The mechanism of volcanic eruptions (a steady …raman/papers2/SlezinJVGR.pdfThe mechanism of volcanic eruptions (a steady state approach) Yu.B. Slezin Institute of Volcanic Geology

The mechanism of volcanic eruptions(a steady state approach)

Yu.B. SlezinInstitute of Volcanic Geology and Geochemistry FED RAS, Piip Blvd. 9, 683006 Petropavlovsk-Kamchatsky, Russia

Received 13 December 2001; accepted 17 September 2002

Abstract

A theory of a volcanic eruption as a steady magma flow from a magma chamber to the surface is described.Magma is considered as a two-component two-phase medium and magma flow is steady, one-dimensional andisothermal. Mass and momentum exchange between condensed and gaseous phases is assumed to be in equilibriumwith nucleation of bubbles as a fast process at small oversaturations. Three basic flow structures in a conduit arerecognized and three resulting types of eruption are described. (1) In the discrete gas separation (DGS) flow, melt withdiscrete gas bubbles erupts. This type includes lava eruptions and Strombolian activity. (2) In the dispersion regimethere is continuous eruption of a gas-pyroclastic suspension. Catastrophic explosive eruptions are included in thistype. (3) Here a partly destroyed foam is erupted with permeable structure that allows gas escape. Near the surfacethis structure evolves with an increase in permeability and a decrease in the total porosity. This type corresponds tolava dome extrusions. A criterion dividing DGS flow from the other two regimes is obtained: Di=URn1=3a/c0, whereU is the magma ascent velocity, R is magma viscosity, n is number of bubble nuclei in a unit mass of magma, a is thesolubility coefficient for volatiles in magma and c0 is the volatile content. The boundary between the dispersion regimeand the partly destroyed foam regime depends on U, c0 and particle size dp. For the dispersion regime the basicgoverning parameters are: magma chamber depth H, conduit conductivity c= b2/R (b is the characteristic cross-dimension of the conduit), and excess pressure in the magma chamber. Instability of the dispersion regime isinvestigated numerically. The dependence of magma discharge rate on the governing parameters is described as acombination of equilibrium surfaces with cusp catastrophes. Magma chamber depth is a ‘splitting’ parameter in mostcusps and controls the existence of unstable magma flow with sharp regime changes. Catastrophic rise of eruptionintensity is possible if H6Hcr. The critical chamber depth Hcr depends approximately linearly on c0 and is about 15km at c0 = 0.05. Some scenarios of eruption evolution are described in a steady state approximation, and themechanism of catastrophic explosive eruptions is proposed. The theory is applied to well documented eruptions(Tolbachik, 1975^1976, and Mount St. Helens, 1980). The theory can also describe changes in a volcanic system, suchas destruction of the conduit or chamber walls, destruction of the edifice, changes in magma composition, and calderasubsidence due to magma evacuation.= 2002 Elsevier Science B.V. All rights reserved.

Keywords: volcanic eruption; steady state; two-phase £ow regime; critical conditions

0377-0273 / 02 / $ ^ see front matter = 2002 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 7 - 0 2 7 3 ( 0 2 ) 0 0 4 6 4 - X

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Journal of Volcanology and Geothermal Research 122 (2003) 7^50

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1. Introduction

Volcanism is a global phenomenon and studyof it includes many aspects and approaches. Inthis paper the physical mechanism of eruption isdiscussed. I have used only a steady state ap-proach, which cannot describe all the details andcomplexities of volcanic processes. However, thisapproach is an important starting point as it helpsto describe and to understand the general featuresof eruptions, including abrupt transitions fromone steady regime to another. The objective ofthis extensive paper is to review and summarizea large body of research results on the theory ofvolcanic eruptions developed by the author overthe last two decades. Principally it concerns thenature of eruption instability and abrupt changesin eruptive style. This work has largely been pub-lished in Russian and is not well known within theinternational community. Although there aresome similarities and parallel developments inthe western scienti¢c literature there are funda-mental aspects of the research results, conceptsand approach which are di¡erent. Some of theseconcepts are emerging outside Russia and so it istimely to publish a synopsis both to familiarize abroader community with interesting results andperhaps to avoid the reinvention of the wheel. Iwill consider some studies (mostly made aftermine), which are partly related to my work. Afull comparison cannot be done properly in thelimits of this paper, and I apologize to thosewhose investigations escape this discussion.

1.1. De¢nitions and formulation of the problem

A volcanic eruption is a dynamical process ofout£ow that is characterized as follows: (1) me-chanical properties, composition, and structure ofthe erupted products; (2) their mass rate of erup-tion; (3) their velocity; (4) the £ow regime,namely the way these characteristics change intime. An eruption is also an event, separatedfrom other similar events by a rest (or dormant)interval. Rest is the second of two possible statesof an active volcano. Long-term volcanic activity isan alternating succession of those two fundamen-tal states. Eruptions can be treated in two re-

spects : as a process of out£ow and the evolutionof this out£ow over the course of the event.

The direct way to obtain quantitative informa-tion about an eruption is by measuring £ow char-acteristics of volcanic eruptions. These measure-ments are very di⁄cult to obtain, especially inintensive catastrophic eruptions. Most informa-tion can only be obtained indirectly, such as byinterpretation of measurements made after aneruption, observations at a great distance, andmeasurements of disturbances and changes inthe environment caused by the eruption. Interpre-tation of such information requires theoreticalmodels. Eruption is the ¢nal stage of processesat depth, including origin, formation and trans-port to the surface of magma; and starts the pro-cesses of dispersal of the products. Consequentlythe study of eruption mechanisms can be dividedinto two problems: internal and external. The ex-ternal problem includes the description of thetransport of volcanic products from the momentof emerging to the moment of their primary de-position or dissipation in the atmosphere. The in-ternal problem concerns describing the dependenceof the eruption on the characteristics of the mag-ma system and on external in£uences. The inter-nal problem involves construction of a theoreticalmodel of these processes.

This work concerns the internal problem. Basicregimes of eruption and their evolution, with con-ditions of their realization, stability limits, and thein£uence of external factors are investigated usinga steady state approach. An approach to predict-ing hazards of a volcano according to the basicinformation on the structure and state of its mag-ma system is suggested. Theoretical results areapplied to two of the most completely studiederuptions: those of Tolbachik 1975^1976 in Kam-chatka, Russia and Mount St. Helens 1980 inWashington State, USA. These eruptions are ofquite di¡erent types with di¡erent magmas anddi¡erent scenarios. The author took part in thestudy of the Tolbachik eruption and observationsmade on this eruption motivated this work.

1.2. A brief review of research

Early studies of volcanic eruptions were con-

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Yu.B. Slezin / Journal of Volcanology and Geothermal Research 122 (2003) 7^508

cerned with classi¢cation. Lacroix (1904), Mercal-li (1907) and Jaggar (Luchitsky, 1971) classi¢edthe main types of eruption on the basis of theconcentration of volatiles in magma and magmaviscosity. Escher (1933) and Sonder (Luchitsky,1971) introduced the magma chamber depth asa control on eruptive type. Early model conceptswere only qualitative and largely speculative. Thesystematic development of the physical conceptsof eruption mechanisms began in the middle ofthe 20th century. Graton (1945) ¢rst formulatedthe physical problem and introduced the termmagma system of a volcano. Graton recognizedthat under some conditions in a volcanic conduita foam of close-packed bubbles must form, whichmust be treated physically as a new phase. Ver-hoogen (1951) consistently described the processof degassing of ascending magma and obtainedsome quantitative relations.

The contemporary period of theoretical model-ing of volcanic eruptions was initially stimulatedby planetary research (McGetchin and Ullrich,1973). A model describing magma £ow in a con-duit as a steady one-velocity (homogeneous) two-phase £ow in a ¢ssure with elastic walls was pre-sented by Wilson et al. (1980). The problem ofbubble rise through ascending magma was ¢rstinvestigated by Slezin (1979), who identi¢ed dif-ferent two-velocity £ow regimes. Wilson andHead (1981) considered the same problem usingdi¡erent assumptions. Subsequently, numerousstudies have developed more elaborate and com-plicated models. Investigations proceeded in thewestern countries and in Russia independentlyand Russian (my) works were not known in thewest, leading to duplication of e¡orts. This papersummarizes some of the author’s works on thedynamics of explosive volcanic eruptions pub-lished in Russian from 1979 to 1995. These inves-tigations anticipated later research in the west,and also included new concepts. I refer the readerto reviews by Sparks et al. (1997), Papale (1998)and Melnik (2000). I cite only work that is directlycomplementary or overlapping with mine.

Model developments have included new sub-stantial parameters and new approaches. Someof those new approaches and additional parame-ters were ¢rst considered in my investigations

published in Russian. These included two-velocity(non-homogeneous) and quasi-steady problems,multiple solutions for the steady £ow, discoveryof £ow instability and recognition of sharpchanges in the eruption intensity and style (Slezin,1979, 1983, 1984, 1991). These and other relatedstudies were summarized by Slezin (1998). Somelater studies in the west partly overlapped withthese studies (Wilson and Head, 1981; Jaupartand Allegre, 1991; Dobran, 1992; Woods andKoyaguchi, 1994; Melnik and Sparks, 1999; Mel-nik, 2000). Only in recent publications (Melnikand Sparks, 1999; Melnik, 2000; Barmin et al.,2002) have preceding Russian results been used.

2. Volcanoes as dynamic systems

2.1. The magma system of a volcano

Volcanism is the result of melt generation in theupper mantle and crust and its transport to thesurface. For eruption modeling it is su⁄cient toassume that magma is generated locally undercertain conditions, which depend on the geother-mal gradient, thermal properties of the systemand geodynamical setting. Magmas are typicallygenerated by partial melting at depths of severaltens of kilometers and are di¡erentiated duringascent. Melting and di¡erentiation decrease den-sity typically by 5^15% relative to the originalrock (Clark, 1966; Lebedev and Khitarov, 1979).The di¡erence in density between magma and itssurroundings causes ascent of magma, its storageand development of excess pressure in magmachambers. In the relatively plastic mantle, asthe-nosphere melts with relatively low density and lowviscosity rise as discrete jets or columns building aperiodical structure (e.g. Fedotov, 1976a,b; Ram-berg, 1981). Density di¡erences can cause excesspressures at the head of magma column up to 100MPa. Because of the mechanical properties ofmost of the lithosphere and especially of theEarth’s crust, the most probable way of magmaascent is through fractures propagating in rigidmedia by the mechanism of hydrorupture inducedby an excess magma pressure of no more than 20MPa (e.g. Popov, 1973; Lister, 1990).

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Yu.B. Slezin / Journal of Volcanology and Geothermal Research 122 (2003) 7^50 9

The inhomogeneous structure of the crust al-lows the formation of intracrustal magma cham-bers, particularly at horizontal strata boundariesand density interfaces. The shallowest of suchchambers are here called peripheral. Direct evi-dence for shallow magma chambers is providedby geological and geophysical data (e.g. Luchit-sky, 1971; Balesta, 1981; Ryan, 1987). Because ofthe mechanism of formation, these chambers areinitially expected to be horizontally arranged. Re-lations between the dimensions of magma cham-bers and volcanic conduits are such that, withrespect to magma £ow, the ¢rst can be treatedby analogy to electric circuits as capacitors withnegligible resistance, and the second as elementsof resistance with very small capacitance.

2.2. Erupting volcano systems

2.2.1. The system geometry and magma propertiesHere the upper part of the system consists of

the peripheral magma chamber with a conduitconnecting it to the surface. The gaseous phasecan exsolve and sharp changes in substance prop-erties and in £ow structure can take place. Thesystem is strongy non-linear, and alternation oferuptions and dormant periods is characteristic.Average magma discharge rate is approximatelyconstant over a prolonged period of a volcano’slife (Kovalev, 1971; Tokarev, 1977; Crisp, 1984).The total energy of an eruption and duration ofthe cycle, including the eruption and the preced-ing dormant period, are correlated positively. Thecorrelation becomes much stronger if groups oferuptions (cycles of activity) are concerned insteadof the individual eruptions (Kovalev et al., 1971).This suggests that intermittence of volcanic activ-ity is the result of episodic (periodical) dischargeof some reservoir (magma chamber), which wassteadily re¢lled from the deeper parts of the mag-matic system (Kovalev, 1971; Kovalev and Slezin,1974).

A volcanic eruption is typically short comparedto the duration of dormant periods. Dormant pe-riods are on average 30 times longer than erup-tion durations for the island arc volcanoes and onaverage 60 times more for other volcanoes (Sim-kin and Siebert, 1984). The magma discharge rate

during an eruption must surpass the average feed-ing rate of the magma chamber in similar propor-tions, and so the latter in most cases can be ne-glected. So an erupting volcano will be de¢ned asa ‘chamber-conduit’ system ¢lled with magmawith some extra pressure and isolated in all direc-tions except at the upper end of the conduit,which is open to the atmosphere. Relations be-tween the characteristic dimensions of the magmachamber and the volcanic conduit and betweencharacteristic times of the processes of the magmasystem during eruption permit analysis of an evensimpler system, and the study of quasi-steadystate scenarios of magma £ow through the con-duit caused by the constant (slowly changing)pressure di¡erence. There are a limited numberof qualitatively di¡erent steady states of such asystem, which correspond to the di¡erent stylesand modes of eruption. The main characteristicsof an erupting system are as follows.

2.2.1.1. Magma chamber. Its shape is approxi-mated as an ellipsoid or cylinder nearly isometricor £attened in the vertical direction (Fedotov,1976a,b). Linear dimensions of the horizontalprojection are estimated to be from a few kilo-meters to a few tens of kilometers (Luchitsky,1971; Farberov, 1979; Balesta, 1981); there is lit-tle reliable information about vertical dimensions.Chamber volume can be from a few to thousandsof cubic kilometers. The top of the chamber canbe situated at depths of a few to a few tens ofkilometers (Zubin et al., 1971; Balesta, 1981).

2.2.1.2. Conduit. Although the initial form of aconduit in the lithosphere is a ¢ssure, after sometime focusing of £ow can change it to a cylinder.At depth this process is slow, but it can be muchfaster near the surface, where gas exsolves and thevelocity and eroding ability of magma increase.Direct observation of the Tolbachik eruption1975^1976 shows that transformation of the ini-tial ¢ssure to a circular cylinder (near the surface)took only a few days (Fedotov, 1984). In a dor-mant period a conduit can become partly solidi-¢ed starting from its upper part ; it may becometotally closed and eruption through it would notnecessarily resume.

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The central conduit of a large stratovolcano is astable, long-lived structure. This is con¢rmed bythe shape of symmetric stratovolcanoes made byinnumerable volcanic eruptions, emerging fromthe same central crater. However, nearly all indi-vidual eruptions inside the central crater occurthrough outlets with dimensions much smallerthan the crater £oor diameter and are located indi¡erent places on this £oor. In eroded volcanoesa large number of intersecting dykes of di¡erentages are observed (Luchitsky, 1971; Sheymovich,1975; Sheymovich and Patoka, 1980). So, even inthe case of many successive eruptions through asingle crater, the upper part of the conduit foreach eruption initially had the form of a newlyborn ¢ssure.

Factors in£uencing dyke propagation includeinternal magma pressure, magma viscosity andtectonic stresses. Two end member models arecommonly assumed: (i) an elastic ¢ssure in whichinternal pressure is equal to external pressure (e.g.Wilson et al., 1980; Wilson and Head, 1981) and(ii) a rigid ¢ssure with behavior only dependenton hydrostatic and magma dynamics (Slezin,1983; Giberti and Wilson, 1990; Dobran, 1992).For the latter case a constant ¢ssure geometry iscommonly assumed. Here the second model wasused. The conduit is considered as a ¢ssure atdepths where magma £ow is taken as a continu-ous liquid and as a cylinder near the surfacewhere £ow consists of a gas^pyroclastic mixture.The model is appropriate for the Great Tolbachikeruption from observations of lava bombs withcores of sedimentary xenoliths (Slobodskoy,1977) derived from a depth of at least 2 km(Shanzer, 1978) and covered with droplets ofmagma stuck to them. So the depth of fragmen-tation had to be no less than 2 km, which wouldnot be so if the ¢ssure were elastic rather thanrigid. In the elastic model the hydrostatic pressurein the conduit would be too high to allow magmaand wallrock fragmentation at depths of morethan a few hundreds of meters. At greater depthsthe elasticity of ¢ssure walls could play a moresigni¢cant role.

