PLUME-MoM 1.0: A new integral model of volcanic plumes based on the method of moments

Transcription

1 Geosci. Moel Dev., 8, , 5 oi:.594/gm Authors) 5. CC Attribution. License. PLUME-MoM.: A new integral moel of volcanic plumes base on the metho of moments M. e Michieli Vitturi, A. Neri, an S. Barsotti Istituto Nazionale i Geofisica e Vulcanologia, Pisa, Italy Icelanic Meteorological Office, Reykavík, Icelan Corresponence to: M. e Michieli Vitturi mattia.emichielivitturi@ingv.it) Receive: 7 March 5 Publishe in Geosci. Moel Dev. Discuss.: 5 May 5 Revise: 4 July 5 Accepte: 8 July 5 Publishe: 6 August 5 Abstract. In this paper a new integral mathematical moel for volcanic plumes, name PLUME-MoM, is presente. The moel escribes the steay-state ynamics of a plume in a -D coorinate system, accounting for continuous variability in particle size istribution of the pyroclastic mixture eecte at the vent. Volcanic plumes are compose of pyroclastic particles of many ifferent sizes ranging from a few microns up to several centimeters an more. A proper escription of such a multi-particle nature is crucial when quantifying changes in grain-size istribution along the plume an, therefore, for better characterization of source conitions of ash ispersal moels. The new moel is base on the metho of moments, which allows for a escription of the pyroclastic mixture ynamics not only in the spatial omain but also in the space of parameters of the continuous size istribution of the particles. This is achieve by formulation of funamental transport equations for the multi-particle mixture with respect to the ifferent moments of the grainsize istribution. Different formulations, in terms of the istribution of the particle number, as well as of the mass istribution expresse in terms of the Krumbein log scale, are also erive. Comparison between the new moments-base formulation an the classical approach, base on the iscretization of the mixture in N iscrete phases, shows that the new moel allows for the same results to be obtaine with a significantly lower computational cost particularly when a large number of iscrete phases is aopte). Application of the new moel, couple with uncertainty quantification an global sensitivity analyses, enables the investigation of the response of four key output variables mean an stanar eviation of the grain-size istribution at the top of the plume, plume height an amount of mass lost by the plume uring the ascent) to changes in the main input parameters mean an stanar eviation) characterizing the pyroclastic mixture at the base of the plume. Results show that, for the range of parameters investigate an without consiering interparticle processes such as aggregation or comminution, the grain-size istribution at the top of the plume is remarkably similar to that at the base an that the plume height is only weakly affecte by the parameters of the grain istribution. The aopte approach can be potentially extene to the consieration of key particle particle effects occurring in the plume incluing particle aggregation an fragmentation. Introuction In the past ecaes, numerical simulation of volcanic eruptions has greatly avance an moels are now often able to eal with the multi-phase nature of volcanic flows. This is the case, for example, with moels escribing the ynamics of pyroclastic particles in a volcanic plume, or that of bubbles an crystals isperse in the magma rising in a volcanic conuit. Despite this, in numerical moels, the polyispersity associate with the multi-phase nature of volcanic flows is often ignore or largely simplifie Valentine an Wohletz, 989; Neri at al., ; Dartevelle, 4; Dufek an Bergantz, 7; Esposti Ongaro et al., 7; e Michieli Vitturi et al., ). For instance, in most of the existing conuit moels, crystals an bubbles are treate as simple flow components an escribe by volume fractions only, while in plume ynamics an ash ispersal moels the grain-size istribution GSD) of pyroclasts is iscretize in a finite number of classes i.e., phases). Both approaches make proper Publishe by Copernicus Publications on behalf of the European Geosciences Union.

2 448 M. e Michieli Vitturi et al.: PLUME-MoM treatment of the continuous variability of the imension of pyroclastic particles an gas bubbles ifficult. Literature results Llewellin et al., ; Pal, ; Costa et al., ) clearly show that this variability can largely affect relevant physical/chemical processes that occur uring the transport of the isperse phase such as, for example, the nucleation an growth of bubbles an the coalescence/breakage of bubbles an crystals in the conuit or the aggregation of pyroclastic particles in a volcanic plume. A theoretical framework an the corresponing computational moels, namely, the metho of moments for isperse multi-phase flows, have been evelope in the past ecaes, mostly in the chemical engineering community Hulburt an Katz, 964; Marchisio et al., ), to track the evolution of these systems not only in the physical space but also in the space of properties of the isperse phase calle internal coorinates). Accoring to this metho, a population balance equation is formulate as a continuity statement written in terms of a ensity function. From the ensity function some integral quantities of interest namely, the moments, i.e., specific quantitative measures of the shape of the ensity function) are then erive an their transport equations are formulate. In this work we present an extension of the Eulerian steay-state volcanic plume moel presente in Barsotti et al. 8) erive from Bursik, ) obtaine by aopting the metho of moments. In contrast to the original works where pyroclastic particles are partitione into a finite number of classes with ifferent sizes an properties, the new moel is able to consier a continuous size istribution function of pyroclasts, f D), representing the number or the mass fraction of particles per unit volume) with iameter between D an D + D. Accoringly, conservation equations of the plume are expresse in terms of the transport equations for the moments of the ash particle size istribution. In particular, in the following we present the new multi-phase moel formulation base on the implementation of the quarature metho of moments McGraw, 6) an we investigate the sensitivity of the moel to uncertain or variable input parameters such as those escribing the grain-size istribution of the mixture. To quantify an incorporate this epistemic uncertainty affecting the input parameters characterizing lack-of-knowlege) into our application of the moel, we teste two ifferent approaches, a moification of the Monte Carlo metho base on Latin hypercube sampling LHS) an a stochastic approach, namely, the generalize polynomial chaos expansion gpce) metho. This paper is organize as follows: in Sect. we present the metho of moments applie to two ifferent escriptions of particle istribution. In Sect. the equations of the moel for the two formulations are escribe. Section 4 is evote to the numerical iscretization of the moel an the numerical implementation of the metho of moments. Section 5 presents the application of the moel to three test cases with a comparison of the moel results for ifferent formulations of the plume moel, an finally an uncertainty quantification an a sensitivity analysis are applie to moel results. Metho of moments. Moments of the size istribution In contrast to previous works, where the soli particles are partitione in a finite number of classes with ifferent sizes Barsotti et al., 8), we introuce here a continuous size istribution function representing the number or mass) concentration of particles per unit volume) as a function of particle iameter. In general, this particle size istribution PSD) is a function of time t, of the spatial coorinate an of the iameter of the particles. First, we present the metho of moments for a particle size istribution f D), representing the number concentration of particles particles per unit volume) with iameter between D an D + D, where D is expresse in meters. When more than one family of particles are present, for example lithics an pumice, we will use the subscript to istinguish among them. Consequently, the function f D) will enote the number concentration of particles of the th family. Given a particle size istribution f D), we observe that its shape can be quantifie through the moments M i) Hazewinkel, ), efine by M i) = D i f D)D. ) The particular efinition of f D) we aopt, expressing the number concentration of particles of size D, allows for the following physical interpretation of the first four moments: M ) is the total number of particles of the th family per unit volume). M ) is the sum of the particle s iameters of the th family per unit volume). M ) is the total surface area of particles of the th family per unit volume). is the total volume of particles of the th family per unit volume) or the local volume fraction of the th isperse phase, also enote with α s,. The multiplying factor π 6 is obtaine assuming spherical particles. For particles with ifferent shape, if volume scales with the thir power of length, we can still relate the particle volume V with the particle length D through a volumetric shape factor k v such as V = k v L. π 6 M) We also note that the central moments i.e., those taken about the mean) can be expresse as function of the raw Geosci. Moel Dev., 8, , 5

