Effect of Wind on the Rise Height Volcanic Plume

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GEOPHYSICAL RESEARCH LETTERS, VOL. 28, NO. 18, PAGES 3621-3624, SEPTEMBER 15, 2001

Effect of Wind on the Rise Height of Volcanic PlumesM. BursikDepartment of Geology, University at Buffalo, SUNY

Abstract. A theoretical model of a volcanic plume, basedon applying the equations of motion in a plume-centeredcoordinate system, suggests hat the interaction between avolcanic plume and wind causesenhanced entrainment of airand horizontal momentum, plume bending, and a decreasein plume rise height at constant eruption rate. Because ofrapid dilution in the high windspeedsof the polar jet, plumesthat vary over more than one order of magnitude in masseruption ate (106 o 107 108kg/s), if injectednto thepolar jet, may all attain rise heights only slightly differentfrom that of the core of the jet, - 10 km, as opposed to17- 33 km in a still atmosphere.1. Introduction

Volcanic plumes interact with the wind at all scales. Onthe smallest scales, he wind shapes ocal eddy structure. Atlarger scales,wind affects he entire plume trajectory. Small,troposphericplumes are distorted or "bent-over" Wright,1984]even n moderatebreezesby the addition of horizon-tal momentum, while large plumes hat penetrate the strato-sphere re bent-overonly when windspeeds high [Sparks tal, 1997].The polar jets or jetstreams are regionsof high windspeedthat span the globe at latitudes from 30to 60. The jetsmark the convergence zone between warm subtropical airand cold polar air. They are geostrophicwinds and thereforeare associatedwith a rapid change in vertical pressuregra-dient and tropopause height. The height of the tropopauseis considerably lower on the poleward side of the jet than itis in the mid- and low-latitudes, averaging 9 km in winter(occasionallys ow as 5 km in troposphericoldingevents)and 10 km in summer, as opposed to 16 km at the equa-tor [Glaze and Baloga,1996]. Windspeedswithin the jetsaverage 40 m/s in winter, and 10 m/s in summer.Max-imum speedsestimated in the core of the strongest jets areas high as 130 m/s. In crosssection,windspeeddecreaseswith height and horizontal distance from the core of the jet,which occursat 10 km altitude. Jet width (in planform)varies up to hundreds of kilometers, whereas et thickness sas little as one kilometer.Even large volcanic plumes that are injected into the at-mosphere at high latitudes can be expected to sometimesshow profound effectscausedby the interaction of the plumewith the jet. In fact, jet speedscan approach and even sur-pass local plume speeds, resulting in ingestion of unusualamounts of air during plume rise, bending over of the plumein the windfield, and subsequenteffects on maximum plumeheight and tephra dispersal. Becauseof this possibility of in-

Copyright 001 by the AmericanGeophysical nion.Papernumber2001GL013393.0094-8276/01/2001GL013393505.00

teraction of even arge plumes with wind in eruptions at highlatitude, the present contribution develops a plume modelthat incorporates some of the potential effects of wind ingeneral, and of the plume-jet interaction in particular.2. Model of Plume Motion in Wind

The relative importance of wind entrainment to plumebending is a complex issue that has been addressed by nu-merous workers interested in the movement of plumes gen-erated by smokestacksand other engineering outflows intowindy air [Wright,1984]. The following nalysis uildsuponthese engineering results and adapts their findings to themuch more vigorous volcanic case.2.1. Coordinate System

We can construct a quantitative integral model for plumesthat entrain mass and momentum rom the wind [Hewettet al, 1971; Wright, 1984], by tracking the developmentofthe plume in a plume-centered coordinate system. In thefollowing analysis, x is the horizontal downwind direction,z is the vertical direction, and s represents a local coordi-nate tangential to the plume axis. Theta, 0, is the coor-dinate expressing he inclination of the plume centerline tothe horizon. The equations expressing the coordinate trans-formationbetween x,z) and (s, 0) are given by:

and,z -- /sinOds (la)x - /cos0ds (lb)

2.2. Equations of Plume MotionIn plumes that are significantly affected by a crosswind,the entrainment velocity, U, must be a function of wind-speed, V, as well as axial plume speed, U. Numerousworkers have investigated the use of different entrainment-velocity elationshipsWright,1984]. Reasonable orrespon-dence between one suchentrainment relationship and exper-imental data has been obtained [Hewett et al, 1971]:

V = clV -- V cos 1 +/3IV sin01 (3)where clU- V cost91s entrainment by radial inflow minusthe amount swept tangentially along the plume margin bythe wind, and filV sin 91 s entrainment rom wind; c s theradial entrainment parameter, and/3 the wind-entrainmentparameter. Equation 3 assumes hat the magnitude of thehorizontal wind component is much larger than the verticalcomponent. Based on laboratory experiments, c and/3 have

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3622 BURSIK: PLUME RISE HEIGHT IN WIND50000

