JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. B10, PAGES 24,283-24,299,OCTOBER 10, 1998
Bubble
collapse
as the source
of tremor
at Old
Faithful
Geyser
Sharon Kedar
and Hiroo Kanamori
SeismologicalLaboratory, California Institute of Technology,Pasadena
Bradford
S•ur•evan•
Graduate Aeronautics Laboratories, California Institute of Technology,Pasadena
Abstract.
Old Faithful Geyser, Yellowstone, was used as a natural laboratory
for fluid-flow-inducedseismicactivity. Pressuremeasurementswithin the geyser's
water column, obtained simultaneouslywith seismicmeasurementson the surface,
demonstrated that the tremor observed at Old Faithful results from impulsive
eventsin the geyser. Tremor intensity is controlledby the rate of occurrenceof
these impulsive events. There is no resonanceobservedwithin the water column.
The impulsive events are modeled by a collapseof a sphericalbubble, including
the effectsof residual non-condensible
g•s •nd d•mping. The pressured•t• c•n be
explained by a collapseof a 0•5 cm radius bubble driven by a pressuredifference
of AP = 0.3 x 10s Pa andeffectiveviscosity•,•=0.04 m2/s. Usinga quasi-static
geysermodel, we treat the individual bubble collapsesas coolingeventsthat occur
when the water column reachesa critical temperature. Their rate of occurrenceis
controlledby the heatingrate dT/dt of the water column.As a result,the intensity
of the hydrothermal and seismicactivities is determinedby the heat and massinput
rate into the geyser. It is demonstratedthat a sharp widening of the conduit can
causethe numberof eventsper unit time to drop (as observed)while the water
level is still rising and heat is being input, and thus the tremor intensity can be
modulated by variations in the conduit shape.
1.
Introduction
thermodynamics.Kieffer [1984]first 'pointedout the
similaritiesbetweengeyserseismicityand volcanicseisOld Faithful is probably the most studied geyserin
the world. Located in the Upper Geyser Basin of Yel- micity and the possiblerelevanceof geyserstudiesto the
interpretation of volcanic tremor, the nonearthquake
lowstoneNational Park, its surfaceexpressionis a 4 m
signalswhich precedeand accompanyvolcanicactivity.
high, 60 m wide mound with an approximately2 mx 1
Thesestudiesprovideda goodoverallunderstanding
m opening at the top. The conduit extendsdownward,
of the geyser'sbehavior. However,sinceall past studies
successively
narrowing and openinginto larger spaces, used seismometers without simultaneous time-resolved
as describedby Birch and Kennedy[1972]and as in-
ferred from the spectacularvideo recordingsmade by
Hutchinsonet al. [1997].The geyser'seruptionsare 25 min long with intervalsbetweenthem rangingfrom 30
to 100 min, and their time of occurrenceis predictable
to within
15 min.
measurementsof pressureand temperature, the source
of tremor
and its interaction
with the solid medium
at
Old Faithful were not fully understood.
Our work at Old Faithful was designedto establisha
cause-and-effectrelationshipbetweenthe sourceof seismic noise and the observed tremor.
We have carried
J. Rinehart was the first to deploy seismometers
around the geyser, as well as to measuretemperatures out three seismicstudiesat Old Faithful Geyser(field
inside it [Rinehart, 1965, 1967, 1980]. temperature seasons1991, 1992, and 1994), in whichtime-resolved
inside it.- Birch and Kennedy[1972] continuedwith pressurevariationsinsidethe geyser,passiveseismicity
temperature measurements at different depths in the
around the geyser, and seismicresponseto an external sledgehammer sourcehave been measured. Kedar
geyser,and Kieffer [1984]gave an elaboratedescription of the geyser'sbehaviorincludingits seismicityand et al. [1996]showedthat the seismicactivity between
eruptionsat Old Faithful Geyseris composedof a superpositionof discreteeventsoriginatedby pressurepulses
Copyright 1998 by the American GeophysicalUnion.
insideOld Faithful, in analogyto volcaniclong-period
Paper number 98JB01824.
(LP) events,whichare considered
the buildingblocksof
0148-0227/98
/ 98JB-01824509.00
continuousvolcanictremor. The seismicobservations,
24,283
24,284
KEDAR ET AL.: BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC TREMOR
110ø49'35"
110ø49'40"
44ø27'35
-
44ø27'30
1994 L-28
1994100Hz24-chan•l
Vertical
Geophone
line
1992GuralpCMG-3ESP
1992 Vertical 1Hz Geophone
1992HammerShot
Bench Mark
Figure 1. A map of the surveyarea and instrumentsetupsfor the 1992 and 1994 deployments
in relation to Old Faithful and two of its neighboring passivedomes. The orientation of the
geyser'sedificeis indicatedby the dashedline acrossits dome.
ond for periods of a day at a time. In addition, an
instruments,wereusedby Kedaret al. [1996]to demon- arrayof 96 short-period(naturalfrequency1 Hz) verstrate that the harmonic characteristics of the seismictical geophoneswere placed in a tight grid over the
ity could result from reverberationsin a near-surface geyser'sdome. The third field experiment(in Octodeployment
of two
soft layer. This paper is a continuationof the paper ber 1994)involvedthe simultaneous
three-component
broadband
sensors,
three
L-28
threeby Kedar et al. [1996]and concentrates
on the physical
componentshort-periodsensors(natural frequency4.5
processeswithin Old Faithful's water column.
