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3-D numerical simulations of eruption column collapse:
Effects of vent size on pressure-balanced jet/plumes
Y.J. Suzukia,∗, T. Koyaguchia
a
Earthquake Research Institute, University of Tokyo, Yayoi 1-1-1, Bunkyo-ku, Tokyo
113-0032, Japan
Abstract
Buoyant columns or pyroclastic flows form during explosive volcanic eruptions. In the transition-state, these two eruption styles can develop simultaneously. We investigated the critical condition that separates the two eruption styles (referred to as “the column collapse condition”) by performing
a series of three-dimensional numerical simulations. In the simulation results, we identify two types of flow regime: a turbulent jet that efficiently
entrains ambient air (jet-type) and a fountain with a high-concentration of
the ejected material (fountain-type). Hence, there are two types of column
collapse (jet-type and fountain-type). Which type of collapse occurs at the
column collapse condition depends on whether the critical mass discharge
rate for column collapse (MDRCC ) is larger or smaller than that for the generation of a fountain (MDRJF ) for a given exit velocity. Temperature controls
the relative magnitude of MDRCC relative to MDRJF , and hence the type of
collapse. For given magma properties (e.g., temperature and water content),
Corresponding author
Email addresses: [email protected] (Y.J. Suzuki),
[email protected] (T. Koyaguchi)
∗
Preprint submitted to Journal of Volcanology and Geothermal Research January 25, 2012
the column collapse condition is expressed by a critical value of the Richardson number (RiCC ≡ g0′ L0 /w02 , where g0′ is the source buoyancy, L0 is the
vent radius, and w0 is the exit velocity). When the jet-type collapse occurs
at the column collapse condition, RiCC is independent of the exit velocity.
When the fountain type collapse occurs at the column collapse condition, on
the other hand, RiCC decreases as the exit velocity increases. As the exit
velocity exceeds the sound velocity, a robust flow structure with a series of
standing shock waves develops in the fountain, which suppresses entrainment
of ambient air and enhances column collapse.
Keywords: eruption cloud, column collapse condition, numerical
simulation, flow-regime map, turbulent mixing
1
1. Introduction
2
During explosive volcanic eruptions, a mixture of hot ash (pyroclasts)
3
and volcanic gas is released from the vent into the atmosphere. The mixture
4
generally has an initial density several times larger than the atmospheric
5
density at the vent and its ascent is driven only by its momentum. As the
6
ejected material entrains ambient air, the density of the mixture decreases
7
because the entrained air expands by heating from the hot pyroclasts. If
8
the density of the mixture becomes less than the atmospheric density before
9
the eruption cloud loses its upward momentum, a buoyant plume rises to
10
form a plinian eruption column. On the other hand, if the mixture loses its
11
upward momentum before it becomes buoyant, the eruption column collapses
12
to generate a pyroclastic flow. We refer to the condition that separates these
13
two eruption regimes as “the column collapse condition” (e.g., Sparks and
2
14
Wilson, 1976). Because the impact and type of volcanic hazards are largely
15
different between the two eruption regimes, predicting the column collapse
16
condition is of critical importance in physical volcanology.
17
The column collapse condition has previously been predicted by one di-
18
mensional (1-D) steady eruption column models (e.g., Wilson et al., 1980;
19
Bursik and Woods, 1991; Kaminski and Jaupart, 2001; Carazzo et al., 2008;
20
Koyaguchi et al., 2010). First generation 1-D steady models are based on
21
a classical theory of turbulent jet/plumes from a point source, in which the
22
efficiency of turbulent mixing between the eruption cloud and ambient air
23
is expressed by a single parameter, the “entrainment coefficient” (e.g., Mor-
24
ton et al., 1956; Taylor, 1945). These models captured the basic physics
25
of eruption column dynamics and enabled us to calculate the upward mo-
26
mentum and buoyancy of eruption columns as a function of the downstream
27
distance from the vent, and thus, to predict the maximum heights of the
28
eruption column and column collapse condition for given magma properties
29
(e.g., temperature and water content) and source conditions (e.g., vent ra-
30
dius and velocity). The models of column collapse show a quasi-quantitative
31
agreement with field observations of witnessed eruptions (e.g., Carey et al.,
32
1990). Recently, the agreement between the model predictions and field ob-
33
servations has been improved by a more sophisticated 1-D model in which
34
the variation of entrainment coefficient with height is taken into account
35
(Carazzo et al., 2008).
36
Since the 1980s, 2-D and 3-D models have been developed for eruption
37
column dynamics. They have shown a number of complex fluid dynamical
38
features that cannot be described by 1-D models (e.g., Wohletz et al., 1984;
3
39
Valentine and Wohletz, 1989; Neri and Dobran, 1994; Oberhuber et al., 1998;
40
Suzuki et al., 2005; Ogden et al., 2008a,b; Suzuki and Koyaguchi, 2009). Gen-
41
erally, the flow of a gas-pyroclast mixture near the vent is characterized by an
42
annular structure consisting of an outer shear region and dense inner core (re-
43
ferred to as a “potential core” in aerodynamics literature (e.g., Rajaratnam,
44
1976)). The presence of the dense inner core affects the flow pattern dur-
45
ing column collapse (e.g., Suzuki et al., 2005). When the mixture is ejected
46
from a large vent (more than a few hundred meters), the outer shear region
47
cannot reach the central axis before the initial momentum is exhausted, so
48
that the dense inner core generates a fountain structure, called “a radially
49
suspended flow” (Neri and Dobran, 1994). On the other hand, for eruption
50
from a relatively narrow vent, the dense inner core disperses due to turbulent
51
mixing and the eruption cloud collapses without a fountain structure. Ogden
52
et al. (2008a) pointed out that the overpressure at the vent generates another
53
type of annular structure in the flow near the vent, which causes complex
54
flow behavior during column collapse; an oscillatory collapse with a regular
55
periodicity occurs even when a steady condition is applied at the source.
56
In this paper and some accompanying papers, we systematically inves-
57
tigate how these 2-D and 3-D effects modify the column collapse condition
58
on the basis of a number of 3-D numerical simulations of an eruption cloud
59
with a high spatial resolution. We carried out an extensive parameter study
60
to make regime maps of different flow patterns and to determine the column
61
collapse condition for each flow pattern. From the features of the column
62
collapse conditions, as well as those of the boundaries between the different
63
flow patterns, we can infer the governing factors which control the dynamics
4
64
of column collapse. In this particular paper, we focus on the flow whose
65
pressure balances ambient atmospheric pressure (referred to as “a pressure-
66
balanced jet/plume”). We show that even for this simplest case, the 2-D
67
and 3-D effects of flow (e.g., the presence of a potential core with or without
68
shock waves) substantially modify the column collapse condition.
69
2. Model Description
70
Our main concern is to clarify how the variation in the vent radius in-
71
fluences the global features of eruption clouds such as the formation of an
72
eruption column and/or pyroclastic flow. The most essential physics which
73
governs the dynamics of an eruption cloud is the efficiency of turbulent mix-
74
ing between the ejected material and ambient air. In order to minimize
75
complexity and to focus on the problem of turbulent mixing, a pseudo-gas
76
model is applied in this study; we ignore the separation of solid pyroclasts
77
from the eruption cloud and treat the eruption cloud as a single gas whose
78
density is calculated from the mixing ratio of the ejected material and en-
79
trained air. The fluid dynamics model solves a set of partial differential
80
equations describing the conservation of mass, momentum, and energy, and
81
constitutive equations describing the thermodynamic state of the mixture of
82
pyroclasts, volcanic gas, and air. These equations are solved numerically by
83
a general scheme for compressible flow (Roe, 1981; van Leer, 1977). Details
84
of the numerical procedures used in this study are described in Suzuki et al.
85
(2005).
86
The simulations are designed to describe the injection of a mixture of
87
pyroclasts and volcanic gas from a circular vent above a flat surface of the
5
88
earth in a stationary atmosphere with a temperature gradient typical of the
89
mid-latitude atmosphere (Table 1). The vent is located in the center of the
90
ground surface. The physical domain extends horizontally and vertically
91
from a few kilometers to several tens of kilometers. At the ground boundary,
92
the free-slip condition is assumed for the velocities of the ejected material
93
and air. At the upper and other boundaries of the computational domain,
94
the fluxes of mass, momentum, and energy are assumed to be continuous,
95
and these boundary conditions correspond to the free outflow and inflow of
96
these quantities.
