Figure 1:
Carapace
Talus
Massive core
Volcano flank
Schematic of a simplified endogenous lava dome comprised of an outer solidified surface
(termed the talus) that enshrouds the massive hot lava dome core. For the purpose of this paper I
term the carapace as the region of solidified lava that is still attached to the lava core, but it is not
explicitly considered in this paper. The talus is comprised of loose brittle lava and at the freesurface the talus slopes at its angle of repose. There is little observational data to constrain the
dimensions of the talus and core, although it is thought that the talus component can be
extensive.
Figure 2:
Shown schematically are the 4 steps used in the axi-symmetrical lava dome model. The images
also show the applied boundary conditions as engineering symbols; the symbol for no-slip is
shown at the base of the domain, while the symbol for slip along the z-axis is shown at the left of
each of the images. Step a) calculates the displacement and pressure fields of the dome material
(talus and core) using the momentum equations. Step b) using the pressure field the extent of the
talus/core within the dome can be calculated and the interface updated. The dashed line
corresponds to the solidus pressure isobar, and the solid line is for the location of the core
previously. In step c) the dome is grown over one-time step using the calculated velocity field
from the momentum equation (arrows in figure a) to displace the surfaces of the dome and core.
Step d) the talus is adjusted to be at the angle of repose, whilst preserving the shape of the talus
where it is below the angle of repose. The four steps are repeated continuously to allow the lava
dome to grow in time. The level-set method is used to track the free surfaces (core and talus),
with both surfaces advected into the model space.
Figure 3:
The axi-symmetric domain (rotated about r=0 as shown by curved arrow) used in the
computational model with the free-surfaces for lava and talus shown as continuous lines. The
base the domain corresponds to the surface of the volcano and has the boundary condition of
zero velocity. The radius of the conduit is a, and the lava inlet boundary condition in the model is
applied at the conduit inlet as shown by P0. The model is initialised with a mound of lava above
the conduit exit.
Figure 4:
Day 1
Day 5
Day 10
Day 15
Day 20
Day 25
Evolution of a lava dome with a talus from one simulation. A visualisation software package has
been used to rotate the modelled domain to produce a three-dimensional image with a segment
left out of the dome to observe the interior structure. Shown in the first image are the coordinate
axes and this is the same for all the 3D visualised imaged that follow in this paper. The light grey
region corresponds to the lava dome core, while the darker grey region is the talus. The model
parameters are given in Table 2 for the reference model but the extrusion rate is 5m3s-1. Images
show the growth of the lava dome to a final radius of 293 metres and maximum height of 84
metres.
Figure 5:
Day 1, R=78m, H=59m
Day 5, R=138m, H=80m
Day 10, R=201m, H=83m
Day 15, R=230m, H=83m
Day 20, R=261m, H=83m
Day 25, R=293m, H=84m
Evolution of a lava dome from one simulation shown as two-dimensional segments from r=0 to
the maximum radius in the horizontal direction, and from z=0 to the maximum height in the
vertical direction. The light grey region corresponds to the lava dome core, while the darker grey
region is the talus. The black solid lime corresponds to where the pressure is equal to the solidus
pressure. Note that very early on in the evolution of the dome, talus can form from all regions,
while at later times talus only forms from the central region of the dome. As for figure 4 the
model parameters are given in Table 2 for the reference model but the extrusion rate is 5m3s-1.
Images show the growth of the lava dome to a final radius of 298 metres and maximum height of
84 metres. The solidus pressure isobar in image a) is not as smooth as the other images because
the dome is relatively small and the isobar is influenced by the coarse element spacing.
Figure 6:
350
90
80
Dome height (metres)
Dome radius (metres)
300
250
200
150
100
50
70
60
50
40
30
20
10
0
0
0
2
4
6
8
10
0
12
a)
2
4
6
8
Volume (Mega cubic metres)
Volume (Mega cubic metres)
b)
10
12
80
250
70
Core height (metres)
Core radius (metres)
300
200
150
100
60
50
40
30
20
50
10
0
0
0
2
4
6
8
10
12
0
2
4
Volume (Mega cubic metres)
6
8
10
12
Volume (Mega cubic metres)
d)
13
0.4
12.5
Pressure at conduit exit (MPa)
0.45
Fraction core
0.35
0.3
0.25
0.2
0.15
0.1
1
0.9
0.8
12
0.7
11.5
0.6
11
0.5
10.5
0.4
0.3
10
0.2
9.5
0.1
0.05
9
0
0
0
0
2
4
6
8
10
12
Yielding
c)
2
4
6
8
10
12
Dome volume (Mega cubic metres)
Dome volume (Mega cubic metres)
e)
f)
g) Newtonian lava dome.
h) Dome model with a talus.
