evaluating suitability of a tephra dispersal model as part

Evaluating suitability of a tephra dispersal model as part of a riskassessment framework

Gordon N. Keating a,⁎, Jon D. Pelletier b, Greg A. Valentine c, William Statham d

a Earth and Environmental Sciences Division, MS D452, Los Alamos National Laboratory, Los Alamos, NM 87545 USAb Department of Geosciences, The University of Arizona, 1040 E. Fourth St., Tucson, AZ, 85721 USAc Department of Geology, 876 Natural Science Complex, University at Buffalo, Buffalo, NY 14260, USAd AREVA Federal Services, LLC under contract to Sandia National Laboratories, Yucca Mountain Project, 1180 North Town Center Drive, Las Vegas, NV 89144, USA

A B S T R A C TA R T I C L E I N F O

Article history:Received 8 November 2007Accepted 4 June 2008Available online 21 June 2008

Keywords:tephra dispersaltephra redistributionrisk assessmentsurficial processesnumerical modelinguncertainty

In volcanic risk assessment it is necessary to determine the appropriate level of sophistication for a givenpredictive model within the contexts of multiple sources of uncertainty and coupling between models. Acomponent of volcanic risk assessment for the proposed radioactive waste repository at Yucca Mountain(Nevada, USA) involves prediction of dispersal of contaminated tephra during violent Strombolian eruptionsand the subsequent transport of that tephra toward a hypothetical individual via surface processes. We testthe suitability of a simplified model for volcanic plume transport and fallout tephra deposition (ASHPLUME)coupled to a surface sediment-transport model (FAR) that calculates the redistribution of tephra, and in lightof inherent uncertainties in the system. The study focuses on two simplifying assumptions in the ASHPLUMEmodel: 1) constant eruptive column height and 2) constant wind speed and direction during an eruption.Variations in tephra dispersal resulting from unsteady column height and wind conditions producedvariations up to a factor of two in the concentration of tephra in sediment transported to the controlpopulation. However, the effects of watershed geometry and terrain, which control local remobilization oftephra, overprint sensitivities to eruption parameters. Because the combination of models used here showslimited sensitivity to the actual details of ash fall, a simple fall model suffices to estimate tephra massdelivered to the hypothetical individual.

Published by Elsevier B.V.

1. Introduction

Predictivemodels of pyroclasticflows, lavaflows, lahars, and tephrafallout are becoming increasingly important components of probabil-istic volcanic risk assessment (e.g., Baxter et al., 1998; Iverson et al.,1998; Saucedo et al., 2005; Bonadonna et al., 2005; Damiani et al.,2006;Magill et al., 2006). Inmany cases thesemodels are developed asresearch tools, and as such they tend to continually increase incomplexity as volcanologists delve deeper into the underlying physics.Application of the models to specific risk assessments, on the otherhand, may require linking togethermany processes that each involve alevel of uncertainty. In some, if notmost, cases, a point may be reachedwhere there is little payoff in improving the physical fidelity of a givenvolcanic process model because that improved fidelity might havenegligible impact on the risk assessment. In addition, inherentuncertainties in linked models may dwarf the effects of improvedphysical modeling. Here we demonstrate for a particular riskassessment application that a relatively simple model of tephradispersal can be applied where more sophisticated models would

have negligible impact on the final result. The purpose of this study isnot to compare theories or numerical codes of fallout, but simply to testthe impacts of simplifications in our approach to tephra dispersalmodeling, given that the results of the model are subsequently“filtered” through a second model that predicts the redistribution oftephra by fluvial processes, and in light of uncertainties in the overallsystem. Our results provide an example of howmodel adequacy can bedetermined for a complex risk assessment.

2. Volcanic risk assessment for the proposed YuccaMountain repository

The proposed high-level radioactive waste repository at YuccaMountain (Nevada, USA) is sited in a region that has experiencedsporadic basaltic volcanism since the end of major caldera eruptions inmid-Miocene time (Crowe,1986; Fleck et al., 1996; Valentine and Perry,2007). Volcanic risk assessment for YuccaMountain involves estimatingthe probability of formation of a new basaltic volcano that mightintersect the ~300 m deep repository (Connor et al., 2000; Perry et al.,2005; Crowe et al., 2006) and the consequences of such an event, shouldit occur (Crowe et al.,1983;Woods et al., 2002; Dartevelle andValentine,2005; Gaffney and Damjanac 2006; Gaffney et al., 2007; Keating et al.,2008; Valentine and Perry, in press). One class of consequences analysis,

Journal of Volcanology and Geothermal Research 177 (2008) 397–404

⁎ Corresponding author. Tel.: +1 505 667 5902; fax: +1 505 667 1628.E-mail addresses: [email protected] (G.N. Keating), [email protected]

(J.D. Pelletier), [email protected] (G.A. Valentine), [email protected] (W. Statham).

