Reconciling pyroclastic £ow and surge: the multiphase physics of

Reconciling pyroclastic £ow and surge:the multiphase physics of pyroclastic density currents

Alain Burgisser a;�, George W. Bergantz b

a Alaska Volcano Observatory, Geophysical Institute, University of Alaska Fairbanks, Fairbanks, AK 99775-7320, USAb Department of Earth and Space Sciences, University of Washington, Box 351310, Seattle, WA 98195-1310, USA

Received 4 March 2002; received in revised form 7 June 2002; accepted 18 June 2002

Abstract

Two end-member types of pyroclastic density current are commonly recognized: pyroclastic surges are dilutecurrents in which particles are carried in turbulent suspension and pyroclastic flows are highly concentrated flows. Weprovide scaling relations that unify these end-members and derive a segregation mechanism into basal concentratedflow and overriding dilute cloud based on the Stokes number (ST), the stability factor (4T) and the dense^dilutecondition (DD). We recognize five types of particle behaviors within a fluid eddy as a function of ST and 4T : (1)particles sediment from the eddy, (2) particles are preferentially settled out during the downward motion of the eddy,but can be carried during its upward motion, (3) particles concentrate on the periphery of the eddy, (4) particlessettling can be delayed or ‘fast-tracked’ as a function of the eddy spatial distribution, and (5) particles remainhomogeneously distributed within the eddy. We extend these concepts to a fully turbulent flow by using a prototypeof kinetic energy distribution within a full eddy spectrum and demonstrate that the presence of different particle sizesleads to the density stratification of the current. This stratification may favor particle interactions in the basal part ofthe flow and DD determines whether the flow is dense or dilute. Using only intrinsic characteristics of the current, ourmodel explains the discontinuous features between pyroclastic flows and surges while conserving the concept of acontinuous spectrum of density currents. 7 2002 Elsevier Science B.V. All rights reserved.

Keywords: density currents; pyroclastic surges; pyroclastic £ows; turbulence

1. Introduction

Pyroclastic density currents are rapidly movingmixtures of hot volcanic particles and gas that

£ow across the ground under the in£uence ofgravity. These multiphase £ows consist of par-ticles of various sizes and densities, and a stronglybuoyant gas phase. The complex interplay be-tween sedimentation and entrainment, the dif-¢culty of direct observations, and the absenceof a direct record of the internal £ow structuremake the study of pyroclastic density currentschallenging. The resulting geologic literature isextensive, complex, and sometimes contradic-tory.

The deposits of pyroclastic density currents

0012-821X / 02 / $ ^ see front matter 7 2002 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 0 2 ) 0 0 7 8 9 - 6

* Corresponding author. Tel. : +1-907-474-5606;Fax: +1-907-474-7290.E-mail addresses: [email protected] (A. Burgisser),

[email protected] (G.W. Bergantz).

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Earth and Planetary Science Letters 202 (2002) 405^418

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vary from strati¢ed to massive. Strati¢ed faciescommonly exhibit sedimentary bedforms and thedeposit is often weakly controlled by topography,generally mantling the landscape. Massive faciesare poorly sorted, often structureless, and pondinto depressions. The recognition of these facieshas motivated two end-member models of pyro-clastic density currents (e.g. [1,2]). Strati¢ed faciesare proposed to be the products of a dilute sus-pension called pyroclastic surge, in which particlesare carried in turbulent suspension and in a thinbed-load layer. The generally thicker massive fa-cies are the result of highly concentrated pyroclas-tic £ows [3].

Mechanical models for both end-members havebeen developed, based on di¡erent assumptions ofthe physics of the £ow. Surge models are assumedto have negligible particle interactions, particlehomogenization by turbulence, an exponentialsedimentation law, and are often restricted to asingle particle size (e.g. [4^6]). Whereas there islittle debate that deposition in surge occurs byaggradation, in a layer-by-layer fashion, it is un-clear whether pyroclastic £ows freeze en masse orgradually sediment particles. Arguments for enmasse deposition include the poorly sorted natureof deposits and the common presence of coarse-tail grading of lithics and/or pumices [7,8]. Sedi-mentation by freezing implies that the deposit isdirectly representative of the dynamical state ofthe moving £ow, and this has motivated analogiesbetween pyroclastic £ow and hydraulic current orsliding bloc (e.g. [9,10]). Arguments for depositionby aggradation include the existence of composi-tionally distinct units within some massive depos-its and the particle fabric of £ow units [11^13].Considering pyroclastic £ows as rapid granular£ow is consistent with aggradation (e.g. [14])and some granular models have recently been ap-plied successfully (e.g. [15,16]). However, any uni-¢cation of the end-members is di⁄cult becausethe assumptions implicit in each model are incom-patible.

