2 Shock Waves observed during Volcanic Eruptions
Transcript of 2 Shock Waves observed during Volcanic Eruptions
Experimental Laboratory Study on the Formation of Multiple 1
Shock Waves observed during Volcanic Eruptions 2
E. F. Médici1, and G. P. Waite2
1Department of Mechanical Engineering - Engineering Mechanics
Michigan Technological University
Houghton, Michigan, USA.
2Department of Geological and Mining Engineering and Sciences
Michigan Technological University
Houghton, Michigan, USA.
Abstract
Sequences of multiple pressure waves have recently been observed in different explosive 3
volcanic eruptions, but their origin is not clearly understood. Although a pulsating type of 4
eruption with many individual discharges can produce this effect, the formation of multiple 5
shock waves could also result from a single discharge through: 1) the dynamics of the shock 6
wave itself; 2) the coupled shock wave-supersonic jet dynamics; or 3) a combination of these 7
factors. Shock tube experiments were performed to evaluate these potential mechanisms under 8
controlled laboratory conditions. The amount of energy released was varied to achieve different 9
weak shock wave conditions and to promote formation of single or multiple shock waves 10
through both shock wave dynamics and coupled shock wave-supersonic jet interactions. Our 11
experimental results indicate that both mechanisms may play a role in the formation of multiple 12
pressure waves in low intensity eruptions. 13
1. Introduction
Explosive volcanic eruptions are characterized by a sudden release of gases, liquid and 14
solid particles. During the first instants of eruptions, the multiphase mixture expands into the 15
atmosphere and may commonly reach supersonic speeds. The supersonic jet accelerates the 16
gasses in front of it, potentially triggering a shock wave. The jet may continue ascending and 17
expanding behind the initial shock wave and persist while there is sufficient energy in the 18
conduit to sustain it. The shock wave, on the other hand, can travel many kilometers from the 19
vent of the volcano but will eventually decay into a sound wave [Morrissey and Mastin, 2000]. 20
The shock wave, and the eventual sonic pressure wave, carries important information about the 21
energy and dynamics of the eruption. A better understanding of supersonic jet dynamics and 22
shock wave propagation may lead to new methods for assessing vent conditions in real-time 23
using existing infrasound and video monitoring technologies. 24
Supersonic jets, weak shock waves, and pressure waves have been observed and 25
measured in volcanic eruptions for decades [Nairn 1976; Morrissey and Chouet, 1997; Morrissey 26
and Mastin, 2000]. Still, there are many fundamental aspects regarding the formation and 27
expansion of supersonic jets and shock waves that require further study. One intriguing 28
observation is the formation of multiple consecutive shock waves during a single eruption pulse 29
[e.g., Taddeucci, et al., 2014]. Typically, in shock tube testing or controlled detonation 30
experiments, a single shock wave is observed for a single discharge or detonation event. The 31
Friedlander wave profile [Freidlander, 1946], and its approximation equation, provides a simple 32
model of a sharp pressure rise followed by a smooth rarefaction, typical of a detonation test. 33
Although the phenomenon described by the Friedlander equation can accurately describe the 34
initial high-pressure front, more complex gas dynamics can take place behind this initial front. 35
For example, it is well known that a second shock wave might develop as consequence of the 36
rarefaction pressure drop behind the leading shock wave [Landau, 1945]. The pressure in the 37
rarefaction phase can drop far enough below atmospheric pressure to trigger a second shock 38
wave. The early work by Landau [1945] discussed the physical mechanism triggering the second 39
shock wave. This phenomenon has since been observed in experimental tests [Biss, and 40
McNesby, 2014; Gitterman and Hofstetter, 2014] and the theoretical analysis generalized for 41
supersonic aerodynamic and sonic booms [Whitham, 1956]. The idealized shock wave pair 42
consists of a linear decrease from the initial shock discontinuity to a value below atmospheric 43
pressure before the second shock is triggered. Because the pressure time history resembles the 44
letter “N”, this type of wave has historically being referred as “N-wave” [Whitham, 1956; 45
Blackstock, 2000]. Rarefaction pressure decay observed in a Friedlander type of pressure profile 46
is not considered as N wave because it does not result in a second shock. 47
The theory describing the N wave profile can explain the formation of two, and 48
sometimes more shock waves. However, it may not sufficiently explain the multiple consecutive 49
