GEOLOGIC NOZZLES

Susan Werner Kieffer U.S. GeologicalSurvey Flagstaff,Arizona

Abstract. Sonicvelocities of geologicfluids, such as low soundvelocity. The high soundspeed of liquid water volcanic magmas and geothermal fluids, can be as low determines the characteristics of harmonic tremor observed as 1 m/s. Critical velocities in large rivers can be of the at the gyeserduring the rechargeinterval, whereas the low order of 1-10 m/s. Becausevelocities of fluidsmoving in sound speed of the liquid-vapor mixture influencesthe these settingscan exceed these characteristicvelocities, fluid flow characteristicsof the eruption. At the rapidsof sonicand supersonicgas flow andcritical and supercritical the Colorado River in the Grand Canyon, Arizona, the shallow-waterflow can occur. The importanceof the low channel is constrictedinto the shape of a converging- characteristicvelocities of geologic fluids has not been divergingnozzle by debrisflows that enter from tributary widely recognized,and as a result, the importanceof canyons. Both subcriticaland supercriticalflow occur supercriticaland supersonicflow in geologicalprocesses within the rapids. The transportcapacity in the rapidscan has generallybeen underestimated.The lateral blast at be so great that the fiver contoursthe channelto a charac- Mount St. Helens, Washington,propelled a gas heavily teristicshape. This shapecan be usedto interpretthe flood laden with dust into the atmosphere.Because of the low historyof the Colorado River over the past 103-105 years. sound speed in this gas (about 100 m/s), the flow was The unity of fluid mechanics in these three natural internallysupersonic. Old Faithful Geyser,Wyoming, is a phenomena is provided by the well-known analogy converging-divergingnozzle in which liquid water refilling betweengas flow and shallow-waterflow in converging- the conduit during the recharge cycle changesduring divergingnozzles. eruptioninto a two-phaseliquid-vapor mixture with a very

1. INTRODUCTION apparentlyfirst describedby Riabouchinsky[ 1932] (a more recentand accessiblereference is Loh [1969, pp. 1-60]). This paper was written for the symposium The analogy is semiquantitativeand was thus widely "Perspectivesin Fluid Mechanics"held at the California exploredin the earlydays of wind tunneldevelopment. In Institute of Technology, January 10-12, 1985. It is modem times the analogyhas been primarily a teaching specifically an essay of the author's perspective on tool [e.g., Thompson,1972, pp. 517-531] and has never geologicfluid flow problems,rather than an exhaustive been used by geologiststo explain large-scalenatural review. For the sakeof brevity,references in thispaper are phenomena.The purposeof thispaper is to showthe basis limited. The authorhas published derailed analyses of the for invoking nozzle-flow theory for interpretationof threegeologic problems presented, and the readerwill find complexgeologic events and to providea perspectiveon referencesto otherrelevant work in the citedpapers. geologicproblems in which the importanceof supercritical and supersonicflow hasbeen underestimated. 2. GEOLOGIC NOZZLES A major reason that geologic events have not been viewed from the particularperspective of fluid mechanics An eruption of Old Faithful geyser, a flood on the presentedhere is the subdivisionof fluid mechanicsand its Colorado River, and a lateral blast from Mount St. Helens applied fields into the specialtiesof compressibleand do not, at first glance,appear to be related. A geographic incompressibleflow, for example, aeronauticsversus map of the locationsof thesethree places certainly does hydraulics. This subdivisionarises from the need to not reveal any underlying geologic unity (Figure 1). simplifythe complexmomentum and continuityequations However, a fluid-dynamical unity is revealed when the in orderto solve mostpractical problems. In vectorform, "locations"are shown insteadon schematicdiagrams of the momentumequation for a viscousfluid moving in a gas flowing througha nozzle or of shallowwater flowing gravitational field under the influence of a pressure througha flume (Figure 2) (for example,see Liepmann gradient is complex because of dimensionality and andRoshko [1957, p. 127]). The analogybetween the flow nonlinearity: Du fields for compressiblegas and shallow water was P•i- =-VP + [V. x]+ pg (1)

This paperis not subjectto U.S. copyright. Reviewsof Geophysics,27, 1 / February1989 pages 3-38 Publishedin 1989 by the AmericanGeophysical Union. Paper number 89RG00093

e3e 4 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer: GEOLOGIC NOZZLES

where p is the fluid density,D/Dt is the materialderiva- tive, u is the fluid velocity,VP is thepressure force acting on the fluid, V. x is the viscousforce, and g is the accelera- tion of gravity. The continuityequation for mass is (or generally simpler but is still difficult to apply in a (a) geometricallycomplicated problem: Dp_ Dt - -p(V.u) (2) where(V ß u) is the divergenceof velocity. In many casesthese two importantequations can be considerably simplified by considerationof the fluid propertiesor the boundaryconditions of the problem. For example,if pressurechanges are relativelysmall, compres- sibilitycan be neglectedso thatp - constand V ß u = 0. Such an assumptionunderlies all of hydraulics, and geologistswith interestsin hydraulicsor relatedgeomor- phic problemstypically divergeat an early stageof their educationfrom advancedstudies of compressiblefluid dynamics. Alternatively,in manyflows the pressuregradient may be great enough that compressibilityis importantbut gravity is not; g ~ 0. This latter conditionis assumedin mostof gas dynamics,and becauseof the prominentrole of gravity in mostgeologic processes, few geologistsare exposedto a rigorousgas dynamics curriculum.

129 ø 125 ø 121 ø 117 ø 113 ø 109 ø 105 ø Figure 2. Four diagramsof behaviorof gas flowing througha , / /?,...,,-'--4.. • • t converging-divergingflume. Vector velocitiesare indicatedby I "•/"r-•. CANADA arrows. At the exit of the nozzle (or channel),on the right, the

45 ø ;,, structureof the flow field in the departing fluid is shown ';i /-'L.*-o,,./ - schematicallyby the medium shading. Shock and rarefaction waves (alternatively,positive and negativenormal and oblique hydraulicjumps) are indicatedby the lightestshading. 41 ø /"•' "7"-• J• / / ! Althoughthe subjectsof nozzle gas dynamicsand of T--'-- shallow-water hydraulics evolve from very different 37 ø - 'x i i , approximationsto the conservationequations, important '\ /' _,,øy•. I' , conceptscommon to both subjectshave beenrecognized because, when reduced to suitable nondimensional variables,the conservationequations in the two subjects 33 ø becomeidentical. (Readersfamiliar with thisidentity can skip directlyto the last paragraphof this sectionand to section3.) Examinefirst the massand momentumequations for a 29 ø EXICO perfect gas. For simplicity, assume that the flow is quasi-one-dimensionalalong a coordinatedirection x. The equationsof massand momentumconservation for flow of 500 k m'"N. a compressiblegas are 25 ø -'''''' i 121ø 117ø 113ø 109ø 105ø 101ø 8p 8p 8u (3) Figure 1. Index map of the geographiclocations of Crystal •)t+ U•xx+P•xx =0 Rapids (Grand Canyon, Arizona), Old Faithful Geyser (Yel- lowstone National Park, Wyoming), and Mount St. Helens 8u 8u ! 8P= 0 (4) (Washington). Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWSOF GEOPHYSICS ß 5

For a perfectgas and isentropicflow, within the contextof all of the simplifyingassumptions, flow of gas in a nozzle and flow of shallow water in a PV = RT (5) flume are governedby the same conservationequations. Identical flow fields therefore occur when the proper P W = P0V0 •' = const (6) nondimensional variables are considered. The illustrations in Figure 2 represent the flow whereP is pressure,V is volume,R is the gasconstant, T is conditionsat different ratios of upstream(reservoir) and temperature,and ¾ is the ratio of specific heats (the downstream(atmosphere or tailwater) conditions. If the isentropicexponent). The subscriptzero indicatesa figure is interpretedas representinga crosssection of a referencestate (typically one wherethe fluid is at rest with horizontalnozzle throughwhich gas is flowing from left to velocityu = u0 = 0). For a perfectgas, (3)-(6) canbe right, the different parts of the figure representthe flow combinedto give field from differentpressures (P1, P2,'") in the left reservoirsto lower pressuresin the right reservoirs. The 3U 3U •tP0T--2 30 -- 0 (7) low-pressurereservoirs are assumedto be infinitely large •}t+ u•-•+ •0•P • and are therefore not shown to scale. Alternatively,if Figure 2 is interpretedas representing For water flowing from one infinite reservoir into a map view of a horizontalchannel in which shallowwater anotherwith lower head,the equationsof motionthat can is flowing from left to right, the differentparts of the figure be directlycompared with (3) and (7) are representthe flow field from reservoirsof different depth. 3h 3h 3u The driving energy for the flow is the elevated depth of i•t+ u•-• + h• =0 (8) water in the left reservoircompared with the right. The waterhas a potentialenergy H r (calledthe headand generallyexpressed asa depth),indicated as H l, H2 ,..., in 3u+ 3u 3h (9) i•t U•xx+g• =0 the partsof Figure2. With this introduction, let us reexamine the sense in whereh is the water depth. In theseequations, and in the which eachof the geologicproblems mentioned above is a figuresin this paper,it is assumedthat in a verticalcross nozzleproblem. In the spiritof emphasizingthe similarity section(specified by the coordinatez) the bottom of the of the variousflow fields discussedin this paper,the word water is at the channelboundary z = 0 and the water hasa "nozzle" will be used interchangeablywith the words free surface at z = h. The free surface is assumed to be at "flume," "channel," and "conduit," and the word "contour- constantatmospheric pressure Pa' but its elevationcan ing" will be usedinterchangeably with the word "eroding." vary alongthe channeland with time. The velocityof the The nozzle of the Colorado River is the river channel, a watercan alsovary with positionalong the channelbut is converging-divergingnozzle formedby debrisflows that assumedconstant over any vertical cross section. The constrict the main channel, and the fluid is shallow water. viscosity and compressibilityof the water are ignored. The "geologictwist" that complicatessimple application of Again, as with gas flow, quasi-one-dimensionalflow is flume conceptsis that the walls and bed of the channelare assumed: the contours of the channel walls must be erodible,and the channelcan thereforechange shape in gradual, and vertical accelerationsof the water must be responseto changingconditions in the flow. The nozzleof smallcompared with theacceleration of gravity,g. Old Faithful geyseris a fissureof irregular (and largely The shallow-waterconservation equations, (8) and (9), unknown)geometry extending more than 20 m into the are identical to the compressible-gasconservation ground. The geologictwist in thisproblem is thatthe fluid equations,(3) and (7), if waterdepth h is analogousto gas is muchmore complex than a perfectgas: hot, liquid water densityp and if the isentropicexponent ¾ of the gas is standsin the conduitbetween eruptions and then boils and equal to 2. Unfortunately,no real gas has sucha high changesthrough a complex unloadingprocess into a value of ht (the highest value being 5/3 for monatomic droplet-ladensteamy aerosol during an eruption. The gases),so that the mathematicallyequivalent flow fields nozzle of the Mount St. Helenslateral blast was a huge cannotbe quantitativelyrealized. However,the effectof ¾ vent createdwhen a landslidecaused by an earthquake on most flow propertiesis small, so that the analogyis openeda verticalscarp nearly 0.25 km 2 in areaand qualitatively, and even semiquantitatively,useful--as exposeda hot hydrothermaland/or magmatic system. The evidencedby the fact that in the early days of rocket eruptingfluid was a hot vapor heavily laden with ash, design, hydraulic flumes were used to simulate wind rocks, ice fragments,and tree debris. As these three tunnels. Examinationof the conservationequations and examplesshow, the scaleof the geologicnozzles is large, the equationsof statefor a perfectgas showsthat h is also the nozzle shapesare irregular, and the thermodynamic analogousto T andthat h 2 is analogousto P. Hence, propertiesof the flowing fluidsare complex. REVIEWSOF GEOPHYSICS / 27,1 Kieffer: GEOLOGIC NOZZLES

