A numerical study of turbulent mixing in eruption clouds using a three-dimensional fluid dynamics model

Authors


Abstract

[1] The dynamics of eruption clouds in explosive volcanic eruptions are governed by entrainment of ambient air into eruption clouds by turbulent mixing. We develop a new numerical pseudo gas model of an eruption cloud by employing three-dimensional coordinates, a third-order accuracy scheme, and a fine grid size in order to investigate the behavior of entrainment due to turbulent mixing. The quantitative features of entrainment are measured by a proportionality constant relating the inflow velocity at the edge of the flow to the average vertical velocity (i.e., the entrainment coefficient). Our model has successfully reproduced the quantitative features of entrainment observed in the laboratory experiments as well as fundamental features of the dynamics of eruption clouds, such as the generation of eruption columns and/or pyroclastic flows. The value of the entrainment coefficient for eruption clouds is estimated from the column height and critical condition for column collapse by comparing results of our model with those of previous one-dimensional models. It is suggested that the value of the entrainment coefficient for an eruption cloud is approximately constant, although the value estimated from the critical condition for column collapse (k ∼ 0.07) is slightly smaller than that based on the column height (k ∼ 0.1). This difference reflects the vertical change of flow structure in the eruption cloud. The eruption cloud in the upper region exhibits a meandering instability, which leads to efficient mixing, whereas the cloud near the vent maintains a concentric structure with an inner dense core surrounded by an outer shear region. Our model is consistent with previous one-dimensional models for steady eruption clouds supported by the laboratory experiments, and it is also applicable to unsteady and transient features of actual eruption clouds.

1. Introduction

[2] Explosive volcanism is one of the most catastrophic phenomena on the earth. The dynamics of explosive eruption clouds has been the central issue of volcanology and hazard science for a long time [e.g., Bullard, 1962; Woods, 1988; Valentine, 2001]. During explosive volcanic eruptions, a mixture of hot ash (solid pyroclasts) and volcanic gas is released from the volcanic vent into the atmosphere. Such events are characterized by the formation of eruption columns and/or pyroclastic flows. Turbulent mixing in and around eruption clouds is an essential part of the dynamics of eruption clouds because the amount of entrained air controls whether or not the eruption cloud becomes buoyant. When the ejected material entrains sufficient air to become buoyant, a large Plinian eruption column rises up to a height of several tens of kilometers as a turbulent plume. The eruption column exhausts its thermal energy and loses its buoyancy within the stratified atmosphere [Morton et al., 1956; Woods, 1988]. At the neutral buoyancy level where the cloud density is equal to that of the atmosphere, the eruption cloud spreads radially and an umbrella cloud grows. On the other hand, if the ejected material does not entrain sufficient air and its vertical velocity falls to zero before the eruption cloud becomes buoyant, a column collapse occurs and the heavy and hot cloud spreads radially as a pyroclastic flow.

[3] Previous studies using steady one-dimensional (1-D) models of eruption clouds [Wilson, 1976; Sparks, 1986; Wilson and Walker, 1987; Woods, 1988, 1995] described the behavior of turbulent mixing in a simple form based on the theoretical work of Morton et al. [1956]. The steady 1-D models applied the entrainment hypothesis for turbulent mixing that the mean inflow velocity across the edge of turbulent flow (jet and/or plume) was assumed to be proportional to the mean vertical velocity. The proportionality constant represents the efficiency of entrainment and it is called the entrainment coefficient. Indeed, experimental studies [e.g., Papanicolaou and List, 1988] have supported the entrainment hypothesis in the case where there is no chemical reaction and the characteristic dimension of the source is small compared with the height of rise at a sufficient downstream distance from the origin. Because the steady 1-D models were based on the entrainment coefficient determined by the experiments of steady jet and/or plume in a uniform (i.e., without density stratification) environment (hereafter referred to as JPUE), they necessarily reproduced the quantitative features of JPUE such as a shape and dimensions. They could also explain fundamental features of a steady state of eruption clouds and the critical conditions for column collapse. However, it is not clear that the entrainment coefficient of an eruption cloud is always equal to the value for the simple case of the JPUE. Besides, by definition the steady 1-D models could not describe unsteady and transitional features of actual eruption clouds.

[4] Over the past 20 years, the development of two-dimensional (2-D), time-dependent, and multiphase numerical models for eruption clouds have provided new explanations for many features of explosive volcanism [Wohletz et al., 1984; Valentine and Wohletz, 1989; Dobran et al., 1993; Neri and Dobran, 1994; Neri and Macedonio, 1996; Oberhuber et al., 1998; Clarke et al., 2002; Neri et al., 1998, 2002a, 2002b, 2003]. These studies focused on the unsteady and multiphase features of eruption clouds, and demonstrated some features of eruption clouds, including stable column, column collapse, and oscillating behavior. However, their predictions for column collapse were not always quantitatively consistent with those of the steady 1-D models in which the entrainment coefficient based on the laboratory experiments was used [Neri and Dobran, 1994]. One possibility for this inconsistency is that the estimations of efficiency of entrainment depend on the numerical simulation procedure.

[5] The aim of this paper is to present a new multidimensional time-dependent fluid dynamics model that can reproduce the entrainment process of eruption clouds. For this purpose, we develop a three-dimensional (3-D) numerical model with high spatial resolution, which is applicable to time-dependent phenomena in actual volcanological situations, and at the same time, correctly reproduces the quantitative features of turbulent mixing in and around the JPUE under the ideal experimental conditions.

2. Model Description

[6] The model is designed to describe the injection of a mixture of solid pyroclasts and volcanic gas from a circular vent above a flat surface of the earth in a stationary atmosphere. In this paper, because we are particularly concerned with turbulent mixing of eruption clouds, we adopt a pseudo gas model; we ignore the separation of solid pyroclasts from the eruption cloud. The momentum and heat exchanges between the solid pyroclasts and gas are assumed to be so rapid that the velocity and temperature are the same for all phases. These assumptions are valid when the size of solid pyroclasts is sufficiently small (<4 mm) [Woods and Bursik, 1991]. Sparks and Wilson [1976] suggested that over 90% of the solids pyroclasts are less than 5 mm in diameter and over 60% are submillimeter in diameter for typical Plinian or phreatomagmatic eruptions. Pseudo gas models are justified for such types of explosive eruptions. The volcanic gas is assumed to be water vapor.

[7] The fluid dynamics model solves a set of partial differential equations describing the conservation of mass, momentum, and energy (see Appendix A), and a set of constitutive equations describing the thermodynamic state of the mixture of solid pyroclasts, volcanic gas, and air. These equations are solved numerically by a general scheme for compressible flow with high spatial resolution. All the constants used in this study are listed in Table 1.