0377-0273/$ – see front matter. Published by Elsevier B.V.doi:10.1016/j.jvolgeores.2008.06.007

Contents lists available at ScienceDirect

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and the focus of this paper, is eruptive effects where radioactivewaste isentrained into magma and erupted and dispersed onto the Earth'ssurface (Jarzemba et al., 1997). Risk is estimated, as defined byregulations, for a hypothetical individual (a.k.a. RMEI: reasonablymaximally exposed individual) located ~18 km south of the repositoryin an area that corresponds to the upper part of the Fortymile Washalluvial fan (Fig. 1), for a future period of 104–106 years.

Based upon analogy with other Quaternary volcanoes in the region(e.g., Valentine et al., 2006, 2007; Valentine and Keating, 2007), weinfer that a potential new volcanowill consist of a scoria conewith oneor two small lava flows and a total eruptive volume between ~0.01–0.1 km3. Eruptive facies in the Quaternary volcanoes indicate thatexplosive eruptions include substantial components of the violentStrombolian style, consisting of sustained eruption columns of well-fragmented basaltic clasts rising to heights of several km. From a riskassessment perspective these eruptions are important because theycould disperse contaminated fallout tephra over areas of 100 s of km2.Such tephra could impact the hypothetical individual by 1) directdeposition by fallout under certain wind and eruption conditions, 2)deposition upstream in the Fortymile Wash drainage basin andsubsequent transport to the control population by surficial processes,or 3) a combination of 1) and 2). Given the dominant southwesterlywinds in the Yucca Mountain region in the 1 to 13 km altitude range(NOAA, 2004; SNL, 2007a), the most likely direction for a wind-driven

eruptive plume is to the northeast, away from the RMEI. Therefore, asignificant component of risk may be due to redistribution of waste-contaminated tephra across the landscape by fluvial processes.

3. ASHPLUME and FAR models

The coupled component models for atmospheric dispersal/deposi-tion and fluvial redistribution of tephra use the ASHPLUME v. 2.1(Jarzemba 1997; Jarzemba et al., 1997; SNL, 2007a) and FAR v. 1.2(Pelletier et al., 2005, 2008) codes, respectively, which are bothcomponents of the Yucca Mountain Total System Performance Assess-ment (TSPA) risk assessment framework. The values of input parameterfor the ASHPLUME model (see Appendix) are developed within TSPAfrom other analyses and models that consider 1) the number andgeometry of dikes and conduits intersecting the repository in a givenmodel realization and resultingwaste mass available for transport (SNL,2007b), and 2) eruption characteristics such as eruption power,duration, tephra and waste particle size distributions (SNL, 2007a).The output fromASHPLUME is a two-dimensional distribution of tephramass on the landscape that provides the initial conditions for FAR, ascour-dilution-mixing model. FAR calculates the fluvial (bedload)transport of tephra that is mobilized from steep slopes and depositedon active channels, and then routed through the drainage system to theRMEI area, located in the upper areas of the FortymileWash alluvial fan.

Fig. 1. Physiographic map of the Yucca Mountain region, including the proposed high-level nuclear waste repository at Yucca Mountain (bright yellow), Fortymile Wash watershedand upper alluvial fan (light green), the location of the RMEI (⁎), and a typical ASHPLUME tephra distribution with 1 km grid spacing (N1 cm thickness, pale orange to brown). Insetmap shows location relative to southwestern U.S. Map grid: UTM zone 11, NAD27.

398 G.N. Keating et al. / Journal of Volcanology and Geothermal Research 177 (2008) 397-404

Areal concentrations of tephra andwaste (g/cm2) are passed fromFAR tothe final TSPA component model, which converts these mass concen-trations to activity concentrations and uses biosphere dose conversionfactors to calculate radiological dose to the RMEI for each Monte Carlomodel realization.

The ASHPLUME numerical model (Jarzemba, 1997; Jarzemba et al.,1997) incorporates the mathematical model of basaltic tephra dispersal

from a steady-state eruption column developed by Suzuki (1983). SeeAppendix for a summary of the ASHPLUME mathematical model andrepresentative input values used in this study. A line source of tephraparticles (density-adjusted for waste content) is transported downwindby 2-D (horizontal) advection and turbulent diffusion. Particle deposi-tion is calculated for each observation point on the ground by aprobabilistic equation that balances particle fall time vs. height in theplume. The model assumes that volcanic activity during an eruption isconsistently energetic (violent Strombolian), with constant eruptivepower and columnheight for the duration of the event.Wind conditionsare held constant throughout the eruption, with speed and directionassigned for conditions at the top of the eruption column.