3. SOUND VELOCITIES AND CRITICAL VELOCITIES: parameters,the flow field can have dramaticallydifferent THEIR INFLUENCE ON THE FLOW FIELD properties,as illustratedin Figure2. Consider,first, that Figure2 representsthe flow of gas The most importantresult from the aboveanalogy is througha nozzle.When the pressure P• in thereservoir is the recognitionthat characteristicvelocities control flow "low," the fluid accelerates from the reservoir into the behavior in shallow water and gas flow. For small constriction and deceleratesin the diverging section disturbancesthe equationsof momentum((7) and (9)) can (Figure 2a). This is the classicventuri tube, and the flow be linearized and written as is everywheresubsonic. In this contexta "low" pressure ratiomeans that the upstream reservoir pressure is lessthan Perfectgas about 2 times the downstreamreservoir pressure(for 02P 2 02P 3t2 a0 -•Yx2=0 (10) example,see the tablesof isentropicflow variablesgiven by Zucrowand Hoffman[1976]). Shallow water If the pressureratio is higher (Figure 2b), the fluid acceleratesfrom the reservoirinto the convergingsection 32 h 32 h • - gh0 -•- =0 (11) and can reach sonic (choked) conditions(M = 1) in the throat;it canbe rigorouslyshown that sonicconditions can These are the well-known wave equations,from which it occur only in the throat (for example, see Zucrow and can immediatelybe seenthat smalldisturbances propagate Hoffman [1976, p. 166]). At one particularpressure ratio with characteristicvelocities proportionalto the square the flow can decelerate back to subsonic conditions in the root of the coefficientof the secondterm. In compressible divergingsection, but for highervalues it will accelerateto gasflow the characteristicvelocity is the soundvelocity a, supersonicconditions in the divergingsection. Strong the velocity at which small perturbationsin densityor nonlinear waves (shock and rarefaction waves) can be pressurepropagate through the fluid: presentand are, in fact, usuallyrequired to deceleratethe flow back to ambient conditions in the exit reservoir. At higherpressure ratios for whichsupersonic flow conditions a2=(•p) s (12, are obtained, a so-called "normal shock" stands in the where the derivative is taken at constantentropy S. In diverging section, and the deceleration to ambient shallow-waterflow the characteristicspeed is the critical conditionsoccurs within the nozzle throughthe shockand velocity, that is, the velocity of a gravity wave of long to the exit plane(Figure 2b). At still higherpressure ratios wavelengthand infinitesmalstrength: (Figures2c and 2d) the shockis "blownout" of the nozzle, and a complicatedflow field consistingof oblique and c2=gh (13) normal shocks and mixed regions of subsonic and supersonicflow existswithin the exiting jet. Becausethe In both casesthe nature of the flow field dependson deceleratingwaves are nonlinear, the jet "overshoots" the magnitudeof the fluid velocity comparedwith the ambient conditions,and multiple shock and rarefaction characteristicvelocity. The Mach numberM of a compres- wavesare requiredto achievethe pressure balance. sible gasflow is the ratio of the meanflow velocityto the Consider alternatively that Figure 2 represents soundspeed: shallow-water flow. When the head difference between m = u/a (14) the reservoirson the left and right is "small,"the flow is subcritical everywhere: the fluid acceleratesin the The Froude number of shallow-water flow, Fr, is the ratio converging section and through the constriction,and of the meanflow velocityto the criticalvelocity: deceleratesin the divergingsection (Figure 2a). The flow field is analogousto that in a venturi tube. The word Fr = u/c (15) "streaming"is often used for subcriticalflow. In this context a "small" head difference means that the elevation The local flow variables are determined by these differencebetween the upstreamand downstreamreser- dimensionalessratios which, in turn, dependon reservoir voirs shouldnot exceedapproximately one third of the conditionsand geometry. For gas flow the important head of the source reservoir. parametersare the ratio of the pressuresin the drivingand If the headratio is greater,as indicatedin Figure 2b, receivingreservoirs, the area ratio along the axis, and the the flow acceleratesfrom the reservoir through the gas equationsof state(particularly R and ¾ for a perfect convergingsection and can reach critical (Fr = 1) condi- gas). For shallow-waterflow the importantparameters are tions in the constriction. At the critical value of head ratio the ratio of upstreamto downstreamenergy and the area the flow can decelerate to subcritical conditions in the ratio of the channel. Dependingon the values of these diverging section, but for other higher values it will Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß 7 accelerateto supercriticalconditions in the diverging massand latent heataccompany passage of a soundwave; section. The word "shooting"is often usedfor supercriti- theseexchange processes also decrease the soundvelocity. cal flow. Strong nonlinear waves, in this case called Sound speedsas low as 1 m/s are possiblefor boiling oblique (or slanting) and normal hydraulicjumps, are water(reviewed by Kieffer [1977]). generallyrequired to decelemtethe flow back to ambient The dependenceof soundspeed on phase,pressure, and conditionsin the downstreamreservoir. Dependingon the temperaturecan be shown on an entropy-density(S-p) head ratio and the severityof the constriction,waves can phasediagram, in this casefor H20 (Figure3) (more standin (Figure2b) or downstreamof (Figures2c and2d) extensivediscussion of this diagramand a similar diagram the divergingsection. for CO2 is givenby Kiefferand Delany [1979]). This The flow fields shown in Figure 2 are a subsetof representationis suggestedby the definition of sound possibleflow conditions,for they do not show possible speedgiven in (12): on a graphof densityversus entropy, wave structuresthat ariseif fluid entersthe constrictedpart sound speed is inversely proportional to the vertical of the nozzle in a supersonicor supercriticalstate. gradient of isobars. Such a graph can be read as an Although such conditions can, in fact, be obtained ordinary topographicmap (shown schematicallyin the geologically,this complexity is ignored in this paper insetof Figure 3) in which the steepnessof the topography becausethe conceptscan be extrapolatedfrom the simpler is inverselyproportional to the spacingof the contours. analysespresented here. The S-p representation,with its widely spacedisobars in Supersonicor supercriticalconditions are amazingly the two-phase region and the steep gradients in the easyto obtainin geologicsettings. If the ratio of reservoir one-phaseregion, graphically illustrates the wide rangeof pressureto atmosphericpressure in a gas nozzle is more sound speeds characteristicof simple one-component than about2, sonicand supersonicflow will occur in the substances. nozzle;for comparison,the ratio of pressurein a volcanic If volatile fluids like water flow from high to low reservoirto atmosphericpressure is oftenaround 100:1. If pressure(e.g., in eruptionsor in geothermalwells), the shallow water flows from one reservoir to another that has phaseof the fluid can changefrom liquid to liquid plus less than two thirds of the head of the source reservoir, vapor,or from vaporto vaporplus liquid. A hypothetical critical conditions can be obtained in the throat; for decompressionpath appropriate to Old Faithful (and comparison,backwater depths on the ColoradoRiver may discussedlater in section5) is shownas the verticalline (a) exceeddownstream tailwater depths by a factorof 2. in Figure 3. Note that along this decompressionpath the Our intuition,however, generally fails to prepareus for soundspeed can change by severalorders of magnitude.If the possibilityof such flows in the natural world. We the fluid is in a two-phasestate, flow velocitiesof only a commonlythink of supersonicflow in terms of modem few tens of metersper secondcan give a wide range of aeronautics:objects obtain high Mach numbersby moving Mach numbers,including sonic (M = 1) and supersonic very fast through air, which has a high sound speed. (M > 1) flow. Geologicfluids rarely move at the speedscharacteristic of Mass loadingof vapor with solid fragmentsor liquid modem aircraft (except in some volcaniceruptions), but dropletsalso produces fluids with low soundspeeds. No the entire spectrumof flow behaviorfrom subsonicto data or theories exist for the fluids encountered in volcanic supersonic(and subcriticalto supercritical)can occurin problems,where, for example,the massratio of solidsto geologic flows becausethe fluids can have very low vapor can exceed 100 and where particle sizes within a characteristic velocities. Fluids with low sound velocities singleflow can rangefrom micronsto meters. At present, can developinternally supersonic flow fields while still we can only apply a simple theory in which the mass- moving subsonicallywith respect to the surrounding loadedgas is mimickedby perfectgas laws (the so-called atmosphere. That is, there can be standingshock or "pseudogas"theory) to obtaincharacteristic sound speeds rareficationwaves internal to the flow but no standing (Figure 4). Soundspeeds of 50-100 m/s are plausible. waves in the external medium. Flow velocitiesin gassyvolcanic eruptions are commonly Fluids in geothermaland volcanic settingstypically of the order of 100 m/s and can exceed 500 m/s. Therefore have low soundspeeds: water that containsgas or steam a wide rangeof Mach numbers,including M > 1, can be bubbleshas a very low soundspeed, because the bubbles obtained. dramaticallyincrease the compressibilityof the mixture Finally, note that the critical velocityin shallow-water 0Cs).An alternativeform of thedefinition of soundspeed, flow playsthe samerole as the soundspeed in determining a = (1/•Csp)m,shows this dependence clearly. The sound transitions between linear and nonlinear flow regimes. speedin an air-watermixture can be aslow as20 m/s. The Critical velocities in rivers can be of the order of the flow soundspeed is furtherdecreased in a mixturein which the velocities,even in majorrivers where large depths increase bubblesare of the same compositionas the liquid (e.g., the critical velocity(equation (13)). In the ColoradoRiver steam bubbles in boiling water), becauseexchanges of in the Grand Canyon, for example,water depthsof the 8 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer: GEOLOGIC NOZZLES 10o (L) order of 10 m are common;the correspondingcritical 500 velocity is 10 m/s. In most calm stretchesof the river, flow velocities are of the order of 1 m/s, and Froude numbersare less than 0.1. In major rapids, however, where the water becomes shallow and fast, the flow TICAL velocity can exceedthe critical velocity (Fr > 1). Super- POINT critical flow is not commonin rivers [Richards,1983, p. 58], but whenit doesoccur, the geologicconsequences can be great;the situationin the ColoradoRiver, discussedin the followingsection, is onesuch occurrence. t0-t Many simplificationshave been made in the foregoing discussion,and these,as well as others,will be used in the analyses to follow, e.g., thermodynamicequilibrium; isentropic,quasi-one-dimensional flow; steadyflow; and perfectgas or pseudogasbehavior. One additionalmajor

•75 simplificationin the following analysesis that the flow 150 fields are assumedto be either compressibleand gravity- 25 free (M > 1, Fr > 1) or gravity-dominatedand incompres- I00 sible (Fr < 1, M < 1). The assumptionof incompressible '5 shallow-water flow is good for the Colorado River. However, compressibilityand gravity are probablyboth importantfor the flow fieldsof Old Faithfuland the Mount St. Helenslateral blast. Usingparameters given by Kieffer bar • [1981a,b,1984a], onecandemonstrate thatinsome parts

- 2• '"••;•I 10,000)'m=Cpg+mCsC;vg'•-•'• Pseudogas, Hm- --I•-•-m SoundRg Speedsa;•7'm Rm T (L+V)NXNx '

Pure H20 Liquid 10001.-- -

S,ope:(•-)y 'Slope"= (•--•)•a• ,oo- , , , Entropy,Jouleg K ß ';.,, , lO I I o.1 1 lO lOO

m (mass ratio solids/vapor) Figure3. Enropy-density(S-piphase diagram for H20 (details are given by Kieffer and Delany [1979]). Entropyis relativeto Figure 4. Soundspeed a of pseudogasversus mass ratio m of the triplepoint of H20. The label1(I) marks the single-phasesolids to vapor (steam). The pseudogasequations are shownat field, and the label 2(I) marks the two-phasefield (L, liquid; V, the top of the graph. The isentropicexponent of the mixture is vapor). Contoursof constantpressure (isobars) are shown in denotedby ?,,,,the heatcapacities of the gasphase at constant 25-bar increments.In the two-phasefield, contoursof constant pressureand volume are denoted by C•g and C,,g, and the heat mass fraction of vapor (isopleths)are shown. The inset is a capacityof thesolid phase is Cs . Thegas constant of themixture comparisonof a standardtopographic map of a conicalhill (left) isR,,,; the gas constant ofthe gas is Rg. Curves for three different with the S-p diagram(fight). The "footsteps"indicate that the temperaturesspanning the range of geothermaland volcanic appropriatedirectional derivative to be examinedin eachcase is interest are shown. The soundspeed of pure liquid water is in the vertical direction. indicated. Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWSOF GEOPHYSICSß 9 of the flow field, M ~ 1 and Fr ~ 1. In differentparts of a flow field, eithercompressibility or gravitymay dominate, and in the discussionsthat follow the authorwill point out somelimits of the assumptions.The complexproblems of flow in which both compressibilityand gravity forcesare of comparablemagnitude and of the magnitudeof the second-ordereffects when one force is dominantare only beginningto be addressedas the capabilitiesof modem supercomputersare beingturned toward the problem.

4. CRYSTAL RAPIDS: SUBCRITICAL AND SUPER- CRITICAL FLOW IN AN ERODIBLE CHANNEL

GeologicSetting and the Eventsof 1983 The ColoradoRiver is the largestof the greatrivers in westernAmerica. In the 400-km stretchthrough the Grand s N A .•__::A' .-:' Sand Canyon, numerousdebris fans have been depositedby /•' o•o//// BouldersBedrock flash floodsin tributarycanyons (Figure 5). Flash floods • Old Terrace in the tributaries can carry boulders many meters in diameterinto the path of the ColoradoRiver becausethe B••,oo••••_o0 ø B' -- WaterSurface gradientof the tributariesis quite steep. When emplaced, the large debrisfans temporarilyobstruct the path of the river, dammingit until the debrisdeposit is breachedand a new channelcarved. The major rapids on the Colorado River are located where the river passesthrough these debris fans. ':'/ EXPLANATION The channel of the Colorado River resembles a . .;•:•.•'• Qd - 1966 debris fan converging-divergingnozzle in the vicinity of thesedebris Qr - Older debris fan ...... • '•i Qt - Older terrace fans (note the constrictionof the channelin Figure 5). ß .. Typically, the channel narrows from a characteristic ". Sch - Schist X - Rock or rock-caused upstreamwidth of about100 m to a narrowestpoint in the wave (I) '"throat" of the rapid, and then diverges back to a --- Oblique wave (11) downstreamwidth aboutequal to the upstreamwidth. The ex% /ds,.teCreek Landslld.ef• -- Normal wave (111) ratio of the width of the river at the throat to the width at Sch\ • \ ",,( I Shad• •bb •/Rock Ledge G Eddy an averageupstream section is hereincalled the "constric- A ------A',etc., Cross sections tion" of theriver or its "shapeparameter." ., " --L Stranded log In (b) Y•.' •'• •.•'• Od-"• •-• • .... Althoughthe shapeparameter is conceptuallysimple, it bb - Boulder-bar •'_•_vur ? ) "x • is difficult to obtain values that are meaningfulfor both •'• Tamarisks field and theoreticaluse because the shapeparameter is a functionof the dischargerate andbecause the river channel is complexin three-dimensions.For example,the rise of Figure 5. Crystal Rapids at two dramaticallydifferent dis- the river shorelinefrom the lower (southem) end of the charges. (a) June 16, 1973 (U.S. GeologicalSurvey Water debris fan in Figure 5a to the base of the old alluvial ResourcesDivision air photograph),at a dischargeof about283 terrace(see Figure 5b for key) representsa stagechange of m3/s. (b) Keysto featuresin Figure5a. (c) Schematiccross about5.5 m and occurswhen the dischargeincreases from sections,with arbitraryvertical exaggeration. (d) June27, 1983, about283 m•/s (10,000 cfs) to about2550 m•/s (90,000 at a dischargeof about 2600 m3/s. (e) Keyto features in Figure 5d. cfs); thus,if the shapeparameter were measuredat these two extreme discharges,it would have quite different values. The valuesof constrictionshown in Figure 6 were which plots as the leftmostblock in Figure 6. However, obtainedby measuringthe width of thesurface water in the for purposesof hydraulicmodeling later in the discussion, photographseries of the 1973 U.S. GeologicalSurvey it is necessaryto assumean idealizedcross section for the Water Resources Division, one of which is shown in channel. A rectangularcross section is assumed.In this Figure5a. For example,measurements of surfacewidth at simplificationthe "average"constriction used for modeling CrystalRapids from Figure 5a givea constrictionof 0.33, is generallyless than that measuredfrom air photographs, 10 ß REVIEWS OF GEOPHYSICS / 27,1 Kieffer: GEOLOGIC NOZZLES

because the shallow, slow flow across the debris fan, cal power and water storageat the dam, rather than by which shows in air photographsof the water surfacebut naturalflooding. About4 yearsafter the dam was closed, accountsfor only a small fractionof the total discharge,is a flash flood down Crystal Creek emplaceda large debris ignored. In the caseof CrystalRapids the model value fan acrossthe fiver (Figure 5) about 175 km below the used is 0.25, substantiallylower than the 0.33 obtained dam. There were no wimessesto this event; by the time from surface width measurements. observationswere made, the Colorado River had carved a The debris fans measured to form Figure 6 are channelthrough the distal (south)end of the debrisfan. generallymuch older than Glen CanyonDam, although When first measured(on the 1973 air photographshown in they are episodicallyreplenished by flash floodsfrom the Figure5a), the constrictionof thefiver channelthrough the fan was roughly0.33, substantiallymore severethan the 30 constrictionsof the more maturefans along the Colorado River. From 1966 to 1983 the constriction remained at about0.33. During this time, dischargesthrough the dam wereheld below 850 m3/s. The water surfaceof the ColoradoRiver becamevery 20- roughand turbulentas it passedthrough the Crystaldebris fan; this stretchof water, nearly 1 km long, is known as Crystal Rapids. The boulders, waves, and eddies in Crystal Rapids made it one of the two most difficult stretchesof the fiver for raft navigation,even at the normal 10- levels of controlleddischarges between 1966 and 1983 (140-850m3/s). The rapid is a majorhazard for recrea- tional rafting, an activity in the Grand Canyoninvolving about10,000 people each year. The schematiccross sections in Figure 5c show the 0 I I I I I i I cause of the navigational difficulties from another 0.30 0.40 0.50 0.60 0.70 perspective.In the backwaterand convergentreach (cross RATIO OF RIVER WIDTH, W2/Wo

Figure 6. Histogramof constrictionvalues (shape parameter) for the ColoradoRiver as it passes54 of the largestdebris fans in the LAKE POWELL 400-km stretch below Lees Fen'y, Arizona. These values are LEES based on the widths of the surface water in the channel on 1973 FERRY GLEN air photographs(such as shown in Figure5a). CANYON DAM

tributaries. Constrictions at these debris fans are remark- ably uniformat a value of about0.5 (Figure6), and the tight constrictionof CrystalRapids (at 0.33 in thisgraph) is almostunique. Thereis no a priorireason to believethat the debris fans themselveswere emplacedin such a way that,by coincidence,half of the main channelwas blocked. What, then,is the significanceof this characteristicnozzle shape?It mustbe telling us somethingabout the ability of //BRIGHT [ ANGEL the Colorado River to erode its own channel, i.e., to contour its own nozzle. EKTAL•• REEK Becausethe debrisfans are generallyvery old (of the ,GAGE orderof 103-105years) and because the flash floods that STATION create and renew them are rare, we have little hope of CRYSTAL RAPIDS observingthe processesthat create the balancesbetween tributary floods and main channelerosion. However, a 0 10 20 Kilometers unique series of events spanningthe two decadesfrom I I I ARIZONA 0 5 11• 15 Miles 1966 to 1986 hasgiven us a glimpseof theseprocesses. In i i I January1963, Glen CanyonDam (Figure 7) was closed, and the dischargeinto the ColoradoRiver throughthe Grand Canyonbecame controlled by demandsfor electri- Figure 7. Index map for locationsnear Crystal Rapids. Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWSOF GEOPHYSICS ß 11