Table 1. List of Material Properties and Values of Physical Parameters
VariableValueUnitsMeaning
g9.81m/s2gravitational body force
Rg462J/(kg K)gas constant of volcanic gas (water vapor)
Ra287J/(kg K)gas constant of atmospheric air
Cvs1617J/(kg K)specific heat of solid pyroclasts
Cvg1155J/(kg K)specific heat of volcanic gas (water vapor) at constant volume
Cva717J/(kg K)specific heat of atmospheric air at constant volume
γ1.40ratio of specific heat for gas phase
Ta0273Ktemperature of air at z = 0 km
pa01.013 × 105Papressure of air at z = 0 km
ρa01.29kg/m3density of air at z = 0 km
H111kmheight of the tropopause
H220kmheight of the stratopause
μ16.5K/kmtemperature gradient in the troposphere
μ22.0K/kmtemperature gradient in the stratosphere

2.1. Constitutive Equations

[8] The most essential physics which governs the dynamics of eruption clouds is that the density of eruption clouds varies nonlinearly with the mixing ratio between the ejected material and air (Figure 1; see Appendix B). Generally, the ejected material has an initial density of several times as large as the atmospheric density since it contains more than 90 wt% solid pyroclasts at the vent [e.g., Sparks and Wilson, 1976]. As the ejected material is mixed with ambient air and the mass fraction of the ejected material decreases, the density of the mixture drastically decreases and becomes less than the atmospheric density because the entrained air expands by heating from the hot solid pyroclasts. The density of the mixture is also a function of magmatic temperature. As the magmatic temperature decreases, the critical mass fraction at which the density of the mixture is equal to that of air decreases. In the case that the magmatic temperature is less than 400 K, the mixture is always heavier than air. Such situations may be relevant to phreatomagmatic eruptions [e.g., Koyaguchi and Woods, 1996].

Figure 1.

Variation of the mixture density of the ejected material plus ambient air as a function of the mass fraction of the ejected material in the mixture. The density is normalized by the atmospheric density. Curves are shown for an initial mass fraction of volcanic gas of ng0 = 0.05 with an initial temperature of 400 (solid curve), 550 (dotted curve), and 1000 K (dashed curve). Atmospheric air has a temperature of 273 K.

[9] We reproduce the nonlinear features in Figure 1 by changing the effective gas constant of the mixture in the equation of state for ideal gases. On the assumption that the differences of velocity and temperature between solid pyroclasts and gas are zero, the equation of state for the mixture of the ejected material and air is

equation image

where ρ is the density of the mixture, σ is the density of the solid pyroclasts, Rg and Ra are the gas constants of volcanic gas and air, respectively, T is the temperature, and p is the pressure. The mass fractions of solid pyroclasts (ns), volcanic gas (ng), and air (na) satisfy the condition of ns + ng + na = 1. The mass fraction of the ejected material in the eruption cloud is given by ξ = ns + ng. The initial mass fraction of volcanic gas (i.e., volatile content in the magma) is given by ng0 = ng/(ns + ng) using these notations. The subscripts s, g and a refer to solid pyroclasts, volcanic gas and air, respectively. The first term of the right-hand side of equation (1) represents the volume of the solid phase in a unit mass of the mixture, and the second term represents the volume of the gas phase. The first term is negligible relative to the second term when the pressure is close to atmospheric pressure (∼105 Pa), because the density of the solid pyroclasts is 103 times as large as that of the gas phase. Therefore equation (1) can be approximated by the equation of state for an ideal gas as

equation image

where the subscript m in Rm refers to the mixture of solid pyroclasts, volcanic gas, and air.

[10] In practice, the change in Rm with mixing ratio is taken into account in the following way. The change of internal energy is proportional to the change of temperature:

equation image

where Cvm is the average specific heat at constant volume, which is defined using the specific heats of solid pyroclasts (Cvs), volcanic gas (Cvg), and air (Cva) as

equation image

When the specific heats of each component (Cvs, Cvg, and Cva) are constant, we can define the internal energy of the mixture, e, in the equation of energy conservation (equation (A4)) such that

equation image

Since the ratio of specific heat at constant volume and constant pressure of the mixture can be defined as

equation image

we can calculate the pressure at the position with an arbitrary mixing ratio using equations (2), (5), and (6) as

equation image

2.2. Numerical Procedure

[11] The calculation is performed on 2-D and 3-D domains. In the 2-D model, we assume axial symmetry and use a uniform rectangular grid. In the 3-D model, we use a uniform grid in a Cartesian coordinates system. Initial and boundary conditions are provided in Appendix C.

[12] The partial differential equations (A1)(A4) are solved numerically for ρ, ρu, e, and ρξ by the Roe scheme [Roe, 1981] in space, which is a general total variation diminishing (TVD) scheme for compressible flow and can simulate a generation of shock waves inside and around the high-speed jet correctly. The Roe scheme can be applied to the present problem of the dynamics of eruption clouds, because the above assumption of the equation of state (i.e., equation (2)) makes it possible to derive analytically the eigenvalues and eigenvectors for the governing equations of two fluids (i.e., the ejected material and air) [Wada et al., 1989]. The MUSCL method is applied to interpolate the fluxes between grid points [van Leer, 1977], and therefore our numerical model solves the Euler equation to third-order accuracy in space. These equations are solved using the time splitting method. We treat the gravitation term of the equations of momentum and energy conservation (equations (A3) and (A4)), and the additional terms due to the curvature of axisymmetric 2-D coordinates as source terms. The density, velocity, total energy, and the mass fraction of the ejected material are solved fully explicitly. After the mixing ratio is calculated, the temperature and pressure are updated employing equations (5) and (7), respectively. The present numerical code is based on the astronomical work of Hachisu et al. [1990], who reproduced most of the observational indications of mixing in SuperNova 1987A.

3. Turbulent Mixing

[13] A number of theoretical and experimental studies on a turbulent jet or plume which is ejected from a nozzle into a uniform environment (i.e., JPUE) have revealed that such a JPUE is characterized by the self-similarity that the radial length scale is proportional to the distance from the nozzle (or a virtual point of origin). This means that the evolution of the JPUE is determined solely by the local scales of length and velocity [Tennekes and Lumley, 1972]. This feature theoretically accounts for the fact that the rate of entrainment at the edge of the JPUE is proportional to a certain characteristic velocity at each height [Morton et al., 1956]. Experimental studies suggest that the proportionality constant, that is, the value of the entrainment coefficient is ∼0.08 for a pure jet that is driven by the initial momentum, and ∼0.12 for a pure plume that is driven by buoyancy [e.g., Fischer et al., 1979]. The numerical models of eruption clouds must reproduce these features of entrainment qualitatively and quantitatively under the ideal condition of JPUE, because flow of an eruption cloud is also a kind of free boundary shear flow with a very high Reynolds number (Re > 108).

