is clear that thicknesses above 5 cm lead to
unlikelyvaluesof overpressure,
above107Pa, andhigh
amplitudesA, above1, for which resultsbecomevery
sensitive to the choice of initial conditions. For values
of h between0.5 and 2 cm, the radiusRo variesonly
by 15%,whilethe bubblelengthL variesby a factorof
4.5, from about 7 m to about 30 m. The overpressure
Table 1. Influence of Equilibrium Thicknesson Bubble Characteristics
Error bars correspondto one standard deviation.
heq(cm)
0.5
1.0
2.0
5.0
10.0
/to (m)
1.05
1.05
0.90
0.80
0.45
44444-
0.15
0.15
0.30
0.30
0.10
L (m)
AP (10sPa)
33 4- 12.5
15.6 4- 6.0
7.3 4- 3.0
1.9 4- 1.0
0.7 4- 0.9
0.32 4- 0.2
0.69 4- 0.3
2.0 4- 1.8
8.4 4- 4.0
107 4- 70.0
A
0.31
0.32
0.37
0.59
1.13
44444-
0.02
0.02
0.03
0.14
0.25
VERGNIOLLE AND BRANDEIS. STROMBOLIANEXPLOSIONS,I
20,442
variesby a factor of 6, between0.32 x l0 s and 2 x l0 s burn et al., 1976]. The scatterin bubbleradiusvalues
Pa. The appropriate thicknessesof the magmalayer are might reflect the existenceof severalventswith different
in goodagreementwith thoseof Chouetet al. [1974], sizes. The maximum radial velocity in the hemispherwho observedthat 90% of ejecta had sizesbetween0.6 ical capranges
between8 and 60 m s-1 (Figure8b),
and 4.5 cm. Despitesomevariation(Table 1), the re- values compatible with independent measurementsof
sults are rather robust, with a bubble radius around 1 velocities
of around40 m s- 1 for ejectaandof 80 m s- •
m, a bubble length between a few and tens of meters, for gas [Chouetet al., 1974; Weill et al., 1992]. The
and an overpressure
around10s Pa. Their robustness initial overpressure
AP variesbetween2 x 104Pa and
is confirmed by the low values of A, between 0.3 and
6 x 105Pa (Figure8b), muchlargerthan the previ-
0.4 (Table 1), whichensuresan only mild nonlinearity ous estimate of 600 Pa, derived from ballistic measureand a weak sensitivity of results to initial conditions. ments at Stromboll, but comparable to that calculated
Hence, for the sakeof simplicity, in the followingwe fix at Heimaey,2.5 x 104 Pa, althoughthis later volcano
the equilibriumthicknessheqat 2 cm, keepingin mind is considered
to be moreexplosive[Chouetet al., 1974;
that slight variationsshouldbe expected.
Blackburnet al., 1976]. A strikingfeatureapparenton
Despitevariations,the valuesof initial bubblelength Figure 8b is the good correlationbetweenmaximum veare between
2 and
30 times
those of the initial
ra-
dius(Figure8a), characteristic
of a well-developed
slug
flow [Wallis, 1969].The bubbleradiusitselfhasvalues
0.9 4-0.3 m (Figure8a) in goodagreement
with observationsmade at Stromboli[Chouetet al., 1974;Black-
I
OI
I
I
locity and initial overpressure.This is the consequence
of the mild departure from linear, small oscillations.
The correspondingradial accelerationsare of 550 4- 300
m s-2 forthepositive(outward)onesandof 13004. 600
m s-2 for the negative(inward)ones,somewhat
higher
7O
I
Equilibriumthickness= 2 cm
Equilibriumthickness= 2 cm
o
6O
1.5
I
5O
iI 0 0
o
I
I
-
1
o
4O
0
o
o
600
I
I
0
I
30-
0 0 0
I
0
•o
o
0
o
o
_1
o
I
o
o
oO
•
10-
5
I
I
I
10
15
20
BUBBLE LENGTH
I
o
o oCDg
ooo
o
o
o
20-
o
0.5 iI o
oo%%
o
0
I
I
I
(a)
i
0
i
i
i
i
i i I
104
25
i
i
(b)
i
i
i
i
i
]0 s
(m)
I
i
INITIAL BUBBLE OVERPRESSURE (Pa)
