Testing the accuracy of a 1-D volcanic plume model in estimating mass eruption rate
by Larry G. Mastin, U.S. Geological Survey, Cascases Volcano Observatory ([email protected])
v=~0.34
2
10
4/4a
0.04
1
0
0
Fuego
15
10
g
vº
N
g
=increase in wind speed
per m altitude
N=Brunt-Vaisala
frequency
Hekla
1974
v=0.11
0
10
10
5
5
1980
1970 v=0.06
0
0
v=0.12
St. Helens
20
15
10
0.10
0.09
6/12
5/25
5/18
5
7/22
40
Pinatubo
v=0.05
30
20
10
Hudson
15
0.06
0.03
When v<~0.35,
Hmax=Hnowind , indicating
weak winds (Fig. 1b).
21 of these 25
eruptions are in this
category.
10
Ruiz
10
5
5
0
0
Redoubt
15
Reventador
25
10
v=0.38
g
10
20
10
0
20
40
8/18
Figure 2
9/17
6/27
0
0
20
40
0
0
20
40
0
0
20
40
wind speed, m/s
Eruptions examined
ù
æ
ö
dE d é
u2
a dM
=
M
+
gZ
+
h
=
gZ
+
h
()
ê
ú
ç
÷
ds ds ë
2
ds
è
ø
û
The model considers ambient wind, temperature variations in the atmosphere, and
enthalpy of phase changes of water & ice.
Input conditions: (1) Mass eruption rate based on assumed magma temperature (900
C), magma gas content (3 wt%), ejection velocity (~150 m/s), and vent diameter
adjusted to give a specified eruption rate. (2) Atmospheric inputs, including temperature,
humidity and wind vector as a function of elevation.
Solution method: The conservation equations are integrated upward using a Cash-Karp
Runge-Kutta method with automatic step-size adjustment.
Information sources: 1[Hill et al., 1998],2[Carey and Sigurdsson, 1986],3[Scollo et al., 2007],4[Bonis and Salazar, 1973],5 [Rose et al., 1973],6 [Rose et al., 2008],7[Thorarinsson and Sigvaldason,
1971],8[Gronvold et al., 1983],9[Naranjo S. et al., 1993],10[Scasso et al., 1994],11[Tupper et al., 2004],12[Nakada et al., 2005],13[Geshi et al., 2002],14[Sarna-Wojcicki et al., 1981],15[PaladioMelosantos et al., 1996],16[Hoblitt et al., 1996],17 [Pallister et al., 1992],18[Koyaguchi, 1996],19[Koyaguchi and Ohno, 2001],20[Holasek et al., 1996],21[Miller and Chouet, 1994],22[Scott and McGimsey,
1994],23[Smithsonian Institution, 2002],24[Prata and Grant, 2001],25[Bonadonna and Houghton, 2005],26[Naranjo et al., 1986],27[Neal et al. [1995],28[Eichelberger et al., 1995],29 [McGimsey et al.,
2001]
real plumes
eq
obs
max
Model representation
max
min
max
min
max
nowind
max
max
max
nowind
min
max
max
min
Figure 1
max
i
2
4
6
8
10
Mmax
Mmin
s
=0.53
m
=0.07
0
-2
bent-plume model
0
10
20
30
40
0
-2
plume model ignoring wind
0
10
20
30
40
?
Low s
indicates better predictive
s
=0.50
m
=-0.07
2
s
=0.62
m
=-0.25
2
accuracy
?
Positive (negative) m
suggests
tendency to over- (under-) predict
0
empirical curve
-2
10
20
30
40
For eruptions overall:
?
Neither the empirical curve (Mempir) nor 1-D model (Mavg) systematically over- or
underpredict eruption rate relative to Mobs.
?
Standard error for the 1-D model (0.53) is not better (more accurate) than for the
empirical curve (0.50).
?
But the data are few, observations are poor, and most plumes occurred during low-wind
conditions.
Modeling plume height in real winds
obs
2
Figure 5
Observed plume height, km
40
Calculated plume height, km
obs
Mavg
2
0
Model limitations:
Each step in the model is assumed to represent the center of a control volume whose
mass, momentum, and energy flux is being tracked.
?
In the absence of wind (Fig. 1d), this results in a vertical stack of control volumes that
don’t overlap. The maximum height (Hnowind) is the point where ascent velocity-->0.
Strong wind
?
In high-winds (Fig. 1f), the
No wind
Weak wind
model produces a bending
a
b
c
stack of control volumes
H
H
whose maximum height is
H
between the maximum
H z
r
height of the plume axis
(zmax) and that elevation plus
the plume radius (zmax+rmax).
?
In low winds (Fig. 1e),
control volumes overlap, and
sometimes zmax+rmax>Hnowind,
H =H
f
producing an unrealistically
H =z +r
H
straight
high plume.
