Dynamics of two-phase conduit flow of high-viscosity gas-saturated magma: large variations of sustained explosive eruption intensity

Transcription

1 Bull Volcanol 2000) 62 : 153±170 Springer-Verlag 2000 RESEARCH ARTICLE O. Melnik Dynamics of two-phase conduit flow of high-viscosity gas-saturated magma: large variations of sustained explosive eruption intensity Received: 25 March 1998 / Accepted: 11 December 1999 Abstract The ascent of gas-saturated magma in a conduit can lead to the transition from the laminar flow of a bubble-rich melt to the turbulent flow of particlerich hot gas in upper part of the conduit. This process is investigated with a help of a disequilibrium twophase flow model for steady and unsteady conditions. For the description of different gas-magma flow regimes in the conduit separated sets of equations are used. The main difference from previous conduit flow models is the consideration of the pressure difference between a growing bubble and the surrounding melt. For a bubbly liquid to evolve into a gas-particle flow a critical overpressure value must be exceeded. This transition region is simulated by means of a discontinuity, namely by a fragmentation wave. The magma flow calculations are carried out for a given conduit length and overpressure between the magma chamber and the atmosphere. The steady solution of the boundary problem is not unique and there are up to five distinct steady regimes corresponding to fixed eruption conditions. Within the framework of the quasi-steady approach governing parameters being varied monotonically) the transition from one regime to another can take place suddenly and is accompanied by the fundamental restructuring of the conduit flow, resulting in rapid or abrupt changes in the intensity of explosive eruptions. Abrupt intensification of an explosive eruption occurs when the chamber pressure becomes sufficiently less than saturation pressure, and therefore corresponds to the case of shallow-depth Editorial responsibility: D.A. Swanson Oleg Melnik Institute of Mechanics, Moscow State University, b, Mitchurinski pr., , Moscow, Russia Fax: melnik@inmech.msu.su Present address: Oleg Melnik Department of Earth Sciences, University of Bristol, Wills Memorial Building, Queen's Road, Bristol, BS8 1RJ, UK magma chamber and high initial water content. A regime with minimum flow rate may relate to the growth of a lava dome following the explosive phase of eruption. These models can explain geological observations that imply large and sudden changes of discharge rate in large-magnitude explosive eruptions, particularly at the transition between plinian phase and ignimbrite formation. Key words Volcanic eruption Conduit flow model Magma chamber Bubbly melt Gas-particle dispersion Fragmentation Bubble overpressure Discharge rate Introduction During ascent of gas-saturated high-viscosity magma from a magma chamber to the Earth's surface exsolution of dissolved gas leads to bubble formation and subsequent growth. Under some conditions the viscous bubbly flow transforms into a turbulent gas-particle mixture accelerating to velocities up to several hundred metres per second. Figure 1 illustrates the flow regimes and pressure distribution, corresponding to this process and the nomenclature used in this paper. The typical range of physical parameters for such sustained explosive eruptions is wide. Key factors that must be considered are the mass concentration of dissolved gas and the high viscosity of the magma. In this paper both dimensional and dimensionless variables are used. We highlight dimensional variables with apostrophe to distinguish them from dimensionless. The parameters considered in this paper are: initial weight concentration of dissolved gas mostly H 2 O) c o = 3±7 wt. %, magma ascent velocity V9 = 0.1±5 m/s recalculated for the density of unvesiculated magma), magma chamber depth L9 = 3±20 km and viscosity of magma m9 =10 4 ±10 8 Pa s, with higher values of viscosity corresponding to lower concentration of dissolved gas. In the case of a volcanic conduit being a fissure,

2 154 Fig. 1 Schematic view of the flow in the conduit of the volcano, corresponding to sustained explosive eruptions. On the left ± typical pressure distribution along the conduit thick curve), mixture weight dotted curve) and conduit resistance dashed curve). For the homogeneous flow the weight and conduit resistance are constant; therefore, pressure drops down linearly with depth. In bubbly flow weight decreases but conduit resistance increases dramatically due to the exsolution of dissolved gas, and pressure drops down faster than in homogeneous regime. In gas-particle dispersion flow weight is small and conduit resistance is negligible. Pressure drops down weakly. Position of the fragmentation level determines average weight and conduit resistance, and, therefore, the discharge rate the typical surface vent width, d9, is approximately a few metres to a few tens of metres. For a cylindrical conduit d9 is in a range of tens to hundreds metres. The values of governing parameters in some well-characterised eruptions are reproduced in Table 1. The simplest steady-state one-dimensional conduit models of sustained explosive eruptions Wilson et al. 1980; Wilson 1980; Slezin 1983; Woods 1995; Sparks et al. 1997) consist of the equations of mass conservation for liquid and gaseous components, the momentum equation for the mixture as a whole, and the equations of state in the bubble melt and gas-particle dispersion zones. The principal differences between published models arise concerning the pressure distribution along the conduit and description of the transition mechanism from a bubbly fluid to a gas dispersion. Thus, in Wilson et al. 1980) and Wilson 1980) lithostatic pressure distribution is assumed. These studies also assume that fragmentation occurs when the gas bubbles coalesce on reaching some critical close packing state. In the model of Slezin 1983) the walls of the conduit are assumed to be rigid; therefore, pressure depends only on flow dynamics. Instead of a fragmentation surface Slezin 1983) introduces a region of so-called ªbreaking down foamº, which is simulated by a liquid porous medium, in which gas and melt have they own velocities. The frictional resisting force within