The FAR model uses a digital elevation model (DEM) of the terrainin Fortymile Wash watershed to determine contributing areas forstreams and slopes greater than a threshold value, fromwhich tephrais mobilized. The tephra blanket from ASHPLUME is processed in FARto exclude tephra deposited outside the watershed and to accumulatethe tephra mass that was deposited on steep slopes and in activechannels. This bedload-transported tephra is mixed with sediment tothe scour depth and routed through channels to the apex of thealluvial fan, simulating the cumulative effects of many flood events(Pelletier et al., 2008). Comparison of the size distribution of tephraparticles expected from an eruption at Yucca Mountain with channelsediments in Fortymile Wash indicates that nearly all of the tephrafallout will be transported as bed-material load in Fortymile Wash(Pelletier et al., 2008). Eolian transport is not included because it isexpected to reduce the concentration of tephra (and associatedradionuclides) at the RMEI area (by removal and dilution) and becausethe range of parameter values for the fluvial model accounts for theeffect of migration of sediment from the channel onto drainagedivides on the fan (SNL 2007c).

The limitations inherent in the Ashplume code have recently beentranscended in other tephra dispersal models that utilize time- andaltitude-dependent wind conditions or unsteady column height (e.g.,Folch and Falpeto, 2005; TEPHRA — Bonadonna et al., 2005; ASHFALL/HYPACT —Turner and Hurst, 2001). The variation of the wind field

Fig. 2. Tephra isopachs (cm) for the base-case eruption modeled by ASHPLUME overlaidon results of FAR tephra redistribution model. Isopach contours are 0.01, 0.1, 1.0, 10,100 cm. Background colors represent tephra concentration (fraction of tephra) instream channel sediment after fluvial redistribution. The point of comparison of tephraconcentrations in sediment at the outlet of Fortymile Wash is located at the bottom ofthe plot, at approximately 7500 m east and 0 m north. In this base-case run (W1), theconcentration of tephra in channel sediment is 0.062.

Table 1Summary of ASHPLUME model runs to evaluate the effect of constant column height and variable wind conditions on concentration of tephra in sediment at the outlet of FortymileWash

ASHPLUME runnumber

Representative input values Composite model runcharacteristics

Fraction of tephra insediment at Wash outlet

Concentration changevs. base case

Volume(km3)

Duration(days)

Power(W)

Column height(km)

Wind speed(m/s)

Base case (W1) 0.035 1.5 2.7×1011 5.9 7 – 0.062 –

H2 0.002 1 2.5×1010 3.3 5 Variable column height 0.058 0.940.010 1 1.3×1011 4.9 6.50.023 1 2.9×1011 6.0 9

H3 0.023 10 2.5×1010 3.3 5 Variable column height 0.071 1.150.010 1 1.3×1011 4.9 6.50.002 0.1 2.9×1011 6.0 9

W4a 0.012 1 1.5×1011 5.1 7 30° wind direction spread 0.063 1.02W3a 0.012 1 1.5×1011 5.1 7 60° spread 0.065 1.05W5a 0.012 1 1.5×1011 5.1 7 90° spread 0.051 0.82W6b 0.0175 1 2.2×1011 5.6 7 90° wind direction

divergence0.139 2.24

W7b 0.0175 1 2.2×1011 5.6 7 120° divergence 0.125 2.02W8b 0.0175 1 2.2×1011 5.6 7 150° divergence 0.078 1.26W9b 0.0175 1 2.2×1011 5.6 7 180° divergence 0.034 0.55W1Nc 0.035 1.5 2.7×1011 5.9 7 90° N rotationd 0.045 0.73W1N30Ec 0.035 1.5 2.7×1011 5.9 7 60° N rotationd 0.160 2.58W1N60Ec 0.035 1.5 2.7×1011 5.9 7 30° N rotationd 0.114 1.84W1S60Ec 0.035 1.5 2.7×1011 5.9 7 30° S rotationd 0.057 0.92W1S30Ec 0.035 1.5 2.7×1011 5.9 7 60° S rotationd 0.055 0.89W1Sc 0.035 1.5 2.7×1011 5.9 7 90° S rotationd 0.012 0.19

a Runs W3, W4, W5 included three individual model runs with identical input parameter values except wind direction.b Runs W6, W7, W8, W9 included two individual model runs with identical input parameter values except wind direction.c Base case, single-eruption plume rotated across the landscape in 30° increments.d Angles of rotation are relative to due east.

399G.N. Keating et al. / Journal of Volcanology and Geothermal Research 177 (2008) 397–404

with height and time during an eruption simulation produces spreadand divergence in the tephra sheet that often matches observedtephra distribution better than a simple diffusion plume. However,because the ASHPLUME code is coupled to the FAR code in our study,the effects of ASHPLUME's limitations on the tephra/waste concentra-tion at the RMEI location after atmospheric dispersal and fluvialredistribution are not obvious and require additional analysis.

