#### 4.2. Theory of Fallout From Column Margin

[11] Some recent studies argue that proximal deposit in an intensive eruption can be attributed to fallout mode that were not described in the current plume models [e.g., *Fierstein et al.*, 1997; *Valentine et al.*, 2005; *Sable et al.*, 2006]. Indeed, the physics of gas thrust region, where expands rapidly and the particle concentration is high, is poorly understood because analogue or numerical investigation of the region is very difficult. In present plume model, gas thrust region tentatively assumes almost same physics as that of convective region. In this study, we examined whether the observed particle fractionation of the 1986 Izu-Oshima B is consistent with fallout as predicted in the present simple models of fallout and eruption plume.

[12] We adopted the modified version of the model by *Bursik et al.* [1992], which is an extension of the theory for sedimentation rate by *Martin and Nokes* [1988] and succeeded to depict fallout from the umbrella region [*Bursik et al.*, 1992; *Koyaguchi and Ohno*, 2001; *Mannen*, 2006]. Here, particle fallout is assumed to take place at the margin of eruption plume in the manner of

where *M*(ϕ, *h*) is the mass of particle of a given size fraction ϕ and height *h*, *A*_{eff} is the effective area at plume margin for particle fallout, *C* is the mass concentration of particle, *v*_{t} is the settling velocity and *u* is the upward velocity of uprising column. When we consider a horizontal slice of the uprising plume with thickness of *dh*, these parameters are expressed as follows;

where *L* is the characteristic radius of eruption column defined as *L*^{2} = *A* and *A* is the cross-sectional area of the plume [*Woods*, 1988].

[13] As shown in equations 1 to 3, the fallout strongly depends on plume geometry. In this study, the plume model of *Woods* [1988] for the column of the 1986B eruption is adopted. Parameters are selected to reproduce the eruption column of 14 km high (magma discharge rate is 8.1 × 10^{5} kg/s) and initial upward velocity and water content of the column is selected based on the observation and petrological analysis (140 m/s and 1 wt% respectively) [*Mannen*, 2006]. Effect of particle fallout is not considered.

[14] The entrainment coefficient ɛ is defined as 0.09 in convective plume and 0.09√*β*/*α* in gas thrust region, where *α* and *β* are densities of atmosphere and the plume respectively. The mass change as a function of height is numerically calculated using equation 1 and terminal velocity of particle [*Kunii and Levenspiel*, 1960], assuming density of all particles is 1000 kg/m^{3}.

[15] We also considered the effects of radial inflow due to entrainment of the uprising plume. The radial inflow velocity (*v*_{e}) is expressed as a function of distance from axis of the uprising plume (*r*) as,

[16] Here, the velocity at the column margin *v*_{e}(*L*, *h*) is defined as ɛ*U*(*h*), where *U*(*h*) is the upward velocity of the plume at the column axis. The radial inflow affects the particle once it is released at the column margin. At certain conditions, the released particle is recycled into the column.

[17] Using the given equations, the mass and trajectory of particles that are released from a certain height are calculated for the whole size range and height from the vent to the neutral buoyancy height [640 and 10000 m above sea level respectively]. For numerical calculation of particle trajectory, the Euler method was used with a step size for vertical axis of 5 m. When particle that released at a given height (*h*_{d}) is re-entrained into the plume in the lower height (*h*_{r}), particle release at *h*_{d} is not counted to simplify the calculations. Bias arising from the omission does not affect the calculations since *h*_{d} and *h*_{r} are similar in most cases. Since the area of CB is roughly 0.5 km^{2}, the calculated particle size distribution of the cone is represented by the composition of particles settled within 400 m from the vent. Firstly, we calculated for the two cases, with and without considering the radial inflow effect (model 1 and 2 respectively).

#### 4.3. Particle Fractionation of CB

[18] The calculated particle fractionation of the cone is shown in Figure 1. The observed particle fractionation shows broad agreement with the prediction of model 1, however, it shows excessive fallout of lapilli sized particles. The finer particles rather appear to be consistent with the model 2. In other word, the finer particles seem to be less affected by lateral inflow. This observation is inconsistent because finer fraction is affected by weaker lateral inflow. Partial column collapse or some discontinuity of the uprising plume such as ‘radially suspended flow’ [*Suzuki et al.*, 2005] may cause such fractionation. However, such have never been observed in the eruption. Here we would like to attribute particle transport within the plume.