Results for the growth of the lava dome with a talus presented in figures 4 and 5 and for a purely
Newtonian lava dome model. The model parameters are given in Table 2 for the reference model
except the extrusion rate is 5m3s-1. Image a shows the maximum radius for the lava dome with a
talus (black line) against dome volume, also shown is the modelled maximum radius of a purely
Newtonian lava dome (grey line) with a viscosity of 1010 Pa s and extrusion rate of 5m3s-1. Image
b shows the modelled maximum height of the lava dome with a talus (black line) and the
modelled maximum height for the Newtonian dome (grey line). Images c and d show the
modelled lava dome core radius and height, respectively, for the lava dome with a talus. Image e
shows the fraction of lava dome core within the dome for the dome with a talus against lava
dome volume. Image f shows the modelled pressure at the conduit inlet for the lava dome with a
talus (black line) and the Newtonian lava dome (grey line). The final dome volumes are different
because the simulation was stopped when the domes reached a radius of 250 metres. The dashed
lines show when the talus region has elements on the yield surface (when equal to 1). Show in
images g and h are the final lava domes visualised in 3D with a segment left out of the dome to
observe the interior structure for the lava dome with a talus and the Newtonian model,
respectively.
Figure 7:
100
350
90
250
1
200
3
150
5
7
100
Dome height (metres)
Dome radius (metres)
300
50
80
70
1
60
3
50
5
40
7
30
20
10
0
0
0
2
4
6
8
10
0
2
Volume (Mega cubic metres)
4
6
8
10
Volume (Mega cubic metres)
a)
b)
250
90
1
150
3
5
100
7
50
Core height (metres)
Core radius (metres)
80
200
70
60
1
50
3
40
5
30
7
20
10
0
0
0
2
4
6
8
10
0
Volume (Mega cubic metres)
c)
2
4
6
8
Volume (Mega cubic metres)
d)
10
0.5
8
0.45
Volume talus (Mega cubic metres)
7
0.4
Fraction core
0.35
0.3
0.25
1
3
5
7
0.2
0.15
0.1
6
5
4
1
3
5
7
3
2
1
0.05
0
0
0
2
4
6
8
10
12
14
0
Dome volume (Mega cubic metres)
e)
20
40
60
80
100
Time (days)
f)
Results for the growth of 4 lava dome models with different extrusion rates. The model
parameters are given in Table 2 for the reference model and the extrusion rate ranges from 1 m3s1
to 7 m3s-1, as shown by the numbers 1 to 7 in the key in the images. Image a shows the
maximum radius of the lava dome with the key showing the extrusion rates modelled and image
b shows the modelled maximum height of the lava domes. Images c and d show the modelled
lava dome core radius and height, respectively. Image e shows the fraction of lava dome core
within the domes and image f shows the volume of talus generated in time.
Figure 8:
a)
b)
c)
d)
Show in images a) to d) are the final lava dome shapes visualised in 3D for the lava dome model
results presented in Figure 7. The light grey region corresponds to the lava dome core, while the
darker grey region is the talus. The images have not been rotated fully in order to show the
interior dome structure. From a to d the extrusion rate increases from 1 m3s-1 to 7 m3s-1, by 2
m3s-1 each image. The final maximum radii of the domes are 298 meters and the final maximum
heights of the domes are a) 57 metres, b) 74 metres, c) 84 metres and d) 90 metres.