3 M. e Michieli Vitturi et al.: PLUME-MoM 449 moments i.e., those taken about zero as in Eq. ), an in this way it is possible to relate the moments of the istribution with the mean, variance, skewness an kurtosis. Furthermore, a mean particle size can be efine as the ratio of the moments M i+) /M i) for any value of i. For example, the Sauter mean iameter efine as the ratio between the mean volume an the mean surface area) is obtaine by setting i =, giving L, = M ) /M ). Similarly, it is possible to efine the mean particle length average with respect to particle number ensity L, = M ) /M ), i.e., the sum of the lengths of particles per unit volume) ivie by the number of particles per unit volume), an the mean particle length average with respect to particle volume fraction L,4 = M 4) /M ). The motivation for the introuction of the moments is to minimize computational costs by avoiing the iscretization of the size istribution in several classes, an nevertheless to capture the polyispersity of the flow through the correct escription of the evolution of the moments Carneiro, ). The moments approach also allows one to treat interparticle processes such as particle aggregation an fragmentation that strongly epen on an affect the GSD of the mixture Marchisio et al., ). The moments an the corresponing transport velocities appear naturally in the mathematical formulation as a irect consequence of the integration of the Eulerian particle equations over the iameter spectrum, as will be shown in the next section.. Moments of other quantities In the plume moel, several quantities characteristic of the particles, such as settling velocity, ensity an specific heat capacity, are also efine as functions of the particle iameter, an thus we can efine their moments as was carrie out for the particle size istribution f D). In general, for a quantity ψ that is a function of the iameter D, we efine its moments as ψ i) = M i) ψ D)D i f D)D. ) As a first example, we consier here the moments of particle ensity ρ s. In particular, following Bonaonna an Phillips ), ensity of lithics is assume to be constant, whereas ensity of pumices ρ s,pum D) with iameter D < D here equal to mm) is assume to ecrease an to reach the lithic ensity value when the fragment iameter ecreases below D here equal to 8 µm). Substituting the expression for the particle ensity of the th particle family in Eq. ), we obtain the moments of the ensity as ρ i) s, = M i) ρ s, D)D i f D)D. ) We remark that moments of ifferent orer are generally ifferent, they will only be equal ρ l) s, = ρm) s,, l = m) in two limiting cases: for a monoisperse istribution with iameter D an ensity ρs, i.e., f D) = δd D ) where δ is the Dirac-elta function) an ρ s, D ) = ρs ; or if all particles have the same ensity, i.e., ρ s, D) = ρs,, D. In both cases, ρ i) s, = ρ s,, i. Otherwise, there is no reason, e.g., for ρ) s, an ρ ) s, to be the same, as they represent length- an volumeweighte ensity averages, respectively. For our application, we are intereste mostly in the volumetric-average ensity ρ ) s,, i.e., the average mass per unit volume of particles from now on enote with ρ s,. The moments efine by Eq. ) can also be use to efine other properties of the gas-particles mixture. For example, it follows from the efinition of the moments that if we have a mixture of a gas with ensity ρ g an a family of polyisperse istributions of particles with ensity ρ s, = ρ s, D), the mixture ensity is given by ρ mix = α s, ρ s, + α s, )ρ g = π 6 M) ρ ) s, + π 6 M) )ρ g, 4) an consequently the mass fraction of the th soli phase with respect to the gas-particles mixture is given by x s, = α s, ρ s, ρ mix = π 6 M ) ρ ) s, π 6 M ) ρ ) s, + π 6 M ) )ρ g. 5) We also remark here that the gas phase is a mixture of atmospheric air, entraine in the plume uring the rise in the atmosphere, an a volcanic gas component, generally water vapor. In the following, we will use the subscript atm to enote the atmospheric air an wv for the volcanic water vapor. In contrast to the approach use in Barsotti et al. 8), where a constant settling velocity for each class is provie by the user, here several moels have been implemente in the coe Pfeiffer et al., 5; Textor et al., 6a, b). For the application presente in this work, the settling velocity is efine as a function of the particle iameter an ensity as in Textor et al. 6a): ) D ρatm k ρ s, D) D µm ρ atm ) D ρ w s, D) = atm k ρ s, D) < D µm ρ atm D ρs, D) ρatm k D > µm, C D ρ atm 6) where k =.9 5 m kg s, k = 8 m kg s an k = 4.8 m kg / s. The rag coefficient C D is a parameter accounting for the particle surface roughness, an Geosci. Moel Dev., 8, , 5