[14] The above features of JPUE result from the two processes of turbulent mixing: (1) engulfment process [Mathew and Basu, 2002] and (2) diastrophy and infusion processes [Dimotakis, 1986]. The engulfment of ambient fluid is caused by the large-scale structures of turbulence. Subsequently, turbulent straining of the entrained fluid reduces its spatial scale to a small enough value at which viscous diffusion dominates (diastrophy). Finally, because of viscous diffusion, the inducted fluid is mixed at the molecular level with the turbulent flow (infusion). Therefore the diastrophy and infusion processes are associated with smaller-scale vortices than those of the engulfment process. In order to reproduce these processes of turbulent mixing numerically, we consider three factors of numerical procedure: (1) three dimensionality, (2) spatial resolution, and (3) subgrid-scale model of turbulence. Since the large-scale structures of turbulence are 3-D in general, the engulfment process should be reproduced on 3-D coordinates. In addition, the grid size should be small enough to resolve the large-scale structures of the engulfment process [Moin and Kim, 1982]. Because, generally speaking, all the small-scale structures of the diastrophy and infusion phases cannot be resolved with a given grid size, the subgrid-scale models of turbulence such as the large eddy simulation [Smagorinsky, 1963; Deardorff, 1970] are commonly applied to the numerical simulation of turbulent flow.

[15] In the following, we systematically evaluate the effects of the above three factors on the entrainment process. For this purpose, we carried out supplementary simulations for the case of the JPUE, whose qualitative and quantitative features were experimentally investigated by a number of previous workers [e.g., Fischer et al., 1979]. In these supplementary simulations, it is assumed that both the ejected and surrounding fluids are air of the same temperature (T = 273 K) and the same ratio of specific heat (γm = 1.40).

3.1. Effects of Three Dimensionality

[16] We simulate the turbulent jet with axisymmetric 2-D and 3-D coordinates and compare our results with the experimental studies from the viewpoint of self-similarity. In the 3-D simulations, the ejected fluid exhibits a meandering instability where the axis of the flow varies with height, which causes efficient turbulent mixing of the ejected and surrounding fluids (Figure 2b). As a result, the radius of jet increases linearly with height. These features are consistent with the laboratory experiments (Figure 2a) [Papanicolaou and List, 1988]. The spreading rate of the jet is explained by the entrainment coefficient of ∼0.08 (Figures 2b and 3). On the other hand, in the axisymmetric 2-D simulations, the ejected fluid rises along the central axis and the spreading rate of the jet is substantially smaller than the results of the experimental measurements and the 3-D simulations (Figures 2c and 3). This difference implies that the efficiency of turbulent mixing is significantly reduced because of the boundary condition at the centerline of the axisymmetric coordinates.

Figure 2.

Comparison of jets ejected into the same fluid for the previous experimental studies and the present model. (a) Laser sheet illumination photograph of experimental high-Re jet (Re ∼ 104) (reprinted with permission from Dimotakis et al. [1983, Figure 9], copyright 1983, American Institute of Physics). The remaining plots illustrate the cross-sectional distribution of the mass fraction of the ejected fluid on the basis of the results of our model. The horizontal distance from the centerline and the vertical distance from the nozzle are represented by x and z, respectively. The contour levels are 10−4, 10−3, 10−2, 10−1, and 5 × 10−1. (b) Simulation of the third-order accuracy scheme with Δx = L0/6 in 3-D coordinates where Δx is the grid size. (c) Simulation of the third-order accuracy scheme with Δx = L0/6 in 2-D coordinates. (d) Simulation of the first-order accuracy scheme with Δx = L0/6 in 3-D coordinates. (e) Simulation of the third-order accuracy scheme with a coarse grid size (Δx = 2L0) in 3-D coordinates. (f) Simulation of the third-order accuracy scheme with Δx = L0/6 and the LES in 3-D coordinates. (g) Simulation of the first-order accuracy scheme with Δx = L0/6 and the LES in 3-D coordinates. Dashed lines in Figures 2b–2g indicate the shape of spreading jet on the basis of the previous experimental studies with k = 0.08 [e.g., Papanicolaou and List, 1988].

Figure 3.

Velocity profiles across a turbulent jet. Vertical axis represents the velocity (u) normalized by the centerline value (uc). Horizontal axis represents dimensionless displacement (x/z). Curves b, c, d, f, and g are the time-averaged velocity profiles at fixed cross sections for the simulations of Figures 2b, 2c, 2d, 2f, and 2g, respectively. The heights of the cross sections are shown by arrows in Figure 2. Curve b illustrates the simulation of the third-order accuracy scheme in 3-D coordinates. Curve c illustrates the simulation of the third-order accuracy scheme in 2-D coordinates. Curve d illustrates the simulation of the first-order accuracy scheme in 3-D coordinates. Curve f illustrates the simulation of the third-order accuracy scheme with the LES in 3-D coordinates. Curve g illustrates the simulation of the first-order accuracy scheme with the LES in 3-D coordinates. In all the calculations, grid size is set to be Δx = L0/6. Curve a is the Gaussian profile (u/uc = exp [−84(x/z)2]) which gives the entrainment coefficient with respect to a top hat profile k = 0.08 (i.e., the entrainment coefficient with respect to a Gaussian profile aJ = 0.055). This value of the entrainment coefficient is based on the experiments of fully developed turbulent jets [e.g., Papanicolaou and List, 1988].

3.2. Effects of Spatial Resolution

[17] Generally speaking, the efficiency of turbulent mixing is a function of the Reynolds number [e.g., Dimotakis and Catrakis, 1999]. At Re < 104, even though the flow may be unsteady, the resulting turbulent flow cannot be described as fully developed and the efficiency of mixing increases with Re. On the other hand, at Re > 104 the turbulence is fully developed, and the large-scale structures of the engulfment process and the efficiency of entrainment no longer depend on Re. We call this transition to the fully turbulent flow (Re ∼ 104) “mixing transition.” Because the flow of eruption clouds is considered to be fully turbulent, the simulations of eruption clouds should be carried out at high “numerical Reynolds number” (defined as Re*) above the mixing transition in order to reproduce the turbulent mixing correctly.