H =z
d
cylinders e
Thus, for specified input
conditions, the model
gives a range of possible
H =H
plume heights, Hmin to Hmax .
H
bent
H =z +r
If zmax+rmax>Hnowind, Hmax=Hnowind.
cylinders
H
g
Otherwise, Hmax=zmax+rmax.
h
4
Below are residuals between calculated eruption rates and the observed (Mobs), with
standard error s
and mean m
.
v=0.21
5
10
6
Modeled vs. empirical: which is more accurate?
Spurr
0.13
Mnowind
plume model w/o wind
M
max
plume model
Mmin
with wind
8
vent elevation
5
0
empirical curve
top of observed
plume
15
5
Mempir
Log(M ), kg s-1
Main Findings:
obs
6
?
For small eruptions (Mobs< ~10 kg/s), the empirical relationship (Mempir) consistently
underestimates eruption rate; however the 1-D plume model (Mmin to Mmax) mostly
overestimates it. (but there are only four eruptions in this category!)
?
For larger eruptions there is no clear tendency for any method of calculation to underor overestimate.
v=0.07
0.12
v=0.03
5
Miyakejima
20
15
30
v=0.69
1
wind soundings
every 12 hrs
10
0
Ruapehu
v=~1.03
20
0.31
0
2
15
v=0.22
6/12
6/15
3
0
20
15
15
0.06
5
0
20
20
1971
0.03
Log calculated eruption rate, kg s-1
4/4b
10
log(Mnowind/Mobs)
n
n 1/ n
dM
a é
ù when r
=
2p
r r
r
a
uvs1 )
+
b
v^
(
(
)
>=r
a
ë
û
ds
d M x dM
=
vx
s=distance along the curving plume path
ds
ds
u=plume ascent (axial) velocity
a
r
=plume bulk density, r
=ambient air density
d M y dM
M
=
vy
vx, vy are x & y components of wind vector
ds
ds
a
and b
are radial & crossflow entrainment coeffificients
vP
=wind velocity component parallel to plume axis
dMz
2
a
=
p
r ()
r
r
g
v^
=wind velocity component perpendicular to plume axis
ds
3
Eruption rate
estimates are
compared to those
obtained by mapping.
4
v=0.03
20
Etna 2001
5
3/29
4
km asl
v=~0.14
El Chichon
log(Mavg/Mobs)
n
n 1/ n
dM
a é
ùwhen r
=
2p
r
r
a
uvs1 )
+
b
v^
(
(
)
<r
a
ë
û
ds
total energy E
5
Cerro Negro 1995
5
30
Hnowind
Each plume is modeled using the
observed eruption rate Mobs to
obtain Hnowind, Hmin, and Hmax.
Results are compared with the
observed height Hobs
Hmax
30
Hmin
in order to improve estimates of eruption rate through 1-D plume models, we need
more accurate observations of plume height.
plume model w/o wind
plume model with wind
20
10
References:
Figure 3
0
0
10
20
Hobs, km
30
40
Estimating eruption rate
Mass eruption rate is estimated by running 5 model
simulations (Fig. 4) and finding the range of eruption rates
(Mmin to Mmax) for which Hmin<Hobs<Hmax. Simulations were also
repeatedly run ignoring wind, to find the eruption rate
(Mnowind) at which the calculated height agreed with Hobs.
Plume height, km
momentum
wind soundings
every 12 hrs
0
In this study I modify the 1-D plume model Plumeria (Mastin, 2007) to account for
crosswinds, using the following conservation equations:
mass M
10
Cerro Negro 1992
Residuals
Model formulation
I examine
atmospheric wind,
temperature, and
humidity during the
25 historical
eruptions below.
Conditions are
obtained from the
NCEP/NCAR
Reanalysis 1 2.5
degree model.
For each eruption I
calculate a
dimensionless wind:
Elevation, km asl
During volcanic eruptions, empirical relationships are used to estimate mass eruption
rate from plume height. Although simple, such relationships can be inaccurate and can
underestimate rates for eruptions in windy conditions. 1-D plume models can incorporate
atmospheric conditions and are hypothesized to give potentially more accurate
estimates. Here, I present a 1-D model for plumes in cross wind and use it to simulate 25
historical eruptions where plume height Hobs was well observed, and where mass
eruption rate Mobs could be calculated from mapped deposit mass and observed duration.
The simulations considered wind, temperature, and phase changes of water.
Atmospheric conditions were obtained from the NCEP/NCAR Reanalysis 2.5 degree
model. Simulations calculate the minimum, maximum, and average values (Mmin, Mmax ,
and Mavg) that fit the plume height. Eruption rates were also estimated from the empirical
formula Mempir=140Hobs4.14 (Mempir is in kg s-1, Hobs is in km). For these eruptions, the standard
error of the residual is about 0.53 for Mavg and 0.50 for Mempir. Thus for this dataset, the
model is slightly less accurate at predicting Mobs than the empirical curve. The empirical
curve however tends to underestimate eruption rate for low plumes during small
eruptions. Improved eruption-rate estimates using this (or other) 1-D plume model(s)
may require more accurate plume-height observations and improvements in model
formulation.