2.2.1.3. Magma. Magma in the conduit is atwo-component, two-phase medium. The compo-

nents are volatile and non-volatile ; the phases arecondensed and gaseous. The non-volatile con-densed component consists of silicate matter.The volatile component, assumed to be water,can exist both in condensed (i.e. dissolved in thesilicate melt) and in gaseous phases. A secondvolatile component, CO2, di¡ers from water byhaving a larger density and being less soluble inthe melt (Kadik and Eggler, 1976; Mysen, 1977;Mysen and Virgo, 1980). These properties in£u-ence eruption dynamics (through the position ofthe fragmentation level) in opposite ways. My es-timates (Slezin, 1982a) show that the e¡ect of thegas density increase is stronger than the e¡ect ofthe decrease of solubility. Substitution of water byCO2 decreases the volume fraction of gas (whenmass fraction is constant) and magma dischargerate, and is approximately equivalent to a de-crease of the mass fraction of water, a resultconsistent with ¢ndings by Papale and Polacci(1999).

The water content of magmas is typically a fewweight percent (up to 6%). However, magma canerupt outside this range. For example, water canbe lost by surface degassing prior to eruption andphreatic water can be added to magma. Addition-al external water was suggested to explain the ex-tremely water-rich basalts erupted by the North-ern Cones of the Tolbachik eruption 1975^1976(Menyaylov et al., 1980; Fedotov, 1984).

The condensed phase is treated as a liquid whenit is the continuous component of the £ow and assolid matter when it forms pyroclastic particles.In£uence of suspended crystals is not consideredexcept for the inability to dissolve volatile compo-nents. Density of the condensed phase is approxi-mated as independent of the concentration of thedissolved volatiles. Magma is assumed to be aNewtonian liquid with constant viscosity. Insome studies the strong dependence of viscosityon dissolved water concentration is considered(Barmin and Melnik, 1990; Dobran, 1992; Slezinand Melnik, 1994). Dobran (1992) investigatedviscosity dependence on bubble concentration.The dependence of viscosity and some other prop-erties of magmas on the composition, pressure,water content and temperature are now wellunderstood (Shaw, 1972; Lebedev and Khitarov,

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1979; Epelbaum, 1980; Persikov, 1984; Dingwellet al., 1993).

The solubility of the volatile component (water)in the melt is considered as dependent on pressurep only (Sparks, 1978):

c ¼ ap1=2 ð1Þwhere c is the mass part of volatile component inliquid phase and a is an approximately constantcoe⁄cient, with values typically in the interval0.0032^0.0064 MPa31=2. The lower ¢gure corre-sponds to ma¢c magmas and rather large pres-sures (about 400^600 MPa) whereas the upper¢gure corresponds to silicic magmas and the pres-sure interval from 0 to 100^200 MPa. In manystudies a=0.0041 MPa31=2 was assumed (Sparks,1978; Wilson and Head, 1981). Wilson and Head(1981) modi¢ed Eq. 1 for basalts with the expo-nent 0.7.

During decompression due to magma ascent,equilibrium between the volatile component inthe melt and in the gaseous phase is assumed.This assumption is usually reasonable for thebubbles in the melt, but invalid in the gas^pyro-clast mixture, where speeds are high. But in thelatter case volatiles dissolved in the particles areusually low and one of two extreme assumptionsprovides good approximations: equilibrium or nogas emission at all. For high-viscosity magmasand rapid £ow the assumption of equilibriummay not be valid even in the bubble zone, andin this case di¡usion delay of gas emission cantake place, which is equivalent to a decrease ofthe volatile content. An attempt to take into ac-count the di¡usion delay was made by Slezin andMelnik (1994). Threshold oversaturation neededfor heterogeneous bubble nucleation is not large(about 0.1 MPa; Sparks, 1978; Hurwitz and Na-von, 1994), so this is neglected. For magma £owdynamics heterogeneous nucleation not only low-ers the threshold of new gas phase generation, butalso induces generation of very large numbers ofbubble nuclei in a unit volume.

2.2.2. Physical conditions in the systemThe temperatures of erupting magmas are typ-

ically 1000^1200‡C for basalts and 700^1000‡Cfor silicic magmas. During magma ascent temper-

ature can be in£uenced by the following pro-cesses : (1) heat loss through the conduit walls ;(2) exsolution of volatiles; (3) adiabatic expan-sion; (4) viscous dissipation; (5) crystallization;(6) exothermic reactions. The ¢rst and the thirdprocesses involve cooling, the second process canhave either sign (Kadik et al., 1971), and the lastthree processes involve heating. These e¡ects canbe approximately estimated as follows.

In the rather ‘rigid’ case of a ¢ssure conduit 10km long and 2 m wide and magma ascent velocity0.2 m s31 the temperature decrease due to max-imum heat loss through the walls will be severaltens of degrees after 24 h and only a few degreesafter 1 month of continuous eruption, accordingto my calculations. For a wider ¢ssure or for theequivalent cylindrical channel the temperature de-crease would be less. The thermal e¡ect of waterexsolution is negative when pressure is less than300 MPa (Kadik et al., 1971) and for albite meltcomposition is about 310‡C per 1% H2O. Above300 MPa the e¡ect changes sign. Wilson andHead (1981), referring to Burnham and Davis(1974), stated that the e¡ect of water exsolutionis positive and is quantitatively no less than+10‡C per 1% H2O. Expansion and accelerationof magma £ow transform thermal energy into ki-netic energy. Viscous friction transforms kineticenergy into thermal energy. The resulting e¡ectmust be negative and proportional to the resultingkinetic energy of eruption products per unit massof magma. For equal temperatures of gas andpyroclastics and an average velocity at the con-duit exit of 100 m s31, my estimates give a tem-perature decrease of about 4‡C, and 16‡C for 200 ms31. The heat of crystallization of silicate meltvaries from 2U105 to 4U105 J kg31 (Clark, 1966;Dudarev et al., 1972). In an explosive eruptionlittle or no melt can crystallize, but this processcan be important in slow extrusions (Sparks et al.,2000). The thermal e¡ect of crystallization is esti-mated be no more than a few tens of degrees.Exothermic reactions occur on contact with atmo-spheric oxygen (Macdonald, 1963; Fedotov,1984), but this e¡ect is negligible for conduit£ow. The total change of magma temperatureover the conduit length typically is negative andestimated as no more than a few tens of degrees.

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These estimates justify the isothermal approxima-tion.

2.2.2.1. Pressure. A magma chamber is fed ulti-mately from a deeper zone of magma generationdue to hydrostatic forces. The density di¡erencebetween magma and country rocks can providean upper limit of magma pressure at the head ofmagma column that exceeds lithostatic pressureby a few hundreds of MPa. However, this limitcan never be reached because dykes form andpropagate to the surface. Lithostatic pressure forthe chambers at di¡erent depths (4^30 km) shouldbe from 100 to 700 MPa. An excess pressure of 20MPa is taken as a typical pressure di¡erence atthe initiation of dyke propagation (Popov, 1973;Stasiuk et al., 1993). After some evolution of theconduit this pressure di¡erence can increase dueto the di¡erence between the lithostatic pressureof the country rocks and the hydrostatic pressureof the magma column in the conduit. Foamingand fragmentation of magma in the conduit de-creases its density and so can increase total staticpressure di¡erence up to several hundreds ofMPa. Real dynamic values are less because offriction and acceleration losses. This driving pres-sure di¡erence allows large velocities and massdischarge rates of magma. In a steady regimethe £ow is mostly compensated by the hydraulicresistance of the conduit.

In constructing hydrodynamical models ofmagma £ow, instead of using the pressure at theentrance of the conduit p0, it is convenient tode¢ne an excess pressure pex as the di¡erence be-tween p0 and the hydrostatic pressure of magmawithout bubbles :

pex ¼ p03b lgH ð2Þ

In Eq. 2 H is the magma chamber depth (lengthof the conduit) ; g is gravity acceleration and bl ismagma density. The hydrostatic pressure of themagma column is in most cases less than thelithostatic pressure. The typical average densitiesfor a crustal section near the Avachinsky volcanoin Kamchatka are illustrated in Fig. 1. So pex canbe much more than 20 MPa at the beginning ofan eruption and can be negative at its end due tothe low density of the gas^magma mixture. If the

magma chamber walls are quite rigid and stablepex could change from +100 MPa to 3200 MPa.In practice the range is expected to be less, ap-proximately from +40 to 340^60 MPa.

2.3. Flow structure in a conduit

Two-phase gas^liquid £ow structures have beenstudied in detail as applied to water^vapor mix-tures, oil with gas, and industrial mixtures inchemical technology (e.g. Deich and Philippov,1968; Wallis, 1969; Boothroyd, 1971; Sternin,1974; Nigmatullin, 1987). However, in all thesecases £ow parameters are quite di¡erent fromthose of magma £ows, and corresponding resultsshould be applied to magma £ow with consider-able caution.

Magma has the following speci¢c features: highviscosity of the liquid phase; large di¡erencein density between the condensed and gaseousphases at all pressures; large di¡erences in the

Fig. 1. Average density of crustal layers for di¡erent depthsin the region of the Avachinskaya volcano group calculatedusing data of Zubin and Kozyrev (1989). To the left is thedepth scale in km; numbers in the middle above the horizon-tal lines are the values of the average density (kg/m3) of thecrustal layer; to the right the levels are marked where theaverage densities of the overlying layers are equal to the typ-ical densities of magmas with named compositions (afterMurase and McBirney, 1973). Magma of the named compo-sition and indicated density can reach ground surface by hy-drostatic forces if sourced from beneath the marked level.

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physical and chemical nature of the condensedand gaseous phases; small mass fraction of thevolatile components (a few percent) ; increase ofviscosity of the non-volatile component due tovolatile exsolution; heterogeneous bubble nucle-ation on a large number of centers; and largevertical length of the conduit. There are someunique features of £ow structures in a volcanicconduit. Rising magma bubbles can reach thestate of close packing without bubbles movingrelative to the liquid; the transition of a liquidwith bubbles into the state of a foam is possible.A dispersion of solid particles in gas (gas^pyro-clastic £ow regime) can develop directly from amagmatic foam. The transition between plug£ow and annular £ow is not usually expected,except for some types of Hawaiian eruption(Vergniolle and Jaupart, 1990).

There is a £ow structure, named ‘partly de-stroyed foam’ by Slezin (1980a, 1995a,b), whichhas not been considered in previous eruptionmodels. In this regime both phases of the two-phase, two-speed £ow are continuous. Gas movesthrough the interconnected pores in a partly de-stroyed foam faster than the permeable foamymagma. Such permeable systems in silicate meltshave been experimentally studied by Eichelbergeret al. (1986), Letnikov et al. (1990), Letnikov(1992) and Balyshev et al. (1996). A sharp directtransition from bubble £ow to gas^pyroclasticmixture is often assumed. This sharp transitionwas supposed to result from increasing pressurewithin bubbles due to pushing viscous meltthrough thin ¢lms dividing bubbles needed forfurther bubble growth (Sparks, 1978). In fact,an expanding bubble in a close-packed ensembleis surrounded by a foam medium which is de-formed not by pushing of viscous liquid throughthin ¢lms, but with stretching and bending ofthose ¢lms. This stretching and bending becomeseasier when ¢lms become thinner, and the resis-tance of the foam to deformation become lower(I.I. Goldfarb, private communication). Graton(1945) wrote that a magmatic foam should betreated as a new phase, but his advice was forgot-ten. The problem is to describe partly destroyedfoam analytically. Here it is approximated by amass of loose particles partly stuck together. The

empirical formulas of Nigmatullin (1987) can beapplied to the dynamics of gas £ow in loose ¢lls.

There can be from two to four successive zones(Fig. 2) with di¡erent £ow structure in a volcanicconduit for steady £ow. The changes of structureresult from the increase of volume fraction of gas.In an homogeneous zone (zone 1) there is onlysilicate liquid in the conduit. In a bubble zone(zone 2) there is gas^liquid dispersion £ow inwhich liquid is the continuous phase. At largevolume concentrations of bubbles (s 50%) thiscan be de¢ned as a foam. In zone 3 three casesare possible, two of which are illustrated in Fig.2a,b. First, the increase of gas volume during as-cent results in the increase of gas bubble sizes.Their upward speed through the liquid results incoalescence due to bubbles overtaking one anoth-er (zone 3 in Fig. 2a). Second, the increase of gasvolume during ascent results in the gas bubblesgrowing ‘in situ’ and rising only with the liquid(Fig. 2b). Bubbles eventually make a foam thatcan disrupt and create a porous permeable sub-stance of partly destroyed foam through which

Fig. 2. Possible structures of two-phase £ow in a volcanicconduit. (a) The discrete gas separation (DGS) regime: Zone1, magma without bubbles; Zone 2, single-velocity bubbling£ow; Zone 3, DGS (two-velocity bubbling) £ow. (b) The dis-persion (steady-jet) regime: Zone 1, magma without bubbles;Zone 2, bubbling single velocity £ow; Zone 3, partly de-stroyed foam (PDF, porous, channeled medium); Zone 4,gas^pyroclastic dispersion £ow.

VOLGEO 2551 7-2-03


gas escapes without entraining particles (Slezin,1979, 1980a). This structure can be illustrated bypumice, in which most pores are interconnected(Witham and Sparks, 1986). Third, the zone ofpartly destroyed foam is reduced to zero and theparticles become £uidized after fragmentation,provided that the £ow velocity and the melt vis-cosity are both large enough. In dispersed £ow(zone 4) the continuous phase is gas and particlesare the dispersed phase, behaving typically as sol-ids on the time scale of the £ow. A relatively largezone containing a dense gas^pyroclast mixture (azone of £uidized £ow) is a characteristic feature ofconduit £ows.

Three of the zones have liquid as the continu-ous phase, and the liquid viscosity is responsiblefor £ow resistance. These three zones are treatedas one zone of liquid £ow that is separated fromthe zone of gas^pyroclast £ow by the fragmenta-tion level. Volcanic two-phase £ows di¡er fromsimilar £ows in technical systems because of thevery sharp decrease of resistance at the fragmen-tation level. The average £ow resistance in gas^pyroclast mixtures per unit length of conduit is atleast 5^6 orders of magnitude less than the £owresistance in a zone of liquid £ow. In the gas^pyroclast zone gravitational forces are large andinertial forces are very small.

2.4. Classi¢cation of eruptions based on £owstructure

Slezin (1979, 1995a) proposed a classi¢cation ofthe basic eruption regimes on the basis of £owregimes (two-phase £ow structure) at the conduitexit.

(1) Liquid with gas in the form of separatedbubbles appears at the surface. The discrete gasseparation (DGS) regime can take place only ifbubble sizes are large enough for their speed tobe much greater than the speed of liquid. Thisregime corresponds to all eruptions with gas emis-sion in discrete portions, including nearly puree¡usive and nearly pure explosive (Strombolian)activity. Observations at Tolbachik (1975^1976)show that during Strombolian activity gas emis-sion on average is directly proportional to magmadischarge rate (Slezin, 1990).

(2) A continuous eruption of gas^particle £owappears at the surface. The dispersion regime cor-responds to all gas^pyroclast eruptions includinghigh intensity, catastrophic explosive eruptions(CEE). In this regime two di¡erent zones of liquid£ow and of gas^pyroclast £ow exist in the con-duit, separated by the fragmentation level. Thepossibility of major changes of total £ow resis-tance after very small displacements of thisboundary make such £ows unstable and makepossible sharp and unexpected changes in theeruption character and intensity.