4. Analysis

This paper describes an analysis of the effects of 1) unsteadycolumn height, and 2) variable wind conditions on the concentrationof waste-containing tephra in sediment at the outlet of FortymileWash (or fan apex), prior to distribution to the RMEI location andsubsequent soil migration. The observed effects of variations in theseindividual eruption parameters are placed in context of effects due tovariations in terrainwithin thewatershed onwhich the tephra blanketis deposited. An alternative approach would be to use the othermodels cited above to conduct comparative simulations. We chose notto use the latter approach because it introduces additional factors,such as different assumptions, formulations of the governingequations, numerical solution algorithms, and grid effects thatwould complicate the testing of the two physical simplificationsalone. (Such a comparison, between the ASHFALL and ASHPLUMEcodes, has been performed, with good agreement between the twocodes; SNL, 2007a). For purposes of comparison, we use as a base casethe results of a coupled ASHPLUME-FAR simulation where the tephraplume is east directed, consistent with prevailing winds. A base-case0.035 km3 tephra deposit results in a tephra concentration (fraction oftephra) of 0.062 in the stream sediment at the fan apex (Fig. 2).

We assessed the effect of variable column height during aneruption by dividing the base-case eruption volume among threesmall eruptions, each with constant wind direction but variable

power, column height, wind speed, and duration (runs H2, H3;Table 1). The sets of small-volume model runs were combined to formcomposite tephra sheets with total volume equal to that of the single-eruption base case. The composite results were similar to theinstantaneous, constant-column height base case (run W1), despitevarying the component eruption volume by an order of magnitude,the column height by a factor of 2, and the wind speed by 30%. Afterrouting through FAR, the tephra concentrations in sediment at theoutlet of Fortymile Wash for all the cases fell within ±15% of the basecase (Table 1).

The effect of variation in wind direction during an eruption wasassessed in two separate cases: 1) spread, caused by minor variationsin wind direction throughout an eruption, and 2) divergence, causedby two distinct wind directions during the eruptive period. Sets ofthree small component eruptions (with combined total volume equalto the base case) simulated varying wind direction (spread) over 30°,60°, and 90° compared to the base case, east-directed run (Fig. 3,Table 1). The concentrations of tephra in sediment at the watershedoutlet resulting from mobilization of these tephra sheets werenormalized vs. the base-case value. As the degree of spread in theplume increased from due east toward the southeast quadrant, theoutlet concentration increased slightly as a greater proportion of thetephra fell in the active channel area just upstream of the watershedoutlet; however, at 90° spread the concentration dropped off, as asignificant portion of the tephra was deposited due south, outside theFortymile Wash watershed (Fig. 4A). Bi-lobate composite tephrasheets resulted from the sets of two small eruptions used to evaluatethe effect of diverging wind directions during eruption (Table 1, Fig. 3).The effect of divergence in wind direction was assessed for 90°, 120°,150°, and 180° separation (Fig. 4B). The local maximum in normalizedtephra concentration at 90° divergence indicates that this is anoptimal plume orientation for depositing tephra on steep slopes andactive channels, with one lobe directed into the heart of thewatershed

Fig. 3. Composite tephra sheets produced by column height variation andwind spread and divergence during eruption, compared to the base-case simple plume (runW1, upper left).Red outlines denote the 1-cm isopach for the component tephra sheets. Contour spacing is logarithmic: 0.001, 0.01, 0.1, 1.0, 10, 100 cm. Run numbers are designated in the uppercorner of each plot.


to the northeast and one onto the large active channel to thesoutheast. The normalized tephra concentration decreases withfurther increasing divergence of the plume, so that by 180° much ofthe tephra is deposited outside the watershed. The general increase inconcentration of the “divergence” simulation results (Fig. 4B) is partlyan artifact of the selected “base case” (Fig. 1), which deposits tephraover an area where remobilization will be relatively less thandeposition in other directions that have steeper terrain or more activedraining channels.

These studies indicate that the geometry and terrain of awatershed play a significant role in the outcome of the coupledtephra dispersal–redistribution model system. In order to investigatethe control of watershed geometry and terrain alone, we rotated the

base-case tephra plume over the landscape in 30° increments fromnorth to south and then routed the resulting tephra sheets throughthe FAR model. This 180° band captures 95% of the wind directions forthe 3 to 13 km altitude (SNL, 2007a). The effect of the simple change inazimuth for the base-case plume (constant-column height, constant-wind conditions) is greater than the effects from the other, morecomplex variations (Fig. 4C). As the plume is rotated from east tosouth, the normalized tephra concentration at the watershed outletdecreases as the tephra deposits fall increasingly on gentle slopes oroutside the watershed. As was seen in the spread and divergence cases,the increase in azimuth north of east produces initial increases intransported tephra concentration as more of the tephra mass isdeposited in the center of the watershed. With a due north azimuth,however, much of the tephra mass is deposited outside the watershedor experiences increased dilution due to its long travel times in thechannels. This trend is also evident in the results of twenty stochasticASHPLUME-FAR realizations in which the effect of terrain andwatershed geometry dominate over other eruption and redistributionvariables (Pelletier et al., 2008).