sectionA-A') thewater is relatively deep and slow. In the surgedonly between3 and 5 m. Becausetypical river rafts constrictedsections (cross sections B-B' and C-C') the are 5-11 m in lengthand 2-4 m in width, the wave was a flow is shallow and fast, thus exposingmany rocks (in severeobstacle to boating. One rafter was drowned,and cross section C-C' the wave discussedbelow is shown dozensof otherswere seriouslyinjured. The NationalPark muchexaggerated vertically). Even in the divergentreach Service closedthe rapid to commercialboating until the (crosssection D-D') wherethe river is somewhatwider dischargeswere decreased. The existenceof this large and the gradientis less steep,the water still flows fast. wave at high discharges,and its evolutionwith changing Becausethe channel bottom is rocky at the downstream discharge,provided clues about the relation between the end of many rapids(see discussion below), this sectionof Colorado River and its debris fans. the fiver channelis also difficult for raft navigation,and many a fiver runnercan tell tales of spendinghours, or evenan unpleasant night, stranded on the rocks near D-D'. (a) Between 1966 and 1983 the major navigational obstacleoccured where water poured over a large rock into a deephole and emergedthrough a sharp-crestedwave in the narrowestpart of the rapid. (The preferrednavigation routeduring this time is shownby theline P-P' in Figure 5a.) This featurewas known as the "CrystalHole" (the locationof the CrystalHole is shownin Figures5b and 5e by the notationN.W., which standsfor "normalwave," but the featureitself is too smalland too blurredby turbulence to showat the scaleof the photographsin Figures5a and 5d). At low discharges,a rock about2 m high was seenat Figure 8. River raft (a) enteringand (b) trappedin the large the CrystalHole, and mostobservers believed that the rock wave at Crystal Rapids on June 25, 1983, when the discharge wasthe solecause of the wave. The relativeimportance of wasapproximately 1700 m3/s. Pontoons on the raft are each 1 m the rock to the hydrauliceffect of the severeconstriction in diameter; the midsection is about 3 m in diameter. More than was, in hindsight,overestimated. Many waves in the 30 passengersare on board (one head is visible on the lower left rapidsof the ColoradoRiver are indeedcaused solely by sideof the raft in Figure8b). To aid the readerin distinguishing betweenthe water surfacenear the boat and in the foreground largerocks. It is well knownfrom fiver guideswho mn the from the turbulentwater in the background,a white line hasbeen fiver beforeGlen CanyonDam was closedthat when large naturalfloods reached 2300-3600 m3/s annually, the drawn alongthe approximatesurface of the waterupstream from and throughthe hydraulicjump in Figure 8a. For explanationof waves in most rapidsbecame very weak or even disap- N.W., seeFigure 5. From the scaleof theraft, the trough-to-crest peared("washed out"). The wash-outoccurred at high height of the wave can be estimated to exceed 5-6 m. dischargesbecause the obstaclescausing them become (Photographscopyrighted by Richard Kocim; reprinted with submerged("drowned"). There was, however,no record permission.) of the behaviorof waves in Crystal Rapids at discharges exceeding850 m3/s,because the rapid, in its modem (b) servere form, did not exist before constructionof Glen CanyonDam. In 1983, rapid snowmelt in the headwatersof the Colorado River forced the Bureau of Reclamation to increasedischarges through Glen Canyon Dam to 2600 m3/sto prevent Lake Powell from flowing over the dam. Asdischarges increased above 850 m3/s (a levelthat had not been exceeded for two decades)the waves in most rapidsdisappeared, as expected.The rapids"drowned out" and the fiver ran smoothand fast through most of the GrandCanyon. This was not the case at Crystal Rapids: as the dischargereached 1700-2000 m3/s, a wavereported by experiencedfiver guidesto havebeen as high as 9 m, and Resultsof Analysisof ShallowWater Flow photographicallydocumented to haveexceeded 5 m, stood Certain aspects of the flow in the rapids of the acrossmuch of the river channel(Figure 8). At greater ColoradoRiver can be analyzedin terms of conservation dischargestheheight diminished: at2600 m3/s the wave of massand momentumfor flow in a converging-diverging 12 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer: GEOLOGIC NOZZLES

channel[Kieffer, 1985]. As discussedin the contextof required to account for a sloping channel bottom by Figure 2, the shapeof the channel,and the upstreamand referencingboth the channelbottom and the water surface downstreamreservoir heads, must be specified,and the to a datum plane as in Figure 9b. For simplicity, and propertiesof the flow field in differentparts of thechannel becauseof a paucityof data, the energychange due to the can then be calculated. Six different flow zones can be changein bedelevation (z• - z2)is assumedto becompen- identified,as shownin Figure9a: (0) an upstreamstate of satedby energydissipation in the flow, E. This assump- unconstricted,uniform flow, (1) the convergentsection of tion allows the flow to be consideredat constantspecific the channel upstream from the constriction,(2) the energy(D + u2/2g),except across hydraulic jumps. For the constriction,(3) the beginningof the divergence,(4) the analysisthe dischargevariation of the specificenergy of end of the divergence,and (5) a downstreamstate of the unconstrictedflow upstreamof Crystal Rapids was uniformflow not influencedby the constriction.Regions estimated from measurementsat the U.S. Geological (3) and(4) maybe separatedby a hydraulicjump. Surveygage station 16 km upstreamat BrightAngel Creek Water flows throughthe rapid becausethe upstream (Figure10a, heavyline, Hr). The flow belowCrystal reservoir is higher than the downstreamreservoir. Rapidswas assumedto returnto thissame specific head. The flow entering the rapid can have the ambient a specifichead • if, andonly if, all of thedischarge can be accommodatedthrough the constriction.If the constriction s is toosevere, the ambient head of theflow, H r, maynot be i sc MAPVIEW sufficient to allow the dischargeto be accommodated throughthe constriction.In suchcases, critical conditions occur in the constriction,and a backwater is required upstream. The deepeningof the backwaterincreases the specifichead of the flow over that in the unconstrictedpart of the channeland permits a greaterdischarge per unit area throughthe constriction. The calculatedbackwater head H• comparedto theambient fiver head H r is shownas a b function of dischargeQ and constrictionvalue in Figure 10a. Note from thisfigure that for a constrictionof 0.25 a U2/2g CROSS SECTION backwaterhead is requiredfor all dischargesover about 300m3/s; the backwater above Crystal Rapids is sobig ws that it has been affectionatelydubbed "Lake Crystal"by river runners. This calculationmeans that supercritical flow will occurin the rapidat dischargesabove a valueof approximately300m3/s. Solutions of the shallow-waterflow equationsare shownin Figures10 and 11. In eachpart of Figure 10 a calculatedor measuredparameter is shownas a functionof Figure 9. (a) Schematicmap view of theriver andthe debrisfan dischargeon the abscissaand the shapeparameter (labeled configurationat Crystal Rapids. HJ indicates a possible on eachcurve separately). In Figure 10a the specifichead hydraulicjump. (b) Schematiclongitudinal profile, showing of theunconstricted fiver (Hr) and the backwater head (Ht,) notationused in the energyrelation given in the text and in that developupstream of CrystalRapids for conditionsof Figure10. WS indicateswater surface. critical (choked) flow are shown. In Figure 10b the calculatedand observedheights of the hydraulicjump Applicationof the Bernoulli equationto any two cross (normal wave, N.W., in Figures 5 and 8) are shown. sections1 and2 givesthe totalenergy balance as Figure 10c showsthree importantvariables: (1) the flow 2 2 velocityin region3 just upstreamfrom the hydraulicjump U1 U2 (upper,solid curves),(2) the flow velocityin region4 just z•+D• +•-• =z2 +D2 +•-• +E (16) downstreamfrom the hydraulic jump (lower, dashed whereE is the energydissipated between sections 1 and2. curves),and (3) a heavystraight line at 9 m/s that indicates The notationhere has been slightly changed from that used the velocity chosenas a thresholdvelocity at which the in the generalequations of section2 to be consistentwith large bouldersof the Crystal debrisfan would be moved Figure 9; the changein conventionfrom the variable h (seeKieffer [1985] for a discussionof thisand more details used in section2 for water depth to the variable D is on this figure). Figure 10d showsthe velocityin region2, Kieffer: GEOLOGIC NOZZLES 27,1 /REVIEWS OF GEOPHYSICS ß 13

18 14 (a) Specific head (b) Height of jump

16 12

14 10

- '• - i - 12 (c)u3 (solid), u,(dashed) 0j (d) u, O'?"•11 18 10

16 - / 14 - •.;,:,,,,'.,,',,'_,,,' :/ _ 12 _ (e) •u th'roughjump 10

10 _ // '' / ( ''•• o.•

2 i I I I 0 500 1000 1500 2000 25000 500 1000 1500 2000 2500

Discharge(m3/s) Discharge(m3/s)

Figure 10. Summaryof the shallow-waterflow calculationsfor CrystalRapids given by Kieffer[1985]. 14 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer:GEOLOGIC NOZZLES

the constriction, and the reference line at the threshold relativeto the bed of C and D, possiblyto a fixed (bed- velocityof 9 m/s. Figure10e shows the change in velocity rock?) position. Curve A showssubcritical flow condi- (%- u4)through the hydraulic jump. In all partsof Figure tions. CurvesB-E all showsupercritical flow conditions. 10 thecurve appropriate to thegeometry of therapid prior For theseconditions, the height of thehydraulic jump and to 1983(shape parameter, 0.25) is shownas a heavyline. the velocitychange across it are givenbeside the vertical Careful inspectionof the differentparameters in the linerepresenting the jump. partsof Figure10 showsthat Crystal Rapids went through At dischargesbelow about 300 m3/sthe flow was the entire spectrum of nozzle behavior shown in the essentiallysubcritical (curve A in Figure11). At higher sketchesof Figure2 as dischargesincreased during the dischargesthe flow becamecritical and then highly 1983 flood. For example,at low dischargesand small supercritical(curves B-E in Figure11). The readershould shapefactors (tight constrictions) the flow wasessentially note the specificuse of the past tensein the last two subcritical;this can be seenin Figure10a by thefact that sentences,because the eventsof 1983 modifiedCrystal thereis nobackwater head at low valuesof discharge,and Rapids,and thesecalculations are not appropriateto its in the otherfigures by theabsence of hydraulicjumps and currentconfiguration. theirassociated velocity changes at low discharges.These A hydraulicjump is requiredin thediverging section of ideasare summarized more graphically in Figure11, which therapid to deceleratethe supercritical flow and to dropits showsschematic longiiudinal water surface profiles and energyfrom the backwater head H b backto theambient channelbottom configurations.The bottom level of the downstreamhead H r . Thusthe modelsuggests that the channel changed with dischargein 1983 becauseof largewave that stoodin the divergingsection of Crystal erosion; the decreases shown are those measured 16 km Rapidsat highdischarges can be interpretedas a normal upstreamat the GrandCanyon gage station. In thisfigure hydraulicjump arisingfrom the severeconstriction of the thefive curvesand bed elevations represent the following channel(Figure 2). In hindsight,this wave can be conditions:A, 283 m3/s and moveable bed A; B,850 m3/s recognizedas having been present all through the andsame bed level as A; C, 1400m3/s with bed lowered 1966-1983 phaseof CrystalRapids. However,because by 1 m relativeto thebed of A andB; D, 1700m3/s with the largerock in thisvicinity acted as a small-scalenatural samebed as C; andE, 2550m3/s with the bed lowered 1m weir, the energy changeof the flow around the rock

lO

8

w2/wo =0.2

& u=8.5 m/s

& D=4.5 m

-2 A•B c,D -4 • ...... :...... F•xed bed ...... :...... -5 0 1 2 3 4 5 1 2 3 4 5

Region Region

Figure 11. Schematiclongitudinal water profiles at Crystal the textand in Figure9 areshown schematically by the labels Rapidsfor the1983 discharges up to 2550m3/s, showing the below the graph. Dischargeconditions for curvesA-E are effectof channelwidening and bed erosion on theheight of the discussed in text. hydraulicjump. The partsof the rapid(regions 0-5) definedin Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß 15 contributedsubstantially to the energyof the wave at low dischargeswere kept relatively high (of the order of 1400 discharges,so that the role of the constrictionwas not m3/s)for nearlya yearafter the peak discharges in the recognized. Only when the wave strengthenedwith summer of 1983. Nevertheless,photographic evidence increasingdischarge, rather than washing out, was the role (Figure 12) suggeststhat this erosiondid occurand that it, of the constrictionrecognized. in fact, occurred in and downstream of the constriction, As the dischargethrough Glen CanyonDam rosefrom just where this model suggestsvelocities should have been 850m3/s to about 1700 m3/s in June 1983, the height of the the highest. In Figure 12, note a large rock, indicatedby wave increased,as would be expectedfor a hydraulicjump an arrow in both photographs,and the widening of the in a channelof fixed geometry(in Figure 10b,compare the channel immediately downstream(right) of the rock in left four data bars with the heavy curve). At higher Figure 12b, taken after recession of the high water. discharges,however, the height of the wave decreased, rather than increasingas predictedby the calculations(in Figure 10b, comparethe right data bar with the heavy line). •.•.-•-•.x•:.:-- ' •, '.•:•:.%...:-•-•.,.-•.-:•:.-•:., * }.:;...... •.....:..: This puzzling observationcan be explained if the magnitudesof the flow velocitiesin the constrictionand upstreamof the hydraulicjump are examined(Figures 10c and 10d; seealso the profilesin Figure 11). In supercriti- cal flow, water acceleratesin the convergingsection of the .....5': .:' ...... -,..::•:: nozzle,reaching critical velocity u 2 in the throat. The water continuesto accelerateout the divergingside of the constriction,reaching a maximumvelocity u 3 immediately upstreamof the hydraulicjump. A suddendeceleration to velocityu 4 occursacross the hydraulicjump as the flow deepens.For example, ata dischargeof 1400 m3/s with a constrictionof 0.25, the calculatedvelocity in the constric- tion,u 2, is 9 m/sand the velocity increases to u3 = 14 m/s Figure 12. Comparisonof the shorelineat Crystal Rapids (a) just upstreamfrom the hydraulicjump (Figures 10c and before and (b) after the 1983 high discharges.The text discusses erosion along the shorelinebetween the rock indicatedby the 10d). The velocitydecreases to u4 - 3 m/s just down- arrow and the boatsindicated by V' s in Figure 12b. streamfrom thejump. Considerationof the Hjfilstromcriterion for particle movement [Hjiilstrom, 1935] and of unit streampower showsthat water movingat 9 m/s can move bouldersthat (Comparisonof thesephotographs is a bit difficult because are 1-2 rn in diameter,the characteristicsize of the large thedischarge inFigure 12a is about 283 m3/s and in Figure boulders of the Crystal debris fan. Therefore when 12bis about 170 m3/s. Even though the discharge islower velocitiesreached this magnitude(at dischargesin the in Figure 12b than in Figure 12a, boats(indicated by V's) rangeof 1400-2000m3/s as shown in Figure10), large- can be seenin an "alcove"in Figure 12b, a positionthey scaleerosion began; that is, the river was able to begin could not have obtainedat the dischargeshown in Figure contouringits own nozzle. Material was erodedfrom the 12a, eventhough there is a higherstage in Figure 12a that sides of the channel and from the river bottom. Vertical should have submergedthe rocks in this vicinity. This erosion scoured the channel in an upstream direction suggests that rocks were indeed removed from this (headwarderosion); lateral erosionincreased the width of vicinity.) Additionally,in the field it can be observedthat the throat. An observerstanding on the shorecould not see thereis a freshcut banknearly 2 rn in heightalong the area this erosiontaking place but could hear loud, bassbooms that is indicatedschematically as regions2 and 3 in Figure as boulders moved in the current. 9; this cut bank is visible as a faint line to the left side of Channel widening at the throat can accountfor the the boatsin Figure 12b. Similar cut banksare now being observed decreasein height of the hydraulic jump. recognizedas signsof fiver channelerosion at a numberof Comparison of the observed wave height with that rapidswhere there have been fresh debrisflows (Granite predictedfor a normal hydraulicjump as the discharge Rapids,Elves Chasm) and indicatethe powerfulability of changedfrom 1700 to 2600 m3/s suggests that the channel the river to cut its own channel. widened from a shape parameter of 0.25 to about In a channelwith a nonerodiblebed the flow velocity 0.40-0.42, a widening of 12-13 rn (Figure 10b). The would be expectedto increasewith discharge,but in this proposed erosion was difficult to document because situation where the fiver channel was erodible, the flow 16 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer:GEOLOGIC NOZZLES velocity becomesfixed at the thresholdfor erosion. This extrapolationof calculationsat CrystalRapids can be used number was chosensomewhat arbitrarily on the basis of to estimatethat a floodof 11,000m3/s might have been the Hjiilstrom criterion as discussedabove, but it is requiredto enlargethe constrictionsto the value of 0.5 consistentwith the only available measuredvelocities. observedfor mostdebris fans [Kieffer, 1985]. This is not Threekayaks were filmed goingthrough the rapidon June an unreasonablevalue, becauseit is known that a flood of 27,1983, when the flow was 2600 m3/s. The trajectory for 8500m3/s occurred in 1884. the one kayak for which the best data were obtainedis The calculationsdescribed here attribute all changesin shownin Figure 5d. Measuredvelocities were as follows: flow regimeto lateralconstriction, because of the assump- immediatelyupstream of point 1, 8.5 m/s;point 1 to point tion of constantspecific energy. In all rapidsthere are 2, 9.8 m/s; point 3 to point4, 8.7 m/s. The kayak stalledto changesin bed elevationthat affect the total and specific an averagevelocity of 3.3 m/s betweenthe troughand crest energy of the flow and, therefore,affect the transitions of the hydraulicjump in this region before accelerating from subcriticalto supercriticalconditions. The words down the backsideof the wave to a velocity of about 8.5 "subcritical"and "supercritical"as usedin this section,and in the following summary,therefore apply to a large-scale In summary,a fascinating,and often tense,feedback descriptionof the rapid,not to local details,because these process involving meteorology, fiver hydraulics, and additionaleffects are not accountedfor. Most rapidsare engineersbegan in June1983 andcontinued into early July weakly supercriticalbecause of changesin bed elevation, as the dischargeincreased; this process can be followedon evenin theregime called "subcritical" in thisPaper. There the curvesof Figure 10. As snow melted in the Rocky are also substantialsmall-scale irregularities in bed Mountains,engineers raised the discharge through the dam topography,such as ledges and rocks, that cause local higherthan the 850 m3/s released during the previous two supercriticalflow. The behaviorof the fiver aroundsuch decades. At the tightly constrictedspot in CrystalRapids, obstaclesis not includedin thisgeneralized discussion. a large hydraulicjump formed becausethe flow became The calculations indicate that when constrictions of the highly supercritical.As the dischargesapproached 1700 Grand Canyon debris fans reach the value of 4).45, the m3/s,the fiver began eroding its channelthrough the flow will be essentially subcritical at all discharges. Crystal Creek debrisfan. In responseto the widening,the Although some standing waves and local regions of flow velocitiesdecreased. If the dischargehad been held supercriticalflow exist in mostof the rapidsof the Grand constant,the channel and hydraulic featuresof the flow Canyon, the wave at Crystal Rapids was unique: other would have stabilized when the channel became wide rapids that are less tightly constricteddo not have strong enough to reduce flow velocities in all sectionsbelow normalwaves (river raftersmight disagree,because many about 9 m/s. However, more snow melted in the heM- of the existing waves are strong enough to flip rafts, watersof the ColoradoRiver, and engineerswere forcedto but--at the scale of convergent-divergentconstrictions increasedischarges through the dam toward 2600 m3/s. In consideredhere--these are local features). responseto the increaseddischarge, flow velocitiesagain Over geologic time, the flow in Crystal Rapids can increased,and erosion of the channelcontinued; by the changefrom subcriticalto supercritical(or vice versa)in time of peak discharge,enough lateral erosion of the two ways: (1) as dischargechanges from seasonto season channelhad occurredthat the heightof the hydraulicjump in a channel of fixed constriction, or (2) as channel had decreased.It is not clear at this time whetherthe high constrictionchanges with time becauseof erosionduring flows were sustainedlong enoughfor the channelto take large floods or becauseof tributary debris flows. This on a shapein equilibriumwith the highdischarge. evolutionis summarizedin Figure 13, a versionof Figure 2 that showsexplicitly the responseof channelshape and flow structuresto changesin discharge. The sequence Implicationsfor GeomorphicEvolution in the Grand shownrepresents but one cycle in recurringepisodes in Canyon which debrisfans are enlargedby floodsin the tributaries Even after the channel was widened by the high andthen modified by floodsin the main channel. dischargesof 1983, the constrictionof 0.40-0.42 at The beginningof the sequenceis arbitrarilychosen as a CrystalRapids is still significantlybelow the valueof 0.5 time when the main channel is relatively unconstricted characteristicof the maturedebris fans along the Colorado (Figure 13a). The fiver is suddenlydisrupted and ponded River (Figure6), and the rapid is significantlydifferent in by catastrophicdebris fan emplacement(Figure 13b), hydraulic characterfrom rapids at locales where the forminga "lake"behind the debrisdam. The surfaceover constrictionis 0.5. This observationsuggests that most which water poursacross the freshlyemplaced debris fan debrisfans in the Grand Canyonhave been subjectedto is calleda "waterfall"in this model. As the pondedwater floodslarger than the 1983 flood. With properrecognition overtopsthe debrisdam, it erodesa channel,generally in of the simplicityof the model and the paucityof data, the distal end of the debris fan (Figure 13c). This is the Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß 17

