Effects of the crater on eruption column dynamics

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009jb007146, 2010 Effects of the crater on eruption column dynamics Takehiro Koyaguchi, 1 Yujiro J. Suzuki, 2 and Tomofumi Kozono 3 Received 18 November 2009; revised 4 February 2010; accepted 1 March 2010; published 16 July [1] During explosive eruptions, a mixture of pyroclasts and volcanic gas forms a buoyant eruption column or a pyroclastic flow. We systematically investigate how the condition that separates these two eruption styles (column collapse condition) depends on crater shape and magma chamber conditions by integrating the theoretical models for conduit flow, flow inside a crater, and eruption column dynamics. The results show that previous model predictions of column collapse condition based on the relationship between magma discharge rate ( _m) and water content (n f ) strongly depend on crater shape (depth D and opening angle ). When a crater is present, the decompression and/or compression of gas pyroclast mixture inside and just above the crater result in two distinct types of column collapse: collapse with increasing _m (HM side collapse) and that with decreasing _m (LM side collapse). HM side collapse is caused by an increase in conduit radius during the waxing stage of an eruption. LM side collapse is associated with a decrease in magma chamber pressure during the waning stage. The value of _m for HM side collapse varies by two orders of magnitude depending on crater shape for fixed n f. This estimate also depends on assumed models for decompression into the atmosphere. The value of _m for LM side collapse is <10 6 kg s 1 for a shallow crater with a small opening angle, whereas it can be >10 8 kg s 1 when D tan >10 2 m. These results are consistent with the field observations from the St. Helens 1980 and Pinatubo 1991 eruptions. Citation: Koyaguchi, T., Y. J. Suzuki, and T. Kozono (2010), Effects of the crater on eruption column dynamics, J. Geophys. Res., 115,, doi: /2009jb Introduction [2] During explosive volcanic eruptions, a mixture of hot ash (pyroclasts) and volcanic gas is released from the vent into the atmosphere at a high speed. The mixture generally has an initial density several times larger than atmospheric density at the vent. As the ejected material entrains ambient air, the density of the mixture decreases because the entrained air expands by heating from the pyroclasts. If the density of the mixture becomes less than the atmospheric density before the eruption cloud loses its upward momentum, a buoyant plume rises to form a plinian eruption column. On the other hand, if the mixture loses its upward momentum before it becomes buoyant, the eruption column collapses to generate a pyroclastic flow. Because the impact and type of volcanic hazards are largely different between the two eruption styles, it has been a central subject of volcanology to quantitatively predict the condition where an eruption column collapses to generate a pyroclastic flow; we refer to this condition as the column collapse condition [e.g., Sparks and Wilson, 1976]. 1 Earthquake Research Institute, University of Tokyo, Tokyo, Japan. 2 Japan Agency for Marine Earth Science and Technology, Yokohama, Japan. 3 National Research Institute for Earth Science and Disaster Prevention, Tsukuba, Japan. Copyright 2010 by the American Geophysical Union /10/2009JB [3] According to the one dimensional (1 D) steady model [e.g., Woods, 1988; Bursik and Woods, 1991], the dynamics of eruption columns is governed by magmatic properties (water content and temperature), magma discharge rate, and velocity at the vent. Previously, because the velocity at the vent was considered to primarily depend on water content [e.g., Wilson et al., 1980], the column collapse condition has been expressed by a relationship between magma discharge rate ( _m) and water content (n f ); a relatively small scale eruption of water rich magma tends to form a buoyant plume, whereas a large scale eruption of water poor magma tends to generate a pyroclastic flow [e.g., Wilson et al., 1980; Kaminski and Jaupart, 2001; Carazzo et al., 2008]. On the basis of these theoretical studies, a typical scenario of column collapse has been proposed: a column collapse occurs in the course of an explosive eruption as the magma discharge rate increases with increasing conduit radius owing to erosion of conduit wall, and/or as the water content of ejected magma decreases because of compositional zonation in the magma chamber [e.g., Sparks and Wilson, 1976; Wilson et al., 1980; Kaminski and Jaupart, 2001]. [4] The model prediction of column collapse based on the _m n f relationship can be tested by field observations. The value of _m of an explosive eruption is estimated from column height [e.g., Settle, 1978] or from total amount of ejecta divided by duration of the eruption. The value of n f is estimated from petrological data. The field data indicate that the magma discharge rates of pyroclastic flow eruptions are 1of26

2 Figure 1. Schematic illustration of the conduit crater system under consideration. v: velocity, p: pressure, r: density, q(= rv): mass flow rate, _m: magma discharge rate, A: cross sectional area, r: radius,h: height from the crater base, D: depth of crater, L: length of conduit, L + D: depth of magma chamber, : opening angle, p 0 : magma chamber pressure, p a : atmospheric pressure just above the crater top, and v a : vertical velocity at the time when the pressure of the gas pyroclast mixture is equilibrated with p a. The subscripts b and t represent the quantities at the base and top of the crater, respectively. systematically greater than those of plinian eruptions and support the above model prediction [e.g., Kaminski and Jaupart, 2001; Carazzo et al., 2008]. In spite of the qualitative agreement, there remain several problems as to quantitative aspects of the column collapse condition. Carazzo et al. [2008] have pointed out that the column collapse condition is affected by efficiency of turbulent mixing between eruption columns and ambient air. Numerical results of 2 D models for eruption column dynamics suggest that the stability of eruption columns depends on over pressure at the vent [e.g., Valentine and Wohletz, 1989; Neri and Dobran, 1994; Ogden et al., 2008b]. Woods and Bower [1995] have shown that the presence of a crater largely affects the fluid dynamical features of the compressible gas pyroclast mixture and, hence, the column collapse condition; column collapse may be induced by a decrease in magma discharge rate because of a shock generated inside the crater. Furthermore, the vent conditions (e.g., velocity and pressure) that control the column collapse condition depend on conduit shape and magma chamber conditions (e.g., depth and pressure) [e.g., Koyaguchi, 2005; Mitchell, 2005]. To