Hence, whether pyroclastic £ows and surgesrepresent two truly distinct phenomena remainsunresolved. The density discontinuities repro-duced in experiments of £uidization [17] andhigh-speed two-phase £ow decompression [18],

as well as the marked facies diversity of the de-posits, are cited in support of a discontinuity be-tween £ow and surge. However, deposits com-posed of a mixture of the two facies, such as theMt. Pele¤e 1902 nue¤e ardente or the Mt. St. Helens1980 blast, motivated a reconsideration of the re-lationship between the two types [2]. Advocatesfor a continuous spectrum of density currentsproposed that surges are density strati¢ed [7,19^21]. They hypothesize that the concentrated baseof such a strati¢ed surge can sometimes generatedense under£ows that produce the massive depos-its characteristic of pyroclastic £ows [11,22]. Forexample, Druitt [21] explains the whole spectrumof facies observed in the 1980 Mt. St. Helens lat-eral blast deposit using the continuum approach.Recently, visual observations of £ow separationat Montserrat [23,24] and Unzen [25] helped toconnect processes and related deposits.

Recognizing the paradox inherent in the con-cept of a continuous spectrum between pyroclastic£ows and surges and the basic assumptions com-monly used in their modeling, we propose a uni-fying mechanical model that identi¢es £ows andsurges as two entities coexisting in pyroclasticdensity currents. Our approach accounts for thecomplexity in the dynamics of multiphase £owintroduced by turbulence, and is based on scalingrelations of the dominant mechanisms that occurin the currents. Our model focuses on the inter-play between particles and turbulence in the ab-sence of particle^particle interaction, and pro-poses a threshold criterion between dense anddilute conditions, from which the coexistence ofsurge and £ow is derived. We adopt a Lagran-gianÊulerian approach in the dilute regime, how-ever a complete development of a mechanicalmodel of dense granular £ow is beyond the scopeof this paper.

The idea of linking £ows and surges has alreadybeen proposed in the literature. While most au-thors present conceptual models based on geolog-ical evidence (e.g. [21,25^29]), few have addressedthe £uid mechanics aspect of the problem (e.g.[20,30,31]). Mechanical models of surges assumethat turbulence homogenizes the vertical distribu-tion of pyroclasts [5,32,33]. The sedimentation ofeach class size of particle within the £ow/surge is

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described by the ratio of the particle terminal fallvelocity, UT, and some Eulerian time scale of the£ow. The time scale could be given by the hori-zontal speed of a given volume within the surge[20,33], or by the £ow thickness if no velocitygradient within the £ow is assumed [5,32]. Coarseparticles are calculated to sediment faster than¢ner ones, and their increased concentration atthe base of the surge generates dense under£ows[20].

Our approach relaxes the ad hoc assumptionthat particles are homogenized and de¢nes dimen-sionless numbers based on the Lagrangian char-acteristics of the £ow, which allows a re¢nementof the understanding of particle gathering anddispersal by turbulence. The homogenization ofparticle distribution is a consequence of the gasphase and the pyroclasts being in dynamic equi-librium when the particles are su⁄ciently ‘small’.Noting that no quantitative estimate of criticalparticle size has been given, we question the as-sumption of homogenization when applied to thewhole spectrum of pyroclastic density currents.We expect that the largest clasts can signi¢cantlya¡ect the current dynamics and can decouplefrom the gas phase. By invoking a Lagrangianformulation, we quantify the critical size abovewhich turbulence segregates particles and organ-izes them within the density current. Turbulencegenerates unsteady variations of the £ow ¢eldwhile gravity sets a steady downward forcing onparticles ; they cannot be considered as two sepa-rate mechanisms that add linearly : their simulta-neous consideration is necessary [34,35].

Neri and Macedonio [31] recognized the cruciale¡ect of particle size on the dynamics of the £owusing a three-phase model of collapsing volcaniccolumns. They point out that introducing twoparticle sizes (10 and 200 Wm) changes dramati-cally the behavior of the £ow. Motivated by thefact that pyroclastic deposit grain size distribu-tions commonly encompass from 36 to 6 P (6.4cm to 156 Wm), we feel there is a need to assessthe role that the whole range of particles size hasin the dynamics of the pyroclastic density cur-rents. The proposed model is based on simpledimensionless numbers and is viewed as a ¢rstapproach to these complex £ows.