of rocks,labeled "rock garden," below the divergentstretch (a) Initial channel (b) Side-canyon flood of the rapid. Rocks in the rock gardenin Crystal Rapids geometry •o•6e6 arevisible at 285 m3/s,cause substantial waves at 850

o "•j•waterfall" ,, •.. m3/s,and are submerged at2600 m3/s (as shown inFigure 5d and its indexFigure 5e). Rock gardenslike thiscan be found below a number of major rapids in the Grand Canyon (for example, below Granite Rapids and Bright (d) Erosion: moderate flood, Angel Rapids). This theory suggeststhat the rocks that (c) Erosion:small floods supercriticalflow form the rock gardenshave beentransported there from the • w2,--,O.4wo originaldebris fan depositsunder a rangeof differentfluid HJ flow conditionsas the ColoradoRiver has progressively enlargedits channelduring big floods(Figures 13c- 13]). Mature debris fans that once blocked the Colorado n River and now have channels cut through them have (e) Erosion:large flood, (f) Longitudinalcross-sections progressedwith time from the bottomto the top of Figure subcriticalflow 'lake"••'waterfall" 2 (or, cyclically,from Figure 13a to 13e), becausenatural (b) .• w•O'5wø..... floods have been large enough to create subcritical ,,. channels from the initially supercritical constrictions. Controlleddischarges through Glen Canyon Dam will not permit CrystalRapids to evolveto the configurationof the older debris fans. However, the controlled releases from thedam (1415 m3/s when only the generatom and bypass Figure 13. Schematicillustration of emplacementand modifica- tubesare used,greater values if the spillwaysare used)are tion of debris fans in the Grand Canyon, modeled after the sufficient to significantlyalter the channel of the fiver processesobserved at Crystal Rapids during 1966-1983. HJ indicatesa hydraulic jump; w indicatesriver width, with the shouldit becomeblocked or tightlyconstricted by tributary floods in the future. subscripts0 and 2 standingfor regions0 and 2 as defined in Figure9a. What perspectivedoes this interpretationof the history of the hydraulicsand geomorphologyof the Colorado River give us? First, the interpretationof a major wave as beginningof evolution of a "rapid" from a "waterfall." a hydraulicjump arisingnot from a rock but from large- Observationsof naturally emplacedearth dams suggest scalechannel geometry provides a differentperspective for that the breachingof the Crystal debris dam probably monitoringand predicting events at newly formedrapids, a happenedwithin hoursor daysof its emplacement. perspectivevaluable for National Park Service officials Unless the debris dam is massivelybreached by the concernedwith navigationalsafety. Any rapid newly first breakthroughof pondedwater (that is, unlessenough formed by a debris flow may exhibit hydraulic features material is removed that the shapeparameter is initially differentfrom thoseseen in channelsthrough older debris greaterthan 0.5), the constrictionof the river is initially fansbecause of the severeconstriction that can be present. severe. Floods of differing size and frequencyerode the Suchrapids should be monitoredclosely if unusuallyhigh channelto progressivelygreater widths (Figures 13c, 13d, dischargesare put throughGlen CanyonDam. and 13e). Small floodsenlarge the channelslightly, but Second, the interpretation of critical flow in the constricted,supercritical flow is still present(for example, constrictionand supercriticalflow downstreamfrom the as in CrystalRapids from 1966 to 1983). Moderatefloods constrictionpredicts erosion in quite differentplaces than enlargethe channel further (perhaps to a constrictionw2/w o would be found in a subcriticalnozzle, and the interpreta- ~ 0.4). The geometryobserved at any instantreflects the tion of the shapeof the fiver channelallows modelingof largestflood in the historyof the fan (for example,floods sizesof ancientfloods. Interpretationof the flow in this of 11,000rn3/s appear toenlarge the constrictions along the way also allows a mechanismfor transportinglarge ColoradoRiver in the Grand Canyonto a shapeparameter boulders a significant distance downstreamfrom the of about0.5). At any time, depositionalevents in the main original debris depositinto rock gardensbecause of the channel or the tributariesmay complicate this simple high velocitiesthat can occurdownstream of the constric- descriptionof the erosionalhistory. tion in supercriticalflow. Without such a mechanism, Rocks from the debrisfan are transportedas far as 1 geologistsare faced with the dilemmathat tributarydebris km downstreamby the high-velocitywater in the conver- flows from side canyonsare building weirs acrossthe gent,constricted, and divergentregions. Notice, in Figure canyon,and downcuttingthrough these weirs is difficult 5a (and in its explanation,Figure 5b) thatthere is a deposit becauseonly local abrasionor chemicalsolution can be 18- REVIEWS OF GEOPHYSICS / 27,1 Kieffer: GEOLOGIC NOZZLES

invokedto get rid of the rocks. The relativeroles of local abrasion, solution, and downstreamtransport of large particlesmust be understoodbefore quantitative models of infilling versusdowncutting of the GrandCanyon can be formulated. Finally, the framework of supercriticalto subcritical evolutionof the rapidswith time and throughflood events of differentsizes suggests new directionsfor geologicand hydraulicobservations at the rapids: searchesfor geologic evidenceof theestimated 11,000 ma/s prehistoric flood (eventraces of thehistoric 8500 m3/s flood are difficult to find); evaluationof the relative roles of lateral constriction versus vertical topography on the channel bottom; documentationof wave behavior as dischargechanges; laboratoryexperiments on the three-dimensionalshapes of hydraulic jumps in flumes of converging-diverging geometry;development of criteria for transportof large boulders(>1 m diameter)by fluids (a researchtopic that will be discussedagain in the concludingsection of this paper on the Mount St. Helens lateral blast); and evalua- tion of the relative frequenciesof tributaryversus main stemfloods in determiningthe rate of downcuttingby the ColoradoRiver within the Grand Canyon. A new data base for these studies is available in the form of detailed maps of the topographicand hydraulic features of the Figure 14. Old Faithful geyser,Yellowstone National Park, in Colorado River channel in 10 important locations: at eruption. The column is about 30 m high. Note the discrete HouseRock, 24.5-mile, Hance, Cremation-BrightAngel, elementsof fluid in the eruptioncolumn. Theseare the "surges" referredto in the text. Photographby G. Mendoza. Horn Creek, Granite, Hermit, Crystal, Deubendorff,and Lava Falls Rapids[Kieffer, 1988]. Faithful led McKee et al. [1981] to suggestthat under- groundgeyser eruptions were occurring at Karkar. 5. OLD FAITHFUL GEYSER: A TWO-PHASE NOZZLE Harmonictremor is a relativelymonochromatic seismic motion that often precedesor follows volcaniceruptions; GeologicSetting as a precursor it has been invaluable in forecasting Reportsof geysersand hot springsin the land that is eruptions,even thoughno theoryhas adequately explained YellowstoneNational Park began in the early part of the its origin. Harmonictremor is also an importantcompo- nineteenthcentury. Old Faithful Geyserhas sincebecome nent of the seismic noise characteristicof geothermal a familiar symbolof the westernlands of the United States fields, and it is thus potentiallyan importantprospecting and their nationalparks (Figure 14). More recently,Old tool for geothermalenergy sources,sources that may Faithful became an important focus of scientific studies provideonly a few percentof the total energyrequirement when striking resemblancesbetween a peculiar type of of the United Statesbut that could provide a substantial volcanic seismicityknown as "harmonictremor" (Figure and critical portion of the energy requirementsof many 15a) and the seismicityof the geyser were noted and countriessurrounding the Pacific basin. In this sectionI explained(Figure 15b) [Kieffer,1984a]. Figure 15b shows summarizemy observationsand theory for the origin of a seismic record of about 1 day of eruptions at Old harmonictremor at Old Faithful, the importanceof fluid Faithful, showing 11 completeeruption cycles (a higher- propertiesin interpretationof the data, and my ideasabout resolutionrecord of one completeeruption cycle is shown the complexprocess that occurswhen the geysererupts. in Figure20 andis discussedin detaillater in this section). For perspective,a fluid dynamicistmight imagine Old Each activeperiod of seismicityof Old Faithful endswith Faithful as a vertical, open-ended,two-phase shock tube an eruption and a characteristiceruption coda (a good with variable crosssection. In addition to the compres- example of an eruption coda is on line 11, beginning sibility effects that normally dominate shock tube betweenthe first and secondtime marksand extendingto dynamics, gravitational (hydrostatic) effects strongly the fourth). The similarity of the record from Karkar influencethe fluid propertiesin the geyser,because each Volcano(Figure 15a) to geyserrecords such as that of Old meter of liquid water (providing approximately0.1 bar Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß 19

Study of this complex shocktube must be done under very restrictedconditions. Whereas a majorproblem in the study of the ColoradoRiver discussedin the preceding sectionis inaccessibility,an equally major problem in studyingOld Faithful geyseris its accessibilityand public visibility. Observationsclose to the vent mustbe madeon the few days of the year when work will not detractfrom tourists' enjoymentof the geyser (namely, when Yel- lowstonePark is closedfor snowplowingof the roads in late winter), and experimentsor observationsmust be designedto avoid even the slightestdamage to the geyser (for example,no hole can be dug to allow positioningand anchoringof a seismometer). Thus the challenge in Figure 15. (a) A seismicrecord from Karkar volcano,Papua studyingOld Faithful is to learnas muchas possibleabout New Guinea. The white bandsare strongseismic activity at a the innerworkings of a complexnozzle from very limited frequencyof 2-4 Hz. The patternrecurs at intervalsof about70 observations. The basic data set consists of float and min. (b) A seismicrecord of about 1 day of eruptionsat Old thermocouplemeasurements made in 1949 [Birch and Faithful. Time marks in both records indicate 1-min intervals; Kennedy,1972], seismicand movie data taken from 1976 invertedV's in Figure 15a emphasizefaint time marks. (Figures to 1984 [Kieffer, 1984a], and unpublishedpressure and courtesyof C. McKee and R. W. Johnson[see McKee et al., temperaturedata that J. Westphaland I obtainedin 1983 19811.) and 1984 (summarizedby Kieffer and Westphal[1985]).

The Rechargeand EruptionCycle Old Faithfulerupts into a tall, continuousvertical jet of water and steam(Figure 14). The maximumheight of the geyserranges between 30 and 50 m, dependingmostly on wind velocity and, to a certain extent, humidity (which affectsthe visibility of vapor in the eruption). Eruptions lastfor ~1.5-5.5 min andoccur every 40-100 min (Figure 16). Measurementsof total dischargesuggest that about 0.114m•/s (114 kg/s; 1800 gal/min) iserupted during the initial and steady-flow stages of the eruption (E. Robertson,U.S. GeologicalSurvey, private communica- tion, 1977). The conduitof the geyser,or the nozzle, is a fissure pressure)changes the boiling temperatureof the fluid by that is flared at the surface, narrows to a constriction at about 2øC. Thus, in contrast to a shock tube in which about 3-5 m depth, and probablydiverges into one or initial conditionsare usuallyisothermal and isobaric,the more caverns below this depth. (Figure 17 shows a initial pressureand temperatureconditions in Old Faithful possibleconfiguration based on calculationsdescribed by are not uniform. The rechargecycle of the geyser is Kieffer [1984a]; available data permit many other con- analogousto the processof filling a shock tube driver figurations,so the shapeshown in this figure shouldbe sectionwith a volatile liquid, suchas liquid Freon,that can consideredhighly schematic.)About 0.5 m below the rim boil upondecompression. At Old Faithful, however,there of the geyserthe fissureis 1.52 x 0.58 m, and I will take is no physicaldiaphragm to containthe fluid as there is in these as the dimensionsof the exit plane of the fluid a laboratory shock tube. The natural diaphragm is the becausethe actual geyseritesurface is very irregular highestwater in the conduitthat maintainsa temperature (includinga petrifiedtree stumpvisible as the knobon the of 93øC (the boiling temperatureat the 2200-m elevation left side of the cone in Figure 14). Probe work by J. of Old Faithful) and sufficientpressure to keep deeper Westphal, R. Hutchinson,and S. Kieffer (unpublished water from massiveboiling and eruption. Finally, whereas data, 1983, 1984) suggeststhat the constrictiondimensions in the laboratoryan experimenterwould worry about the are approximately0.1 x 1.5 m. The depthof the conduit "cleanliness"of eruption initiation over time scales of that can be reachedby a probeis 22 m (this regionis called microsecondsto milliseconds,at an eruption of Old the "immediatereservoir"); it is plausiblethat water from Faithful we may be lucky to forecastthe initiationtime of greaterdepths is ejectedduring a long eruption. Water fills an eruptionto within 10 min! the conduitonly to within 6 m of the surface(Figure 17). 20 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer: GEOLOGIC NOZZLES

(a) (b)

I ERUPTIONS

40 60 80 100 2.0 3.0 4.0 5.0 6.0 DURATION IN MINUTES Figure 16. (a) The relativefrequency of intervalsbetween eruptionsof OldFaithful. (b) Tl•e relative frequency of eruption. All dataare for 1979(unpublished data provided by R. Hutchin- son, U.S. NationalPark Service,from analysisof 3308 eruptions).