[18] In a numerical simulation of fluid dynamics, Re* increases by means of increasing spatial resolution. High spatial resolution can be attained by (1) high-order accuracy schemes and (2) fine grid sizes. We evaluate these two effects here. Figure 2b indicates that the third-order accuracy scheme with a fine grid size reproduces the turbulence containing the various scale of vortices, and that the spreading rate of the jet is consistent with laboratory experiments for a fully developed turbulent jet (Re ∼ 104) (Figure 2a). On the other hand, in the simulations of the first-order accuracy scheme with the same grid size (Figure 2d) or those of the third-order accuracy scheme with a coarse grid size (Figure 2e), the vortical structures of the JPUE are not correctly reproduced. In the simulation using the first-order accuracy scheme, the spreading rate of the jet is smaller than laboratory experiments, suggesting that the efficiency of entrainment is substantially reduced in this simulation (Figure 3). These results indicate that the condition above the mixing transition is achieved only when the third-order accuracy scheme with a fine grid size is applied in the present model.

3.3. Effects of Subgrid-Scale Model

[19] We carried out numerical simulations of jets with and without the large eddy simulation (LES), and compared them in order to investigate the effects of the small-scale structures that cannot be resolved on a given grid size. Simulation results show that when spatial resolution is sufficiently high using the third-order accuracy scheme and fine grid size, both the numerical results with and without the LES correctly reproduce the spreading rate of jets observed in the experiments (Figures 2b, 2f, and 3). On the other hand, when spatial resolution is insufficient using the first-order accuracy scheme, the spreading rate of jets in both the numerical results with and without the LES is substantially smaller than that of the experiments (Figures 2d, 2g, and 3). These results indicate that spatial resolution is the essential factor, but the subgrid-scale models play only a secondary role in reproducing the global features of turbulent mixing and the efficiency of entrainment.

[20] The above results are explained by the fact that the efficiency of entrainment is determined by the kinematic evolution of the largest eddies, and that the major function of the subgrid sizes is only to dissipate the kinetic energy provided by the large eddies. Therefore the conclusion that the subgrid-scale models play only a secondary role in reproducing turbulent mixing is considered to be robust regardless of the details of subgrid filters, at least for free boundary shear flow such as the JPUE. Note that the subgrid-scale models may play an essential role in simulating the detailed structures of pyroclastic flows, because the kinetic energy produced in the small-scale structures within the wall-bounded shear flow can affect the mean flow of turbulence [Tennekes and Lumley, 1972].

3.4. Summary of Treatment for the Turbulent Mixing

[21] Judging from the numerical simulations for the JPUE, the global features of turbulent mixing and efficiency of entrainment depend mainly on the engulfment process, and not on the diastrophy and infusion processes. These results are consistent with the interpretation by Turner [1986] that the key step in the entrainment process is the one that controls the rate at which ambient fluid enters into the turbulent region, i.e., the engulfment process. In order to correctly reproduce the engulfment process, it is essential to apply 3-D coordinates with a sufficiently high spatial resolution; simulations should be carried out above the mixing transition where the efficiency of entrainment is independent of Re* (i.e., independent of grid sizes). In the following analyses, we have applied a third-order accuracy scheme, and also have carefully performed sensitivity tests with different grid sizes to find the condition where the efficiency of turbulent mixing no longer depends on the grid size.

4. Application to the Dynamics of Eruption Clouds

[22] Turbulent jets and plumes of eruption clouds differ in several ways from the ideal situations of the JPUE. In eruption clouds the magnitude of buoyancy drastically changes with the amount of entrained air due to the nonlinear feature of the equation of state (Figure 1), whereas the relationship between ρ and ξ can be approximated by a linear function in the case of the JPUE. Secondly, the surrounding atmosphere is not uniform but stratified. Thirdly, the length scale of source cannot be ignored near the vent in comparison with the downstream distance from the vent. Because of these differences, the assumption of self-similarity is not always valid for the flow of eruption clouds. This means that in this case a constant entrainment coefficient is not guaranteed. In the preceding section we have developed a numerical model which can reproduce the quantitative features of entrainment in the JPUE (i.e., constant entrainment coefficient). We will apply our numerical model to the dynamics of eruption clouds, and investigate how these features of entrainment are modified.

[23] Table 2 lists the initial conditions for the simulations of the 3-D model. In most simulations, an initial temperature of T0 = 1000 K and initial mass fraction of volcanic gas of ng0 = 0.05 are assumed (group P), while some simulations use a lower temperature condition (T0 = 550 K and ng0 = 0.05 in group Q; T0 = 400 K, ng0 = 0.05 in group S) and are performed for comparison. The simulation time of each run is 200–400 s, which is sufficient to establish the steady state of the column and of the pyroclastic flow.

Table 2. Summary of the Input Parameters of the Simulations of Group P, Run Q1, and Run S1
SimulationTemperature, KMass Discharge Rate, kg/sVelocity, m/s
Group P
Run P11000107.6150
Run P21000108.2150
Run P31000108.6150
Run P41000108.65150
Run P51000108.7150
Run P61000108.8150
Run P71000109.2150
Run P81000109.2200
Run P91000109.4200
Run P101000109.6200
Run P111000109.6250
Run P121000109.8250
Run P1310001010.0250
Run P141000109.8300
Run P1510001010.0300
Run P1610001010.2300
Run P1710001010.6300
 
Group Q
Run Q1550106.2120
 
Group S
Run S1400106.2120

4.1. Qualitative Features of Eruption Clouds

[24] Our simulations have successfully reproduced the behavior of eruption clouds including eruption columns and/or the formation of pyroclastic flows. The flow patterns can be classified into three regimes: stable column, partial column collapse, and full column collapse regimes. We present qualitative features of turbulent mixing for these regimes on the basis of representative results.

4.1.1. Stable Column

[25] Run P1 shows the typical result of stable column (Figure 4). The eruption cloud is ejected from the vent with a large density (4.39 kg/m3) relative to the atmospheric density (1.29 kg/m3). As it rises and entrains ambient air from the edge of the eruption cloud, the density of the cloud becomes less than that of air and then a buoyant plume rises up to a height of 12 km at 180 s (Figure 4b). The buoyant plume is highly unstable as it ascends, and undergoes a meandering instability that induces efficient mixing so that its radial scale gradually increases with height. At 270 s the plume has reached its maximum height (about 18 km), and the density at the top of the column (above 12 km) becomes larger than ambient air (Figure 4c). Subsequently, the umbrella cloud begins to spread radially near the top of the column, and reaches 8 km from the central axis horizontally by 360 s, while the top of the column is kept at 18 km high (Figure 4d). These results clearly indicate that the flow pattern of stable column is characterized by the 3-D structures such as a meandering instability, which have not been observed in the previous 2-D model [Oberhuber et al., 1998] or axisymmetric 2-D models [Valentine and Wohletz, 1989; Neri and Dobran, 1994; Neri et al., 1998].