4 45 M. e Michieli Vitturi et al.: PLUME-MoM for this work we use a value of.75 as in Carey an Sparks 986). As carrie out for particle ensity, it is possible to evaluate the moments w i) s, of the settling velocity w s, D), efine as w i) s, = M i) w s, D)D i f D)D 7) an representing weighte integrals of the settling velocity over the size spectrum. Again, moments of ifferent orer are generally ifferent. There is no reason, e.g., for w ) w ) s, s, an to be the same, as they represent surface an volumeweighte averages, respectively. Finally, it is possible to efine the moments C i) s, of the particles specific heat capacity C s, : C i) s, = M i) C s, D)D i f D)D. 8) We observe that for the specific heat capacity, we are generally not intereste in a volumetric average but in the mass average, enote here with the notation C s, an given by the following expression: C s, = C s, D) ρ s, D)D ρ s, M ) f D)D = ρ s, [ Cs, ρ s, ] ). 9). Mass fraction istribution While in chemical engineering, where the metho of moments is commonly use, the particle number istribution f D) is generally preferre to escribe the polyispersity of the particles; in volcanology it is more common to use a mass fraction istribution γ φ), efine as a function of the Krumbein phi φ) scale: φ = log D D, where D is the iameter of the particle expresse in meters, an D is a reference iameter, equal to mm to make the equation imensionally consistent). In this case, the istribution γ φ) represents the mass fraction of particles mass per unit mass of the gas-particles mixture) of the th family with iameter between φ an φ + φ. Again, the shape of the istribution γ φ) can be characterize by its moments i, efine by i) = φ i γ φ)φ. ) Geosci. Moel Dev., 8, , 5 Also in this case the particular efinition of γ φ) allows for a physical interpretation of the moments; for example, the moment ) is the mass fraction of the th soli phase x s, with respect to the gas-particles mixture. As carrie out for particle number istribution, it is possible to efine a mean particle size in terms of the moments of the mass fraction istribution as i+) / i) ; this ratio, for i =, gives the massaverage iameter, corresponing to the volume-average iameter L,4 = M 4) /M ) when the ensity ρ s, φ) is constant. Again, it is possible to efine the moments of other quantities ψ φ) in terms of the continuous istribution of mass fraction γ φ) as ψ i) = i) ψ φ)φ i γ φ)φ. ) For example, when the mass fraction istribution γ φ) is use, the mass-average heat capacity C s, is given by the following expression: C s, = x s, C s, φ)γ s, φ)φ = C ) s, ) an the volumetric-average ensity, i.e., the mass of particles per unit volume, can be evaluate from ρ s, = x s, Plume moel γ s, φ) ρ s, φ) φ = [ ρ s, ] ). ) In this section we escribe the assumption an the equations of the moel. As in Bursik ), the moel assumes an homogeneous mixture of particles an gases with thermal an mechanical equilibrium between all phases. Aggregation an breakage effects are not consiere an consequently ensity oes not change with time. Finally, the moel oes not consier effects of humiity an water phase changes. The equation set for the plume rise moel is solve in a -D coorinate system s, η, θ) by consiering the bulk properties of the eruptive mixture see Fig. ). The plume is assume with a circular section in the plane normal to the centerline traectory with curvilinear coorinate s, a top-hat profile of the velocity along the centerline, an inclination on the groun efine by an angle η between the axial irection an the horizon, an an angle θ in the horizontal plane x,y) with respect to the x axis. These angles are neee to escribe the evolution of weak explosive eruptions that are strongly affecte by atmospheric conitions. Following Bursik et al. 99) an Ernst et al. 996), the conservation of flux of particles with size D of the th family

5 M. e Michieli Vitturi et al.: PLUME-MoM 45 R are taken into account partitioning the size spectrum in a finite number N of soli classes, the set of Eq. 6) replaces the N mass conservation equations for the N particulate classes. From Eq. 4), if we multiply both the terms by the mass of the particles of size D, given by π 6 D ρ s, D), we obtain the aitional equation 9 8 θ U sc η Air Entrainment f D) π ) 6 D ρ s, D)πr U sc = πrpw s, D)f D) π 6 D ρ s, D) 7) z km) y km) 4 Particles Loss x km) Figure. Schematic representation of the Eulerian plume moel. The ashe black line represent the axis of the curvilinear coorinate s. is given by ) f D)πr U sc = πrpw s, D),f D) 4) where r is characteristic plume raius, U sc represents the velocity of the plume cross section along its centerline a tophat profile is assume) an p is the probability that an iniviual particle will fall out of the plume, efine as a function of an entrainment coefficient α as ) α p = ). 5) α + Equation 4) states that the number of particles of the th family with size D lost from the plume is proportional to the number of particles at the plume margin, given by f D) πr, to the settling velocity w s, D) an to the probability factor p. Now, multiplying both the sies of Eq. 4) for D i an then integrating over the size spectrum [, ], we obtain the following conservation equations for the moments M i) : M i) U scr ) = rpw i) s, Mi) ) If we compare our formulation with that presente in Barsotti et al. 8), where the effects of a polyisperse soli phase an, integrating over the size spectrum, U sc r π ) 6 M) ρ ) s, = rp π 6 M)[ ] ), w s, ρ s, 8) where on the left-han sie the term π 6 M) s, represents the volume average bulk ensity of the particles of the th family i.e., the mass of particles of the th family per unit volume of gas-particles mixture, enote with the superscript B, ρs, B ), while on the right-han sie the term [ ] ) w s, ρ s, represents the mass-average settling velocity of the particles of the th family multiplie by the volume-average particle ensity. Equation 8) is the mass conservation equation for the th family of particles, relating the variation of the mass flux of particles within the plume with the loss at the plume margin. Now, following the same proceure, we reformulate the other conservation equations escribing the steay-state ascent of the plume in terms of the moments of the continuous istributions of sizes, ensities an settling velocities instea of the averages over a finite number of classes of particles with ifferent size. First of all, we erive the conservation equation for the mixture mass. As in the plume theory, we assume that the entrainment, ue to both turbulence in the rising buoyant et an to the crosswin fiel, is parameterize through the use of two entrainment coefficients, α ɛ an γ ɛ. The theory assumes that the efficiency of mixing with ambient air is proportional to the prouct of a reference velocity the vertical plume velocity in one case an the win fiel component along the plume centerline in the other), by α ɛ an γ ɛ Morton, 959; Briggs, 975; Wright, 984; Weil, 988). Thus, following Hewett et al. 97) an Bursik ), we efine the entrainment velocity U ɛ as a function of win spee, U atm, as well as axial plume spee, U sc : ρ ) U ɛ = α ɛ U sc U atm cosφ + γ ɛ U atm sinφ, 9) where α ɛ U sc U atm cosφ is entrainment by raial inflow minus the amount swept tangentially along the plume margin by the win, an γ ɛ U atm sinφ is entrainment from win. With this notation, the total mass conservation equawww.geosci-moel-ev.net/8/447/5/ Geosci. Moel Dev., 8, , 5