I
I
Burstingtime> Durationof onecycle
hmax= 9.7 + 6.1 cm
rl
O
o
O
O
O0
O
o
OOO
o
0
o
0
o
O
o
oo
Rmin
0
O0
0
"•-øø O oO
OOO
0.1
o o
0.01
•
•
0
0
O0
• o.o.6_o_O oo_O_._o_O
• • o_o_o_
__.
Rmax---
o
hmin= 1.2 + 0.2 cm
•0.001
Equilibrium
thickness
=2cm
I
I
I
I
10
20
I
I
30
I
(c)
40
0.01 '•
''
0.01
CHRONOLOGYOF EVENTS
•
........
ursting
time< Duration
of onecycle
•
'
' ' .....
0.1
DURATION OF THE MAIN EVENT (s)
Figure 8. Resultsof the analysisof the 36 explosionsrecordedin April 1992 from the eastern
vents. (a) Initial radiusRo versusbubblelengthL showingthat lengthis largerthan radius,
trademarkof a well-developed
slugflow. (b) Radialvelocityversusinitial overpressure
insidethe
bubble. (c) Liquidthicknesses
at minimumand maximumradii. Ripplewaveswill preferentially
developwhen radiusis maximum. (d) Burstingtime versusdurationof one cycle. For most
explosions,bursting occursbefore one cycle is completed.
(d)
VERGNIOLLE AND BRANDEIS: STROMBOLIAN EXPLOSIONS, 1
thanprevious
estimates
at Heimaey,
from250to 600
i
m s-2 [Blackburn
et al., 1976],in agreement
with the
ß
higherinitialoverpressure
at Stromboli
i
i
i
I
i
20,443
i i l
Equilibriumthickness= 2 cm
Fromthegeometry
ofthebubble
andthethickness
of
themagma
above
thebubble,
wehave
calculated
the
volume of gas and ejecta emitted during one explosion.
o
o
o
•ooøo
The volume of gas, at atmosphericpressure,variesbe-
tween
2 and100m3 (Figure
9),ingood
agreement
with
o
airbornemeasurements,
between44 and 178m3 ejected
persecond
of explosion
[Allardet al., 1994].The volume
o oo
0 oO
o
o
o
ooCb o
00
0
ofliquidabove
thebubble
varies
between
0.07and1ms
(Figure
9),compatible
with
previous
estimates
forthe
easternvents, obtainedindependently,
from9 x 10-s ms
I
I
conduit
below.
I
I
I I I [
I
I
I
I
I
• ] •
l0
[Blackburn
et al., 1976]to 1.9ms [Ripepe
et al., 1993].
This suggeststhat ejecta comemostly from the magma
layer above the bubble and not from the walls of the
I
100
VOLUME OF GAS EJECTED PER EXPLOSION (m3)
Figure 9. Volurnesof gas and magmaejectedduring
the 36 explosions.
Sensitivity of Results to Input Parameters
We havesofar keptthe liquiddensityPliqandviscosity/• constant,2700kg m-• and400Pa s, respectively.the air-magmainterface. However,if the magma above
If the magma layer abovethe large bubble containstiny
bubbles,its densitycandecrease
to about2000kg m-s
the bubble is still draining, the bottom of the bubbleis
still rising, thus increasingthe pressurein the bubble.
This could explain why the calculatedacousticpressure
is larger than the measuredone closeto the maximum
for a 30% vesicularity. We have verified that this decrease in density has only minor consequences
on our
results: the initial bubble radius Ro does not change radius(Figures5b and 7d).