2. Segregation model : principles and assumptions

A pyroclastic density current is a fully turbulentparallel shear £ow of gas with a signi¢cant load ofparticles with a wide range of sizes and densities.The £ow is bounded by the ground at the bottomand by a free surface at the top, and the turbu-lence generates eddies of various sizes and speeds.In the fully turbulent regime, scalar quantitiessuch as chemical components or temperature arewell mixed, but separate phases in the £ow suchas particles are not necessarily well mixed, form-ing what has been recognized as ‘mesoscale struc-tures’ [36]. To understand the interplay betweenthese particles and the turbulence, consider onlyone given eddy within this spectrum. The acceler-ation of a sphere in a non-uniform £ow is givenby the Basset^BoussinesqÔseen (BBO) equationderived by Maxey and Riley [37], which is thesummation of the various forces acting on theparticle (see Appendix). Following the truncationof the BBO equation by Raju and Meiburg [34],the Lagrangian formulation of a particle motion,in dimensionless form, is:

dvdt

¼ uðtÞ3vðtÞST

þ egF2

R

ð1Þ

where u(t) is the gas velocity, v(t) the particle ve-locity, and eg the unit vector in gravity direction(see Table 1 for symbol de¢nition). The Stokesnumber ST and the particle Froude number FR

are given by:

ST ¼ tvfvUN

¼ 1fvbd2

18WvUN

ð2Þ

FR ¼ vUffiffiffiffiffiffigN

p ð3Þ

where vb is the density di¡erence between theparticle and the gas (vbWbp, with bp being theparticle density), d is the particle diameter, W is thegas dynamic viscosity, vU is the eddy rotationspeed, N its diameter, tv is the response time ofparticles (Eq A2 in Appendix), f is a drag factorfunction of the particle Reynolds number Rep(Eq. A3), and g is the acceleration of gravity.

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Our approach is predicated on the statement thatthe interaction of particles with this eddy can beunderstood with two concepts: the Stokes number(ST) and the stability factor (4T), which is a ratioof Stokes and Froude numbers.

ST measures the coupling between gas and par-ticles and is the ratio of the response time of par-ticles tv (particle reaction to unsteady forcing bygas turbulence), and a time scale of gas motion(eddy rotation time in turbulent £ows). ST con-

trols a self-organization of the particles within aneddy, concentrating or dispersing particle as afunction of their density and/or size; smallenough particles follow the eddy motion whereaslarge enough particles are not be a¡ected by theeddy [35]. If STI1, particles couple with the gas.If STV1, particles tend to travel at the eddy pe-riphery, possibly escaping from its gyratory mo-tion. Thus, particles with ST near unity tend togather at the eddy periphery [38]. If STE1, par-ticles decouple from turbulence, and particle mo-tion is not governed by the gas phase.

4T assesses the steady gravitational forcing onparticles and is a measure of the particle residencewithin an eddy. We de¢ne 4T as the ratio of theterminal fall velocity UT and the eddy rotationvelocity vU :

4 T ¼ b pgd2

18W fvU¼ ST

F2R

¼ UT

vUð4Þ

If 4TE1, particles are in£uenced by gravityand tend to sediment from the eddy. If 4TI1,particles are in£uenced by the eddy motion andtend to stay within it (R. Breidenthal, unpublishedexperimental results). The stability factor predictsthe migration towards the base of the eddy oflarge and/or dense particles.

2.1. Eddy mechanisms

The simultaneous consideration of ST and 4T

with the conditions listed in Table 1 leads to therecognition of ¢ve regions within a continuumof particle behaviors (Fig. 1). In the Fall zone(4TE1, ST s 1), particles sediment from theeddy. We de¢ne that the lower boundary of theFall zone is reached when vU is 30% superior toUT (log(4T) = 0.5). In the Unroll zone (4TV1,ST s 1), particles are preferentially settled outwhere the vertical component of vU is maximaldownward but can be carried during the upwardmotion of the eddy. Particle transport becomesasymmetric; the eddy ‘unrolls’ the range of par-ticle sizes lying in this zone. We de¢ne that thelower boundary of this asymmetric transportis reached when UT is 30% inferior to vU(log(4T) =30.5). In the Margins zone (4T 6 1,

Table 1Symbols and constants

u(t) gas velocity (m/s)v(t) particle velocity (m/s)eg unit vector in gravity directionb £ow bulk density (kg/m3)bp particle density (kg/m3)bg gas density (kg/m3)vb density contrast between particles and gas

(kg/m3)d particle diameter (m)P length unit de¢ned as 3log2 (mm)UT particle terminal fall velocity (m/s)vU eddy revolution speed (m/s)N eddy diameter (m)t time (s)tv particle response time (s)tc time between particle collisions (s)n particle number density (m33)W gas dynamic viscosity (1.5U1035 Pa s for

air at 300‡C)e gas kinematic viscosity (3U1035 m2/s for

air at 300‡C)g gravity acceleration (9.81 m/s2)Rep particle Reynolds numberf drag factor function of RepST Stokes numberFR particle Froude number4T stability factorDD dense^dilute conditionE kinetic energy per unit massO turbulence decay rate (m2/s3)U eddy wave number (m31)U i, vUi, N i quantities evaluated at the spectrum

injection pointUrms gas velocity root mean square (m/s)Vrms particle velocity root mean square (m/s)N Brunt-Va«isa«la« frequency (m31)Ufl mean £ow speed (m/s)FRflow £ow Froude numberH upstream £ow height (m)HD dimensionless obstacle heighthobs obstacle height (m)