(3) A two-phase medium erupts in which bothphases are continuous and move with di¡erentvelocities. This medium consists of high viscositymagma with a system of interconnected pores andchannels, through which gas rapidly escapes. Thisregime is named the regime of partly destroyedfoam (PDF). The third regime corresponds to ex-trusive eruptions. This is possible only as a veryslow movement of high viscosity magma, such asin lava dome eruptions.

3. Conditions for the basic regimes of volcaniceruptions

The boundaries between the three £ow regimesare de¢ned on the basis of six main parametersfor conduit £ow (Slezin, 1979, 1995a): magmaascent velocity, melt viscosity, number of bubblenuclei per unit volume of magma, mass fractionof volatile components, volatile solubility coe⁄-cient and mean size of the pyroclastic particles.They depend on many properties of the magmaand the eruption system. For example, mean size,porosity, and the shape of the particles can de-pend on some small constituents modifying sur-face properties of magma (Kovalev and Slezin,1979a,b). The number density of bubble nucleidepends on many factors, including volatile con-tent, composition of melt, temperature and speedof decompression.

3.1. Discrete gas separation and dispersion regimes

For the fragmentation of magma foam a rela-tively large volumetric fraction of gas bubbles

VOLGEO 2551 7-2-03


should be attained near the state of close packing.At close packing, the interbubble melt ¢lms be-come thin and unstable enough for disruption tobegin due to further foam expansion. The ¢lms’stability depends not only on the volumetric frac-tion of gas, but also on the small amounts ofsome magma components known as ‘foam stabil-izers’ (Kovalev and Slezin, 1979a,b). So the foamdensity before disruption can vary between di¡er-ent magmas.

The attainment of the close-packed state can beprevented by either low gas content (6 0.06 wt%)or gas removal by rising bubbles. The latter isplausible, provided the primary bubbles are ableto overtake and coalesce with one another, in the‘discrete gas separation’ regime. The necessarycondition for this can be written as follows:

vlsk1d ð3Þ

where d is the mean distance between bubbles ;vl = lL3lS is the di¡erence between the paths trav-eled by large (lL) and small (lS) bubbles as theyrise through the melt, and k1 is a coe⁄cient ac-counting for the non-uniform distribution of bub-bles in space. The distance between the bubbles isno larger than the distance between their centers :

d6n31=3 ð4Þ

where n is the number of bubble nuclei in a unitmagma volume. The value of n is taken as con-stant after a very short nucleation interval (Tora-maru, 1989). The nucleation of new bubbles ismost likely at the greatest distance from existingbubbles where maximum supersaturation is ex-pected (Blower et al., 2001). Consequently, bythe end of nucleation the distribution of the bub-ble nuclei in space must be approximately uni-form, and we can put:

k1 ¼ 1 ð5Þ

The path, l, traveled by the bubble as it rises atspeed u relative to magma, can be approximatelyde¢ned as:

l ¼ uT ð6Þ

T is the time of bubble rise. If it does not coa-lesce with its neighbors each bubble can grow to amaximum close-packing size, approximately equal

to n31=3. Then the average speed relative to mag-ma, de¢ned approximately as an arithmetic meanof the initial and the ¢nal rate, is :

u ¼ b lg36R

n32=3 ð7Þ

where bl is the liquid density, g is the accelerationof gravity, and R is the liquid viscosity.

A bubble can rise through magma during thetime T that magma takes to rise from the level ofthe bubble generation h0 to the level hp, where aclose-packing state is attained. Here the ratio be-tween the gas and liquid volumes Vg/Vl becomesapproximately 3. Expressing the di¡erence be-tween the levels through a pressure di¡erence(p03pp) :

h03hp ¼ p03pp

bg þ GUR

b2

ð8Þ

Here b is the average density of the melt^bub-ble mixture, and U is the average £ow velocity onthe h03hp interval, b is the characteristic cross-dimension of the conduit and G is the factor char-acterizing its cross-sectional geometry. The timeT can be found as the distance h03hp dividedby U :

T ¼ p03pp

Ubg þ GU2

R

b2

ð9Þ

From continuity of the £ow we get:

Ub ¼ Ub l ; U ¼ Ub l

b

ð10Þ

where U and bl are the £ow speed of magmawithout bubbles and its density, respectively.The value of b (and hence of U) over the distance(h03hp) can be calculated assuming equilibriumfor volatiles between liquid and gas phases. Fortypical volatile contents, the ratio U/U ranges be-tween 2.5 and 1.7 and is equal to 2 with a volatilecontent of slightly less than 5%. Putting U=2U inEq. 9 and taking into account Eq. 10, we obtain:

T ¼ p03pp

Ub lg1þ G

4UR

gb lb2

� �31

ð11Þ

The pressures p0 and pp can be found from the

VOLGEO 2551 7-2-03


magma volatile content, c0, and Henry’s law (Eq.2). Assuming the gas to be ideal, one can write :

Vg

Vl¼

b lpaðc03affiffiffip

p Þb gap

ð12Þ

where pa is atmospheric pressure and bga is thegas density at this pressure and magma temper-ature. Vg/Vl =0 at p0 and Vg/Vl =3 at pp. Substi-tuting these values into Eq. 11 and then substitut-ing Eqs. 11 and 7 into Eq. 6, we can ¢nd thedistance that an average bubble can rise relativeto magma l= l.

Let us de¢ne the ‘large’ and the ‘small’ bubbleand express vl through l. The radii of large andsmall bubbles (rL and rS, respectively) are chosenso that the probability that the ‘small’ bubble isahead of the ‘large’ bubble in the ‘capture zone’is unity in any conduit cross section after nucle-ation ceases. The capture zone is assumed to bea vertical circular cylinder with the same radiusas the ‘large’ bubble. This approximation is basedon the tendency of small bubbles to follow £ow-lines around large bubbles, but neglects coales-cence.

The probability calculation indicates that, fortypical conduit widths, any number of bubble nu-clei per unit volume and any bubble size distribu-tion, we can use the maximum radius rmax for the‘large’ and the minimum radius rmin for the ‘small’bubble. For instance, for a uniform size distribu-tion (dN(r)/dr=Constant between rmin and rmax),the condition that the probability of at least onepair of a ‘small bubble ahead of a large bubble inthe capture zone’ in any cross section of the con-duit is unity is as follows:

vr rLsvrmaxffiffiffiffiffiffi

6Sp n32=3 ð13Þ

where vr= rmax3rL = rS3rmin, vrmax = rmax3rmin,and S is the cross-sectional area of the conduit.For example, let S=100 m2, vrmax = 50 Wm35Wm=4.5U1035 m, and n=1011 m33. Then Eq.13 is rewritten as:

vr rLs8:5U10314 m2 ð14Þ

Because rL 6 rmax = 5U1035 m, the conditionnecessary forEq. 14 to be valid isvrs 1.7U1039 m.

That is, rL can di¡er from rmax, and rS from rmin,by not more than a hundredth of a percent.

The bubble size distribution depends on the ki-netics of bubble nucleation and growth. After ini-tial nucleation, oversaturation can decrease due tofurther nucleation, and gas di¡usion into existingbubbles. The competition of these two processesinvolves a rapid cessation of nucleation and thestabilization of the number of bubbles in a unitmass of magma. Theoretical calculations indicatethat the nucleation interval is of the order of 0.1MPa (Toramaru, 1989). My approximate esti-mates gave values of about 1 MPa, that is lessthan 1% of the total pressure di¡erence in theconduit. The initial bubble size distribution isthus formed, which changes during further riseof magma due only to bubble growth. New bub-ble nuclei can be generated if the pressure de-crease accelerates, probably in the last stages ofmagma ascent when the melt^bubble mixture isnear to close packing, and is not expected toplay a signi¢cant role in controlling the £ow re-gime.

Bubble growth in ascending magma proceedsby two mechanisms: mass di¡usion and decom-pression. Some time after the end of nucleationthe ¢rst mechanism prevails, but soon its in£uencebecomes unimportant. Sparks (1978) showedthese e¡ects for a single bubble; for a bubblegrowing in an ensemble of other bubbles theyshould be more prominent. The e¡ect of the dif-fusion depends on the dynamics of magma ascentand tends to restrict the bubble size range; de-compression tends to keep the ratio of the bubbleradii constant. Hence, if there is no coalescence,the change of the ratio of the largest and smallestbubble sizes should be small, and its value, mea-sured in pyroclastic particles, should give the low-er limit of this ratio in the conduit.

The measurement of bubbles in pyroclastic par-ticles (Heiken and Wohletz, 1985) and in Tolba-chik scoria (the bubbles whose form suggested acoalescence origin were discarded) gave rmax/rminV10. Obviously, this ratio cannot be smallerin the conduit. Considering the above estimatesone also can put rL/rSV10. Since lVr2, one canput approximately vl= lL3lSVlL, and alsorLV2r and lLV4l. Consequently:

VOLGEO 2551 7-2-03


vl ¼ 4l ð15Þ

Taking into account Eqs. 5, 6 and 15, the start-ing condition 3 can be rewritten as:

ls14n31=3 ð16Þ

The value of l can be found using Eq. 6 throughthe substitution of Eqs. 7 and 11 into it. Perform-ing appropriate substitutions in Eq. 16 and simplemanipulations, we obtain the condition necessaryfor the discrete gas separation regime:

URn1=3a2

c20FðQÞð1þ F1Þ60:11 ð17Þ

where

FðQÞ ¼ 13Q2

ffiffiffiffiffiffiffiffiffiffiffiffi1þ 2

Q

r31

� �2

� �31

;

Q ¼ b lpaa2

6b gac0; F 1 ¼ 4G

UR

gb lb2

All quantities that enter the expression for Q,except c0, have a possible variation range of 20^30%; c0 may vary several-fold. My estimate sug-gests that Q varies within 0.05^1, with F(Q) vary-ing only two-fold (1.08^2.15). Considering thegeneral validity of this derivation and the accura-cy of the initial parameter values, F(Q) can bereplaced safely by 1.6 as a typical value.

The quantity F1 is a ratio between the contri-butions of friction and magma weight to the pres-sure loss over the interval of the bubble £ow. Inthe discrete gas separation regime, when the zoneof liquid suspended in gas is absent, its value isalways smaller than one and is approximately(bl3b)/bl, where b is the average density of mag-ma in the conduit. Near the boundary betweenthe discrete gas separation and dispersion regimesFl 6 0.5; it tends to this value only if all the con-duit is ¢lled with a bubbling liquid and its lengthis small. Taking the maximum value F1 and put-ting it into Eq. 17, together with the average valueof F(Q), we obtain the inequality:

Di ¼ URn1=3a2

c2060:05 ð18Þ

The dimensionless number Di (Slezin, 1979,

1995a) has a critical value Dicr =0.05. WhenDi6 0.05 the discrete gas separation regime oc-curs; when Dis 0.05 one of two other regimes(dispersion or extrusive) is possible. Slezin (1979,1995a) proposed that the condition DisDicr isnecessary, but not su⁄cient, for the dispersionregime.

A change to the dispersion regime is promotedby the increase of magma ascent velocity, viscos-ity, and the number of bubble nuclei per unitmass. Less obvious is the dependence upon thevolatile content and the solubility coe⁄cient a.Both the increase of the volatile content and thedecrease of solubility result in the appearance ofbubbles at a greater depth, lengthening their £oat-ing time, and, accordingly, increasing the proba-bility of their coalescence and the rapid removalof large gas volumes from the £ow. This criterioncan be used where the magma volatile content issu⁄cient for the bubbles to attain close packingat least at atmospheric pressure, and this condi-tion holds for typical gas contents (c0 s 0.0006).

Let us consider the variables that control theDi criterion more generally. Velocity U can betreated as a dynamic parameter. The volatile con-tent, c, characterizes the main components of thesystem that control the £ow dispersion. The othervariables, R, n, and a, describe system properties,mainly of magma. They enter the Di criterion asco-factors and because they vary unidirectionally,as magma changes from basic to silicic, the com-bination

h ¼ Rn1=3a2 ð19Þ

can be taken as a generalized characteristic ofmagma, and Di can be written as a function ofthree variables:

Di ¼ U h

c20ð20Þ

where hV1032 s m31 for silicic magma, andV1035 s m31 for basic magma.

After substituting Dicr for Di in Eq. 20, weobtain an equation connecting three variables U,h and c0, which describes a curved plane. All thepoints below this plane represent combinations ofparameters corresponding to the discrete gas sep-

VOLGEO 2551 7-2-03


aration (DGS) regime. Some cross sections of thiscurved plane for constant values of h are shown inFig. 3. The DGS regime should prevail when ba-sic magmas erupt, but has a very small probabil-ity for silicic eruptions.

Magma ascent velocity U is a convenient vari-able to investigate evolution of eruptions. Conse-quently, the condition for the realization of theDGS regime can be de¢ned as:

U9Ucr ¼Dicrc20h

; Dicr ¼ 0:05 ð21Þ

Wilson and Head (1981) made their calcula-tions of bubble rise and coalescence for basaltmagmas, using the magma velocity as the onlygoverning parameter on the evolution of an erup-tion. Their results are in good agreement with theresults obtained using formula Eq. 21.

3.2. Dispersion and PDF (extrusive) regimes

When DisDicr the state of close packing isreached due to bubble growth in the ascendingmagma, and produces foam, which begins to dis-rupt. To create the dispersion regime this disrup-tion must be complete, and the velocity of thedischarged gas £ow must be su⁄cient for £uid-ization of the resulting particles. In a steady£ow immediately after the beginning of foam dis-

ruption these conditions as a rule are not ful¢lled.Foam disrupts only partly. A porous substance(partly destroyed foam) is formed. This is justi¢edby the interconnected character of pumice (With-am and Sparks, 1986; my own measurements inbasaltic slags of Tolbachik), and by the permeablenature of extrusive domes (Sparks et al., 2000).

With further magma ascent, the volume £owrate and the velocity of gas increase, and £uid-ization and transformation of the foam to agas^pyroclast mixture can occur. To describe theprocess of gradual foam disruption I treat thepartly destroyed foam as a mass of separate par-ticles characterized by de¢nite sizes and partlystuck together (Slezin, 1980a, 1995a,b). Gas can£ow through the channels between particles and£uidize them if it has su⁄cient velocity not onlyto suspend them, but also tear them from eachother. If the gas velocity is not su⁄cient for par-ticle £uidization even at the exit of the conduit,partly destroyed foam (PDF) extrudes.

The gas velocity needed for £uidization dependson the particle size, density, and on the stickingforces. The magnitudes of the sticking forces arenot known, so we must consider several possiblecases. The particle size distribution forms in theprocess of foam disruption, and is assumed tochange little in the gas suspension. Particlesshould be smaller when viscosity of magma islarger and eruption is more intensive because ofboth larger pressure gradients and more brittleresponse of magma due to faster deformation.These factors also lead to larger pressures in bub-bles before disruption. As a limit, pyroclasts canconsist only of ‘shards’ that are remnants of in-terbubble walls. The density of shards is bl , butthey have complicated shapes and large drag co-e⁄cients in the gas £ow. Larger particles includepores and their density is less than bl , but theirshape is more isometric and the drag coe⁄cient isnot so large (Wilson and Huang, 1979). In furthercalculations a polydispersed particle suspensionwas substituted by a monodispersed one (Slezin,1980b), consisting of particles with an appropriatedensity and with a drag coe⁄cient equal to unityin agreement with experiments (Walker et al.,1971).

The volume fraction of free gas needed for £u-

Fig. 3. Critical velocity of magma ascent plotted against vol-atile content. Curves labeled from 1 to 5 describe critical ve-locities of magma ascent against volatile content, c0, corre-sponding to h values from 1031 to 1035 s m31, respectively.For a given value of h the DGS regime plots to the right ofand below the curve of critical velocity.