the conduit in this foam is calculated using the Poiseuille law where the viscosity is that of the melt. The slip velocity between the particles and a gas is determined from the condition of dynamical equilibrium, as in dilute gas dispersions. In Barmin and Melnik 1990) a more complete model was developed which takes into account disequilibrium effects in mass transfer between a growing bubble and surrounding melt due to the low value of gas diffusion coefficient. The model also takes account of magma viscosity dependence on concentration of dissolved gas, the temperature and the volume fraction of bubbles, as well as heat transfer processes. The inertia of the particles in the gas-dispersion regime was also considered. The slip velocity of free gas in the fragmentation zone is calculated from Darcy's law for a porous medium with variable permeability. In Slezin 1983), Slezin 1991) and Barmin and Melnik 1990) the dependence of magma discharge rate on the eruption parameters was considered for a fixed chamber pressure and conduit length. Up to three different regimes of magma ascent were recognised corresponding to the same values of chamber pressure and other fixed parameters of eruption. In a Table 1 Dimensional parameters for eruptions of Mount St. Helens and Santorini Eruption c o L9 km) d9 m) m9 Pa s) Q9 kg/s) Mount St. Helens a 18 May 1980) ~ Santorini BC 1650) 5.5 b 5±6 c d e a Carey and Sigurdsson 1985) b Sigurdsson et al. 1990) c Cottrell et al. 1999) d Kaminski and Jaupart 1997), value calculated for 1.8 wt. % H 2 O, in the approximation Eq. 3)) m9 0 is 10 8 Pa s e Sparks and Wilson 1990), value of discharge rate for initial Plinian phase

3 155 low-intensity regime the conduit is filled by bubbly melt. The effect of frictional resistance can be neglected in comparison with that of the mixture weight, so the pressure drop between the chamber and the atmosphere is controlled by the high average weight of the mixture. In an intensive regime the bubbly liquid occupies only the lowermost portion of the conduit and the conduit resistance is large, due to the high mixture velocity. In an intermediate regime both conduit friction and mixture weight are important. The gradual variation of the parameters can lead to a sudden intensification or weakening of the eruption, due to the transition from one regime to another. These studies demonstrated nonuniqueness in the solutions for conduit flows. Nonunique solutions for conduit flow were also obtained in Jaupart and Allegre 1991) and Woods and Koyaguchi 1994) where the effect of gas loss to the surrounding rock was investigated. In the numerical model of Dobran 1992) the distribution of key parameters along the conduit was obtained for constant cross section of the conduit and sonic flow velocity at the outlet of the conduit choked condition). The velocity disequilibrium between gas and liquid were taken into account both in bubble-melt and gas-particle dispersion regimes. Dobran 1992) demonstrated that the pressure distribution could be far from the lithostatic, especially in the gas-particle flow regime. In the models described above pressures in the growing bubble and the surrounding melt were assumed to be the same. The applicability of these models is limited to relatively slow melt motions and low magma viscosity. The pressure in the growing bubble can be obtained from the Rayleigh-Lamb equation see Scriven 1959): p 0 g ˆ p0 m 2s0 a 0 4m 0 _a 0 a 0 3 r0 m 2 _a02 a 0 a 0 1 Here p g 9 and p m 9 are internal bubble and ambient pressures, respectively, m9 is the dynamic viscosity, s9 is the interfacial tension coefficient, a9 is bubble radius and r m 9 is the density of surrounding melt. The inertia term fourth) is negligible in almost all cases of volcanological interest, and the surface tension term second) is large only for bubble radius close to an embryo radius. Therefore, the main cause of pressure disequilibrium is the viscous resistance of surrounding melt. McBirney and Murase 1971) discussed the possibility of accumulating the potential energy of a compressed gas during the process of bubble growth in magma. Sparks 1978) investigated the ascent of a single bubble in magma and demonstrated that the viscous term in bubble growth becomes large in sustained explosive eruptions for viscosities of 10 8 Pa s or above. In Barclay et al. 1995) the analytical solutions for single bubble growth in suddenly decompressed magma both in cases of infinite volume of surrounding melt and a bubble surrounding by a viscous shell. It was shown that pressure relaxation time depends strongly on the relative thickness of the shell. For viscosity 10 7 Pa s an analytical solution gives the relaxation time approximately 0.8 s for the ratio of bubble radius to the shell thickness equal 20 and approximately 10 3 s in cases of infinite shell thickness. The shell solution is based on a finite velocity in the external shell boundary; therefore, it is not applicable to the case of high bubble volume concentration, because, from the condition of symmetry, the liquid velocity on the external shell boundary must be equal to zero. As a result, the response times predicted by shell theory are smaller than real times. Only solution of the three-dimensional problem can clarify the situation completely. There are several more complicated bubble growth models Toramory 1989; Proussevitch et al. 1993; Hurwitz and Navon 1994). The nonequilibrium diffusion profiles and melt viscosity dependence on the concentration of dissolved gas were taken into account there in the case of radial symmetry of flow. Bennett 1976) proposed that the eruption process be considered qualitatively as the propagation of a rarefaction wave into the volcanic conduit by analogy with processes taking place in the high-pressure chambers of shock tubes. Alidibirov 1987) considered the propagation of a fragmentation wave through a solidified magma containing gas bubbles at high pressure, following the approach of Khristianovich 1979), who applied it to blasts in coal banks. In Alidibirov 1988) the conditions leading to the formation of a magma containing overpressurised bubbles and its explosive fragmentation were examined. In the first stage viscous magma ascends rapidly in comparison with gas diffusion time) to refill a