Figure 4.

Numerical results of stable column at (a) 90, (b) 180, (c) 270, and (d) 360 s from the beginning of eruption in run P1. Parameters used and conditions at the vent are listed in Tables 1 and 2, respectively. Cross-sectional distributions of the density difference relative to the stratified atmospheric density at (left) the same vertical position ρ/ρa − 1 and (right) the mass fraction of the ejected material ξ are shown in x-z space. The contour levels in plots on the right are ξ = 10−4, 10−3, 10−2, 10−1, and 5 × 10−1.

Figure 4.

(continued)

[26] The flow near the vent is characterized by a concentric structure consisting of an outer shear region and inner dense core (Figure 5b). In the outer shear region, the ejected material and ambient air are efficiently mixed by the eddy due to shear so that the density of the mixture becomes less than that of air (Figure 5a). In the inner dense core, the ejected material is not mixed with ambient air (the mass fraction of the ejected material is about 1.0) (Figure 5b) and its density remains larger than that of air (Figure 5a). As the eruption cloud ascends, the inner dense core disperses because of erosion by the outer shear region. In the case that the vent radius is sufficiently small (140 m in run P1), the density of the resultant mixture becomes less than that of ambient air before the initial momentum at the vent is exhausted (Figure 5a).

Figure 5.

Representative numerical results of the stable column regime (run P1) showing the detail flow structures near the vent. Parameters used and conditions at the vent are listed in Tables 1 and 2, respectively. (a) Cross-sectional distribution of the density difference relative to the stratified atmospheric density at the same vertical position (ρ/ρa − 1) in x-z space at 190 s. (b) Cross-sectional distribution of the mass fraction of the ejected material (ξ) in x-z space at 190 s. We call the region where the mass fraction of the ejected material is 1.0 the inner dense core. It is surrounded by the outer shear region, where the ejected material entrains ambient air. In this calculation, the initial momentum is exhausted at about 1.5 km height.

[27] In the case that the vent radius is large (280 m in run P2), the outer shear region cannot reach the central axis before the initial momentum is exhausted (Figure 6). The inner dense core is maintained up to a height of about 1.5 km and the top of the inner dense core subsequently spreads radially. This structure is called as “the radially suspended flow” [after Neri and Dobran, 1994]. The inner dense core and outer shear region are mixed by the large-scale eddy of the suspended flow (Figure 6b). Consequently, the resultant mixture becomes buoyant and produces another type of stable column (Figure 6a).

Figure 6.

Representative numerical results of the stable column regime which is characterized by the suspended flow of the inner dense core (run P2). Parameters used and conditions at the vent are listed in Tables 1 and 2, respectively. (a) Cross-sectional distribution of the density difference relative to the stratified atmospheric density at the same vertical position (ρ/ρa − 1) in x-z space at 230 s. (b) Cross-sectional distribution of the mass fraction of the ejected material (ξ) in x-z space at 230 s. The contour levels in Figure 6b are ξ = 0.001, 0.1, 0.5, 0.8, and 0.9. The suspended flow develops when the inner dense core remains at the height where the initial momentum is exhausted (1.5 km in this calculation).

4.1.2. Partial Column Collapse

[28] When the vent radius is still larger (460 m in run P4), a buoyant plume and pyroclastic flow develop simultaneously (Figure 7). We define such a flow pattern as “partial column collapse.” In run P4, the inner dense core remains when the initial momentum is exhausted (Figure 7b). It forms suspended flow and mixes with the outer shear region because of a large-scale eddy (cf. run P2). As the vent radius increases, the ratio of the entrained air against the ejected material in the eruption cloud (i.e., 1 − ξ in Figure 1) decreases, so that the average density of the mixture in the suspended flow increases. Some parts of the mixture remain heavier than air, whereas the others become lighter (Figure 7a). The heavier parts collapse to the ground and spread radially as a pyroclastic flow. The lighter parts of the mixture continue to rise as a buoyant plume and form an umbrella cloud. The upper region of the pyroclastic flow entrains air and forms coignimbrite ash clouds which subsequently join the buoyant plume.

Figure 7.

Representative numerical results of the partial column collapse regime (run P4). Parameters used and conditions at the vent are listed in Tables 1 and 2, respectively. (a) Cross-sectional distribution of the density difference relative to the stratified atmospheric density at the same vertical position (ρ/ρa − 1) in x-z space at 270 s. (b) Cross-sectional distribution of the mass fraction of the ejected material (ξ) in x-z space at 270 s. (c) Trajectories of marker particles which are initially placed above the vent superposed on the result in Figure 7b. (d) Heights of the marker particles as a function of time. The contour levels in Figures 7b and 7c are ξ = 0.001, 0.1, 0.5, 0.8, and 0.9. In order to avoid the effect of initial expansion and to focus on the patterns of semisteady flow, the marker particles in Figures 7c and 7d are placed above the vent at 100 s. In this calculation, the initial momentum is exhausted at about 1.5 km height.

[29] These features of partial column collapse are represented by trajectories of marker particles (Figures 7c and 7d). Some particles fall to the ground after moving along the suspended flow at 1.5 km high, whereas the others continue to ascend from the suspended flow. The fallen particles move along the ground surface, and then some of them ascend again when the coignimbrite ash cloud develops.

[30] When the initial temperature is low and the vent radius is small (run Q1 with T0 = 550 K and L0 = 23 m; the result of this run is not shown), the partial column collapse occurs without the inner dense core developing a suspended flow; collapse occurs after the inner dense core (i.e., the high-concentration part of the ejected material) disperses because of erosion by the outer shear region. When the initial temperature is low, the amount of entrained air necessary to generate a buoyant plume is large (see Figure 1). Therefore the whole mixture cannot become buoyant, even though a jet with a small vent radius entrains a relatively large amount of air.

4.1.3. Full Column Collapse

[31] When the vent radius is extremely large (880 m in run P7), the eruption column fully collapses to spread radially as a pyroclastic flow (Figure 8). Only a small amount of air is entrained into the eruption cloud in the case of large vent radius, so that, the most parts of the mixture due to the suspended flow remain heavier than air and collapse to the ground (Figure 8a). After downflow of the collapsing column develops, the downflow prevents mixing between newly ejected materials and ambient air (see arrows in Figure 8b). Consequently, the eruption column continuously collapses and forms a dense pyroclastic flow in which the mass fraction of the ejected material is maintained about 1.0 [cf. Valentine and Wohletz, 1989].