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STV1), particles concentrate on the periphery ofthe eddy. Following Hogan and Cuzzi [38], we setthe boundary at ST = 1. In the Turbulent Sedimen-tation zone (4TV1, ST 6 1), particle sedimenta-tion is modi¢ed by the turbulence structure. Par-ticles settling can be delayed or ‘fast-tracked’ as afunction of the eddy spatial distribution [39]. Inthe Homogenous Transport zone (4TI1, STI1),particles remain homogeneously carried withinthe eddy. Since particles in the HomogenousTransport zone are dynamically ‘attached’ to thegas, we can assume that the £ow satis¢es the cri-teria for the application of the mixture theory.The £ow can be considered as a heavy gas, witha total density equal to the gas density plus theparticle load of the Homogenous Transport zone.We de¢ne this condition as ‘particle homogeniza-tion’. According to the boundaries de¢ned above,the conditions ST = 1 and 4T 6 1 de¢ne the critical

size above which the homogenization assumptionno longer holds. Fig. 1 shows the pattern of par-ticle behavior de¢ned by a 10-m wide eddy spin-ning up to 50 m/s. We choose this relatively smallsize to account for the reducing e¡ect of the den-sity strati¢cation on eddy sizes (see Section 2.3).Using the limits between the domains in a quan-titative fashion, these conditions would de¢ne amedian critical size of 0.75 P (0 P at 20 m/s and1.5 P at 50 m/s). The change of eddy size of anorder of magnitude will in£uence the median crit-ical size by a factor 3.5 (Fig. 2).

2.2. The kinetic energy spectrum

The concepts developed for one eddy can beextended to a fully turbulent £ow by using a pro-totype of the kinetic energy distribution within afull eddy spectrum. Eddies generated by turbu-lence are represented by a kinetic energy spectrumin Fourier space. The dimension of one eddy canbe expressed as a wave number U and its rota-tional speed as the kinetic energy per unit mass:

EðU ÞU ¼ 12vU2ðU Þ ð5Þ

U ¼ 4Z=N ð6Þ

where E(U) corresponds to the kinetic energy spec-trum in Fourier space integrated over a three-di-mensional vortex of radius U and vU(U) is thecharacteristic speed of this vortex [40]. This spec-trum describes how the energy is transferred fromthe injection frequency Ui to (1) the smaller scale(higher wave numbers) at a rate OVU

35=3 (Kol-mogorov’s law of decay) and (2) to the largerscale at a rate VU

4 [40]. The largest possible scaleis on the order of the £ow height for incompres-sible £ows, and the smallest scale we consider ison the order of the particle size. The total kineticenergy is related to E(U) by:

12U2

rms ¼Z

EðU ÞdU ð7Þ

where Urms is the root-mean-square velocity of thegas. Given a prototype of E(U), the ¢ve domains

Fig. 1. The interaction of 1000 kg/m3 particles of varioussizes (x-axis) with eddies of rotation speed vU (y-axis) and adiameter N=10 m. Thin curves are Stokes numbers log(ST)and thick curves are stability factors log(4T). See text for thesigni¢cance of Fall, Unroll, Margins, Homogenous Transport,and Turbulent Sedimentation zones.

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of particle behavior de¢ned above can be trans-posed from a vU3N space to an E3U space forany speci¢c particle size (Fig. 3). In a ¢rst ap-proach, we use the prototype of energy spectrumfor a free decaying three-dimensional isotropicturbulence described by Me¤tais and Lesieur [41],noting that spectrum prototypes for shear £owhave a similar form [42] :

EðU Þ ¼ AU 8 expð34ðU =U iÞ2Þ ð8Þ

Eq. 7 gives Aw44U2rms/U

9i and the space trans-

position can be done with Eqs. 5^8. The dynamicbehavior of particles of a speci¢c size within a£ow can henceforth be characterized for a givenkinetic energy spectrum (Fig. 3B).

In Section 2.1, we have characterized particlebehavior in a single eddy. The extension to afull spectrum of eddies allows us to relate thesize of eddies to their speed (Eq. 8 and Fig. 3B),and enables us to understand the behavior of allparticle sizes in a full spectrum of turbulence (Fig.4, with the same turbulent conditions as Fig. 3B).Integration of Eq. 8 shows that 90% of the kineticenergy is contained between (3/2)U i and (2/3)Ui(bold part of the N-axis in Fig. 4). The largest

Fig. 3. Dynamic behavior of 33 P (8 mm) particles in twocharacteristic eddy spaces. Patterns code the particle behav-ior, thin curves are log(ST) and thick curves are log(4T). (A)vU^N space. (B) EÛ space. The kinetic energy spectrumE(U) is calculated for U i = 0.25 m31 and Urms = 35 m/s. Ar-rows show the e¡ect of density strati¢cation on the shape ofthe spectrum.