5. Discussion and conclusions

Unsteadyeruption columnheight simulated in a series of ASHPLUMEmodel runs produced only slight change in the distribution of tephradeposited in the Fortymile Wash watershed. The aspect ratio for thecomposite tephra sheet changed by ±20% vs. the simple diffusive plumebase case.When the composite tephramass from these variable columnheight eruptions was routed through the FAR sediment-transportmodel, the concentration of tephra in sediment at the outlet of FortymileWash varied by ±15% or less compared to the base case. The effect ofwind variation during an eruption (spread or divergence of the eruptiveplume) produced a maximum increase in sediment tephra concentra-tion 2.24 times that of the base case. The variation in amount of tephramobilized off steep slopes and through active channels varied non-monotonically with azimuth in these studies, suggesting that factorsother than unsteady eruption or variable wind conditions werecontrolling tephra remobilization efficiency.

The controlling effect of watershed geometry and terrain wasidentified in an exercise in which the simple single-eruption, con-stant-wind tephra sheet was rotated in 30° increments within an 180°azimuthal band downwind of the vent area. Notably, the span of tephraconcentrations in sediment at the Fortymilewatershed outlet calculatedfor the simple azimuthal rotation case (0.2 to 2.6 times the base case)encompasses all variation resulting from the more complex eruptivecases involving variation in column height and wind conditions duringan eruption (Table 1, Fig. 4).

Is this variation in tephra sediment concentration at the watershedoutlet significant compared to other uncertainties in volcanic riskpredictions for the Yucca Mountain repository? The overall tephraconcentration varied by less than a factor of three in these studies. Incontrast, the uncertainty is much higher in other risk parameters usedto develop the inputs to the ASHPLUMEmodel; for instance, the rangeof likely values for eruptive conduit size (which directly determinesthe quantity of radioactive waste entrained into the eruption),eruptive volume, and eruption duration vary over one to two ordersof magnitude (SNL, 2007b). Monte Carlo modeling techniques arenecessary to capture the range in uncertainty of model inputs derivedfrom these data. Therefore, from a tephra dispersal and depositionmodeling perspective, variations in tephra concentration at theFortymile Wash outlet due to unsteady eruption or variable windsduring an eruption are not significant in the context of the uncertaintyin other eruption parameters.

Despite its relative simplicity compared to recently developedtephra dispersal models, we consider the ASHPLUME model to beadequate for its intended purposewithin the YuccaMountain Project'sTSPA risk assessment framework. The analysis of risk due to tephra

Fig. 4. Summary plot of tephra concentration variation in sediment at the outlet ofFortymile Wash due to variation in wind direction during eruption that produces A)spread in the tephra sheet, B) divergence of tephra sheet lobes, and C) rotation of theplume through the range of likely downwind direction. Variation due to unsteadycolumn height denoted by (+). Concentrations are normalized to the base-case value of asimple plume directed due east (Fig. 3).


remobilization is highly dependent on the nature of the specificgeometry and terrain in the study area, and these effects outweighsensitivities to eruption parameters in the coupled-model riskassessment. Because the combination of tephra dispersal/depositionand redistribution models used here shows limited sensitivity to theactual details of ash fall, a simple fall model suffices to estimate tephramass delivered to the RMEI. In general, the effect of sensitivities orlimitations of a given process model may be moderated by othercoupled models. Similar assessments can aid in the prioritization ofmodel development and/or the selection of models for volcanic riskassessments elsewhere.

Acknowledgments

This work was funded by U.S. Department of Energy's YuccaMountain Project, which is managed by Bechtel SAIC, LLC and SandiaNational Laboratories. We thank Mike Cline for his support of thiswork. The comments of Jean Younker, Shane Cronin and threeanonymous reviewers strengthened the text.

Appendix

This appendix provides background on the ASHPLUME computa-tional code and the underlying mathematical model of tephradispersion and deposition. This information has been previouslypublished in part— the mathematical model in Suzuki (1983) and theASHPLUME code description in Jarzemba (1997) and Jarzemba et al.(1997). The following discussion also includes specifics of theapplication of the ASHPLUME code to the Yucca Mountain TSPA(and the study described in the main body of the paper) as describedin SNL (2007a).

The ASHPLUME code implements the conceptual and mathema-tical model of Suzuki (1983) for estimation of the areal density oftephra deposits on the surface of the Earth following a violentStrombolian-type volcanic eruption. The computer code, developed byJarzemba et al. (1997), includes estimation of the areal density (g/cm2)on the Earth's surface of tephra particles (and, separately, the spentnuclear fuel particles transported with the tephra) due to an eruptionthat intersects the repository at Yucca Mountain. Areal densities canbe converted to deposit thickness by dividing the areal density by thevalue of settled (deposit) density: 0.3–1.5 g/cm3 (Sparks et al., 1997),approximated by 1.0 g/cm3 (SNL, 2007b). Note that the terms nuclear“fuel” and “waste” are used interchangeably in this discussion.