make a quantitative prediction of the column collapse condition, the diverse effects of these factors have to be systematically evaluated. [5] In this paper, we attempt to establish a method to predict the column collapse condition by integrating the theoretical models for conduit flow, flow inside a crater, and eruption column dynamics. For this purpose, we posit a conduit crater system in which the conduit has a constant radius between the magma chamber and the crater base, and in which the radius of the crater increases upwards at a constant rate (Figure 1). For simplicity, it is assumed that both the conduit and the crater have circular cross sections. The exit condition of the conduit crater system provides the vent condition of the 1 D model for eruption column dynamics. The column collapse condition for a system like that in Figure 1 is determined by solving the following three problems. [6] The first problem concerns the dynamics of conduit flow. During explosive eruptions, the ascending velocity of magma typically reaches the sound velocity of the gaspyroclast mixture (referred to as the choking condition ) at the base or the top of the crater. In this case, the mass flow rate (i.e., the magma discharge rate divided by the crosssectional area) of conduit flow q b is governed by the conditions in the upstream region, such as the pressure at the magma chamber (p 0 in Figure 1). Generally, p 0 deviates from the lithostatic pressure (r s g(l + D), where r s is the density of the country rock, g is the acceleration due to gravity, and L + D represents the depth of the magma chamber). We, therefore, determine q b as a function of p 0 and L + D on the basis of the 1 D steady model of conduit flow [e.g., Koyaguchi, 2005] in this problem. [7] The second problem deals with the compressible fluid dynamics inside and just above the crater. The velocity and pressure at the crater top (v t and p t in Figure 1) depend on the mass flow rate at the crater base (i.e., q b ) and parameters related to crater shape (e.g., the ratio of the cross sectional areas at the top and the base A t /A b, the depth D, and the opening angle ) [Woods and Bower, 1995]. When p t differs from the atmospheric pressure just above the crater (p a in Figure 1), the ejected material accelerates or decelerates owing to decompression or compression into the atmosphere; it has a certain velocity, v a, when its pressure reaches p a [e.g., Kieffer, 1989; Woods and Bower, 1995; Ogden et al., 2008b]. This velocity is regarded as the vent velocity in eruption column dynamics models [e.g., Woods, 1988]. In this problem, we determine v a as a function of q b, A t /A b, D and. [8] In the third problem, we estimate the column collapse condition on the basis of the 1 D steady model for the eruption column dynamics. When the magmatic properties (n f and T) and the magma discharge rate ( _m) are given, the vent velocity at the column collapse condition (referred to as v crit ) is uniquely determined [Bursik and Woods, 1991; Kaminski and Jaupart, 2001; Carazzo et al., 2008]. Therefore, we can discuss how the column collapse condition 2of26

3 depends on crater shape and magma chamber conditions by comparing v crit with v a in the second problem; namely, a buoyant plume forms when v a > v crit, whereas a pyroclastic flow is generated when v a < v crit. [9] As mentioned above, the column collapse condition has been expressed by an _m n f relationship on the assumption that v a is approximated by a function of n f [Wilson et al., 1980; Wilson and Walker, 1987; Kaminski and Jaupart, 2001; Carazzo et al., 2008]. In this study, because we take the effects of crater shape on v a into account, the column collapse condition cannot be represented by a single _m n f relationship. We attempt to obtain a more generalized column collapse condition in the parameter space of _m and the crater radius (r b or r t ). This approach is an extension of the study by Woods and Bower [1995]. They discussed the effects of the presence of a crater on v a and presented a column collapse condition in the _m r t space; however, their calculations were limited to a single boundary condition at the magma chamber so that the result remained qualitative [cf. Woods and Bower, 1995, Figure 10]. Here, we quantify their result through a comprehensive parameter study. In section 2, we establish a systematic method for analyzing the column collapse condition. In section 3, we quantitatively evaluate the effects of crater shape on the column collapse condition using this new method. In section 4, the present results are compared with field data. [10] The main purpose of this study is to understand the framework of the complex relationship among the column collapse condition, crater shape, chamber conditions, and magmatic properties by integrating the above three problems. Details on the individual problems will be discussed elsewhere. We follow the basic equations of the standard 1 D steady models for conduit flow and eruption column dynamics. The derivations of mathematical relationships used in the present analyses from those basic equations are given in Appendices A and B. 2. Method for Analyzing the Column Collapse Condition [11] In this section we solve each problem in Figure 1. For convenience of presentation, we first estimate v a in the _m r b space on the basis of the fluid dynamics inside craters (Problem 2). Secondly, we determine the boundary of the regions of v a > v crit and v a < v crit in the _m r b space on the basis of the eruption column dynamics (Problem 3). Finally, we discuss how _m depends on r b and magma chamber conditions on the basis of the dynamics of conduit flow (Problem 1). The combination of the results of these three problems comprises a complete set for analyzing the column collapse condition Flows Inside Craters [12] As is shown in Appendix A, the relationships that describe the fluid dynamical features inside a crater are expressed by functions of the mass flow rate and the ratio of the cross sectional areas at the top and base of the crater (i.e., q b /q* and A t /A b in equation (A13)). Therefore, we classify flows inside craters in the q b /q* A t /A b space and analyze how v a varies with these quantities. Subsequently, we discuss the variation of v a in the _m r b space Classification of Flow Types Inside Craters [13] The compressible fluid dynamics of quasi 1D flow through a converging and/or diverging nozzle as in Figure 1 are well established for an ideal gas without gravity [e.g., Shapiro, 1953]. In this study we extend this theory to the case of gas pyroclast mixtures with gravity. The main difference of the present results from those of the classical theory is in the position where flow velocity reaches the sound velocity so that the flow changes from subsonic to supersonic flow (i.e., the choking condition; see