shock waves observed in some volcanic eruptions, such as those captured by Taddeucci, et al. 50
[2014] at Yasur in 2011. If the rarefaction between subsequent shocks is not far enough below 51
atmospheric pressure to trigger a shock, they must be explained by another mechanism. The 52
complex gas flow dynamics taking place between the leading shock wave and perhaps, the 53
expanding jet head might produce these pressure waves or weak shock waves. Supersonic jets 54
are well known to produce lateral shock waves known as Mach wave radiation and jet crackle, 55
the cause of which is unclear [e.g., Nichols et al., 2014]. Mach wave radiation [Tam, 1995] and 56
jet crackle [Williams, et al., 1975] have been historically described in association with heated jet 57
engine exhaust, but recently, have been suggested to occur during volcanic eruptions as well 58
[Fee, et al., 2013]. The field measurements presented by Taddeucci, et al. [2014] show images of 59
both lateral and upward sequence of pressure/weak shock waves generated after an eruption 60
pulse. In this letter, the described sequences of pressure waves and model are ahead of the jet, so 61
are distinct from crackle. 62
While a number of studies have been conducted with detonations and shock tubes as 63
explosive eruption analogs, there is an important difference between these two types of analog 64
models. In detonations, the event consists only of the initial shock wave (Friedlander profile) and 65
perhaps, if the energy and observation time are long enough, a second shock wave (N wave). On 66
the other hand, in shock tube experiments, the initial shock waves will be followed by a 67
relatively long duration supersonic jet. Although the flow system generated in shock tube 68
experiments is more complex when compared to detonations, they tend to more closely resemble 69
the dynamics occurring during an explosive volcanic eruption. For instance, the multiple 70
pressure waves associated with the supersonic jet that are intended to reproduce experimentally 71
in this article are unattainable in a detonation experiment. 72
Shock tube experiments allow for study, not only of the formation and propagation of the 73
initial shock wave, but also the dynamics of jet expansion and the possible interaction of the jet 74
with the leading shock wave. The interaction of shock waves and the supersonic jet may be at 75
least partially responsible for the formation of the multiple shock waves observed in some 76
volcanic eruptions. In fact, the expansion and deceleration of the jet from supersonic to subsonic 77
speed alone has also been shown to be responsible for acoustic emission in actual volcanic 78
eruptions [Genco et al., 2014; Taddeucci, et al., 2014]. In this paper, shock tube experiments 79
designed to provide insight into the generation of multiple shock waves occurring ahead of an 80
expanding supersonic jet are described. To the best of the authors’ knowledge, the sequence of 81
pressure waves occurring between a leading shock wave and a supersonic jet have not been 82
previously studied. 83
2. Experimental Testing
A series of shock tube tests were performed to demonstrate the formation of multiple shock 84
waves in the open space under a single discharge of compressed dry nitrogen. The number of 85
shock waves and their intensity was measured with video imaging and pressure records. The 86
shock tube is made of a driver section with annular cylindrical shape of volume 8.51 10-3 m3, 87
followed by a 1.2 m long driven section, followed by open space [Médici et al., 2014]. The 88
driven and driver section are separated by a piston which is activated by a rapid decompression 89
of the rear chamber. After the piston opens the driver section, nitrogen expands through the 90
driven section, generating a shock wave, and then out through the open end. 91
92
93
94
95
96
Figure 1:
tests: 517
the first f
the first s
M
The disch
shock tub
respectiv
nitrogen
: Image sequ
70 kPa in a a
four noticeab
shock wave
Multiple tests
harge energy
be. These pr
vely. The tota
is calculated
uence of the
and b; and 34
ble shock wa
and the jet fr
s at two diffe
y was set by
essures, also
al energy,
d from a ther
multiple pre
450 kPa in c
aves are labe
front. The fie
erent dischar
controlling
o referred as
, stored in th
rmodynamic
essure waves
c and d. For t
eled, but add
eld of view i
rge energies
the loading
discharge p
he driver sec
c analysis ass
s observed at
the 3450 kPa
ditional shoc
s approxima
were perfor
pressure on
ressure, wer
ction when c
suming an is
t different di
a discharge p
ck waves are
ately 15.2 x1
rmed to ensu
the driver se
re set at 3450
ompletely fi
sothermal pr
ischarge pre
pressure, on
e present betw
13.0 cm.
ure repeatabi
ection of the
0 and 5170 k
illed with dry
rocess:
essure
nly
ween
ility.