VOLGEO 2551 7-2-03


idization N0 equals 0.4 for spheres (Boothroyd,1971). A similar value (0.44) was measured onash from Bezymyanny volcano (Alidibirov andKravchenko, 1988). The dispersion regime beginswhen £uidization conditions are ful¢lled at thevent exit into the air at pressure pa (atmospheric)and gas density bga. Assuming that all gas hasbeen exsolved, the required speed can be foundfrom the continuity condition:

ug ¼ Ub lc0b gaN 0

ð22Þ

The speed of the particles up can be found as adi¡erence between ug and the terminal speed ut ofthe particles of average size dp :

ut ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gb pdp

3b ga

skðN 0Þ ð23Þ

The subscript p is used with the variables thatcharacterize the particles, and bp = bl(13L), whereL is the porosity of particles. The ‘£ow tightness’(i.e. the in£uence of adjacent particles) is takeninto account by multiplying the expression bythe coe⁄cient k(N). The speed of the particles ina stable £ow must satisfy the continuity conditionfor non-volatile components:

up ¼ Ub l

b pð13N 0Þð24Þ

Denoting the gas speed su⁄cient to surpass thecohesion of the particles as u(Fp), where Fp is thecohesive force needed to tear one particle fromanother, the condition for the dispersion regimecan be written as:

ugvupðN 0Þ þ utðN 0Þ þ uðFpÞ ð25Þ

The three terms on the right hand side of Eq.25 are not equivalent. The second term dependson the size and density of the particle only, butthe ¢rst term depends on the magma upward ve-locity U and is in nearly all cases much less thanthe second term. The third term cannot be esti-mated: it could be much smaller or much largerthan the second term. So, as a ¢rst approximationwe neglect the ¢rst term and consider possiblelimiting cases for the third term.

Assuming that the ¢rst term in the right side of

Inequality 25 is equal to 0 and substituting in itEqs. 22^24, then:

UvN 0b ga

b lc0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gb pdp

b ga

skðN 0Þ þ uðFpÞ

!ð26Þ

If the cohesive force is much less than the forceneeded for £uidization of loose particles, and theparticles are not stuck together after disruption,we can put u(Fp) = 0 and rewrite Eq. 25 as:

UvN 0kðN 0Þ

c0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gb gað13L Þdp

3b l

sð27Þ

If the cohesive force is much larger than theforce needed for £uidization we can write :

UvN 0b ga

b lc0uðFpÞ ð28Þ

If condition 26 or one of the conditions 27 and28 is not ful¢lled, the PDF regime can take place.Illustrative values corresponding to the momentof £uidization at the exit in Eq. 27 are:N0 = 0.44; k(N0) = 0.1 (Alidibirov and Kravchenko,1988), bl =2500 kg m33, bga =0.2 kg m33 andL=0.5. As a result we obtain:

Uv0:0017

c0

ffiffiffiffiffidp

pð29Þ

where dp is in m and U in m s31. Substituting inEq. 29 values of c0 about 0.05 and dp =1034^1032

m, we ¢nd that, if particles do not stick to eachother, for the extrusive regime to be realized thevelocity of magma ascent should be about 0.001 ms31 or less. Magma ascent velocities correspond-ing to the extrusive regime on Mount St. Helens(Slezin, 1991, 1995a) give values of more than0.001 m s31. So the e¡ect of cohesion of particlesmay be not small. Equating the drag force and Fp,an expression for u(Fp) is obtained through Fp

and an approximate inequality analogous to Eq.29 is found:

Uv0:00012

c0

ffiffiffiffiffiffiFp

pdp

ð30Þ

The cohesive force Fp increases when the par-ticle size dp increases, so the dependence of theright hand side of Eq. 30 as well as the right

VOLGEO 2551 7-2-03


hand side of Eq. 29 on dp is not strong. So for anapproximate estimate one can write instead ofEqs. 29 or 30:

Ud ¼ Ac0

ð31Þ

where Ud is in m s31 and A can be from V1035 ms31 for the small, non-cohesive particles toV1033 m s31 for the case when the cohesive forceis greater than particle weight.

3.3. Relations between three basic regimes

The conditions for each basic regime are sum-marized in Table 1 and illustrated in Fig. 4. Forthe small non-cohesive particles, the curve Ud

would be lower than is shown in Fig. 4; when

the particle cohesive force is large, the curve couldbe much higher. The curve Ucr can have di¡erentinclinations (Fig. 5). In principle any regime canbe realized for any magma (Fig. 5), but the rela-tive probabilities of the regimes are quite di¡er-ent. For basalt eruptions the DGS regime pre-vails; the dispersion regime can be realized onlywhen the volatile content is small and the speed ofmagma ascent is large; and the PDF regime can-not practically be realized. For silicic eruptionsthe dispersion regime strongly prevails ; the PDF(extrusive dome) regime is feasible for a widerange of parameters, but the DGS regime hasvery low probability. All this corresponds wellto observations.

Table 1The conditions necessary for di¡erent regimes to occur

Condition: UsUcr U6Ucr UsUd U6Ud

Regime

DGS ^ + 0 0PDF + ^ ^ +Dispersion + ^ + ^

‘+’ means the condition must be ful¢lled; ‘^’ means the condition must not be ful¢lled; ‘0’ means the condition is not critical.

Fig. 4. Map showing eruption regimes on a plot of magmavelocity versus volatile (water) content bounded by criticalcurves. The curve for Ucr divides the DGS regime fromothers, calculated using Eq. 21 with h=5U1033 s m31 ; curveUd divides extrusive and dispersion regimes calculated usingEq. 31 with A=0.01.

Fig. 5. The dependence of eruption regimes on the parameterh. Other parameters are the same as in Fig. 4. Curves withnumbers from 1 to 5 correspond to the values of h from1031 to 1035 s m31, respectively (from rhyolites to basalts).For a given value of h the DGS regime plots to the right ofand below the curve of critical velocity. The dispersion andextrusive regimes plot above the curve (see Fig. 4).

VOLGEO 2551 7-2-03


3.4. Some estimates for well known eruptions

3.4.1. Tolbachik 1975^1976This eruption is described in a monograph (Fe-

dotov, 1984). It includes two stages that werenamed the North and South Breakthrough. Thesestages di¡er from each other both in time and inspace (10 km distance) and in regime. At theSouth Breakthrough there was only the DGS(classic Strombolian) regime. At the North Break-through the dispersion regime mostly occurredand was very stable for several days. Some aver-age characteristics of both stages are shown inTable 2. For the South Breakthrough Di=0.005was obtained, which is an order of magnitude lessthan the critical value and corresponds to the sta-ble DGS regime. For the North BreakthroughDi=0.1, which corresponds to the dispersion re-gime. In the latter case the value of Di is only alittle greater than critical, and instability of massdischarge rate related to changing regimes can beexpected. Such an instability was observed nearthe end of the North Breakthrough eruption.

3.4.2. Mount St. Helens 1980The ¢rst stage of the Mount St. Helens erup-

tion included mild phreatic explosions and defor-mation of the volcanic edi¢ce (Lipman and Mul-lineaux, 1981). Then a large landslide triggered adirected blast. A gas^pyroclast steady-jet stagefollowed, which lasted for 9 h in the stable regimeand then stopped abruptly. After a 3.5 weekpause a dome started to extrude. During thesteady-jet (Plinian) phase 0.25 km3 of juveniledense magma with a density of 2600 kg m33

was erupted, so the average volume dischargerate was about 8000 m3 s31 (Carey et al., 1990).At the beginning of the extrusive dome growththe volume discharge rate was about 8 m3 s31

(Swanson and Holcomb, 1990). Slezin (1991) esti-mated the magma ascent velocity during thesteady-jet stage and consequently the conduitcross-sectional area using characteristics of thegas^pyroclastic jet and the theoretical model de-scribed in the next section. This velocity was alittle more than 1 m s31. So the magma ascentvelocity during the extrusive stage should beabout 0.001 m s31.

The original content of the dissolved water inthe melt phase of the magma, calculated usingmineral-phase relations, was c0 = 0.046 (Ruther-ford et al., 1985). Using granulometric data andspace-size distribution of pyroclastics from Sarna-Wojcicki et al. (1981), the average size of particles(total volume divided by the total number of par-ticles) was estimated as dp =0.001 m. Substitutingappropriate values in Eq. 21, we ¢nd that Ucr ismuch less than 1 m s31 with any feasible value ofh for Mount St. Helens magma. The theory pre-dicts the dispersion regime. Calculation of Ud ,dividing the dispersion and PDF regimes (Eq.29), yields 0.0007 m s31, which is similar to ob-servations on the dome growth (Moore et al.,1981).

3.4.3. Bezymyanny 1956The eruption of Bezymyanny 1956 followed the

same scenario as Mount St. Helens 1980 (Belou-sov and Bogoyavlenskaya, 1988). Quantitativedata are not so detailed as for Mount St. Helens.The dissolved water content is between 0.05 and0.06 (Kadik et al., 1986) and the average particlesize is approximately the same as on Mount St.Helens. Hence the value of Ud must also be thesame. The average magma discharge rate duringthe ¢rst 3 months of dome extrusion was 3 m3

s31. The observations on Bezymyanny also con-¢rm theoretical results.

Table 2Average values of the parameters typical for the Tolbachik eruption in 1975^1976

U (m s31ÞR (Pa s) n (m33) c0 a (Pa31=2) dp (m) bp (kg m33)

Northern Breakthrough 0.15 105 3U1010 0.09 3U1036 0.01 1500Southern Breakthrough 0.02 103 3U1010 0.01 3U1036 ^ ^

VOLGEO 2551 7-2-03


4. Theory of the dispersion regime

4.1. Formulation of the problem and solutionmethod

4.1.1. Physical formulationThe problem is as follows: magma £ow moves

through a vertical conduit under a certain pressuredi¡erence between its ends. The eruption type,mass discharge rate, £ow structure, and velocityof the £ow at the exit of the conduit must be foundfor any combination of system parameters. In thedispersion regime there are both main forms oftwo-phase £ow in a volcanic conduit : liquid andgaseous (gas^pyroclast mixture). The followingsimplifying assumptions were made: (1) £ow issteady-state, isothermal and one-dimensional ; (2)mass and momentum exchange between phases isan equilibrium process; (3) the gas^pyroclast mix-ture is monodisperse and non-colliding; (4) thedensity of the condensed phase is independent ofthe dissolved volatile content; (5) the condensedphase is incompressible and the gas is ideal ; (6)nucleation of bubbles starts at gas saturation con-ditions and stops after a very small interval oftime, which is neglected, and then the number ofbubbles per unit mass remains constant.

4.1.2. The system of equationsThe mathematical description of conduit £ow

includes: (1) equations describing conservationlaws: for the mass of each component, momen-tum and energy; (2) equations of state for eachphase and for the mixture; (3) equations describ-ing interaction between phases: mass, energy andmomentum exchange. Boundary conditions arerequired at the ends of the conduit and the con-ditions on the boundaries between zones with dif-ferent £ow structure.

Assumptions of isothermal £ow and imperme-ability of the conduit walls for heat and matterallow equations of energy conservation and onesdescribing energy exchange between phases to beneglected. In its general form the systems of equa-tions, with simplifying assumptions, can be writ-ten as:

ðbuÞnv ¼ Const ð32aÞ

ðbuÞv ¼ Const ð32bÞ

dp ¼ dpst þ dpu þ dpd ð33Þ

b g ¼ b gðpÞ ð34aÞ

b l ¼ b lðpÞ ð34bÞ

b ¼ b ðp; b g; b lÞ ð34cÞ

c ¼ ap1=2 at p6c20a2

ð35aÞ

c ¼ c0 at pvc20a2

ð35bÞ

ug3ul ¼ FðPiÞ ð36Þ

In the equations, p is pressure, u is velocity, andb is density; when used without subscripts thesequantities relate to the two-phase mixture as awhole; c is dissolved volatile component contentin the condensed phase; c0 is total volatile con-tent; a is the gas solubility constant; Pi are pa-rameters of the £ow on which momentum ex-change between phases depends. Subscripts st, u,and d are used to describe static, dynamic anddissipative pressure losses, respectively; nv de-scribes the non-volatile component; v describesthe volatile component; g refers to the gas phase;and l to the condensed (liquid) phase.

Eqs. 32 are continuity equations; Eq. 33 is theequation of momentum written as a balance ofpressure increments and describes the change ofpressure along the conduit ; Eqs. 34 are the equa-tions of state for each phase and for the mixture;Eqs. 35 describe mass exchange between phases(solubility of the volatile component in themelt); Eq. 36 characterizes momentum exchangebetween phases. Speci¢c forms of the equationsdepend on the £ow structure and are not thesame in the di¡erent parts of the conduit. Equa-tions use real velocities and densities of the phasesrather than normalized values. In the zones whereall the phases have the same velocity, we write onecontinuity equation:

bu ¼ b lU ð37Þ

If the velocities of the phases are di¡erent in the

VOLGEO 2551 7-2-03


zones of partly destroyed foam and dispersion, wewrite two continuity equations, one for each com-ponent:

ulð13cÞð13L Þð13N Þ ¼ Uð13c0Þ ð38aÞ

b gugN þ b gulL ð13N Þ þ b lulcð13L Þð13N Þ ¼

b lUc0 ð38bÞ

Here L is the gas volume fraction enclosed with-in the particles and N is the volume fraction of freegas in the £ow. In Eq. 38b, written for the volatilecomponent, the ¢rst term on the left hand sidedescribes the mass £ux of free gas, the secondterm describes the mass £ux of gas in the poresof the particles, and the third term describes themass £ux of the volatile component dissolved inthe condensed phase. The second term of Eq. 38bis much smaller than the others and in most casescan be neglected.

If the vertical coordinate h, is directed down,and the origin is placed at the exit of the conduit,one can write the terms of Eq. 33 in the followingway:

dpst ¼ bgdh ð39Þ

dpu ¼ 3b uudu ð40Þ

The inertial term, Eq. 40, is relevant to the gas^pyroclast zone only, because in other zones it isnegligibly small. The third (dissipative) term inthe right hand side of Eq. 33 in the region ofthe liquid £ow (i.e. ¢rst three zones from the mag-ma chamber) has the form:

dpd ¼ Gul

c

dh ð41aÞ

c ¼ b2

R

ð41bÞ

where c is called the conduit conductivity, b is thecharacteristic cross-dimension of the conduit (fora ¢ssure it is the width, for a cylinder it is thediameter), R is the constant magma viscosity, Gis the coe⁄cient relating to the form of the con-duit cross section (for a ¢ssure G=12 and forcylinder G=32).

In the zone of the gas^pyroclast mixture:

dpd ¼ kfu2

4Rbdh ð42Þ

where kf is the friction coe⁄cient and R isthe conduit radius in cylindrical form. For typi-cal velocities of gas^pyroclast mixtures and con-duit radii, kf depends only on the wall rough-ness (Chugaev, 1975). Slezin (1982a, 1983) esti-mated the friction coe⁄cient as 0.02 or a littleless.

If temperature is about 1000‡C and pressure isno more than a few hundreds of MPa, the melt isan incompressible liquid and the gas is ideal. Sothe equations of state of the phases are:

b l ¼ Const ð43aÞ

pb g

¼ Const ¼ pa

b gað43bÞ

where pa and bga are, respectively, the atmospher-ic pressure and the gas density at the atmosphericpressure and the temperature of magma.

For the bubbling £ow:

1b

¼ c03cb g

þ 13ðc03cÞb l

ð44Þ

For the gas^pyroclast mixture and partly de-stroyed foam:

b ¼ b gN þ b pð13N Þ ð45aÞ

b p ¼ b lð13L Þ þ b gLWb lð13L Þ ð45bÞ

The di¡erence between gas and pyroclast veloc-ities can be found from the equation:

ðug3ulÞ2 ¼gdpb lð13L Þ

Cf ðN Þb gð46Þ

where dp is the diameter of the particle and Cf (N)is its drag coe⁄cient. In the dilute gas^particlemixture the drag coe⁄cient is assumed to be 1,in accordance with experimental data of Walkeret al. (1971). In the zone of PDF the drag coef-¢cient increases due to high concentration and isestimated using empirical formulae (Nigmatullin,1987).