Fig. 2. Evolution of the particle critical size with eddy size.The critical size is the upper limit of validity of particlehomogenization by turbulence. The upper and lower valuesof the ‘transition’ region are de¢ned by ST = 1 andlog(4T) =30.5 within the interval 5^50 m/s of eddy spin ve-locity.

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eddies contain therefore most of the kinetic en-ergy and will dominate the particle transport.

When using the spectrum prototype (Eq. 8), weassume negligible momentum exchange betweengas and particles, and no particle^particle inter-actions. However, the spectrum of turbulence islikely to be modi¢ed by the particles. It has beenshown that large particles with high Rep create awake that increases the amount of turbulence,

whereas small particle dampen turbulence (e.g.[43]), and Elghobashi [44] proposed that this tur-bulence modulation is a function of ST. Since themodulation is generated over the length scale ofthe particle, O will depart from the Kolmogorovdecay. Unfortunately, no generalized prototypefor inhomogeneous, particle-laden £ow is yetavailable, but a coupled LagrangianÊulerian ap-proach would allow modulating the spectrum infunction of the particle load.

2.3. The density pro¢le

Consider again a fully turbulent parallel shear£ow of gas with a random load of pyroclasts.Given both (1) the self-organization process con-trolled by ST (unsteady e¡ect of the turbulence)and (2) the gravity-driven strati¢cation of concen-tration predicted by 4T (steady forcing of thegravity), the emergence of density strati¢cationwithin the pyroclastic density current is expected.Whereas large particles with a Fall behavior areexpected to concentrate at the base of the currentrapidly, small particles with a Homogenous Trans-port behavior are homogenized within the currentand produce a constant density pro¢le. The gen-eral average density pro¢le of the £ow is a sum-mation of each particle size characteristic pro¢ledetermined by their dynamic behavior.

Density gradients within horizontally strati¢ed£ows hinder vertical energy transfer and limit themaximum internal waves frequency to the Brunt-Va«isa«la« frequency N. In our case, the strati¢cationis mainly caused by particles with large 4T.Among these particles, those with small ST arethe most e¡ective in hindering the energy transfer.Hence, the density pro¢le can be used to connecta concentration gradient within the £ow to a max-imum eddy size Ni and speed vUi (Fig. 5) :

N ¼ 12Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

gb 0

dbdz

s¼ ZvUi

N ið9Þ

From the self-organization process (ST) and thegravity-driven particle migration (4T), the concen-tration of (coarse and/or dense) particles is higherat the base of the £ow and the concentration gra-dient tends to be steepest at the base. Eq. 9 pre-

Fig. 4. Particles behavior in the full spectrum of eddiesshown in Fig. 3 as function of eddy size (y-axis) and particlesize (x-axis). Bold parts of the y-axis correspond to 90% ofthe total kinetic energy. Patterns code the particle behavior,thin curves are log(ST) and thick curves are log(4T). ArrowsI designate the maximum size transported and arrows II themaximum size homogenized (see text). (A) Particles arepumices (1000 kg/m3). (B) Particles are lithics (2500 kg/m3).

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dicts that eddies tend to be faster and smaller instrong concentration gradients (arrows in Fig. 5).Eddies formed at the base of such a turbulentdilute £ow have therefore an enhanced carryingcapacity compared to eddies higher above thebase. The strati¢cation process allows the £owto accommodate its loading and increases itstransport capacity.

Eq. 9 gives the largest possible scale of eddies inthe kinetic energy spectrum for density-strati¢ed£ows. In consequence, we expect the spectrum ofturbulence given by Eq. 8 for a non-strati¢ed £owto be modi¢ed by the density gradient (Fig. 5).The amount of shear within a strati¢ed £owmodi¢es also the turbulence spectrum. Qualita-tively, the increase of shear raises Ui and dimin-ishes the turbulent decay rate towards the higherwave numbers [45]. In other words, more energyis dissipated by the larger wave number andsmaller eddies take more importance in the £owdynamics, modifying the spectrum shape (arrowsin Fig. 3B).

2.4. Interaction with topography

Salient parameters to describe the encounter ofa density-strati¢ed £ow with an obstacle are the£ow Froude number FRflow of the fastest mode ofthe undisturbed £ow (upstream) and the dimen-sionless obstacle height HD [46] :

FRflow ¼ U fl

4NHand HD ¼ hobs

Hð10Þ

where Ufl is the mean £ow speed, H the upstream£ow height, and hobs the obstacle height. FRflow

indicates the £ow hydraulic regime, sub- or super-critical. The FRflow3HD space de¢nes three main£ow behaviors : crossing, blocking, and hydraulicjump. In the ¢rst case, the £ow strata maintaintheir integrity during the crossing. In the secondcase, blocking of the lower parts of the £ow oc-curs. In the third case, a regime change occursand a hydraulic jump separates the two re-gimes.