97
In order to correctly reproduce the general feature of turbulent mixing
98
that the efficiency of entrainment is independent of the Reynolds number
99
(Dimotakis, 2000), it is essential to apply 3-D coordinates with a sufficiently
100
high spatial resolution (Suzuki et al., 2005). In this study, the calculations
101
were performed on a 3-D domain with a non-uniform grid. The grid size
102
was set to be sufficiently smaller than L0 /8 near the vent, where L0 is the
103
vent radius. In order to effectively simulate the turbulent mixing with high
104
spatial resolutions both far from and close to the vent, the grid size increases
105
at a constant rate (by a factor of 1.01 for the vertical coordinate and 1.02
106
for the horizontal coordinate) up to L0 with the distance from the vent, such
107
that the grid size is small enough to resolve the turbulent flow far from the
108
vent (cf. Suzuki and Koyaguchi, 2010).
109
We assume steady conditions; for each run, magma temperature T0 , water
110
content ng0 , vent radius L0 , and exit velocity w0 are fixed. The mass discharge
111
rate ṁ0 is determined from the relationship of ṁ0 = πρ0 w0 L20 , where ρ0 is
112
a function of T0 and ng0 . In this paper, for simplicity we assumed that the
6
113
pressure at the vent is equal to the atmospheric pressure. In general, the
114
pressure of a gas-pyroclast mixture deviates from the atmospheric pressure
115
during explosive eruptions; the mixture is accelerated and/or decelerated
116
because of decompression and/or compression within a short distance from
117
the vent (5–20 vent radius) (Woods and Bower, 1995; Ogden et al., 2008b;
118
Koyaguchi et al., 2010). Another extensive parameter study focusing on this
119
effect is in progress and the results will be reported elsewhere.
120
In order to classify different flow patterns and determine the column col-
121
lapse condition in pressure-balanced jet/plumes, we performed a parameter
122
study involving 95 numerical simulations. The conditions for these simula-
123
tions were divided into four groups according to magma temperature (T0 ) and
124
water content (ng0 ); group H with a high magma temperature (T0 = 1000
125
K and ng0 = 2.84 wt %), group I with an intermediate magma temperature
126
(T0 = 800 K and ng0 = 2.84 wt %), group L with a low magma temperature
127
(T0 = 550 K and ng0 = 2.84 wt %), and group N with a small water content
128
(T0 = 1000 K and ng0 = 1.23 wt %). These magma properties are con-
129
sidered to represent those of typical explosive eruptions (see Carazzo et al.,
130
2008; Koyaguchi et al., 2010). Low temperature mixture of group L can be
131
ejected during eruptions involving the interaction of ground water and/or
132
country rock with magma (e.g., Koyaguchi and Woods, 1996). In order to
133
cover the conditions of typical explosive eruptions, we set the mass discharge
134
rate to be 104 –109 kg s−1 and the exit velocity to be 0.5c0 –3c0 , where c0
135
136
137
is the sound velocity of the gas-pyroclast mixture at the vent, defined as
p
c0 = 0.01ng0 Rg T0 . Generally, exit velocity, magma properties (i.e., T0 and
ng0 ), vent radius, and hence, magma discharge rate are coupled such that
7
138
they are consistent with the fluid dynamics in the conduit and crater (Wil-
139
son et al., 1980; Koyaguchi et al., 2010). In this study, we assessed whether
140
or not the above ranges are physically realistic on the basis of Koyaguchi
141
et al. (2010). The wide range of exit velocity allows us to investigate fea-
142
tures of subsonic, sonic, and supersonic flows. A summary of the simulation
143
parameters for the four groups is listed in Table 2.
144
3. Results
145
3.1. Classification of Flow Patterns
146
The flow patterns in the simulation results are classified into four flow
147
regimes: an eruption column with a jet structure (e.g., run 99), an eruption
148
column with a fountain structure (e.g., run 33), jet collapse (e.g., run 78),
149
and fountain collapse (e.g., run 49). The representative features of each flow
150
regime are described below.
151
3.1.1. Eruption Column Regimes
152
In run 99 (T0 = 1000 K, ṁ0 = 1.0 × 107 kg s−1 , L0 = 49 m), a stable
153
eruption column with a typical jet structure develops (Fig.1). In this run,
154
the mixture of volcanic gas and pyroclasts is ejected from the vent as a
155
dense, high speed jet. After traveling a short distance from the vent, the
156
shear flow at the boundary between the jet and the ambient air becomes
157
unstable. The jet entrains ambient air by this shear instability; it forms an
158
annular mixing layer which surrounds an unmixed core flow (Fig.1a). The
159
unmixed core is eroded by the annular mixing layer and disappears at a
160
certain level (at a height of ∼ 3 km (60 times L0 ) in run 99). As the eruption
8
161
column further ascends, the flow becomes highly unstable and undergoes a
162
meandering instability that induces more efficient mixing (e.g., Suzuki et al.,
163
2005). The stream-wise evolution of the flow results in a complex density
164
distribution in the eruption cloud (Fig.1b). The unmixed core is denser than
165
the ambient air, whereas the annular mixing layer is lighter than the ambient
166
air and gains buoyancy owing to expansion of the entrained air. After the
167
meandering instability develops, the dense unmixed core disappears before
168
it exhausts its initial momentum. We refer to the eruption column that is
169
characterized by the disappearance of the unmixed core before it exhausts
170
its initial momentum as “the jet-type column”.
171
When the gas-pyroclast mixture is ejected from a larger vent (run 33;
172
L0 = 154 m), a stable eruption column with a fountain structure develops
173
(Fig.2). In this run, because the vent radius is relatively large, the erosion
174
by the annular mixing layer does not reach the central axis at the height
175
where the unmixed core exhausts its initial momentum (Fig.2a). The dense
176
unmixed core around the central axis spreads radially at this height (Fig.2b).
177
Such a radially suspended flow is commonly observed in the fountain which
178
results from the injection of a dense fluid upwards into a less dense fluid (e.g.,
179
Neri and Dobran, 1994; Lin and Armfield, 2000). The large-scale eddy of this
180
radially suspended flow causes an intensive mixing between the ejected mate-
181
rial and ambient air. After this mixing, the resultant mixture becomes buoy-
182
ant and generates an upward flow from the large-scale eddy; this produces
183
another type of stable column. The eruption column that is characterized
184
by the radially suspended flow is referred to as “the fountain-type column”.
185
Fig.3 shows the time evolutions of the vertical profiles along the central
9
186
axis for the jet-type column (run 99) and fountain-type column (run 33). In
187
the jet-type column, the mass fraction of the ejected material, ξ, gradually
188
decreases with height (Fig.3a). The high-ξ region intermittently rises at a
189
time interval of ∼ 10 s; the flow of the unmixed core is highly unsteady. In the
190
fountain-type column, on the other hand, ξ remains at 1.0 below a height of
191
2.2 km (Fig.3b) and decreases rapidly at this level, which corresponds to that
192
of the large-scale eddy at the top of the fountain. The height of the rapid
193
change of ξ is constant with time and does not show oscillatory behavior,
194
which suggests that the flow of the unmixed core is almost steady.