Appendix A for its definition). Without gravity, the transition from subsonic to supersonic flow is possible only at the throat of a converging diverging nozzle. When the effect of gravity is present, on the other hand, such a transition occurs at a point where the conduit radius slightly increases upwards (see equation (A7)). Because of this difference, a flow of gaspyroclast mixture can reach its sound velocity at the base and/or top of the crater depending on q b and A t /A b (>1); as a result, flows inside craters are classified into 6 types according to these parameters (Figure 2). [14] When q b is small, the magma ascends as a subsonic flow through both the conduit and the crater (Figure 2a). As q b increases, the flow reaches the choking condition at the base or top of the crater (Figures 2b 2f). When A t /A b 1, the flow is choked at the crater top. It has an exit pressure of p t p a and is freely decompressed into the atmosphere. We refer to this type of flow as free decompression flow (Figure 2b). When A t /A b exceeds a certain value, the flow reaches the choking condition and changes from a subsonic to a supersonic flow at the crater base. Flow that is choked at the crater base is subdivided into four types (Figures 2c 2f). When A t /A b is relatively small or q b is large, the gaspyroclast mixture issues from the crater top as a supersonic flow with p t > p a. The flow is decompressed and generates rarefaction waves just above the crater. We refer to this type as underexpanded flow (Figure 2f) following the fluid dynamics convention. As A t /A b increases or q b decreases, the exit pressure (p t ) decreases. When p t = p a, the expansion of the gas pyroclast mixture in the crater is most efficiently transferred to upward momentum. This flow type is referred to as correctly expanded flow (Figure 2e). For greater A t /A b or smaller q b, the gas pyroclast mixture erupts as a supersonic flow with p t < p a, and is then compressed and decelerated by oblique shocks just above the crater. This flow type is referred to as overexpanded flow (Figure 2d). As A t /A b further increases or q b decreases, a shock forms inside the crater. In this instance, the gas pyroclast mixture is compressed and decelerated by the shock and issues from the crater top as a subsonic flow (Figure 2c). Velocity and pressure profiles inside the crater for some of these flow types are presented by Woods and Bower [1995]. We determine the q b /q* A t /A b relationships that represent the boundaries of these flow types below. [15] Let us start with the boundary between the free decompression flow choked at the crater top (Figure 2b) and the underexpanded flow choked at the crater base (Figure 2f). In order to determine this boundary, we consider a hypothetical crater where the flow satisfies M T = 1 at all heights. Here M T is the Mach number defined in Appendix A. When M T = 1, the first equation of equation (A1) and equation (A6) yield a relationship: pa = constant. Substituting this rela- 3of26

4 Figure 2. Classification of flows inside craters. (a and c) Subsonic eruptions, (b) sonic eruption, and (d f) supersonic eruptions. The boundaries of different flow types are given by equations (2), (3), (5), and (6). tionship into equation (A13), we obtain the cross sectional area of the hypothetical crater as a function of height from the crater base, A(h), as h ¼ ln Ah ð Þ þ 2 V L q b l sc A b V Ga q* 1 A b Ah ð Þ þ 1 ( V L q 2 b 1 A ) 2 b : ð1þ 2 V Ga q* Ah ð Þ From this equation and the definition of cross sectional area (A pr 2 ) we can calculate the radius of the hypothetical crater. The hypothetical crater has an opening angle of tan 1 {r b g/(2a 2 )} at the base (see equation (A7)), and it increases with h (Figure 3; see Table 1 for parameters used in the calculations). [16] Whether the flow is choked at the top or base of the crater depends on whether the radius of the crater top (r t )is smaller or greater than the radius of the hypothetical crater. In Figure 1, we set a conduit crater system where a conduit with constant r b smoothly changes into a crater with an opening angle, ; the radius of the crater at h is expressed by r b + h tan. When < tan 1 {r b g/(2a 2 )}, r t is always smaller than the radius of the hypothetical crater. In this case, the flow is choked at the crater top (Figure 3). On the other hand, when > tan 1 {r b g/(2a 2 )}, the line representing the crater wall (i.e., r b + h tan ) intersects the curve representing the hypothetical crater wall at P in Figure 3. If P is located below the crater top, the flow is choked at the crater top, whereas if P is located above the crater top, the flow is choked at the crater base and changes from a subsonic to a Figure 3. Schematic diagram showing the positions of crater walls with constant opening angles (dashed lines) and that of a hypothetical crater in which the flow satisfies M T = 1 at all the heights (solid curve). When > tan 1 {r b g/(2a 2 )}, the line of the crater wall intersects the curve of the hypothetical crater wall at P. In the calculation of the curve for the hypothetical crater wall, the parameters for Magma 1 withn 0 = 0.04 in Table 1 and q b /q* = 10 are used. 4of26

5 Table 1. Parameters Used in the Calculations Common Parameters Value g, ms crit 0.8 supersonic flow there. The boundary between Figures 2b and 2f is given by the condition that M T = 1 at the top and base of the crater simultaneously, namely the condition for the level of the crater top to coincide with P in Figure 3. The q b /q* A t /A b relationship for this condition is obtained by substituting h = D and A = A t into equation (1) as D ¼ ln A t þ 2 V L q b l sc A b V Ga q* 1 A b A t þ 1 ( V L q 2 b 1 A ) 2 b : ð2þ 2 V Ga q* A t [17] When the flow is choked at the crater top, the boundary of the subsonic flow (Figure 2a) and the free decompression flow (Figure 2b) is given by the condition that the magma issues from the pffiffiffiffiffiffiffiffiffiffi crater top at its sound velocity with p a : namely q t = q* p a / n f RT (see equation (A6)). The q b /q* A t /A b relationship for this condition is obtained from the mass conservation as q b q* ¼ A t A b : [18] When the flow is choked at the crater base, the magma issues from the crater top as a subsonic flow (Figure 2c) or a supersonic flow (Figures 2d 2f). The boundary between them is given by the condition where a shock is located at the crater top. When a shock is located at the crater top, p + in equation (A19) in Appendix A is equal to p a, and hence, A t A b Value Magma Properties Magma 1 Magma 2 T, K r l,kgm s, Pa 1/ R, Jkg 1 K C m,jkg 1 K C s,jkg 1 K h, Pas 10 6 n 0 (n f ) 0.02 (0.0123) (0.023) 0.04 (0.0284) 0.06 (0.0455) (0.027) ð3þ 2 ¼ p ap t : ð4þ p 2 b Value Atmosphere Properties ML 1 ML 2 TR 1 T a at vent, K dt a /dz, Km R a,jkg 1 K C a,jkg 1 K The q b /q* A t /A b