2.5. The dense^dilute condition

The frequency of particle interactions is a keyfactor in the £ow dynamics. The dense^dilute con-dition (DD) is a measure of the importance ofparticle interactions within the £ow [35] :

DD ¼ tc ftv

¼ 3W fbV rmsd

ð11Þ

where b is the £ow bulk density, Vrms the rootmean square of the particle speed, f is the Rey-nolds number factor based on this velocity (Eq.A3), and tc the characteristic time between par-ticle collision given by:

tc ¼1

nZV rmsd2 ð12Þ

where n is the number density of particles. Theright-hand side of Eq. 11 is obtained using thatnZd 3

bp =KbpWb, with K being the volume frac-tion of particles. If DD 6 1 (dense £ow), particlesdo not have time to respond to the gas dynamicforces before the next collision, and the dynamicsof the £ow is dominated by particle^particle in-

Fig. 5. Maximum eddy size Ni and speed vUi for a givenconcentration gradient db/dz in a turbulent strati¢ed £ow.Arrows show that an increase of the concentration gradientshortens and/or accelerates the more energetic eddies.

EPSL 6318 29-8-02


teractions. DD s 1 (dilute £ow) implies non-zerointerparticle distance.

Given the density strati¢cation and the physicallimit between a dense and a dilute £ow as de¢nedby DD, a concentration threshold may be reachedin the basal part, where granular motion willdominate. It is therefore likely that the currentsegregates into a basal concentrated, granular£ow and an overriding dilute, turbulent, and den-sity-strati¢ed cloud (Fig. 6). Short-living collision-al interactions dominate the resistance stresses forthe rapid granular £ow regime [15], and high par-ticle concentration suppresses turbulence-gener-ated segregation. We therefore expect the granular£ow to be composed of particles with a small DD,either because they are not sustained by turbu-lence (Fall region), or because they are likely togather (Unroll and Margin regions). The gatheringbeing controlled by the transient nature of turbu-lence, the latter case is expected to play a minorrole in the average location of the boundary.Since particles from the Homogenous Transportproduce a constant average vertical density pro-¢le, they will be trapped in the granular £ow aswell, producing a poorly sorted £ow. Althoughbased on a given particle size, DD quanti¢es the

boundary between dense and dilute parts of the£ow, because the particle size that features thelowest DD is likely to control this boundary. Fur-ther links between collisional interaction and DD

may validate the idea that the density gradientmight be so important at the boundary that adiscontinuity would be formed.

3. Discussion

3.1. General implications

Our model predicts the maximum particle sizethat a turbulent £ow can carry. For example, apumice of 3.2 cm needs 10-m wide eddies to befaster than 30 m/s to travel within a dilute surge(Unroll zone, Fig. 1) and cannot be transportedhomogenously by a turbulent £ow of gas travelingat subsonic velocities (Fig. 2). Products of largeash-£ows can be examined using these criticalsizes to assess their possible mode of transport.

We expect the segregation process caused bythe interplay of ST and 4T to occur whether thedensity current is initially in£ated, as it probablyis the case during a column collapse, or de£ated,

Fig. 6. Schematic cross-section of a pyroclastic density current perpendicular to the £ow direction with characteristic values ofthe three dimensionless numbers that govern the dynamics of the dilute part (ST, 4T, and DD). The end-member ‘surge’ is ob-tained if the £ow consists essentially of the dilute part, whereas the end-member ‘pyroclastic £ow’ has a very thin dilute portion.The right part shows a schematic density pro¢le for three particle sizes in the dilute part. Note the total density is strati¢ed dueto the distribution of the coarse material, whereas the ¢nest particles are homogenized throughout the £ow thickness. The leftpart of the ¢gure illustrates the overbanking of the dilute part. In this scenario, the dilute part overrides the obstacle, leavingstrati¢ed deposits on the topographic high.

EPSL 6318 29-8-02


like in a dome collapse. In the latter case, thecurrent is entirely granular with DDI1 duringits initiation and may in£ate by incorporatingair or exsolved gases. If the mixture of particleand gas becomes such that DD s 1, segregationprocesses will take place, without the need of anupward £ux of gas (e.g. [25]).