Figure 8.

Representative numerical results of the full column collapse regime (run P7). Parameters used and conditions at the vent are listed in Tables 1 and 2, respectively. (a) Cross-sectional distribution of the density difference relative to the stratified atmospheric density at the same vertical position (ρ/ρa − 1) in x-z space at 200 s. (b) Cross-sectional distribution of the mass fraction of the ejected material (ξ) in x-z space at 200 s. (c) Trajectories of marker particles which are initially placed above the vent superposed on the result in Figure 8b. (d) Heights of the marker particles as a function of time. The contour levels in Figures 8b and 8c are ξ = 0.001, 0.1, 0.5, 0.8, and 0.9. In order to avoid the effect of initial expansion and to focus on the patterns of semisteady flow, the marker particles in Figures 8c and 8d are placed above the vent at 100 s. The downflow of the collapsing column develops from the suspended flow at the height where the initial momentum is exhausted (1.5 km in this calculation; see arrows in Figure 8b).

[32] The boundary between partial and full column collapse is not well defined. Trajectories of marker particles show that almost all the marker particles fall to the ground from the suspended flow and move along the ground surface (Figures 8c and 8d), but a few particles continue to ascend from the suspended flow. We tentatively define the full column collapse regime as a flow pattern where less than 5% of marker particles ascends from the suspended flow.

[33] When the initial temperature is significantly low and the vent radius is small (run S1 with T0 = 400 K and L0 = 30 m; the result of this run is not shown), the full column collapse can occur after the inner dense core disperses because of erosion. Although a jet with a small vent radius efficiently entrains ambient air before the initial momentum is exhausted, the mixture of the ejected material and air is always heavier than air when T0 is as low as 400 K (Figure 1). As a result, all the eruption cloud collapses to the ground and cannot generate a coignimbrite ash cloud from a pyroclastic flow [cf. Koyaguchi and Woods, 1996].

4.2. Flow Regime Map

[34] As described above, when T0 and ng0 are given, the flow pattern of the eruption cloud varies depending on the exit velocity and vent radius. The flow patterns observed in the present study are summarized by a flow regime map shown in Figure 9. If exit velocity is fixed, because the vent radius is proportional to the square root of mass discharge rate, we may follow a path from left to right in Figure 9 to see the results with increasing vent radius. A column with a small mass discharge rate is likely to be in the stable column regime. As mass discharge rate increases, the full column collapse regime is attained via the partial column collapse regime. The critical value of mass discharge rate between those three regimes increases as the exit velocity increases. At a fixed mass discharge rate, a column with a large exit velocity tends to become a stable column, whereas a column with a small exit velocity tends to collapse. The partial column collapse regime lies between the stable column and full column collapse regimes.

Figure 9.

Flow regime map for the results of group P (T0 = 1000 K and ng0 = 0.05). Squares represent the stable column regime. Triangles indicate the column collapse regime. Diamonds represent the partial column collapse regime. Solid curve is the critical condition for column collapse based on the results of our 3-D model. Dashed curves are the critical conditions for column collapse which are predicted by the steady 1-D model of Woods [1995] with variable entrainment coefficients k. The values in parentheses are the assumed values for the entrainment coefficient in the 1-D model. Normally, the value of k is assumed to be 0.1 in the 1-D models [e.g., Woods, 1995].

[35] The critical condition between the stable column and partial column collapse regimes (solid curve in Figure 9) would correspond to the critical condition for column collapse suggested by the steady 1-D model of Woods [1995] (dashed curves in Figure 9). The features of this critical condition estimated by the present 3-D model are qualitatively consistent with the prediction by the steady 1-D model in the sense that the mass discharge rate necessary for column collapse increases as the exit velocity increases.

[36] In order to quantitatively determine the critical condition for column collapse in Figure 9, we systematically examined the effects of grid size on the numerical results. Since vent radius is the minimum characteristic length scale in the boundary condition of the present problem, we attempt to find an adequate number of grid points in vent radius, L0x. We found that the quantitative features of the dynamics of eruption clouds such as the value of the critical mass discharge rate between stable column and column collapse regimes are independent of grid size when L0x exceeds 5 (Figure 10). It is suggested that grid size smaller than L0/5 is required to reproduce the quantitative features of an eruption cloud when our numerical model with the third-order accuracy scheme is applied. In the calculations for Figure 9, Δx is set to be L0/10 to L0/15.

Figure 10.

Critical conditions for column collapse as a function of the number of grid points in vent radius. Solid curve is the critical mass discharge rate between the stable column (squares) and column collapse (triangles) regimes on the basis of the present 3-D simulations (u0 = 150 m/s, T0 = 1000 K, and n0 = 0.05). The scale at the top shows the grid size. In the steady 1-D model [e.g., Woods, 1995], the critical mass discharge rate for column collapse is a function of assumed value of k for a given exit velocity (see dashed curves in Figure 9). The scale at the right shows the value of k determined by the relationship between the assumed value of k and the critical mass discharge rate for u0 = 150 m/s in the steady 1-D model by Woods [1995].

[37] In Figure 10, the vertical axis (i.e., the critical mass discharge rate for column collapse) represents the efficiency of turbulent mixing because the amount of entrained air controls whether or not the eruption cloud becomes buoyant. The horizontal axis (i.e., the grid size) is related to Re*. Therefore the feature of Figure 10 suggests that the efficiency of turbulent mixing is independent of Re*, in other words, the condition above the mixing transition is considered to be achieved in the dynamics of eruption clouds when L0x > 5.

[38] The previous 1-D model [Bursik and Woods, 1991] demonstrates that the dynamics of eruption clouds is classified into three regimes: a simple buoyant column regime generates a stable column which decelerates at any height, a superbuoyant column regime generates a stable column which accelerates within some height range (dashed curves in Figure 11), and a column collapse regime produces a pyroclastic flow. Some stable columns in the present 3-D model show the superbuoyant behavior (runs P1 and P2; see Figure 11). The full column collapse regime corresponds to the column collapse regime of the steady 1-D model (run P7). The partial column collapse regime (run P4) is a new transitional regime between the superbuoyant column and column collapse regimes, because a buoyant plume and pyroclastic flow develop simultaneously. It should be noted that the eruption cloud accelerates to become a stable column even after the vertical velocity along the central axis reaches zero in run P2 (dashed curve in Figure 11). In the transition regimes (runs P2 and P4), 2-D or 3-D nature of flow, such as the large eddy due to suspended flow, plays an essential role in the dynamics of eruption clouds.