195
In Fig.4, the time-averaged mass fraction of the ejected material, ξ, and
196
normalized upward velocity, w/w0, along the central axis are presented as
197
functions of the normalized vertical position, z ∗ (≡ z/(w02 /2g)) for the jet-
198
type column (run 99) and fountain-type column (run 33); normalized quan-
199
tities are used for quantitative comparison between results of different vent
200
conditions in these diagrams. In the jet-type column, ξ gradually decreases
201
with height from 1.0 to 0.8 below z ∗ = 1 and rapidly falls to 0.15 from z ∗ = 1
202
to 2 (solid curve in Fig.4a). The averaged upward velocity, w/w0, gradually
203
decreases from 1.0 to 0.5 below z ∗ = 1 and slightly increases above it (solid
204
curve in Fig.4b). The height of z ∗ = 1 coincides with the level where the
205
unmixed core disappears (z = 3 km). These profiles of ξ and w/w0 are ex-
206
plained by the gradual growth of the annular mixing layer and the erosion of
207
the unmixed core below z ∗ = 1. In the fountain-type column, on the other
208
hand, ξ remains 1.0 below z ∗ = 0.7, rapidly decreases to 0.1 at the fountain
209
top (z ∗ = 0.7), and remains 0.1 above it (dashed curve in Fig.4a). The aver-
210
aged upward velocity, w/w0, decreases from 1.0 to −0.1 in the fountain below
10
211
z ∗ = 0.7, and rapidly increases above it (dashed curve in Fig.4b). These pro-
212
files of ξ and w/w0 reflect the features of entrainment in the fountain-type
213
column: extensive entrainment occurs at the fountain top, whereas the en-
214
trainment is inefficient in the fountain. It is noted that w/w0 has a negative
215
value just above the fountain top (at z ∗ = 0.7–0.9). This negative value of
216
w/w0 is caused by vortical motion associated with the large-scale eddy at the
217
fountain top. The extensive entrainment by the large-scale eddy at the foun-
218
tain top provides new insight into the mixing process in eruption columns,
219
because the mixing at the vertical edge of column has been assumed in the
220
1-D models.
221
In the simulation results of the fountain-type column regimes, we ob-
222
served that the qualitative feature of pressure distribution inside the foun-
223
tain changes with the exit velocity (Fig.5). When the exit velocity is sonic
224
(the Mach number is 1.0 in run 34), the time-averaged pressure inside the
225
fountain is uniform and almost equal to the atmospheric pressure (Fig.5a).
226
When the exit velocity is supersonic (the Mach number is 1.5 in run 33), a
227
spatially periodic pattern of higher and lower pressure is observed in the foun-
228
tain (Fig.5b). This periodic pattern results from the generation of standing
229
shock waves in the fountain; like in an overexpanded jet (i.e., a jet with exit
230
pressure lower than the surrounding fluid), compression waves (i.e., oblique
231
shock waves) are produced in the supersonic jet, and these compression waves
232
reflect from the symmetry axis and generate expansion waves (Dobran et al.,
233
1993).
11
234
3.1.2. Column Collapse Regimes
235
In the present simulations, column collapse occurs when the vent radius is
236
large. For given vent radius and exit velocity, column collapse tends to occur
237
as magma temperature decreases. The flow pattern of the column collapse
238
regimes depends on the magma temperature and vent radius.
239
When a low-temperature gas-pyroclast mixture is ejected from a narrow
240
vent (run 78; T0 = 550 K, L0 = 11m), the eruption column collapses from
241
the jet (Fig.6). In this run, the jet entrains ambient air by the shear instabil-
242
ity; the annular mixing layer develops and the unmixed core disappears at a
243
height of 0.5 km (Fig.6a). Because the initial temperature is low, the erup-
244
tion cloud remains heavier than air even though it entrains a large amount of
245
air. As a result, the low-concentration mixture generates a pyroclastic flow
246
(Fig.6b). The column collapse regime that is characterized by the disappear-
247
ance of the unmixed core before it exhausts its initial momentum is referred
248
to as “the jet-type collapse”.
249
In contrast to the jet-type collapse, when a high-temperature gas-pyroclast
250
mixture is ejected from an extremely large vent (run 49; T0 = 1000 K,
251
L0 = 403 m), the eruption column collapses from a fountain (Fig.7a). As was
252
observed in the case of the fountain-type column (Fig.2), when the vent ra-
253
dius is large, the unmixed core reaches a height where the initial momentum
254
is exhausted and generates a radially suspended flow which causes mixing by
255
a large-scale eddy. As the vent radius exceeds a certain value, the large-scale
256
eddy of the radially suspended flow cannot entrain sufficient air for the whole
257
mixture to become buoyant (Fig.7b). As a result, the unmixed part in the
258
radially suspended flow collapses and generates a pyroclastic flow. The col-
12
259
umn collapse regime that is characterized by the radially suspended flow is
260
referred to as “fountain-type collapse”. When the dense eruption cloud hits
261
the ground and spreads horizontally, it efficiently entrains ambient air. The
262
entrained air expands by heating from the pyroclasts so that the mixture
263
of the ejected material and ambient air forms a buoyant co-ignimbrite ash
264
cloud.
265
Fig.8 shows the time-averaged vertical profiles of physical quantities along
266
the central axis for the jet-type collapse (run 78) and fountain-type collapse
267
(run 49). In the jet-type collapse, ξ and w/w0 gradually decrease with height
268
and fall to zero at z ∗ ∼ 1 (solid curves in Fig.8). The gradual decreases in
269
ξ and w/w0 represent the dispersion of the unmixed core due to the gradual
270
growth of the annular mixing layer. In the fountain-type collapse (dashed
271
curve in Fig.8a), ξ remains 1.0 in the fountain (z ∗ < 1.2) and rapidly de-
272
creases to 0.1 at the fountain top (z ∗ = 1.2). The value for z ∗ > 1.2 (ξ = 0.1)
273
represents the mass fraction of the ejected material in the co-ignimbrite ash
274
cloud. The upward velocity decreases with height in the fountain and falls to
275
zero at the fountain top (dashed curve in Fig.8b). In contrast to the case of
276
the jet-type collapse, the unmixed core itself is the main source of pyroclastic
277
flows in the case of the fountain-type collapse.
278
3.1.3. Transitional Regimes
279
The boundaries of the above four flow regimes (the jet-type column,
280
fountain-type column, jet-type collapse, and fountain-type collapse regimes)
281
are determined by two types of transition. One is the transition between the
282
jet-type and fountain-type regimes; we call the condition for this transition
283
the jet-fountain (JF) condition. The other is the transition between the col13
284
umn convection and collapse regimes; we call the condition for this transition
285
the column collapse (CC) condition.
286
Around the JF condition, some intermediate features between jet-type
287
and fountain-type regimes were observed. Run 46 (T0 = 1000 K, L0 = 82 m)
288
shows an intermediate feature between the jet-type column and the fountain-
289
type column regimes (Fig.9). In this run, the unmixed core is eroded to a
290
considerable extent by the annular mixing layer, while relatively large-scale
291
eddies develop at the fountain top (z ∼ 4 km). In run 92 (T0 = 550 K,
292
L0 = 38 m), the type of column collapse changes from the jet-type to the
293
fountain-type with time (Fig.10). At the initial stage of eruption (t < 50 s),
294
the unmixed core disappears at z = 0.6 km (see the contour of ξ = 0.95),
295
whereas the total height of the negatively buoyant jet reaches 1.4 km high.
296
Subsequently, the height of the unmixed core increases with time (50 s < t <
297
80 s) and reaches the top of the jet so that the fountain structure (i.e., the
298
radially suspended flow) develops (t > 80 s).
299
Around the CC condition, some intermediate features between column
300
and collapse regimes are observed. In run 27 (T0 = 1000 K, L0 = 345 m), a
301
fountain-type column and pyroclastic flow develop simultaneously (Fig.11).
302
After mixing due to the large-scale eddy at the fountain top, most of the
303
cloud becomes a buoyant plume, whereas the side of the fountain partially
304
flows down to form a pyroclastic flow. This pyroclastic flow is less energetic
305
than the pyroclastic flow of the fountain-type collapse regime which falls
306
down directly from the fountain top (cf. Fig.7). Around the CC conditions
307
for groups L and I, a jet-type column becomes unstable with time and can
308
collapse to generate a dilute pyroclastic flow; for example, in run 120 (T0 =
14
309
800 K, L0 = 18 m), the transition from the jet-type column to the jet-type
310
collapse regimes occurs at t ∼ 150 s from the beginning of the eruption (the
311
result is not shown).
312
3.2. Flow-Regime Map
313
On the basis of extensive parameter studies, we made flow-regime maps
314
in the parameter space of the mass discharge rate (ṁ0 ) and exit velocity (w0 )
315
for 550 K < T0 < 1000 K and 1.23 wt % < ng0 < 2.84 wt % (Fig.12).
316
When the temperature is low (group L) or intermediate (group I), the
317
flow changes from the jet-type column to the jet-type collapse regimes, and
318
from the jet-type collapse to the fountain-type collapse regimes as the mass
319
discharge rate increases for a given exit velocity (Figs.12a,b). The boundary
320
between the jet-type column and the jet-type collapse regimes represents the
321
CC condition (solid curves), and that between the jet-type collapse and the
322
fountain-type collapse regimes represents the JF condition (dashed curves).