6 45 M. e Michieli Vitturi et al.: PLUME-MoM tion solve by the moel becomes ρ mix U sc r ) = rρ atm U ɛ rp π [ ] ), 6 M) ws, ρ s, ) stating that the variation of mass flux left-han-sie term) is ue to air entrainment first right-han-sie term) an loss of soli particles secon right-han-sie term), as obtaine from Eq. 8). From Newton s secon law an the variation of mass flux, we can erive also the horizontal an vertical components of the momentum balance solve by the moel as an ρ mix U sc r u U atm ) r ρ mix w U atm z ρ mix U sc r w ) = ) = upr gr ρ atm ρ mix ) wpr π [ ] ), 6 M) ws, ρ s, ) π [ ] ), 6 M) ws, ρ s, ) where the two components of plume velocity along the horizontal an vertical axes are u an w, respectively, an they are linke by the relation U sc = u + w. In the right-han sie of Eq. ) the terms relate to the exchange of momentum ue to the win Barsotti et al., 8) an to momentum loss from the fall of soli particles appear. Similar contributions are evient in the right-han-sie term of Eq. ) where the vertical momentum is change by the gravitational acceleration term an the fall-out of particles. Now, following the notation aopte above an enoting with T the mixture temperature, the equation for conservation of thermal energy solve by the moel is written as ρ mix U sc r C mix T r wρ atm g Tpr ) = rρ atm U ɛ C atm T atm π [ ] ). 6 M) Cs, w s, ρ s, ) The first term on right-han sie escribes the cooling of the plume ue to ambient air entrainment, the secon one takes into account atmospheric thermal stratification, an the thir term allows for heat loss ue to loss of soli particles. Again, this last term is obtaine writing the heat loss for the particles of size D, an then integrating over the size spectrum. A thermal equilibrium between soli an gaseous phases is assume. In Eq. ) C atm an C mix are the heat capacity of the entraine atmospheric air an of the mixture, respectively, the latter being efine as C mix = ) x s, C p,g + x s, C s, 4) or, in terms of the bulk ensities ρatm B = x atmρ mix, ρwv B = x wv ρ mix an ρs, B = π 6 M) ρ s,, as ρatm B C atm + ρwv B C wv + ρs, B C s, C mix = ρatm B + ρwv B + ρ B. 5) s, From this expression, if we multiply all the terms at the numerator an the enominator of the right-han sie by U sc r an we ifferentiate with respect to s, we obtain after some cancellation an algebra manipulations the following equation for the variation of the mixture specific heat with s: C mix = ρ mix U sc r [ C atm C mix ) ρatm B U scr ) + ) Cs, C mix ρs, B U scr )] + ρs, B C s, ρs, B U scr ) ρ mix ρs, B U scr C s, ρs, B U scr ) ρs, B U. scr 6) Now, substituting the expressions for the erivatives appearing in each term on the right-han sie, we obtain the equation for the variation rate of mixture specific heat in terms of the moments: C mix = ρ mix U sc r pr pr [C atm rρ atm U ɛ C mix rρ atm U ɛ [ ] ) ) ws, ρ s, π 6 M) ] π [ ] ) 6 M) ws, ρ s, C s,. 7) Similarly, a gas constant R g can be efine as a weighte average of the gas constant for the entraine atmospheric air R atm an the gas constant of the volcanic water vapor R wv R g = ρb atm R atm + ρwv B R wv ρatm B + ρwv B, 8) an a conservation equation can be erive, knowing that the variation of gaseous mass fraction with height is solely ue to entraine air: R g = R atm R g ρ mix x s )U sc r rρ atmu ɛ. 9) This formulation reuces, for particular cases, to the expressions of Woo 988) an Glaze an Baloga 996). Equations 7) an 9) are neee in orer to close the system of Geosci. Moel Dev., 8, , 5

7 M. e Michieli Vitturi et al.: PLUME-MoM 45 equations an recover the new values of the temperature an the mixture ensity once the system of orinary ifferential equations is integrate. Otherwise, without the solutions of Eqs. 7) an 9), we shoul use the ol values of ρ mix an C mix at s to obtain the values of the temperature at s + from the lumpe term ρ mix U sc r C mix T ) obtaine integrating Eq. ). Finally, as in Bursik ), the equations expressing the coorinate transformation between x, y, z) an s, η, θ) are given by z = sinη, x = cosη cosθ, y. Mass fraction istribution = cosη sinθ. ) Similarly to the istribution of particle number f D) an the moments M i), it is possible to erive a set of conservation equations in terms of the moments i) of the mass fraction istribution γ φ) expresse as a function of the Krumbein scale. In this case, the conservation of mass flux of particles with size φ of the th family is written as ) ρ mix γ φ)πr U sc = πrpw s, φ)ρ mix γ φ). ) Multiplying both sies of the equation by φ i an integrating over the size spectrum [, ], we obtain the following conservation equations for the moments of the continuous istributions γ φ): i) ρ mixu sc r ) = rpρ mix w i) s, i). ) For i =, the equations of conservation of the moments give x s, ρ mix U sc r ) = rpρ mix w ) s, x s, ) expressing the loss of mass flux of the particles of the th family an thus we can write the total mass conservation equation as ρ mix U sc r ) = rρ atm U ɛ rpρ mix w ) s, ). 4) From the variation of mass flux, as was carrie out for the istribution of particle number f D) an the moments M i), we erive the horizontal an vertical components of the momentum balance: ) ρ mix U sc r u U atm ) r ρ mix w U atm z = uprρ mix ) ρ mix U sc r w = gr ρ atm ρ mix ) wprρ mix w ) s, ) w ) s, ), 5). 6) The equation for conservation of thermal energy is ) ρ mix U sc r C mix T = rρ atm U ɛ C atm T atm r [ ] ) ) wρ atm g Tprρ mix Cs, w s, 7) an the equation for the variation rate of mixture specific heat in terms of the moments of the mass fraction istribution is written as C mix = [C ρ mix U sc r atm rρ atm U ɛ C mix rρ atm U ɛ ) rpρ mix w ) s, ) ] [ ] ) ) prρ mix Cs, w s,. 8) The formulation of the equations for the gas constant R g an the coorinates of the x,y,z) remain unchange. 4 Numerical scheme The plume rise equations are solve with a preictor corrector Heun s scheme Petzol an Ascher, 998) that guarantees a secon-orer accuracy, keeping the execution time on the orer of secon. If we rewrite the system of orinary ifferential equations with the following compact notation: y = f s,y), ys ) = y, 9) where y is the vector of the quantities in the left-han sies of the conservation equations presente in the previous section, then the proceure for calculating the numerical solution by way of Heun s metho Süli an Mayers, ) is to first calculate the intermeiate values ỹ i+ an then the solution y i+ at the next integration point ỹ i+ = y i + f s i,y i ), preictor step 4) y i+ = y i + f si,y i ) + f s i+,ỹ i+ ) ), corrector step. Geosci. Moel Dev., 8, , 5