Figure 11.

Velocity profiles of runs P1 (equation image = 107.6 kg/s) and P2 (equation image = 108.2 kg/s) along the central axis. Dashed curves are velocity profiles predicted by the 1-D model for equation image = 107.6 and 108.2 kg/s.

5. Entrainment Coefficient

[39] The entrainment coefficient is a measure of the degree of turbulent mixing. The results of the steady 1-D models [e.g., Woods, 1995] depend on an assumed value for the entrainment coefficient; the value of the entrainment coefficient controls (1) the critical condition for column collapse and (2) column height. The critical condition for column collapse is determined by the competition between the negative buoyancy due to the dense ejected materials and positive buoyancy due to expansion of entrained air. Therefore the steady 1-D model predicts that the column with a small entrainment coefficient tends to collapse, whereas the column with a large entrainment coefficient tends to become a stable column (see dashed curves in Figure 9). On the other hand, the column height is principally determined by the balance between the thermal energy ejected at the vent and the work done in transporting the ejected material plus entrained air through the atmospheric stratification [Settle, 1978]. Therefore the steady 1-D model predicts that the column height increases as the entrainment coefficient decreases (see dashed curves in Figure 12).

Figure 12.

Maximum heights of eruption columns in group P(T0 = 1000 K and ng0 = 0.05). Exit velocity is assumed to be 150 m/s. The heights calculated from the steady 1-D model of Woods [1995] with variable entrainment coefficients k are shown by dashed curves. The values in parentheses are the assumed values for the entrainment coefficient in the 1-D model. Normally, the value of the entrainment coefficient is assumed to be 0.1 in the 1-D models [e.g., Woods, 1995].

[40] The present 3-D model can determine the critical condition for column collapse and the column height without any assumptions regarding turbulent mixing (i.e., an a priori entrainment coefficient). We can estimate effective entrainment coefficients for eruption clouds of the present 3-D model by comparing the column height and the critical condition for column collapse based on the 3-D model with those based on the steady 1-D model. Because the critical condition for column collapse is controlled by the amount of air entrained before the initial momentum is exhausted, the effective entrainment coefficient estimated from the critical condition for column collapse represents an efficiency of entrainment near the vent; we call it kv. On the other hand, the column height is controlled by the amount of the entrained air integrated from the vent to the top of the column. Therefore the effective entrainment coefficient estimated from the column height represents an efficiency of entrainment for the whole region of the column; we call it kw.

[41] First, we estimate the effective entrainment coefficient on the basis of the critical condition for column collapse in Figure 9. Transitions from a stable column to column collapse regimes in the results of the 3-D simulations are consistent with the critical condition based on the 1-D model for kv ∼ 0.07. Next, we estimate the value of the entrainment coefficient on the basis of the column height in Figure 12. Column heights in the results of the 3-D simulation are consistent with column heights based on the 1-D model for kw ∼ 0.1. According to previous experimental studies, the value of the entrainment coefficient of the JPUE ranges from 0.07 to 0.16 [Kaminski et al., 2005]. The estimations on the basis of our 3-D model suggest that both kv and kw are within the range of these experimental results, although kv is slightly smaller than kw.

[42] We consider that the above small difference between kv and kw is substantial because the flow structure changes with height as described in the preceding section. The eruption column in the upper region exhibits a meandering instability that induces efficient mixing with ambient air. On the other hand, near the vent, the inner dense core prevents the large-scale meandering instability. As a consequence, entrainment near the vent is less efficient than that in the upper region of the eruption column.

[43] There are several experimental studies which support the idea that the entrainment coefficient of an eruption cloud changes with height. Bhat and Narasimha [1996] reported that the efficiency of entrainment is reduced (k ∼ 0.05) when the buoyancy is created away from the source of flow due to expansion by chemical reaction (e.g., gas burning in the jet) and latent heat release (e.g., latent heat release during the condensation of water vapor in a cumulus cloud). The addition of buoyancy disrupts the large-scale vortical structures, which are largely responsible for the engulfment of ambient fluid. In eruption clouds, buoyancy is also created away from the vent, because the density of the mixture decreases because of expansion of entrained air. This effect would be particularly important near the vent because the temperature of solid pyroclasts is high and buoyancy is created rapidly. Kaminski et al. [2005] have pointed out that entrainment coefficients of a jet are generally smaller than those of a plume, and have proposed, on the basis of their experimental results using a mixture of EEG (ethanol and ethylene glycol) and water, that the efficiency of entrainment of ambient fluid increases as the magnitude of the buoyancy force increases. In the case of eruption clouds, the magnitude of the buoyancy force also increases dramatically as the eruption cloud rises, which may account for the increase in entrainment coefficient.

[44] Field observations also support the idea that the value of the entrainment coefficient varies slightly with height. Column heights observed for eruptions from some volcanoes showed a good quantitative agreement with the prediction of the steady 1-D models with k = 0.1 [Settle, 1978; Wilson et al., 1978; Woods, 1988]. On the other hand, Kaminski and Jaupart [2001] have pointed out that the critical mass discharge rate for column collapse reconstructed from the field data is much lower than the prediction on the basis of the 1-D model with k = 0.1. They have proposed that this inconsistency is caused by the overestimation of mass fraction of volcanic gas at the vent; because the large amount of volcanic gas remains trapped in bubbles within the solid pyroclasts, the effective amount of the gas phase is significantly smaller than the total amount of the volcanic gas in the magma. Our results suggest that the inconsistency between observation and prediction is partially attributed to the smaller entrainment coefficient near the vent.

[45] The small value for kv has also been reported by the previous numerical simulations of Neri and Dobran [1994]. Figure 13 illustrates the flow regime map based on their simulation results and the critical conditions based on the steady 1-D model. Transitions from a stable column to column collapse regimes in their results (solid curve in Figure 13) are accounted for by the critical condition based on the 1-D model for kv ∼ 0.03. This small value of kv may be partially attributed to their numerical procedure using a first-order accuracy scheme. In the steady 1-D model, the critical mass discharge rate for column collapse is a function of assumed value of k; it decreases as the assumed value of k decreases (Figures 9 and 13). On the basis of this relationship, we can estimate the value of k corresponding to the critical mass discharge rate in the steady 1-D model (the right-hand axis in Figure 10). The solid curve in Figure 10 indicates that the simulation with a low spatial resolution (L0x < 2 in our 3-D model) results in a low value of effective entrainment coefficients (k < 0.05).

Figure 13.