part of conduit, emptied in the previous explosion. Then solidification of the interstitial melt occurs due to the loss of gas to growing bubbles. As the interstitial melt solidifies, gas flows into the bubble leading to an increase in pressure in the bubble. When the pressure rises to a level higher than a critical value determined by the tensile strength, fragmentation occurs and propagates down along the conduit as a shock wave. The potential energy, accumulated in overpressured bubbles, is released by fragmentation and leads to rapid evacuation of the disintegrated magma. This process then repeats itself. This theory predicts separate volcanic explosions alternating with pauses during which the potential energy of gas accumulates in pressuring bubbles. Although this model is more applicable to vulcanian eruptions than sustained explosive eruptions, some of these principles can be developed in the context of sustained explosive activity. Decompression of the ascending magma can result in overpressures developing in the bubbles and a similar kind of fragmentation wave. Recently, a new approach to fragmentation was developed by Papale 1999). He assumed Maxwell behaviour of magma and obtains a fragmentation

4 156 criterion that the elongation strain rate must be higher than structural relaxation time of the fluid. This criterion can be written in the form: dv 0 1 G 0 1 dx 0 > k r t 0 ˆ k r r m 0 Here dv9 /dx9 is the elongation strain rate; t r 9 magma structural relaxation time; G? 9 the elastic modulus at infinite frequency; and k r is a proportional coefficient determined from experiments by Webb and Dingwell 1990). Mathematically this criterion represents the same relation between the critical value of the elongation strain rate and magma viscosity as those, developed by Barmin and Melnik 1993), but uses a different physical approach to fragmentation. Papale 1999) shows the possibility of different steady-state solutions which correspond to the same set of governing parameters with discharge rate difference of more than two orders of magnitude. The difference rises from different assumption on boundary condition on the outlet of the conduit: low-intensity regime corresponds to subsonic flow conditions; for high-intensity regime flow is choked. In a series of papers presented by Papale and coauthors Papale and Dobran 1994; Papale et al. 1998; Neri et al. 1998; Papale and Polacci 1999) significantly large variations in magma discharge rate were explained as a consequence of gas content and chemical composition variation in ascending magma. For example, in Papale and Dobran 1994) change in composition of erupting magma resulted in changing magma viscosity and volatile solubility. Consequently, the discharge rate changes from to kg/s without changes in conduit diameter. In Papale and Polacci 1999) the role of carbon dioxide in the dynamics of magma ascent was studied. They showed that increase in CO 2 content from 0 to 50 % from total volatile content decreases discharge rate by 2.8 times as solubility of CO 2 and H 2 O mixture is strongly different from pure water solubility. In Papale et al. 1998) the role of magma composition, water content, and crystal content on the dynamics of explosive eruptions was investigated. The results of the modelling show complex dependence of the flow variables on governing parameters. The common compositional trend in explosive eruptions is characterised by chemically evolved, water-richer and crystal-poorer magma erupted first followed by more mafic crystal-rich magmas. The increase in mass flow rate results from the increasing density of erupting products. The role of chemical composition and amount of dissolved volatiles on discharge rate and other parameters of explosive eruption is studied by Papale et al. 1998) in detail. The increase in amount of initially dissolved water from 1.5 to 6 wt. % leads to an increase in discharge rate of approximately an order of magnitude. Change of the chemical composition from rhyolite to dacite causes an increase in the discharge rate of more than two times. The influence of crystals is in decreasing discharge rate for fixed chemical composition. Neri et al 1998) took input parameters from calculations done by Papale et al. 1998) and studied the influence of magma chemical composition on dispersion of pyroclastic in the atmosphere. The new contribution of the model presented in this paper is the consideration of pressure disequilibrium between growing bubbles and interstitial melt. A new condition for fragmentation is proposed based on the critical overpressure between bubbles and liquid. Both steady state and unsteady cases are investigated. The implications of nonunique character of the solutions for interpreting sustained explosive eruptions and pyroclastic sequences are discussed. Formulation of the problem of high-viscosity gas-saturated magma flow Physical properties of magma As magma ascends through the conduit, the pressure falls down from a value of the order of several kilobars to atmospheric. This process causes exsolution of a dissolved gas, bubble nucleation and subsequent growth. The relation between the equilibrium dissolved gas mainly H 2 O) concentration c and the pressure p9 is approximated by the following experimental formula: p c ˆ k 0 p p 0 ; k 0 p ˆ 4:11 6: Pa 1:2 ; 2 The coefficient k p 9 depends on the type of magma and the composition of the dissolved gas. The temperature dependence of the solubility is much weaker for the pressures up to 200 MPa Lebedev and Khitarov 1979; Burnham 1979; Stolper 1982). General discussion on the role of solubility law on the eruption dynamics is given by Tait et al. 1989). The rheology of a magmatic melt is complex and depends on many factors: temperature; pressure; composition; and dissolved gas content. The magma is assumed to be a Newtonian fluid with the viscosity depending on dissolved gas concentration and temperature: m 0 ˆ m 0 0 exp A 0 R 0 T 0 exp B 0 c 1 ; 3 m Pa s Here m 0 9 is the viscosity of the ªdryº magma, T9 temperature, R9 gas constant, A9 and B9 constants, depending on the magma type. In particular we choose A9/R9T9 = 15.6, B9 = 11.3 as typical of rhyolitic melts. Equation 3) expresses that the logarithm of viscosity has an exponential dependence on dissolved

