International Journal of Geosciences, 2011, 2, 326-335
doi:10.4236/ijg.2011.23035 Published Online August 2011 (http://www.SciRP.org/journal/ijg)
Copyright © 2011 SciRes. IJG
Pyroclastic Flow from Soufrière Hills Volcano, Montserrat:
Solid Block Model
Irina Nikolkina1,2,3, Narcisse Zahibo1, Tatiana Talipova3, Efim Pelinovsky1,2,3
1Physics Department, University of the French West Indies and Guiana, Pointe-à-Pitre, Guadeloupe, France
2Department of Applied Mathematics, Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia
3Department of Nonlinear Geophysical Processes, Institute of Applied Physics, Nizhny Novgorod, Russia
Received March 5, 2011; revised June 7, 2011; accepted July 11, 2011
The solid block model is applied to describe the motion of the pyroclastic flow under the joint action of
gravity and Coulomb friction. Special attention is paid to characteristics of the pyroclastic flow generated by
Montserrat volcano in likely directions. The critical friction angle of the flow propagation is evaluated em-
pirically. Characteristic parameters of the pyroclastic flow (travel time and impact velocity) are well ap-
proximated by linear regressions. Proposed estimations of the parameters of pyroclastic flow are useful for
the rough and express evaluation of its characteristics.
Keywords: Landslide Dynamics, Solid Block Model, Soufrière Hills Volcano
The Soufrière Hills Volcano, Montserrat situated in the
Caribbean (Figure 1) has been erupted since 1995. The
activity of volcano has been carefully studied by [1-6].
Since 1995 three events have triggered the tsunami
waves in December 1997 [7,8], July 2003 , May 2006
Various mathematical models were developed to de-
scribe the motion of the avalanche and the landslide dy-
namics [12-18]. The solid block model has been widely
used [19-22] for rough estimations of landslide charac-
teristics. Real and potential explosions of the Soufriere
Hills Volcano were modeled using different models, see
In this paper special attention is paid to the character-
istics of the pyroclastic flow generated by the Soufriere
Hills Volcano. The travel time and the impact velocity of
the pyroclastic flow are computed applying the solid
block model in order to estimate potential debris ava-
lanche hazard associated with the eruption of the Sou-
frière Hills Volcano.
2. Mathematical Model
According to the solid block approach, the pyroclastic
flow is modeled as a solid block of a mass m sliding
down a titled plane with a constant slope angle α, under
the action of gravity mg, where g is acceleration of grav-
ity, the friction F, and the normal force N. Let us con-
sider a two-dimensional coordinate system XZ, Figure 2.
The normal force N equals the component of the gravita-
The simple Coulomb friction law applied here is based
on a constant basal friction angle δ that is an empirical
property of the contacting materials:
The motion equation along x-coordinate is given by
the second Newton’s law:
If we have the complex path of the pyroclastic flow
which consists of several approximately constant slopes
and the constant friction, the Equation (3) transforms into
the set of equations
d, where cossincos
and i is a number of the path. Previously, the equation of
I. NIKOLKINA ET AL.327
Figure 1. Montserrat Island, Lesser Antilles, Caribbean Sea.
Figure 2. Model geometry: A block slides down a slope
tilted α under the action of gravity force mg, friction F, and
normal force N.
motion was solved numerically for paths of complex
geometry for the snow-avalanche motion . After in-
tegrating the Equation (4) we obtain the velocity of the
pyroclastic flow in the end of the i-part:
where Ti is the travel time and vi-1 is the initial velocity
on the start point of the i-part. The travel time is deter-
mined after the second integration of the Eq.(4)
In case tan 0
the pyroclastic flow reaches
the sea, otherwise the pyroclastic flow stops if the initial
velocity is not too high
Of course, the Equation (3) can be easily integrated
numerically for any mountain profile. In fact, all avail-
able digital maps have a resolution of several hundred
meters, and the linear approximation of the profile is a
natural spline. The approximation discussed here can be
applied to study the dynamics of the pyroclastic flow if
its horizontal scale of the pyroclastic flow is less than the
length of each path.