relationship representing this boundary is, therefore, calculated from equations (4) and (A13) as 2 1 q 2 ( b A 2 t V L þ q* 2 ) 3 A 2 2 t 2 q* A b V Ga q b A VL þ q* b V Ga q b ( þ V ) ( L q b A 2 ) t q b V Ga q* A b q* 1 A 2 t q b þ ln þ D ¼ 0: A b q* l sc ð5þ Because p t = p < p + = p a at this boundary, the supersonic flow next to this boundary is the overexpanded flow (i.e., Figure 2d). [19] Supersonic flow which is choked at the crater base is overexpanded or underexpanded flow depending on whether p t < p a or p t > p a (Figures 2d and 2f). The boundary of these two flow types is defined by the correctly expanded flow with p t = p a (Figure 2e). The q b /q* A t /A b relationship for the correctly expanded flow is obtained by substituting p = p a, A = A t and h = D into equation (A13) as ( 1 q 2 b A 2 t V 2 L þ 1 V L þ q* ) 2 2 q* A b V Ga V Ga q b þ V L V Ga 1 q b q* þ ln q* q b þ D l sc ¼ 0: Equation (6) has two solutions of q b /q* for given A t /A b. The solution with larger q b /q* represents the relationship for the correctly expanded flow. The solution with smaller q b /q* represents the boundary between the subsonic flows with and without a shock inside the crater (i.e., that between Figures 2a and 2c). [20] Figure 4 shows a regime map of the flow types inside the crater in the q b /q* A t /A b space. The boundaries of the different flow types are expressed by equations (2), (3), (5), and (6). Equation (3) represents a flow that reaches the choking condition at the crater top and issues at p t = p a. Equation (6) represents a flow that reaches the choking condition at the crater base and issues at p t = p a. Equation (2) represents a flow that reaches the choking condition at both the top and base of the crater simultaneously at p t p a. Furthermore, when a shock is located at the crater top, the jump of physical quantities across the shock approaches zero as v t approaches the sound velocity (i.e., a a ). Consequently, when a flow reaches the choking condition at both the top and base of the crater simultaneously at p t = p a, all the boundaries of the different flow types converge to one point in this diagram; we call this condition the CA condition. We also refer to the conditions of equations (2), (5), and (6) at q b = q f as the M1F, VSF, and CEF conditions, respectively. [21] The positions of the CA M1F, CA VSF, and CA CEF curves depend on the depths of craters D as well as on magmatic properties; for example, the mass flow rate at the crater base under the CA condition (q b ) CA increases with D for a given magma. When (q b ) CA = q f, the CA, M1F, and CEF conditions coincide. This implies that D has a maximum value D max in the present framework. The dependencies of the CA M1F, CA VSF, and CA CEF curves on crater shape and the possible range of D will be discussed in section 3. ð6þ 5of26

6 flow, p t /p a = 1 for the correctly expanded flow and p t /p a >1 for the underexpanded flow. For all cases, the value of v t can be calculated from p t using the mass conservation and the equation of state in equation (A1) as v t ¼ q ba b t A t ¼ q ba b A t n f RT p t þ 1 n f l : ð9þ [24] Generally, the estimation of v a from v t and p t is difficult, particularly for the cases of p t p a (i.e., the free decompression, underexpanded, and overexpanded flows) because the velocity and the pressure fluctuate just above the crater as a result of the generation of the shock and/or rarefaction waves. Woods and Bower [1995] proposed a formula to estimate a first order approximation of v a on the basis of the conservation laws in the control volume just above the crater as v a ¼ v t þ p t p a v t n f RT p t þ 1 n f l : ð10þ Figure 4. The relationships between A t /A b and q b /q* showing the boundaries of the different flow types inside craters. The parameters for Magma 1 with n 0 = 0.04 in Table 1 and D = 500 m are used. All the curves of equations (2), (3), (5), and (6) converge at the CA condition, and equations (2), (5), and (6) at q b = q f represent the M1F, VSF and CEF conditions, respectively. The boundary between the subsonic flows with and without a shock inside the crater is shown by the dashed curve. The values of A t /A b used in the calculations of Figure 5 are shown by arrows at the top Estimation of v a [22] For the present purpose of predicting the column collapse condition, we are concerned with the velocity at atmospheric pressure (v a in Figure 1). The value of v a is obtained from pressure and velocity at the crater top (p t and v t in Figure 1) for each flow type. Let us estimate p t (in practice p t /p a ) and v t first. [23] For the subsonic flow, p t /p a is unity. For the free decompression flow, p t /p a is obtained from equation (A6) and the mass conservation in equation (A1) as p t p a ¼ q ba b q*a t : From Figure 4 it is obvious that the right hand side of this equation exceeds unity for free decompression flow. When the flow is choked at the crater base, p t /p a is obtained by substituting h = D, q c = q b, A c = A b, A = A t and p = p t into equation (A13) as ( 1 q 2 b A 2 t V L þ p 2 a V L þ q* ) 2 2 q* A b V Ga p t V Ga q b þ V L V Ga p t p a q b q* þ ln q* p t q b p a þ D ¼ 0: l sc The value of p t /p a can be determined by numerically solving this equation; by definition p t /p a < 1 for the overexpanded ð7þ ð8þ For subsonic and correctly expanded flows, because p t = p a, we obtain v a = v t from this formula. For the correctly expanded flow in particular, equation (9) with p t = p a and the definition of the sound velocity (see Appendix A) yield a useful relationship: M Ta v a a a ¼ q ba b q*a t ¼ p ba b p a A t : ð11þ [25] On the basis of the formula (equation (10)), Woods and Bower [1995] emphasized that v a for free decompression flow is nearly independent of the magma discharge rate and primarily depends on water content n f. For free decompression flow, because v t = a t, we obtain v a from equation (10) and the definition of the sound velocity as v a ¼ ffiffiffiffiffiffiffiffiffiffi p 1 n f RT t þ 1 p a p t ; ð12þ where t is the gas volume fraction at the crater top. Considering that t 1p near the surface, v a in equation (12) is approximated by 2 ffiffiffiffiffiffiffiffiffiffi n f RT under the conditions of p t p a, which leads to the conclusion by Woods and Bower [1995]. This p approximation of v a for free decompression flow (v a 2 ffiffiffiffiffiffiffiffiffiffi p n f RT) and/or its simplified formula (v a 1380 ffiffiffiffi n f ) have been widely used in previous model predictions of the column collapse condition [e.g., Kaminski and Jaupart, 2001; Carazzo et al., 2008]. [26] Recently, Ogden et al. [2008b] performed a series of numerical simulations on the decompression of axisymmetrical jets just above volcanic vents and have pointed out that the effective values of v a for jets with p t > p a are substantially smaller than the approximation based on equation (12) because of the loss of momentum accompanying expansion. According to their numerical results, the radius of the jet following decompression to atmospheric pressure just above the vent is well approximated by r a ¼ p 1=2 t : ð13þ r t p a 6of26