8 454 M. e Michieli Vitturi et al.: PLUME-MoM 4. Quarature metho of moments We observe that to calculate the right-han sie for both the preictor an corrector step we nee not only the moments M i) but also the aitional moments [w s ] i, [w s ρ s ] i) an [w s ρ s C s ] i). As in Marchisio an Fox ), the integral in the efinition of these moments is replace by a quarature formula an the moments, for a generic variable ψ = ψd), are approximate as ψ i) = M i) ψd)f D)D i D N ψd l )Dl i ω l. 4) Here ω l an D l are known as weights an noes or abscissae ) of the quarature, respectively, an the accuracy of a quarature formula is quantifie by its egree. The egree of accuracy is equal to if the interpolation formula is exact when the integran is a polynomial of an orer less than or equal to an there exists at least one polynomial of an orer + that makes the interpolation formula inexact. In particular, an N point Gaussian quarature rule, is a quarature rule constructe to yiel an exact result for polynomials of egree N or less by a suitable choice of the noes D l an weights ω l for l =,...,N Golub an Welsch, 969). The Wheeler algorithm, as presente in Marchisio an Fox ), provies an efficient ON ) algorithm for fining a full set of weights an abscissas for a realizable moment set. The resulting noes D l are always within the support an therefore represent realizable values of the particle size), an the weights ω l are always positive, ensuring that, when the quarature is use, accurate results are obtaine Marchisio an Fox, ). Nevertheless, these properties are respecte only if the moment set is realizable, meaning that there exists a particle size istribution resulting in that specific set of moments. A strategy that might overcome the problem of moment corruption i.e., the transformation uring the integration of the moment-transport equations of a realizable set of moments into an unrealizable one) is replacing unrealizable moment sets as soon as they appear. An algorithm of this kin was evelope by McGraw McGraw, 6). The algorithm first verifies whether the moment set is realizable by looking at the secon-orer ifference vector or by looking at the Hankel Haamar eterminants; Gautschi, 4). If the moment set is unrealizable it procee with the correction. In the numerical moel presente here, the implementation of the correction algorithm of Wright 984) is erive from the version presente in Marchisio an Fox ). Thus, in both the preictor an corrector step, the following algorithm is use: l= The noes D,l an weights ω,l are calculate with the Wheeler algorithm for l =,...,N. The quarature formula Eq. 4) is use to evaluate the moments [w s ] i), [w sρ s ] i) an [w s ρ s C s ] i). Geosci. Moel Dev., 8, , 5 The right-han sie of the ODE s system Eq. 9) is evaluate explicitly. The solution is avance with the preictor or the corrector) step of the Heun s scheme. For each particle family, the moments M i) i =,...,N ), if require, are correcte with the Mc- Graw or Wright) algorithm. We observe that if the th family of particles is monoisperse with iameter, the Wheeler algorithm fe with the first two moments only gives a result of a single quarature noe D, = with weight ω, =. This allows us also to use the moel for the simplifie case where the soli particle istribution is partitione in a finite number of classes with constant size, assigning to each class a monoisperse istribution. 4. Initial conition Initial conitions at the vent inclue the initial plume raius r ), mixture velocity U sc, ) an temperature T ), gas mass fraction n ) an the particle size istribution through the initial moments M i). In the next section we erive analytically the moments of a specific initial istribution a normal istribution in the Krumbein scale) for both the formulations base on the number of particles as a function of the particle iameter expresse in meters an the formulation base on the mass concentration expresse as a function of the phi scale. Lognormal istribution For the application presente in this work, the initial istribution f D) at the base of the plume is efine as a function of the particle iameter expresse in meters m), in orer to give a corresponing normal istribution with parameters µ an σ for the mass concentration expresse as a function of the Krumbein phi φ) scale when all the particles have the same ensity): γ φ) = K σ φ µ) π e σ, 4) where K is a parameter that has to be chosen in orer to satisfy the initial conition on the soli mass fraction. Given the parameters µ an σ, the initial istribution f D) is then written as f D) = [ lnd) µln] 6C σ ln)d 4 π e σ ln), 4) where C, analogously to K, is a parameter that has to be fixe in orer to satisfy the initial conition prescribe for the mass or volume) fraction of particles.

9 M. e Michieli Vitturi et al.: PLUME-MoM 455 We observe that if we introuce the following re-scale variables for the iameter, the mean an the variance: D = D, µ = µln, σ = σ ln, 44) then it is possible to rewrite the particle istribution f D) in terms of a lognormal istribution in the variable D with parameters µ an σ : f D) = 6 C πd σ D e [lnd) µ] σ π = 6 C πd lognormd,µ,σ ). 45) Consequently, we can evaluate the moments M i) of f D) analytically from the moments of the lognormal istribution as M i) = 6C π i) exp an we obtain, for the thir moment, [i )µ + i ) σ ], 46) M ) = 6C π C = α s, 47) where αs is the initial volume fraction of the particles in the soli-gas mixture. From the expressions of the moments it follows also that, if the mass concentration expresse as a function of the Krumbein scale has a normal istribution, the Sauter mean iameter D A expresse in meters can be evaluate as D A = L = M) M ) = exp µ ) σ, 48) or, if expresse in φ, as D φ A = Lφ = µ + σ ln). 49) Processes involving the mutual interaction between particles an the interaction between the particles an the carrier flui friction an cohesion between the particles; viscous rag; chemical reactions between flui an soli components) operate at the surface of the particles. For this reason the Sauter mean iameter, base on the specific area of the particles, is a convenient escriptor an it is important to remark that it iffers from the mean µ of the lognormal istribution by a factor proportional to the variance σ. For numerical moels escribing the multi-phase particulate) nature of the matter an which approximate the particle size istribution with an average size, it is hence more appropriate to use, as particle size representative of a lognormal istribution, the Sauter mean iameter than the mean iameter µ. The ifference between the two approximations is smaller the narrower the particle size istribution. We must also remark that, for particles in the inertial-ominate regime e.g., Re p > ), Loth et al. 4) showe that the Sauter mean iameter is the effective iameter, regarless of particle shape, particle size istribution, particle ensity istribution or net volume fraction; for particles in the creeping flow regime Re p ) the effective mean iameter is the volume-with iameter. When the Sauter mean iameter is use, also the variance an the stanar eviation SD shoul be base on the specific surface area Rietema, 99). Hence, σa = D D A ) π 6 D f D)D, 5) or expresse as a function of the moments: σ A = M) M ) M ) ) M ) ). 5) Finally, we note that if the particle ensity is constant an the mass concentration expresse as a function of the Krumbein scale has a lognormal istribution an both the Sauter mean iameter L = M ) /M ) an the mean particle length average with respect to particle number ensity L = M ) /M ) or if the first 4 moments) are known, then we can solve for the re-scale mean an variance µ an σ the following system: L = exp µ 5 ) σ L = exp µ ). 5) σ Once the re-scale mean an variance are known, we can obtain µ an σ in the Krumbein φ scale. When the initial istribution is expresse for the mass fractions instea of the particle number, an the mass fraction written as a function of the Krumbein scale has a normal istribution with mean µ an variance σ, then the continuous istribution is given by Eq. 4). We observe that this expression of the istribution is not base on the assumption of constant ensity for the particles of ifferent size. In this case, the moments i) are given by the following expression i/ ) i) i = K )!!σ µ i. 5) = where the symbols an!! enote the integer part an the ouble factorial n!! = m k= n k), where m = n/ ), respectively. Now, as the th moment is equal to the mass fraction of particles, we obtain K = x s. Furthermore, we observe that the mass fraction-average iameter in the φ scale is given by the ratio ) / ), while the variance of the mass fraction istribution can be evaluate as Geosci. Moel Dev., 8, , 5