4.1.3. Boundary conditionsAt the conduit entrance the pressure p0 was

VOLGEO 2551 7-2-03


¢xed as the pressure at the top of the magmachamber:

p=h¼H ¼ p0 ð47aÞ

p=h¼0 ¼ pexit ð47bÞ

The pressure pexit is the atmospheric pressure pa

if the £ow velocity at the exit is no more than thelocal sound velocity. If the velocity of the £owrises to this critical value, then a jump-like changeof pressure and density arises at the conduit exit.This constitutes the choked £ow condition. In thiscase pexit must be ¢xed on the inner side of thepressure jump:

pexit ¼ pa when R6Rcr

pexit ¼ pa þ vps when RsRcrð48Þ

In the latter case the boundary condition de-pends on the £ow characteristics. The pressurejump vps must be set so that the sound velocityon its inner side is equal to the £ow velocity. Thiswas done by iteration. Sound velocity in a gas^particle mixture depends on the mass fraction ofparticles, average particle size distribution, andthe sound wave frequency (steepness of the front).The sound velocity was estimated after Deich andPhilippov (1968):

us ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinp

xb 1þ 13xx

vup

vug

� �vuut ð49Þ

where n is the exponent in the polytropic (isen-tropic) law, x is the mass fraction of free gas, vug

and vup are the average increments of gas andparticle velocities, respectively, after a soundwave front has passed.

The relation vup/vug in Eq. 49 de¢nes the de-pendence of the critical velocity on the sizesof particles and wave frequency. The followinglimit cases are considered: (1) the wave front isvery steep and the particles are large; (2) the wavefront is gently sloping and the particles are small.In the ¢rst case, a passing wave cannot change theparticle velocity (cannot pass momentum and en-ergy to large particles), vupV0, and Eq. 49 isreduced to:

umaxs ¼

ffiffiffiffiffiffiffinpxb

rW

ffiffiffiffiffiffinpb g

rð50Þ

In the second case, particles exactly follow thegas motion and they exchange momentum andenergy, vup/vugV1 and Eq. 49 is reduced to:

umins ¼

ffiffiffiffiffinpb

rW

ffiffiffiffiffiffiffiffinpxb g

rW

ffiffiffiffiffiffixpb g

rð51Þ

In the ¢rst case, the critical velocity equals thatin a pure gas phase, in the second case it equalsthe critical velocity in the gas having the sameelastic properties, but approximately 1/x timeslarger density.

In Eq. 51 we have put nV1, because in a mix-ture containing a small mass fraction of gas rapidheat exchange between particles and gas allows anisothermal approximation of expansion and com-pression when a sound wave passes. If in Eqs. 50and 51 values approximately correspond to thatof the gas^pyroclast jet of the First Cone of theTolbachik 1975 eruption (gas is H2O, TV1100‡C,x=0.05), we obtain: us

max = 900 m s31, usmin =

175 m s31.The pyroclasts are polydisperse, and some par-

ticles can be de¢ned as ‘small’ and some as ‘large’.If the mass fraction of small particles in the py-roclasts is designated as fs, the approximate for-mula for the velocity of sound in a gas^pyroclastmixture can be written as:

us ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip

b g 1þ f s

x

� �vuut W

ffiffiffiffiffiffiffiffiffiffixp

f sb g

rð52Þ

The zone boundaries with di¡erent £ow struc-ture are calculated from the volume relationshipsbetween phases. On the lower boundary of thebubbling £ow zone (zone 2 in Fig. 2b) pressureis de¢ned by Eq. 12 at Vg =0, or by Eq. 2, sub-stituting c= c0. The upper boundary of the bub-bling zone (the beginning of foam disruption) isde¢ned by the gas volume fraction (Vg/VlV3).The boundary between zones of partly destroyedfoam and gas^pyroclast dispersion volume frac-tion of free gas phase, N0, is de¢ned with a value0.44.

4.1.4. Numerical solutionThe solution includes numerical integration of

the di¡erential equation of momentum, Eq. 33along the conduit. Pressure was used as the vari-

VOLGEO 2551 7-2-03


able of integration, because all the characteristicsof the £ow depend on it and it is de¢ned at bothends of the conduit. Geometrical and substantialparameters of the system and mass discharge rateof magma were ¢xed at the start of integration.Integration started at one end of the conduit withthe corresponding boundary condition and pro-ceeded to the other end. By varying magmamass discharge rate one can obtain the secondboundary condition. Taking into account theboundary condition at the conduit exit, it is con-venient to start integration from the upper enddown, opposite to the direction of magma £ow.

4.2. Quantitative characteristics of magma £ow ina volcanic conduit

4.2.1. Pressure changes along the conduitCalculated pressure changes along the conduit

for two di¡erent eruptions are shown in Fig. 6. Asilicic magma and shallow magma chamber (curve1) approximately correspond to Mount St. Helens1980. Basalt magma and a deep magma chamberapproximately correspond to the First Cone atNorthern Breakthrough at Tolbachik. In both

cases pex (see Eq. 2) was assumed equal to +20MPa at the start of the gas^pyroclastic eruption.These two examples di¡er in the relative and ab-solute length of the dispersion zone. For MountSt. Helens it is about 5 km and occupies muchmore than half of the conduit ; in the Tolbachikcase it is 2.6 km and occupies only W10% of theconduit length. In both cases the pressure in theconduit is less than the hydrostatic pressure of amagma column without bubbles (which is near tolithostatic). The pressure di¡erence between mag-ma in the conduit and the pressure in the sur-rounding rock is largest at the fragmentation level(points A and B in Fig. 6). It is about 105 MPafor Mount St. Helens and 60 MPa for Tolbachik.For such large pressure di¡erences destruction ofthe conduit walls by a process similar to miningburst (Kravchenko, 1955) is probable. This de-struction was indicated by lava bombs with xeno-lith cores at the Northern Breakthrough (Slobod-skoy, 1977; Kovalev and Kutyev, 1977).

Fig. 6. Pressure variations with height along conduits of vol-canoes with di¡erent types of magma. (1) Model of theMount St. Helens 1980 eruption (H=7.2 km, UV1 m s31,a=0.0060 MPa31=2, dp =0.001 m, c=0.046); (2) model ofthe First Cone at the Northern Breakthrough of the Tolba-chik 1975 eruption (H=25 km, UV0.1 m s31, a=0.0032MPa31=2, dp =0.01 m, c=0.05). Points A and B indicatefragmentation levels for these eruptions.

Fig. 7. Map of the relative lengths of the zones with di¡erent£ow structures, hi/H versus magma ascent velocity for thedispersion regime. Curves are the boundaries dividing thezones with di¡erent £ow structures; hi is the length of zonei. Zone 1 (i=1) is the homogeneous melt; zone 2 is the bub-bling £ow; zone 3 is the partly destroyed foam (PDF); zone4 is the dispersion of a two-phase gas^pyroclastic mixture.Parameters used in the calculations: c0 = 0.046; a=0.0060MPa31=2 ; c=1034 m2/Pa s; dp =0.001 m; pex =0.

VOLGEO 2551 7-2-03


4.2.2. Length of zones with di¡erent £ow structuresThe relative lengths of zones with di¡erent £ow

structure calculated for pex =0 and other param-eters corresponding to the eruption of Mount St.Helens (except H and U) are shown in Fig. 7. Atvalues of U of about 0.001 m s31, nearly thewhole conduit is ¢lled with the homogeneousmelt ; at UW0.02 m s31, the lengths of the bubbleand PDF zones reach their maximum values ofabout 0.2 and 0.2, respectively, of the conduitlength and the zone of dispersion £ow appears;at Us 0.1 m s31, the bubble and PDF zones be-come narrow and the dispersion zone enlargesdramatically.

4.2.3. Some estimates for the Tolbachik eruptionField data from the Tolbachik eruption (Fedo-

tov, 1984) allow calculation of magma viscosity inthe conduit using the model and comparison withthe measured viscosity of lavas. The width andthe horizontal length of the ¢ssure, the depth ofthe magma chamber, the mass discharge rate andthe volatile content of the magma were estimatedby direct measurements. A feasible range of mag-ma ascent velocities (0.10^0.16 m s31) and watercontents (0.05^0.09, assuming a phreatic originfor the latter value (Menyaylov et al., 1980))were used. Values of c= b2/R and viscosity R

were then calculated (Table 3). The viscosity esti-mates range from one to one and a half orders ofmagnitude less than 3U105 Pa s, measured on theinitial lava £ow at the base of the First Cone(Fedotov, 1984; Vande-Kirkov, 1978)). Di¡eren-ces between calculated and measured viscositiesare reasonable, as the basalt is expected to in-crease in viscosity due to degassing during ascent.

5. Controls of the magma mass discharge rate ondynamic stability and the nature of catastrophicexplosive eruptions

5.1. Basic governing parameters

There are numerous parameters that in£uence avolcanic system, but not all of them are indepen-dent and their in£uences are not of equivalentvalue. Appropriate sets of the basic governing pa-rameters need to be chosen. The in£uence of otherimportant parameters can then be studied (seeSection 5.4). In general, the magma mass dis-charge rate depends on the driving pressure, thatis, on the di¡erence between the values of thepressure at both ends of the conduit minus thehydrostatic pressure of the magma in the conduit,and on the total hydraulic resistance of the con-duit. These characteristics depend on the conduitlength, H, and the £ow resistance in the region ofliquid £ow. Conduit resistance depends on itscharacteristic cross-dimension b and magma vis-cosity R, parameters which appear in the pertinentequations in a ¢xed combination (e.g. conduitconductivity de¢ned as c= b2/R in Eq. 4). Thepressure di¡erence between the conduit ends is

Table 3Values of the conduit conductivity (c) and magma viscosity(R) calculated for two possible magma ascent velocities (U)and di¡erent possible water contents

U (m s31) c0 c (m2 Pa31 s31) R (Pa s) (b=1.2 m)

0.10 0.05 0.00018 8U103

0.07 0.00010 1.4U104

0.09 0.00006 2.4U104

0.16 0.05 0.00024 6U103

0.07 0.00014 1U104

0.09 0.000075 1.9U104

Fig. 8. Magma ascent velocity, U, plotted as a function ofthe conduit conductivity for di¡erent conduit lengths in kilo-meters. Figures near the curves represent values of theconduit lengths; other parameter values are: c0 = 0.05;a=0.0041 MPa31=2 ; bl =2700 kg/m3 ; bga =0.16 kg/m3 ;dp =0.01 m; L=0.3; pex =40 MPa, and correspond to typicalvalues for the Tolbachik eruption, 1975.

VOLGEO 2551 7-2-03


mostly dependent on magma chamber pressure.However, it is more convenient to use excess pres-sure pex, (Eq. 2) as a governing parameter. Bothpex and the driving pressure di¡erence have pos-itive values just before the beginning of an erup-tion and later pex monotonically decreases to neg-ative values, but driving pressure can change upand down, in some cases dramatically. It is usefulto consider the in£uence of two parameters withthe third ¢xed. Let us ¢x pex ¢rst and consider thedependence of magma mass discharge rate onconduit length and conductivity.

5.2. Magma mass discharge rate as a function ofthe length and conductivity of the conduit

Sets of curves expressing the magma ascent ve-

locity, U, as a function of conduit conductivitywere calculated at di¡erent values of chamberdepth (Fig. 8). Here magma ascent velocity is pro-portional to magma discharge rate. Two types ofcurves are found: (1) every value of c correspondsto a single value of U (curves for Hs 13 km inFig. 8); (2) every value of c corresponds to threevalues of U (curves for H6 13 km in Fig. 8).These two types of curves are separated by somecritical conduit length Hcr, which equals 13 km inthe models in Fig. 8. If magma ascent velocity asa function of two variables (H and c) is put on athree-dimensional graph, a curved plane (equilib-rium surface) is obtained with a singularity ofcusp type (Fig. 9), which describes the standard‘catastrophe’ of two-parameter families of func-tions (Poston and Stewart, 1978; Arnold, 1979;

Fig. 9. Magma ascent velocity as a function of governing parameters H and c. The curved plane is an equilibrium surface of thestandard catastrophe of two-parameter function with cusp-like singularity (two folds converging onto a cusp). It is drawn as anillustration without calculated ¢gures on the axis. Point ‘a’ in the c^H plane has three images in the equilibrium surface (‘b’, ‘c’and ‘d’) showing three feasible states of an eruption. The line with arrows on the equilibrium surface is the trajectory of the im-age point showing changes due to the conductivity increase at the beginning of an eruption with a sharp increase of the eruptionintensity. Two lines connecting points ‘o’ and ‘a’ on the c^H plane indicate two possible smooth evolution paths leading to dif-ferent points (‘b’ and ‘d’) on the equilibrium surface.

VOLGEO 2551 7-2-03


Gilmore, 1981). Every point on the equilibriumsurface indicates a steady state and changing ofthe steady states can be described by moving suchan indicative (or image) point over this surface.‘Catastrophe’ here indicates a sharp, dramaticchange of the state of a system caused by thesmall and smooth change of a governing param-eter. In the case of a volcano this mathematicalterm coincides with the practical (common) mean-ing of catastrophe.

Projection of the singularity on the plane of thetwo governing parameters or the ‘map of the cat-astrophe’ is shown separately (Fig. 10). It has theform of an angle made by two lines connected tothe cusp, which are projections of the folds of theequilibrium surface and represent the ‘separatrixof the catastrophe’. Every point on the plane in-side this angle has three images on the equilibriumsurface, and outside only one. If the image pointon the plane passes across the separatrix, thefunction (magma ascent velocity) can undergoan abrupt change or ‘catastrophe’. Such an ‘evo-lutionary track’ is shown in Fig. 9. As a ‘standardcatastrophe’ of two-parameter families of func-tions a cusp singularity has structural stability in

that is it is not changed as a qualitative feature ifall the parameters are changed by some small but¢nite amount. This indicates some standard stablebehavior of an erupting volcano.

In Fig. 11 a vertical section of the equilibriumsurface is shown. It has three branches (limbs),and three points of intersection of any verticalline with these branches re£ect three di¡erent re-gimes corresponding to one value of conductivity.The upper and lower branches describe stable re-gimes, whereas the middle branch (between A andC in Fig. 11) describes an unstable regime. If thepoint describing the state of the system is on thelower branch of the graph, and the conductivityof the conduit gradually increases, at the reversalof the curve at A the conditions must jump to theupper branch (from point A to point B in Fig.11). The magma velocity, and therefore the massdischarge rate, increases by an order of magni-tude. In some cases the increase can be up to threeorders of magnitude and more. If the conditionsmove to the left on the upper branch (conductiv-ity decreases), it must eventually jump to the low-er branch (from C to D) and magma mass dis-charge rate decreases abruptly. This decreasecould be from one order of magnitude to in¢nity,when the eruption stops entirely.

The set of curves in Fig. 8 corresponds to thestart of the eruption (pex maximum). The decrease

Fig. 10. Projection of the ‘cusp’ on the plane of the govern-ing parameters (chamber depth and conduit conductivity) ormap of catastrophe. Calculated using the same parameters asfor Fig. 8. Inside the angle, between lines 1 and 2, everypoint corresponds to three values of magma velocity: twostable regimes with low and high intensity (‘lower’ and‘upper’ regimes) and one unstable intermediate intensity re-gime. Region A corresponds to the ‘lower’ regime only; re-gion B, to the ‘upper’ (much more intensive) regime.

Fig. 11. Magma ascent velocity as a function of conductivityof the conduit in the region of the ‘cusp’ (qualitative pic-ture). The points A, B, C and D are discussed in the text.

VOLGEO 2551 7-2-03


of pressure does not change the qualitative pic-ture, but moves the whole set of curves to theside of higher conductivities and stretches the re-gion of multiple states in the horizontal direction.

A governing parameter is called splitting if anychange moves the image point through the cusppoint in the direction of the cusp axis. A changeof splitting parameter along the cusp axis leads tobifurcation of the process at the cusp point. Be-hind it there are two possible paths for the point:the system can progress along either the upper orlower limb (sheet) of the folded equilibrium sur-face (Fig. 9). The value of the splitting parameterde¢nes the feasibility of two values of U (low andhigh) and of abrupt changes between them. Aparameter which drives the point indicating thestate of the system perpendicular to the cuspaxis is called normal. The moment of abruptchange of U depends on its value. In the volcaniccase none of the chosen governing parameters canbe called splitting, but the conduit length is clos-est. Conduit length changes very little duringeruptions, although conduit length can vary be-tween volcanoes. Conduit conductivity is a nor-mal parameter, which varies greatly during aneruption and determines conditions for abruptchange if H6Hcr at constant pex. The criticalconduit length Hcr corresponds to the point ofthe cusp.