[19] The present model for eruption column discusses the mass and heat balance of the ascending plume to introduce ‘control volume’ dimension of which is *L*^{2}*dh*. The characteristic column radius *L* does not correspond to the ‘visible’ column radius (*L*_{v}). It is noteworthy that the material released at *L* can be transported by the turbulent eddies to some distance up to *L*_{v} (>*L*). Here, we assume that smaller particles released at *L* drift laterally to *L*_{d} (<*L*_{v}) of the same height loss their support by upward gas flow and start to fall out due to the effect of atmospheric radial inflow (model 3). In this paper, the coefficient defined as *L*_{d}/*L* is given as *ω*. The maximum *ω* correspond to *L*_{v}/*L* and can be obtained from the numerical result and the observation. In Figure 2, the increase rates of *L* and *L*_{v} as a function of height are shown. From numerical analysis *L*/*h* is known to be 0.14 and from photographic analysis maximum *Lv*/*h* is obtained to be 0.35. Using them, maximum *ω* is assumed to be 2.5. In the model 3, *ω* is thus assumed to be 1.2, 1.4 and 2.5 for particles of >8 mm, 8-4 mm and <4 mm, respectively. The model 3 seems to be consistent with the observation (Figure 1).

#### 4.4. Fallout of Cone-Forming Column

[20] *Ernst et al.* [1996] argued that fallout from the uprising column is very limited and major fallout takes place at the transition between the column and the umbrella region. It is also argued that the particles that fallout at the transition are swept back to form near vent fall deposits. The model 1 and 3 in this study also predict that the fallout from the column margin is nearly absent in almost the whole range of the column height (Figure 3). However, our calculation indicates that the contribution of fallout from the column-umbrella transition is minor while the fallout from the base of the uprising column is most significant. Extensive fallout at the base of the column is attributed to the slope of column edge.

[21] We may derive a conceptual criterion for particle re-entrainment to understand the reason of extensive fallout at the base of the column. The slope of the column edge is expressed as dL/dh. Provided the settling velocity and air inflow velocity are constant, the condition for re-entrained particles is expressed as *dL*/*dh* < *v*_{e}/*v*_{t}. This relationship qualitatively indicates that rapid expansion of column as a function of height tends to avoid particle re-entrainment.

[22] In the case of the 1986B eruption, our calculation predicts *dL*/*dh* = 0.38 for *h* < 500 m and 0.15 for > 500 m (Figure 3). Since the heat transfer from the pyroclasts to the entrained air is significant, rapid expansion takes place at the bottom part of the column [*Woods*, 1988]. Such rapid expansion is not predicted in the classical column model based on scaling approach and was not reproduced by analogue experiments that do not replicate great heat transfer from magma to air [*Sparks*, 1986; *Ernst et al.*, 1996]. The effect of rapid expansion to extensive fallout near the source vent has not been considered in previous studies. In addition to the rapid expansion, high particle density is also essential for extensive fallout as known from equation [1].

[23] Scoria cones seem to be developed on relatively flat ground. As such, the occurrence of scoria cones may be attributed to this ideal condition for particle fallout as predicted in the model presented here. In certain situation such as the source vent located in the bottom of a large crater, the plume may sufficiently expand until it rises to the crater rim. Indeed, we often observe plume that is issued from very small sources vent occupies entire width of the large crater. In such case, significant fallout of uprising plume beyond the crater rim could be absent. Construction around the source vent could be impeded due to vigorous mixing within the crater or the steepness of the slope of the crater prevents cone formation.

[24] Cone-forming eruptions could be hazardous in certain situations since they may generate pyroclastic flows [*Yamamoto et al.*, 2005], however, physics of basal part of the eruption column is poorly understood. Quantitative observation of particle fractionation as shown in this study will be one of a few observational constraints to the modeling such as high resolution 3D calculation [e.g., *Suzuki et al.*, 2005], which is a promising approach to understand processes in gas thrust region.