e
kPa
y
= (1) 97
where is the initial pressure (atmospheric pressure), is the loading, or discharge pressure, 98
and is the volume of the driver section. Using the discharge pressures of 3450 and 5170 kPa, 99
the discharge energies become 1.04 105 and 1.74 105 J, respectively. Video images were 100
recorded at 20,000 fps using Photron FASTCAM-APX RS high-speed camera mounted on a 101
shadowgraph array illuminating the open space at the front of the shock tube. Figure 1 shows 102
image sequences of the expanding spherical shock waves for the two discharge pressures. 103
At the lower energy test (1.04 105 J), a series of shock waves was observed in the open 104
space following the leading shock front (Figure 1c and d). After the shock wave travels through 105
the driven section, it starts expanding spherically in the open space. This first shock wave is 106
weak enough that only a fraction of the spherical shape develops. Behind this shock wave, the jet 107
head decelerates as it expands into the open space and a series of shock waves are generated in 108
its wake. Only the first four clearly visible shock waves are marked, although more were 109
generated. 110
At the higher discharge energy test of 1.74 105 J, the shock front is stronger and a shock 111
wave leading the discharge was observed to fully expand into a half spherical shape (Figure 1a 112
and b). As with the lower energy test, a sequence of weaker shock waves following the main 113
shock was observed. 114
115
116
117
118
Figure 2:
for three
panels a,
indicates
The notic
repeatabi
P
imaging
placed on
The wedg
: Pressure sig
tests at each
c, and e, an
the shock w
ceable simila
ility of the ex
ressure mea
of the comp
n a wedge ou
ge used to d
gnals record
h of the two t
d the lower p
wave labeling
arity in press
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surements w
lete evolutio
utside the dr
ivert half of
ded 5.8 cm fr
test pressure
pressure set
g sequence c
sure profile u
were made in
on of the sho
riven section
f the spherica
rom the end
es. The highe
(3450 kPa)
correlated wi
under the sa
n separate tes
ock waves. T
n, centered 5
al shock wav
of the driven
er pressure t
in the right p
ith the video
ame testing c
sts to avoid i
The pressure
.8 cm from t
ve away left
n section on
test set (5170
panels b, d,
o imaging of
conditions in
interference
sensor (PCB
the end of th
an unperturb
the open sp
0 kPa) in th
and e. Part b
f Figure 3c a
ndicates the h
with the vid
B 113B22) w
he shock tube
bed pressure
ace
e left
b
and d.
high
deo
was
e.
e
recording on the center line for two reasons. First, the shock wave can be described as a 119
hyperbolic problem, which guarantees that the shock wave will remain unaffected by anything 120
located in front such as the wedge [Hoffmann and Chiang, 2000]. Second, the spherical 121
symmetry of the shock means splitting the wave in half does not lead to losing generality. 122
However, since the jet is not hyperbolic problem, using a wedge in this way is not valid for 123
studying jet dynamics. 124
The signal from the pressure transducer was recorded at a 250,000 samples per second. 125
Figure 2 shows the pressure traces for both discharge experiments. Figure 3 show the shock 126
waves as they passed over pressure sensor. The instantaneous dimensionless shock wave speed, 127
Mach number, was obtained for every test from the video images (Figure 4). While in this 128
manuscript these pressure waves are referred to as shock waves, their Mach numbers are close to 129
1, meaning they are near the transition between supersonic and subsonic wave speed. This can 130
explain the relatively slow rise time of the shock waves observed in Figure 2, compared with 131
strong shock waves. 132
133
134
135
136
137
Figure 3:
pressure
d. The fir
approxim
R
multiple
passes ov
labeled in
first shoc
: Image sequ
sensor at dif
rst eight noti
mately 15.2x
Repeated test
shocks is rep
ver the senso
n the pressur
ck wave sepa
uence of the
fferent disch
iceable wave
13.0 cm.
ts with each
plicated. Fig
or for the low
re plot of Fig
arates from t
multiple pre
harge pressur
es are labele
of the discha
gure 2b show
wer discharg
gure 2b and
the jet head d
essure waves
re tests: 5170
ed for the 345
arge pressur
ws the pressu
ge energy tes
the correspo
due the incre
s observed a
0 kPa in a an
50 kPa test.