The maximumlength of the water columnin the immedi- lengthof the reposeinterval, I, following an eruptionwas atereservoir prior to an eruptionis thereforeabout 16 m. quite closely related to the durationof the eruption,D. For nearly a centuryafter its discovery,Old Faithful The empiricalformula maintaineda fairly regular pattern of eruptions,with intervalsbetween eruptions averaging 60-65 min. During ! = 10D + 30 (17) mostyears, Old Faithfulexhibited two typesof eruptions: "shorts,"which were 2.5-3.5 min in duration,and "longs," (I and D in minutes)proved very usefulto the National whichwere about5 min in duration.During the 1970sthe Park Servicefor predictingwhen eruptions would occur.

SCHEMATIC PLUMBING AND FILLING SEQUENCE MINUTES 15 30 60 t > 32 min 32

6 z

0 8 <

o-10 ? ? ? (D) •6 i "r' ,-- 1_18 ! Q. i ...i oo•B)

i 22

a b c d e Figure 17. (a) Rate of filling of Old Faithful,as inferredfrom diagramsof theconduit of Old Faithful.The solidline represents reportedfloat measurementsin 1948 [Birchand Kennedy,1972]. the width versusdepth in the widestdirection (approximately Time t = 0 was taken when the probewas lowered. Recent east-west),and the dashedline representsestimated narrowest (unpublished)work by the author and J. Westphal shows dimensions. The filling rate shown is based on the float differencesin detailsof the filling rate but doesnot changethe measurementsin Figure 17a and an assumptionof constant generaldiscussion chosen for this paper. (b)-(d) Schematic rechargerate. Time until thenext eruption is indicatedat thetop. Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß 21

In the pastfew yearsthe duration-intervalbehavior has [Kieffer, 1984a, p. 66]. A workingmodel for rechargeof changeddramatically, although changes in the observable fluids and heat to the geyseris basedon measurementsof characteristicsof the geyserduring an eruption(such as depthand temperatureversus time by Birch and Kennedy height versus time) have not been apparent. Intervals [1972]. Water rises slowly up the conduit during the averaged over a month commonly exceed 75 min, and rechargeinterval, as shownby the levels A-I in Figures individual intervals have sometimes exceeded 100 min. 17b- 17e. During the rise, temperaturesrange from 93øC The interval-durationequation no longer applies to the at the surface to 116 ø- 118øC at the bottom of the immedi- samestatistical accuracy; in 1987, shorteruptions ceased atereservoir. The temperatureat any depthfluctuates with for a while, and only long eruptionsoccurred (R. Hutchin- time, as can be seen, for example, by examining the son,National Park Service,private communication, 1987). temperature-depth-timecurves in Figure 18; note espe- The mysteriesof geyser eruptionshave long intrigued cially the temperaturebehavior at 18 m depthduring the scientists(the original theory of the inner workings of period 35-25 min before the eruption. However, the geyserswas publishedby Bunsenin 1846), but questions overall effect is a generalheating of the water relative to about the inner workingsof Old Faithful have taken on a the referenceboiling curve, shown on each temperature- new urgency because of these dramatic changes in depth curve in the sequencein Figure•18. The hottest behavior. water at the bottom is about 7ø-9øC below the boiling temperaturefor the totalpressure at the bottom,-0.23 MPa The RechargeProcess: Clues to GeothermalSeismicity (2.3 bars) (0.08 MPa (0.8 bar) atmosphericpressure plus After an eruption ceases,the conduit is empty (or 0.14 MPa (1.4 bars) hydrostaticpressure). Becausethe nearly so) and mustbe rechargedwith both water and heat. deep water is hotter than the shallow water, heat for the Estimatesof total volumeerupted and conduitdimensions rechargecycle is most likely suppliedby the additionof give an approximaterecharge rate of 6 kg/s (liquid water) hot water or steam at the base of the immediate reservoir,

I IITI•, IlllllU[f•5 45 40 35 30 25 20 15 10 7.5 5 2.5 1.5sec

e f o• 6 E .E 12 b c %, ,•d ', ',

c3 18 ,,.a "•'-.. •' -.. " -.. "•'x-.. '"'Nt.. '- '"' • " " \".. /'-.. 1"--. '\" \' k" ",,' "'., '",, 93 116øC 93 I 16øC 104 104 Figure 18. Relationsbetween seismicevents, level of filling, (Bottom) A histogramof the number of seismic events per and temperaturein the conduitof Old Faithful. (Top) Data on minute througha rechargeinterval at Old Faithful. Letters a-f depthof water in the conduitat varioustimes (graphsabove the show approximatecorrelations between temperature-depth curves histogram)versus the temperatureof the water and versus the and the histogram. referenceboiling curve (the top curve in each small graph).

1.0 I I I I I I I I I I I Z

• 0.9

E 0.8

(d?) • 0.7

> 0.6

c o 0.5

E 0.4

'o 0.3 N • 0.2 E o z 0.1

l i i 1 20 30 50 60

Time, minutes 22 ß REVIEWSOF GEOPHYSICS / 27,1 Kieffer: GEOLOGIC NOZZLES

indicated by flow arrows and bubbles at the bottom of correlationis not great becausethe seismicdata and the Figures 17b- 17e. temperaturedata were obtained40 yearsapart (seeKieffer Two processesprobably contribute to the mixing of the [1984a] for details). In Figure 18 the initial time (t = 0) for hot, deep water with the coolersurface water: convection the histogramof seismicevents shown was taken at the and migrationof steambubbles, illustrated schematically beginningof an eruptionthat was about4.5 min long. A in Figures 17b and 17d. It is not clear that the two 66-min interval followed before the next eruption. The processescan be distinguishedby usingavailable measure- first appreciableseismicity started at about21 min (that is, ments. Much of the deeperwater is too cool to boil under 45 min before the next eruption), and the associated the total pressureat any given time (Figure 18). However, temperature-depthconditions are indicatedby the frame on since the temperatureof the deeper water exceeds the the left of the top line at 45 min and by the data at the two atmosphericboiling temperatureof 93øC, this water is letters"a." Successivegraphs in this figure labeled with superheatedrelative to the boiling point at atmospheric decreasingtimes (40, 35, 30 min..... down to 15 s) show pressure. If suchsuperheated water is convectedupward, the gradual filling and heating of the geyser and the steam bubblesform if the pressuredecreases below the correlations with seismic details. The cause of the saturationpressure of the water. For example,3% of the seismicitywill be summarizedin thispaper, but an equally deepwater at 116øCwill transformto steamwhen the total fascinatingproblem is the causeof the two periodsof pressuredecreases to 0.1 MPa (1.0 bar). As steambubbles seismicquiet that occur 1 min and about 10 min prior to rise, however, they encountercooler water and may the eruption(see Kieffer [1984a] for theoryregarding this collapse(shown schematicallyby the asteriskin Figures phenomenonthat precedes all eruptions). 17b and 17d). Althoughthis processcannot be directly The collapseof a vapor bubble in a liquid of its own observedin the depthsof Old Faithful, it is easilyobserved compositioncan occur within milliseconds[Fujikawa and in a pot on a stove, as well as in othergeysers, suchas Akamatsu, 1978, 1979] and pressuresin the collapsing Strbkker in Iceland, where the steam bubbles rise into a cavity can be as large as a few to tensof megapascals(tens diverging surface pool and can be observed both to to hundredsof bars) [Fufikawaand Akamatsu,1978; Biasi collapse before reaching the surface and to reach the et al., 1972]. The high pressuresgenerated by the bubble surface and explode into a beautiful fragmentingshell collapsedecay quickly with distance,becoming seismic- (Figure 19). In geyserswhere the bubblescan be directly level disturbancesat distancesof only a few bubbleradii. observed,they frequentlyoccupy the full diameterof the The acousticnoise of collapsingsteam bubbles can be conduit,which may be of the orderof, or more than, 1 m. detected by seismometersplaced around Old Faithful (Figures15a and 20). The seismiccodas of two eruptions, labeledA and B, are shownin Figure 20. These codasare primarily due to water falling back from the top of the -.;:,:;. eruptingcolumn onto the ground (this fallback was just beginningto occurwhen Figure 14 was taken;it is visible on the fight sideof the photograph).Short, discrete bursts of seismicity occur throughoutmost of the recharge interval (four such events, indicated in Figure 20, are shown enlargedin Figure 21). The number of seismic eventsper minuteincreases as the geyserfills and the fluid becomesgenerally hotter (Figure 18, histogram). Two characteristicsof this seismicityare the relatively high frequencycontent of the individualevents (a few tens of hertz) and the characteristicdamping time of a few Figure 19. Bubble eruptingfrom the vent of Str6kker geyser, tenths of a second. The envelope of these signals is Iceland, through a surface pool of water. Bubble diameter is about 2 m. Note fine-scale structure on bubble surface. remarkably similar in shape to results obtained by (Photographby H. Kieffer.) Hentschel [1979, 1980] in laboratory experiments on collapsingbubbles, although the geysersignals are much Collapseof the bubbles,and releaseof their latent heat,is longer in duration (tenths of secondscompared with probablya major processby which heat is transferred hundredsof microseconds),presumably because of the upward in the water column [White, 1967], and the muchlarger size of the geyserbubbles. collapseof thesebubbles is believedto causethe individ- A collapsingbubble could cause the observedseis- ual seismic events observed. micity by generatingwaves within the fluid column, as The numberof seismicevents per minutean be crudely illustratedin Figures 22a and 22b. These waves set the correlatedto the temperature-depth-timeevolution of the fluid in the conduit into resonance;that is, the conduit is an water in the conduit (Figure 18); the accuracy of this organpipe filled with liquid waterand is setinto resonance Kieffer:GEOLOGIC NOZZLES 27,1/REVIEWS OF GEOPHYSICS ß 23 by the occasional(or frequent)collapse of a largesteam bubble.Detailed treatment of thedisturbance caused by a singlebubble as a hydraulictransient [Kieffer, 1984a] accountsfor the distensibility(approximately the elas- o ticity)of theconduit walls and allows the damping of the signalsto be calculated.

0 I 2 3 4

0720 ...... -- '

07:20:50 GMT O725 Begineruption A Enderuption A I

O73O

0735 TIME a b Event 0740 Figure22. Schematicdrawing of wavespropagating in a 0745 standingcolumn of waterof depthL. (a) Thecollapse of a steam

0750 bubble(asterisk) in thebase of thecolumn. (b) Onecycle of compression(C) and rarefaction(R) waves due to bubble 0755 collapsein the bottomof the conduit.

Evenin a fissurelikepipe with the irregular shape of 0805 Old Faithful,the characteristic fundamental frequency of

0810 oscillationinduced should be closeto the longitudinal , Event resonanceof a cylindricalpipe. The geometryof the 0815 conduit determines whether the conduit should be

O82O consideredopen-ended orclosed-ended. Even though the conduitmay flare with depth, the bottom seems relatively 0825 well defined,and so, in the absenceof otherdata, the Event4 0830 conduitis hereintreated as a closedpipe with resonant Eruption frequenciesgiven by O835 ;;:4..;.:-•,,g.::;;:•.;).1•-•:.: :::::;::, ,,, ' ' ' ...... I f = a/4L (18) Till IN IINUTIS

Figure 20. A seismicrecord of one eruptioncycle of Old whereL is thelength of thefluid columnin theconduit. In Faithful. Details of the seismicevents labeled 1-4 are shownin this equationthe soundspeed a is the effectivesound Figure21. To obtainthe histogram of Figure18, thenumber of speedof the fluid modifiedfor the distensibilityof the theseevents exceeding a peak-to-peak amplitude of 4 cm/swere counted. conduitwalls. Theresonant frequency for an openpipe wouldbe twice this value. Whereas a is 1440m/s for pure water,it decreasesto 1385 m/s if thedistensibility of the i i walls of the conduit is accountedfor. The lengthof the fluid columnin the conduit,L, Event 1 dependson the time in therecharge cycle (note in Figures -- i i 17 and18 howslowly the recharge process occurs): L is estimatedto be 8 rn whenthe seismicity is firstdetected Event 2 and 16 rn when the conduitis full near the time of an eruption(Figures 17c-17e and Figure 18). The cor- respondingresonant frequencies obtained from (18) are43 Event 3 and20 Hz, valuesthat approximately span the range of measuredfrequencies during the recharge interval (Figure 21). Fromhydraulic transient theory the characteristic Event 4 1 second dampingtime is 0.12-0.46s, in goodagreement with the durationof theobserved pulses (Figure 21). Figure 21. Detailsof thefour impulsive seismic events from Old The resonantfrequency associated with individual FaithfulGeyser as indicated in Figure20. Notethat the dominant seismicevents, and the duration of eachevent during the frequenciesrange from ~20 to ~40 Hz. pre-emptionseismicity at Old Faithful,can be explained, :24 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer: GEOLOGIC NOZZLES

to first order, as arising from hydraulic transientsin a eruptioncolumn developsand rises in a seriesof bursts slowly rechargingcolumn of liquid water. If the fluid in (typically one to eight) to a maximumheight of about the conduit were, for example, a boiling, two-phase 30-50 m. This part of the eruptionis typically 20-40 s mixture,the soundspeed would be dramaticallylower, and long. Two minor burstsand four major ones occurred the observedseismicity could not be explainedso simply duringthe first 20 s of the eruptiondocumented in Figure asarising from resonances of theliquid column standing in 23. the immediate reservoir. In the next section I describe the The third stage of the eruption, "steadyflow," is an transformationof the rechargedliquid in the conduitas an intervalof about30 s duringwhich the columnstays near eruptionoccurs and show how the associatedchange in maximumheight. During this time, surgesare observedin soundspeed of the fluid influencesthe resonantfrequen- the eruptioncolumn at a frequencyof about2 Hz in short ciesof the conduitduring eruption. eruptionsand 1-1.5 Hz in long eruptions(Figure 24b). The surgesare shownphotographically in Figure 14 and EruptionDynamics and Thermodynamics graphicallyfrom 20 to 45 s of the eruptiondocumented in An "eruptioncycle" of Old Faithful consistsof one Figure23. eruptionand one rechargeinterval. By convention,the The fourth stageof an eruption,the "decline,"begins start of an eruption cycle is taken as the onset of an rather dramatically,after approximately30 s of steady eruption,although the first part of an eruptionactually flow, with a drop in columnheight. The declinecan last beginsbelow groundlevel and cannotbe monitored(see up to 3 or even 4 min; during this time the heightof the the discussionof preplaybelow). The visible flow field column dropsto about 10 m, and play continuesat low and its variationwith time during an eruptionare collec- levels. tively referredto as the "play" of the geyser. The play Differences between long and short eruptions are consistsof four parts which can be distinguishedon a threefold: the frequencyof surgingduring the steadyflow graphshowing height of the eruptioncolumn versus time stage,the durationof the declinestage, and the seismic (Figure23). patternfollowing the eruption. Analysisof height-time The first part of an eruption (the part that begins data(such as thoseshown in Figure23) for manyeruptions undergroundand fails to maintain a height above the reveals that there is no correlationof maximum height surfaceof the groundfor more than a few seconds)is obtained,duration of maximumheight, or numberof bursts "preplay,"the ejectionof waterintermittently prior to the in the initiationstage with eruptionduration (Figure 24a). actualeruption (the lastfew episodesof preplaybefore the The only measurabledifferences in eruptionplay between eruption,from -15 s to 0 s, are shownin Figure 23). long and shorteruptions are the frequencyof the surges Episodesof preplaylast for a few seconds,and wateris (Figure 24b) and the durationof the declinephase, which thrown to a heightof a few metersor occasionallya few is simplytruncated at about2-3 min for a shorteruption. tensof meters. During preplay,water ejectedupward into Seismically,a period of about20 min of quiet follows a the atmosphereis cooledby expansionand entrainment. long eruption (Figure 20), whereas seismicity begins When this cooled water falls back into the conduit, it immediatelyafter a shorteruption. After shorteruptions mixes with the heatedconduit fluid. The resultingoverall we are able to hear water splashingin the bottom of the cooling of near-surfacewater can delay the onsetof an immediate reservoir,which suggeststhat it is not com- eruptionuntil the fluid has been reheatedto initiation pletely emptied during short eruptions and that the conditions. seismicityarises within the immediatereservoir. The second stage of the eruption is called herein Water near the surfaceof the rechargingcolumn boils "initiation and unsteadyflow." During this stage, the nearly continuously,and vigorousboiling can accelerate

09 30 n,' -Th,•,,ne-Max,lmum he,gh:• ' • •..• '• ' ' OLDbAITHFUd SHORT •:RUPTIOI•I -

:2:2 20 Xz O- _Heavyline-In•iu•isCltt•a•l ]•1[• '7 -xv •-•_ •A•k.UGUg7T:•20, pIM977 _ •..(• lo

o _Pre• jUn!l!ewady IStf•øawdY I I Declile •,V,vT Y, ,•i - 10 0 10 20 30 40 50 60 70 80 90 100

TIME IN SECONDS

Figure 23. Height versustime for a short eruptionof Old Figure 14, in order to obtain the frequency of surging Faithful. The light line is the heightof the water-streamcolumn; (approximately1-2 Hz). the heavier line traces individual pulses of water, visible in Kieffer:GEOLOGIC NOZZLES 27,1 /REVIEWS OF GEOPHYSICSß 2,5

© 50 • 4

ß

ß• 40 ß

ß

E ß E 30 ß ß ß ***dJl 6 ß ß E ß _ .,.. 20 ß ß ß o ß ß ß

0 I I I I I 0 I 2 3 4 5 u. 0 I 2 3 4 5

Duration (min.) Duration (min.)