If conductivity increases from a small value andthe indicative point moves from region A to reachline 1 (Fig. 10), a jump to the upper part of thefolded plane occurs (magma discharge rate in-creases abruptly). The physical mechanism ofthis jump is as follows. An increase of magmadischarge rate results in two e¡ects : (1) increaseof magma ascent velocity and a correspondingincrease in the resisting friction forces; (2) in-crease of the region of gaseous £ow with a corre-sponding decrease of the liquid £ow region andaverage magma density in the conduit, resulting ina decrease of the total £ow resistance and an in-crease of the driving pressure di¡erence. If theabsolute length of the liquid £ow region is smallenough (as it is when H6Hcr) the second e¡ectprevails with positive feedback. As a result, mag-ma mass discharge rate increases dramatically upto the state when total resistivity in the liquid £owregion becomes less than that in the gaseous £owregion (the length of the liquid £ow region shouldbe much less than 1% of the total for this) or thecritical £ow velocity at the exit is reached (thechoking condition).

These principles show how an eruption canabruptly change states between explosive and ef-fusive activity. Using three principal governingparameters the in£uence of other parameters onthe evolution of an eruption can be studied.

Fig. 12. The dependence of magma ascent velocity on the excess pressure pex. (a) Conduit conductivity c=1034 m2 Pa31 s31 is¢xed. Numbers on the curves are chamber depth in km. (b) Chamber depth H=12 km is ¢xed. Numbers on the curves are con-ductivity of the conduit in 1034 m2 Pa31 s31.

VOLGEO 2551 7-2-03


5.3. Magma mass discharge rate as a function ofchamber pressure

The dependence of the magma discharge rateon any other pair of governing parameters includ-ing pex has the same form of cusp catastrophe. pex

is a normal parameter. In Fig. 12 such dependen-cies are calculated for an eruption similar to thatof Mount St. Helens 1980 with volatile fraction0.046, average particle size 0.001 m, average par-ticle porosity 0.3, melt density 2500 kg m33, gasdensity 0.2 kg m33 at atmospheric pressure andmagma temperature, and coe⁄cient a=0.0060MPa31=2. In Fig. 12a the conductivity of the con-duit is ¢xed and the chamber depth is varied, andvice versa in Fig. 12b. These two ¢gures illustratethe similarity of the in£uence of conduit lengthand conductivity on magma ascent velocity. Whenthe region of liquid £ow is large enough, a sev-eral-fold change of H has nearly the same e¡ect asa change of c in the same proportion. However,when the relative length of the liquid £ow zone issmall, a change of H is more e¡ective than achange of conductivity (compare Fig. 12a withFig. 12b).

5.3.1. Change of the position of the cusp pointMagma discharge rate changes are described by

the indicative (image) point movement over the

equilibrium surface of the catastrophe. Instabilityand discharge rate jumps can appear when theindicative point intersects the line of a fold onthe equilibrium surface. Such an intersection canoccur either as a result of the indicative pointmoving or the cusp singularity moving. The posi-tion of the cusp singularity (Fig. 9) depends onthe coordinates of the cusp point (Hcr and ccr),which depend on the value of pex and other mag-ma system properties. Both critical values increasesigni¢cantly when pex decreases (Fig. 13). The in-crease of Hcr leads to an increase of the feasibilityof mass discharge rate jumps: at a value of pex alittle under zero a discharge rate jump becomesfeasible at any magma chamber depth. The in-crease of the critical conduit conductivity, in con-trast, decreases the feasibility of a discharge ratejump. Such opposite tendencies do not excludethe theoretical possibility of jump-like increasesof magma discharge rate due to decrease of pex

when the magma chamber is situated at ratherlarge depths. But in this case implausibly largeincreases of conductivity of the conduit at depthare needed near the end of eruption.

5.3.2. Change of the cusp boundariesWhen H6Hcr, the function ccr(H) is repre-

sented by the upper boundary line of a cusp atwhich a jump up can occur (the axis line of the

Fig. 13. Change of the position of the cusp point on the co-ordinate plane Hcr3ccr as a result of a pex change calculatedfor the parameters corresponding to the Tolbachik eruption(see Fig. 8 for values). The value of pex in MPa is shownnear the points on the graph.

Fig. 14. The dependence ccr(H) at H6Hcr calculated atdi¡erent values of pex. (1) pex =40 MPa (Hcr =10 km),(2) pex =20 MPa (Hcr =20 km), (3) pex =5 MPa (Hcr =30km), (4) pex =0 (Hcr =50 km), (5) pex =35 MPa (HcrE50km).

VOLGEO 2551 7-2-03


lower fold in Fig. 9, line 1 in Fig. 10), and is astraight line. Sets of such straight lines are shownin Fig. 14, calculated for di¡erent values of pex.The function ccr(pex), calculated for Mount StHelens (H=7.2 km), is shown in Fig. 15. Thereis a very steep increase of ccr when pex becomesless than zero. The greater the chamber depth orthe less the excess pressure, the larger the valueof conduit conductivity required for the jump tothe upper (catastrophic) regime. These conditionsare contradictory: naturally, conduit conductivitytends to decrease if chamber depth increases orexcess pressure decreases. This places limits onthe chamber depth at which catastrophic jumpsof eruption intensity can occur.

The lower boundary of the cusp (Fig. 10) is theprojection of the axis of the upper fold (Fig. 9).When the indicative point on the projection inter-sects this line, its origin on the equilibrium surfacefalls from the upper to the lower limb of thefolded surface and magma ascent velocity de-creases abruptly. The critical conductivity at thisjump is designated as cs. The position of theboundary line changes slightly when chamberpressure decreases (Fig. 16). At small chamberdepths all the lines converge and intersect atcs=0 with H approximately equal to 7 km.The physical meaning of this intersection pointis that at these (and lower) chamber depths thejump to the lower (extrusive) regime cannot occurat any c up to 0. The upper regime should ceaseonly after total evacuation of the chamber or de-struction of the chamber roof or conduit walls.Such an eruption would not have an extrusion(PDF) stage (if the model assumptions are notchanged).

The dependence cs(pex) is a straight line on asemi-logarithmic scale at a constant chamberdepth (Fig. 17). When the conduit conductivityis less than 1.5U1035 m2 Pa31 s31 a jump-likedecrease of the magma discharge rate shouldtake place at pex =+60 MPa, that is, the upperhigh-intensity regime is improbable. When theconduit conductivity is more than 2.5U1035 m2

Pa31 s31 the transition to the lower regime cantake place only if pex falls to 340 MPa or less.

Fig. 15. The dependence ccr(pex) calculated for Mount St.Helens (H=7.2 km, a=0.0060 MPa31=2, c=0.046).

Fig. 16. The dependence cs(H) calculated for a volcanowith parameters of Mount St. Helens (see Fig. 15) and di¡er-ent values of pex, which are written near the curves in MPa.

Fig. 17. Log(cs) as a function of excess pressure pex calcu-lated for the Mount St. Helens volcano (H=7.2 km).

VOLGEO 2551 7-2-03


When the negative pressure becomes so large, thedestruction of conduit walls and chamber roofbecomes very likely, causing the regime change.Suitable values of c lie in a narrow interval.This fact can be used for indirect estimates ofconduit dimensions and magma properties.

For shallow magma chambers and volatile-richmagmas the fragmentation level can migratedown into the magma chamber. As the horizontalcross-sectional area of the chamber is several or-ders of magnitude larger than that of the conduit,the velocity of gas up£ow in the magma chamberis not su⁄cient for £uidization of the pyroclastparticles and so ejection of the condensed materialmust stop. Consequent subsidence of the pyro-clasts and foam in the chamber can cause aneruption to stop and the chamber roof to collapsebecause of the lack of support. This mechanism ofstopping an eruption should occur in most cal-dera-forming eruptions. Further considerationsof magma discharge rate variations in large mag-nitude explosive eruptions can be found in Slezinand Melnik (1994) and Melnik (2000).

5.3.3. Amplitude of the magma mass discharge ratejump on the cusp boundaries

The amplitude of the discharge rate jump isde¢ned as the absolute value of the ratio of themagma mass £ux magnitudes before and after the

jump and is denoted as u/uu and u/us. This ratioincreases when chamber depth decreases with re-spect to critical depth and when pex decreases.Near Hcr, when H diminishes, the jump amplitudeimmediately reaches a factor of a few and thenrises slowly at ¢rst but accelerates thereafter. Thevalue of u/uu increases close to exponentially andfor shallow chambers can reach more than threeorders of magnitude. The value of u/us increases

Fig. 18. Relative amplitude of the magma mass rate jump as a function of magma chamber depth calculated for a volcano simi-lar to Mount St. Helens and the di¡erent excess pressures, which are indicated at the curves in MPa. (a) u/uu (jump upward).(b) u/us (jump downward).

Fig. 19. Amplitude of the magma mass rate jump (U) as afunction of pex calculated for a volcano similar to MountSt. Helens (H=7.2 km). Here U is the change in velocity atthe jump. Line A, U= u/uu (jump upward); line B, U= u/us(jump downward).

VOLGEO 2551 7-2-03


much faster and becomes in¢nite at the value ofH at which the eruption stops entirely. Results ofsome calculations made for a volcano similar toMount St. Helens are illustrated in Figs. 18 and19.

5.4. The dependence of magma mass discharge rateon the magma properties

The dependence of cusp point coordinates Hcr

and ccr on magma properties is now investigated(Slezin, 1994). Calculations of Hcr and ccr using arange of typical values of the volatile content c0,solubility coe⁄cient a, size of pyroclast particlesdp and their porosity L are shown in Figs. 20 and21. For a ¢xed chamber pressure, Hcr dependspredominantly on the volatile content c0. The ef-fect of the other parameters is relatively small andirregular (Fig. 20). All the points plot in a narrowzone close to a straight line, described by theequation:

HcrðkmÞ ¼ H�ðc03c�Þ ð53Þ

where H*= 356 km and c*= 0.01. The character-istic value c0 = 0.01, at which Hcr became 0, ap-proximately corresponds to the no-dispersionzone. An increase of c0 leads to an increase of

Hcr and Hcr/H and so increases the probabilityand amplitude of the sharp, jump-like increaseof magma discharge rate. The limiting chamberdepth at which instability and mass dischargerate jumps are possible is obtained by substitutingH=Hcr into Eq. 53. For c0 = 0.06 as a maximumvalue, the limiting value of HV18 km is calcu-lated.

The value of ccr strongly depends on the sizeand porosity of particles and less strongly on thevolatile content (Fig. 21). ccr decreases and theprobability of magma discharge rate jump in-creases when volatile content and porosity of theparticles increase and the particle size decreases.All the listed changes lead to an increase of par-ticle velocity in the £ow and consequently to adecrease of the £ow density and an increase ofthe driving pressure di¡erence.

5.5. Nature of catastrophic explosive eruptions

5.5.1. Types of catastrophic explosive eruptionsand stages of their development

In catastrophic explosive eruptions (CEE) largemasses of pyroclastic material are ejected in ashort interval of time. Historic examples includeTambora (1815), Krakatau (1883), Bezymyanny(1956) and Mount St. Helens (1980) and prehis-toric examples include Santorini, Greece (3400BP) and Toba, Sumatra (V74 000 BP). In allthese events the volume of erupted materialgreatly exceeded 1 km3 and in the Toba eruptionwas more than 2000 km3 (Gushchenko, 1979;Simkin and Siebert, 1994). The main and some-times the only substage of a large CEE is asteady-jet stage. There can be £ow unsteadinessdue to external factors, such as collapse of con-duit walls or chamber roof, and hydrodynamicalpulsations. In this de¢nition the steady-jet stageproduces both pyroclastic fallout and pyroclastic£ows, as they are generated by the same conduit£ow structure. Pyroclastic £ows can result fromcollapse of a vertical gas^ash column (Wilson,1976; Sparks et al., 1978) fed by a steady conduit£ow.

In some cases the steady-jet stage is precededby a ‘directed blast’ (another substage of the cata-strophic stage), which is usually caused by defor-

Fig. 20. Value of Hcr as a function of volatile (water) con-tent in magma c0 at pex =20 MPa. For other parameters thefollowing values were used: (1) dp =0.001 m, L=0.75,a=0.0041 MPa31=2 ; (2) dp =0.01 m, L=0.75, a=0.0032MPa31=2 ; (3) dp =0.0001 m, L=0, a=0.0041 MPa31=2 ; (4)dp =0.001 m, L=0.3, a=0.0064 MPa31=2 ; (5) dp =0.01 m,L=0.75, a=0.0041 MPa31=2.

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mation of a volcanic edi¢ce and landslide. A di-rected blast partly destroys the volcanic edi¢ceand ejects much old material. The catastrophicstage is sometimes preceded by a period of mod-erate activity, which could last from hours anddays to several years and can provide signi¢cantamounts of juvenile pyroclastic products (Kraka-tau 1883 ^ 3 months, Bezymyanny 1956 ^ 5months, Tambora 1815 ^ 3 years; (Gushchenko,1979)). After the catastrophic stage in most casesan extrusive stage develops. After the end of orduring the catastrophic stage of the largest of theCEE, caldera subsidence takes place. The startand end of the steady-jet stage occur abruptlywith a sharp increase or decrease of the magma

discharge rate by more than an order of magni-tude (typically 2^3 orders and sometimes more).

5.5.2. Mechanism of transition of eruption intocatastrophic stage

The theory described here can explain the prin-cipal features of CEE and their relationship to thecharacteristics of the magmatic system. The tran-sition of an eruption into a catastrophic stage isdescribed as an abrupt jump-like increase of themagma mass discharge rate, which takes placewhen conduit conductivity reaches its critical val-ue. It can take place when the conduit length H isless than Hcr. This mechanism inevitably impliesmoderate activity preceding the catastrophic

Fig. 21. Critical conductivity of volcanic conduit ccr as a function of di¡erent parameters. (a) The volatile (water) content. (b)The size of pyroclastic particles. (c) The particle porosity. In a, pex =20 MPa, d=0.01 m, porosity= 0.75. In b, pex =20 MPa, vol-atile content= 0.05, porosity= 0.75. In c, pex =20 MPa, volatile content = 0.05, d=0.01 m.

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stage. When an eruption starts, the upper part ofthe conduit is newly created. Magma intrudes intothe new ¢ssure in relatively cold rock and resis-tance to the magma £ow is initially large. Theimage point of the system is expected to be onthe lower branch (limb) of the curve in Fig. 11.As an eruption proceeds, the active £ow of degas-sing magma erodes ¢ssure walls and the magmaheats the walls. These processes cause an increasein the cross-dimension of the conduit and a de-crease of heat loss through the walls and so adecrease in magma viscosity. These processes in-crease conduit conductivity and so lead to anabrupt transition to an intensive regime relatedto the upper limb of the curve in Fig. 11. Theamplitude of the mass discharge rate jump de-pends on the relationship between critical and ac-tual length of the conduit Hcr/H. The greater thisratio, the greater is the amplitude of the jump(Fig. 22). The intense gas^pyroclast £ow can en-large the cross-sectional area of the upper part ofthe conduit, making a nozzle that allows super-sonic £ow. In the latter case the amplitude of themass discharge rate jump can be up to several-fold larger.

Magma chamber depths are reliably known foronly a few volcanoes where CEEs have takenplace. However, all available data suggest thatdepths are shallow. Geophysical studies have lo-cated magma chambers under young calderas atdepths of 5^6 km (Yokoyama, 1969; Zubin et al.,1971; Gorshkov, 1973; Zlobin and Fedorchenko,1982; Sanders, 1984; Ferrucci et al., 1992; Smithand Braile, 1994; Miller and Smith, 1999). Calcu-lated critical values of magma chamber depths forthe corresponding volcanoes are about 15 km(three times larger than the estimated depths).These high values of Hcr/H correspond to massdischarge rate jump amplitudes of more thantwo orders of magnitude and the onset of veryintensive steady-jet regimes.