5 157 gas concentration. Equation 3) was obtained as an approximation of viscosity measurements data for rhyolitic gas-rich magma Lebedev and Khitarov 1979). More recent data Dingwell et al. 1996; Richet et al. 1996) show that melt viscosity increases rapidly when the concentration of dissolved gas c decreases from 1 wt. % to zero. However, when c>1 wt. % typical concentration before the fragmentation; Eq. 3)) has a good agreement with these data. Equation 3) does not take into account the influence of crystals and bubbles on the viscosity of mixture. High crystal content in magma can increase the viscosity by several orders of magnitude Lejenue and Richet 1995), but the influence of crystals on volcanic eruption dynamics is beyond the scope of this paper. The influence of bubbles is much smaller. The increase in volume concentration of bubbles from 0 to 50 % decreases viscosity only by a factor of 3 Lejeune et al. 1999). Physical approach to magma flow description The problem of magma ascent along a volcanic conduit from a chamber located at a depth L9 is considered. At the chamber outlet pressure, temperature and the bubble volume concentration in the escaping magma are given. If the chamber pressure is greater than the saturation pressure p 0 9, then nucleation occurs in the conduit; otherwise, the bubble volume concentration in the chamber is calculated assuming equilibrium. During magma ascent along the conduit the pressure decreases and the velocity of the bubble-rich melt tends to increase due to expansion of decompressing bubbles. Magma viscosity also increases due to the decrease of dissolved gas see Eq. 3)). As a result, the time for pressure relaxation between growing bubbles and magma increases with distance up the conduit; therefore, magma pressure decreases faster than the pressure in growing bubbles. It is assumed that the bubbly mixture fragments, when the overpressure in the bubbles exceeds a critical value Dp9 *.The particular values of Dp9 * is discussed below. Melt partitions between bubbles break down and the medium transforms into a gas-particle dispersion. It is shown later that the length of the region where the pressure difference between bubbles and surrounding melt is significant is small in comparison with the conduit length; thus, fragmentation is visualised as taking place in a thin flow zone, which is investigated as a discontinuity: a fragmentation wave. After the fragmentation, the gas-particle mixture accelerates to velocities comparable to the velocity of sound. If the outgoing flow is subsonic, the outlet pressure is equal to atmospheric pressure; otherwise, for conduit of constant cross-section area the equality of the flow velocity to the local velocity of sound must be taken as the boundary condition the choked flow condition; see Shapiro 1954; Woods and Bower 1995). For conduits becoming larger with height the transition to a supersonic regime can occur inside the conduit Shapiro 1954; Wilson et al. 1980). The bubbly flow model The following assumptions are made after comparison of values of the terms in general multiphase flow system Nigmatulin 1979): 1. The velocity difference between magma and gas is negligible in comparison with the ascent velocity. 2. Temperature variations in the bubbly regime are small due to the high thermal capacity of magma and low expansion velocity. 3. Nucleation is assumed to be instantaneous Sparks 1978; Hurwitz and Navon 1994) and no new bubbles appear after an initial nucleation event. 4. The mass transfer between dissolved gas and bubbles maintains the system at equilibrium. This assumption is not applicable to the conditions close to fragmentation level, because diffusion coefficient becomes small when the melt viscosity increases dramatically. However, the contribution of diffusion to the volume changes in growing bubble is small in comparison with decompression in the upper part of the bubbly flow. Pressure nonequilibrium between bubble and melt phase also changes the intensity of mass transfer. As is shown below, pressure difference is negligibly small in almost all flow except the vicinity of fragmentation level; therefore, it does not contribute significantly in mass transfer rate. The one-dimensional equations for bubbly flow are developed in dimensionless form. As a characteristic density we take the density of magma without bubbles r 0 9; as a characteristic pressure, the saturation pressure p9 0 as a viscosity, viscosity of dry magma, and as a velocity the typical ascent velocity of unvesiculated magma V 0 9 = Q 0 9 r 0 9 S c 9) ±1, where Q 0 9 is average mass discharge rate, S c 9 is the cross-sectional area of the conduit. The following dimensionless variables are introduced as: r m ˆ r0 m r 0 ; r g ˆ r0 g 0 r 0 ; p ˆ p0 g0 p 0 ; x ˆ x0 0 x 0 ; 0 V ˆ V0 V0 0 ; a ˆ a0 a 0 ; n ˆ n0 0 n 0 ; m ˆ m0 0 m 0 0 p9 0 = c 0 /k p 9) 2 ; r9 g0 = p 0 9/R9T9; x 0 9 = p 0 9/r 0 9g9; a 0 9 = 3n 0 9/4p) ± 1/3 Here g9 is gravitational acceleration, and the characteristic length of the conduit x 0 9 corresponds to the depth equivalent at which the saturation pressure is reached for a column of unvesiculated magma. In these notations the system of bubble flow equations is written in the following form:

6 @rv ˆ m ˆ C ˆ 0 a ˆ Eu r larmv b a 4m p g p m p s ˆ ap g 1 a p m ; p g ˆ r 0 g ; c ˆ c p 0 p g; a ˆ a 3 n; d r m ˆ 1 a 1 c ; r g ˆ dr 0 g a 1 a c; Ar ˆ m0 0 V0 r 0 0 g0 d 02 ; Eu ˆ p0 0 r 0 0 V0 0 2 ; d ˆ p0 0 r 0 0 R0 T0 0 ; C a ˆ p0 0 x0 0 m 0 ; e 0 V0 0 c 4) This system of equations Eq. 4)) consists of the continuity equations for the components of the mixture and number density of bubbles a), the momentum equation for the mixture as a whole b), as well as the Rayleigh-Lamb equation for bubble growth c); d) represents definitions for pressures and densities and e) the notations for nondimensional parameters. Gravity forces and the conduit resistance in Poiseuille form, l is a shape factor of the conduit equal 32 for cylindrical conduit and 12 for a fissure) in momentum equation are taken into account. As a consequence of the bubbles and viscous melt having equal velocities, it is possible to express densities of components in such a way as to avoid mass transfer terms in the continuity equations a). All simplifications and assumptions in