As on any air-liquid interface, surface wavescan deand the bubblelength L increasesby only 30%, while
velop
at the gas-magmaand air-magmainterfaces.Bethe initial overpressure
decreases
by also 30%. If the
cause
the magma layer is thin, lessthan a few tens of
viscosityis below 1000 Pa s, 10 times the valuegener-
(Figure8c),ripplewaves,controlled
by surally adoptedat Stromboli[Blackburn
et al., 1976],re- centimeters
face
tension
[Lighthill,
1978],
are
more
likely
than
pure
sultsare independentof its exact value,simplybecause
the damping is small. Higher valuesof viscositycan gravity waves. They developeither in an antisymmetway (snakemode)or in a dispersive
be discarded
(seediscussion
by Vergniolle
et al., [this ric, nondispersive
symmetrical
way
(varicose
mode)[Taylor, 1959]. Beissue]).
causethe film is thinner, <_ 2 cm (Figure 8c), when
the bubble radius is maximum, these ripple waveswill
be stronger and therefore will more affect the magma
Discussion
layer closeto the maximum radius. The growth of these
waveswould take energyfrom the radial volume mode,
Differences
Between
Measurements
and Model
which could accountfor the exagerationin the calcuAlthough most waveformscan be reasonablywell re- lated negative acousticpressure.
produced(Figure5a), it is sometimes
difficultto fit the
Instabilities developingat the gas-magmaand airdetails of the first negative peak of acousticpressure magma interfacesmight also modify the sphericalge(Figure 5b) and consecutives
oscillations.Thesedis- ometry of the bubble cap. A Rayleigh-Taylor instacrepanciesbetween observedand synthetic waveforms bility developsalong a planar interface when two fluids of different densities are accelerated toward each
are probably related to our simplifyingassumptions.
The main assumptionof the model is that the bubble other [Chandrasekhar,
1961]. However,the stabilityof
cap is perfectly hemispherical, whatever the velocities a sphericalsurfacedependsnot just on its acceleration
and accelerations
are. Experiments[TaylorandDavies, but on its velocity. During bubble growth, from Rminto
1963]have shownthat bubblestend to becomespher- Rmax,the streamlinesdivergeand producea stabilizing
ical during expansionwhile, in contrast, they tend to effecton the innerinterface[Prosperetti,
1986;Leighton,
loosetheir symmetryduringcontraction.Consequently, 1994]. Viscosityalsoslowsdown the growth on instadeparture from sphericity would thereforebe expected bilities, but calculationsbecome exceedinglydifficult.
closeto the minimum radius, when the acousticpres- However, close to the maximum radius, the film of
sure is zero. This could explain the mild discrepan- magma is less than 2 cm and surface tension tends to
cies between observedand synthetic acousticpressure stabilize the growth of these instabilities. Thus deparduringthe first contraction(Figure5b). We havealso ture from perfect sphericity would be moderate, inducassumedthat the rising motion of the bubble stops at ing a equally moderate discrepancybetween observed
20,444
VERGNIOLLE AND BRANDEIS: STROMBOLIAN EXPLOSIONS,
and calculated acoustic pressure. The interaction of
all these factor is very complex, which prevents a real
quantitative assessmentof their effectson the shapeof
the bubble and therefore on the acousticpressure.
where•uis theliquidviscosity
and/• is theradialvelocity of the bubble-magmainterfacefor steadygrowth
[Sparks,
1978;Leighton,
199.4].Sincebubblegrowthis
due to its upward motion, R scalesas the rise speed,•.
1.5 rn s-•. For a magmaof 400 Pa s, the viscous
overpressurereachesat mosta valuearound3 x 10a Pa, far
Bursting
It is clear that our model fails to reproduce the observed acoustic pressurejust before the bubble radius
belowourestimates
(• 105Pa). Therefore
thismecha-
nism is probably not efficientat depth wherethe rising
velocity is steady.
When the bubble getscloserto the top of the magma
first cycle,wherehigherfrequencies
appear(Figures5
column,
its upward motion becomes unsteady, with
and8d). This is a strongindicationthat the bubblecap
large accelerationsaway from the interface when the
stops to oscillate and bursts at this moment.