323
Both the critical mass discharge rate for the CC condition (MDRCC ) and
324
for the JF condition (MDRJF ) increase as the exit velocity increases. For a
325
low temperature, MDRCC is substantially smaller than MDRJF ; as a result,
326
the jet-type collapse occurs in a wide range of parameters in this diagram
327
(see Fig.12a). As the temperature increases, the distance between these two
328
boundaries decreases so that the region of the jet-type collapse regime shrinks
329
(Fig.12b).
330
When the temperature is high (group H), MDRCC becomes greater than
331
MDRJF for a given exit velocity; as a result, the flow changes from the
332
jet-type column to the fountain-type column regimes, and from the fountain-
333
type column to the fountain-type collapse regimes as the mass discharge rate
15
334
increases (Figs.12c,d). In these cases, the CC condition is represented by the
335
boundary between the fountain-type column and the fountain-type collapse
336
regimes, and the JF condition is represented by the boundary between the
337
jet-type column and the fountain-type column regimes. The results of group
338
N (T0 = 1000 K and ng0 = 1.23 wt %) shows that the distance between
339
these two boundaries decreases as the water content decreases for a given
340
temperature (Fig.12d).
341
4. Physical Meanings of the Two Critical Conditions
342
The variation in flow-regime maps suggests that which flow regime can
343
occur is determined by the positional relationship between the JF and CC
344
conditions. This positional relationship depends on the magma properties;
345
for a given exit velocity, MDRCC is smaller than MDRJF when the tem-
346
perature is relatively low, whereas MDRCC is larger than MDRJF when the
347
temperature is high. In this section, we discuss how each critical condition
348
depends on the magma properties.
349
In order to understand the physical meanings of the two critical condi-
350
tions, we redraw Fig.12 in a non-dimensional space, namely in the space
351
of the Richardson number (Ri) and the Mach number (M) (Fig.13). The
352
Richardson number is defined by
Ri =
g0′ L0
,
w02
(1)
353
where the source buoyancy, g0′ , is defined using the density of the ejected
354
material, ρ0 , and the atmospheric density, ρa , as g0′ = (ρ0 − ρa )g/ρ0 . The
355
Mach number is defined as the ratio of the exit velocity and the sound velocity
16
356
of the ejected material, c0 :
M=
w0
.
c0
(2)
357
Fig.13 indicates that both the two critical conditions are determined
358
mainly by the Richardson number for given magma properties (cf. Valentine
359
and Wohletz, 1989). We define the Richardson number at the CC condi-
360
tion as RiCC and that at the JF condition as RiJF , respectively. For a given
361
Mach number, RiCC increases as T0 and/or ng0 increases (see gray stars in
362
Fig.13), whereas RiJF is almost independent of T0 and/or ng0 (see white
363
stars in Fig.13). Whether RiCC < RiJF or RiCC > RiJF determines the type
364
of the intermediate flow regime. In the low-temperature case (Figs.13a,b),
365
RiCC < RiJF so that the intermediate flow regime is the jet-type collapse.
366
In the high-temperature case (Figs.13c,d), RiCC > RiJF so that the inter-
367
mediate flow regime is the fountain-type column. The physical mechanisms
368
that cause the dependencies of RiJF and RiCC on magma properties will be
369
discussed below.
370
4.1. Physical mechanisms to control RiJF
371
The JF condition is determined by whether the unmixed core is eroded
372
by the annular mixing layer or not before the initial momentum is exhausted.
373
This problem has two length scales: the height where the unmixed core dis-
374
appears (Hcore ), and the height where the initial momentum is exhausted
375
(Hmmt ). When Hcore > Hmmt , the fountain is generated. According to ex-
376
perimental studies on high-speed jets (e.g., Nagamatsu et al., 1969), Hcore is
377
a function of the vent radius, L0 , and M:
Hcore = (a1 M + a2 )L0 ,
17
(3)
378
where a1 and a2 are positive constants. On the other hand, Hmmt is estimated
379
from the balance between the kinetic and potential energy:
Hmmt =
380
w02
.
2g0′
These two equations yield
2(a1 M + a2 )g0′ L0
Hcore
= 2(a1 M + a2 )Ri.
=
Hmmt
w02
381
(4)
(5)
Consequently, we obtain the JF condition (i.e., Hcore = Hmmt ) as
RiJF =
1
.
2(a1 M + a2 )
(6)
382
This equation successfully explains the simulation results in Fig.13 that RiJF
383
is independent of the magma properties for a fixed M, and that RiJF decreases
384
as M increases.
385
The above result that the transition between the jet and fountain occurs
386
at a certain critical Ri is supported by laboratory experiments of negatively
387
buoyant jet/fountains (e.g., Kaye and Hunt, 2006). In these experiments,
388
when Ri at the source is smaller than a certain value of O(1), the flow behaves
389
as a highly forced jet where the injected material efficiently mixes with the
390
ambient fluid (i.e., the jet-type). When Ri is larger than the critical value,
391
on the other hand, the flow becomes a fountain where there is little mixing
392
between the injected material and the ambient fluid (i.e., the fountain-type).
393
In order to extend the conclusion derived from these experimental results for
394
subsonic flows to the case of supersonic flows, we carried out supplementary
395
simulations (Appendix A).
18
396
4.2. Dependency of RiCC on the Magma Properties
397
The CC condition is determined by whether the eruption cloud becomes
398
buoyant or not at z = Hmmt . When the mass fraction of the ejected material,
399
ξ, is smaller than a certain value, ξcrt , the mixture of ejected material and en-
400
trained air is lighter than the ambient air. On the other hand, when ξ > ξcrt ,
401
the density of the mixture is larger than the atmospheric density. Therefore,
402
the column collapse occurs when the mass fraction of the ejected material
403
at z = Hmmt , ξmmt, is larger than ξcrt . The value of ξmmt is approximately
404
estimated by
ξmmt =
ṁ0
,
ṁ0 + ṁair
(7)
405
where ṁ0 is the mass discharge rate at the vent, and ṁair is the mass of air
406
entrained below z = Hmmt per unit time. By definition, ṁ0 is represented by
ṁ0 = πρ0 w0 L20 .
(8)
407
Because ṁair is proportional to the surface area of the eruption column and
408
the exit velocity, w0 , it is expressed by
ṁair = 2πkρa w0 L0 Hmmt
(9)
409
under the assumption that the efficiency of turbulent mixing (i.e., the en-
410
trainment coefficient, k (Morton et al., 1956)) is constant. Substituting Eqs.
411
(8) and (9) into Eq.(7), we obtain
ξmmt =
πρ0 w0 L20
.
πρ0 w0 L20 + wπkρa w0 L0 Hmmt
(10)
412
From Eqs. (1), (4), and (10), we obtain the CC condition (i.e., ξmmt = ξcrt )
413
as
RiCC = k
ρa ξcrt
.
ρ0 1 − ξcrt
19
(11)
414
This result (i.e., Eq.(11)) successfully explains the simulation result that
415
RiCC depends on magma properties even for a fixed M; because 1/ρ0 and ξcrt
416
increase with T0 and ng0 , RiCC increases with T0 and ng0 . This result is also
417
consistent with the predictions based on the 1-D model of eruption column
418
by Woods and Caulfield (1992) (Appendix B).
419
4.3. Dependency of RiCC on the Mach Number
420
The results of our 3-D simulations show that RiCC is independent of
421
the Mach number when RiCC is smaller than RiJF (Figs.13a,b).
When
422
RiCC > RiJF , on the other hand, RiCC decreases as the Mach number in-
423
creases (Figs.13c,d). According to Eq.(11), RiCC is independent of the Mach
424
number under the assumption that the efficiency of turbulent mixing (i.e.,
425
the entrainment coefficient) is constant. The variation of RiCC in Figs.13c,d,
426
therefore, means that the efficiency of turbulent mixing changes with the exit
427
velocity for RiCC > RiJF .
428
The efficiency of turbulent mixing for the results of 3-D simulations (the
429
effective entrainment coefficient) can be estimated by comparing the CC con-
430
dition based on the 3-D simulations with that predicted by the 1-D model (cf.