the development of Eq. 4) are quite usual for conduit bubbly flows, except the usage of Reyleigh-Lamb equation for bubble overpressure growth. This equation is obtained from equations describing spherically symmetrical individual bubble growth in an infinite incompressible liquid. In Hurwitz and Navon 1994) and Navon and Lyakhovski 1998) the modification of this equation is proposed to model the presence of other neighbouring bubbles for the case when foam has a high volume concentration of bubbles. They solve 1D spherically symmetric problem of the growth of a bubble surrounded by a shell of viscous liquid. Unfortunately, this solution does not represent the real situation, because the radial velocity on the outer shell boundary does not tend to zero, which is needed from the conditions of symmetry of the bubbly grid; therefore, this modification gives a wrong relation between the bubble overpressure and its growth rate. To get more accurate pressure distributions around the growing bubble the full 3D equation for momentum should be solved and averaged. Gas-particle mixture flow Writing the system of equations for the gas-particle dispersion the following simplifying assumptions are made. 1. The particle temperature is assumed to be constant and equal to the temperature of the surrounding gas. This is justified by a high heat capacity, large mass concentration and small size of particles. Second, t 2. The variation of the gas temperature is neglected since there is an efficient heat exchange between the gas and the particles; therefore, the flow in the conduit is assumed to be isothermal. At the outlet of the volcanic conduit, where the particle concentration is low and the expansion rate of the gas is substantial, the gas temperature will have decreased by not more than 3±5 % Barmin and Melnik 1993). A single particle size dispersion is considered. The gas dispersion dynamics is described by the continuity and momentum equations for each component with allowance for the force of interaction Fgm between gas and particles and the equation of state for the gas phase. Introducing the partial densities r m and r g of the particles and the gas, respectively, this system of equations is given in the following m m g V p mv 2 ˆ gv g ˆ EuF gm r gv 2 g r g d 1 r m ; F gm ˆ C m r m ; r g r m ˆ 1 a ; r g ˆ ar 0 g ˆ EuF gm 2 Vm V g 2 5 No further gas exolution occurs from particles formed in the fragmentation process. This is reasonable because the residence time of particle in the conduit is small in comparison with gas diffusion time. The interaction force between gas and particles depends on the volume concentration of particles Nigmatulin 1979). Fragmentation wave The thickness of the transition zone between the bubbly melt and the gas-particle dispersion is estimated as follows: the lower boundary of fragmentation zone is defined as the level at which the pressure difference between growing bubbles and melt exceeds a critical value. The upper boundary of the fragmentation zone is defined by a gas velocity such that particles become just supported by the upward gas flow F gm = r m ). After the excess pressure in the bubble exceeds the

7 159 critical threshold, some of the partitions between the bubbles are considered to break down, and gas pressure falls down to a value between bubble pressure p g and melt pressure p m. Consequently, gas density decreases and gas velocity increases. Gas acceleration causes additional partitions to break. As a result, the medium evolves into a gas-particle dispersion. From Eq. 4) we obtain an equation which describes the growth of overpressure between a bubble and the magma: p ˆ C 1 a 4m 3a da a 7 Coefficient C a is large, so in most of flow the pressure difference is small. During magma ascent along the conduit viscosity increases strongly with exsolution of dissolved gas. Therefore, most of the pressure drop is accommodated in a narrow region just below the fragmentation zone. In this region magma has large accelerations and bubble overpressures become significant. The fragmentation condition is defined by Eq. 6) when the value of Dp reaches some threshold value and shows that fragmentation occurs when the mixture reaches a critical value of acceleration. Papale 1999) suggested a mechanism for fragmentation based on a threshold strain rate which is similar mathematically to Eq. 6). It is based on consideration of Maxwell type magma rheology and experimental data of the critical strain rate for silicate melts of different composition from Webb and Dingwell 1990), and it describes the transition form viscous to brittle behaviour. The choice of a critical Dp will be at a value when the magma breaks apart due to Dp exceeding the tensile strength Alidibirov and Dingwell 1996) or when the accelerating forces tear the magma apart in a ductile manner. The details of the fragmentation mechanism are still not well established Mader 1998), but the calculations are not strongly sensitive to the choice of Dp9 * due to the narrowness of the region where large bubble overpressures develop in comparison with the total length of the conduit. For example, in this paper Dp9 * is always chosen as 1 MPa. The results of calculations give an exponential increase in Dp9 over a distance of approximately 10 m. A choice of 10 MPa would only increase the distance by a few metres, so the changes of the overall flow resistance would be small. The experiments by Alidibirov and Dingwell 1996) imply that a few MPa is a typical tensile strength of high-temperature magmatic materials. Formally, the vertical velocity gradient can be estimated from the momentum equation because in the steady case the velocity gradient is proportional to the inertia term. The contribution of the inertia term is given by 1/Eu. Its value for the magma flow in the region close to saturation is only approximately 10 ±4, so the inertia of the medium is not important. Near the fragmentation level Eu is approximately 10 ±2. Therefore, during the ascent of gas-saturated magma the overpressure value increases both due to the rise in melt viscosity and due to increase of vertical acceleration. The results of the calculations show that the overpressure in a growing bubble increases exponentially with height, and that the length of the zone, where