7 This relationship and the mass conservation (pr a 2 r a v a = pr t 2 r t v t ) yield v a ¼ p a t p t a v t : ð14þ For free decompression flow, because v t = a t in equation (14), we obtain v a ¼ a a ¼ a p a p ffiffiffiffiffiffiffiffiffiffi : ð15þ n f RT This estimation of v a results in approximately 1/2 of the value resulting from equation (12) [see Ogden et al., 2008b, Figure 14]. Although Ogden et al. [2008b] have removed some of the simplifications in the control volume estimate of Woods and Bower [1995], a number of effects that likely affect the velocity after decompression into the atmosphere (e.g., those of separation of pyroclasts and of gravity) are not included in their numerical model. Because of the uncertainty of these additional effects, it is difficult to judge which model is more appropriate for the present problem at this stage. In this study, we adopt the formula by Woods and Bower [1995] (i.e., equation (10)) for flows with p t p a, whereas we tentatively use the new relationship based on the numerical results by Ogden et al. [2008b] (i.e., equation (14)) for flows with p t > p a. For flows with p t > p a, we also present the results based on the model of Woods and Bower [1995] (equations (10) and (12) and their approximations) when necessary, and discuss how the choice of models for decompression into the atmosphere influences the predictions for column collapse. [27] Figure 5 shows the variations of velocity at atmospheric pressure (v a ), of velocity at the crater top (v t ), and of pressure at the crater top (p t ) as a function of mass flow rate (q b ) for the same parameters related to magmatic properties and crater shape as for Figure 4. The qualitative features of the results in Figure 5 change depending on whether A t /A b is greater or smaller than that of the CA condition (A t /A b ) CA (see the arrows in Figure 4 for the values of A t /A b used in the calculation of Figure 5). [28] When A t /A b >(A t /A b ) CA, the flow at the crater top changes from subsonic flow through overexpanded flow with p t < p a and correctly expanded flow with p t = p a to underexpanded flow with p t > p a as q b increases (Figure 5c). The value of v t discontinuously increases from subsonic flow (v t < a a ) to overexpanded flow (v t > a a ) (Figure 5b). For overexpanded flow, the value of v a is substantially smaller 7of26 Figure 5. Diagrams showing (a) the velocity at atmospheric pressure just above the crater v a,(b)thevelocity at the crater top v t, and (c) the pressure at the crater top p t as functions of q b for A t /A b varyingfrom1.02to100.the same parameters as those in Figure 4 are used. For these parameters, (A t /A b ) CA is 1.45 (dashed and dotted curves). Equation (14) is used in calculating v a for p t > p a. The values of v a for the correctly expanded flow (CE) and the flow with a shock at the crater top (VS) are shown by short dashed curve in Figure 5a.

8 Figure 6. The boundaries of the different flow types inside craters in (a) the _m r b space and (b) the _m r t space. The boundary between the subsonic flows with and without a shock inside the crater is shown by the dashed curve. The same parameters as those in Figure 4 and = 10 are used. Dotted lines show q b = q* and q b = q f. than v t because of the deceleration accompanying compression by the oblique shock just above the vent; note that v a < a a for overexpanded flow with small q b. The value of v a rapidly increases with q b in the range of overexpanded flow, and that rate of increase is suppressed in the range from correctly expanded flow to underexpanded flow (Figure 5a). [29] When A t /A b <(A t /A b ) CA, the flow at the crater top changes from subsonic flow to free decompression flow; v a increases with q b in the range of subsonic flow and reaches a constant value calculated from equation (15) (or equation (12)) in free decompression flow. When A t /A b (A t /A b ) CA, the flow type changes from free decompression flow to underexpanded flow as q b increases, which leads to a slight increase in v a with q b. [30] When q b is fixed, the flow type typically changes as A t /A b increases from free decompression flow through underexpanded flow, correctly expanded flow, and overexpanded flow to subsonic flow (see Figures 2 and 4). Figure 5a shows that v a increases with increasing A t /A b in the range of underexpanded flow from free decompression flow to correctly expanded flow; v a decreases with the further increase in A t /A b in the range of overexpanded flow from correctly expanded flow to subsonic flow. Consequently, v a has a maximum value in correctly expanded flow for a fixed q b Behavior in the _m r b Parameter Space [31] So far, we have analyzed fluid dynamics inside a crater in the q b /q* A t /A b parameter space. For the present purpose of comparing model predictions with geological data, however, this parameter space is inconvenient because the relationship between A t /A b and the crater shape is not straightforward; A t /A b depends on the opening angle (), the depth of the crater (D), and the conduit radius (r b )as A t A b ¼ 2 ¼ 1 þ D tan 2 : ð16þ r t r b We are interested in how the flow types and v a vary in a parameter space that is directly related to observable quantities such as the magma discharge rate ( _m) and the radius of the crater (r b or r t ). [32] Using equation (16) and the definition of magma discharge rate (i.e., _m pr 2 b q b ), we can transform the boundaries of the different flow types in Figure 4 into those in the _m r b and _m r t spaces (Figures 6a and 6b). The isopleths of q b are expressed by parallel lines in Figure 6a, whereas they are represented by a series of curves in Figure 6b. In the present framework, we are concerned with the range of q b < q f (i.e., the range where the choking pressure is less than the pressure at the fragmentation surface, p f ; see equation (A6)) in this diagram. Figure 6 indicates that the free decompression flow and the underexpanded flow occur during eruptions with high magma discharge rates, whereas the overexpanded flow occurs during relatively small scale eruptions. The subsonic flow tends to occur during smallscale eruptions or ones with large r b and r t. Unlike in Figure 4, the boundaries of the different flow types in Figure 6 depend on, as well as on D and magmatic properties (n f and T). This dependence on will be discussed in section 3. [33] Through a similar transformation, we also obtain the variation of v a in the _m r b space (Figure 7). The value of v a varies from less than 1 to 360 m s 1 in this diagram. The r b 8of26