Hz), and a probedesignedto measurepressure
inside
which were made with both broadband and short-period
2.
the water column. All three sensortypes were record-
Data
2.1.
Fieldwork
ing simultaneously
and continuouslyat 250 samplesper
and Setup
Three scientific excursions to Old Faithful Geyser
weremade(Figure1). Thefirst(notshownin Figure1)
second.
2.2.
Eruption
Cycle
Figure 2 displays6-hour-longunfilteredverticaland
sensors(GURALP CMG-3ESP, fiat velocityresponse north-southvelocityrecordsat station WY000 (Fig0.0333-50Itz) and REF-TEK six-channel
portabledata ure 1). Theserecordsincludeseveraleruptioncycles
loggerswith 16-bit digitizers. This servedas a pilot (definedas the time intervalbetweenthe end of one
study for a secondand a more elaborate seismicsur- eruptionto the endof the next). The generalpatternof
vey in October 1992, which incorporatedbroadband the eruptioncyclestartswith a quietperiod,of 1/2 hour
recordingsof both geyser-generated
signalsand sledge duration. Seismicitythen increasesin intensityand amhammer pulses. This survey includedsix broadband plitude and then gradually decaysuntil the final mosetups identical to the 1991 instruments. These were mentsbeforean eruption.A short(~ 50 min) eruption
placedaround the geyserin differentconfigurationsfor cycle showsa similar behaviorwithout the initial pecontinuousrecordingsat 100 and 200 samplesper sec- riod of seismicsilence.Kieffer [1984]explainedthe amwas carried out in 1991 and deployedtwo broadband
KEDAR ET AL.' BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC TREMOR
WYOO0
24,285
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for 6 hoursat stationWY000. Eruptionsare
marked by E. Six 1-min time windowsmarked by a - f are displayedin Figures 3.
plitudedecayby an increased
acousticimpedance
mis- these discreteevents as well as their amplitude and that
the pulsesget strongerand more frequent as the eruption cycleprogresses.As shownin Figure 3, in the early
in the final stagesof the eruption cycle. Although the stagesof the eruption cycle infrequent low-amplitude
interval between eruptions fluctuates over a period of pulsesare observed. The pulsesget stronger and more
several years, it is known to have a bimodal distribu- frequent until their amplitude and rate becomesteady.
tion with peaksat ~ 50 min and ~ 75 min [Rinehart, Minutes before the eruption the amplitude decreases
due to the poor acousticimpedancematch between the
1980].
2.2.1. Discrete events. Figure 3 demonstrates bubblyfluid and the geyserwall [Kieffer, 1984],while
that the seismic activity depicted in Figure 2 is com- the rate stays fairly constant.
2.2.2. Event rate. A quantitative analysisof the
posed of discrete events with a sharp onset and a characteristic decay time. It is evident that the intensity of numberof eventsper minute (Figure 4) illustratesthe
the activity is dependent on the rate of occurrenceof overall seismic behavior described above. The events
match between a two-phase water-steam mixture and
the conduit
walls due to an increased
amount
of steam
24,286
KEDAR ET AL ß BUBBLE COLLAPSEAS THE SOURCE OF HARMONIC TREMOR
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were countedby a computerizedcounterdesignedto de- for another 30-40 min. We will describe a model which
tect eventsenergeticenoughto appear distinctly above accountsfor the generalbehaviorobserved,namely,the
the noiselevel (signalto noise~10). The eruptionit- initial rise in event rate, the suddendecrease,and the
selfgeneratesa distinctbut weakseismicsignalas can final plateau as depictedby dashedlines in Figure 4.
2.2.3. The eruption. Although we will focuson
be observedin Figures2 and 5. The eruptionsignalis
characterizedby a 1-3 min long codageneratedmainly the activity betweeneruptionsin this paper, here we
briefly describethe eruptionsthemselves.The seismic
by the ejectedwater falling on the ground.
It is apparentfrom Figure4 that the numberof events signalsfrom severaleruptionsare presentedin Figure5.
per minute increasesover a time scaleof about 20 min, An eruption occurswhen a massof water is removed
then decreases
suddenly,then increasesagainto plateau temporarily from the top of the superheatedcolumn,
KEDAR ET AL.' BUBBLE COLLAPSE
AS THE SOURCE OF HARMONIC
E
E
n.
,J
E
,
i
0
TREMOR
24,287
E
. , Ja
.
i
2
4
6
Timo
Figure 4. Eventrate at WY000inferredfroma 12-hourlongrecord.The generaltrendto be
modeledis outlinedby a dottedline. Eruptionsaremarkedby "E." The dropin eventrate toward
the end of the cycleis an artifact of the automatedeventcounter,becausethe signal-to-noise
ratio decreasesas the eruption nears.
10 minute•
Figure 5. A 10-min windowof the vertical componentat WY000 aroundnine eruptions. In some
cases(fourout of the nineeruptions)thereis a distincteventwhichseemsto initiatethe eruption.
The significanceof this observationis debatable,as it appearsonly for someeruptions. There is
no other signal which may be used as a prediction tool, other than the overall reduction in the
backgroundnoiseamplitudein the minutesprior to an eruption. In most cases,the eruption's
seismicsignal emergesfrom the backgroundnoisewith no clear precursor.
24,288
KEDAR ET AL.: BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC TREMOR
a
b
Depth
Steam
slug
h ,.
T(h•
p(O)
p(h)
Temperature
T(h•
Temperature
II pressure
I
beloV•
h p(O)l•*..