most of the overpressure increase takes place, is only approximately 10 m for Dp = 1 MPa. After the start of fragmentation, particles and gas have the same velocities. We now consider the equation of momentum for a single particle with radius r p 9 in dimensional form to estimate the length where the drag force of gas just supports the particle. dv 0 m dt 2 V 0 ˆ g 0 z 0 g V0 m 2 t 0 ˆ 0; V 0 g ˆ V0 m ; V0 g ˆ const ; z 0 ˆ 3C mr 0 g 0 8r 0 0 r0 p 7 Here C m is a drag coefficient, for high Reynolds number of particle C m " 0.5. The solution of this equation is given by the following formula: s r! Vm 0 t0 ˆ Vg 0 2g 0 z 0 g z 0 tanh 0 2 t0 8 We now estimate the characteristic length over which particle velocity reaches a value of 90 % of the velocity V9 F = V9 g ± 2g9/z9) 1/2, needed to keep particle in dynamic equilibrium with gas flow. The characteristic time t9 F from Eq. 8) is equal to s tf 0 ˆ 2 z 0 atanh g 0 0:9 0:1V 0 g s! z 0 2g 0 9 The estimate for r9 p = 1 cm, r9 0 g = 10 kg/m 3, r9 m = 2500 kg/m 3 V9 g = 2 m/s gives t9 F approximately 1.8 s; therefore, the characteristic length L9 F = V9 g t9 F is approximately a few metres. Since the thickness of the transition zone between the bubbly liquid and the gas-particle dispersion is small in comparison with the characteristic dimension of the problem, this region is simulated by a discontinuity, namely a fragmentation wave. Under the assumptions made above concerning the flow in the bubbly liquid and the gas-particle dispersion, the conservation laws for the fragmentation wave can be written in a following form: 1-a ± ) S ± V ± )=r + m S ± V + m) rg 0 ± a ± S ± V ± )=r + g S ± V + g) Eup s ± + r ± V ± S ± V ± )=Eup + 10) +r + gv + g S±V + g)+r + mv + m S±V + m) F + gm = r + m; p g ± ± p m ± = Dp* Here superscripts ± and + correspond to bubbly liquid and gas-particle mixture, respectively, and S is the velocity of the fragmentation wave. The first two rela-

8 160 tions express the mass conservation laws for the liquid and gas phases. The third expression is the momentum conservation for the mixture as a whole, and the fourth is the particle support condition at the outlet of the fragmentation zone written in place of the momentum equation for one of the phases). The fifth expression is the condition for the fragmentation onset the overpressure in the bubble reaches the critical value). The boundary conditions and numerical methods As the lower boundary condition, either the chamber pressure or the chamber pressure variation with time in the unsteady case must be specified. At the conduit outlet we assign the pressure to be equal to atmospheric pressure if the flow is subsonic, or the velocity to be equal to the local velocity of sound for chocked conditions. Supersonic regimes are not considered in this paper. It is necessary to define the sound velocity for the gas-particle dispersion described by Eq. 5). Equation 5) is a hyperbolic fourth-order system with characteristic velocities: V g @p g ; V g ; V m ; V m. Therefore the velocity of sound for this system is d 1±r m )) ± 1/2. In the steady case the calculations were carried out using the ªcut and tryº method. In each iteration the set of differential equations Eqs. 4) and 5)) were solved by means of fourth-order Runge-Kutta method with automatic step correction. For a given flow rate and the calculated conduit length was compared with the chosen conduit length. Depending on the magnitude and sign of this difference, a new value of the flow rate was chosen. For unsteady cases the numerical method was described in Barmin and Melnik 1996). In each zone Eqs. 4) and 5) were solved on a uniform mesh using the purely implicit compact difference code proposed by Tolstiykh 1990). Then the conservation laws Eq. 10)) were solved at four points in the neighbourhood of the disintegration wave, and values of variables and the velocity of the fragmentation wave were determined. For the purely implicit code the calculation accuracy is of the order of [Dt, h 3 ], where Dt and h are mesh steps of time and coordinate, respectively. The code is stable for any time-step value. The particular time step was chosen to keep the Curant-Fridrix- Levi number less then unity. See the Appendix for the details of the numerical method. Magma flow rate for the steady case Influence of chamber depth and conduit resistance on eruption dynamics By varying pressure in the chamber we obtain a sequence of different steady-state regimes and thus simulate the quasi-steady process of eruption. This is valid when considering gradual changes of chamber pressure, i.e. for a prolonged eruption. In a magma chamber which is closed to new influxes the chamber pressure must fall with time Druitt and Sparks 1984); thus, the chamber pressure can be regarded here as a proxy for time. In order to estimate the typical nondimensional values see Table 1), dimensional parameters are chosen from the well-documented 1980 eruption of Mount St. Helens and of the Bronze Age Minoan eruption at Santorini. In order to illustrate the process of calculation of nondimensional parameters we consider the case of the Mount St. Helens eruption. The initial concentration of dissolved gas in the melt phase c 0 " 4.6 wt. % the magma contains ca. 30 % of crystals) for this eruption. Taking into account the value of the saturation coefficient k9 p = ±6 Pa ± 1/2 we obtain a saturation pressure p9 0 " 125 MPa. For an average melt density r9 0 = kg/m 3 melt density kg/m 3, crystal density kg/m 3 and 30 wt. % of crystals) the value of the characteristic length in this model x9 0 " p9 0 /r9 0 g9 is 5.4 km. As the chamber depth L for this eruption was estimated at approximately 7 km Carey and Sigurdsson 1985), H = L9/x9 0 = The ratio of densities d = r9 g0 /r9 0 = r9 0 / R9T9r9 0 )=r9 0 r9 a / r9 a r9 0 ) r9 0 r9 a = ±2 using r9 a = 0.16 kg/m 3 for pure H 2 O vapour at 1000 C) and p9 a =10 5 Pa ± atmospheric gas density and pressure, respectively. The estimate of Eu number is for a magma ascent speed V9 0 " 1 m/s, Ar = ±1 for a pre-exponential viscosity coefficient m9 0 =10 7 Pa s see Eq. 3)) and conduit diameter d9 =30 m. The parameter C a which corresponds to the pressure disequilibrium is approximately For the Bronze Age Minoan eruption at Santorini dimensional parameters are also summarised in Table 1. Using the same procedure as described above, the set of nondimensional parameters represented in Table 2 are obtained. The sets of steady solutions for discharge rate as a function of chamber pressure are investigated. Dimensionless parameters depend on velocity and therefore on the discharge rate. To fix values of the eruption parameters along the solution curves dimensionless parameters are calculated using a characteristic velocity V9 0, the value of which gives the characteristic discharge rate Q9 0 = V9 0 S9 c r9 0. For each point of solution curve we can calculate the actual set of dimensionless parameters using Q9/ S9 c r9 0 ) as a characteristic velocity. However, all these parameters can be reduced to ini-