Contrasting with what happensnear the maximum bubble is expanded and small accelerationstoward the
1994].
radius, both outward acceleration and inward velocity interfacewhenthe bubbleiscontracted[Leighton,
contribute to enhancethe developmentof the Rayleigh- This will create a significantoverpressure,difficult to
Taylor instabilitiesat the inner interface[Prosperertl, quantify, inside the bubble, as pressure and accelera1986; Lei#hton,1994], when the bubbleradiusis be- tion are directly related.
From experimental work, it has been suggestedthat
tweenR•q and Rmin.The alestabilization
of the spherthe
bubbles at Stromboli result from the coalescence at
ical shape is maximum near the point where the indepth
of a foam layer of small bubbles, from i mm to
ward motion is rapidly stoppedand reversed[Plesset
i
cm
[Jaupart
and Vergniolle,1988, 1989]. This proand Mitchell, 1956; Prosperetti,1986]. Althoughthis
cess
suddenly
releases
an excesspressurerelated to the
theory has been developed for small bubbles, experisurface
tension
of
each
bubble[Jaupartand Vergniolle,
ments have shownthat it applies also to large bubbles
1989].
At
the
estimated
depth of the magmachamwith strong accelerations,like •hose producedby unhas returned
to its minimum
value at the end of the
derwaterexplosions[Taylor and Dames,1963]. Since ber, a few hundredof meters[Gibertiet al., 1992],the
Strombolian bubbles undergo also large accelerations excesspressurein the new gas pocket is of the order
(Figure6c), we believethat the samemechanism
is at
work at Stromboll.
The exponentialgrowth of the Rayleigh-Taylorinstability will ultimately lead to the rupture of the magma
layer into fragments. Although the magma layer had a
small inward radial velocity, these dispersedfragments
are now subject only to the strong pressuredifference
between the volcanic gas and the atmosphere. They
will thereforebe propelledoutward(Figurelb). This
could be further amplified by the presenceof an extra
of the lithostaticpressure,
• 107Pa. This coalescence
can trigger the oscillationsin volume of the new gas
pocket. The superpositionof theseoscillationswith the
bubble rise is complex;thereforeit is difficult to quantify the overpressureat the surface from its value at
depth. Although, at this stage,we cannotdiscriminate
betweena deep and a shalloworigin of overpressure
in
the bubble when it reachesthe surface, our model is
able to reproducethe recordedacousticpressurewith
good accuracy.
pressuredueto the still risingbottomof the bubble(see
above).
Alternate
Models
Origin of Overpressure
This s•udy suggests•ha• S•rombolian bubbles are
overpressurizedwhen •hey break a• •he surface of •he
lava column. We have sugges[edabove four possible
sourcesfor •his overpressure.
Firs[, surfacelension is known •o allow excesspressure within bubbles. Surface•ensionbetweengas and
Although our model is satisfactoryin reproducing
acousticpressure,other mechanisms
can be envisaged.
Azbel[1981]hasproposedthat whenthe liquidlayer
reachesa critical thickness,it vibrates in resonancewith
the volumemodeof the bubblebeneath,whichtriggers
its rupture. For Strombolian bubbles, the radian frequency of a membrane of 2 cm in thicknessis around
magmais about0.4 N m-• [WalkerandMullins,1981] 0.3 tad s-•, muchbelowour acousticmeasurements.
showingthat this effect is small for metric bubblesand
Becauseresonanceis knownto concentrateenergyin a
very narrow frequencyrange, this mechanismdoesnot
When a bubble risesin a viscousliquid, the viscosity explain why a significantamount of energyis released
prevents the pressure in the bubble to adjust instan- at the measuredradianfrequency,•. 60 tad s-•. The
taneously to the decreasingexternal pressure. For a observationthat soundis producedwhen small bubbles
sphericalbubble growingin an infinite liquid, the over- burst at the surfaceof the oceanhas led Spiel [1992]
pressure APr between a bubble of radius R and the to suggestthat these bubbles behave like a Helmoltz
liquid directly above it is equal to
resonator,i.e., a rigid cavity with a small hole through
which the gas escapesand producessound. Applied to
4•u/•
metric bubbles,suchat Stromboli,this mechanismpreAPr- R
(18) dicts a internal overpressureof i Pa. Sincethe acoustic
can be ruled out.