I, ,.,o,,,ered
_p(h•..
_•'_-•___
Pressure
.
Pressur'
e
Figure 6. A schematicillustrationof the processof instantaneous
boilingof a water columnas
a result of a formation of a steamslug. As a slugof steamis formedinsidethe pipe, the water is
pushedup into the tank. Sincethe massof the steamslugis negligible,mostof the water which
occupiedthe volume(now occupiedby vapor)is spreadovera widerspace,thus loweringthe
hydrostaticpressurein the fluid below the slug at depth h. If the fluid is near boiling, a slight
reductionof the hydrostaticpressurewill causeinstantaneousboiling of the fluid belowthe slug.
It shouldbe pointedout that the samephenomenontakesplacefor any shapeof wideningpipe
(suchas a funnel).
causinga sufficientdrop in the hydrostatic head to resuit in instantaneousboiling of the entire column and
subsequentevacuation of the conduit in the form of an
eruption. Two main mechanismsmay be responsible
for the reduction in the hydrostatic head; a temporary
displacement of a mass of water by an explosive process,or a mass of water pushed into a larger diameter
space at the top of the water column due to volume
2.3.
Pressure
Measurements
Based on the observations from 1991 and 1992, a
probewas designedand built to measurepressureinside
the water column simultaneouslywith seismicmeasurements on the geyser'sdome. The instrument consisted
of three sensors3 m apart on a coaxial cable, each containing a KULITE XTC-190, high-frequencypressure
change.The former is knownas "preplay"[Rinehart, transducer(Figure7). In October1994the instrument
1980],a processobserved5-10 rain beforean eruption was lowered into the geyser'sconduit during the quiet
at Old Faithful, which starts like a normal eruption that
period in a long eruption cycle until the bottom sensor
immediatelydies as the displacedwater falls back into was 22 m deep and was below 7 m of water.
the conduit. The latter occurs when a slug of steam
Figure 8 displays the ,,30-min-long pressure data
in a conduit pipe displacesa volume of water upwards recorded inside the conduit of Old Faithful. A ques-
into a largerdiameterspace(Figure6), thuslowering tion could be raised about the possibledepth drift of
the hydrostatic pressurein the fluid below the slug and
causinginstantaneous
boiling[Griffith, 1962].
the sensorsin the turbulent water column. From Figure 8, it is apparent that fluctuations in the hydrostatic
KEDAR ET AL.: BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC TREMOR
24,289
b
12c
Temperature
Pressure
ß
Figure 7. (a) The threepressure
sensors
layingon sinterdepositsat the foot of Old Faithful's
dome.(b) A schematic
diagramof the interiorof the containers.The thermocouple
recordwas
too noisy to be used for analysis.
pressuremeasuredby the pressuretransducersare observed. However, the overall trend of hydrostatic head
increaseis consistentwith Old Faithful's filling history
taking place within the water column and the seismic
signal that follows it. Therefore the seismicactivity's
ing on the short-period discreteevents that compose
the pressureactivity, any possibledepth fluctuations
becomenegligible.
The malfunctioningof the bottom transduceris attributed to water penetratinginto its container;thus its
recordwill be ignored. However,after a periodof thermal adjustment,the top and middle transducersdisplay
vidual pressurepulses. These pressurepulsesare generated near the top of the water column and are strongly
attenuated within it, while no pressure reverberations
intensity (i.e., the numberof seismiceventsper unit
[tttt•chinsone! al., 1997],and sincewe will be focus- time) is determinedby the rate of occurrenceof indi-
are observed
within
the water
column.
This eliminates
any notion of "organ-pipe" type resonancein the water column
as the source of tremor
in the case of Old
Faithful Geyser.
a behavior consistent with their relative locations in the
water and a steady rise in the pressureover a period of
30 minutesas the geyseris filling.
3.
The
Source
Figure 9 displays•
a 2-s-longpressureand seismic In section 2 it was established that tremor at Old
records(stationBD60A) duringwhicha singlepressure Faithful originates from sharp pressure pulses within
pulsein the water is followed•0.1 s later by a seismic the water column. In order to gain further physicalunsignal. This observation(whichis representative
of all derstandingof the nature of these processes,two probthe geyser-generated
activity at Old Faithful Geyseras lems are posed:
1. What arethe characteristics
(i.e., shape,timescale,
wasdemonstrated
by Kedare! al. [1996])establishes
a
cause-and-effectrelationship between a pressurepulse and amplitude)of the impulsivesource?
24,290
KEDAR
ET AL.-
BUBBLE
COLLAPSE
AS THE
SOURCE
OF HARMONIC
TREMOR
0.5
-0.5
I
0
6oo
i
1200
1800
Time (s)
Figure 8. The long-periodbehaviorof the pressuremeasuredinsidethe conduit. The top and
middletransducers
are 3 m apart; thusthe top transducer
is measuring
a -,,3.5x104Pa lower
pressurethan the middle transducer. The bottom transduceris malfunctioningdue to water
leaking into the containerand is not shownhere. The sensorswere put in immediatelyafter a
long eruption and were taken out ,-,40 min later, prior to the followingeruption. The first 30
min are presentedhere, until the middle transducermalfunctioned. After about 5 min of drift
due to thermal effects,the pressuresensorsstablize. The zero pressurevalue is with referenceto
the middle
transducer.
2. What controls the rate of occurrence of pulses?
The rate of occurrence of pulses determines whether
continuousor impulsive tremor is excited. The filling
rates and heating rates, the interaction betweenthem,
and their effect on the rate of pressurepulse excitation
will be discussed.