10 456 M. e Michieli Vitturi et al.: PLUME-MoM Table. Input parameters use for the numerical simulations. Vent height is the elevation of the base of the column above sea level. The values ρ, an D, are use to compute the ensity of the particles as a function of the iameter, accoring to the formulation of Bonaonna an Phillips ). The values reporte for µ an σ efine the range use for the uncertainty quantification an sensitivity analysis. Parameters Units Test case Test case Test case Vent raius m Vent velocity m s Vent temperature K Vent gas mass fraction...5 Vent height m ρ kg m ρ kg m D m D m µ φ [.;.] [.;.] [.;.] σ φ [.5;.5] [.5;.5] [.5;.5] [ ) ) ) ) ] / ) ). These two quantities correspon to the parameters µ,σ ) generally use to escribe the mass fraction when a normal istribution in the φ scale is assume. For this reason, when we want to track the changes of the mass fraction-average iameter an its stanar eviation or variance) in the φ scale uring the plume rise, it is preferre to use a formulation base on the moments i) than the moments M i). 5 Application 5. Simulation inputs We applie the moel to three ifferent test cases with ifferent vent an atmospheric conitions: test case low-flux plume without win; test case low-flux plume with win weak bent plume); test case high-flux plume strong plume). The parameters use for the ifferent test cases are liste in Table, while the atmospheric conitions are plotte in Fig.. For the low-flux plumes, a mass flow rate of.5 6 kg s has been fixe, while for the strong plume the value is.5 9 kg s. The temperature pressure an ensity profiles use for the test case without win test case ) are those efine by the International Organization for Stanarization for the International Stanar Atmosphere Champion et al., 985), while the profiles for the other two test cases come from reanalysis ata. For all the runs presente here, a single family of particles has been use, with a normal istribution with parameters µ an σ ) for the mass concentration as a function of the iameter expresse in the φ scale an with ensity varying with the particle iameter. We first present a comparison of the plume profiles obtaine with the three ifferent escriptions presente in the previous sections an highlighte in the three colore boxes of Fig. for the test case : metho of moments for the particle number that is the function of the size expresse in meters; metho of moments for the particle mass fraction that is the function of the size expresse in the φ scale; iscretization in uniform bins in the φ scale. For this comparison, the mass flow rate at the vent is.5 6 kg s an a rotating win is present, as shown in Fig., while the mean an the stanar eviation of the initial total grain-size istribution are, respectively, an.5, expresse in the φ scale. The results of the numerical simulations obtaine with the three ifferent formulations are presente in Fig. 4 an they perfectly match, showing that the metho of moments otte lines), both applie to the continuous istribution of the particle number re) or to the mass istribution green), gives the same results of the classical formulation base on the iscretization of the mass istribution in bins soli line). For these simulations, we use only the first six moments of the istributions, while bins have been employe with the iscretize formulation. This results in a smaller number of equations to solve for the metho of moments an, espite the aitional cost of the metho of moments ue to the evaluation of the quarature points an formulas through the Wheeler algorithm, in a smaller computational time, with a gain of about %. 5. Simulation results In this section we want to stuy the variation uring the ascent of soli mass flux ue to particle settling) an of the mean an the variance of the mass istribution along the column. As shown in the previous section, there are no significant ifferences in the results obtaine with the three ifferent escriptions of the grain-size istribution. For this reason, in the following we restrict the analysis only to the formula- Geosci. Moel Dev., 8, , 5

11 M. e Michieli Vitturi et al.: PLUME-MoM 457 Particles number istribution Particles volume istribution Particles mass istribution Mass fraction istribution x Diameter m) Diameter m) Diameter m) Diameter φ) Sauter Mean Diameter M ) /M ) ) Mean value μ of the initial mass-fraction istribution expresse as a function of iameter in the phi scale Length-average Mean Diameter M ) /M ) ) Total grain size istribution mass %) Diameter φ) Number-average Mean Diameter M ) /M ) ) Figure. Visualization of a normal initial istribution in the Krumbein φ scale for the soli particles. On the top plot, the particle number istribution expresse as a function of the iameter expresse in meters is plotte. On the secon an thir plots from the top, the corresponing istributions of volume an mass are plotte, these two being ifferent because the ensity is a function of the iameter. On the fourth plot the continuous istribution lognormal) of mass fraction as a function of the φ scale is plotte, while in the last plot the istribution has been iscretize with bins in the range 4;8). On each panel ifferent average raii are also plotte, together with the mean of the initial istribution. The first, fourth an fifth panel are highlighte with ifferent colors, also use in Fig. 4 for the solutions obtaine with the three ifferent representations of the initial grain-size istribution. 4 x 4 4 x 4 4 x 4 4 x 4 Test Case.5 Test Case Test Case z m) z m) z m) z m) Pressure Pa) x Temperature K) Win m/s) Win angle Figure. Atmospheric profiles for the three test cases. The height is expresse in meters above sea level, an for all the test cases the vent is locate at 5 m above sea level. For the win profiles, only the profiles for the two test cases with win are plotte. tion base on the moments of the mass fraction istribution as a function of the iameter expresse in the φ scale. With this approach, the mean, the variance an the skewness of the mass istribution along the column are easily obtaine from the first four moments i) of the mass fraction istribution. In Fig. 5 we present the results relative to the test case for an initial particle size istribution with mean iameter an stanar eviation.5, expresse in the φ scale. In the left an mile panels the mean, the variance an the skew of the mass fraction istribution are shown, while in the right panel the cumulative loss of soli mass flux is plotte as a percentage of the initial value. We observe a ecrease in the mean size of the particles, ue to the ifferent settling velocities of particles of ifferent sizes. A ecrease in the variance of the size istribution with height is also observe from the secon plot. We remark that the particles have a normal istribution only at the base of the column resulting in a null skewness), an the negative skew at the top of the column inicates that the tail on the left sie of the grain-size istribution is longer than the tail on the right sie; i.e., the mass Geosci. Moel Dev., 8, , 5