VERGNIOLLE AND BRANDEIS: STROMBOLIANEXPLOSIONS,1
20,445
pressureis still above several pascalsat 250 m from the 1994]. The vent can be viewedas a pipe of length L,
source(Figure2), we feel that this mechanism
should containinga mixture of air and volcanicgas, driven at
be discarded.
one side by a "piston", i.e., a bubble at the surfaceof
magma, and with an open end at the other side. If
examplesuchas a air gun [Avediket al., 1993], can the velocity at the closedend is u, at the open end it
be envisaged.Recently,Buckinghamand Gavels[1996] becomes
u
Other mechanisms,which call for a deep source,for
have proposedthat the sound originatesfrom a deep
and explosivesource. The calculated power spectrum
matches the observedone with a reasonableaccuracy. where k is the wavenumber and c is the sound speed
However, this model predicts a conduit diameter 6 times [Landauand Lifshitz, 1987]. The amplificationcoeffilarger than the one observedat the surfaceand a pres- cient of the pipe C is equal to
sure excess30 times the lithostatic pressure. Such a
1
large overpressurecould be easily obtained by highly
c =
cos•(kL)
reacting gasspecies,but the chemistryand the amount
of gasespresent at Stromboli are far below the critical
amountfor explosivity(F. Le Guern,personalcommu- Calculationsshowthat during energeticoscillationsof
nication,1995). Althoughthe mechanism
responsible the main event, the velocity of the bubble cap is on the
for such a deep energetic sourcehas still to be dis- order of the velocity of gas and ejecta, showingthat
covered, this model may provide an alternative to our there is probably no significantamplificationbetween
model.
the bubble cap and the exit of the volcanic conduit.
Therefore the amplification coefficientC is of the order
vo-cos(kL)
Conclusion
(A1)
of 1.
Our model, based on physical phenomenathat have Appendix B' Sources of Damping
been observedin many laboratory experiments, provides a simpleexplanationfor the soundproduceddurHere we discussthe possiblesourcesof damping for
ing Strombolianexplosions.When a largebubble,com- a bubblevibrating in a liquid. We evaluatethe imporing from depth, reachesthe top of the magmacolumn, tance of each sourcesby measuringthe ratio between
internal overpressureforces its cap to inflate until it the dampingcoefficientand the angularfrequencyof osreaches a maximum value. Then it deflates back tocillations co. First, the bubble vibration can be damped
ward its initial value. Just before the minimum radius
by viscousforcesof the magma. The viscouscoefficient
is reached, the bubble breaks, probably by instability /•v (12) is equalto
developingon the thin magma layer.
From the fit between observed and synthetic waveforms of acoustic pressure,we predict that the bubble
plRe2q
radiusis of the orderof I m, in agrementwith observed
vent sizes. The length of the bubble is between sev- and is equal to 0.9 for a Strombolian magma with a
eral meters and a few tens of meters, typical of a well- viscosity of 400 Pa s. This is indeed small compared to
developedslugflow. The initial overpressure
in the bub- the measured value of •-, 60 rad s-t for co.
ble is •, 105Pa, muchhigherthan previous
estimates. Acousticdamping by radiation into the liquid finds
This overpressuremay arise from the coalescenceof a its origin in the compressibilityof liquid. For a spherical
foam layer at the top of the magmachamberor be gen- bubble with small oscillations,the coefficientof damperated by the influenceof the air-magmainterfaceon ing/•ac is equalto [Plessetand Prosperetti,1977;Prosthe approachingbubble. The radial velocitiespredicted perertl,1986]
by our model are comparable to independentmeasurements of both gas and ejecta. Since the thicknessof
the magma layer is constrainedby observationsof sizes
of ejecta which vary in a narrowrange,we believethat where Cl is the soundspeedin magma slightly aboveto
our predictions are robust. Measurements of acoustic 2500m s-t [Riversand Carmichael,
1987;Kressand
pressurecan therefore provide valuable informations on
Carmichael,1991] for the shoshonitecompositionat
the physicsof volcanicactivity.