3.1.
Source of Impulses
A commonsourceof repetitive pressureimpulsesin a
ating a pulse. The collapsetime rc is measuredfrom
the moment the pressurestarts increasingto the time
of the pressurepeak as is demonstratedin Figure 10.
The following discussionwill concentrateon attempting to model the recorded pressurepulsesas a bubble
collapse.
3.1.1. Rayleigh collapse. The simplestcollapse
modelwascalculatedby Rayleigh[1917],whoestimated
the collapsetime of an empty sphericalcavity in a body
columnof boilingwater is bubblecollapse[Blake,1986; of incompressiblefluid with constantpressurePooat inThe equation of motion in spherical coordinates
Boureet al., 1972],aswassuggested
by Kieffer[1984] finity.
is then
and Kedar et aL [1996]in the contextof Old Faithful. Bubblecollapsewasalsosuggested
as the sourceof
ß.
_
Poo zxP
volcanictremor by Leer[1988]. When a steambubble
P
P
rises to some cooler, lower-pressurezone in the water
column,the steam vapor condenses
and the bubble col- where R is the bubble radius, p is the fluid density,
lapses,generatinga pressurepulsewith a characteristic and/•- dR/dt.Fora spherical
voidthewallpressure
RR
+•(/•)•-
collapse time re. Sometimes a rebound or several re-
boundsresultfrom an incompletecollapse
of the bubble,
as is shownin Figure 10. This is frequentlyobserved
in the laboratory[Blake,1986].Similarly,
whena large
P(R)-
=
(1)
0 and the collapsetime is
rc- 0.915Rov/P/P•
bubble of the order of the size of the conduit bursts
whereR0 istheinitial (maximum)bubbleradius.Choos-
at the water surface, the displacedmassof water will
ing the driving pressurePooto be atmosphericpressure,
drop backon the emptiedcavity (a processsometimes 10• Pa, andthe densityof water,p=1000kg/m3, gives
referredto as "chugging"
[Boureet al., 1972]),generr for the averageobservedcollapsetime of rc=0.05-F0.02
KEDAR ET AL.' BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC
TREMOR
24,291
TP.
ressure
', Tseismic
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Top
Middle
Bottom
0.7
0.35
0.0
-0.35
0.7
0.35
0.0
-0.35
0.7
0.35
0.0
-0.35
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Vertical
O. 00
--0.05
--0.10
0.05
Radial
0.00
--0.05
--0.10
0.05
Tangential
O. O0
--0.05
--0.10
o .0
0.5
1.5
2.0
Time (s)
Figure 9. A simultaneous
recordof pressureand seismicwavesshowsa distinctcause-andeffectrelationshipbetweenthe impulsivepressuresourceand the impulseresponseof the rock
surrounding
the watercolumn[alsoseeKedaret al., 1996].In addition,the pressure
pulse,which
isstrongest
at the top transducer,
stronglyattenuates
downward,
whilenopressure
reverberations
are apparent within the water column.
s a bubble radius of/•0 ~0.5 m. This result is clearly
not applicableto Old Faithful sincefor the most part
the geyserconduitis muchnarrowerthan 1 m, and at
its narrowestpoint, it is only ~0.1 m wide. Therefore
a morecomplicatedmodelneedsto be explored,which
includesprocesses
that yield smallerbubbleradii from
the observedpressureamplitudes.
3.1.2. The role of noncondensible gases and
damping. The Rayleighcollapseis an idealizedcase.
In reality, noncondensible
gas such as CO2 in water,
whichis trapped insidethe bubble,is actingas a spring
asthe gasis compressed
duringthe collapse
phase.As
can be seenfrom Figure 10, two to three collapsecycles are commonly observedin the laboratory as well
as at Old Faithful. As is also observed,damping plays
an important role in the collapseprocessand is incorporated into the calculationthrough an additional viscous
term. The abovephysicalprocesses
are describedby the
Rayleigh-Plesset
equation[Piesset,1949]
•/•+3•(/•)2
_7 P"+
Pa0 - Poo
]- -•-•
When •, - 0, equation(3) resultsin nonlinearoscillations,while in the limit of smalloscillations,
it yields
linear harmonic
p
oscillations:
Ro
When Pv - Py0= 0 and •, = 0, equation(3) yields
Rayleighcollapse(equation(1)). For water at ~100
øC,• = 10-6 m2/s,resulting
in a negligible
damping
term, sincethe minimumradiusis neversmallenough
to increasethe dampingforceto the magnitudeof the
pressureor inertia forces. Nonetheless,dampingdoes
occur,and someother dampingmechanismneedsto be
pursued.
The calculation
basedonequation(3) isshownin Figure 11, wherean effective
viscosity
•,E=0.04m2/s and
an isothermalprocess
(7- 1) yielda reasonable
bubble
radius(5.5 cm) and a goodfit to the datafrom the onset of collapse.(It shouldbe notedthat using? = 1.4
doesnot significantly
changethe bubbledimensions.)