Let us now apply this model to analyze the character-
istics of the pyroclastic flow from the Soufrière Hills
3. Characteristics of Pyroclastic Flow at Sea
This simplified theoretical model is applied to calculate
the parameters of the pyroclastic flow from the Soufrière
Hills Volcano in directions of Tuitt’s Ghaut, Plymouth,
Tar River and White River valley, hereafter referred as
To study the propagation of the pyroclastic flow, two
parameters should be determined: Coulomb friction and
initial velocity of the pyroclastic flow.
In fact, data of initial velocities of the pyroclastic
flows from the Soufrière Hills Volcano is poorly docu-
mented. We can expect (see below) that the initial veloc-
ity has the same order as the mean velocity. Mean veloci-
ties of pyroclastic flows from the Soufrière Hills Volcano
were measured earlier: 5 m/s - 30 m/s in 1996-1997
[27,28], 15 m/s in 2003 . In fact, Calder et al.  sug-
gested that velocity of dome-collapse flows was up to
10 m/s - 60 m/s. The simplified estimation of initial ve-
locity can be done using the assumption that a dome col-
lapses and falls as a free body in a crater achieving ve-
locity 02vgh, where h is a height of lava dome.
Assuming that it varies from 50 m to 100 m, the initial
velocity attains 30 m/s - 40 m/s. These values argued
with estimates by  are used in further calculations.
As for the basal friction angle, usually it is consid-
ered to be equal to 17˚ - 30˚ for the sand-textured
debris  and 20˚ - 40˚ for the granular debris ava-
lanche . Previously the debris avalanche from Car-
ibbean volcanoes was actively modeled [24,30,31] and
appropriative basal friction angle was suggested to be
rather small (5˚ < δ < 8˚ for Mt Pelée, Martinique and
13˚ < δ < 35˚ for the Soufrière Hills Volcano, Montser-
rat). Earlier Heinrich et al.  suggested that the best
agreement between calculated and theoretical data was
Copyright © 2011 SciRes. IJG
I. NIKOLKINA ET AL.
obtained when the small basal friction angle 13˚ < δ < 15˚
is used. Furthermore, the motion of real landslides is
governed by low values of the friction angle [24-30].
Considering these studies, values 8˚ - 17˚ are chosen to
estimate the characteristics of the pyroclastic flow in the
Below the pyroclastic flow from the Soufrière Hills
Volcano is studied in likely directions.
3.1. Tar River Direction
According to results of the marine geophysical survey on
the flanks of Montserrat, deposits are located 15 km off-
shore the Tar River valley . During a period of heavy
rainfall on 29 July 2001 the low volume pyroclastic
flows in the Tar River Valley continued for five hours
before intensifying into large pyroclastic flows with
surges . Totally, three large lava dome collapses oc-
curred in the Tar River Valley between November 1999
and July 2003 ; in July 2003 the pyroclastic flows
impacted the sea and produced tsunami . Small pyro-
clastic flows also occurred in the Tar River Valley in
November-December 2009 .
Different splines of the volcano profile in the Tar
River direction are demonstrated in Figure 3. The origi-
nal spline represents 3’’ resolution map presented in ,
Figure 3(a). Various splines are used in order to study
the sensitivity of the results to the resolution, see Figures
3(b)-(d). First and second volcano profiles consist of 14
and 7 paths correspondingly, and a zone of an easy slope
(5˚ - 12˚) is clearly observed. The use of the small fric-
tion angle (13˚ - 15˚) recommended previously by
Heinrich et al. , and initial velocity of 30 - 40 m/s
leads to deceleration and stop of the pyroclastic flow in
the area of the top. Then, a long-distance steep zone with
a maximum slope angle of 34˚ begins; and finally, a third
easy zone 2000 m - 2250 m off the volcano peak is
characterized by the moderate slope angle (1˚ - 12˚). So,
the first (original) and the second (approximated) profiles
can be divided into three zones characterized by the dif-
ferent average slope angle, Figures 3(a) and (b).
The third profile (Figure 3(c)) is rather homogeneous
in terms of the slope angle; it varies smoothly from
15˚ to 5˚. The last profile (Figure 3(d)) represents a lin-
ear approximation of the original one; the slope angle is
equal to 14˚. Calculations with the use of the initial ve-
locity (30 m/s - 40 m/s) and the moderate slope angle
(α < 17˚) show that neither third nor fourth profiles “ob-
struct” the propagation of the pyroclastic flow; and it
reaches the sea.