The movement of air mass in the atmosphere is relatively randomwithin the scale of eddy motions in wind currents (Suzuki 1983).Therefore, the dispersion of the ash-waste particles in the atmosphereis treated as having a random (diffusion-like) component with anadvection component representing the mean horizontal wind.Particles disperse in the atmosphere in both vertical and horizontaldirections; however, the scale of horizontal turbulence ismuch greaterthan the scale of vertical turbulence. Therefore, in Suzuki's develop-ment of the mathematical model, particle diffusion is considered to betwo-dimensional in the horizontal x–y plane. Particlemovement in thevertical (z) direction is accounted for by settling velocity.

The underlying two-dimensional partial differential equationrelating the change in concentration, ∂ξ, at a point x–y (with xdownwind) to wind velocity, u, and an eddy diffusivity constant, K,follows Suzuki (1983, Eq. 1):

∂�∂t

¼ −u∂�∂x

þ ∂∂x

K∂�∂x

� �þ ∂∂y

K∂�∂y

� �: ðEq: A:1Þ

By selecting an appropriate value for the diffusivity constant, K,Eq. (A.1) is appropriate for estimating the two-dimensional diffusionof particulate matter in the atmosphere downwind from a source.Because the x direction is assumed to be aligned with the wind, the ycomponent of the advective term in Eq. (A.1) is zero.

To derive a solution to Eq. (A.1) suitable for application tocalculation of tephra dispersion in the atmosphere after a volcaniceruption, Suzuki (1983) used the following boundary and initialconditions (summarized by Jarzemba 1997): 1) the erupted material(the source) consists of a finitemass of volcanic tephra particles; 2) thesource of tephra particles is described by the distribution of thediameter of the released particles, and the distribution has a singlemode; 3) tephra particles have a probability to diffuse out of theeruption column during upward travel in the column as well as duringtransport of the plume downwind; and 4) all particles fall at theterminal velocity and accumulate on the ground.

The solution to the mathematical model described in Eq. (A.1) isprovided by Suzuki (1983), modified by Jarzemba et al. (1997), and canbe summarized by the following equation that describes the arealdensity of erupted ash deposited on the Earth's surface:

X x; yð Þ ¼Z ρmax

ρ¼ρmin

Z H

z¼0

5Qp zð Þf ρð Þ8πC t þ tsð Þ5=2

exp−5 x−utð Þ2þy2

n o8C t þ tsð Þ5=2

24

35dzdρ

ðEq: A:2Þwhere X(x, y) is the mass of ash per unit area (g/cm2) accumulated atlocation (x, y); ρ is the common logarithm of particle diameter d (cm);z is the vertical distance (km) of particle from ground surface; H is theheight (km) of eruption column above vent; x and y are the coordinateson the surface of the Earth (cm) oriented parallel and perpendicular,respectively, to the prevailing wind; Q is the total quantity (g) oferuptedmaterial; p(z) is the distribution function for particle diffusionout of the column within ±dz of height z; f(ρ) is the distributionfunction for log-diameter of particles within ±dρ of ρ normalized perunit mass; C is the constant (cm2/s5/2) relating eddy diffusivity andparticle fall time; t is the particle fall time (s); ts is the particle diffusiontime (s) in the eruption column; and u is the wind speed (cm/s).

The probability density distribution function for particle diffusionout of the eruption column p(z) is given by Suzuki (1983, Eq. 7) withparameters modified by Jarzemba et al. (1997):

p zð Þ ¼ βW0Ye−Y

V0H 1− 1þ Y0ð Þe−Y0f g ðEq: A:3Þ

where Y ¼ β W zð Þ−V0ð ÞV0

; Y0 ¼ β W0−V0ð ÞV0

; β is a dimensionless constantcontrolling diffusion of particles in the eruption column; W0 is theinitial particle rise velocity (cm/s) of the convective part of theeruptive column; V0 is the particle terminal velocity (cm/s) at meansea level; and W zð Þ ¼ W0 1− z

H

� �is the particle velocity (cm/s) as a

function of height.The particle terminal velocity at mean sea level is given by (Suzuki

1983, Eq. 4′; modified by Jarzemba et al., 1997, Eqs. 2–3):

V0 ¼ ψpgcd2

9ηaF−0:32 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi81η2aF−0:64 þ 3

2ψpψagcd3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:07−F

pq ðEq: A:4Þ

where ψa, ψp represent the density (g/cm3) of air and of particles,respectively; gc is the gravitational acceleration constant (here:980 cm/s2); ηa is the dynamic viscosity of air (g/(cm s)); F is theshape factor for particles (for an elliptically shaped particle withprincipal axes a, b, and c, F=(b+c)/2a, where a is the longest axis);and d is the mean particle diameter (cm).

Jarzemba et al. (1997) define particle density, ψp (g/cm3), to beroughly an inverse function of the particle log-diameter, ρa (cm), toaccount for variations in density due to vesicularity as follows:

Ψp ¼ Ψhighp for ρa < ρlow

a

Ψp ¼ Ψlowp þ Ψhigh

p −Ψlowp

� �ρhigha −ρa

� �= ρhigh

a −ρlowa

� �for ρlow

a < ρa < ρhigha

Ψp ¼ Ψlowp for ρaNρ

higha

ðEq: A:5Þ


where ψphigh, ψp

low, ρahigh, and ρa

low are defined by user inputs(Table 2).