Figure 24. (a) Durationof steadyflow stage(maximum height) correlation. (b) Frequencyof surgesversus duration of eruption versusduration of eruptionof Old Faithful geyser,showing no showinginverse correlation. fluid severalmeters upward. Early in the rechargecycle, (suchejection can occur when the fluid is withinabout 6 m the fluid is too deep in the conduitfor vigorouslyboiled of the top of the conduit) and (2) the underlyingwater water to be ejected over the rim. Thus even though mustbe hot enoughso that the massunloading triggers a vigorousboiling and splashingoccur within the conduit, positivefeedback process. the pressuredistribution at depth remains unchanged A simplifieddiagram of Old Faithful modeledas a because fluid is not removed from the conduit and an layeredshock tube is shownin Figure25. In the example eruptioncannot begin. An eruptiontherefore begins when shown I have arbitrarily divided the fluid in the column two criteria are met: (1) the water must have risen high into sevencells of differentlength. Cell A haspressure- enough in the conduit so that vigorous boiling can temperatureconditions that follow the referenceboiling dischargesome water over the rim, therebyreducing the curve. I chosethe lengthof 1 m for this zone somewhat hydrostaticpressure on the fluid by removalof somemass arbitrarilyafter watchingthe onsetof about200 eruptions

90 100 110 120 130

90 100 110 120 130 TEMPERATURE (øC) Figure 25. Old Faithful modeledas a shockrobe. The initial statesin the unsteadyunloading process of an eruptionof Old configuration(Figure 25a) shows layers of fluid at different Faithful. The segmentsshown with large "bubbles"are boiling; initial pressuresand temperaturesin theirposition relative to the the segmentsshown as a fine "mist" are erupting from the referenceboiling curve in Figure25f. In Figures25a and25f the conduit;and the threesegments shaded black are liquidH20 initial temperatureis assumedconstant within six convection becauseof the relationsof parts of cells D, E, and F to the cells (labeled B, C, D, E, F, and G). The fluid in cell A is reference boiling curve during the unloading process. The assumedto be at pressure-temperatureconditions on the boiling thermodynamicpath of the unloadingprocess is discussedin the curvein orderto triggerthe eruption.Figures 25a-25e showfive text. 26 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer:GEOLOGIC NOZZLES andestimating the amountof waterejected. Cells B, C, D, may play a prominentrole in ejectingthe differentlayers E, F, and G are each assigneddifferent, but uniform, of water [Kieffer, 1984a], or chokedflow may be more temperatures;they might be thought of as simplified importantin controllingthe massflux rate (see following convectioncells. The overall temperaturedistribution of discussion). Resolutionof this questionmay only arise the cells mimicsthe temperaturecurve measuredby Birch from detailedtheoretical or laboratorymodeling because of andKennedy [1972] at 15 s beforethe onsetof an eruption the extreme difficulty of field measurementsof conduit (see Figure 18, last temperature-depthgraph). The top geometryand time historyof thefluid flow. zone, A, 1 m in length,has been assumed to be the massof Becausethe initial pressure-temperaturecurve at the waterthat is ejectedout of the conduitto startthe eruption onsetof an eruptionlies no farther than 0.05-0.07 MPa (Figure25a). (0.5-0.7 bar) below the referenceboiling curve, the whole When A is ejectedout of the conduit,the pressureon columnof fluid in the conduitlies on the referenceboiling the underlyingwater is everywherereduced by the weight curveafter the unloadingof only 5-7 m of thewater. Thus of the ejectedfluid (assumethis to be about 0.1 bar; $I whenabout half of the verticallength of the watercolumn unitsare abandonedtemporarily here for easein following has been unloaded,the pre-emption"organ pipe" filled Figure 25). Relative to the reference boiling curve, with liquid water has been transformedinto a steamy therefore, the initial temperature distribution curve is two-phasenozzle. (Note that if the conduitis perceivedas elevated toward the boiling curve by 0.1 bar. If it is a nozzle,the originalstanding water occupied only 16 m of assumedthat no large-scaletemperature inversion exists, length, whereasthe eruptingfluid fills the full 22 m and such as J (Figure 25f), some underlyingwater is now extendsinto a jet 30-50 m high outsidethe conduit.) It is superheatedwith respectto the boiling temperatureat the likely that the "bursting"observed during the unsteady new pressure,specifically, any water that was originally initiation of the eruption representsthe eruption of within 0.1 bar of the referenceboiling curve. In the progressivelyhotter parcels of waterand that the transition exampleshown, the water between B andB' will boil from unsteady bursting flow to steady surging flow (Figure25b). When thiswater is erupted,more underlying observedin the behaviorof the eruptioncolumn (Figure water will boil: any water that was originally within 0.2 23) occurswhen the whole immediate reservoir is filled bar of thereference boiling curve. In Figure25 this is only with a two-phasefluid. thewater between B' andB", althoughthe water at the top As the water in the base of the reservoir is of cell C is now very closeto the referenceboiling curve decompressedto atmospheric pressure, some of the (Figure 25c). When this water has been unloaded,the enthalpy stored in the hot fluid is convertedto kinetic pressureis everywhere0.3 bar lessthan the initial pressure, energy. The hottest water, at 116øC, has an enthalpy and water between B" and B'", as well as between C and (relativeto the triplepoint) of 486.72 kJ/kgand an entropy C', will boil. Notethat because of theweight of B"-B"', of 1.4842 kJ/kg K. If this fluid decompressesisentropi- the fluid between B'" and C cannot boil until some of the cally to 93øC, at 0.08 MPa (0.8 bar) pressure,4 wt % of fluid in B"-B'" eruptsfrom the conduit. Presumably,as the liquid is convertedto vapor. The final enthalpyof the thewater between B" andB'" erupts,and as'C-C' boils mixture is 482.83 kJ/kg, so 3.89 kJ/kg is available for and expands,the liquid water betweenB'" and C will be kinetic energy. This is sufficientenergy to acceleratethe pushedup andwill boil as thepressure decreases. Slugs of fluid isentropicallyto a velocityof 88 m/s. liquid water trappedbetween slugs of boilingfroth should Althoughthe velocityat the exit planeof the geyserhas be expectedbecause of the differentpossible relations of not beendirectly measured, the fact that the densityof the the fluid temperatureto the reference boiling curve eruptedfluid is large comparedwith atmosphericdensity (Figures25c-25e). suggeststhat a simple ballistic calculationof velocity Whenfluid has been erupted down to levelC', 6 m of basedon the heightof the eruptioncolumn might be used water will have been ejected, and the pressurein the to estimateejection velocity. This calculation uses x m = column will be everywhere0.6 bar less than the initial Uo2/2g,where xm is themaximum height, u0 is theexit pressure.Except for smallamounts of fluidbetween E' velocity,and g is theacceleration of gravity. For x m = 50 m andF andbetween F' andG (Figure25e), the fluid will be this equationyields u 0 = 31 m/s. If the accelerationis everywhereon thereference boiling curve, and the conduit presumedto beginat thebase of theconduit, xm could be will becomenearly completelyfilled with a two-phase about70 m, andu 0 fromthis model is 37 m/s. mixture. The fluid is probablya boiling liquid at depth, However,because an eruptionof Old Faithfulproduces gradingupward into the steamyaerosol that emerges at the a jet rather than a "ballistic billiard ball," the simple surface. ballistic equationmay not accuratelyestimate the exit Further detailsof this unloadingprocess are unknown; velocity. Experimentaldata on thedynamics of negatively dependingon the constriction,shock and rarefaction waves buoyantplumes [Turner 1966;Chen and Rodi, 1980] give Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß :27 the relation durationcan be interpretedas representingan effective changein the length L of the resonatingcolumn. The x,•= 1.85FInD (19) possibilitythat multiple chamberscontaining fluid exist cannotbe excluded,and emptyingof differentnumbers of where F is the densimetric Foude number.. suchchambers could accountfor an effective changeof L. Another possibilityis that water is vaporizedto different Po F= uo2 [ (p. _po )gO ] (20)levels in the eruptionsof differing durations. During a long eruption,for example, all water in the immediate In thisequation, P0 is thefluid density, p. is thedensity of reservoir is convertedinto a two-phasemixture, and the the fluid into which the jet is emerging, and D is the immediatereservoir is completelyemptied. For the long diameterof thejet, assumedaxially symmetric. (This F is eruptionsthe bottom of the conduitis probablytruly the comparableto (Fr) 2, withFr asdefined in (15).) The geyserirebottom reached by probe,and L is the measured absolutevalue of the densitydifference is used in (19), 22 m. During a shorteruption, only part of the waterin the wherethe squareroot of F is needed.For the exit planeat immediatereservoir appears to be discharged. There is Old Faithful (1.5 x 0.6 rn fissure),take D = 1.1 m. Assume thereforeprobably a level below which water does not thatP0 = 11.23kg/m 3 (4% vapor) and p. = 0.7kg/m 3. vaporize during short eruptions. In these eruptionsthe The aboveequations give u 0 = 78 m/s,a valuein surpris- surfaceof the unvaporizedwater would be the "effective ingly goodagreement with thevelocity of 88 m/spredicted bottom"of the reservoirbecause of the large differencein simply from the enthalpy change of the fluid. The acousticimpedance between boiling and liquid water;that differencemay be within the uncertaintiesof the field is, the lengthL would not be the physicalconduit length measurementsor may suggestthat the enthalpyconversion but a shortervalue equal to the length of water column is roughly90% efficient. vaporized. This could account for the higher surge The low-frequencysurging (-1-2 Hz) observedwhen frequenciesobserved during short eruptions (Figure 24b). the geyseris in steadyflow (Figures14, 23, and24) could The mechanismwhereby the water, which is initially at be the resonancesof the conduitfilled with the two-phase 116øC when an eruption begins, is prevented from mixture.The equilibrium sound speed of an H20 mixture vaporizing under pressurereduction remains a mystery. composedof 4% vaporis 57 m/s at 1 bar pressure.The The most common speculationis that cold water can soundspeed does not vary significantlybetween 0.8 bar occasionallyenter the reservoirduring an eruption,but atmosphericpressure and 0.18 MPa (1.8 bars) vapor thereis no proof of this,and it is perhapsequally plausible pressurefor the fluid boilingat 116øCat the bottomof the that local temperatureinversions in the fluid couldquench vent. The sound speed would decreasedramatically, eruptions. however,in the zone wherethe vapor fractionapproaches zero (that is, right in the region where bubblesbegin to Speculationsand Summary nucleatein the liquid water phase),as shownin Figure 3, Is the flow from Old Faithful choked? Is there verticalline (a). However,I assumethat the verticalrange supersonicflow anywherein the eruptionjet? There are over which the soundspeed is substantiallylower than 50 too few data to permit firm conclusions.A few specula- m/s is small and that the soundspeed of 57 m/s is typical tions can be offered, although calculation of choking of the fluid in mostof the conduit.The resonantfrequency conditionsin two-phaseflow is a notoriouslydifficult of a 22-m closedpipe with thissound speed would be 0.64 problem even in well-designedpipes [Wallis, 1969, and Hz, correspondingto a period of 1.5 s. Although this referencestherein]. frequencyis abouta factor of 2 lower than that measured The highestvapor pressuresin the reservoirwill be for along eruption(1-1.5 Hz), it is certainlycorrect in generatedas the hottestwater boils: 0.175 MPa (1.75 bars) order of magnitudeand indicatesthe dramaticchange in as the 116øC water boils. Equilibrium expansionto resonantfrequencies as the fluid goesfrom a single-phase atmosphericpressure of 0.08 MPa (0.8 bar) producesa liquid to a two-phasemixture. Given the uncertaintiesin fluid that is about4% vapor. Given a maximumreservoir the model (the relativelyunknown geometry of the conduit pressureof 0.175 MPa (1.75 bars), the chokepressure can and whetheror not it is best approximatedas an open- or be calculatedto be about 0.13 MPa (1.3 bars) [Moody, closed-endedpipe) and theextreme difficulty of measuring 1965]. Experimentalevidence [Fauske, 1962] suggests each individualeruption surge, the calculatedand meas- that this calculatedchoke pressure is too high and that the uredfrequencies are probably in satisfactoryagreement. choke pressurecould be as low as about 0.55 of the Because we have no reason to believe that the sound reservoirpressure, about 0.09 MPa (0.9 bar). The near speedof the fluid is any different in long versusshort similarity of atmosphericpressure and estimatedchoke eruptions,the changein surge frequencywith eruption pressuresuggests that the flow couldbe chokedwhen this 28 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer: GEOLOGIC NOZZLES

hottestwater is flowing but that supersonicflow in the divergingpart of the conduitwill be weak. The discussion above about the decompressionprocess suggeststhat chokedflow would be most likely during the steady-flow stage,beginning 20-30 s into the eruption. Assumethat chokingoccurs at conditionsvery closeto those seen at the exit plane, that is, at 0.1 MPa (1 bar) pressurewhen the fluid is about3% vapor. The massflux rn is givenby

rh= p* A* a* (21)

whereA* is the area of the choke plane and a* is the sound speed,45 m/s for equilibrium at 0.1 bar pressure. The densityp* of thefluid is 19.68kg/m 3 (3%vapor), and the chokearea is about0.15 m2. The calculatedtotal mass r. flux is 132 kg/s. Of this,96%, or 127 kg/s, is liquid water. This value is satisfactorilyclose to the measuredvalue of water in the runoff channelsfrom Old Faithful (114 kg/s), given the extreme difficulty of measuringthe discharge accuratelyand the relativelylarge uncertainty of the choke

If, as calculatedabove, the exit plane velocity is --80 m/s and the equilibriumsound speed at the exit planeis 57 m/s, the implied Mach numberat the exit plane is --1.5, barely supersonicwithin the uncertaintiesof the modeling. It is thereforenot surprisingthat shock featuressuch as Prandtl-Meyer expansion,visible shock waves, or noise

originatingfrom shockswithin the plume are not observed a b at Old Faithful. In contrast,Beehive geyser (Figure 26a), which sitsjust a few hundredyards from Old Faithful and Figure 26. (a) Beehivegeyser in eruption. The coneis about 1 emptstoo erraticallyto be easily monitored,sounds like a m high. The arrow points toward the secondof three white jet engineand, with a little imagination,can be envisioned diamondsin the center of the flow, interpretedas shock wave to containinternal shockwaves (the arrow in Figure 26a structureswithin supersonicflow. In the backgrounda low cloud points to three diamond-shapedstructures that may be of steam marks the vent of Old Faithful, and the column of steam shockwaves). These shocksare similar to thoseobserved ascendingtoward the left of this area, behind the plume from Beehive, is probablyfrom an eruptionof Old Faithful that had at weakly supersonicgeothermal well heads(Figure 26b). just ended when this photographwas taken. Although no In summary, Old Faithful is a complex two-phase physical connectionbetween Old Faithful and Beehive (which nozzle,possibly sonic or weakly supersonic,and certainly are separatedby the Firehole River) has been proven, their large enoughin scalefor both gravity and compressibility eruptionsare sometimescoupled. (Photographby J. Schmidt to be important. Although available data still do not [seeFuller and Schmidt,1984], reprintedwith permission.) (b) permit a derailed model for the eruptiondynamics, they Geothermal well MG-5, Tongonan, Philippines, showing have servedto point out new directionsfor experiments diamond shocksthat can be comparedwith those in Beehive andobservations, some of whichare now in progress.One geyser (Figure 26a). (Photographby C. Darby, KRTA Ltd., of the mostimportant directions of researchfocused on by Auckland,New Zealand.) these discoveriesrelates to the similaritiesin seismicity between geysers and volcanoesthat exhibit harmonic tremor(Figure 15). Harmonictremor has for decadesbeen attributedto magmamotion and/or fracture propagation in thismechanism are now in progress[Leet, 1988]. Because volcanoes,but the quantitativenature of the mechanism weknow neither the dimensions of the conduits containing causingit hasbeen elusive. The geyserstudy suggests that the fluid nor the natureof the fluid thatis causingthe bubblesin groundwatercontained in fissuresor pockets seismicityat volcanoes,this is a verydifficult problem. surroundinghot magmacould be the sourceof tremor in The studyof Old Faithful,where at leastconstraints can be somesettings [Kieffer, 1984a], and quantitative studies of put on the fluid and on the conduitdimensions, is thus Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß 29

...... ??:i•?';'-•'•;i{•:, "..?..i$? $".".iii•?.,•.71:.:,($.?•......