Stromboli[Capaldiet al., 1978]. For radian frequen-
/•v- 6•u
(B1)
flac-Reqco2
2el
(92)
ciesaround60 rad s-t, as measuredfor Strombolian
Appendix A' Amplification by a Tube
We discussthe amplification of the sound which can
result of superimposinga tube full of air above the
source. It is the casefor the eastern vents, even if the
explosions,the damping coefficientfiac is equal to 1,
again small comparedto co.
Third is the thermal damping inside the gas. The
coefficientof thermal damping/•th for small oscillations
[Prosperertl,1986]dependson the thermaldiffusivityof
lava is closeto the surface[Vergniolleand Brandeis, gasXg and on the ratio of specificheat '7
20,446
VERGNIOLLE
AND BRANDEIS: STROMBOLIAN
]•th 9'7(71)pg
eq Xg
-- 2plRe3q
(2--•2)
•/2
(B3)
EXPLOSIONS, 1
Francalanci, L., P. Manetti, and A. Pecerillo, Volcanological and magmatological evolution.of Stromboil volcano
(Aeolian islands): The rolesof fractionalcrystallization,
magma mixing, crustal contamination and sourceheteroTakinga diffusivity
equalto 8 x 10-7 m2s-• [Sparks, geneity, Bull. Volcanol., 5I, 335-378, 1989.
1978]and7 equalto 1.1for hotgases
[Lighthill,
1978], Giberti, G., C. Jaupart, and G. Sartoris, Steady-state opwe obtaina dampingc6efiicient/3th,equalto 7 x 10-s
eration of Stromboil volcano, Italy: Constraints on the
feeding system, Bull. Volcanol.,54, 535-541, 1992.
for a bubble of 1 m, exceedinglysmall comparedto w.
Finally, for a bubble closeto solid walls, like a Strom- Jaupart, C., and S. Vergniolle, Dynamics of degassingat
Kilauea volcano, Hawaii, Nature, 331, 58-60, 1988.
bolian bubble vibrating •in a tube, damping by radia- Jaupart, C., and S. Vergniolle, The generationand collapse
tion of seismicwaves into the ground is also possible.
of a foam layer at the roof of a basaltic magma chamber,
Measurementssuggestthat seismicwavesare much less
J. Fluid Mechanics, 203, 347-380, 1989.
energeticthan acousticwavesproducedin air [Braun Kay, J. M., and R. M. Nedderman, Fluid Mechanics and
TransferProcesses,
602 pp., CambridgeUniv. Press,New
and Ripepe,1993]and hencehavebeenneglectedin our
York, 1985.
model. In summary, amongthe different damping coef- Kress, V. C., and I. S. E. Carmichael, The compressibility
ficients, the largest is the viscousdamping.
of silicateliquidscontaining
Fe203 andthe effectof comAcknowledgments.
We thank J.-C. Mareschal, J.-L.
Chemin•e, X. Hill, A. Davaille, A. Woods, V. Ferrazini, P.
Allard, J. D•verch[re, Y. Gaudemer, M.-F. Le Cloarec, F.
Le Guern, D. Snyder, T. Trull. Thorough reviews by R. S.
J. Sparks, M. J. Buckingham, and T. Koyaguchigreatly improved the manuscript. This work was supported by CNRS-
INSU-DBT Terre profondeand EEC grantsSCl*CT005010
and EV5V-CT92-0189.
INSU-DBT
contribution
This is IPGP
764.
Contribution
1441 and
References
Allard, P., J. Carbonelle, N. Metrich, and H. Loyer, Sulphur output and magma degassingbudget of Stromboil
volcano, Nature, 368, 326-330, 1994.
Avedik, F., V. Renard, J.-P. Allenou and B. Morvan, "Single-bubble" air-gun array for deep exploration, Geophysics, 58, 366-382, 1993.
Azbel, D., Two-Phase Flow in Chemical Engineering, 311
pp., Cambridge Univ. Press, New York, 1981.