res demonstr
ure fluctuatio
st. The first e
onding imag
easing relati
s they passe
nd b; and 34
Field of view
rate that the
ons as every
eight shock w
ge in Figure 3
ive velocity d
d over the
450 kPa in c
w is
formation of
y shock wave
waves are
3c and d. As
difference, t
and
f
e
s the
the
pressure behind the shock decreases. If the pressure decreases to a value far enough below the 138
atmospheric pressure, it will initiate a second shock wave [Landau, 1945]. This process may 139
repeat, generating additional shock waves. Between the second shock wave, labeled as 2a, and 140
the third shock wave there are two more shock waves, labeled as 2b and 2c that can be traced in 141
the pressure profile and the video imaging. Because they are very weak they are considered as 142
part of the second shock wave, generated by the rarefaction that followed the leading shock. The 143
pressure drop behind the second shock wave may have been large enough to generate a third 144
shock wave in the same way. However, five more shock waves were observed and the pressure 145
drop between shock wave pairs 3-4, 4-5, 5-6, 6-7, and 7-8 are not as prominent as between 1-2 or 146
2-3. This leads to the notion that these later shock waves could have been generated by a 147
different mechanism than the rarefaction of the N-wave, perhaps through the deceleration of the 148
jet. 149
Behind the third shock, pressure decreased as it separated from the jet head, which was 150
still moving at supersonic speed. As the jet head decelerated, the pressure ahead of it decreased 151
and it emitted a new shock wave. This newly-generated shock wave then separated from the jet 152
head, as the jet continued losing speed. This process repeated until the jet lost most of its 153
supersonic speed and was no longer able to sustain the formation of pressure waves. The region 154
in space at which the jet no longer generated pressure waves is out of the field view in the video 155
imaging. 156
The test performed at higher discharge energy, (Figure 3a and b), has similar flow 157
characteristics, but the formation of the multiple shock waves can all be attributed to the 158
dynamics associated with the N-wave pattern. Between every pair of shock waves the pressure 159
decreased below atmospheric pressure (Figure 2a). However weaker pressure waves similar to 160
those observed in the lower discharge energy test may have formed outside the field view. 161
Figure 4: Non dimensional shock wave speed, Mach number, versus position from the end the
shock tube for both experiments at 3450 and 5170 kPa. For the 3450 kPa experiment, the Mach
numbers of the 1st, and 2nd shock waves shock waves were measured. The relative position of the
pressure transducer is indicated for comparison with the Mach number obtained from the
pressure measurements shown in Figure 2.
The pressure difference across a shock wave is correlated with the Mach number through 162
the Rankine-Hugoniot relationship [Shapiro, 1953]: 163
= (2) 164
where is the initial pressure (atmospheric pressure), is the discharge pressure, and is the 165
specific heat ratio (1.4 for air). At the lower discharge energy test, the measured pressure reached 166
by the first shock wave was 20 kPa, which gives 120 kPa absolute for an atmospheric pressure at 167
the time of the experiments of 100 kPa. This gives a pressure ratio of 1.2, which according to the 168
Rankine-Hugoniot relationship corresponds to a Mach number of 1.08. For the second shock 169
wave of the same test the peak pressure was 13 kPa gauge (113 kPa absolute) which corresponds 170
to a Mach number of 1.05. At the higher discharge energy test, the pressure reached by the first 171
shock was 27 kPa gauge (127 kPa absolute) which corresponds to a Mach number of 1.11. These 172
Mach numbers are in good agreement with shock-wave speed measurements obtained through 173
the video imaging despite the uncertainties in speed calculations using the images. The 174
uncertainty in the Mach number calculation due to ±1 pixel error estimation on the shock wave 175
position is 0.06. Figure 4 shows the Mach number of the first two shock waves for the low 176
energy test and the leading shock for the high energy test as a function of the position relative the 177
end of the shock tube. Figure 4 also indicates the relative position of the pressure where the 178
Mach number from the pressure recording and the video imaging are compared. 179
3. Discussion
Different mechanisms can trigger the formation of multiple shock waves for a single discharge 180
event, all of which could resemble potential scenarios of explosive volcanism. The particular 181
case of pairs of shock waves was explained as consequence of the natural expansion of weak 182
shock waves by Landau [1945] and later by Whitham [1956]. In this case, the formation of this 183
second shock wave is due to the shock wave dynamics and does not involve the jet. The 184
existence of the double shock wave can be identified wherever the pressure trace follows the 185
patterns described by Landau [1945] and is normally referred as N-wave. Both shock waves will 186
eventually dissipate into sound waves. For example, the experimental observations presented by 187
Biss and McNesby [2014] clearly show the N-wave pressure profile of the double shock wave, 188
which were also captured on the video imaging. This type of N-wave pressure profile has also 189
been observed in infrasound recordings from different volcanic eruptions [Marchetti et al., 2013; 190
Fee et al., 2014; Matoza et al., 2014]; from detonation tests at the surface [Gitterman and 191
Hofstetter, 2014] and shallow depth [Goto et al., 2001; Bowman et al., 2014]; as well as on both 192
tests reported here. 193
But in addition to the classic N-wave double shock wave, formation of multiple shock 194