Figure 9:
90
350
80
250
30
200
35
40
150
45
100
Dome height (metres)
Dome radius (metres)
300
50
70
60
30
50
35
40
40
45
30
20
10
0
0
0
5
10
15
0
Volume (Mega cubic metres)
5
10
15
Volume (Mega cubic metres)
a)
b)
70
300
60
200
30
35
150
40
45
100
Core height (metres)
Core radius (metres)
250
50
50
30
40
35
30
40
45
20
10
0
0
0
5
10
0
15
5
10
15
Volume (Mega cubic metres)
Volume (Mega cubic metres)
c)
d)
11
0.5
0.45
10.5
0.4
Pressure (MPa)
Fraction core
0.35
0.3
30
35
40
45
0.25
0.2
0.15
0.1
10
30
35
40
45
9.5
9
8.5
0.05
0
8
0
2
4
6
8
10
12
0
Dome volume (Mega cubic metres)
e)
2
4
6
8
Dome volume (Mega cubic metres)
f)
10
12
g)
h)
Results for the growth of 4 lava domes with different friction angles for the talus. The model
parameters are given in Table 2 for the reference model and the friction angle is modelled to be
between θ = 30º and θ = 45º. Image a shows the maximum radius of the lava dome with the key
showing the friction angles modelled in degrees and image b shows the modelled maximum
height of the lava domes. Images c and d show the modelled lava dome core radius and height,
respectively. Image e shows the fraction of lava dome core for the domes and image f shows the
modelled pressure at the conduit inlet. Images g and h show the final lava dome shapes
visualised in 3D for a friction angle of 30º and 45º, respectively. The final maximum radii of the
domes are 298 meters and the final maximum heights of the domes are all 78 metres.
Figure 10:
350
100
90
80
250
Dome height (metres)
Dome radius (metres)
300
200
0.2 M Pa
150
0.4 M Pa
0.6 M Pa
100
0.8 M Pa
1.0 M Pa
50
70
60
50
0.2 M Pa
40
0.4 M Pa
0.6 M Pa
30
0.8 M Pa
20
1.0 M Pa
10
0
0
0
2
4
6
8
10
12
0
Volume (Mega cubic metres)
a)
2
4
6
8
Volume (Mega cubic metres)
b)
10
12
300
80
0.2 M Pa
0.6 M Pa
Core height (metres)
Core radius (metres)
70
0.4 M Pa
250
0.8 M Pa
200
1.0 M Pa
150
100
60
50
0.2 M Pa
40
0.4 M Pa
30
0.6 M Pa
0.8 M Pa
20
1.0 M Pa
50
10
0
0
0
2
4
6
8
10
12
0
2
4
Volume (Mega cubic metres)
6
8
10
c)
d)
0.6
12
0.2 M Pa
0.4 M Pa
0.6 M Pa
0.8 M Pa
1.0 M Pa
11.5
0.5
0.2 M Pa
0.4 M Pa
0.6 M Pa
0.8 M Pa
1.0 M Pa
0.4
0.3
0.2
Pressure (MPa)
11
Fraction core
12
Volume (Mega cubic metres)
10.5
10
9.5
9
0.1
8.5
0
8
0
2
4
6
8
10
12
0
Dome volume (Mega cubic metres)
e)
2
4
6
8
10
12
Dome volume (Mega cubic metres)
f)
Results for the growth of lava domes with a talus. The model parameters are given in Table 2 for
the reference model and the solidus pressure is modelled to be between 0.2 MPa and 1.0 MPa
(including atmospheric pressure). Image a shows the maximum radius of the lava dome with the
key showing the solidus pressure modelled and image b shows the modelled maximum height of
the lava domes. Images c and d show the modelled lava dome core radius and height,
respectively. Image e shows the fraction of lava dome core for the domes and image f shows the
modelled pressure at the conduit inlet.
Figure 11:
a)
b)
c)
d)
e)
Show in images a to e are the final lava dome shapes visualised in 3D for the lava domes
presented in Figure 11 with different values for the solidus pressure. The light grey region
corresponds to the lava dome core, while the darker grey region is the talus. From a to e the
solidus pressure modelled is from 0.2 M Pa to 1.0 M Pa, increasing by 0.2 M Pa in each image.
The final lava dome heights are 75 metres, 78 metres, 83 metres, 87 metres and 92 metres for the
images from a to e, respectively.