Modeled vs. observed eruption rates
Wind during real eruptions
log(Mempir/Mobs)
Abstract
Run
1 2
Hmax
3
4
5
Hobs
Hmin
Figure 4
Mmin
Mmax
Log mass eruption rate
Finally, mass eruption rate is calculated using an empirical formula rearranged from
Mastin et al. (2009, eq.1):
Mempir=140Hobs4.14
where Mempir is in kg s-1, Hobs is in km. This equation assumes a magma density of 2500
kg m-3
Bonadonna, C., and B. F. Houghton (2005), Total grain-size distribution and volume of tephra-fall deposits, Bull. Volcanol., 67, 441-456.
Bonis, S., and O. Salazar (1973), The 1971 and 1973 eruptions of Volcan Fuego, Guatemala, and some socio-economic considerations for the volcanologist, Bulletin Volcanologique, 37(3), 394-400.
Carey, S., and H. Sigurdsson (1986), The 1982 eruptions of El Chichón volcano, Mexico (2): Observations and numerical modelling of tephra-fall distribution, Bull. Volcanol., 48(2/3), 127-141.
Eichelberger, J. C., T. E. C. Keith, T. P. Miller, and C. J. Nye (1995), The 1992 eruptions of Crater Peak vent, Mount Spurr Volcano, Alaska: Chronology and summary, in The 1992 Eruptions of Crater Peak
Vent, Mount Spurr Volcano, Alaska. U.S. Geological Survey Bulletin 2139, edited by T. E. C. Keith, pp. 1-18, U.S. Government Printing Office, Washington, D.C.
Geshi, N., T. Shimano, T. Chiba, and S. Nakada (2002), Caldera collapse during the 2000 eruption of Miyakejima Volcano, Japan, Bull. Volcanol., 64, 55-68.
Gronvold, K., G. Larsen, P. Einarsson, S. Thorarinsson, and K. Saemundsson (1983), The Hekla eruption 1980-1981, Bulletin Volcanologique, 46(4), 349-363.
Hill, B., E., C. B. Connor, M. S. Jarzemba, P. C. La Femina, W. Navarro, and W. Strauch (1998), 1995 eruptions of Cerro Negro volcano, Nicaragua, and risk assessment for future eruptions, Geological
Society of America Bulletin, 110, 1231-1241.
Hoblitt, R. P., E. W. Wolfe, W. E. Scott, M. R. Couchman, J. S. Pallister, and D. Javier (1996), The Preclimactic eruptions of Mount Pinatubo, June 1991, in Fire and Mud: Eruptions and Lahars of Mount
Pinatubo, Philippines, edited by C. G. Newhall and R. S. Punongbayan, pp. 457-511, University of Washington Press, Seattle, WA.
Holasek, R. E., S. Self, and A. W. Woods (1996), Satellite observations and interpretation of the 1991 Mount Pinatubo eruption plumes, J. Geophys. Res., 101(B12), 27635-27656,doi 10.1029/96JB01179.
Koyaguchi, T. (1996), Volume Estimation of Tephra-Fall Deposits from the June 15, 1991, Eruption of Mount Pinatubo by Theoretical and Geological Methods, in Fire and Mud: Eruptions and Lahars of
Mount Pinatubo, Philippines edited by C. G. Newhall and R. S. Punongbayan, pp. 583-600, University of Washington Press, Seattle.
Koyaguchi, T., and M. Ohno (2001), Reconstruction of eruption column dynamics on the basis of grain size of tephra fall deposits: 2. Application to the Pinatubo 1991 eruption, J. Geophys. Res., 106(B4),
6513-6534.
Mastin, L. G. (2007), A user-friendly one-dimensional model for wet volcanic plumes, Geochemistry, Geophysics, Geosystems, 8(Q03014),doi 10.1029/2006GC001455.
Mastin, L. G., et al. (2009), A multidisciplinary effort to assign realistic source parameters to models of volcanic ash-cloud transport and dispersion during eruptions, J. Volcanol. Geotherm. Res., 186, 1021.
McGimsey, R. G., C. A. Neal, and C. Riley (2001), Areal distribution, thickness, volume, and grain size of tephra-fall deposits from the 1992 eruptions of Crater Peak vent, Mt. Spurr volcano, Alaska, in
U.S. Geological Survey Open-File Report 01-0370, edited, p. 38, U.S. Government Printing Office, Washington, D.C.