431
Suzuki et al., 2005). In Fig.12, the column collapse conditions predicted by
432
the 1-D model with different assumed entrainment coefficients are also illus-
433
trated by the dotted curves. For jet-type collapse (RiCC < RiJF ), the column
434
collapse condition based on the 3-D simulations coincides with that of the 1-
435
D model with a constant entrainment coefficient (∼ 0.03 in Figs.12a,b). For
436
fountain-type collapse (RiCC > RiJF ), the effective entrainment coefficient of
437
the 3-D simulations decreases from 0.10 to 0.03 with the increasing Mach
438
number (Figs.12c,d).
20
439
We attribute the systematic decrease in the efficiency of turbulent mixing
440
with the increasing exit velocity to the formation of shock wave structures
441
inside the fountain. When the gas-pyroclast mixture is ejected from the
442
vent at high speed, the standing shock waves are generated in the fountain
443
even if the pressure at the vent is balanced with the atmospheric pressure
444
(Fig.5). These shock wave structures suppress efficient mixing between the
445
ejected material and ambient air. Recent experimental results by Saffaraval
446
et al. (in review) also indicates that the efficiency of entrainment is reduced
447
when shock wave structures are formed in high-speed jets. As the Mach
448
number increases, the flow with a series of shock waves reaches a higher
449
level, which causes less efficient mixing. Thus, the region of the fountain-
450
type collapse regime expands with the increasing Mach number in the Ri–M
451
space (Figs.13c,d).
452
5. Geological Implications
453
The present 3-D numerical simulations of eruption clouds show two types
454
of column collapse (CC) conditions. The CC condition is characterized by
455
the jet-type collapse when the initial temperature of the ejected material is
456
relatively low, whereas the CC condition is characterized by the fountain-type
457
collapse when the initial temperature of the ejected material is high.
458
The flow patterns in the jet-type and fountain-type collapse regimes
459
are considered to affect depositional structures in pyroclastic flows. When
460
the jet-type collapse occurs, the eruption cloud contains a large amount of
461
air, generating a dilute pyroclastic flow. The column collapse during low-
462
temperature eruptions, therefore, tends to generate a dilute pyroclastic flow.
21
463
Generally, eruptions with low temperatures are attained by the interaction
464
between the magma and water (e.g., Koyaguchi and Woods, 1996). The
465
present numerical results are consistent with the fact that surge deposits
466
from dilute low-temperature pyroclastic flows are commonly observed in
467
the deposits of phreatomagmatic eruptions (e.g., Moore, 1967). When the
468
fountain-type collapse occurs, on the other hand, high-concentration and
469
high-temperature pyroclastic flows are generated. Such pyroclastic flows ac-
470
count for the occurrence of highly welded pyroclastic flow deposits (e.g.,
471
Branney and Kokelaar, 1992). During the fountain-type collapse, a large
472
amount of pyroclasts fall to the ground directly from the fountain top, which
473
generates a chaotic flow around the vent as well as outward pyroclastic flows
474
(see Fig.6). The chaotic flow around the vent may explain complex features
475
of proximal facies in some pyroclastic flow deposits (e.g., Druitt and Sparks,
476
1982; Walker, 1985).
477
Another implication is concerned with the CC condition predicted by
478
the 1-D models. The efficiency of entrainment estimated in the present 3-D
479
numerical simulations (k ∼ 0.03 for the jet-type CC condition and k = 0.03–
480
0.10 for the fountain-type CC condition) is substantially smaller than the
481
value commonly assumed in the 1-D models (e.g., k = 0.09 in Woods (1988)).
482
This means that the eruption column collapse is likely to occur for smaller
483
mass discharge rates than the CC condition predicted by those 1-D mod-
484
els. Our results also suggest that the efficiency of entrainment decreases as
485
the exit velocity increases along the fountain-type CC condition (Figs.12c,d).
486
Small values of k have been proposed in some 1-D models (Carazzo et al.,
487
2008, 2010; Koyaguchi et al., 2010) on the basis of recent laboratory exper-
22
488
iments (Kaminski et al., 2005) and direct measurements in numerical simu-
489
lations (Suzuki and Koyaguchi, 2010). The effect of the exit velocity on the
490
efficiency of entrainment (e.g., Saffaraval et al., in review), however, has not
491
been taken into account in any 1-D models. We suggest that this effect may
492
play an important role in column collapse, especially during high-temperature
493
eruptions.
494
On the basis of the 3-D numerical simulations, we have shown that the 3-
495
D structures of flow largely affect the column collapse condition. Our results
496
indicate that the differences between the jet-type and fountain-type collapses
497
are closely related with the unsteady vortical structures (e.g., Suzuki and
498
Koyaguchi, 2010). For better understanding of column collapse condition,
499
further analyses based on 3-D flow visualization are required.
500
Appendix A. Heights of Negatively Buoyant Jet and Fountain
501
In section 4.1, we presented a simple analysis for the JF condition on
502
the assumption that the fountain is generated when Hcore > Hmmt (Eq.(6)).
503
The analysis is based on previous experimental studies on the transition
504
between negatively buoyant jets and fountains for subsonic flows (e.g., Kaye
505
and Hunt, 2006). According to the experimental results, the JF condition of
506
negatively buoyant jet/fountains is closely related to the relationship between
507
the maximum height (Hmax ) and Ri. When Ri is sufficiently small so that
508
Hcore < Hmmt (see also Eq.(5)), Hmax of the negatively buoyant jet increases
509
with Ri for a fixed exit velocity. This is because the ratio of the mass of
510
entrained air to that of the ejected material decreases as Ri increases, and
511
hence, Hcore /Hmmt increases. When Ri is large so that Hcore > Hmmt , on the
23
512
other hand, Hmax of the negatively buoyant fountain remains Hmmt for a fixed
513
exit velocity and is independent of Ri. These results indicate that, at least
514
for subsonic flows under the experimental conditions, the JF condition can
515
be determined by the critical value of Ri at which the Hmax –Ri relationship
516
changes.
517
Here, we attempt to extend the above idea to the JF condition of super-
518
sonic flows; we carried out supplementary simulations for negatively buoyant
519
jet/fountains of a gas-pyroclast mixture for various Mach numbers. In order
520
to avoid unnecessary complications (e.g., rapid expansion of the cloud due
521
to heating of the entrained air) and to extract the effect of the Mach number
522
on the JF condition, the temperature of the gas-pyroclast mixture was set
523
to be equal to the ambient temperature (273 K). Under this condition, the
524
cloud density is always larger than the ambient density.
525
Fig.A.1 shows the Hmax –Ri relationship of the negatively buoyant jet/
526
fountains for M= 1–3. For sonic flows (M= 1), negatively buoyant jets de-
527
velop and Hmax increases with Ri when Ri < 0.6. When Ri > 0.6, negatively
528
buoyant fountains with a constant Hmax develop. The results show that the
529
critical Richardson number for the JF condition (RiJF ) based on the change
530
in the Hmax –Ri relationship decreases with the increasing Mach number (ar-
531
rows in the figure). From this RiJF –M relationship, we obtain the values
532
of a1 and a2 in Eq.(6); the estimated values are a1 = 1.25 and a2 = 0.42,
533
which are consistent with those estimated from Fig.13 (a1 = 1.3–3.4 and
534
a2 = 0.16–4.7).
24
535
Appendix B. Derivation of a Formula for the Column Collapse
536
Conditions by 1-D Model on the Basis of Analogue
537
Experiments
538
Woods and Caulfield (1992) carried out a series of analogue experiments
539
simulating column collapse and proposed a simple formula for the CC condi-
540
tion. In their experiments, mixtures of methanol and ethylene glycol (MEG)
541
were injected as a downward propagating jet into fresh water. The MEG is
542
lighter than fresh water, whereas the MEG-water mixture can be denser than
543
fresh water. The density difference between the fresh water and MEG-water
544
mixture was approximated by a function of the volume fraction of MEG in
545
the mixture, W , as
∆ρ ≡
β−α
W 2
=1− 1−
,
βm − α
Wm
(B.1)
546
where β is the MEG-water mixture density and α is the water density. The
547
MEG-water mixture has a maximum density of βm when W = Wm . If the
548
MEG entrains sufficient water before its initial momentum is exhausted, the
549
MEG-water mixture becomes denser than the ambient water and continues
550
to flow downwards as a negatively buoyant “plume”. If the MEG does not
551
mix with sufficient water, the mixture “collapses” and rises back toward the
552
top of the tank. Although the direction of the flow is upside-down, the
553
experiments reproduce fundamental features of eruption column collapse.