9 161 Table 2 Values of dimensional parameters for eruptions of Mount St. Helens and Santorini Parameter Responsible for Mount St. Helens Santorini c 0 Initial mass concentration of dissolved gas 4.6 wt. % 5.5 wt. % Eu Inertia terms scale in momentum equation reciprocal value) Ar Viscous conduit resistance ± d Initial ratio of densities ± C a Pressure nonequilibrium reciprocal value) H Non-dimension chamber depth ±0.75 * Notations a Volume concentration of gas Dp* Fragmentation overpressure r Density 0 r g Density of pure gas l Conduit resistance coefficient, dimensionless m Viscosity of magma t Pressure relaxation time in magma chamber A,B Coefficients in viscosity equations a Bubble radius C m Particle drag coefficient c Mass concentration of the dissolved gas d Conduit diameter or width of fissure F gm Gas drag force in gas-particle dispersion regime g Gravity acceleration H F Coordinate of fragmentation level k p Solubility coefficient L Conduit length L F Particle dynamical equilibrium length n Numerical concentration of the bubbles p 0 Solubility pressure p a Atmospheric pressure p ch Chamber pressure p Pressure Q Magma discharge rate R Gas constant J kg ±1 K ±1 ) r p Particle radius m) S Fragmentation wave velocity S c Conduit cross section area m 2 ) T Temperature t Time t F Particle relaxation time V Velocity x Vertical coordinate, dimensionless Dimensional parameters c 0 Initial mass concentration of dissolved gas 2 Eu = p9 0 /r9 0 V9 0 Inertia terms scale in momentum equation reciprocal value) Ar = m9 0 V9 0 /r9 0 g9d9 2 Viscous conduit resistance d = p9 0 /r9 0 R9T 9 Initial ratio of densities C a = p9 0 x9 0 /m9 0 V9 0 Pressure nonequilibrium reciprocal value) H = L9/x9 0 Non-dimensional chamber depth Subscripts and superscripts 9 Dimensional parameter, all units in CI m Magma property g Gas property 0 Initial or characteristic value a For chamber depth 5 and 6 km, respectively tial calculated with velocity V9 0 ) parameters by multiplication on the corresponding power of Q9/Q9 0 ). For example, Ar actual = Ar 0 Q9/Q9 0 ), Eu actual = Eu 0 Q9/Q9 0 ) ±2, etc. Therefore, along the whole solution curve all the dimensionless parameters with subscript ª0º remain constant. This guarantees that eruption parameters are also constant except the chamber pressure and the discharge rate. In both figures and text, however, the subscript ª0º is omitted for simplification of notations. We focus on the influence of the Ar number, the relative length of the conduit H = L9/x9 0 and initial gas concentration c 0 on the eruption dynamics. The ratio of the real discharge rate Q9 to Q9 0 is shown on all plots. We have fixed the basic set of dimensionless parameters as following: c 0 = 5 wt. %, Ar = 1.5, Eu =10 4, Ca =10 5 and d = Using these parameters wide range of dimensional values can be studied. For example, to obtain the set of dimensional parameters

10 162 listed above the following set of dimensional values can be used: p9 0 = 147 MPa, m9 0 = Pa s, T9 = 1458K, V9 0 = 2.31 m/s, r9 0 = 2738 kg/m 3, d9 = 28.3 m and x9 0 = 5.47 km. Corresponding value of discharge rate Q9 0 is kg/s. In order to compare results of calculations with observational data, dimensional values corresponding to this set of dimensional governing parameters are shown simultaneously with dimensionless data, but dimensionless results cover much wider area because they correspond to a wide range of possible dimensional parameters. With reference to Fig. 1 the aim of the calculations is to establish the relationship between chamber pressure, p ch, and the different regimes of flow. Of principal interest are the magma discharge rate Q, the position of the fragmentation level, and how these parameters vary with chamber pressure. The position of the fragmentation level determines the average mixture weight and conduit resistance resulting in a major influence on magma discharge rate. When the chamber pressure p ch decreases in a sustained explosive eruption and falls below the saturation pressure p 0, the bubbly zone extends into the chamber. The proportion of bubbles at the top of the chamber increases as the p ch decreases with the consequence that the overall magma density decreases in the bubbly zone. For a given flow rate the velocity of the mixture increases together with the melt viscosity due to gas exsolution. These factors lead to an increase in the pressure relaxation time between growing bubbles and surrounding melt, and therefore the bubble overpressure increases more rapidly. The interplay of this process is now explored for Ar = 1.5 and H = 0.6. The fragmentation level position and the discharge rate as a function of chamber pressure are shown in Fig. 2 a and b correspondingly) for this shallow chamber case, when chamber pressure p ch < p 0. Real chamber depth for this case is 3.28 km, lithostatic pressure is 83 MPa and corresponding dimensionless pressure is At point A chamber is overpressurised with overpressure value of 33 MPa, the discharge rate is approximately kg/s m 3 /s). Each point on this diagram represents a steady solution of the boundary problem. For the same values of the chamber pressure there can be several up to five) steady solutions, and the topology of the diagram indicates that there can be sudden jumps from one solution to another. The physical consequences of these multiple solutions can be best illustrated by following a curve from the left to the right in the direction of decreasing chamber pressure. In this case all other eruption parameters are fixed and chamber