Figure 12:
90
70
60
70
Core height (metres)
Dome height (metres)
80
60
50
40
1 G Pa s
30
2 G Pa s
5 G Pa s
20
10 G Pa s
10
50
40
30
1 G Pa s
2 G Pa s
20
5 G Pa s
10
0
10 G Pa s
0
0
2
4
6
8
10
12
0
2
Volume (Mega cubic metres)
4
6
8
10
12
Volume (Mega cubic metres)
a)
b)
11
0.45
0.4
10.5
Pressure (MPa)
Fraction core
0.35
0.3
0.25
1 G Pa s
2 G Pa s
5 G Pa s
10 G Pa s
0.2
0.15
0.1
10
9.5
1 G Pa s
2 G Pa s
5 G Pa s
10 G Pa s
9
8.5
0.05
8
0
0
2
4
6
8
10
0
12
c)
2
4
6
8
10
12
Dome volume (Mega cubic metres)
Dome volume (Mega cubic metres)
d)
Results for the growth of lava domes with a talus when varying the viscosity of the core. The
model parameters are given in Table 2 for the reference model and the viscosity is varied for the
core from 109 Pa s to 1010 Pa s, and the viscosity of the talus is 1011 Pa s. Image a shows the
maximum modelled height of the lava domes and images b show the modelled lava dome core
height. The lava dome and core maximum radial extent are the same for all the model runs and
show the same results as that obtained in figures 9a and c, respectively. Image c shows the
fraction of lava dome core for the domes and image d shows the modelled pressure at the conduit
inlet.
Figure 13:
a)
b)
c)
d)
Show in images a to d are the final lava dome shapes visualised in 3D for the lava domes
presented in Figure 13 when varying the core viscosity. The light grey region corresponds to the
lava dome core, while the darker grey region is the talus. The models have not been rotated fully
to allow the interior structure to be observed. The viscosity in the core and talus is modelled to
be a) 109 Pa s, b) 2x109 Pa s, c) 5x109 Pa s and d) 1010 Pa s. The viscosity of the talus for all the
models is 1011 Pa s and the yield strength is 5 M Pa.
0.45
12
0.4
11.5
0.35
11
Pressure (MPa)
Fraction core
Figure 14:
0.3
1 M Pa
2 M Pa
5 M Pa
10 M Pa
0.25
0.2
0.15
10.5
10
9.5
9
0.1
8.5
0.05
8
0
0
2
4
6
8
10
0
12
2
4
6
8
10
12
Dome volume (Mega cubic metres)
Dome volume (Mega cubic metres)
a)
1 M Pa
2 M Pa
5 M Pa
10 M Pa
b)
c)
Results for the growth of lava domes with a talus when varying the yield strength. The model
parameters are given in Table 2 for the reference model and the yield strength is modelled to
vary from τY = 0.5 M Pa to τY = 10MPa. Model results show no variation in radial extent or
maximum height against dome volume. Image a shows the fraction of lava dome core for the
domes and image b shows the modelled pressure at the conduit inlet. Image c shows the final
lava dome shape visualised in 3D for a yield strength of 1MPa. The light grey region
corresponds to the lava dome core, while the darker grey region is the talus. The radius of the
dome is 298 metres and the maximum height is 78 metres.
Figure 15:
a)
b)
c)
d)
e)
f)
1.2
80
Dome and core height (metres)
1
Plastic
0.8
1 M Pa
10 M Pa
0.6
0.4
0.2
0
70
60
50
40
30
20
10
0
0
2
4
6
8
10
0
Dome volume (Mega cubic metres)
g)
1 M Pa
10 M Pa
0.2
0.4
0.6
0.8
1
Volume (Mega cubic metres)
h)
For images a to f, the vertical and horizontal axes are in metres and the black continuous line
corresponds to the lava dome surface, while the grey continuous line is for the talus/core
interface. The arrows and shading are for the velocity field in metres per second. Image a and b
are at a time of 0.25 days for models with yield strengths of 1 M Pa and 10 M Pa, respectively.
Image c and d are at a time of 1 day for models with yield strengths of 1 M Pa and 10 M Pa,
respectively. Image e and f are at a time of 3 days for models with yield strengths of 1 M Pa and
10 M Pa, respectively. Note how the flow is more pronounced above the conduit exit for the
model with the lower yield strength. Image g shows an indication of when the talus domain has
elements that are on the yield surface, i.e. when corresponding to a value of 1. When the value is
zero, there are no elements within the talus domain that are on the yield surface. Image h shows
the height of the lava dome (solid line) and core (dashed line) against the volume of the dome,
where a volume of 1Mm3 is approximately 3 days.