The particle fall time (s) is given by (Suzuki 1983):

t ¼ 0:752� 106 1−e−0:0625z

V0

0:926ðEq: A:6Þ

Suzuki assumed that the eruption column radius is equal to 0.5z(z is height in km). The particle diffusion time in the eruption column,ts, is given by:

ts ¼ 5z2

288C

� �2=5

ðEq: A:6aÞ

where C is the constant relating eddy diffusivity and particle fall time.The height of the eruption column or plume, H, used in Eq. (A.2),

follows buoyant plume theory applied to volcanic eruptions byWilson et al. (1978) and discussed by Jarzemba et al. (1997). InASHPLUME, height in km is given as:

H ¼ 0:0082P0:25 ðEq: A:7aÞ

where the eruption column power, P, in watts, is determined by theeruption mass flux and heat content:

P ¼:Q CpΔTE� � ðEq: A:7bÞ

The parameters in parentheses in Eq. (A.7b) represent the heatcontent and its efficiency in adding buoyancy, fixed by magma andtephra characteristics. The mass flux Q̇, can be evaluated by assuminga constant eruptive mass flux over the duration of the eruption, whichis related to the erupted-ash settled volume by Eq. (A.7c). In thatequation, the transformation, for purposes of power calculation,neglects the smaller mass and heat contribution from gas.

:Q ¼ Q

Td¼ Vψs

Td¼ ψmW0π

dc2

� �2

ðEq: A:7cÞ

Where (for Eqs. (A.7a)–(A.7c)) Cp is the heat capacity of magma(J/kgK); ΔT is the temperature difference (°C) between magma and

ambient; E is the efficiency factor of heat usage; Q is the total mass(kg) of erupted material; V is the volume (m3) of erupted ash; Td isthe eruption duration (s); ψs is the density (kg/m3) of settled ash(deposit density); ψm is the bulk density (kg/m3) of erupting magmaand gas mixture; W0 is the initial particle rise velocity (m/s); and dcis the effective conduit (vent) diameter (m). The units listed are forEqs. (A.7a)–(A.7c) only. The ASHPLUME input parameters of initial risevelocity, power, and duration are linked in Eqs. (A.7b) and (A.7c) anddetermine the plume height in Eq. (A.7a); initial rise velocity alsocontributes to the probability density distribution function(Eq. (A.3)). The value for the efficiency factor (E) is assumed to beequal to 1 in this analysis, given the uncertainties in values for Cpand ΔT. As already noted, the calculation neglects the mass andthermal content of gas in the plume. In the ASHPLUME code, thetotal mass of erupted material, Q, is calculated from input values forpower, P, and eruption duration, Td.

In the Suzuki mathematical model, the volcanic ash size isdistributed log-normally:

f ρað Þ ¼ 1ffiffiffiffiffiffi2π

pσd

exp −ρa−ρa

mean

� �22σ2

d

" #ðEq: A:8Þ

where f(ρa) is the probability distribution for log diameter of ash(tephra); ρa is the log-diameter of ash particle size (cm); ρmean

a is themean of log-diameter of ash particle size (cm); and σd is the standarddeviation of log particle size.

The TSPA analyses for Yucca Mountain require an estimate of spentfuel per unit area on the ground surface as a function of locationrelative to the volcanic vent after a hypothetical eruption through theproposed waste repository. It is assumed that the transport mechan-ism for fuel particles is by combination with tephra particles byrelative size according to an incorporation ratio. Note that theASHPLUME code can be used to simulate the transport of tephraalone, without the addition of spent fuel.

Fuel mass is defined by Jarzemba et al. (1997) as following a log-triangular distribution function of the log-diameter of fuel particles(specifically, a log-triangular distribution for fuel mass within ±dρf ofρf normalized per unit mass):

m ρf� � ¼ k1 ρf −ρf

min

� �for ρf

min < ρf ≤ ρfmode

¼ k1 ρfmode−ρ

fmin

� �−k2 ρf −ρf

mode

� �f or ρf

mode < ρf ≤ ρfmode

¼ 0 otherwise;ðEq: A:9Þ

where m(ρf) is the log-triangular distribution of fuel particlesize; ρf is the log-diameter of fuel particle size (cm); k1 ¼