Thus, results are sensitive to the resolution of the pro-
file, and the use topographic maps of low resolution can
lead to improper conclusions.
Figure 3. Profile of Tar River valley in different resolutions:
original (a), small (b), moderate (c), linear approximation
The use of small friction angle (8˚ - 15˚) and the initial
velocity of 40 m/s permit us to observe some peculiari-
ties of the pyroclastic flow: in terms of the spatial and
temporal variation of the velocity, it is described rather
fairly when the first and second profiles are used (250 m
and 500 m); the effect of the easy slope that decelerate
the pyroclastic flow, is observed 500 m - 700 m from the
volcano peak, Figure 4. Then, the velocity decreases
slightly hundred meters off the shore.
As for the travel time, it does not exceed 2 minutes,
Figure 4. The pyroclastic flow stops on the top of the
volcano when the initial velocity of 30 m/s - 35 m/s is
Copyright © 2011 SciRes. IJG
I. NIKOLKINA ET AL.
Copyright © 2011 SciRes. IJG
where v is in m/s, hereinafter R is a coefficient of corre-
used. A case of maximum initial velocity is specially
studied here because it permits to estimate the minimum
travel time that is important in terms of time alert. As
mentioned, results are sensitive to the resolution of the
profile, and the moderate (250 m and 500 m) and the
original resolution (250 m) is used in order to obtain
Two parameters of the propagation of the pyroclastic
flow (the travel time and the impact velocity) are spe-
cially studied for the original resolution. Calculations are
produced for different initial velocity (30 m/s, 35 m/s
and 40 m/s), and a wide range of the friction angle
(8˚ - 17˚), Figure 5(i). In general, the travel time does
not exceed 2 minutes. The travel time is can be approxi-
mated by the following equation:
Let us study the critical parameters of the pyroclastic
flow for moderate resolutions (250 m - 500 m). As dis-
cussed earlier, the pyroclastic flow stops when the mod-
erate resolution is applied. The proposed range of un-
known parameters (the initial velocity and the friction
angle) is used to identify the “behavior” of the pyroclas-
tic flow. The critical values of the friction angle vary
from 12˚ to 15.5˚. The pyroclastic flow stops when the
critical value is exceed (other conditions being equal).
For example, the pyroclastic flow does not reach the sea
when the initial velocity is 33 m/s and friction angle ex-
ceeds 13˚ in case of moderate resolution, see Figure 5(e).
The critical friction angle is evaluated empirically from
the performed calculations for the moderate resolution,
and the linear regression has the following form:
where T is in seconds, the friction angle δ is in degrees. In
all cases the impact velocity does not exceed 100 m/s. A
minor rise of the basal friction angle reduces significantly
the velocity of the pyroclastic flow. For example in a case
when the initial velocity is 40 m/s, the sought parameter
decreases twice (from 80 m/s to 40 m/s) while basal the
friction angle increases from 10˚ to 14˚, Figure 5(m).
The impact velocity υ can be fitted by
, (8) where υ is in m/s.
Figure 4. Velocity of the pyroclastic flow along the Tar river profile for the original (a,b), small (c,d) and moderate resolution
(e, f): Velocity versus distance (left column) and velocity versus time (right column).
I. NIKOLKINA ET AL.
Figure 5. Parameters of the pyroclastic flow in likely directions: the original profile (a–d); the critical friction angle of the
propagation (e–h); travel time (i–l); impact velocity (m–p).
3.2. Tuitt’s Ghaut Direction
Since 1995 the northern side of the volcano has been
strongly affected by the surge, by main and derivative
pyroclastic flows . Debris torrents occurred in Tuitt’s
Ghaut in mid-November, 2009 when a moderate flow
was observed; followed by numerous torrents in the end
of November. In December the runout distance of pyro-
clastic flows reached 2 km from lava dome; after several
weeks of large pyroclastic flows, helicopter observations
showed that the head of Tuitt’s Ghaut was full of pyro-
clastic deposits .