The modelsuccessfully
describes
the generalcharacteristics of the process,namely, the period of oscillation
(3) and the overall damping. However, the effectiveviswhere Pv is the vapor pressureof condensiblegasesin- cosityis much higher than the viscosityof the water,
side the bubble, Py0 is the initial noncondensible
gas implying that someother mechanismis responsiblefor
pressure,? is the specificheatsratio, and • is the kine- the damping.
matic viscosityof the surroundingfluid. The last term
As was shownby Chapmanand Plesset[1971], at
on the right-hand side is the dampingterm.
largebubbleradii, the effectiveviscosityis significantly
24,292
KEDAR ET AL ß BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC TREMOR
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V O LUM
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'• c =0.05s
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x
•
oo
1101
1102
1103
11;04__ ' •
t105
m 0.0
1100
Middle
1101
1102
1103
1104
1105
Time (s)
Figure 10. (a) Characteristics
of bubblecollapse.Comparison
betweena laboratorymeasurement[Blake,198•]and (b) the pressure
pulse.The collapse
time vcis measured
from the moment
the pressurestarts increasingto the time of the pressurepeak.
larger than the liquid viscositydue to energy lost by
[ChapmanandPlesset,1971],andfor smalloscillations,
acoustic radiation
it is
and thermal
diffusion.
v• = vt + VA+ vv
Therefore
(5)
where vz is the effective viscosity and vt, VA, and vT
are the liquid, acoustic,and thermal viscosities,respectively. It follows that at radii of a few centimeters
for water-air mixtures the acousticviscositydominates
=
4ct
(6)
where f is the bubble oscillationfrequencyand ct is the
speedof soundin the liquid. For the observedfrequency
of f - 1/rc ~20 Hz with c•=1500 m/s and a collapse
radiusof 0.05 m, VA~ vZ ~ 10-5 m2/s, 3 ordersof
KEDAR ET AL.: BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC TREMOR
24,293
transducers
is ~0.002 s+0.002s (i.e., onesampleinter-
0.3
val), whichmay constrainthe speedof soundin the
two-phasewater-steamfluid in the geyserto a lower
bound of about 1500 m/s, the speedof soundin water. In this case,the main sourceof dissipationwould
be scatterby irregularitiesin the conduitwallsrather
than the fluid compressibility.
Figure 12 presentsthe time recordas well as the
corresponding
Fourierspectrafor 5 s of pressuredata
recordedat the top and middle pressuretransducers.
I
From the time record,it is clearthat very little disperi
sion, if any, is observedas the acousticsignal prop0.0 , I , I , I , 0.00
f , 0.05
I , 0.10
I , I ,
agatesthroughthe water. However,the dissipation
Time (s)
is frequencydependent. As Figure 12 demonstrates,
('>10 Hz) loseenergywhilepropagatFigure 11. Timeseriesof a collapse
of a bubblewith highfrequencies
compared
to
an initial radius /•0=5.5 cm, internalpressureP•0 = ing from the top to the middletransducer
0.2x 105Pa,witha drivingpressure
AP -- 0.3x 105Pa lowerfrequencies.A mechanismthat might have the
•.0.1
and z/z=0.04m2/s. The drivingpressure
is equivalent above characteristics is the effect of viscous forces felt
to the weightof 3 m of water.Notethat thezerolevel by a cloudof bubblesundercompression.
In this case,
was chosenarbitrarily, sinceonly the collapseprocess the fluid viscositywill play an important role as liq(nottheinflation)is illustrated
here.
uid is squeezedbetweenthe bubblesduringexpansion
and compression
of the cloud. For a speedof soundof
~20 m/s, the wavelengthat 20 Hz is ~1 m. For pure
magnitude
smallerthantheviscosity
usedin Figure11. water at i Hz, the wavelengthis 1500 m. If the bubIf the regionof collapse
contains
10-3 massfraction ble cloud is near the top of the water column at the
of steam,cl canbe aslow as 20 m/s [tfieffer,1977], time of collapse,then wavelengthsof the order of the
andthenyz ~ 10-3 m2/s,closerto the observed
value clouddimensionwill be stronglydissipatedwithin that
but still an order of magnitudetoo small. Therefore region,whereaswavelengths
severaltimeslongerthan
we must concludethat mechanismsother than acoustic, the bubbly zone will be dominated by the portion of
thermal,or viscous
damping
arerequired
to explainthe
fluid in the column where the volume fraction of steam
strongdampingobserved.
is significantlylower.
A hint for the possibledissipationmechanismmay
be foundin the largeattenuationin the water column 3.2. Event Rate
(Figure9). Analysis
of arrivaltimesof the pressure
Severaldynamicprocesses
can causeperiodicpulsapulses
at thetoptwosensors
indicates
that mostof the
pressure
pulsesarriveat the top sensor
first,implying tion in a superheatedpipe full of coolant,as was dis-
studies[Boure
that mostoriginatenearthe top of the watercolumn. cussedin numerousnuclearengineering
The average
arrivaltimedifference
between
thetoptwo et al., 1972]. Suchregularinstabilitiesare generated
o.o
1ø-1
• o.1
lO
-2
did'le
'
I-•
,,, ,,.
0.0[•4"
I•/, i ''•
I":
• , , ,• I , , , 't• , •
•00
1102
Time (s)
1104
10-3•
10-1
•00
•01
•0:
Frequency(Hz)
Yigare 12. A 5-s-longtime windowof •he lop (solidline) •nd middle(d•shedline) pressure
sensors•d
•heir corresponding
•ourier •mp]itude spectra. Hole [he differencein spec[r• •
frequencies•bove 10 H•.
24,294
KEDAR ET AL.: BUBBLE COLLAPSEAS THE SOURCEOF HARMONIC TREMOR
when the fluid in the system fluctuates between two
thermodynamic states, i.e., when a small thermodynamic perturbation can result in an instantaneousphase
change.The excitation of this type of two-phaseinsta-
bilities is stronglydependenton the heat and massflux
in and out of the system and the geometry of the system.