554
Woods and Caulfield (1992) showed that, when the ∆ρ–W relationship is
555
explicitly given as Eq.(B.1), a simple formula of the CC condition (i.e., the
556
region of collapse regime) for the MEG-water system can be derived on the
25
557
basis of the 1-D model with a constant entrainment coefficient (k) as:
5/2
2
4kαṀ0
W0 W0
,
−1 ≥
Wm 2Wm
5(βm − α)g ṁ30
(B.2)
558
where W0 is the initial volume fraction of MEG, Ṁ0 (≡ w02 L20 ) is the initial
559
momentum flux, and ṁ0 (≡ w0 L20 ) is the initial mass flux. It should be noted
560
that this formula is applicable only to the MEG-water system, because it is
561
based on the fact that the density of the mixture is equal to the ambient
562
density when W = 2Wm (≡ Wcrt ) in Eq.(B.1). Here, we extend Eq.(B.2) to
563
the general case and show that it is consistent with our result (i.e., Eq.(11)).
564
In order to obtain the CC condition for the general case, Wm in Eq.(B.2)
565
must be given as a function of general quantities such as the critical vol-
566
ume fraction (Wcrt ) or the critical mass fraction (ξcrt ) of ejected materials.
567
Considering that Eq.(B.1) and Wcrt = 2Wm are the relationships for the
568
MEG-water system, Eq.(B.2) is rewritten using the general quantities as
−Ri ≥
8k
Wcrt
.
5 W0 − Wcrt
(B.3)
569
Here, the definition of Ri is given by Eq.(1) (note that Ri is negative because
570
the direction of the flow is upside-down). Furthermore, because the mass
571
fraction is expressed by
ξ=
572
W β0
,
W β0 + (1 − W )α
(B.4)
where β0 is the initial density of MEG, Eq.(B.3) is rewritten as
−Ri ≥ Ricrt ≡
8k α ξcrt
.
5 β0 1 − ξcrt
(B.5)
573
This has the same form as our formula for the CC condition (i.e., Eq.(11)).
574
These two formulae have different proportionality constants (1.0 in Eq.(11)
26
575
and 8/5 in Eq.(B.5)). This difference is attributed to the fact that the mass
576
of entrained air is assumed to be constant at any heights in our analysis
577
(see Eq.(9)), whereas it is calculated on the basis of the 1-D model with a
578
constant k in Woods and Caulfield (1992).
579
Acknowledgments
580
The manuscript was improved by comments from L. Wilson, D. E. Ogden
581
and an anonymous reviewer. Computations were carried out in part on Earth
582
Simulator at the Japan Agency for Marine-Earth Science and Technology,
583
Altix 4700 at the Earthquake Research Institute, University of Tokyo, and
584
Primergy RX200S6 at Computer System for Research, Kyushu University.
585
Part of this study was supported by the ERI cooperative research program
586
and by KAKENHI (No. 21340123).
587
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31
679
Table 1: List of material properties and values of physical parameters
680
Table 2: Input parameters and flow regimes
681
Fig.1. Representative numerical results of the eruption column with a jet
682
structure (run 99). Parameters used and conditions at the vent are listed in
683
Tables 1 and 2, respectively. (a) Cross-sectional distribution of the mass frac-
684
tion of the ejected material (ξ) in the r–z space at 600 s. (b) Cross-sectional
685
distribution of the density difference relative to the stratified atmospheric
686
density at the same vertical position (∆ρ = ρ/ρa − 1) in the r–z space at 600
687
s.
688
Fig.2. Representative numerical results of the eruption column with a foun-
689
tain structure (run 33). Parameters used and conditions at the vent are
690
listed in Tables 1 and 2, respectively. (a) Cross-sectional distribution of the
691
mass fraction of the ejected material (ξ) in the r–z space at 190 s. (b)
692
Cross-sectional distribution of the density difference relative to the stratified
693
atmospheric density at the same vertical position (∆ρ = ρ/ρa − 1) in the r–z
694
space at 190 s.
695
Fig.3. Timewise distributions of the mass fraction of the ejected material
696
(ξ) along the central axis (spatial average within the range of |r| < L0 ) for
697
(a) run 99 and (b) run 33.
32
698
Fig.4. Time-averaged vertical profiles of physical quantities along the central
699
axis (spatial average within the range of |r| < L0 ) for run 99 (solid curves)
700
and run 33 (dashed curves). (a) The mass fraction of the ejected material,
701
ξ. (b) The upward velocity normalized by the exit velocity at the vent,
702
w/w0. The vertical axis represents the vertical position normalized by the
703
characteristic height (z ∗ = z/(w02 /2g)).
704
Fig.5. Time-averaged distribution of the pressure difference relative to the
705
stratified atmospheric pressure at the same vertical position, ∆p = p/pa − 1,
706
where pa is the atmospheric pressure. Cross-sectional distributions for (a)
707
run 34 and (b) run 33. Parameters used and conditions at the vent are listed
708
in Tables 1 and 2, respectively.
709
Fig.6. Representative numerical results of the column collapse from the jet
710
structure (run 78). Parameters used and conditions at the vent are listed in
711
Tables 1 and 2, respectively. (a) Cross-sectional distribution of the mass frac-
712
tion of the ejected material (ξ) in the r–z space at 120 s. (b) Cross-sectional
713
distribution of the density difference relative to the stratified atmospheric
714
density at the same vertical position (∆ρ = ρ/ρa − 1) in the r–z space at 120
715
s.
716
Fig.7. Representative numerical results of the column collapse from the
717
fountain structure (run 49). Parameters used and conditions at the vent
718
are listed in Tables 1 and 2, respectively. (a) Cross-sectional distribution of
719
the mass fraction of the ejected material (ξ) in the r–z space at 500 s. (b)
33
720
Cross-sectional distribution of the density difference relative to the stratified
721
atmospheric density at the same vertical position (∆ρ = ρ/ρa − 1) in the r–z
722
space at 500 s.
723
Fig.8. Time-averaged vertical profiles of physical quantities along the central
724
axis (spatial average within the range of |r| < L0 ) for run 78 (solid curves)
725
and run 49 (dashed curves). (a) The mass fraction of the ejected material,
726
ξ. (b) The upward velocity normalized by the exit velocity at the vent,
727
w/w0. The vertical axis represents the vertical position normalized by the
728
characteristic height (z ∗ = z/(w02 /2g)).
729
Fig.9. Cross-sectional distribution of the mass fraction of the ejected mate-
730
rial (ξ) in the r–z space for the result of run 46 at 190 s. Parameters used
731
and conditions at the vent are listed in Tables 1 and 2, respectively. The
732
contour levels are ξ = 0.8 and 0.9.
733
Fig.10. Timewise distributions of the mass fraction of the ejected material
734
(ξ) along the central axis (spatial average within the range of |r| < L0 ) for
735
run 92. Parameters used and conditions at the vent are listed in Tables 1
736
and 2, respectively. The contour levels are ξ = 0.9 and 0.95.
737
Fig.11. Cross-sectional distribution of the mass fraction of the ejected ma-
738
terial (ξ) in the r–z space for the result of run 27 at 590 s. Parameters used
739
and conditions at the vent are listed in Tables 1 and 2, respectively.
34
740
Fig.12. Flow-regime maps for the results of (a) group L, (b) group H, (c)
741
group I, and (d) group N. Parameters used are listed in Table 1. The condi-
742
tions at the vent for simulations are listed in Table 2. Pluses represent the
743
jet-type column regime. Circles indicate the fountain-type column regime.
744
Diamonds are the jet-type collapse regime. Triangles represent the fountain-
745
type collapse regime. Solid curves are the column collapse condition. Dashed
746
curves are the jet-fountain condition. Dotted thin curves are the column col-
747
lapse condition which are predicted by the 1-D model of Woods (1988) with
748
variable entrainment coefficients. The values at the edge of these curves are
749
the assumed values for the entrainment coefficient.