9 _ b Figure 7. The variation of the velocity at atmospheric pressure just above the crater (va) in the m r space. (a) The result using equation (10) for pt > pa. (b) The result using equation (14) for pt > pa. These diagrams show the values of va between qb = 0.5q* and qb = qf in Figure 6a. The same parameters as those in Figure 6 are used. value of va is below aa for a portion of the overexpanded flow region near the CA VSF curve, as well as for the region of subsonic flow. It increases from the CA VSF curve to the CA CEF curve and forms a steep slope in the region of overexpanded flow (see also Figure 6a for the positions of the CA VSF and CA CEF curves). The value of va decreases from correctly expanded flow to free decompression flow, which forms another slope in the region of underexpanded flow. Between the two slopes, va of correctly expanded flow forms a ridge and has a maximum value of 3aa at the CEF condition. The features of the slope from underexpanded flow to free decompression flow differ between the cases where equation (10) or equation (14) is used in calculating va (Figures 7a and 7b). When equation (10) is used, the variation of va forms a gentle slope from correctly expanded flow (va 3aa) to free decompression flow (va 2aa) (Figure 7a). 9 of 26

10 Figure 8. The critical velocity for column collapse (v crit )as a function of magma discharge rate based on the eruption column model. Thick solid curve: the result based on the model by Carazzo et al. [2008]. Short dashed curves: the results for constant entrainment coefficients ranging from k = 0.03 to The parameters for Magma 1 with n 0 = 0.04 and ML 1 in Table 1 are used. When equation (14) is used, on the other hand, the variation of v a forms a steep slope in the region of underexpanded flow and forms a plateau with a horizontal plane of v a = a a in the region of free decompression flow (Figure 7b). This difference influences the model predictions of column collapse, particularly those for free decompression flow Comparison Between v a and v crit by the Eruption Column Dynamics Model [34] According to the 1 D steady model of eruption column dynamics, when the parameters of magma properties (e.g., T and n f ) are provided, the column collapse condition of an explosive eruption with a magma discharge rate _m is determined by the critical velocity at atmospheric pressure just above the crater, v crit [Woods, 1988; Bursik and Woods, 1991; Kaminski and Jaupart, 2001; Carazzo et al., 2008]. Generally, the value of v crit depends on the efficiency of turbulent mixing between the eruption column and ambient air (i.e., the assumed value of the entrainment coefficient k in the 1 D model). Carazzo et al. [2008] have shown that the values of k for turbulent plumes and jets systematically change with their buoyancy on the basis of published experimental data and proposed a new 1 D steady model in which the value of k is allowed to change with height as a function of buoyancy. The value of v crit based on the model by Carazzo et al. [2008] roughly coincides with that of k 0.05 for a wide range of _m (Figure 8). In this study, we discuss the column collapse condition on the basis of v crit calculated from the 1 D steady model by Carazzo et al. [2008]. We also investigate the sensitivity of the column collapse condition to the efficiency of turbulent mixing by changing the assumed value of k from 0.03 to [35] The region where a buoyant plume or a pyroclastic flow forms in the _m r b (or _m r t ) space is determined by comparing v a in Figure 7 with v crit in Figure 8; a buoyant plume forms in the region where v a > v crit, whereas a pyroclastic flow is generated in the region where v a < v crit (Figure 9). In Figure 9 we have superimposed the boundaries of different types of flow (see Figure 6). The boundary between the buoyant plume region and the pyroclastic flow region is divided into two parts: the part in which _m is lower than that of the correctly expanded flow (LM side) and the part in which it is higher (HM side). The LM side boundary corresponds to the points of v a = v crit in the slope of the overexpanded flow in Figure 7. This boundary is located along a slightly higher _m side of the CA VSF curve and is nearly independent of the assumed value of k. This feature reflects the fact that v a of overexpanded flow rapidly increases along the high _m side of the CA VSF curve. On the other hand, the HM side boundary corresponds to the points of v a = v crit in the slope of the underexpanded flow region (or the region of free decompression flow) in Figure 7; it shifts to the high _m direction as the assumed value of k increases. These results suggest that the positions of the LM side and HM side boundaries are controlled by the positions of the slopes of v a in Figure 7; in other words, the column collapse condition sensitively depends on the relationships in Figure 6. Figure 7 also indicates that the choice of models for decompression into the atmosphere influences the estimate of v a, and hence, the predictions of column collapse. We will discuss the governing factors of column collapse condition in relation to these points in section The _m r b Relationship Constrained by the Conduit Flow Model [36] In section 2.2 we have determined the regions of buoyant plume and pyroclastic flow in the _m r b and _m r t spaces on the basis of the analyses of fluid dynamics inside the crater (Problem 2 in Figure 1) and of eruption column dynamics (Problem 3 in Figure 1). In order to complete the problem of the column collapse condition, we must understand how the _m r b relationship during explosive eruptions depends on magma chamber conditions on the basis of the dynamics of the conduit flow (Problem 1 in Figure 1). [37] When a conduit like that in Figure 1 is assumed, the 1 D steady model of conduit flow allows us to calculate mass flow rate in the conduit (q b ) from magmatic properties (temperature T, water content n f, liquid density r l, and viscosity h), conduit radius (r b ), depth of the magma chamber (L or L + D), and pressure at the magma chamber (p 0 ). Figure 10a shows numerical results based on the basic equations in Appendix A. phere, ffiffiffiffiffiffiffiffiffiffiq b is normalized by the maximum flow rate q f ( p f / n f RT); note that q f is constant for a given magma. [38] The features of the numerical results in Figure 10a can be understood through the semi analytical solution shown in Figure 10b. In Appendix B, the semi analytical solution of the 1 D steady flow model in conduits with constant radii [Koyaguchi, 2005] is extended to the present case with a crater. According to equation (B10), q b /q f is expressed as a function of a dimensionless parameter g, 10 of 26