A directed blast is a natural ‘trigger’ for thesteady-jet stage, because destroying the upperpart of the volcanic edi¢ce shortens and widensthe conduit. A sector collapse also can act as thetrigger of a steady-jet stage, but it is more e¡ec-tive as a trigger of a directed blast.

5.5.3. Final stage of the catastrophic explosiveeruption

The catastrophic steady-jet stage stops abruptly(Gorshkov and Bogoyavlenskaya, 1965; Williamsand Self, 1983; Fedotov, 1984). In some cases, anextrusive stage follows with a magma mass dis-charge rate about three orders of magnitude lessthan that during the catastrophic stage. In othercases the eruption stops entirely. The theory re-lated here predicts this behavior and suggests thatan abrupt stop to a CEE corresponds to the shal-lowest magma chambers.

5.6. General description of the evolution oferuption in a steady-state approximation

The steady-state approximation describes erup-tion evolution as a succession of steady states forchanges that are very slow. Also, the transientperiod of a new steady state is considerably short-er than the time of the system’s existence in a newstate. Though changing of eruption regime frommoderate to catastrophic can be nearly instanta-neous, the formation of the new stable £ow struc-ture in the conduit can take considerable time.

Fig. 22. Amplitude of the jump-like increase of magma mass£ux at the beginning of the catastrophic stage of eruption asa function of the relation Hcr/H. R1 and R2 are the mass£uxes before and after the jump. The value of the criticalconduit length Hcr was taken as 15 km. Other parameters ex-cept H are corresponding to the eruption of Mount St. Hel-ens 1980. The possible development of a supersonic nozzle isnot taken into account.

VOLGEO 2551 7-2-03


Slezin (1997) estimated a time of approximately3 h for the eruption of Mount St. Helens, threetimes less than the duration of the steady-jetstage. For similar eruptions, transient times of acatastrophic stage must be of the same order. Thetransient time from a catastrophic to an extrusivestage can be considered as the time interval be-tween the end of the steady-jet stage and the ap-pearance of the extrusive dome.

5.6.1. Evolution of the dispersion regimeThe evolution of an eruption involves three

governing parameters: conduit conductivity, con-duit length and the excess chamber pressure. Themagma discharge rate as a function of any two ofthese parameters can be expressed by the equilib-rium surface with a standard cusp catastrophe.The function of all three parameters is a combi-nation of the several cusp catastrophes and isa three-dimensional plane in a four-dimensionalspace. The corresponding catastrophe separatrixis two-dimensional surface in a space of three di-mensions. Projections of the cusp singularities onthe pex^c plane calculated for the di¡erent H are

shown in Fig. 23. On the three-dimensional graphwith chamber depth as a third coordinate a three-dimensional image of the separatrix of the catas-trophe resembles the edge of a cutter. In practice,the dimensions of this cutter-like separatrixshould be limited by the real values of parametersin volcanic systems.

Available data suggest that maximum magmachamber depths should be about 50 km. WhenH=20 km, instability and jumps of magmamass £ow rate are possible only if cs 0.00015m2 Pa31 s31, and when H=50 km, only ifcs 0.0005 m2 Pa31 s31 (Fig. 23). The likelihoodof the ¢rst value is very small and the secondvalue is practically impossible. So the maximumchamber depth for which catastrophic increases ofmagma mass discharge rate occur must be nomore than approximately 20 km. Direct observa-tions of volcanic conduits at depth are not possi-ble, but the theory discussed here can be used toestimate conduit conductivity. Using parametersfor Mount St. Helens in 1980 (see Section 3.4.2),the sharp decline in discharge rate at the end ofthe steady-jet stage could take place only if theconductivity was in the range (2^3)U1034 m2

Pa31 s31. The initial excess pressure is controlledby the strength of the overlying rock and by hy-drostatics. The minimum excess pressure at theend of eruption depends on the strength of thewalls of the conduit and the chamber roof. Themaximum pressure is limited by the long-livedstrength of material ; the minimum pressure is lim-ited by the instantaneous strength of the roof as astructure. To estimate the latter value is muchmore di⁄cult than the former. The minimum val-ue of pex is assumed to be about 3100 MPa, andin most cases it is no more than 340 to 350 MPa.

In the evolution of an eruption, a change of thegoverning parameters involves a continuous andmonotonic decrease of the chamber pressure. Ini-tially, the conduit conductivity is expected to in-crease and in the ¢nal stage to decrease, due tothe decrease of magma upward velocity and sothe increase of relative heat and volatile losses.Conduit length usually has a constant value.However, in intensive eruptions the erupted prod-ucts are widely dispersed, the supersonic jet canproduce a shallow crater, and thus the length of

Fig. 23. Maps of catastrophe on the plane pex^c calculatedfor a volcano similar to Mount St. Helens and di¡erentchamber depths. Bold lines are the lines of the folds describ-ing the jump-like decrease of magma mass rate; thin linescorrespond to the folds describing the jump-like increase ofmagma mass rate. Figures near the cusps (the point wherethe two lines intersect) are the values of the magma chamberdepth in km. The dashed line shows the location of this in-tersection (cusp) point in pex^c space.

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the conduit decreases. The prehistoric eruption ofTaupo, New Zealand in AD 180 is an example(Walker, 1980; Wilson and Walker, 1985).

Besides these standard smooth changes of gov-erning parameters there are other kinds of varia-tions. The conduit can be shortened by destruc-tion of the volcanic edi¢ce by landslide orexplosion. The conduit and chamber roof maybe destroyed by subsidence. New deeper magmacan enter the chamber. The model can predict thequalitative e¡ects of such processes through theirin£uences on the main governing parameters. Allthese additional factors add complications, but donot a¡ect underlying principles.

The following ‘standard’ scenarios for the evo-lution of an eruption are described:

(1) HsHcr. Magma mass discharge rate in-creases monotonically; then, after a smoothwide maximum, the rate decreases also monotoni-cally but rather sharply. At the end of the erup-tion an extrusion could appear.

(2) H6Hcr. Magma chamber is small and rela-tively deep. Magma mass discharge rate increasesand then decreases monotonically, being small atall stages of eruption.

(3) H6Hcr. Magma chamber is not very deepand of rather large volume. Magma mass dischargerate increases monotonically, then sharply in-creases by several orders of magnitude, changingto an explosive steady-jet stage, and ¢nally de-creases sharply to an extrusive stage.

(4) H6Hcr. Rather small and very shallow mag-ma chamber. A scenario similar to that describedin (3), but with a more rapid increase of intensityat the start and the eruption completely stoppingafter steady-jet stage (extrusive stage is absent).

(5) H6Hcr. Shallow and rather large magmachamber. Scenario is the same as in (3) and (4),

but with caldera subsidence at the end of erup-tion. At the end of subsidence extrusion is possi-ble.

5.6.2. Map of the basic eruption regimesTo describe the general evolution of a volcanic

eruption the region of the catastrophic conditionsis added to a map of the three basic regimes (seeFig. 4). The boundaries of the catastrophic regioncan be found speci¢cally only by means of numer-ical calculations, but for general analysis simpli-¢ed analytic expressions can be obtained from nu-merical calculations.

The value of U before a jump-like increase isuu. It decreases as volatile content increases. Forexample, uu equals 0.025 m s31 at c=0.06 toabout 0.1 m s31 at c=0.02. uu can be approxi-mately derived by equating the driving pressuredi¡erence to the pressure loss due to frictionover the conduit length. After some transforma-tions the expression for uu is (Slezin, 1997):

uu ¼ cu

HFðc; dp; pexÞ ð54Þ

The function F(c, dp, pex) is weakly dependenton its arguments and to a ¢rst approximation canbe considered to be a constant, the value of whichis obtained by numerical calculations.

The value of cu (ccr) is linearly dependent onH (Fig. 14). If H/Hcr replaces H on the abscissain Fig. 14 all the inclined lines on the graph havethe same length and would di¡er only by the an-gle of inclination to the horizontal axis, whichincreases as pexdecreases. All the lines intersectthe abscissa at a point a little to the left of theorigin. Thus:

cu ¼ AðpexÞH

Hcrþ BðpexÞ ð55Þ

Fig. 24. ‘Maps of regimes’ of eruptions for the di¡erent types of magma and chamber depths. Black area gives the region of theextrusive regime; grey area gives the region of the DGS regime; densely dotted area gives the region of the moderate dispersionregime; lightly dotted area gives the region of the transition to the catastrophic dispersion regime. The descending line on eachgraph describes Ud (c) given by Eq. 31, that is, the upper boundary of the extrusive regime; the ascending line describes Ucr(c,h)given by Eq. 21, that is, the upper boundary of the DGS regime; the descending dashed line with the vertical section is the lowerboundary of the region where the catastrophic jump of magma mass discharge rate takes place. Its vertical straight section corre-sponds to Inequality 59, and the hyperbolic curve section corresponds to Eq. 57. On all graphs the hyperbolic part of the lastcurve was calculated at pex =20 MPa, corresponding to the initial stage of eruption. Thin vertical lines on the four left graphsare examples of an evolution path of eruption.

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For a magma system similar to that of MountSt. Helens the value of A(pex) is approximately 20times that of B(pex) and to a ¢rst approximationB(pex) can be neglected. When pex s 0 and thejump-like increase of magma mass discharge rateis feasible, A(pex) can be described as

AðpexÞ ¼a

b þ pexð56Þ

where a=0.0033, b=35.5 MPa and pex is in MPa.Substituting Eq. 56 and Hcr from Eq. 53 into Eq.55, putting B(pex) = 0, and then substituting Eq.55 into Eq. 54, one ¢nally obtains:

uu ¼ Aðb þ pexÞðc03c�Þ ð57Þ

Numerical calculations give for the case ofMount St. Helens A=0.08. The value of uu isindependent of H.

To determine boundaries of the catastrophicconditions on the regimes map, condition 58should be taken into account:

H6Hcr ð58Þ

As H is ¢xed and Hcr is variable the combina-tion of condition 58 with Eq. 53 gives the inequal-ity:

c0sH=H� þ c� ð59Þ

The limit value of c0 :

cmin0 ¼ H=H� þ c� ð60Þ

also gives the boundary of the region for mag-ma discharge rate jumps and can be added toEq. 57.

All possible eruption states are determined inthe map coordinates by relations 21, 31, 57 and60. Variants of maps are shown in Fig. 24 for aninitial pex =20 MPa. The descending hyperbolicline on each graph describes the relation Ud(c)given by Eq. 31, indicating the upper boundaryof the extrusive regime; the ascending line de-scribes the relation Ucr(c,h) given by Eq. 21, in-dicating the upper boundary of the DGS regime;the descending line, Eq. 57, with the vertical sec-tion, described by Eq. 60, is the lower boundaryof the region where a catastrophic jump of mag-

ma discharge rate takes place. The dispersion re-gime can occur at the parameter values lying inthe region above both of these two curves. Achange to the catastrophic regime can take placeabove all three curves (and to the right of thevertical part of the last curve).

Illustrative evolutionary paths of an eruption inFig. 24 follow a vertical line at constant c0 start-ing near the abscissa at U=0 and with velocityincreasing with time. The regime changes if aboundary line is crossed. Consider, for example,H=5 km and c0 = 0.04 (Fig. 24). In the ¢rst (top)graph (h=1035 s m31, corresponding to basalts)the whole path is in the region of the DGS£ow; in the second graph (h=1034 s m31) achange to the catastrophic explosive regime(CEE) from the DGS regime is feasible, but re-quires a high velocity of magma ascent; in thethird graph (h=1033 s m31) most of the pathoccurs in the region of the catastrophic jump,which occurs directly from the DGS regime; onthe last, bottom, graph (h=1032 s m31, corre-sponding to rhyolites) the DGS regime and theextrusive regime both occur for very small magmaascent velocities, then these regimes change to themoderate steady-jet regime and then to CEE. Ifthe catastrophic regime is possible it should startfrom a DGS regime. A moderate steady-jet re-gime between these two can take place in a rathersmall interval of c0 values and small U values atlarge values of h.

Direct transition from the extrusive to the cata-strophic regime as a result of smooth evolutionrequires very large values of h and very smallmagma ascent velocities. This transition can betriggered by other processes. Destruction of anextrusive dome and volcanic edi¢ce with a subse-quent explosion can trigger a CEE, as took placein the eruption of Mount St. Helens in 1980.

Condition 60 limits the region of catastrophicjumps to small chamber depths. At a magmachamber depth of 20 km a catastrophic jump ispossible only if the volatile content exceeds 6.6wt% (Fig. 24). If water contents are typicallyless than 6.6%, then 20 km is an upper limit onthe magma chamber depth at which the cata-strophic regime can occur. In Section 5.5.1 thesame value of about 20 km for the maximum

VOLGEO 2551 7-2-03


chamber depth was obtained by other considera-tions.

All the graphs in Fig. 24 describe conditions atthe beginning of an eruption and help to describethe evolution of the eruption before a catastroph-ic jump and at the moment of the jump. Theamplitude of the jump cannot be deduced fromthese graphs. To describe evolution after a tran-sition to the steady-jet regime these graphs shouldbe supplemented to show the amplitude of magmamass discharge rate jumps before and after thesteady-jet stage. A decrease of the magma massdischarge rate will not follow the same path as itsincrease. The abrupt changes from the steady-jetto the PDF regime or to a complete cessationoccur at much higher discharge rates than thechanges from the DGS to a steady-jet regime.The ¢nal stages of an eruption can be describedby another map. In Fig. 25 the velocity of theeruption is shown at the moment just before thecessation of the steady-jet regime and the follow-ing decrease to a much lower velocity regime.Curves are shown for di¡erent water contents asa function of chamber depth.

6. Some e¡ects connected with eruption

6.1. Evacuation of magma chamber

This section develops a model of a magmachamber evacuation during explosive eruptionscausing caldera formation based on the analysisof Slezin (1987) published in Russian. This workcomplements studies by Druitt and Sparks (1984),Lipman (1984) and Roche et al. (2000).

6.1.1. Theoretical descriptionThe cause of caldera subsidence due to an ex-

plosive eruption is evacuation of a magma cham-ber, which weakens the support of the chamberroof and results in a pressure decrease, with ex-solution of gas in the chamber. To a ¢rst approx-imation the de¢cit of magma volume in the cham-ber is equal to the total gas volume in it.Subsidence of the roof ¢nally stops when the pres-sure needed for its support is restored. Subsidencetakes place when all foaming magma has left thechamber or the conduit becomes blocked. In thelast case the residual gas in bubbles would dis-solve back into the magma as the pressure is re-stored.

Magma chamber evacuation is characterized bythe value of ‘magma drawdown’ v=V/A (Smith,1979; Spera and Crisp, 1981), where V is theerupted volume reduced to dense magma and Ais the area of the caldera, assumed to be equal tothe area of horizontal projection of the magmachamber. Expressing V through erupted mass Mand magma density bl , one can obtain:

v ¼ Mb lA

ð61Þ

The problem is to ¢nd v as a function of thecharacteristics of the magma system.

The magnitude of draw-down must dependstrongly on the relation in time between processesof chamber evacuation and roof subsidence. If theeruption continues during subsidence, nearly totalevacuation of the chamber is possible, and themagma draw-down v would be approximatelyequal to the mean vertical dimension of the cham-ber. If the eruption stopped before the beginningof subsidence, the value of v would depend on the

Fig. 25. Dependence of the velocity of magma ascent just atthe moment of cessation of the steady-jet stage of eruption(us) on magma chamber depth at the di¡erent values of vol-atile (water) content. The velocity here refers to the valuejust before the jump down to a lower velocity. Figures nearthe curves are volatile (water) content in wt%. Curves arecalculated at c=1034 m2 Pa31 s31 and at the values of theother parameters, except H, approximately corresponding toeruption of Mount St. Helens, 1980.