12 458 M. e Michieli Vitturi et al.: PLUME-MoM continuous number istribution continuous mass istribution Height km) 4.5 Height km) 4.5 bins Raius km) Velocity m/s) Height km).5 Height km).5 Height km).5 Height km).5 Figure 4. Height vs. raius left) an velocity right) for a low-flux plume, simulate with three ifferent moels. In blue the profiles obtaine using bins, in re the profiles obtaine using a continuous istribution of the particle number ensity an in green using a continuous istribution of the mass fraction. is more concentrate on the right of the spectrum of particle sizes finer particles). For this reason we o not have to look at the mean an the variance plotte in Fig. 5 as the parameters of a normal an symmetric) istribution. Nonetheless, changes in the mean, the variance an the skewness are observe, we remark that these changes are quite small an for this reason the parameters of the total grain-size istribution at the top of the eruptive column are a goo approximation of the parameters at the base of the column, an vice versa. However, this is true for the specific input conition of this test case an not in general. For this reason, it is important to quantify the uncertainty of this assumption for ifferent initial total grain-size istributions an ifferent atmospheric conitions. 5. Uncertainty an sensitivity analysis When ealing with volcanic processes an volcanic hazar, our unerstaning of the physical system is limite, an vent parameters volatile contents, temperature, grain-size istribution, etc.) are often not well constraine or are constraine with significant uncertainty. These factors mean that it is ifficult to preict the characteristic of the ash clou release from the volcanic column with certainty. An alternative is to quantify the probability of the outcomes for example the grain-size istribution at the top of the column) by coupling eterministic numerical coes with stochastic approaches. It is our goal in this work also to assess the ability to systematically quantify the uncertainty an the sensitivity of the plume moel outcomes to uncertain or variable input parameters, in particular to those characterizing the grain-size istribution at the base of the eruptive column. Uncertainty quantification UQ) or noneterministic analysis is the process of characterizing input uncertainties, propagating forwar these uncertainties through a computational moel, an performing statistical or interval assessments on the resulting responses. This process etermines the effect of uncertainties on moel outputs or results. In particular, in this work we wante to investigate for ifferent test cases the uncertainty in four response functions plume height, soli mass flux lost an mean an variance of the mass fraction µ φ) σ φ) Skew φ) x Soli Mass Flux lost %) Figure 5. Particle istribution parameters mean, variance an skewness) an cumulative loss of soli mass flux for the test case low flux without win), simulate with the formulation base on the moments of the mass fraction istribution. istribution at the top of the eruptive column) when the mean an the stanar eviation of the istribution at the base are ranom variables with a uniform probability istribution in the space µ,σ ) [ ;] [.5;.5]. In volcanology Monte Carlo simulations are frequently use to perform uncertainty quantification analysis. These metho rely on repeate ranom sampling of input parameters to obtain numerical results; typically one runs simulations many times over in orer to obtain the istribution of an unknown output variable. The cost of the Monte Carlo metho can be extremely high in terms of number of simulations to run, an thus several alternative approach have been evelope. LHS is another sampling technique for which the range of each uncertain variable is ivie into N s segments of equal probability, where N s is the number of samples requeste. The relative lengths of the segments are etermine by the nature of the specifie probability istribution e.g., uniform has segments of equal with; normal has small segments near the mean an larger segments in the tails). For each of the uncertain variables, a sample is selecte ranomly from each of these equal probability segments. These N s values for each of the iniviual parameters are then combine in a shuffling operation to create a set of N s parameter vectors with a specifie correlation structure. Compare to Monte Carlo sampling, the LHS has the avantage that in the resulting sample set every row an column in the hypercube of partitions has exactly one sample, an thus a smaller number of samples is require to cover all the parameter space. In the left panel of Fig. 6 an example of LHS with N s = an a uniform istribution probability for both µ an σ is plotte. An alternative approach to uncertainty quantification is the so-calle generalize polynomial chaos expansion metho, a technique that mirrors eterministic finite element analysis utilizing the notions of proection, orthogonality an weak convergence Ghanem an Re-Horse, 999). The polynomial chaos expansion PCE) metho was evelope by Norbert Wiener in 98 an it soon became wiely use because Geosci. Moel Dev., 8, , 5

13 M. e Michieli Vitturi et al.: PLUME-MoM 459 σ φ) µ φ) σ φ) µ φ) Figure 6. Two-parameters Latin hypercube sampling LHS) with points left) an tensor prouct gri using 9 9 Clenshaw Curtis points right). of its efficiency when compare to Monte Carlo simulations. The term chaos here simply refers to the uncertainties in input, while the wor polynomial is use because the propagation of uncertainties is escribe by polynomials. If ζ is the vector of uncertain input variables, the aim of the gpce is to express the response function Y in the form of a polynomial ξ as follows: ξζ ) = ξ + ξ P ζ ) + ξ P ζ ) ξ m P m ζ ), 54) where P,...,P m are polynomials that form an orthogonal basis. The choice of the polynomials basis epen on the probability istribution of the input variables. In particular, for a uniform istribution, the basis of the expansion is given by the Lagrange polynomials. For the application presente in this work the coefficients of the expansion have been evaluate using a spectral proection where the computation of the require multi-imensional integrals is base on the tensor prouct of -D Gaussian quarature rules. In orer to compute the quarature points, the gri use in our work is the Clenshaw Curtis gri Fig. 6, right), representing a goo solution for a multi-imensional Gaussian quarature with a small number of variables Elre an Burkart, 9). We present here the results of several tests performe coupling the plume moel with the Dakota toolkit Aams et al., 9) to investigate systematically the capability of the LHS an the gpce techniques to assess the uncertainty in four response functions plume height, soli mass flux lost an mean an variance of the mass fraction istribution at the top of the eruptive column) when the mean an the variance at the base are unknown. For all the test cases three sets of 5, an simulations have been performe for the LHS, an the results have been compare with those obtaine with three tests for the gpce an respectively 9, 6 an 8 simulations performe for the multi-imensional quarature. The first set of runs for the LHS, consisting of 5 simulations only, was not sufficient to provie accurate results an for this reason in the following we presents only the results obtaine with an simulations. In orer to compare the two techniques, the cumulative istributions of the four response functions obtaine with the LHS an the gpce, have been plotte in Fig. 7 for test case no win). On the x axes we can see the range of the values obtaine for the response functions:.5 for the mean of the total grain-size istribution TGSD) at top of the column expresse in the φ scale;.4. for the stanar eviation;.4.47 km for the column height an 6 % for the percentage of soli mass flux lost. All the uncertainty quantification tests prouce very similar results, with a small ifference in the cumulative istribution observe only in the istribution of the soli mass flux lost obtaine with the gpce technique an 9 an 6 quarature points. Similar results have been obtaine for the other test cases not shown here). Thus, the results highlights that for the moel an the applications presente in this work gpce represents a vali alternative to Monte Carlo simulations, with a number of runs require to prouce the same accuracy reuce by a factor 8 simulations vs. simulations). If more parameters were varie, the computational cost woul increase for both gpce an LHS, although the avantage of gpce woul be reuce. As mentione previously, the aim of the gpce is to express the output of the moels as polynomials an these polynomials can be use to obtain response surfaces for the output parameters as functions of the unknown input parameters through the polynomials efine by Eq. 54). In the four bottom panels of Fig. 7 the contours of the four response surfaces for the output investigate in this work have been plotte, showing the epenence on the uncertain input parameters. The mean an the stanar eviation of the TGSD at the top of the eruptive column clearly reflects the corresponing values at the bottom, with a small effect of the bottom stanar eviation on the mean size at the top, resulting in an increase in the average grain size with increasing values of the initial stanar eviation see the curves in the first panel bening on the left for higher values of σ ). Conversely, the plume height for this test case shows a nonlinear epenency but at the same time a small sensitivity to the initial grain-size istribution, with changes, for the specific conitions consiere here, smaller than % of the average height. This can be explaine by the fact that a large amount of air is entraine in the column uring the ascent an the contribution of the soli fraction to the overall ynamics becomes small compare to that exerte by the gas. Finally, we observe that the loss of particles is mostly controlle by the mean size of the TGSD. In Fig. 8 the same contour plots are shown for the polynomial expansion compute for test case top) an test case bottom) with 8 quarature points. The results show again that the total grain-size istribution at the base of the vent represents a reasonable approximation of that at the top of the column. For these test cases, both the column height an the soli mass lost appear to be mostly controlle by the mean size of the TGSD at the base of the column, with a small sensitivity of the height to the initial grain-size istribution. We also observe that the maximum percentage of loss in the soli mass flux is about 5 % for the strong plume simulations, an it is attaine for larger mean sizes an smaller variance of the Geosci. Moel Dev., 8, , 5