In this section, the interaction between these parameters, and their influenceon the event rate in the water column, will be discussed.The uncertaintiesabout
the dynamiceffectsof the highly turbulent Old Faithful
make any attempt to obtain a deterministic model for
the event rate a highly complicated,poorly constrained
task. Therefore, rather than provide a detailed description of the dynamics of the instabilities, we treat the
problem as quasi-staticwith the followingassumptions:
1. The fluid is well mixed(i.e., isothermal.)
2. Each pulseis a coolingevent in which a quantum Figure 13. A schematicdiagramof a filling model. A
of heat is released.
tank with liquid of densityp at pressureP0 is connected
3. The rate of occurrenceof pulsesis solely depen- to a pipe of crosssectionarea A, through a network of
dent on the heating rate of the fluid in the conduit, n pipesof radius a and length 1.
i.e., a pressurepulse correspondingto a coolingevent
occurswhen the temperature in the fluid reachesa certain threshold of supersaturation.
that the forcingpressurestaysP0 for the durationof
The secondand third assumptionsneed further clarthe filling process.First we treat the time historyof
ification. We assume that each collapse is the prodthe water rise in the pipe. The averagevelocity Ua• in
uct of an impulsive steam release which occurs as the
one of the n thin pipesof radiusa and lengthl, assumwater column reachesa specificthreshold temperature
ing that inertialforcesarenegligible(i.e., that the flow
and thus cools the water column instantaneouslyby a
throughthe pipe is Poisseuille),
is
quantum of heat. As heat and water are continuously
pumped into the system, the temperature rises again
to the threshold temperature and the processrepeats
t
itself. When the time interval between pulsesis short
compared to the time between eruptions, the rate of where p is the fluid density,p is the viscosityof the
Po-pgz
a21
(7)
occurrence
of pulses(coolingevents)is proportionalto fluid (p = pv), and P0 is the drivingpressure.From
the time derivative of the temperature history of the
water
This calculation should be treated as a thought experiment rather than an attempt to exactly model the
dynamic behavior of the geyser. The advantageof
these assumptionsis that the fluid in the conduit is
a well-defined control volume whose temperature history is completelydeterminedby the amount of heat
per unit mass,while no considerationof dynamicprocesseswithin the control volume, such as heat or mass
transport, is necessary.These assumptions
are simplistic, as temperature gradientsare knownto exist in the
column, the pulse amplitudesfluctuate, and dynamic
constraintson the pulse rate are likely to exist. However, as will be demonstrated,significantinsight into
the heatingprocessof the water columncan be gained
with the above assumptions,as simple as they might
be.
conservationof mass, we get
column.
-
A
n•.a 2
(s)
whereA is the pipe(geyser)crosssectionareaand• is
the time derivative of the water level in the big pipe.
Combiningequations(7) and (8), the equationof motion for z is
Po- pgz- c•] = 0
(9)
wherea (the constantdeterminingthe durationof the
fillingprocess)is
a-
8Atzl
•t 7ra 4
(10)
The above discussioncan be similarly formulated in
terms of the medium's permeability, •. Using Darcy's
law for fluid flow in porous media,
3.2.1.
Filling rate. Consider the system disRAp
played in Figure 13. At time to a valve is released,
and the fluid flowsthrough the thin pipesand fills the
larger pipe. We assumethat the tank volumeis suf- where Ap = Po- Pg. Thus the relationshipbetweenthe
ficiently large comparedto the volumeof the pipe, so parameter c• and the medium permeability • is
pl
(11)
KEDAR
ET AL'
BUBBLE COLLAPSE
AS THE SOURCE OF HARMONIC
TREMOR
24,295
data
- - - Equation13
E
/
/
/
/
1 hour
. ........ L...... l ........I .......I ......t.
l
I
I .
I
I
I........l._.
!
l...
I _
I ...... I..... 1.... I_ _. l ......I ........[ ........I ........I.........[.
Time
Figure 14. Pressuredata (1 sampleper second)obtainedby J. Westphal,S. Kiefferand R.
Hutchinson
in 1993,anda calculated
fillingcurve(dashedline) corresponding
to c•- 2 x 10-7
kg/m2 in equation
(13). Themodeldescribed
by Equation13obtainsa goodfit to the data.
pl
atureof the waterenteringthe system,and M(t) is the
total
mass in the column
as a function
of time.
In our
model,
Integratingequation(9), we get
M(t) - pA
Po[+Ce
•dt - pA[z(t)- z0]
(16)
(13)
where C is a constant of integration determined from
the initial conditions.For the initial conditionz(t 0)=0, C=-I.
Comparing the timescalesof the calculated column
height to the pressurehistory measurementobtained
If we assumethe heat flows in through the bottom of
the cylinder only, then
Q(t) = qA(t - to)
(17)
Combiningequations(15), (16), and (17),
by J. Westphalet al. (written communication,
1993)
(Figure14), we find that c• = 2 x 107kg/m2syields
good fit to the data.