Miller, T. P., and B. A. Chouet (1994), The 1989-1990 eruptions of Redoubt Volcano: an introduction, J. Volcanol. Geotherm. Res., 62(1-4), 1-10.
Nakada, S., M. Nagai, T. Kaneko, A. Nozawa, and K. Suzuki-Kamata (2005), Chronology and products of the 2000 eruption of Miyakejima Volcano, Japan, Bull. Volcanol., 67, 205-218,doi DOI
10.1007/s00445-004-0404-4.
Naranjo, J. L., H. Sigurdsson, S. N. Carey, and W. Fritz (1986), Eruption of the Nevado del Ruiz Volcano, Colombia, on 13 November 1985: Tephra fall and lahars, Science, 233(4767), 961-963.
Naranjo S., J. A., H. Moreno R., and N. G. Banks (1993), La eruptcíon del Volcán Hudson en 1991 (46° S), Región XI, Aisén Chile, in Servicio National de Geologia y Mineria--Chile, Boletin No. 44, edited,
p. 50, Servicio National de Geologia y Mineria--Chile, Santiago.
Neal, C. A., R. G. McGimsey, C. A. Gardner, M. Harbin, L., and C. J. Nye (1995), Tephra-fall deposits from the 1992 eruptions of Crater Peak, Mount Spurr, Alaska, in The 1992 eruptions of Crater Peak,
Mount Spurr, Alaska, U.S.G.S. Bulletin 2139, edited by T. E. C. Keith, pp. 65-79, U.S. Geological Survey, Washington, D.C.
Paladio-Melosantos, M. L. O., R. U. Solidum, W. E. Scott, R. B. Quiambao, J. V. Umbal, K. S. Rodolfo, B. S. Tubianosa, P. J. Delos Reyes, R. A. Alonso, and H. B. Ruelo (1996), Tephra Falls of the 1991
Eruptions of Mount Pinatubo, in Fire and Mud: Eruptions and Lahars of Mount Pinatubo, Philippines, edited by C. G. Newhall and R. S. Punongbayan, pp. 513-536, University of Washington Press,
Seattle.
Pallister, J. S., R. P. Hoblitt, and A. G. Reyes (1992), A basalt trigger for the 1991 eruptions of Pinatubo volcano?, Nature, 356, 426-428.
Prata, A. J., and I. F. Grant (2001), Retrieval of microphysical and morphological properties of volcanic ash plumes from satellite data: application to Mt. Ruapehu, New Zealand, Quarterly Journal of the
Royal Meterological Society, 127, 2153-2179.
Rose, W. I., S. Bonis, R. E. Stoiber, M. Keller, and T. Bickford (1973), Studies of volcanic ash from two recent Central American eruptions, Bulletin Volcanologique, 37(3), 338-364.
Rose, W. I., S. Self, P. J. Murrow, C. Bonadonna, A. J. Durant, and G. G. J. Ernst (2008), Nature and significance of small volume fall deposits at composite volcanoes: Insights from the October 14, 1974
Fuego eruption, Guatemala, Bull. Volcanol., 70, 1043-1067,doi DOI 10.1007/s00445-007-0187-5.
Sarna-Wojcicki, A. M., S. Shipley, R. Waitt, D. Dzurisin, and S. H. Wood (1981), Areal distribution, thickness, mass, volume, and grain size of air-fall ash from the six major eruptions of 1980, in The 1980
Eruptions of Mount St. Helens, Washington; USGS Professional Paper 1250, edited by P. W. Lipman and R. L. Christiansen, pp. 577-601, U.S. Geological Survey.
Scasso, R. A., H. Corbella, and P. Tiberi (1994), Sedimentological analysis of the tephra from the 12-15 August 1991 eruption of Hudson volcano, Bull. Volcanol., 56, 121-132.
Scollo, S., P. Del Carlo, and M. Coltelli (2007), Tephra fallout of 2001 Etna flank eruption: Analysis of the deposit and plume dispersion, J. Volcanol. Geotherm. Res., 160, 147-164.
Scott, W. E., and R. G. McGimsey (1994), Character, mass, distribution, and origin of tephra-fall deposits of the 1989-1990 eruption of Redoubt Volcano, south-central Alaska, J. Volcanol. Geotherm. Res.,
62(1-4), 251-272.
Smithsonian Institution (2002), Strong, sudden 3 November eruption; 8-km-long pyroclastic flow, Global Volcanism Program Monthly Reports, 27(11).
Thorarinsson, S., and G. Sigvaldason (1971), The Hekla eruption of 1970, Bull. Volcanol., 36(2), 269-288.
Tupper, A., S. Carn, J. Davey, Y. Kamada, R. Potts, F. Prata, and M. Tokuno (2004), An evaluation of volcanic cloud detection techniques during recent significant eruptions in the western 'Ring of Fire',
Remote Sensing of Environment, 91(1), 27-46.