pressure decreases and can be considered as a proxy for the time in an eruption. The eruption parameters modelled in Fig. 2 correspond to typical plinian eruption conditions from a shallow chamber. In the interval AB the decrease in bubbly zone length is not rapid enough to offset the decrease in p ch ; therefore, the flow rate decreases slightly. From a certain value of the pressure p ch the decrease of the fragmentation level is sufficiently rapid that the flow rate, passing through the minimum point B, begins to increase. This increase in flow rate leads to increase in the rate of bubble overpressure growth, and therefore the fragmentation level goes further down to the conduit. A critical condition is reached at point C in which there is no continuous steady-state solution as the chamber pressure decreases further. The system must adjust to one of three other solutions. The plau- Fig. 2 Fragmentation level position measured from the chamber and related to the length of the conduit) and discharge rate of eruption related to Q9 0 = r9 0 V9 0 S9 c ) as a function of chamber pressure p ch. On the opposite axis typical dimensional values are given. Each point represents a steady solution of a boundary problem for the conduit flow. In the point A the discharge rate is in the range of plinian phase of Santorini BC 1650 eruption. As p ch decreases, fragmentation level goes dipper in the conduit and at critical point C the intensity of eruption increases an order of magnitude while fragmentation level reaches the chamber exit. At the point F eruption abruptly stops with a possibility of caldera collapse

11 163 sible response is that the fragmentation level drops dramatically and flow rate increases by an order of magnitude to point E Fig. 6 presents the calculation when this transition occurs monotonically). Note that the flow rate increases by an order of magnitude and reach the value of approximately kg/s m 3 /s) and that the fragmentation level is at the chamber exit. Regimes CD±DE cannot be physically reached when the pressure in the chamber decreases monotonically. Due to the reduction in length of the bubbly zone, the pressure drop takes place mainly in the gas-particle mixture zone at the expense of its weight interval EF). As p ch decreases further the amount of gas contained in the bubbles before fragmentation increases, and the average weight of the gas-dispersion decreases. Finally, this weight becomes comparable to the conduit resistance in the short bubbly zone. It leads to a rapid decrease in the flow rate as point F is approached, where critical conditions are reached. Further monotonic decrease in flow rate for the pressures lower then p ch F) is impossible, because it requires the increase in fragmentation level, and therefore, sharp increase in conduit resistance. From the point F the system may change to a low-intensity eruption regime with flow rate comparable to lava dome growth flow rates). In this regime the entire conduit is filled with bubble liquid, because the overpressure in the growing bubble does not exceed the critical value and no fragmentation occurs. Of course, due to the elasticity of conduit and chamber walls, the pressures corresponding to the end of the explosive part of the eruption will be higher and explosive activity will stop somewhere in the interval EF. It is anticipated that the very low chamber pressures near point F are not realised, so that the conduit closes or the roof of the chamber collapses. It is clear from Fig. 2 that for fixed chamber pressure up to five steady-state eruption regimes are possible. In Fig. 3 the profiles of pressure along the volcanic conduit are shown for p ch = Curve 1 corresponds to the upper regime EF). Here the length of the bubbly zone is too small to be shown in the plot, and the pressure drop occurs in the gas-particle dispersion. Because the outlet flow is sonic, the outlet pressure is greater than atmospheric. Curve 2 corresponds to B±C branch of solution. In the short liquid zone, the sharp pressure decrease occurs due to the high resistance of the conduit. In the lowest regime G±H, curve 3) most of the conduit is occupied by bubbly melt and the pressure distribution is closer to lithostatic. As the critical overpressure in the growing bubble is used for the fragmentation condition, the volume concentration of bubbles does not have a fixed value at the fragmentation. Values of volume concentration on the fragmentation level are: 0.23 for the upper regime; 0.44 for the intermediate; 0.78 for the lowest regime. Fig. 3 Pressure profiles for fixed chamber pressure p ch = 0.6 and different flow regimes. Jog point corresponds to the fragmentation level. Curve 1 corresponds to the highest intensity regime DF in Fig. 2), where all the conduit is occupied by gas-particle dispersion; curve 2 to initial phase AC); curve 3 to the low-intensity regime GH) In the case of a deeper chamber H = 0.9; Fig. 4) the topology of the solution is more complicated. Three steady solutions correspond to the chamber pressure range between p ch A) and p ch C). They form an S-shaped curve obtained also in Barmin and Melnik 1990, and Slezin 1991). At lower pressures two steady regimes exist, which cannot be reached by monotonic decrease of chamber pressure. On the plane Q ± p ch they form a cyclic curve. Discharge rate at point A is 10 8 kg/s, pressure is equal to saturation pressure 147 MPa), and chamber is overpressurised by the value of 22MPa. The evolution of the steady solution topology as a function of chamber depth is shown in Fig. 5. For deeper chamber location the loop contracts into a point see curves for H = 0.9, 0.95) and the S-shaped curve moves to the higher chamber pressure values. At H = 1.5 the S-shaped dependence of discharge rate on chamber pressure becomes monotonic and singlevalued. These results show that condition for sudden changes in discharge rate can occur in magma chambers that are saturated in volatiles, and therefore for shallow chambers and high gas content. Corresponding values of chamber depth and lithostatic pressures for dimensionless chamber depth are: 0.95±5.2 km and 132 MPa; 1.2±6.6 km and 167 MPa; 8.2 km and 209 MPa for H = 1.5. The effect of Ar number on the dynamics of eruption is now considered. Larger Ar values correspond