3.2.2. Heating.
The temperature of the water as
a function of time can be calculatedby assuminga constant heat source. Thus the total of heat Q pumped
into the water
column
Q-
at time t is
qA(r)dr
(14)
T(t)- To+ Cpp[z(t)q(t-to)
z0]
(18)
Rinehart[1980]estimatedthe heat flux q at 5 x 105
J/m2 s, whichwill be usedin the followingcalculations. Rinehart [1980]assumedthat the volumeof water ejectedfrom Old Faithful duringan eruption(-,•50
ma) isheatedduringoneeruption
cycle(-,•1hour)from
groundwatertemperature of 5øC to 112øC, the temperature of the water ejected out of Old Faithful. The
whereq is the heat flux and A(t) is the surfacearea assumptionthat the water heatsfrom 5øC to 112øCdurthrough which heat flows into the water column. The ing one cycleis probably a grossoverestimationsince
temperature is then calculated using the specificheat
at constantpressure
Cp (for water,Cp - 4187J/kg K)
zxh(t)
- c(r-
-
Q(t)
where h is the enthalpy of the water, To is the temper-
the water might very well be in the systemalreadyand
preheatedto sometemperature betweenthe abovelimits. On the other hand, the systemis losingheat by
meansother than an eruptionvia the numerouscooling
events.
Figure 15a displaysthe temperaturehistorygivenby
24,296
KEDAR ET AL.' BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC TREMOR
o
o
......
...... •
•
• • '
•
0.0
•=5X107
•=10 7
•=2x10 7
__,,__ a=10
e
I
2000.0
.-•
.. •
....
..... ,' •..•.../. •
I
I
4000.0
I
8000.0
I
8000.0
! 0000.0
Time (s)
b
At=10.'
,. ,-'- ,.
"At=5
._AL•2.
At=l
• = 2 X 10 •
a=4X10 7
a=10 s
a=2XIO s
....
ßß ß
I
0.0
,
I
5000.0
I
10000.0
0.0
5000.0
Time (s)
10000.0
Time (s)
/
/
/
.-..'- '..-.
•-,4X10 7
a-10 ß
e-2X10 •
....
I
0.0
2000.0
I
I
4000.0
6000.0
I
8000.0
I
10000.0
Time (s)
....
..
0
I
0.0
8000.0
I
I
4000.0
6000.0
I
8000.0
Time (s)
Figure 15. (a) Temperaturehistoryfor a modelheatedfrom below.The slowerthe fillingrate
(highervaluesof c•), the largerthe temperaturerise per unit time. For the samereason,the
curvesfor smaller values of c• approachlinearity faster. The initial conditionsin all the calculations assumethat there is no massin the pipe at t = 0 and that the water flowingin is at 0øC. It
is alsoassumedthat the heatingof everyunit massis instantaneous,
causinga temperaturejump
at time t = to + dt. Therefore the fluid temperature at infinitesimallyshort time after to = 0
will differsignificantly
betweenprocesses
of differentheat andmassfluxes.(b) Heatinghistory,
water level,and total massas a functionof time, for a pipewideningat z1=7 m. Differentcurves
describethe behaviorfor differentcrosssectionareaA1. Notethat c•is proportionalto A1. (c)
Heating history of a pipe narrowingat z1=7 m.
a-2X10 7
o•-.10 7
•.5X10 8
•-.2X10 8
I
10000.0
KEDARET AL.' BUBBLECOLLAPSE
ASTHE SOURCEOF HARMONICTREMOR
15
10
I
2000
I
I
4000
6000
I
8000
I
1 0000
Time (s)
20000
10000
I
2000
I
4000
I
I
ooo
ooo
oooo
I
8000
10000
8000
10000
Time (s)
600
500
400
o
500
2OO
lOO
•000
4000
6000
Time (s)
0.075
•
0.050
0.025
2000
4000
6000
I
:
Time (s)
Figure 16. A pipeof crosssectional
areaA• - 1 m2 opening
at z• - i0 m to A2 - 10m2, and
narrowing
backto A• at z2- 10.2m. Shown
in thefigurearethewater-column
heightZ(m), the
cumulative
mass,the heatinghistory,andthe derivativeof the heatinghistorywhichdetermines
the rate of events.
24,297
24,298
KEDAR ET AL.' BUBBLECOLLAPSEAS THE SOURCEOF HARMONIC TREMOR
equation
(18) usinga heatflux q = 5 x 106J/m•' s. interval between pulsesis short compared with the inFrom equations(14) and (15) and from Figure 14, it
is apparent that the temperature rise in the column
will asymptotic/filybecomelinear, as the water level
approaches
its hydrostaticequilibriumlevel.
3.2.3. Conduit geometry. The "heating from
below" model resultsin a heating curvethat is approximately linear in the timescaleof ~1 hour in which we
are interested.This impliesthat the rate of occurrence
of heat lossevents,which we assumeto be proportional
to the heating rate, is also linear. Sinceour observation is that there are at least two regimesof eventrates
terval between geysereruptions, as is the case at Old
Faithful, the number of "saw-teeth"per unit time will
be proportional to the time derivative of the temperature. Figure 16 displaysthe unperturbed temperature
history and its time derivative. It is suggestedby this
model that the temperature time derivative, and thus
the event rate, would drop dramatically when the water level reaches the wide section of the conduit.
This
calculationillustrates the importanceof the role played
by the geometryof the pipe in regulatingthe intensity
of the activity.
(Figure4), a constantcross-sectional
pipemodelcannot
explain the observedchangein event rate. A possible 4.