Particles in the Unroll zone (e.g. pumices be-tween 0 and 34 P in Fig. 4A and lithics between1 and 33 P in Fig. 4B) are likely to travel byintermittence, whenever an eddy of the appropri-ate size and spin occurs in the current. The kineticenergy spectrum will evolve in time as the densitycurrent travels across the landscape, modifyingparticle sizes a¡ected by the Unroll zone. Giventhe asymmetric transport of this zone and thatparticle collection is favored at ST near unity,the sedimentation/deposition of these particle islikely to occur in an intermittent fashion. If thecurrent is dilute throughout its entire thickness(surge end-member), the Unroll zone is expectedto control particles in saltation. Since particleswith low ST and 4T are not sedimented, the de-posits of such turbulent £ows will periodically ex-hibit a preferential settling of particles with ST s 1and 4TV1. The layered deposit of surges maytherefore represent the rapid variations of the tur-bulent conditions within the current.

Despite the observation that the ‘¢nes-depleted£ow’ de¢ned by Walker [2] includes elutriationgas pipes, we note that the smallest median sizeis about 1 P, whereas median sizes up to 310 P

have been measured [47]. Whatever processes gen-erate ¢nes-depleted deposits, this smallest value isconsistent with the critical size for tephra homog-enization by turbulent £ow (Fig. 2). In otherwords, particles below the critical size can veryeasily be reentrained by a turbulent cloud andtherefore are less likely to sediment.

Pyroclastic density currents have particles ofdi¡erent densities, ranging commonly from 1000kg/m3 (pumice) to 2500 kg/m3 (lithic). The turbu-lent £ow illustrated in Fig. 4 is able to transportpumice up to V34 P and homogenize (turbulentmixing) pumice smaller than 0 P (Fig. 4A, arrowsI and II). Lithics smaller than 33 P are carriedwhereas lithic smaller than 1 P are homogenized(Fig. 4B, arrows I and II). We note that a simple

‘hydraulic equivalence’ (bpWd) is a good ¢rst-orderapproximation. Widely used to characterize par-ticle suspension in turbulent £ow, the Rouse num-ber is a concept close to 4T, although based on anaverage Eulerian velocity of the £ow (horizontalin our case). If this velocity is on the order of Urms

at Ui, it would predict that the boundary betweentransport and deposition is located at 4TV1.Since this condition is satis¢ed in the middle ofthe Unroll zone (Fig. 4), the Rouse number basedon such a velocity is also satisfying a ¢rst-orderapproximation of the time-averaged behavior ofthe £ow. However, the ST^4T framework is nec-essary to understand transient phenomena such asparticle clustering, which are likely to control par-ticle sorting, sedimentation, and the dense^dilutethreshold.

The density pro¢le of a given particle size de-rived by Valentine [20] is a sole function of thevertical velocity gradient. The Lagrangian ap-proach reveals that the density pro¢le is a com-plex function of Urms, the dusty gas bulk density(as de¢ned by the concentration of particles lyingin the Homogenous zone), the turbulent spectrumshape, and the velocity gradients. Beyond theaverage density pro¢les proposed previously (e.g.[20,21]), our approach highlights the potential fortransient high concentration of particles (Marginzone, STV1), as large eddies are generated anddissipated continuously.

We would like to emphasize that ST and 4T areimportant scaling parameters for experimentalwork. In other words, particle sedimentation can-not be well represented if these dimensionlessnumbers are not properly scaled. Although thecomparison between sedimentary structures oc-curring under water and surge bedforms is tempt-ing, it should be considered that, under equivalentconditions, ST could vary by two orders of mag-nitude depending on the nature of the carrierphase (hot air viscosity is V1.5U1035 Pa s at300‡C whereas water is about 1033 Pa s at20‡C). Moreover, the density contrast with theparticles is greatly reduced with water as a carrierphase. Eq. 1 is no longer valid because terms ofthe BBO equation neglected in the Raju and Mei-burg [34] truncation cease to be negligible, and thefull Eq. A1 has to be used (see Appendix).

EPSL 6318 29-8-02


3.2. Hydraulic jump and blocking

Salient parameters to describe the interaction ofa density-strati¢ed £ow with a relief are FRflow

and HD (Eq. 10). A hydraulic jump generatedby an obstacle or a break in slope causes a dra-matic increase in the current depth and reduces itsvelocity. In the case of surges (DD s 1), the po-tential e¡ect of a hydraulic jump on the currentcan be represented by a sudden decrease of the£ow speed and an increase of £ow depth. Theincrease of depth (VN i) narrows the saltationsize range (Unroll zone in Fig. 7), and the speedreduction (VUrms) lowers the maximum size the£ow can transport (limit FallÛnroll in Fig. 7). Inthis example, both the maximum size of particlescarried and the critical size for homogenizationare approximately reduced by a factor 2 (arrowsI and II in Fig. 7). This should be expressed by anenhanced sedimentation after the jump to readjust

the particle load to the new £ow conditions. Ex-periments involving the interaction of a densitycurrent in a water tank with a ridge con¢rm thisincrease in sedimentation (e.g. [33,48]). We predictfrom Fig. 7 that the load drop occurring at thejump between the two hydraulic regimes generatesa moderately well-sorted deposit coarser thanthe local average. Field studies describe ignim-brite lag breccia as very coarse material with atypical median size 33 P and coarser, generallylithic-rich, and often devoid of ¢nes (e.g. [2]).Although the generation of lithic-breccia by hy-draulic jumps has been evoked by several authors[22,49], they do not consider the complexity intro-duced by the density strati¢cation of the £ow (i.e.Eq. 10).