Flow regime map of the previous axisymmetric 2-D model by Neri and Dobran [1994]. Squares represent the stable column regime, and triangles represent the column collapse regime. Initial mass fraction of volcanic gas (ng0) is 0.008 and magmatic temperature (T0) is 1200 K. The dashed curves are the critical conditions for column collapse predicted by the steady 1-D model of Woods [1995]. The values in parentheses are the assumed values for the entrainment coefficient in the steady 1-D model.

6. Concluding Remarks

[46] In this paper, we have developed a new numerical model of the dynamics of eruption clouds. Employing three-dimensional coordinates, a third-order accuracy scheme and a fine grid size, our model can successfully reproduce the quantitative features of entrainment due to turbulent mixing observed in laboratory experiments without any a priori assumptions. The results of our 3-D model agree well with those of the previous 1-D steady models under the ideal conditions of steady state eruptions, while it is also applicable to transient and multidimensional phenomena of actual volcanic eruptions. For example, our model reveals some 2-D or 3-D structures which characterize the transitional state between stable eruption column and column collapse regimes, such as the partial column collapse and large eddies due to suspended flow.

[47] We estimated the entrainment coefficient for an eruption cloud by comparing the results of our 3-D model with those of the previous 1-D model. The range of our estimation is within the range of the experimental results for the jet and/or plume in a uniform environment. In addition, slight increase in the entrainment coefficient with height is suggested. The small difference in entrainment coefficient is attributed to vertical changes in the flow structure. In the upper region, the eruption column behaves as a turbulent buoyant plume which entrains ambient air because of the large-scale meandering instability. Near the vent, on the other hand, the eruption cloud has an inner dense core, and entrains ambient air in the outer shear region. As a result, entrainment near the vent is less efficient than that in the upper region. These results imply that the self-similarity is not perfectly valid for the flow of eruption clouds. Further experimental and theoretical studies will be necessary to confirm and also to explain the variation in entrainment coefficient [e.g., Bergantz and Breidenthal, 2001; Govindarajan, 2002; Kaminski et al., 2005].

[48] Our model is based on the assumption that the velocity and temperature of solid pyroclasts, volcanic gas, and air are the same at each point (i.e., a pseudo gas model). For a more quantitative understanding of the dynamics of eruption clouds, we should take into account the effects of the presence of solid pyroclasts, such as the effect of separation of pyroclasts from gas phase [e.g., Burgisser and Bergantz, 2002] and that of thermal disequilibrium between solid and gas phases [Woods and Bursik, 1991] in future.

Appendix A:: Governing Equations

[49] The dynamics of eruption clouds is based on the Navier-Stokes equation of a compressible gas. Since the molecular viscosity is negligibly small compared to the eddy viscosity due to turbulence, we can assume that the molecular viscosity is zero and that the equations are reduced to the Euler equation. The mass conservation for the all components (solid pyroclasts, volcanic gas, and air) is

equation image

where ρ is the density of the mixture, u is the velocity vector, and t is the time. The mass conservation equation for the ejected material (i.e., solid pyroclasts plus volcanic gas) from the vent is independently written as

equation image

where ξ is the mass fraction of the ejected material.

[50] The conservation equations for momentum and energy can be written as

equation image
equation image

where p is the pressure, I is the unit matrix, g is the gravitational body force per unit mass, and E is the total energy per unit mass, that is, the internal energy (e) plus kinetic energy (K): E = e + K.

Appendix B:: Density Change of Magma-Air System

[51] In this appendix we derive the density of the mixture of the ejected material and air at constant pressure as a function of the mass fraction of the ejected material, ξ, in Figure 1. When the ejected material with a high temperature, T0, and air with a low temperature, Ta, are mixed and come in to thermal equilibrium at a constant pressure, p, the equilibrium temperature for the mixture, Tm, has the form

equation image

where Cp0 and Cpa are the specific heat of the ejected material and air at constant pressure, respectively. At the same time, this new mixture satisfies equation of state as

equation image

Combining equations (B1) and (B2) and using the equation of state for air (p = ρaRaTa), the mixture density relative to air is obtained as

equation image

Note that this relationship is independent of pressure.

Appendix C:: Initial and Boundary Conditions

[52] Before eruptions, the atmosphere is stationary and stratified in pressure and density. The adopted temperature profile has a linearly decreasing temperature in the troposphere, a constant temperature between the tropopause and stratopause, and a linearly increasing temperature in the stratosphere [Gill, 1982]:

equation image

where Ta is the atmospheric temperature, z is the height, H1 = 11 km is the height of tropopause, H2 = 20 km is that of stratopause, and Ta0 = 273 K is the temperature at z = 0 in the midlatitude atmosphere. The temperature gradients in the troposphere and stratosphere are assumed to be μ1 = 6.5 K/km, and μ2 = 2 K/km, respectively. The atmospheric density (ρa) and pressure (pa) can be calculated from the relationship for the hydrostatic pressure,

equation image

[53] The physical domain involves a horizontal and vertical extent of several tens of kilometers. In the 2-D model, the vent is located in the lower left-hand corner of the computational domain, in which the pressure, exit velocity, mass fraction of volcanic gas, temperature, and density are fixed and constant. The axis of the flow is modeled as a free-slip reflector in order to preserve the symmetry of the system in the 2-D model. In the 3-D model, in contrast, the vent is located in the center of the ground surface, and no boundary condition along the central axis of the flow is required. At the ground boundary, the free-slip condition is assumed for the velocity of the ejected material and air. At the upper and other edges of computational domain, the fluxes of mass, momentum, and energy are assumed to be continuous, and these boundary conditions correspond to free outflow and inflow of these quantities.

[54] We assume that the pressure at the vent (p0) is equal to the atmospheric pressure at z = 0 (pa0), and that the exit velocity (u0) is larger than the sound velocity of the ejected material. When the initial mass fraction of volcanic gas (ng0), magmatic temperature (T0), and p0 are given, the density of the ejected material (ρ0) is calculated from the equation of state (equation (2)). When the vent radius (L0) and u0 are specified as parameters, the mass discharge rate is calculated from the relationship as

equation image

Acknowledgments

[55] Comments by G. W. Bergantz, E. Kaminski, and L. Mastin greatly improved the manuscript. We thank T. L. Wright for helpful comments in improving the manuscript. The discussion with S. Tait was fruitful. Numerical computations were in part carried out on the Earth Simulator at Japan Agency for Marine-Earth Science and Technology, on VPP5000 at the Computing and Communications Center of Kyushu University, and on Altix 3700 at the Earthquake Information Center of the Earthquake Research Institute, University of Tokyo. Part of this study was supported by funds from the Ministry of Education Science and Culture of Japan (14080204 and 14540388).

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