Batchelor, G. K., An Introduction to Fluid Dynamics, 615
pp., Cambridge Univ. Press, New York, 1967.
Blackburn, E. A., L. Wilson, and R. S. J Sparks, Mechanics and dynamics of Strombolian activity, J. Geol. Soc.
London, I32, 429-440, 1976.
Boulton-Stone, J. M., and J. R. Blake, Gas bubblesbursting
at a free surface, J. Fluid Mech., 99, 9415-9422, 1993.
Braun, T., and M. Ripepe, Interaction of seismic and air
waves recorded at Stromboil volcano, Geophys. Res.
Lett., œ0,65-68, 1993.
Buckingham, M. J., and M. A. Garc•s, Canonical model
of volcano acoustics, J. Geophys. Res., 101, 8129-8151,
1996.
Butterworth, D., and G. Hewitt, Two-Phase Flow and Heat
Transfer, 514 pp., Oxford Univ. Press, New York, 1977.
Capaldi, G., et al., Stromboil and its 1975 eruption, Bull.
Volcanol., dl, 1-27, 1978.
Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, 654 pp., Dover, New York, 1961.
Chouet, B., N. Hamisevicz, and T. R. McGetchin, Photobalilstics of volcanic jet activity at Stromboil, Italy, J. Geophys. Res., 79, 4961-4975, 1974.
Cole, R. H., Underwater Explosions, 432 pp., Princeton
Univ. Press, Princeton, N.J., 1948.
Davies, R. M., and G.I. Taylor, The mechanicsof large bubbles rising through extended liquids and through liquids
in tube, Proc. R. Soc. London A, œ00,375-390, 1950.
Drazin, P. G., Nonlinear Systems,317 pp., Cambridge Univ.
Press, New York, 1992.
position, temperature, oxygen fugacity, and pressure on
their redox state, Contrib. Mineral. Petrol., 108, 82-92,
1991.
Landau, L. D., and E. M. Lifshitz, Course in Theoretical
Physics: Fluids Mechanics,vol. 6, 536 pp., Pergamon,
Tarrytown, N.Y., 1987.
Leighton, T. G., The Acoustic Bubble,613 pp., Academic,
San Diego, Calif.,1994.
Lighthill, J., Waves in Fluids, 504 pp., Cambridge Univ.
Press, New York, 1978.
Longuet-Higghins,M., Bubble noisespectra, J. Acoust. Soc.
Am., 87, 652-661, 1990.
Lu, N. Q., H. N. Oguz, and A. Prosperetti, The oscillations
of a small floating bubble, Phys. Fluids A, 1, 252-260,
1989.
Pierce, A.D., Acoustics: An Introduction to its Physical
Principles and Applications, 678 pp., McGraw-Hill, New
York, 1981.
Piesset, M. S., and T. P. Mitchell, On the stability of the
spherical shape of a vapor cavity in liquid, Quat. Appl.
Math., 13, 419-430, 1956.
Plesset, M. S., and A. Prosperetti, Bubble dynamics and
cavitation, Annu. Rev. Fluid Mech., 9, 145-185, 1977.
Prosperetti, A., Bubble dynamics, Proc. Int. School Phys.
E. Fermi, 93, 145-188, 1986.
Richards, A. F., Volcanic sounds,investigationand analysis,
J. Geophys. Res., 68, 919-928, 1963.
Ripepe, M., M. Rossi, and G. Saccarotti, Image processing
of explosiveactivity at Stromboil, J. Volcanol. Geotherm.
Res., 5d, 335-351, 1993.
Rivers, M. L., and I. S. E. Carmichael, Ultrasonic studies of
silicate melts, J. Geophys. Res., 92, 9247-9270, 1987.
Ryerson, F. J., H. C. Weed and A. J. Piwinskii, Rheology
of subliquidus magmas, I, Picritic compositions,J. Geophys. Res., 93, 3421-3436, 1988.
Shaw, H. R., Rheology of basalt in the melting range, J.
Petrol., 10, 510-535, 1969.