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' ' ' ' ' ' ''l ' ' ' ' ' ' ''lWind speed- , No wind V=25 m/s V=50 m/s- V=100 m/s- V =25 m/s, etax- Vmax=50/s,etl--I V =100 m/s, etax

0106 107 108 109Masseruption ate (kg/s)Figure 1. Eruptionolumneightsa functionfmassruptionateunder onditionsfa jet aloft rconstantindspeedithheight.Forclarity, ot all points t which calculationasdone reshown. he modelwindspeedrofilesor he et cases ereV -- Vmax exp(-(z- 10km)2/(lkm)2). hus,he et is assumedo have Gaussianrofile, itha maximumpeedt 10km.Winddecreaseslumeiseheight onsiderably.ith a et, over wide ange f mass ruptionate,plume eightemainselativelyconstant becauseof the rapid ingestion of air aloft.

beenshown o be constants qual o 0.15 and unity, re-spectivelyHewettet al, 1971].Near the vent,whereplumedensitymay be five times that of the ambientatmosphere[Sparks t al, 1997],and plumedecompressionccursn thecrater [Woods nd Bower,1995], he entrainment arame-ters may vary from thesevalues. However, all modelsof ver-tical plumessuggesthe parameters houldapproach hesevalueswithin a kilometer r soof the sourceSparks t al,1997].The equations f motion Hewerr t al, 1971]havebeenrederived or the volcaniccase, o account or realistic sedi-mentation of pyroclastsand for the large densitydifferencesbetween plume and ambient medium that occur in volcanicplumesErnst t al, 1996;Glaze ndBaloga, 996].Formassconservationcontinuity),we have:

Nd .b2pU)2-pabUy dsss (4)i--1whereb is characteristic lumeradius,p is bulk plumeden-sity, pa is ambient atmosphericdensity and Mi representsthe mass flux of pyroclasts of size fraction i within theplume. The first term on the right-handside epresentshegain in mass lux by entrainment of air, whereas he second

term representshe lossof mass lux by fallout of pyroclasts.The equation for conservation of axial momentum is:d (rb2pU2)rb2Apgin+ V cos

N dM+ (5)i=1where he first term on the right-handside representshechangen momentum aused y the component f gravita-tional acceleration, , in the axial direction,and the secondterm represents entrainment of momentum from wind. Oneeffectof entrainmentof wind is to generatea net horizontalplumemomentum hat doesnot exist n a still atmosphere.The conservation f the radial componentof momentum sgiven by:(xb2pU2) (xb2pU) 6)- xb2ApgcosVsindswhere he left-hand side representshe change n causedby the entrainment f momentum t an angle o the plumeaxisby bothgravity first ermonright-hand ide)andwind(seconderm). Note that the small-angle pproximationsmade or de. The conservationf specific nthalpy s givenby: d (b2pUCT)2bUpCTb2Ugsin


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BURSIK- PLUME RISE HEIGHT IN WIND 3623N+ (7)i--1

The first term on the right-hand side is the energy addedby the entrainment of air, the second term is the changein thermal energyby conversiono/from gravitationalpo-tential energy, and the third term is the loss of heat bysedimentation of pyroclasts. Cv is the bulk heat capacityof the material in the plume, T is its temperature, Ca andTa are the heat capacity and temperature respectively ofthe air, and Cp is the heat capacity of the pyroclasts.Theconservation f mass lux of particles or multiple grain sizefractions,Mi, is givenby [Ernst et al, 1996]:dM

ds bu (8)where p is a probability that an individual particle will fallout of the plume, f is an empirical re-entrainment parameterthat is a function of plume strength and particle size, andws is the settling speed of a particle in the given size class(in the current model, -- i to 19 for pyroclastsbetween10 and -8 qbat 1-qbntervals). The probability of fallout,p, is a function of plume geometry, and should have an ap-proximately constant value of 0 0.23. Because of the stronginflow toward the plume caused by wind and atmospheric