750
Fig.13. Flow-regime maps for the results of (a) group L, (b) group H, (c)
751
group I, and (d) group N in the non-dimensional space. Parameters used are
752
listed in Table 1. The conditions at the vent and non-dimensional parameters
753
for simulations are listed in Table 2. The horizontal axis is the Richardson
754
number (Ri) and the vertical axis is the Mach number (M). Pluses repre-
755
sent the jet-type column regime. Circles indicate the fountain-type column
756
regime. Diamonds are the jet-type collapse regime. Triangles represent the
757
fountain-type collapse regime. Solid curves are the column collapse condi-
758
tion. Dashed curves are the jet-fountain condition. Gray and white stars
759
are the critical points for the column collapse condition and those for the
760
jet-fountain condition for M= 1.2, respectively.
761
Fig.A.1. Maximum heights of the negatively buoyant jet/fountains (Hmax )
762
as a function of the Richardson number (Ri) on the basis of the 3-D numerical
35
763
simulations. White and black points represent the jet-type and fountain-type
764
regimes, respectively. Grey plots are the intermediate regime between the jet-
765
type and fountain-type regimes. Circles, diamonds, and triangles indicate the
766
results for M= 1, 2, and 3, respectively. Arrows represent the JF condition
767
based on the Hmax –Ri relationship.
36
Table 1
Click here to download Table: Table1.pdf
Variable Value (Unit)
Meaning
9.81 (m s−2 )
gravitational body force per unit mass
Rg
462 (J kg−1 K−1 )
gas constant of volcanic gas
Ra
287 (J kg−1 K−1 )
gas constant of atmospheric air
Cvs
1100 (J kg−1 K−1 ) specific heat of solid pyroclasts
Cvg
1348 (J kg−1 K−1 ) specific heat of volcanic gas at constant volume
Cva
713 (J kg−1 K−1 )
specific heat of air at constant volume
Ta0
273 (K)
atmospheric temperature at z = 0 km
pa0
1.013 × 105 (Pa)
atmospheric pressure at z = 0 km
ρa0
1.29 (kg m−3 )
atmospheric density at z = 0 km
μ1
-6.5 (K km−1 )
temperature gradient at z = 0.0–11.0 km
μ2
0.0 (K km−1 )
temperature gradient at z = 11.0–20.0 km
g
37
Table 2
Click here to download Table: Table2.pdf
Simulation
ṁ0 (kg s−1 )
w0 (m s−1 ) L0 (m)
Ri
M
Regime1
group H (T0 = 1000 K, ng0 = 2.84 wt %)
run 24
1.0 × 109
115
598
2.2 × 100
1.0
F-CC
run 25
1.0 × 109
230
423
3.9 × 10−1
2.0
F-CC
run 26
1.0 × 109
173
488
8.0 × 10−1
1.5
F-CC
run 27
1.0 × 109
346
345
1.4 × 10−1
3.0
F-EC
run 28
1.0 × 109
288
378
2.2 × 10−1
2.5
F-CC
run 29
1.0 × 108
230
134
1.2 × 10−1
2.0
F-EC
run 30
1.0 × 108
346
109
4.5 × 10−2
3.0
J-EC
run 31
1.0 × 108
115
189
7.0 × 10−1
1.0
F-CC
run 32
1.0 × 108
288
120
7.0 × 10−2
2.5
F-EC
run 33
1.0 × 108
173
154
2.5 × 10−1
1.5
F-EC
run 34
1.0 × 107
115
59.8
2.2 × 10−1
1.0
F-EC
run 35
1.0 × 107
230
42.3
3.9 × 10−2
2.0
J-EC
run 36
1.0 × 107
346
34.5
1.4 × 10−2
3.0
J-EC
run 37
1.0 × 107
57.6
84.6
1.2 × 100
0.5
F-CC
run 39
1.0 × 106
57.6
26.8
3.9 × 10−1
0.5
J-EC/F-EC
run 40
1.0 × 106
115
18.9
7.0 × 10−2
1.0
J-EC
run 41
3.2 × 106
57.6
47.6
7.0 × 10−1
0.5
F-CC
run 42
3.2 × 106
115
33.6
1.2 × 10−1
1.0
J-EC
run 43
3.2 × 106
173
27.5
4.5 × 10−2
1.5
J-EC
run 44
1.0 × 106
80.7
22.6
1.7 × 10−1
0.7
J-EC
38
run 45
3.2 × 106
80.7
40.2
3.0 × 10−1
0.7
J-EC/F-EC
run 46
3.2 × 107
196
81.6
1.0 × 10−1
1.7
J-EC/F-EC
run 47
3.2 × 107
253
71.7
5.5 × 10−2
2.2
J-EC
run 48
1.0 × 108
150
166
3.6 × 10−1
1.3
F-EC
run 49
1.0 × 109
254
403
3.1 × 10−1
2.2
F-CC
run 50
1.0 × 106
69.1
24.4
2.5 × 10−1
0.6
J-EC
run 51
1.0 × 107
80.6
71.5
5.4 × 10−1
0.7
F-CC
run 52
1.0 × 107
92.2
66.9
3.8 × 10−1
0.8
F-EC
run 53
1.0 × 108
138
173
4.4 × 10−1
1.2
F-CC
run 54
3.2 × 108
173
275
4.5 × 10−1
1.5
F-CC
run 55
3.2 × 108
196
258
3.3 × 10−1
1.7
F-CC
run 56
3.2 × 108
219
244
2.5 × 10−1
1.9
F-EC
run 57
3.2 × 108
242
232
1.9 × 10−1
2.1
F-EC
run 67
3.2 × 106
69.1
43.4
4.4 × 10−1
0.6
F-EC
run 68
1.0 × 106
50.0
28.7
5.6 × 10−1
0.4
F-EC
run 69
1.0 × 106
34.6
34.5
1.4 × 100
0.3
F-CC
run 99
1.0 × 107
173
49.0
7.5 × 10−2
1.5
J-EC
run 101
1.3 × 108
288
136
7.5 × 10−2
2.5
F-EC
group I (T0 = 800 K, ng0 = 2.84 wt %)
run 120
1.0 × 106
103
17.9
1.1 × 10−1
1.0
J-EC/J-CC
run 121
4.0 × 106
103
35.7
2.1 × 10−1
1.0
F-EC/F-CC
39
run 122
1.0 × 107
103
56.6
3.4 × 10−1
1.0
F-CC
run 123
2.5 × 107
103
89.7
5.4 × 10−1
1.0
F-CC
run 124
4.0 × 106
206
25.2
3.8 × 10−2
2.0
J-EC
run 125
1.0 × 107
206
40.0
6.0 × 10−2
2.0
J-EC
run 126
2.5 × 107
206
63.4
9.5 × 10−2
2.0
J-CC/F-CC
run 127
4.0 × 105
103
11.3
6.8 × 10−2
1.0
J-EC
run 128
6.3 × 107
206
101
1.5 × 10−1
2.0
F-CC
run 129
1.6 × 108
206
159
2.4 × 10−1
2.0
F-CC
run 130
1.6 × 106
103
22.5
1.3 × 10−1
1.0
J-CC
run 131
2.5 × 106
103
28.4
1.7 × 10−1
1.0
J-CC/F-CC
group L (T0 = 550 K, ng0 = 2.84 wt %)
run 58
1.0 × 106
128
13.3
7.8 × 10−2
1.5
F-EC
run 59
1.0 × 106
145
12.5
5.7 × 10−2
1.7
J-CC
run 60
1.0 × 106
171
11.5
3.8 × 10−2
2.0
J-CC
run 61
1.0 × 107
214
32.6
6.9 × 10−2
2.5
J-CC
run 62
1.0 × 107
231
31.4
5.7 × 10−2
2.7
J-CC
run 64
1.0 × 107
171
36.4
1.2 × 10−1
2.0
J-CC
run 65
1.0 × 107
128
42.1
2.5 × 10−1
1.5
F-CC
run 66
1.0 × 107
85.5
51.5
6.8 × 10−1
1.0
F-CC