Flow segregation between a dilute cloud and agranular basal part is usually not recorded in de-posits because of sedimentation processes occur-ring at the base of the current. However, when apyroclastic current hits a barrier or sudden reliefchange, the lower part of the current may beblocked, whereas the upper part rides the ob-stacle. The dividing streamline proposed by Val-entine [20] as a blocking criterion is based on ex-periments only valid at low Froude number [50].Following Baines [46], we extend the concept ofblocking to high Froude number and proposethat it is controlled by the density gradient-depen-dent FRflow and the ratio of £ow to obstacle heightHD. Although blocking can occur at any level ofthe strati¢ed £ow, the strongest density gradientoccurs at the dense^dilute boundary. The granu-lar part of the £ow is therefore the most likely tobe blocked. On the high side of the obstacle, a‘segregated deposit’ may result, consisting oflayers from speci¢c levels within the strati¢ed£ow. Fig. 6 illustrates the blocking of the granularpart of a density current that produces strati¢eddeposits on the topographic high.

4. Conclusions

We propose a segregation mechanism of pyro-clastic density currents into basal concentrated,granular £ows and a overriding dilute, turbulent,and density-strati¢ed cloud based on the Stokes

Fig. 7. E¡ects of a hydraulic jump from super- to subcriticalregimes on the transport capacity of a turbulent £ow. Pat-terns code the particle behavior, thin curves are log(ST),thick curves are log(4T) and particle density is 2500 kg/m3.Flow conditions for stippled line A: Urms = 15 m/s, Ni = 20 m.Flow conditions for stippled line B: Urms = 7.5 m/s, Ni = 50 m.

EPSL 6318 29-8-02


number (ST), the stability factor (4T) and thedense^dilute condition (DD). Our analysis revealsthe importance of the combined unsteady e¡ectsof turbulence and steady e¡ects of gravity. Thisanalysis is able to explain the discontinuous fea-tures between pyroclastic £ows and surges whileconserving the concept of a continuous spectrum.From limited assumptions, the two end-membersof pyroclastic density current can be derived byusing only intrinsic characteristics of the £ow con-sidered.

Acknowledgements

We are grateful to R. Breidenthal and J. Gard-ner for helpful discussions throughout the courseof this study. Thorough reviews by S. Hughes, S.King, S. Sparks, and two anonymous reviewersgreatly helped improve this manuscript. Fundingwas provided by the NSF grants EAR-9805336and EAR-0106441 to G.W.B. and by the VolcanoHazards Program of the US Geological Survey,through the Alaska Volcano Observatory toA.B.[SK]

Appendix

The BBO equation expresses the acceleration ofthe spherical particle in a non-uniform £ow as[35,37] :

1þ b g

2b p

� �dvdt

¼ ftv½uðtÞ3vðtÞ� þ g 13

b g

b p

� �þ

3b g

2b p_uuþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9b g

2Zb ptv

s Z t

0

_uu� _vvffiffiffiffiffiffiffiffiffit3t0

p dt0 þ ðu3vÞ0ffiffit

p� �

ðA1Þ

where tv is the particle velocity response time giv-en by:

tv ¼vbd2

18WðA2Þ

and f is a drag factor valid over the entire sub-critical range of particle Reynolds number

(Rep 99 105) [51] :

f ¼ 1þ 0:15Re0:687p þ 0:01751þ 42500Re31:16

pðA3Þ

With:

Rep ¼ UTde

ðA4Þ

where ee is the kinematic viscosity of the gas. Par-ticles with a small Rep (6 10) have a drag causedby the gas viscous friction along the particle body,and fVV1. For high Rep, the drag generated byvortices in the particle wake overcomes the vis-cous drag, and fEE1. The right-hand side of Eq.A1 is the sum of the viscous, gravitational, buoy-ancy, virtual mass, and Basset forces acting re-spectively on the particle. In the case of pyroclas-tic density currents, the density ratio betweenparticle and gas exceeds 103. It is therefore possi-ble to truncate Eq. A1 and use only the two ¢rstterms, namely the viscous drag and the gravityforce [34] :

dvdt

¼ ftv½uðtÞ þ vðtÞ� þ g ðA5Þ

Non-dimensionalization by the turbulence timescale (i.e. eddy rotation time) gives Eq. 1 [52].

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