2ρfmax−ρ

fminð Þ ρf

mode−ρfminð Þ ; k2 ¼ 2

ρfmax−ρ

fminð Þ ρf

max−ρfmodeð Þ; ρmin

f the minimum log-

diameter of fuel particle size (cm); ρmaxf is the maximum log-

diameter of fuel particle size; and ρmodef is the mode log-diameter of

fuel particle size.Jarzemba et al. (1997) determined the fuel fraction (ratio of fuel

mass to tephra mass) as a function of ρa by considering that all fuelparticles of size smaller than (ρa−ρc) have the ability to be incorporatedsimultaneously into volcanic tephra particles of sizeρa or larger. The fuelfractionas a functionofρa is determinedby summingall the incrementalcontributions of fuel mass to the volcanic tephra mass from fuel sizessmaller than (ρa−ρc). An expression for the fuel fraction is given as:

FF ρað Þ ¼ UQ

�Z ρ¼ρa

ρ¼−∞

m ρ−ρcð Þ1−F ρð Þ dρ ðEq: A:10Þ

where Q and U are the total mass (g) of tephra and fuel, respectively,ejected in the eruption; m is the probability density function offuel particle size; ρc is the waste incorporation ratio (defined below);and F(ρa) is the cumulative distribution of f(ρa). Eq. (A.10) assumes the

Table 2Summary of ASHPLUME input parameter values used in this study

ASHPLUME parameter(units)

Valuea ASHPLUME parameter(units)

Valuea

Mean ash particle diameter,d (cm)

0.0100 Minimum waste particlesize, ρfmin (cm)

0.0001

Ash particle diameter standarddeviation, σd (log(cm))

0.602 Mode waste particlesize, ρfmode (cm)

0.0013

Minimum ash particle density,ψplow(g/cm3)

1.04 Maximum waste particlesize, ρfmax (cm)

0.2000

Maximum ash particle density,ψphigh (g/cm3)

2.08 Column diffusionconstant, β

0.3

Ash particle size at maximumash density, ρalow (log(cm))

−3 Waste incorporationratio, ρc

0.0

Ash particle size at maximumash density, ρahigh (log(cm))

0 Mass of waste totransport, U (g)

4.01×107

Ash particle shape factor, F 0.5 Initial rise velocity,W0 (cm/s)

10.1540b

Air density, ψa (g/cm3) 0.001134 Eruptive power, P (W) 2.715×1011b

Air viscosity, ηa (g/cm s) 0.000185 Eruption duration, Td (s) 1.3048×105b

Eddy diffusivity constant,C (cm2/s5/2)

400.0

Wind speed (cm/s) and direction (°N of due east) were derived from NOAA (2004) dataat Desert Rock airfield; the data were divided into 1 km altitude bins and mean valueswere used for this study, chosen from the altitude bin corresponding to the calculatedcolumn height for a given model run.Notes:

a Constant values for all model runs except as indicated.b Example value for run W1; parameter value varies among model runs.


resulting contaminated particles have the same size distribution asthe original volcanic tephra particles. This assumption seems reason-able because the total mass of volcanic tephra erupted will be muchgreater than the total mass of fuel available for incorporation.Introduction of a relatively small amount of fuel mass into the tephramass is unlikely to alter the size distribution of the tephra. TheASHPLUME mathematical and computational models do, however,adjust the density of tephra particles to account for the incorporationof fuel. The particle density used in the calculation of the terminalvelocity of a particle is adjusted as a combined particle in thedispersion calculation as ashden=ashden×[1+fuel fraction]. In thisstatement, “ashden” represents the ash (tephra) particle density and“fuel fraction” represents the mass fraction of fuel in the combinedparticle. The integrand of Eq. (A.2) is multiplied by FF(ρa) and thenrecalculated to find the spent fuel density at the (x, y) location.

The incorporation ratio describes the ratio of tephra/waste particlesizes that can be combined for transport. The waste incorporationmodel included in the ASHPLUME code, involves a conceptual modelof the combination of two particle streams. The particles of waste are“instantaneously” homogenized in tephra particles, and the control-ling parameter is related to the relative sizes of the waste and tephraparticle populations:

ρc ¼ log10damin

df

ðEq: A:11Þ

where ρc is the waste incorporation ratio; dmina is the minimum tephra

particle size (cm) needed for incorporation; and df is the fuel particlesize (cm).

Magma and spent fuel conceptually combine in the subsurface(repository level) as a well-mixed suspension before ascending theconduit to the vent, and the waste is already incorporated into themelt prior to magma fragmentation. In this case, Eq. (A.11) is used byprescribing a neutral value for ρc, that would simply allow the naturalsize distributions of tephra (produced in the fragmentation process)and waste particles (retained from the original release of waste fromthe waste packages) to prevail in the tephra-waste mixture. Thus thewaste particles can be treated as refractory xenoliths in the melt,which, upon magma fragmentation, will reappear in the tephra asmixed particles of waste and silicate melt (glass). The appropriate“neutral” value for ρc is one that simply allows the combination of theoriginal particle size distributions of waste and tephra. In this case, avalue of ρc=0 is used in Eq. (A.10) (that is, the ratio of tephra andwasteparticle sizes is 1:1).

Representative values for ASHPLUME input parameters used in thisstudy are listed in Table 2.

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evaluating suitability of a tephra dispersal model as part

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