•-.•-••••:•••:::•:?•..:...... • ...... ;• ,:•a "'•?.•;•*'*•...... *%?t.;••:,:-•? .•./...... :;;• .•..:;;•,•:•:.:::...•..•::•**:'•:•;$•... .:,?**•*:-:--•'...... • ...... :•.::::..: -:•?•:*•*::;+"':"•'*"•*•*•-*•'•:,..x;::,f.•".•:...... •::;•...... ß...... •:•"

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...... •:::•...... ;.:.::::•.::.,...... :...... :..•:;,..::::::::::::::::::::::

Figure 27. A geyserlikeeruption of Mount St. Helens,April 1, 6. MOUNT ST. HELENS: A SUPERSONIC JET 1980. An ash-ladendensity flow rolls down the southwestern slopes (on the right) from the summit crater at about 2930-m GeologicSetting elevation, while steam separatesand rises to ~4500 m. Mount St. Helens became famous as an "active (Photographby H. and S. Kieffer.) volcano"on May 18, 1980. However,that eruption was heraldedby nearly6 weeksof precursoractivity during relatively unique. Fluids proposedfor the source of whicheruptions were strikingly similar in scale,frequency, volcanic tremor (undersaturatedmagma, gassy magma, andfluid dynamicsto eruptionsof majorgeysers like Old water) can have soundspeeds that differ by nearly three Faithful.On March 27, 1980,after a reposeof onecentury orders of magnitude,and the above discussionsuggests and ominous seismicity for a week, an unobserved that therewill be an ambiguityin alecouplingthe effectsof eruption created a small crater in the summit of the conduitdimensions from fluid propertiesin any analysisof mountain.For a few weeks,eruptions of steamand ash volcanic harmonic tremor. This is an active area of emanated from the summit (Figure 27). Studies of researchin volcanologybecause of the regularoccurrence deformationof the mountain,and laterevents, strongly of volcanic tremor at some volcanoes near heavily suggestthat magmawas being intrudedinto the edifice populatedareas where forecastinghas enormousimplica- fromdepth at thistime (Figures 28a and28b). Waternear tionsfor life and economy;for example,at Ruiz volcanoin the magmawas heated and convected upward (indicated Colombia, some seismicity strongly resembled that of by arrowsin Figure28a), emerging in eruptionsthat were Karkarvolcano (Figure 15) andhas now been interpreted geyserlikein scale(hundreds of metersto a few kilometers to indicateunderground geysering and phreatic activity (as high),duration (minutes to tensof minutes),and frequency summarizedby Williamsand Meyer [1988]). Laboratory (everyfew hours). The repeatederuptions created and studiesof the dynamicsof largecollapsing bubbles and of enlargeda conduitand summit crater. Crushedrock, ice, two-phaseflow in long pipesof variablearea are needed. ash,and waterwere intermittently ejected. Someof this In particular, theoreticaland experimentalwork on the materialfell back into the conduitand cloggedit tem- dynamicsof compressibleflow with gravitationaleffects porarily,only to be reworkedand ejectedin successive will be required to deal with problemsinvolving these eruptions.These early eruptions were driven by heated fluids at geologicscales. groundwater: no magmawas involved. 30 ß REVIEWSOF GEOPHYSICS / 27,1 Kieffer: GEOLOGIC NOZZLES

•200

•100

2500 •000

2000 9OO 1500 8OO

s N •_) 700

• 600

3000 2_500 2500

2000 300 15oo (L) (v) 200 OLD I •00 bar-- I • (L+V) I øo 1 2 5 4 5 6 7 8 9 3000(•.._. I Enfropy,Joule K 2500 • Figure29. Temperature-entropydiagram for H20 withisobars 2000 Po,To'Po, m,ao (solid lines except where closely spaced)and isopleths(dashed

1500 lines insidetwo-phase field). Symbolsl(I), 2(I), L, and V are the

I I I I sameas in Figure3. CP, criticalpoint. On the left, an isentrope o I 2 3 4km for an eruptionof Old Faithful;on the right, two isentropesfor Figure 28. (a) South-to-northcross section showing schemati- eruptions from the assumedMount St. Helens (MSH) initial cally the conditionsinside Mount St. Helensduring March, April, conditions. and early May 1980. Magma hasmoved high into the mountain. (b) Reconstruction[Moore and Albee, 1981] of the initiation of vapor is accountedfor by a decreasein the isentropic the lateral blast on May 18. (c) Schematicdrawing of initial exponentof theperfect-gas law. conditionsassumed for the fluid flow model. SymbolsPo' To, The expansionof a mass-loadedvapor is contrasted P0'rn, and a o are discussed in text. with the expansionof a vaporalone, or of a decompressing liquid, in Figure 29. On the left side of this figure an Althoughthese eruptions were more geyserlikethan isentroperepresenting decompression from the hottest "volcanic,"theydiffered thermodynamically from eruptions water of Old Faithful (116øC) to 93øC is shown (refer to of mostgeysers because the eruptingvapor carried a heavy discussionin previous section). On the right side, two load of particulatematerial, crushed rock and ice gouged isentropesfor hotter conditionsmore appropriateto the from the conduitand crater(this particulatematerial gives volcanic environment of Mount St. Helens are shown the lower part of the eruptionplume in Figure 27 a dark (labeledMSH). The particularinitial conditionspicked are color). The mass loading by this material affected the discussed below as lateral blast conditions, but the thermodynamicsin two ways: the entrainmentof solid thermodynamicpoint to be illustratedhere is the qualita- fragmentsincreased the bulk densityof the fluid, and heat tive differencebetween the two isentropes. The vertical transferbetween the solidsand the expandinggas altered line under the MSH label is an isentropeof pure steam. the expansionof the gas from that which would be Note that isentropicexpansion results in condensationof a obtainedby a two-phasemixture or vaporalone. The mass liquid fraction. In contrast,the isentropeslanted toward loading of these eruptionswas probablyrelatively light, the lower fight, and labeled "large m" is the isentropeof especiallyin the dusty steam (Figure 27), and thus the the vaporphase in a mixturethat containssufficient mass pseudogasapproximation discussed in section3 mightbe a loadingrn that heat transferbetween the particlesand gas rather good descriptionof thesefluids. In sucha model, is effective. The heattransfer increases the entropyof the mass loading is taken into account as an increased steamphase throughout the expansion, and condensation is molecularweight of the mixture (see the equationsat the therebyprevented. As can be seenfrom this figure,under top of Figure 4). Heat transferfrom hot particlesto cooler somecircumstances, mass loading can simplify the fluid Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß 31

and to man-madeequipment around the mountain[Lipman andMullineaux, 1981]. Nearly600 km 2 of forestwas devastated,and approximately60 people were killed. Heavy logging equipment was tossed and overturned (Figure 30a). Trees were totally strippedfrom the land over a large area, and where they remained,as muchas 10 cm of bark and wood was abraded, the interiors being impregnatedwith rock shrapnel(Figure 30b). The pattern of tree blowdown (Figure 31) provided a remarkable record of local' flow directions, certainly a flow-field patternto challengegeologists and fluid dynamicistsfor years to come.

Figure 30. (a) Damageto heavy loggingequipment during the May 18 lateral blast. (b) Damageto trees. (Photographsby H. and S. Kieffer.)

Figure 31. Typical patternof tree blowdown. Note that the tops of the trees and most of the small limbs are missing. The contrastin this photographis low becausevolcanic ash mantled treesand slope when it wastaken, shortly after May 18, 1980.

dynamicsby preventingphase changes: the entropyof a In the region closestto the volcano,trees were either gas phaseis increasedby heat conductedfrom solids,and strippedfrom the land or felled subradiallyaway from the thus formation of a condensedphase is suppressed. vent,and the blowdowndirection showed little dependence However, complications of heat transfer, drag, and on the terrain(Figure 32). This zone is called the "direct interparticleinteractions arise. No theoreticalmodels yet blast zone" to emphasizethat the blast traveleddirectly account for these effects realistically for the range of away from the mountain without regard to even major particlesizes, particle shapes, and massloading typical of topographicobstacles. Surrounding the directblast zone is volcaniceruptions. a zone in which topographydid influencethe blowdown The north flank of Mount St. Helens was badly direction,called the "channelizedblast zone"to emphasize fractured and weakenedby the intrusion of magma in that the blast followed channels in the topography. March and April 1980. At 0832 on May 18 a magnitude Surroundingthis region and marking the limits of the 5.2 earthquakeshook the mountain,and at leastthree large devastatedarea is the "singedzone," a zone in which trees landslide blocks broke loose and slid downhill toward the were left standingbut were singedby the heat of the blast North Fork of the Toutle River (Figure28b). Within a few as it becamepositively buoyant and lifted from the ground secondsthe pressureon magma, hot water, and gases into theatmosphere [Kieffer, 198 la]. inside the mountainwas greatly reduced,and their rapid This devastationwas causedby the eruptionof more expansionproduced the devastatingevent that was to than10 TM g of magma,hot water, and entrained glacier ice becomeknown as the "lateralblast." Magma was present and trees. The vent throughwhich the material emerged in each slide block, but the lateral blast appeared to covereda large fractionof the north side of the mountain. originatemost stronglyfrom slideblock 2. The evolution Available evidenceallows many theories,and geologists of this blast was recordedby several eyewitnesses,by do not even agree on initial and boundaryconditions for seismic equipment stationedaround the mountain, by the flow (see, for example, the contrastingmodels of weather barometers,and by damage to the environment Kieffer [ 1981a, b, 1984b],Eichelberger and Hayes [ 1982], 32 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer:GEOLOGIC NOZZLES

Malin and Sheridan [1982] and Moore and Rice [1984]. I instantaneous;this may not be a bad assumptionbecause describebelow my model for the blast, a model that eyewimessphotographs show a rapid and dramaticonset emphasizesthe role of gas expansionand nozzle flow. of the blast from slideblock 2. In comparingFigures 28a Given a plausibleset of simplifyingassumptions, this and28b with the simplifiedmodel in Figure28c thereader model attemptsto define the flow characteristicsand to shouldnote that only the toppart of the sourcemagma was correlatethese predicted characteristics with featuresin the eruptedinto the lateralblast. The remainderwas erupted devastatedarea: tree directions, the transition from direct into a tall verticaleruption column during the severaldays to channelized blast zones, measured velocities and after the lateral blast (see, for example, descriptionsby temperatures,and the general shape of thedevastated area. Lipman and Mullineaux [1981], Scandoneand Malone [1985], andCarey and Sigurdsson[1985]. A SimpleNozzle Model Althoughthe modelcan easily be scaledboth geometri- In my model the lateral blast is simplifiedto the cally and thermodynamically,one set of plausiblevalues problemof the eruptionof a pseudogasfrom a single for initial conditions demonstrates the features of the reservoirunder pressureinto an atmosphereat lower model: an averagereservoir pressure of 12.5 MPa (125 pressure(this simplificationcan be seenby comparing bars,the lithostaticpressure appropriate to 650 m of rock Figures28b and28c). The thermodynamicproperties and overlyingthe reservoir),an initial temperatureof 600 K, reservoirgeometry are alsosimplified accordingly (Figure and a mass ratio of rock to steam of 25:1. The initial 28c). The fluid is describedby its initial (average) temperatureassumed may seemsurprisingly cool if one pressureP0, temperatureT0, andmass ratio m of solidto associatesvolcanic eruptionsonly with red-hot, incan- vaporphases. The reservoir is assumedto haveresembled descentmagma. However, many so-called"volcanic" a convergingnozzle whose exit plane (vent) was the eruptions,such as the one shown in Figure 27, are not landslidescarp left on the northface of the mountainby driven by magma but by heatedwater Cphreaticerup- the removalof the avalanchematerial. Althoughthe time tions"),or by a mixtureof heatedgroundwater and magma historyof evolutionof the landslideswas complex,for Cphreatomagmaticeruptions"). The derailednature of the modelingpurposes the triggeringis assumedto havebeen volcanicgases driving the eruptionis ignoredhere because

CHANNELIZED BLAST ZONE

DIREC BLAST

EXPLANATION

Generolizedtree blow-' downpotterns of directions 4000'

• Standingtimber, green

:,_:.',,:.•Standing timber, singed

:.• Mudflows o I 2 MILES I 5000' Figure 32. Map showinggeneralized direction of tree blow- •øøø• o I 2 KILOMETERS down,with approximateextent of direct,channelized, and singed I I i zones. Mt. St. Helens Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWSOF GEOPHYSICS ß 33 the large-scalefeatures of the fluid mechanicsare probably supersonicflow were discussedin section2 (Figure2) and not sensitiveto the gas source. The chosentemperature are shownin more detail in Figure 33a; the origin and happensto be the saturationtemperature of pure water at significanceof thesefeatures will be discussedbelow. In 12.5 MPa (125 bars), and it is a reasonablenumber to Figure33a, gasis shownemerging in the axialy direction assumea priori if one believes that geothermalwaters froma ventthrough plane x-x'. Thegas is assumed tobe a heatedto saturationconditions drove the eruption(and that pressurehigher than atmosphericand thus producesan the badly fracturedmountain edifice could not sustainany overpressured(underexpanded) jet. Typical rarefaction overpressurein the eruptivefluid). The temperatureof 600 waves (fans), compressionwaves, shock waves, and K can also be thoughtof as an averagetemperature for a discontinuitysurfaces are labeledin Figure 33a. Relative complexmixture which, after it had traveledonly a short values of the Mach number (M) in different flow-field distance, contained material ranging from the melting regions (subsonicand supersonic)are shown, and a temperatureof the magma (-1220 K) to the freezing schematicMach diskshock is alsoshown. The bounding temperatureof glacial ice and snowentrained in the flow. shearlayer of the flow is indicated.The majorfeatures to Reasonablechanges in assumedinitial pressure,tempera- notein thisfigure are the expanded shape of thejet andthe ture, and solid-to-massratio do not qualitativelyalter the existenceof a complicatedinternal flow structure. conclusions.Atmospheric pressure is taken as 0.87 bar. An enlargedview of the expansionof the flow field For scale, the vent diameteris taken as 1 km, the ap- aroundthe comerof a vent is shownin Figure33b, which proximatewidth of the scar left by the avalanches.The couldbe interpreted asa mapview of theflow pattern from eruptionis assumedto be centeredat 2135 m elevation, the east comer of the vent at Mount St. Helens. The flow and the centerline of the flow is oriented about 5 ø east of at Mount St. Helens exited from a vent in the north side of north to match the overall direction of the flow field. the mountain;in three dimensions,therefore, the flow was These are the only variablesin the model: there are no not restrictedby a flat surfaceas is impliedby the way arbitraryfitting parameters. Figure 33a is drawn. In Figure 33b the stippledpattern Nozzlesoperating at pressureratios much greater than indicatesa projectionof a confiningsurface into the plane about2:1 are supersonic.At the ratio of 125:0.87assumed of the flow field, but becauseof the three-dimensionality above for Mount St. Helens, the emergingflow should of the problem,the flow is free to expandbeyond the have been highly supersonic. The general features of half-spaceinto which it emerges. To envisionthis, the

a. Y/'re(coord,notes)

¾re i•/ I • Discontinui,yI :"'"'' ' ' ""i •.77-\ ::iI ' B •.89 •

198

Com press ion"" erc;•ø•[•••2ctiø/\ !