160
4.5
140
4
120
3.5
3
100
2.5
80
2
60
1.5
Height
Radius
Volume
40
20
1
0.5
0
0
0
Extrusion rate (cubic meters/sec)
Volume (Mega cubic metres)
Height/radius of dome
Figure 16:
20
40
Day from start of dome growth
60
a)
2.5
2
1.5
1
0.5
0
0
20
40
60
Days from start of dome growth
b)
a) Height, radius and dome volume measurements of the October – December 1996 lava dome
growth at Soufrière Hills Volcano (MVO data). b) Inferred lava dome extrusion rate (circles)
from observational data in image a. Also shown is the best-fit curve (dashed line) for the
observational data.
Figure 17:
4.5
4.5
Observations
4
Volume (Mega cubic metres)
Volume (Mega cubic metres)
4
100 G Pa s; 1000 G Pa s,
40 degrees
100 G Pa s; 1000 G Pa s,
45 degrees
100 G Pa s; 100 G Pa s,
45 degrees
3.5
3
2.5
2
1.5
1
3.5
3
2.5
2
Observations
1.5
100 G Pa s; 1000 G
Pa s, 40 degrees
100 G Pa s; 1000 G
Pa s, 45 degrees
100 G Pa s; 100 G
Pa s, 45 degrees
1
0.5
0.5
0
0
0
50
100
150
200
0
250
20
40
60
80
100
120
140
Height (metres)
Radius (metres)
a)
b)
0.6
40
100 G Pa s;
100 G Pa s;
100 G Pa s;
100 G Pa s;
100 G Pa s;
100 G Pa s;
35
30
0.4
Pressure (M Pa)
Fraction core
0.5
0.3
100 G Pa s; 1000 G Pa s,
40 degrees
0.2
100 G Pa s; 1000 G Pa s,
45 degrees
1
2
3
4
0.7
25
0.6
20
0.5
0.4
15
0.3
0.2
5
0.1
0
0
0
Dome volume (Mega cubic metres)
c)
0.8
5
0
0
0.9
10
100 G Pa s; 100 G Pa s,
45 degrees
0.1
1
1000 G Pa s, 40 degrees
1000 G Pa s, 45 degrees
100 G Pa s, 45 degrees
1000 G Pa s, 40 degrees
1000 G Pa s, 45 degrees
100 G Pa s, 45 degrees
1
2
3
4
5
Volume (Mega cubic metres)
d)
Results for the growth of lava domes with a talus as a comparison to the lava dome extruded on
SHV Oct. – Dec. 1996. The viscosity of the core, talus and friction angle are shown in the plots.
The solidus pressure is 0.4 M Pa (including atmospheric pressure) and the yield strength is 5 M
Pa for all the models. Image a) shows the maximum radius of the lava dome plotted against the
dome volume along with the observational data. Image b) shows the modelled maximum height
of the lava domes plotted against dome volume with the observational data. Image c) shows the
fraction of core within the dome against dome volume and d) show the pressure at the conduit
inlet as well as an indication of when the dome is behaving plastically in the talus region. Note
that one of the runs does not reach a final volume/time before reaching the edge of the modelled
domain.
Figure 18:
4.5
4.5
Observations
3.5
200 G Pa s
500 G Pa s
3
Observations
4
100 G Pa s
Volume (Mega cubic metres)
Volume (Mega cubic metres)
4
1000 G Pa s
2.5
2
1.5
1
100 G Pa s
200 G Pa s
3.5
500 G Pa s
3
1000 G Pa s
2.5
2
1.5
1
0.5
0.5
0
0
0
50
100
150
0
200
20
40
60
a)
100
120
140
b)
0.5
80
100 G Pa s
0.45
200 G Pa s
70
0.4
500 G Pa s
60
Pressure (MPa)
Fraction core
0.35
0.3
0.25
0.2
100 G Pa s
0.15
200 G Pa s
0.1
500 G Pa s
1000 G Pa s
50
40
30
20
1000 G Pa s
0.05
10
0
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0
4.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Volume (Mega cubic metres)
Volume (Mega cubic metres)
c)
80
Height
Radius (metres)
d)
Results for the growth of lava domes with a talus using a linearly increasing viscosity within the
dome core and talus as a comparison to the lava dome extruded on SHV Oct. – Dec. 1996. The
final viscosity of the core, at day 62, is shown in the plots from starting with a viscosity of 1010
Pa s. The solidus pressure is 0.4 M Pa (including atmospheric pressure) and the yield strength is
5 M Pa for all the models. Image a) shows the maximum radius of the lava dome plotted against
the dome volume along with the observational data. Image b) shows the modelled maximum
height of the lava domes plotted against dome volume with the observational data. Image c)
shows the fraction of core within the dome against dome volume and d) show the pressure at the
conduit inlet.