Based on the topography data presented in , the
splines of different resolution are examined, the original
one is given in Figure 5(b). It is interesting to mention
that Tuitt’s Ghaut is located close to the Tar River profile
described in the previous section. Both profiles have a
characteristic feature that influence the propagation of
the pyroclastic flow: a zone of the steep slope is clearly
observed 500 m from the top. This peculiarity makes the
pyroclastic flow decelerate. In general, the Tuitt’s Ghaut
profile is rather homogeneous, with average slope angle
The study of critical characteristics appears to be very
conclusive, see Figure 5(f) where calculated data for the
original resolution of 250 m is given; the same values are
obtained for the moderate resolution of 500 m. The criti-
cal angle varies from 15˚ to 16.5˚ being almost constant
for the initial velocity more than 34 m/s.
Mean values of the significant parameters are studied
using the same approach as described in the previous
section. It takes about 50 - 100 seconds to cover a dis-
tance of 3200 m, Figure 5(j). It bears repeating that time
alert up to 1 - 2 minutes is rather short to claim the alarm.
The deviation between values of the travel time obtained
for different cases is rather big. The travel time can be
approximated by a simple linear regression:
where T is in seconds, friction angle δ is in degrees.
According to calculations, the impact velocity varies
from 30 to 100 m/s, Figure 5(n). Considering the fact
that during the period of high volcanic activity in No-
vember–December 2009 no pyroclastic flow reached the
sea in this direction, it would appear reasonable to sup-
pose that either real basal friction angle is bigger that 16˚
or initial velocity of the pyroclastic flow is less than
30 m/s. Deviation between obtained values of the impact
velocity for different cases is rather insignificant, and a
linear regression can be used to approximate this pa-
where entrance velocity υ is in m/s, friction angle δ is in
3.3. White River Direction
In December 1997 a debris avalanche and a pyroclastic
Copyright © 2011 SciRes. IJG
I. NIKOLKINA ET AL.331
flow with a volume of 55·106 m3 were produced in White
River valley ; this lateral blast is associated with the
tsunami inundated 80 m inland at the Old Road Bay
[7,8,35], see Figure 1; this event was actively studied
[7,24,36]. This tsunami might have reached the neighbor
island (Guadeloupe) situated about 50 km from Mont-
serrat. Zahibo et al.  suggested that the eruption of
Soufrière Hills Volcano represents a potential danger of
distant tsunami. In October 2009 the pyroclastic flow
deposits extended 3 km down the White River. In No-
vember - December 2009 small pyroclastic flows oc-
curred in the White River valley. In November 2009 py-
roclastic flows moved down Gingoes Ghaut (SSW from
White River valley) to within 200 m of the sea .
Geographically, the White River profile is divided into
2 paths of different slope angle, the first one stretches for
500 - 700 m and is characterized by a big slope angle (up
to 30˚) that is followed by a steep path with a mean slope
angle about 11˚. The latter decelerates significantly the
propagation of the pyroclastic flow , see Figure 5(c).
Computed critical characteristics of the propagation of
the pyroclastic flow (250 m resolution) are presented in
Figure 5(g). According to calculations, the pyroclastic
flow does not stop and reaches the sea when splines of
other resolutions are used. This peculiarity can be ex-
plained by the fact that there are no sudden changes of
the slope angle and the pyroclastic flow is not deceler-
For all profiles in the White River direction mean val-
ues of the travel time and the impact velocity are deter-
mined, Figure 5(k). The travel time can be approximated
by a following linear regression
White River 416, 0.91T
where T is in seconds, the friction angle δ is in degrees.
The impact velocity is also examined, and it is shown
that the impact velocity decreases considerably when the
basal friction angle is slightly modified, Figure 5(o). A
simple regression (14) is characterized by a high value of
the coefficient of correlation R:
White River 11178, 0.97R
where υ is in m/s.
3.4. Plymouth Direction
A former capital of Montserrat, Plymouth was aban-
doned after the eruption of the Soufriere Hills eruption in
1997. In June 1997 pyroclastic debris entered the Belham
Valley, when a portion of the surge cloud detached to
travel westwards of Cork Hill , see Figure 1 for loca-
tions. Since then a partial dome collapse with pyroclastic
flow activity has occurred once: on 8 January 2007 pyro-
clastic density currents were observed in Gages Valley.
The original geographical profile in the direction of
Plymouth  is given in Figure 5(d). It is characterized
by a complicated geometry; it consists of several paths of
different inclination. The first steep path that stretches
for 1000 m is described by a big slope (19˚ - 38˚), then
another path begins (6˚ - 18˚), after that a plain and ex-
tended zone is located. One more path is observed 2000
m from the volcano peak (α < 17˚).