11 Figure 9. Diagrams showing the regions where a pyroclastic flow or a buoyant plume is generated during explosive eruptions in (a) the _m r b space and (b) the _m r t space. Thick solid curve: the boundary based on the model by Carazzo et al. [2008]. Short dashed curves: the boundaries for constant entrainment coefficients ranging from k = 0.03 to The boundaries of the different flow types in Figure 6 are also shown by thin curves. The same parameters as those in Figure 6 and ML 1 in Table 1 are used. Equation (14) is used in calculating v a for p t > p a. For definitions of the LM side and HM side, see text. Figure 10. The mass flow rate normalized by the maximum mass flow rate (q b /q f ) as a function of the conduit radius (r b ) based on the conduit flow model. (a) Numerical results based on the full basic equations. (b) Semi analytical approximation based on Koyaguchi [2005]. The same parameters as those in Figure 6 are used. Solid curves: the results for L + D = 5 km with Dp varying from 0 to 100 MPa. Dashed curves: the results for L + D = 10 km with Dp varying from 0 to 100 MPa. The derivation of the semi analytical solution and the difference between the numerical and semi analytical results are discussed in Appendix B. 11 of 26

12 whose value is determined by magmatic properties and the geological conditions as pffiffiffiffiffiffiffiffiffiffi 2 l gr2 b n f RT : ð17þ 8p f For small and large values of g, equation (B10) has asymptotic solutions of q b /q f as 8 exp L q >< ð! 0Þ b l sc! q f A t exp L þ D l ð18þ h >: ð!1þ; A b where l sc and l h are the characteristic length scales defined by equations (A12) and (B3), respectively. [39] These asymptotic solutions indicate that the governing factor of q b changes with r b : q b /q f is governed by chamber depth for small r b and by chamber pressure for large r b. Let us express the pressure of a magma chamber as the sum of the magmastatic pressure and the deviation from it as l sc p 0 ¼ l glþ ð DÞ Dp; ð19þ where Dp represents the degree of underpressure at the magma chamber. In this case, l h in equation (18) (see equation (B3) for its definition) is expressed by l h ¼ L þ D Dp l g : ð20þ Considering that g is proportional to r b 2 (see equation (17)) and that A t /A b 1asr b (see equation (16)), equation (18) is rewritten as 8 exp gl q >< b n f RT! q f exp Dp >: l n f RT ðr b! 0Þ ðr b!1þ: ð21þ [40] Taking the above asymptotic behavior into consideration, we can summarize the results in Figure 10a as follows. For small g (i.e., high h and small r b ), q b primarily depends on L (or L + D) and is nearly independent of Dp. When L + D is as small as 5 km, q b is greater than q*, and the flow is choked at the base (or top) of the crater. When L + D is greater than 10 km (i.e., a few multiples of l sc ), the magma issues as a subsonic flow with q b < q*. For large g (i.e., low h and large r b ), on the other hand, q b is governed by Dp. In the limit of g, the pressure distribution in the conduit is approximated by the static pressure of the gas liquid mixture (see Appendix B). Because the average density of the gas liquid mixture is generally smaller than that of the surrounding rocks, the flow inevitably has a pressure greater than p a at the top of the conduit when p 0 is around the lithostatic pressure; as a result, it is choked at the base (or top) of the crater with q b > q*. As p 0 decreases from the lithostatic pressure, the pressure at the top of the conduit, and hence, q b decrease. [41] We can judge whether q b is governed by L or Dp from the criterion whether g q b /q f or g q b /q f. This criterion depends on h, n f, and r b (see equation (17)). For a typical silicic magma with a few weight % H 2 O and h 10 6 Pa s, the transition between the two regimes occurs around r b m. It is suggested that the depth of the magma chamber (i.e., L + D or L) becomes the governing factor of q b only for relatively small scale eruptions from narrow conduits ( _m <10 6 kg s 1 ). [42] The relationship between q b and r b in Figure 10 can be transferred to the curves in the _m r b and _m r t spaces using equation (16) and the definition of _m(= pr b 2 q b ). The column collapse condition is given by the intersections of the curves based on the conduit flow model and the boundary between the buoyant plume region and the pyroclastic flow region (Figure 11). Each curve intersects the boundary of the two regions in Figure 9 at two points: the points at the HM and LM sides. This implies that there are two distinct conditions of column collapse. For example, if r b and hence _m increase under a constant condition at the magma chamber, a buoyant plume may become unstable and collapse as r b and _m cross the HM side boundary [e.g., Sparks and Wilson, 1976]. On the other hand, if p 0 decreases under a condition with relatively large g, _m decreases for fixed r b, which can lead to a transition from a buoyant plume to a pyroclastic flow at the LM side boundary [e.g., Woods and Bower, 1995]. [43] Thus far, we have established a method for analyzing the column collapse condition for given magma and crater shape. In the next section, we will perform a systematic parameter study by this method focusing on the problem of how the magma discharge rates at the column collapse conditions of the HM and LM sides depend on magmatic properties and crater shape. 3. Parameter Study [44] In this section we first review the column collapse condition for the case without a crater (the no crater case) in order to clarify the effects of the presence of a crater. Subsequently, we discuss the effects of crater shape in a quantitative fashion Column Collapse Conditions for the No Crater Case [45] For the no crater case, the magma issues as a sonic flow (i.e., free decompression flow) or a subsonic flow. For example, when the model by Ogden et al. [2008b] (i.e., equation (14)) is used for decompression into the atmosphere, the velocity at atmospheric pressure just above the vent is obtained from equation (15) and from _m pr 2 b q b as 8 a a : sonic flow >< v a ¼ _m ð22þ >: rb 2 : subsonic flow: a Note that r b = r t for the no crater case. The boundary of the buoyant plume region and the pyroclastic flow region (cf. Figure 9) is given by the intersections of v a in equation (22) and v crit based on the 1 D model of eruption column dynamics (Figure 12a). Generally, the curves of v a and v crit intersect at two points in this diagram. One is the intersection of v crit and v a = a a, which corresponds to the HMside boundary in Figure 9. Let us refer to _m at this point as 12 of 26