Figure 19:
a)
b)
c)
d)
e)
f)
Final lava dome shapes (a to e) at a time of 62 days after this eruption period began. The
approximate height and radius of the observed lava dome at this time at SHV was 120 metres
and 150 metres, respectively. For images a to e, τY = 5MPa and the solidus pressure is 0.4 M Pa,
including atmospheric pressure. a) Core viscosity is 1011 Pa s and the talus viscosity is 1012 Pa s,
and the friction angle is 40º. The final radius and height are 72 metres and 182 metres
respectively. b) The viscosity of the core is initially 1010 Pa s and increased linearly in time to a
final value of 1011 Pa s. The friction angle is 45º and the final height and radius are 71 metres and
183 metres respectively. c) The viscosity of the core is initially 1010 Pa s and increased linearly in
time to a final value of 2x1011 Pa s. The friction angle is 45º and the final height and radius are
83 metres and 171 metres respectively. d) The viscosity of the core is initially 1010 Pa s and
increased linearly in time to a final value of 5x1011 Pa s. The friction angle is 45º and the final
height and radius are 102 metres and 161 metres respectively. e) The viscosity of the core is
initially 1010 Pa s and increased linearly in time to a final value of 1012 Pa s. The friction angle is
45º and the final height and radius are 116 metres and 156 metres respectively. For images b to e
the viscosity in the talus is one order of magnitude height at all times during the simulation.
Image f shows a photo of the lava dome extruded on Soufrière Hills Volcano on approximately
20 October, reproduced with permission of MVO.
Figure 20:
4.5
4.5
Observations
3.5
500 G Pa s, 45 deg
1000 G Pa s, 35 deg
3
Observations
4
500 G Pa s, 35 deg
Volume (Mega cubic metres)
Volume (Mega cubic metres)
4
1000 G Pa s, 45 deg
2.5
2
1.5
1
500 G Pa s, 35 deg
3.5
500 G Pa s, 45 deg
1000 G Pa s, 35 deg
3
1000 G Pa s, 45 deg
2.5
2
1.5
1
0.5
0.5
0
0
0
50
100
150
200
0
20
40
Radius (metres)
60
80
100
120
140
Height
a)
b)
0.5
80
0.45
70
0.4
60
Pressure (M Pa)
Fraction core
0.35
0.3
0.25
0.2
500 G Pa s, 35 deg
0.15
1000 G Pa s, 35 deg
0.05
40
30
10
1000 G Pa s, 45 deg
0
0
0
0.5
1
1.5
2
2.5
3
3.5
0
4
1
2
Volume (Mega cubic metres)
Volume (Mega cubic metres)
c)
500 G Pa s, 35 deg
500 G Pa s, 45 deg
1000 G Pa s, 35 deg
1000 G Pa s, 45 deg
20
500 G Pa s, 45 deg
0.1
50
d)
3
4
e)
f)
Results for the growth of lava domes with a talus using an increasing viscosity within the dome
core and talus. The final viscosity of the core, from a value of 1010Pa s, is shown in the plots. The
viscosity in the talus is one order of magnitude higher that the core at all times during the
simulation. The solidus pressure is 0.4 M Pa (including atmospheric pressure) and the yield
strength is 5 M Pa for all the models. Image a) shows the maximum radius of the lava dome
plotted against the dome volume with the observational data. Image b shows the modelled
maximum height of the lava domes plotted against dome volume with the observational data.
Image c shows the fraction of core within the dome against dome volume and d show the
pressure at the conduit inlet. Images e and f are the final lava dome shapes at a time of 62 days
after this eruption period began with a friction angle of 35º. Image e has a final viscosity of
5x1011Pa s and image f has a final viscosity of 1012 Pa s.