In Figure 5(h) critical characteristics of the propaga-
tion of the pyroclastic flow for the original resolution are
presented; the same values are obtained for the resolution of
500 m. For a wide range of the initial velocity (30 - 38 m/s),
the critical friction angle is the same (13˚).
Let us discuss briefly peculiarities of the impact veloc-
ity and travel time, related with the geometry of the pro-
file in the direction of Plymouth. The pyroclastic flow
traverses 4000 m, from the top of the volcano to the sea
shore, in a short period of time (50 - 80 seconds), but
calculated values have a wide scatter, Figure 5(l). As for
the impact velocity, it depends significantly on the fric-
tion angle, Figure 5(p). Both parameters can be ap-
proximated by linear regressions:
Plymouth 416, 0.89,TR
Plymouth 13185, 0.97,R
A series of calculations are produced to estimate charac-
teristic parameters of the propagation of the pyroclastic
flow during eruptions of the Soufrière Hills Volcano,
Montserrat. In the framework of the solid model
key-characteristics of the pyroclastic flow are calculated
for likely directions; and these profiles are rather similar
in terms of the length and the slope angle, Figure 6(a).
The Tuitt’s Ghaut and Tar River profiles situated close
one to another, are characterized by a steep zone of top
near volcano peak, and this characteristic feature influ-
ences the propagation of the pyroclastic flow (detailed
description of this phenomenon is given in paragraph 3.1).
As for the Plymouth and White River profiles oriented to
the west, a zone of big slope angle is located near the
peak of the volcano that makes the pyroclastic flow de-
scend with acceleration. Another curios point to mention
is that all profiles are similar over a distance of 2000 m
from the volcano peak.
Critical parameters of the pyroclastic flow with the
moderate initial velocity (30 m/s - 40 m/s) calculated for
original profiles are specially compared, Figure 6(b).
For example, in the Tuitt’s Ghaut direction, the pyroclas-
tic flow stops when the friction exceeds 15˚ - 16˚, such
conclusion seems to be closely related with the mean
Copyright © 2011 SciRes. IJG
I. NIKOLKINA ET AL.
value of the slope angle in this direction (α = 15˚). In the
Plymouth direction, the pyroclastic flow stops when the
friction angle is equal to 13˚ that exceeds the mean slope
angle (α = 12˚). In general, the critical friction angle dif-
fers slightly for various directions being in a range from
12˚ (Tar River) to 16.5˚ (Tuitt’s Ghaut). The range of
critical angles for all profiles is rather wide. In the Tuitt’s
Ghaut direction (homogeneous with average slope angle
α = 15˚), the pyroclastic flow can propagate farther than
in other directions (exactly what happened in Octo-
Principal parameters of the pyroclastic flow are esti-
mated for different profiles (250 m resolution) and mod-
erate initial velocity (30 m/s - 40 m/s), Figure 6(c)-(d).
Although linear regressions are rather similar with the
deviation of 15 - 20 seconds, the pyroclastic flow reaches
the sea faster in case it propagates in the White River
direction. It is notable that Heinrich et al.  who per-
formed 3D simulation of the 1997 avalanche when pyro-
clastic flows were observed in White River valley,
pointed out that the small friction angle (13˚ - 15˚) gives
the best agreement. It correlates with our results as the
solid model approach demonstrates that torrents do not
reach the sea in characteristic direction in case friction
angle is greater than 16˚.
Previously, the travel time of the pyroclastic flow was
assumed to be 60 seconds by Heinrich et al.  who
applied homogeneous fluid model to describe the motion
of the pyroclastic flow in White River valley in Decem-
ber 1997. Later, Heinrich et al.  studied the same
event considering the model of incompressible homoge-
neous fluid under the joint action of gravity and Cou-
lomb friction and showed that the travel time attained
3 minutes that is close to our estimations. However, from
the point of mitigation view, the time available to spread
the alarm, is rather short. Earlier the potential danger of
distant tsunami triggered by pyroclastic flow from the
Soufrière Hills Volcano was discussed by ; recently,
Bellotti et al.  studied the feasibility of Tsunami
Early Warning Systems for small volcanic islands, and
came to the conclusion that time available for detecting
tsunamis and spreading the alarm is of the order of few
Another important parameter studied for all profiles is the
impact velocity fitted by a linear regression, Figure 6(d).