14 46 M. e Michieli Vitturi et al.: PLUME-MoM.9.8 L.H.S. L.H.S. P.C.E., P.C.E. 6,6 P.C.E. 9, Cumulative Probability Cumulative Probability Cumulative Probability Cumulative Probability Bottom TGSD σ φ). 4 Top TGSD Meanφ) Bottom TGSD µ φ) Bottom TGSD σ φ). Top TGSD Stanar eviationφ) Bottom TGSD µ φ) Bottom TGSD σ φ) Column height km) Bottom TGSD µ φ) Bottom TGSD σ φ) Soli Mass Flux Fraction Lost Bottom TGSD µ φ) Figure 7. Cumulative istributions an response surfaces for test case low-flux plume without win). In the top panels the cumulative probability for several variables escribing the outcomes of the simulations mean an variance of the grain-size istribution at the top of the column, column height an cumulative fraction of soli mass lost) are plotte for the uncertainty quantification analysis carrie out with the two ifferent techniques an for ifferent numbers of simulations. The contour plots of the response functions of the four output variables, resulting by the polynomials given by Eq. 54) an obtaine with the PCE with 8 quarature points, are plotte in the bottom panels. The variables contoure in the lower panels are the same as those on the horizontal axes in the upper panels. initial TGSD. This value is noticeably smaller than that obtaine for the weak bent test case 4 %) an for the weak test case without win 6 %). Despite the loss of particles, in both the cases the range of variation of the column height is quite small an, as previously mentione, this is ue to the large amount of air entraine in the volcanic column that reuces the contribution of the soli fraction to the overall ynamics. As an example to unerstan the relevance of the entraine air, for a simulation performe for the low-flux plume without win an with µ = an σ =.5 in the φ scale, the mass flow rate at the top of the column is. 8 kg s, compare to the value at the base of.5 6 kg s. 5.4 Sensitivity analysis With the polynomial chaos expansion it is also possible to easily obtain the variance-base sensitivity inices Saltelli et al., 8) with no aitional computational cost. In contrast with some instances, where the term sensitivity is use in a local sense to enote the computation of response erivatives at a point, here the term is use in a global sense to enote the investigation of variability in the response functions. In this context, variance-base ecomposition is a global sensitivity metho that summarizes how the variability in moel output can be apportione to the variability in iniviual input variables Aams et al., 9). This sensitivity analysis uses two primary measures, the main effect sensitivity inex S i an the total effect inex T i. These inices are also calle the Sobol inices. The main effect sensitivity inex correspon to the fraction of the uncertainty in the output, Y, that can be attribute to input x i alone. The total effects inex correspon to the fraction of the uncertainty in the output, Y, that can be attribute to input x i an its interactions with other variables. The main effect sensitivity inex compares the variance of the conitional expectation Var xi [EY x i )] against the total variance VarY ). Formulas for the inices are S i = Var x i [Y x i )] VarY ) an 55) T i = EVarY X i)), 56) VarY ) where Y = f x) an x i = x,...,x i,x i+,...,x m ). Similarly, it is also possible to efine the sensitivity inices for higher-orer interactions such as the two-way interaction S i,. The calculation of S i an T i requires the evaluation Geosci. Moel Dev., 8, , 5

15 M. e Michieli Vitturi et al.: PLUME-MoM 46 Figure 8. Response surfaces for test case low-flux plume with win, four top panels) an test case strong plume with win, four bottom panels) obtaine with the PCE with 8 quarature points. Please note that the color scale is not consistent between plots. of m-imensional integrals that are typically approximate by Monte Carlo sampling. However, in stochastic expansion metho, it is possible to approximate the sensitivity inices as analytic functions of the coefficients in the stochastic expansion. The results of the sensitivity analysis for the four outputs an the three test cases investigate are presente in the bar plot of Fig. 9. For each of the four groups one for each of the ifferent output functions), the three bars represent the main sensitivity inices for the three test cases test on the left, test in the mile an test on the right), while the ifferent colors are for the sensitivity inices with respect to ifferent variables blue is for the mean of the initial TGSD, green for the stanar variation of the initial TGSD an brown for the secon-orer couple interaction). Again, the sensitivity analysis confirms that the mean an the stanar eviation of the grain-size istributions at the top of the eruptive column are controlle primarily by the respective parameters of at base of the column. The mean of the TGSD also controls the percentage of soli mass flux lost uring the rise of the column an the plume height for the two test cases with win, while for the weak test case without win the ispersion of the istribution an secon-orer interaction also play a maor role in controlling plume height variability. However, as alreay observe with the uncertainty quantification analysis, we remark that the variability in the plume height, when the mean an the stanar eviation of the TGSD vary in the investigate ranges, is extremely small for all the test cases less than % with respect to the average values), an thus the investigation of how the variability in moel output can be apportione to the variability in iniviual input variables Bottom μ Bottom σ Bottom μ+σ Top μ Top σ Column height Soli Mass Flux Lost Figure 9. Sobol main sensitivity inices. For each of the four output parameters the three bars are for the ifferent test cases: test case on the left, test case in the mile an test case on the right. For each test case the ifferent colors of the bars are for the ifferent sensitivity inices: blue for first-orer sensitivity inex with respect to the bottom TGSD mean, green for the first-orer sensitivity inex with respect to the bottom TGSD stanar eviation an brown for the secon-orer combine sensitivity inex. is less relevant for the plume height than for the other output parameters. 6 Conclusions In this work we have presente an extension, base on the metho of moments, of the Eulerian steay-state volcanic plume moel presente in Barsotti et al. 8) erive from Morton, 959; Ernst et al., 996; Bursik, ). Two ifferent formulations, one base on a continuous istribution of the number of particles as a function of the size an a secon base on the continuous istribution of the mass fraction, have been presente. The tracking of the moments of mass istribution, efine as a function of the Krumbein phi scale, Geosci. Moel Dev., 8, , 5