Shaw, H. R., Viscosities of magmatic silicate liquids: An
empirical method of prediction, Am. J. Sci., 272, 870893, 1972.
Sparks, R. S. J., The dynamics of bubble formation and
growth in magmas: A review and an analysis, J. Volcanol.
Geotherm. Res., 3, 1-37, 1978.
Spiel, D. E., Acoustical measurementsof air bubbles bursting at a water surface: Bursting bubbles as Helmholtz
resonators, J. Geophys. Res., 97, 11443-11452, 1992.
Taylor, G.I., The dynamics of thin sheet of fluids, II, Waves
on fluid sheets, Proc. R. Soc. London A, 253, 289-312,
1959.
Taylor, G.I., The vertical motion of a spherical bubble and
the pressure surrounding it, in The Scientific Papers of
VERGNIOLLE
AND BRANDEIS:
G.L Taylor, vol. 3, pp. 320-336, Cambridge Univ. Press,
New York, 1963.
Taylor, G.I., and R. M. Davies, The motion and shape of
the hollow produced by an explosionin a liquid, in The
Scientific Papers of G.I. Taylor, vol. 3, pp. 337-353,
Cambridge Univ. Press, New York, 1963.
Thorpe, S. A., Bubbles and breaking waves, Nature, 283,
463-465, 1980.
STROMBOLIAN
EXPLOSIONS,
I
20,447
Webb, S. L., and D. B. Dingwell, Non-Newtonian rheologyof
igneousmelt at high stressesand strain rates: Experimental resttits for rhyohte, andesite, basalt, and nephehne,
J. Geophys. Res., 95, 15695-15701, 1990.
WeiH, A., G. Brandeis, S. VergnioHe,F. Baudin, J. Bilbille,
J.-F. F[vre, B. Piron, and X. Hill, Acoustic sounder measurementsof the vertical velocity of volcanicjets at Stromboli volcano, Geophys. Res. Lett., 19, 2357-2360, 1992.
Wilson, L., Relationshipsbetween pressure,volatile content
and ejecta velocity, J. Volcanol. Geotherrn. Res., 8, 297-
van der Molen, I., and M. S. Paterson, Experimental deformation of partially melted granite, Contrib. Mineral.
Petrol., 70, 299-318, 1979.
313, 1980.
VergnioHe,S., and G. Brandeis,Origin of the soundgener- Woulff, G., and T. R. McGetchin, Acoustic noise from volated by Strombolian explosions,Geophys.Res. Lett., 21,
canoes: Theory and experiments, Geophys. J. R. A s1959-1962, 1994.
tron. Soc., 45, 601-616, 1976.
Vergniolle,S., and C. Jaupart, Separatedtwo-phaseflow
and basaltic eruptions, J. Geophys.Res., 91, 12842-12860,
1986.
VergnioHe, S., and C. Jaupart, Dynamics of degassingat
Kilauea volcano, Hawaii, J. Geophys.Res., 95, 2793-2809,
1990.
Vergniolle, S., G. Brandeis, and J.-C. Mareschal, Strombolian explosions, 2. Eruption dynamics determined from
acoustic measurements, J. Geophys. Res., this issue.
Walker, D., and O. Mullins Jr., Surface tension of natural silicates from 1200-1500øC and implications for melt
structure, Contrib. Mineral. Petrol., 76, 455-462, 1981.
Wallis, G. B., One Dimensional Two-Phase Flows, 408 pp.,
McGraw-Hill, New-York, 1969.
G. Brandeis, UMR 5562/CNRS, GRGS, Observatoire
Midi-Pyrdn•es, 14 avenue Edouard-Belin, 31400 Toulouse,
France.(e-mail:[email protected])
S. Vergniolle, Laboratoire de Dynamique des Syst•mes
Gdologiques,Institut de Physique du Globe de Paris, 4
place Jussieu, 75252 Paris cedex 05, France. (e-mail:
[email protected])
(ReceivedJune 13, 1995;revisedApril 1, 1996;
acceptedApril 9, 1996.)