The biggest values of the impact velocity are achieved in
case of the Tuitt’s Ghaut’s profile. In general, the impact
velocities are rather important in terms of possible tsu-
nami generation. Previously Heinrich et al.  obtained
that the pyroclastic flow reached the sea in a minute with
velocities of 80 m/s. That is close to our calculations ob-
tained in the framework of the solid model in case of
small friction (9˚ - 10˚). Previously, Heinrich et al. 
Figure 6. Calculated parameters of the pyroclastic flow in
the likely directions: original profiles (a); the critical fric-
tion angle of the propagation (b); travel time (c); impact
studied tsunami simulation from Montserrat, and consid-
ered impact velocities from 25 m/s to 55 m/s. The same
values are obtained in the framework of our approach for
large initial velocity of the dome collapse (35 m/s - 40 m/s)
and moderate values of the friction angle (13˚ - 15˚).
Thus, main characteristic parameters of the pyroclastic
flow associated with possible tsunami, are quite similar
for different likely directions, except the critical basal
friction angle that differs significantly from 12˚ to 16.5˚.
This permits to give rough estimations of main charac-
teristics required to analyze not only the propagation of
the pyroclastic flow, but also tsunami generation.
In fact, the pyroclastic flow events associated with the
small collapse of lava dome have been reported since
Copyright © 2011 SciRes. IJG
I. NIKOLKINA ET AL.333
October 2009, but only few of them reached the sea; and
no tsunami warning was issued . No data about the
volume of pyroclastic flows is available but according to
our calculations, the pyroclastic flow does not reach the
sea when its initial velocity is less than 20 m/s that corre-
spond to collapse of dome 30 m height. So, theoretical
predictions for small events are in agreement with ob-
served data. At the present time due to continuous erup-
tion of the Soufrière Hills Volcano, dome grows and thus
the tsunami danger rises.
The application of the avalanche model, based on the
solid block approach, is discussed here to describe the
motion of the pyroclastic flow from the Soufrière Hills
Volcano, Montserrat. Two parameters are determined to
study the propagation of the pyroclastic flow: Coulomb
friction and the initial velocity of the pyroclastic flow.
The analysis of average characteristics of the pyroclastic
flow is performed for different friction angles (8˚ - 17˚),
and moderate initial velocity of 30 m/s - 40 m/s. The
theoretical model is applied to calculate parameters of
the pyroclastic flow in the likely directions (Tar River,
White River, Tuitt’s Ghaut and Plymouth). Several reso-
lutions of mountain profiles are used, and it is shown that
results are sensitive to spline approximation and the use
of the average slope angle leads to improper conclusions.
At the same time, the original and slightly modified
resolution (250 m - 500 m) make possible to describe the
observed descend of the pyroclastic flow. The spatial and
temporal variations of the velocity in case of the large
initial velocity are specially discussed for the Tar River
profile. It is shown that the propagation of the pyroclastic
flow is described rather fairly in the framework of the
applied theoretical model when small profiles are used
(250 m and 500 m), and the effect of topography (the
steep zone) that decelerates the pyroclastic flow, is ob-
served. For all likely directions, the critical friction angle
is evaluated empirically from performed calculations for
the small resolution; it varies significantly from 12˚ to
16.5˚ for different profiles. Two parameters of the pyro-
clastic flow are particularly discussed: the travel time
and the impact velocity, these characteristics are impor-
tant from the point of view of tsunami generation. It is
shown that both parameters are approximated by similar
linear regressions for all studied directions. Generally,
the impact velocity hardly exceeds 100 m/s. What is
more, the insignificant variation of the friction acts very
much on the impact velocity of the pyroclastic flow. The
travel time does not exceed 2 minutes; and time available
to spread the alarm is rather short; furthermore, volcano
eruptions can represent a potential danger of distant tsu-
nami for coastal regions far from Montserrat Island that
should be taken into consideration. Proposed estimations
of the parameters of the pyroclastic flow in the frame-
work of the solid model are useful for the rough and ex-
press evaluation of the characteristics of the debris ava-
Partial support from the grants: CPER (2007–2013),
ANR project (Vitesss), RFBR (11–05–00216; 11-05-
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