13 Figure 11. Diagram showing the column collapse condition during explosive eruptions. The results of Figures 9 and 10a are superimposed in (a) the _m r b space and (b) the _m r t space. The column collapse conditions are given by intersections between the relationships based on the conduit flow model (solid and dashed curves for L + D = 5 and 10 km, respectively) and the boundary of the buoyant plume and pyroclastic flow regions. The same parameters as those in Figure 9 are used. Figure 12. The diagrams showing the column collapse condition for the case without a crater. (a) The critical velocity for column collapse (v crit ) based on the eruption column model and the velocity at atmosphericpressurejustabovethevent(v a ) as a function of magma discharge rate. The results of v a are shown for r b = 30 m (solid curve) and r b = 10, 100 and 1000 m (dashed curves). Note that v a is fixed at a a and independent of r b when the model of Ogden et al. [2008b] is used for free decompression flow. (b) The column collapse condition in the _m r b space. The _m r b relationships in the conduit flow with Dp varying from 0 to 100 MPa are shown for L + D = 5 km (solid curves) and L + D =10km (dashed curves). The boundaries of the LM and HM sides in Figure 12b are obtained from _m L and _m H for variable r b in Figure 12a, respectively. The parameters for Magma 1 with n 0 = 0.04 and ML 1 in Table 1 are used. For comparison, p the value of _m H based on Woods and Bower [1995] (i.e., the critical magma discharge rate for v a =2 ffiffiffiffiffiffiffiffiffiffi n f RT) is also shown in these diagrams. 13 of 26

14 through a narrow conduit. It is suggested that the column collapse at the LM side boundary is possible only during small scale eruptions with _m L <10 5 kg s 1 for the no crater case. [48] We are concerned with how these features of _m H and _m L are modified by the presence of a crater. In Figures 11 and 12b the values of _m L and _m H are controlled by the positions of the boundaries of the buoyant plume and pyroclastic flow regions rather than the _m r b curves of the conduit flow. We, therefore, concentrate on the positions of the boundaries that are determined in Problems 1 and 2 below. Figure 13. The relationship between the critical magma discharge rate of the HM side ( _m H ) and the gas content (n f ). The shaded zone represents the range of _m H for v a /a a = 1 to 3. Thick solid curve: _m H for the free decompression flow with v a /a a =1.Thindashedcurves: _m H for v a /a a =2 and 3. p Thick dashed and dotted curve: _m H for v a = 1380 ffiffiffiffi n f. The eruption column model by Carazzo et al. [2008] p is applied for these curves. The results for v a = 1380 ffiffiffiffi n f with constant entrainment coefficients (k = 0.03 and 0.11) are also shown by dotted curves for comparison. The parameters for Magma 1 and ML 1 in Table 1 are used. _m H. Because both v crit and a a are functions of magmatic properties, _m H has a fixed value for a given magma (Figure 12b). The other point is the intersection of v crit and v a for the subsonic flow, which corresponds to the LM side boundary in Figure 9. The value of _m at this point (referred to as _m L ) depends on r b (Figure 12a). For the no crater case, because _m L < pr 2 b q*, the LM side boundary is located in the region of q b < q* in Figure 12b. [46] The position of the HM side boundary for the nocrater case is strongly influenced by the choice of models for decompression into the atmosphere. When the model of equation (10) is used, the velocity at atmospheric pressure just p above the vent for sonic flow is approximated by 2 ffiffiffiffiffiffiffiffiffiffi p n f RT (or 1380 ffiffiffiffi n f ; see equation (12)). In this case _mh is estimated to be one to two orders of magnitude greater than the estimate based on equation (15) (Figures 12a and 12b). [47] The column collapse condition for the no crater case is obtained from the intersections of these boundaries and the _m r b curves of the conduit flow obtained in section 2.3. Because the HM side boundary is vertical in Figure 12b, _m H is independent of the conduit flow dynamics. On the other hand, the _m r b curves of the conduit flow do not intersect the LM side boundary in Figure 12b. The results of section 2.3 indicate that q b is less than q* only when p 0 is much lower than the lithostatic pressure or when the magma ascends from a deep magma chamber (L + D > 10 km) 3.2. Effects of Crater Shape on _m H and _m L [49] The presence of a crater results in two major changes in the positions of the boundaries of the buoyant plume and pyroclastic flow regions (compare Figures 9a and 12b). First, the HM side boundary shifts to the higher _m direction (see the arrow in Figure 12b) because v a can be several times larger than a a as a result of the acceleration inside the crater (see Figures 5 and 7). Second, when a shock forms inside the crater, v a can be less than a a even when q b > q* [Woods and Bower, 1995], which results in a shift of the LM side boundary to the higher _m and smaller r b direction (see the arrow in Figure 12b). These shifts of the boundaries of the HM and LM sides lead to increases in _m H and _m L, respectively. [50] Before estimating _m H and _m L by the present method, let us review some previous studies on these quantities to clarify the problems that should be solved here. The position of the HM side boundary critically depends on v a of the underexpanded flow. In previous studies, it was considered that v a of the underexpanded flow mainly depends on water content n f and that its variation depending on crater shape is limited to around v a 2a a to 3a a [e.g., Wilson et al., 1980; Wilson and Walker, 1987; Woods and Bower, 1995; Kaminski and Jaupart, 2001; Carazzo et al., 2008] (see Figure 7a). For example, Kaminski and Jaupart [2001] and p Carazzo et al. [2008] assumed that v a 1380 ffiffiffiffi n f for typical explosive eruptions on the basis of the model for free decompression flow by Woods and Bower [1995] (i.e., equation (12)). On this assumption, the value of _m H is fixed in Figure 12b for given n f so that the column collapse condition is represented by a single _m n f relationship (Figure 13). In this study, we have shown that v a of underexpanded flow may strongly depend on crater shape when the model of Ogden et al. [2008b] is used for decompression into the atmosphere; v a for free decompression flow is approximated by a a (see equation (15)), whereas v a for correctly expanded flow is as much as 3a a (see Figures 6 and 7b). This variation in v a between the free decompression and correctly expanded flows leads to a variation of two orders of magnitude in _m H (Figure 13). This variation in _m H is greater than those caused by other factors such as the variation of the entrainment coefficient (see Figure 13 for the results with k = ). When the model of Ogden et al. [2008b] is used for decompression into the atmosphere, it is essential to specify the condition for underexpanded flow to occur in the _m r b space in order to estimate _m H. [51] Studies on _m L are scarce at present. Woods and Bower [1995] found that the presence of a crater can induce a new type of column collapse accompanied by the generation of a 14 of 26