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eruption dynamics and the £ow structure in theconduit. Geological data suggest that both alter-natives exist. Lipman (1986) demonstrated thatsimultaneous eruption and collapse occur in cal-dera-forming eruptions (102^103 km3). In smaller-magnitude eruptions subsidence begins after theeruption has stopped. In some eruptions evacua-tion of the magma chamber was not accompaniedby caldera subsidence at all.

Evacuation of a chamber begins when the levelof bubble nucleation reaches chamber depths andstops when magma movement through the con-duit becomes impossible. This approximation re-sults from the model assumptions of an incom-pressible condensed component of magma anda rigid chamber. Evacuation of the magma cham-ber can stop in one of the following three cases:(1) when the excess pressure is not su⁄cient tomove magma through the conduit ; (2) when theconduit is blocked; (3) when the fragmentationlevel is in the chamber. In the last case out£owof pyroclast particles stops because in a chamber(where the cross-sectional area is much largerthan in a conduit) upward velocity of gas is notsu⁄cient to suspend particles. When the fragmen-tation level reaches the magma chamber, supportof the chamber roof will have decreased dra-matically and its collapse becomes inevitable.So, it follows that the eruption stops when thevolume fraction of gas bubbles at the roof ofthe chamber, L0, corresponds to the limit ofthe foam stability. This volume fraction is ex-pected to be near to the state of close packingequal to 0.74.

The value of pressure and the volume fractionof the gaseous phase at the lower end of a conduitis obtained by numerical solution of Eqs. 32^36.In a chamber these values can be obtained by astatic approximation. The vertical coordinate isdenoted z and is directed downward, with theorigin at the roof of the magma chamber. Atz=0, p= p0 and L= L0 ; at z= z1 the pressurep= p1 at which the gaseous phase disappears :L1 = 0. The volatile content c0 is assumed to beconstant throughout the magma chamber withsolubility dependent only on pressure (Eq. 2).

In the static one-dimensional case the followingexpression for magma draw-down can be written:

v ¼ zl3p13p0

gb lð62Þ

To evaluate p0, p1 and z1, ¢rst the dependenceof L on the pressure should be found. Per unitmass of magma this can be written as:

L ¼ Vg

Vg þ Vlð63Þ

For the static problem, and neglecting surfacetension e¡ects, we can write :

Vg ¼ c03cb g

ð64Þ

Vl ¼13ðc03cÞ

b lð65Þ

Substituting Eqs. 65 and 64 into Eq. 63 andtaking into account dependencies of the solubilityand density of gas on pressure, we obtain:

L ¼ 1

1þ Bp13ðc03cÞ

c03c

ð66Þ

where B= bga/(pabl).The typical values of (c03c) are of order 1032,

so Eq. 66 can be simpli¢ed to:

L ¼ 1

1þ Bpc03c

ð67Þ

From Eq. 67 the values of the pressure corre-sponding to the upper (p0) and lower (p1) bound-aries of the zone with decreased magma densitycan be obtained (substituting L0 or 0 correspond-ingly instead of L) :

p0 ¼14

a2

B2

13L 0

L 0

� �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4Bc0L 0

a2ð13L 0Þ31

s !2

ð68aÞ

p1 ¼c20a2

ð68bÞ

Density depends on porosity L in the followingway:

b ¼ b gL þ b lð13L Þ ð69Þ

Substituting Eq. 67 and expressions for c and b

as a function of p into Eq. 69, after simple trans-formations the following expression for b can beobtained:

VOLGEO 2551 7-2-03


b ¼ b lBpðc03a

ffiffiffip

p þ 1Þc03a

ffiffiffip

p þ Bpð70Þ

The depth where the ¢rst bubbles appeared canbe calculated as the integral :

z1 ¼Z p1

p0

dpgb

ð71Þ

where b is determined by Eq. 70. After sometransformations one obtains:

z1 ¼1

Bgb l

c0c0 þ 1

lnp1p0

þ 21

ðc0þÞB3c0 þ 1

a2

� �

lnc03a

ffiffiffiffiffip1

p þ 1c03a

ffiffiffiffiffip0

p þ 132að ffiffiffiffiffi

p1p

3ffiffiffiffiffip0

p Þ�

ð72Þ

Eq. 72 can be simpli¢ed a little if expandedin series logarithms using the small parameterc03a

ffiffiffip

pand leaving only the ¢rst members of

the series :

z1 ¼1

Bgb l

c0c0 þ 1

lnp1p0

þ 2ðc0 þ 1ÞB

a3

ac0

� �ð ffiffiffiffiffi

p1p

3ffiffiffiffiffip0

p Þ �

ð73Þ

Substituting Eqs. 68 and 73 into Eq. 62, thedependence of the magma draw-down on the ini-tial volatile content in the magma c0, the solubil-ity coe⁄cient a, the gas density at atmospheric

pressure bga and the maximum porosity at thechamber roof L0 is obtained.

6.1.2. Results of calculationsMaximum magma draw-downs for the di¡erent

combinations of parameters are shown in Fig. 26.Every curve starts at some minimum volatile con-tent, which increases as the chamber depth in-creases and solubility coe⁄cient, a, increases.For typical water contents the maximum chamberdepth at which partial evacuation is possible isabout 15 km when a=0.0041 MPa31=2 and about10 km when a=0.0064 MPa31=2. At greaterdepths slight evacuation is only likely if the watercontent exceeds 5%. Every curve has a break atthe water content corresponding to the fragmen-tation level reaching the top of the chamber. Afterthis break the curve transforms into a straight lineparallel to the abscissa. A curve drawn throughthe break-points describes some optimum combi-nation of the volatile content and the depth of themagma chamber roof for e¡ective evacuation. Itshows a minimum value of volatile (water) con-tent at which the maximum value of v corre-sponding to a ¢xed value of H can be reached.If the volatile content is more than this optimumthe additional gas has no in£uence on magmadraw-down for a ¢xed chamber depth, becauseafter destruction of the foam in the chamber the

Fig. 26. The dependence of magma draw-down on the volatile (water) content calculated for the di¡erent depths of magmachamber. The curves are marked for the depth of the magma chamber in km. (a) Solubility coe⁄cient a=0.0041 MPa31=2. (b)Solubility coe⁄cient a=0.0064 MPa31=2.

VOLGEO 2551 7-2-03


out£ow ceased. The range of c in which the mag-nitude of magma draw-down v changes from zeroto its maximum value is about 0.01. These depen-dencies put de¢nite limits on the values of H, c,and v. The increase of v is possible due to in-crease of H only if at the same time the minimumand optimum values of c also increase.

6.1.3. Comparison with observationsWater content is typically estimated between

0.04 and 0.06 for the most catastrophic eruptions.Magma chamber roof depths estimated by vari-ous geophysical methods are between 5 and 7 km(Balesta, 1981; Zlobin and Fedorchenko, 1982;Walker, 1984; Denlinger and Riley, 1984). Themaximum magma draw-down for such parame-ters of the magma system is calculated as about3 km. Approximately the same value of 3 kmwas found as a maximum draw-down by Speraand Crisp (1981) from observational data. Forthe Mount St. Helens eruption (18 May 1980)c=0.046, H=7.2 km, magma chamber diameterDchV1.5 km, V=0.25 km3 (Sarna-Wojcicki etal., 1981; Rutherford et al., 1985). Using thesedata, draw-down is calculated as vV0.12 km. Allthese characteristics correlate well for aV0.0060MPa31=2, a typical value for rhyolitic melts (Fig.26).

6.2. Variations of intensity and pauses in thedispersion regime of eruption

Signi¢cant variations of magma mass dischargerate and relatively short-lived periods of inactivitycan take place during the dispersion regime. Thelargest instability of the £ow regime and pausesoccur near critical points of eruption before thesharp changes. On the First Cone of the Tolba-chik eruption 1975^1976, numerous pauses withdurations of minutes to a few hours (with sub-sequent restoring of £ow intensity) precededthe appearance of lava £ows at the South of theFirst Cone and the start of the Second Cone erup-tion.

New magma boccas, which developed near theactive cone in the Tolbachik eruption, are thoughtto be connected to new magmatic ¢ssures branch-ing from the main conduit. Every new lava break-

through was preceded by shallow volcanic earth-quakes due to propagation of new ¢ssures(Fedotov, 1984). A new ¢ssure requires magmaout£ow from the main conduit. Thus magmamass discharge rate in the main conduit is de-creased and is then gradually restored before thenext impulse. The change of magma mass dis-charge rate in£uences the £ow structure and erup-tion regime. Decrease of the magma mass dis-charge rate causes a decrease of the dispersionzone length, and the average density of magmain the conduit increases. The density increase re-quires additional magma, so at the conduit exitthe discharge rate decreases much more than atdepth, which can cause a pause. The new averagedensity and additional magma mass required forit can be calculated. The maximum possible dura-tion of the pause can be calculated as a relation ofthe additional mass in the conduit to the magmamass discharge rate in the new £ow state.

This maximum duration can be achieved in thecase of instantaneous reaction of the £ow at thestep-like mass discharge rate change. The actual£ow reaction is not instantaneous. A gradual de-crease of magma mass discharge rate before thepause and an increase after it would shorten thepause. If the time of mass discharge rate restora-tion is less than the characteristic time of £owresponse on disturbance, a pause would not take

Fig. 27. Maximum duration of the pause in a steady-jet gas^pyroclastic eruption as a function of the magma ascent ve-locity. Figures on the curves are the values of the step-likepercentage decrease of magma mass discharge rate.

VOLGEO 2551 7-2-03


place at all. The characteristic time of the £owresponse on disturbance, tr, is the sum of thetime needed for the disturbance to propagatethrough the conduit to the vent, tc, and the timeof relaxation of the particle velocity after decreaseof the velocity of gas, trel. For the Tolbachik erup-tion the time of £ow response to an instantaneouschange of magma mass £ux at the fragmentationlevel is estimated as 30 s (Slezin, 1982b). It is nota large time and so the proposed cause of a pauseseems reasonable.

Flow response in the vent to the partial out£owat depth depends on the location of the out£ow.Out£ow at the entrance to the conduit is equiva-lent to enlargement of its cross section and in-crease of the total magma mass discharge ratefrom the chamber, but magma mass dischargerate through the conduit above the out£ow levelremains unchanged. If out£ow takes place nearthe fragmentation level there is little in£uenceon the total magma mass discharge rate fromthe chamber, because the total resistance of a con-duit depends on the conditions in the region ofliquid £ow below the fragmentation level. So themagma mass discharge rate above the level ofout£ow would be less by the value of out£ow,which a¡ects the £ow density and magma massdischarge rate at the exit. The generation of thenew ¢ssures during the Tolbachik eruption and

accompanying seismic events suggests that out-£ow was not far below the fragmentation level.The possible maximum duration of pauses tmax

was estimated, supposing that the out£ow tookplace just below the fragmentation level. Calcula-tions of tmax for the First Cone of the Tolbachikeruption are shown in Figs. 27 and 28.

The relative length of the zone of gas^pyroclast£ow and the average magma density in a volcanicconduit strongly depend on the volatile contentand on the e¡ective diameter of particles. A sharpchange of these parameters could take place ifmagma composition were changed, for examplefrom a strati¢ed magma chamber (Spera andCrisp, 1981). Calculations con¢rm that underideal conditions, when the compositional changeand £ow response are nearly instantaneous, thereshould be pauses of the same duration as in thecase of £ow diversion at depth (Slezin, 1982b).However, in this case the £ow response time isnot so small as in the case of diversion. The dis-turbance propagates with the velocity of the £owand so a complete pause is unlikely.

6.3. Problem of prediction of volcanic eruptions

Prediction is crucial in the mitigation of erup-tions. There are two principal issues: (1) predic-tion of when the eruption starts; (2) prediction ofthe characteristics of the eruption. The secondpart includes prediction of magnitude, intensityand violence of eruption and prediction of theduration of eruption and changes of regime. Thetheory describes the dependence of an eruption onthe characteristics of the magma system and socan constrain the characteristics of future erup-tions. The theory makes this task easier becauseit de¢nes a small number of principal parametersof the magma system and provides criteria forpredicting catastrophic changes of intensity.

7. Concluding remarks

Volcanic eruptions are treated as a quasi-steadyprocess of magma out£ow from a chamberthrough a narrow vertical conduit. Magma is con-sidered as a two-component two-phase medium.

Fig. 28. Maximum duration of the pause in a steady-jet gas^pyroclastic eruption as a function of the values of the step-like percentage decrease of magma mass discharge rate. Fig-ures on the curves are the magma ascent velocities (m s31).

VOLGEO 2551 7-2-03


‘Quasi-steady’ means that changes of the £ow pa-rameters very little disturb the equilibrium be-tween phases: equilibrium is restored much fasterthan the parameters change. Such an approach isusual in volcanological models, which vary in as-sumptions and methods of analysis. This theorywas developed in Russia and was not known inthe west, where analogous but not identical resultswere obtained independently and later.

The type of eruption depends on the two-phase£ow structure at the exit of the conduit. Thisstructure depends on many characteristics of themagma system, including its geometry, the mag-ma properties and the dynamical characteristics ofthe process. Three principal types of magma £owstructure and consequently three principal typesof eruption are proposed: (1) discrete gas separa-tion (DGS) regime ^ bubble £ow; (2) dispersionregime ^ gas^pyroclast suspension £ow; (3) re-gime of partly destroyed foam (PDF) ^ porouspermeable £ow. Criteria for the occurrence ofeach regime have been obtained, and depend onseveral parameters : magma ascent velocity; mag-ma viscosity ; number of bubble nuclei per unitvolume of magma; volatile content; volatile sol-ubility; and particle size and cohesive forces be-tween particles.

The analysis of the dispersion regime demon-strates multi-state conditions and £ow instabilityleading to an abrupt transition to another steady-state £ow regime, changing magma discharge rateby several orders of magnitude. The principalcharacteristic of magma £ow in a volcanic conduitthat makes it unstable is the great di¡erence in£ow resistance between the regions with liquidviscous £ow and gas-suspension £ow, which candi¡er by 6^10 orders of magnitude. Steady-statetheory cannot describe jump-like changes in de-tail, but it can characterize steady regimes beforeand after the jump, and de¢ne the conditionsneeded for the change of regimes. For the latterpurpose the right choice of basic governing pa-rameters incorporating all other parameters isnecessary and these are identi¢ed as magmachamber depth, conduit conductivity and excesschamber pressure.

As a function of the basic governing parame-ters, magma discharge rate is described as a com-

bination of cusp catastrophes. The main splittingparameter, which de¢nes the position of the cusppoint and the possibility of a catastrophe, is inmost cases magma chamber depth. Normal pa-rameters are conduit conductivity and chamberpressure. Magma discharge rate as a function ofgoverning parameters creates an equilibrium sur-face. A point on this surface describes the state ofthe erupting volcano. Movement of this pointover the surface describes changes in the statewhen governing parameters change. Intersectionsof the fold lines on this surface indicate sharpchanges of the system state to a new stable re-gime, which can be predicted using the steady-state approach.

The jumps of magma discharge rate, describedby the steady-state theory, are the mechanism ofcatastrophic explosive eruptions, in which sharpincreases of intensity (‘explosions’) usually takeplace after moderate activity. The principal re-quirement for an increase of intensity is relativelysmall depth of the magma chamber. So the theorycan be used as a basis for the prognosis of thepotential danger of a volcano. The steady-statetheory can also be used for the analysis and pre-diction of many other sharp changes in eruptionbehavior induced by internal dynamics as well asby external factors. This was illustrated by exam-ples of caldera subsidence and pauses in gas^py-roclast jets. Steady-state theory can describe theevolution of an eruption as a succession of steadystates with sharp transitions between them.

Acknowledgements

This work was supported by the Russian Foun-dation for Basic Research. The visit to BristolUniversity to prepare this paper was supportedby the Royal Society of London. The author isgrateful to Steve Sparks for useful discussions andadvice, which have helped to improve the paperand for his enormous work in correcting the Eng-lish text. The author also thanks O. Melnik forhelp connected with computer processing. The au-thor is grateful to Elisabeth A. Par¢tt and Karl L.Mitchell for detailed reviews of the manuscriptwith many helpful comments.

VOLGEO 2551 7-2-03


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