run 70
1.0 × 106
214
10.3
2.2 × 10−2
2.5
J-EC
run 71
1.0 × 107
248
30.2
4.7 × 10−2
2.9
J-CC
40
run 74
1.0 × 106
197
10.7
2.7 × 10−2
2.3
J-EC/J-CC
run 75
1.0 × 106
205
10.5
2.4 × 10−2
2.4
J-EC
run 77
1.0 × 106
188
11.0
3.0 × 10−2
2.2
J-CC
run 78
1.0 × 106
179
11.2
3.4 × 10−2
2.1
J-CC
run 80
1.0 × 105
111
4.52
3.5 × 10−2
1.3
J-CC
run 81
1.0 × 105
120
4.35
2.9 × 10−2
1.4
J-CC
run 85
1.0 × 106
85.5
16.3
2.2 × 10−1
1.0
J-CC/F-CC
run 86
1.0 × 106
42.7
23.0
1.2 × 100
0.5
F-CC
run 87
1.0 × 106
59.8
19.5
5.3 × 10−1
0.7
F-CC
run 88
1.0 × 106
68.4
18.2
3.8 × 10−1
0.8
F-CC
run 89
1.0 × 105
128
4.21
2.5 × 10−2
1.5
J-EC
run 90
1.0 × 105
137
4.07
2.1 × 10−2
1.6
J-EC
run 91
1.0 × 107
145
39.5
1.8 × 10−1
1.7
J-CC/F-CC
run 92
1.0 × 107
154
38.4
1.6 × 10−1
1.8
J-CC/F-CC
run 93
1.0 × 107
162
37.4
1.4 × 10−1
1.9
F-CC/F-CC
run 94
1.0 × 107
137
40.7
2.1 × 10−1
1.6
F-CC
run 95
1.0 × 107
94.0
49.1
5.4 × 10−1
1.1
F-CC
run 96
1.0 × 107
103
47.0
4.3 × 10−1
1.2
F-CC
run 97
1.0 × 107
111
45.2
3.5 × 10−1
1.3
F-CC
run 98
1.0 × 107
120
43.5
2.9 × 10−1
1.4
F-CC
41
group N (T0 = 1000 K, ng0 = 1.23 wt %)
run 103
1.0 × 107
75.6
48.6
1.1 × 100
1.0
F-CC
run 104
1.0 × 108
75.6
154
3.4 × 100
1.0
F-CC
run 105
1.0 × 109
75.6
486
1.1 × 101
1.0
F-CC
run 106
1.0 × 107
151
34.4
1.9 × 10−1
2.0
F-EC
run 107
1.0 × 108
151
109
6.0 × 10−1
2.0
F-CC
run 108
1.0 × 109
151
344
1.9 × 101
2.0
F-CC
run 109
4.0 × 106
75.6
30.7
6.8 × 10−1
1.0
F-CC
run 110
4.0 × 106
151
21.7
1.2 × 10−1
2.0
J-EC
run 111
4.0 × 105
75.6
9.70
2.1 × 10−1
1.0
J-EC
run 112
4.0 × 105
151
6.86
3.8 × 10−2
2.0
J-EC
run 115
1.0 × 106
75.6
15.4
3.4 × 10−1
1.0
J-EC
run 116
1.6 × 106
75.6
19.4
4.3 × 10−1
1.0
F-EC/F-CC
run 117
2.5 × 106
75.6
24.4
5.4 × 10−1
1.0
F-CC
run 118
1.6 × 107
151
43.3
2.4 × 10−1
2.0
F-CC
run 119
2.5 × 107
151
54.5
3.0 × 10−1 2.0
1
J-EC: jet-type eruption column, J-CC: jet-tyle column collapse,
F-CC
F-EC: fountain-type eruption column, F-CC: fountain-type column collapse
42
z [km]
3
z [km]
3
Figure 1
Click here to download Figure: figure1.pdf
(b) a
3
0
0
0
(a) 
0
r [km]
0.5
3
1
3
-0.5
0
r [km]
0
3
0.5
z [km]
3
z [km]
3
Figure 2
Click here to download Figure: figure2.pdf
3
0
0
(b) a
0
(a) 
0
r [km]
0.5
3
1
3
-0.5
0
r [km]
0
3
0.5
1
10
Figure 3
Click here to download Figure: figure3.pdf
0
0
0.5
z [km]
5
(a) run 99
0
200
400
600
200
300
1
10
Time [sec]
0
0
0.5
z [km]
5
(b) run 33
0
100
Time [sec]
Figure 4
Click here to download Figure: figure4.pdf
2
z*
z*
2
1
1
(a)
0
0
(b)
0.2
0.4

0.6
0.8
1
0
-0.2
0
0.2
0.4
0.6
w/w0
0.8
1
1.2
z [km]
2
z [km]
2
Figure 5
Click here to download Figure: figure5b.pdf
(b) run 33
2
-0.2
0
0
(a) run 34
0
r [km]
0
2
0.2
2
-0.2
0
r [km]
0
2
0.2
z [km]
0.5
z [km]
0.5
Figure 6
Click here to download Figure: figure6.pdf
0.5
0
0
(b) a
0
(a) 
0
r [km]
0.5
0.5
1
0.5
0
r [km]
-0.5
0
0.5
0.5
z [km]
10
z [km]
10
Figure 7
Click here to download Figure: figure7.pdf
10
0
0
(b) a
0
(a) 
0
r [km]
0.5
10
1
10
0
r [km]
-0.5
0
10
0.5
Figure 8
Click here to download Figure: figure8.pdf
2
z*
z*
2
1
1
(a)
0
0
(b)
0.2
0.4

0.6
0.8
1
0
-0.2
0
0.2
0.4
w/w0
0.6
0.8
1
z [km]
4
Figure 9
Click here to download Figure: figure9.pdf
0.8
0
0.9
4
0
0
r [km]
0.5
4
1
1
2
Figure 10
Click here to download Figure: figure10.pdf
0.5
0
0.95
0
z [km]
1
0.9
0
50
100
Time [sec]
150
0
z [km]
10
Figure 11
Click here to download Figure: figure11.pdf
10
0
0
r [km]
0.5
10
1
Jet-type Collapse
100
5
6
7
10
10
10
6
10
200
0
5
10
Exit velocity [m/s]
Exit velocity [m/s]
300
Jet-type
Column
Fountain-type
Collapse
100
(c) group H
(T0=1000 K
n0 =0.0248)
6
10
7
10
8
10
9
10
10
10
Mass discharge rate [kg/s]
03
0.
0.1 0.0 05
7
0
8
10
9
10
Fountain-type
Column
Fountain-type
Column
200
7
10
Mass discharge rate [kg/s]
0.0
3
0.0
5
0.0
7
0.1
0
1
0.0
01
(b) group I
(T0=800 K
Jet-type
n0 =0.0248)
Collapse
Mass discharge rate [kg/s]
400
Fountain-type
Collapse
100
0 5
10
8
10
Jet-type
Column
3
0.0
5
0.0
7
0.1
0
0
(a) group L
Fountain-type (T0 =550 K
n0 =0.0248)
Collapse
200
0.0
200
Jet-type
Column
07
0. 0 300
1
0.
Exit velocity [m/s]
Exit velocity [m/s]
300
0.
03 04
0. 0.
0.
Figure 12
Click here to download Figure: figure12.pdf
11
10
Jet-type
Column
Fountain-type
Collapse
100
(d) group N
(T0=1000 K
n0=0.0123)
0
5
10
6
10
7
10
8
10
Mass discharge rate [kg/s]
9
10
Figure 13
Click here to download Figure: figure13.pdf
3
3
(b) group I
(T0=800 K
n0 =0.0248)
Mach number
Mach number
(a) group L
(T0 =550 K
n0 =0.0248)
2
Fountain-type
Collapse
1 Jet-type
2
Jet-type
Column
Fountain-type
Collapse
1
Column
Jet-type
Collapse
0
0.01
0.1
Jet-type
Collapse
1
0
0.01
10
Richardson number
1
10
Richardson number
3
3
(c) group H
(T0=1000 K
n0 =0.0248)
Mach number
Fountain-type
Column
Mach number
0.1
2
Fountain-type
Collapse
Jet-type
Column
1
0
0.01
0.1
1
Richardson number
10
Fountain-type
Column
(d) group N
(T0=1000 K
n0=0.0123)
Jet-type
Column
Fountain-type
Collapse
2
1
0
0.01
0.1
1
Richardson number
10
Height of Negatively Buoyant Jet [m]
Figure A1
Click here to download Figure: figureA1.pdf
104
M=3
M=2
103
0.01
M=1
0.1
1
Richardson Number
10