Conclusions
model that can accountfor the observedrate changeis
The source of continuous harmonic
that of a pipe with a (sudden)changeof diameter.It
tremor
at Old
has beenobservedthat sucha changedoesoccurin Old Faithful Geyser can be modeled by discretebubble colFaithful. Accordingto Birch and Kennedy[1972]and lapse events occurring near the top of the superheated
Hutchinsonet al. [1997],thereis an areaof wideningat water column, each generatinga seismicsignal. The
about the 10 m depth from the top of the hole, whose number of seismiceventsper unit time, or the intensity
of the tremor, is determined by the rate at which bubextent is not known.
Assumethat a pipe has a cross-section
area A0 that bles collapse, which in turn dependson the heat and
opensat somedepth Zl to a cross-section
areaA1 • A0. massflux and is strongly controlledby the geometryof
Since the water level rise • is slow, we can regard the the conduit.
A bubble collapsemodel with pressuredifferenceAP =
time when the water level reaches the widened conduit
internalgaspressure
Pg0=
at Z1, as a starting point for a new filling processof 0.3 x l0s Pa, residual
0.2
x
10
•
Pa,
and
an
effective
viscosity
yE--0.04
m•'/s,
a pipe of crosssectionA1, but now with a new initial
which
implies
a
bubble
radius,
R0
~5
cm,
is
in
agreeconditionz(t -- O) - z• which resultsin a constant
C - z•(pg/Po)- i in equation(13). Thereforethe ment with our pressuremeasurementsand is consistent
temperature is given by
1
Qo -• qoAot
with the measured conduit's narrowest spot, 0.1 m in
diameter. Although at this point we cannot identify
any damping mechanismwhich can fully account for
the large effective viscosity needed to match the observations,the strong frequency-dependentdissipation
suggestsa zone of bubbly liquid at the top of the water
column in which high frequenciesare attenuated and
which doesnot affect longer periodsas much.
A constant heat flux from below and a filling process
driven by pressureequalizationbetweenthe aquifer and
the conduit through a porous medium give rise to a
heating rate which asymptotically approachesa linear
temperature increaseas the water columnexponentially
nears its equilibrium level. Neglectingdynamic effects,
V- To.n
t-•pp
No.n
t-pAl[z(t)- z1] (19)
where Q0 and M0 are the heat and massaccumulated
in the narrow(crosssectionA0) pipe, respectively.
This result is shown in Figure 15b with the correspondingwater levelriseand cumulativemass.As soon
as the water level reachesthe z• level, the massflux of
fluid into the pipe increases,causingthe temperature
rise or, equivalently,the cooling-eventrate to go down.
The wider the pipe, the faster coldwater is pushedinto
the system and the slower the heating rate. This effect can be understoodif we considerthe limiting case
when at z• the pipe opens indefinitely, causinga con- when the event rate is solelydependenton the heating
stant mass flux. Since we assumeconstant heat flux,
the massof water will asymptoticallyapproachconstant
temperature. This in turn will causethe coolingevents
to ceasealtogether. Similarly, if the pipe narrowsat the
z• level, then the massflux goesdown and the heating
rate, the time-dependent event intensity can be modu-
lated by widening and narrowingof the conduit, thus
controlling the mass flux and consequentlythe heating rate and the event rate. The modulation of tremor
intensity by simple geometricvariationsmay be significant in the context of volcanictremor. This result sugrate increases
(Figure 15c).
gests
that an apparent relaxation in tremor intensity
Figure 16 illustrates the behavior of a pipe with a
does
not
necessarilymean a pause in the geothermal
wide section in the middle. The actual temperatureactivity.
time curveshouldappear as a "saw-tooth" shapedcurve
where every time the temperature reachesthe threshold
Acknowledgments.
The authors wish to thank Yeltemperature, it drops by an amount correspondingto
lowstoneNational Park Authorities, the ScienceOffice, and
one heat quantum releasedby one bubble collapseand the late Park Geologist Rick Hutchinsonfor invaluable help
heats again to the threshold level following the slope in carrying out the experiments. Doug Dreger at U C Berkeof the heating curve at that instant. As long as the ley; Blair Zajac, Craig Scrivner, Timothy Melbourne, and
KEDAR ET AL.: BUBBLE COLLAPSE AS THE SOURCE OF HARMONIC TREMOR
24,299
Miriam Jackson at Caltech; John Holt at JPL; Alessandro Kedar, S., B. Sturtevant, and H. Kanamori, The origin of
Pino at the Istituto Nazionale di Geofisica, Italy, for hard
harmonic tremor at Old Faithful Geyser, Nature, 379,
work in the laboratory and in the field; Jim Westphal and
708-711, 1996.
Joe Shepherdat Caltech for sharedexpertisein the design Kieffer, S. W., Sound speed in liquid-gas mixtures: Waterof the probe; and Susan Kieffer for help in planning and
air and water-steam, J. Geophys. Res., 82, 2895-2904,
1977.
analysisof the 1991 experiment. Special thanks to Start
Cincera, Dave Johnson,Wayne Miller, JoseNunez-Anzueto, Kieffer, S., Seismicity of Old Faithful Geyser: An isolated
Vick Nenow, and Bob Taylor at Caltech for invaluabletechsource of geothermal noise and possibleanalogue of volnical support. We would alsolike to thank the reviewersfor
canic tremor, J. Volcanol. Geotherm. Res., 22, 59-96,
1984.
a thorough,constructivecritique. Contribution 8492, Divisionof Geologyand Planetary Sciences,CaliforniaInstitute Leet, R. C., Saturated and subcooled hydrothermal boilof Technology.
ing in groundwater flow channelsas a sourceof harmonic
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