0 5000 10000

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2500-

-2500 0 2500 5000 7500Distance frn vent (m)I

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entrainment, pyroclasts


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3624 BURSIK- PLUME RISE HEIGHT IN WINDplumes,orwhichhemass f air entrainedy wind epre-sents significantroportionf the otalentrainedir masseven close to the vent. For plumes with a larger radius, theproportionf air added y wind s smaller, enceheeffecton plume eights less ronounced.evertheless,givenobservedlume eight annot e directly sed o calculate Fmass eruption rate without taking into consideration wind-speed.For argeeruptionsnd owwindspeeds,he errorintroducedy not consideringind s small.However,ven Tfor largeeruption olumns ith riseheights f 30 km, forexample,ealistic ut highmeanwindspeedshrough he Uatmospheref 25m/s could ielderuptionateestimatesUthat are an orderof magnituden error. bAt sufficientlyighwind speeds r low mass ruption frates, a plume can not only ingest significant quantities of gair by wind,but also hecenterlinef theplume anbecome hdistorted or bent-over n the wind field (Fig. 2). A plume Pbends over because of the addition of horizontal momentum rby the wind (equation6). A typical bent-overplume does snot have a constant radius of curvature along its entire path, wseven in a wind that is uniform with height. The radius of xcurvature increases with height in a uniform field because zthe fractional increase in the horizontal component of mo-mentum flux decreases as more of the wind's momentum istaken into the plume.No plumes are erupted into atmospheres having a uni-form wind speed with height. The most extreme case isperhaps the polar jet, from which most of the horizontalcomponent of momentum is added to the plume in a rela-tively narrow height range aloft. The radius of curvature ofthe plume decreaseswith height into the jet, then increasesas the plume traverses he jet on the upper side. Becauseofthe high wind speed within the jet, and the low plume speedat height, the sudden encounter of the jet by the plume canresult in a rapid ingestion of air and dilution of buoyancy.Plume height can thus be lowered dramatically, and in fact,plumes rise to heights about equal to the height of the jetover a range of masseruption ates (Fig. 1).4. Conclusions

The main results of the interaction between plumes andwind are a decrease in plume rise height at constant erup-tion rate, and plume bending. Especially for conditions ofhigh windspeed or low mass eruption rate, wind should beconsideredamong the several mportant variables that affectcolumnheightand tephradispersal Sparks t al, 1997]. Foreruptions into the polar jet, plume height remains approx-imately constant over a wide range of mass eruption rates.Jet speed and geometry vary widely; nevertheless, for rea-sonable aluesof theseparameters Fig. 1), volcanic lumesin the rangebetween 106kg/s and 107 o 108kg/sall rise between 9 and 11 km. In a still atmosphere, the riseheight over this range of mass eruption rate increases rom 17 km to 33 km.

heat capacityof air, 1000J kg K -mass concentration f particles n i *h grain-sizefractionheat capacity f pyroclasts, 100J kg- K- bulk heat capacity of plume materialspecifichermal lux, b2UCvTspecificmomentumlux, b2Umass lux of i *h particle-sizeractiontemperature of plumetemperature of airaxial plume speedentrainment speedcharacteristic plume radiusparticle re-entrainment parametergravitational cceleration,.807ms2fall-out height of a pyroclastprobability of falloutradial coordinatedownstream coordinateparticle settling speedhorizontal coordinatevertical coordinate

Acknowledgments. This researchwas supported n partby grants from the National Aeronautics and Space Administra-tion, the California Institute of Technology, The National ScienceFoundation and Science Applications International Corp. I wouldlike to thank the Center for Computational Research at SUNYBuffalo, and especially Jeff Tilson, for expertise and supercom-puter time. Aaron Burns and Vassili Kouznetsov programmedparts of the model. Two reviewers are thanked for useful sugges-tions about the improvement of the presentation.ReferencesErnst, G.G.J., Carey, S.N., Sparks, R.S.J., Bursik, M.I., Sedi-mentation from turbulent jets and plumes, J. Geophys. Res.,101, 5575-5589, 1996.Glaze, L.S. and Baloga, S.M., Sensitivity of buoyant plumeheights to ambient atmospheric conditions: Implications forvolcanic eruption columns, J. Geophys. Res., 101, 1529-1540,1996.Hewett, T.A., Fay, J.A. and Hoult, D.P., Laboratory experimentsof smokestack plumes in a stable atmosphere, Atmos. Env., 5,769-789, 1971.Morton, B., Turner, J. and Taylor, G., Gravitational turbulentconvection from maintained and instantaneous sources, Pro-ceedingsRoyal Soc. London Ser. A23, 1-23, 1956.Sparks, R.S.J., Bursik, M.I., Carey, S.N., Gilbert, J.S., Glaze,

L.S., Sigurdsson, H. and Woods, A.W., Volcanic Plumes, J.Wiley and Sons, Chichester, UK, 574 pp., 1997.Woods, A.W. and Bower, S.M., The decompression of volcanicjets in a crater during explosive volcanic eruptions, EarthPlanet. Sci. Lett., 131, 189-205, 1995.Wright, S., Buoyant jets in density-stratified crossflow, Jour.Hydr. Eng., 110, 643-656, 1984.M. Bursik, Department of Geology, 876 Natural SciencesCom-plex, University at Buffalo, SUNY, Buffalo, NY, 14260. (e-mail:[email protected])

5. Notationa radial entrainment coefficient, 0.15fi wind-entrainment coefficient, 1.0e entrainment coefficientp bulk plume densitypa atmospheric density angle between plume axis and horizon (ReceivedMay 3, 2001; revised June 23, 2001;acceptedJune 27, 2001.)

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Effect of Wind on the Rise Height Volcanic Plume