Conduitor•5 { • X surface Figure 33. (a) Schematicdiagram of the structureof an underexpanded supersonic jet [JANNAF, 1975]. Further craterwall• • Ii discussionof this figure is given in the text and by Kieffer

Overpressured [1981a, b, 1984b]. (b) Detail of a corner rarefaction for flow conditionsappropriate to the Mount St. Helensmodel. 34 ß REVIEWSOF GEOPHYSICS/ 27,1 Kieffer:GEOLOGIC NOZZLES readermight note that when gases emerge from supersonic rest in the reservoirto sonicvelocity at the vent, 100 m/s. rocketengines high in theatmosphere or in space,they can Laboratorystudies [Kieffer and Sturtevant,1984] have expandbackward around their exhaustplane and, if the shown that in the absenceof gravitationaleffects the rocketis notproperly designed, the exhaustgases actually flow-front velocity of dense fluids remains at ap- can cause erosion on the sides of the rocket vehicle that proximatelythe sonicvelocity for manysource diameters, theyare propelling. becauseentrainment of the light surroundingatmosphere In Figure33b, "characteristic lines" of theflow (usually causesvery little deceleration.At Mount St. Helensthe justcalled "characteristics") areshown as the thin lines that flow wouldhave accelerated as it droppeddown the face of radiate from the comer A of the vent. Characteristic lines the mountaininto the valleyof the North ToutleRiver, but satisfysecond-order, partial differential equations of the it would have decleratedas it rose back up into the high hyperbolictype, with two independentvariables [Shapiro, countrynorth of the ToutleRiver (the regionnow called 1953]. For thisanalysis of MountSt. Helensthe equations JohnstonRidge). Becausethe combinedeffects of gravity solved for the theoretical model are those for two- andcompressibility are not includedin the model,it cannot dimensionalsteady flow [e.g., Thompson,1972, pp. be accurate to this level of detail. Measured velocities of 443-482]. In steadytwo-dimensional flow thecharacteris- the flow, averagedover substantialtopographic relief to tics are "Mach waves." If the characteristicsare straight, JohnstonRidge, were about 100 m/s, in goodagreement the flow is termed "simple,"and the characteristicsare with thecalculated sonic velocity. lines along which certain propertiesof the flow are Accordingto the model, the fluid emergedfrom the constant. In Figure 33b, for example,the problemis a volcano as an underexpandedsupersonic jet whose simple centeredrarefication problem, and the Mach calculatedflow parametersare shown in Figure 34. In numberM is constanton eachcharacteristic; other physical Figure 34 all lengthdimensions are normalizedto vent variables can be calculated from the Mach numbers. diameter. To easenumerical computations, the exit Mach Streamlines of the flow can be calculated from the number of the flow is assumed to have been 1.02 instead of propertiesof thecharacteristics because the turningof the the sonic Mach number 1.00. The boundaryof flow is flow as it crosseseach characteristicis given by the assumedto have been at a constantpressure of 0.87 bar methodof solvingthe hyperbolicequations. For example, and was determinedfrom empiricalformulas in JANNAF A-A' andB-B' are streamlines. [1975]. The peripheralintercepting shock formed by To relate the model to field data, I have assumedthat at reflectionof the expansionwaves from this boundaryis any point in the flow field, trees were blown down shownas a dashedline. Becausethe model is symmetric tangentiallytoa streamline;C-C' is a tangenttoB-B'. The aboutthe ventaxis, the plane of symmetry(centerline) acts small arrows d, e, f, and g representcalculated local as a rigid planeboundary, and theresults can be illustrated directionsof flow, as would be recordedin directionsof usingthe two halvesof the flow field to conveydifferent tree blowdown. Arrows e, f, and g are particularly information for conciseness. significantbecause if theseare assumedto lie alongflow On the left half of Figure 34 the characteristicsare streamlinesand are extrapolatedlinearly backward, a flow shownas light linesradiating from the comerof the vent. sourcesignificantly in front of the mountainwould be The characteristicshere take on a much morecomplicated inferred.In compressibleflow, linearextrapolations of the meaning than the "simple" set shown in Figure 33b, streamlines cannot be made because the curvature of the becausethroughout most of the flow field the expansion flow throughexpansion waves would not be properly characteristicsfrom one vent comer (shown emanating accounted for. from the comerof the vent on the left) are influencedby The mostimportant dynamic parameter of the erupting characteristicsemanating from the other vent comer fluid is its soundspeed, 105 m/s for the reservoirfluid (shown in the left half of Figure 34 as originatingat the postulatedabove, accordingto pseudogascalculations. plane of symmetry). The fields in which the flow is This soundspeed is aboutone third of the value of the "simple"in the mathematicalsense are too smallto show atmosphericsound speed. Thereforethe flow field of the on this diagram,and throughoutmost of the flow field the volcanicpseudogas can be internallysupersonic but still Mach number changes continuouslybecause of the subsonicwith respectto the surroundingatmosphere. interaction of characteristics. Therefore contours of Thus there is no contradictionbetween the postulated constantproperties (e.g., Mach number) must be con- supersonicflow and the notableabsence of atmospheric structed from numerical solution of the interacting shock waves during the lateral blast (see eyewitness characteristics.The arrowson the left side of Figure 34 accountsin worksby Lipmanand Mullineaux[1981] and representtangents to local streamlines. Kieffer [ 1981a]). The contours of constant Mach number M are shown Considerfirst the initial velocityof the fluid. Accord- on the right side of Figure 34; velocities are given ing to theproposed model, the fluid wouldaccelerate from implicitly by the Mach numbers. By implication,these Kieffer: GEOLOGIC NOZZLES 27,1 /REVIEWS OF GEOPHYSICS ß 3.5

Figure 34. Map of flow field accordingto model of blast M, P/Po,T/To, and P/P0 are given. In thesupersonic region, these dynamics[Kieffer, 1981a, b]. (Left) Characteristicsof the flow valuesare (2.23, 0.087, 0.91, 0.095); (2.49, 0.047, 0.89, 0.053); (light lines) and tangentsto streamlines(vectors). (Right) (2.75, 0.025, 0.87, 0.029); (2.88, 0.018, 0.86, 0.021); (3.01, Contoursof nondimensionalvalues of parameters,labeled by 0.013, 0.85, 0.016); and(3.14, 0.009, 0.83, 0.011), respectively. Mach number. From the innermost contour outward, values of See text for detailed discussion. contoursare alsocontours of constantpressure (P/Po), The fluid inside the jet expandsand acceleratesas it temperature(T/To), and density(P/P0)- Valuesof these passesthrough the expansionwaves, obtaining, according ratiosare givenin the figurecaption. to the model, a Mach number of more than 3 on the According to the model calculations,the pressure centerline and velocities in excess of 300 m/s. Internal would have decreasedto 7.5 MPa (75 bars) as the fluid velocitiescan be locallyhigher than the flow-front velocity accelerated from the reservoir to the vent. Further decrease because of the internal rarefaction and shock waves. of the fluid pressureto ambientatmospheric pressure was Pressure,temperature, and densitydecrease through the accomplishedthrough a series of complex rarefaction expansions.The pressurebehavior is particularlyinterest- wavesand shockwaves within thejet outsidethe volcano, ing and illustratesthe nonlinearityof the supersonic as shown schematicallyin Figure 33a and in detail in expansionprocess: as the fluid expands,the pressure Figure34. Becauseof thehigh pressure in thejet as it left decreasesbelow atmosphericpressure, and a largezone of the vent, it wouldhave spread laterally through a charac- subatmosphericpressure develops inside the supersonic teristicangle known as the Prandtl-Meyerangle (Figure zone (see the shadedarea in Figures 34 and 35). The 33b). For the initial conditionspostulated, the Prandtl- numericalmodel discussedby Kieffer [1984b, p. 155] Meyer angleis 96ø . Thus flow that initially was directed suggeststhat the pressurein thisregion could be as low as northwardby the geometry of the vent would have 4% of atmosphericpressure. The existenceof such a divergedto the eastand west. The predictedflow zoneis low-pressurecore has someinteresting volcanic hazards superimposedon top of the map of the devastatedarea in implications:for example,plastic components on vehicles Figure35. I suggestthat the devastatedarea has a southern in or nearthis part of thedevastated area were degraded by boundarythat actually curvessouth of an east-westline the formationof largevapor bubbles [Davis and Graeber, near the volcano becausethe initial Prandtl-Meyer 1980]. Laboratorystudies demonstrated that the vapor expansiondrove gas around in thesedirections. formationwas causedby exposureof the plasticto high In an underexpandedsupersonic jet, rarefactions temperaturesduring the blast. Efforts to duplicatethe crisscrossthe flow and reflect off the flow boundary, degradationby heatingsimilar plastics in the laboratory assumedto be at a constantpressure equal to ambient underatmospheric pressure produced general similarities atmosphericpressure (Figure 33a). Upon reflectionthey but failed to producethe largesize of the bubblesfound on turninto weak compressive shocks called "intercepting" or thecomponents from thevehicles. I speculatethat bubbles "barrel" shocks (Figure 33a). The reflection of the may have grown unexpectedlylarge becausethe external rarefactionsfrom the flow boundarytums the diverging pressurewas temporarilylower than atmosphericin the flow back towarda moreaxial direction. I suggestthat supersoniccore of the lateralblast. thesereflections are responsiblefor focusingthe direct In an underexpandedjet the intercepting shocks blastzone so stronglyto the north(Figures 32 and 35) and strengthenwithin the flow and coalesceacross it into a for limiting the extentof east-westdevastation. strongshock standing perpendicular to the axisof the flow. 36 ß REVIEWSOF GEOPHYSICS / 27,1 Kieffer: GEOLOGIC NOZZLES

:z

0 I 2 KILOMETERS I I I Mt. St. Helens Figure 35. The modelof Figure34 superimposedon the map of the author that the direct blast zone was a zone of strong the devastated area. The coincidence of the Mach disk with the supersonicflow. directblast zone boundary in the northerlydirection suggested to

This shockis calledthe "Machdisk" (Figure 33a). As gas A computermodel that produceda somewhatsmaller flows throughthe shock,it deceleratesfrom supersonicto supersoniczone becausethe flow boundarywas assumed subsonicconditions; to a first approximation,the pressure to be inviscidrather than viscous(as in the above model) on the downstreamside of the Mach disk is atmospheric. was run for the author by R. A. O'Leary (Rocketdyne) Inertia of the heavy debrisentrained by the blast at the [Kieffer, 1984b]. Differencesbetween the two modelsare Mach disk (the overturnedlogging vehicle in Figure 30a not significantin terms of our lack of knowledgeof the and the debrisaround it give someimpression of the size real complexitiesof the eruption,e.g., materialemerging of the debrisload near the Mach disk) would propelthe from two moving landslidesinstead of from a single particulatematter through a "gasshock," so that the Mach vertical vent. No existingmodel is adequatefor calculat- diskshould, in thisgeologic case, be thoughtof as a Mach ing propertiesbeyond the Mach disk shock (compare disk "zone,"perhaps of the orderof 1 km in thickness.As discussionsby Kieffer [1981a, 1984b]). I believe that the the fluid decelerates into the subsonic zone downstream of most plausibleassumption is that the flow is returnedto the Mach disk, flow velocitiesdecrease, pressure rises ambientatmospheric pressure beyond the Mach disk (and from subatmosphericback toward atmospheric,and the perhapsbeyond the peripheralintercepting shocks) and densityof the fluid increases.According to the calcula- that compressibilityplays a muchless important role in the tions, t•e Mach risk would have stood about 11 km north flow field beyondthese features. of the vent. The calculatedposition and, to a lesserextent, Because of the dramatic deceleration of the flow at the the position of the lateral interceptingshocks coincide Mach disk, gravity(which wasnot a dominantforce within roughly with the boundariesbetween the direct and the directblast zone) dominatedflow mechanicsoutside of channelizedblast zones(Figure 35). I proposethat these this zone, that is, in the channelized blast zone. Thus the two zones correspondroughly to the boundarybetween flow streamlines,as indicated by the tree blowdown supersonicand subsonicflow regimeswithin the lateral patterns, are more influenced by topography in the blast. channelized, subsoniczone. The devastatedarea therefore Kieffer: GEOLOGIC NOZZLES 27,1 / REVIEWS OF GEOPHYSICS ß 37 consistsof two parts: the inner directblast zone, in which exciting discoveriesof modem times--the existenceof gas dynamicseffects and supersonicflow were probably eruptingvolcanoes on anotherplanet, Io (seethe summary dominant,and the surroundingchannelized blast zone, in by Morrison [1982] and the interpretationby Kieffer which downhill flow driven by gravity was probably [1982]). Becauseof the large ratios of conduitand vent dominant. This is an oversimplification,because both pressureto atmosphericpressure on Io (and perhapsin past effects were probably importantthroughout much of the volcanism on Mars and the Moon), geologists and devastatedarea (for example,the most highly supersonic planetary scientistswill continue to pose problems of zone and the Mach disk happento coincidewith a region scale,chemical complexity, and geometrythat will present of very steep topography),and quantitativemodeling challengingfluid dynamicsproblems for decadesto come. includingboth effects is requiredin thefuture. Temperaturesthroughout a particle-ladenflow like the ACKNOWLEDGMENTS. This paper is dedicatedto Hans lateral blast are remarkablyhigh and uniform becauseof W. Liepmmm,a pioneerin fluid dynamics,on theoccasion of his the high massratio of solidsto vapor(this is the effectof ), seventiethbirthday. The studiesdescribed here have extended over 12 yearsand have involvedmany colleaguesto whomI owe - 1; seeFigure 4 and equation(2)). Calculatedtempera- thanks, including those referencedor mentioned as "private tureschanged only from 600 K to 480 K at the limits of the communication" in this article, and numerous others who could devastated area; these temperaturesare in excellent not be referencedbecause of spacelimitations: colleagueswho agreementwith temperatures measured in the deposits work within the National ScienceFoundation, the U.S. Geologi- immed!atelyafter the eruption (see articles in thework by cal Survey,and the Bureau of Reclamationto help fund the work; Lipmanand Mullineaux [ 1981]). and, EugeneShoemaker and BradfordSturtevant who interested me in, and taughtme, fluid dynamicsas a studentfrom the text Severalother properties of the blastcan be calculated by Liepmannand Roshko. Finally, specialthanks axe owed to a from this model. For example,the maximum massflux is supportivehusband, who has encouragedall of my work, and to calculatedtohave been 10 4 g s-• cm-2, and the thermal flux our son--a valuable field assistant. tohave been 2.5 MW/cm 2. Thecalculated total energy of The editors for this paper were W. R. Peltlet and M. the blast was 24 megatons(Mt), of which 7 Mt was Neugebauer.They thankan anonymousreferee for his assistance dissipatedduring the blastitself, and the remaining17 Mt in evaluatingthe technical content of this paper and P. L. Richardsonfor servingas a cross-disciplinaryreferee. was dissipatedduring the almostsimultaneous condensa- tion of steamin the blast and the subsequentcooling of steam and rock to ambient temperaturein the weeks REFERENCES followingMay 18. As mentionedabove, the supersonicflow modelfor the Biasi, L., A. Prosperetti,and A. Tozzi, Collapseof a condensing lateral blast has been controversial. Nevertheless, features bubble in compressibleliquids, Chem. Eng. 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