waves during volcanic eruptions, could also be a consequence of a series of individual, closely-195
spaced pressure release events (discharges) that resemble a pulsating phenomenon. The multiple 196
shock waves formed under these conditions will be associated with a pulsating jet and not result 197
from the shock wave dynamics described in the previous discussion. Although this could 198
certainly be the case in many eruptions, such us the eruptions of Sakurajima described by Fee et 199
al. [2014], there are other cases that call for a mechanism such as those observed in the shock 200
tube experiments. In some eruptions where multiple shock waves were present, such us the 201
Eyjafjallajökull eruptions in August 2010, the periodicity of the events (eruptions) was slower 202
than the time observed between each shock wave of the couple. Ripepe et al. [2013] using 203
thermal imaging of the Eyjafjallajökull eruption plume, showed that it consisted of a series of 204
puffs at an average period of 20s. The infrasound measurements showed a wider range of 205
pulsating events in the order of 2 to 50 s. Although the spacing between many of the discharges 206
(puffs) was quite long, the close spacing of a subset of the discharges could be explained by 207
shock wave fluid dynamics in the upper conduit and above the conduit. This gives rise to the idea 208
that the multiple shock waves are the outcome of a single discharge rather than the result of 209
multiple discharges. Some of the video footage taken during the eruption shows the 210
coupled/multiple shock waves followed by a single jet/puff indicating that they are the result of a 211
single event. 212
While the scenarios described previously could be associated either with the shock wave 213
or the jet dynamics, at least some of the later multiple shock waves formed during the shock tube 214
tests are due to the coupled shock wave-jet interaction dynamics. Regarding the Eyjafjallajökull 215
eruptions in August 2010, a question still remains: are the observed of shock wave couples a 216
consequence of a pulsating jet or due to the coupled shock wave-jet interaction dynamics 217
observed in the experiments presented here. Clearly more work must be done before this can be 218
decisively answered. For example, an important aspect of multiple shock formation that has not 219
yet been explored is the influence of vent shape. Preexisting vent or crater shape is known to 220
cause changes in the jet characteristic and the associated pressure trace when compared with a 221
no-crater eruption [Taddeucci, et al., 2013]. 222
The vent for this particular eruption is particularly interesting. According to Magnússon 223
et al. [2012], the subglacial vent of Eyjafjallajökull melted cauldrons into the overlying ice that 224
acted as an elongation of the conduit. The shock wave propagation inside the constrained space 225
of the cauldron could be comparable to the shock wave propagation inside the driven section of 226
the experimental shock tube. For a relatively low energy discharge, the shock front could 227
dissipate some of its energy inside the cauldron while the jet was still advancing at supersonic 228
speed giving rise, as the in the experiment, to a shock wave couple in the open atmosphere. On 229
the other hand, for discharges with relatively higher energy, single shock wave events would 230
have been produced; single shock wave events were also observed during this eruption. The most 231
probable explanation of the observed events may be a combination of the factors described 232
herein. 233
In addition to the role of vent geometry, the influence of particles, and other complexities 234
related to the formation of shock waves, there are challenges associated with comparing 235
laboratory measurements with field data. Shock waves can be captured in volcanic eruption 236
pulses using normal and high-speed video imaging, but only under certain environmental 237
conditions [e.g., Nairn, 1976; Yokoo and Ishihara, 2007; Taddeucci et al., 2014]. The relatively 238
low energy, low Mach number, shock waves produced in laboratory experiments decay to sound 239
waves at a short distance from the source. Similarly, by the time comparable pressure waves 240
from a volcanic eruption are recorded at a microphone station in the field, they likewise will 241
have decayed into sound waves. Although estimates of the source pressure can still be made 242
[Medici et al., 2014], this aspect remains a challenge. 243
Clearly the multiple shock waves presented in this experimental work will not be 244
observed in every type of eruption. Based on our experimental evidence, they will only be 245
observable in low intensity eruptions. More intense eruptions will still show the traditional N-246
wave pattern discussed by Marchetti et al. [2013]. 247
4. Conclusion
A series of experimental tests were performed with an open-ended shock tube to model the 248
generation of multiple, coupled shock waves for single discharge volcanic eruptions by at least 249
two distinct mechanisms. The experimental observations at lower discharge energies, show that 250
the first weak shock wave was followed by a series of N-wave pattern shock waves produced by 251
shock wave dynamics. This first set of shock waves was then followed by additional shock 252
waves generated by the decelerating jet. The formation of multiple shock waves was also 253
observed in higher discharge energy tests, although they are more difficult to distinguish in the 254
available field of view. This implies that while moderate and large eruptions will exhibit the 255
traditional N-wave pressure pattern, multiple shock waves might be more common on low 256
intensity eruptions. A combination of different the factors such as vent shape and the amount of 257
energy being released in every puff may play a role in the formation of multiple shock waves. 258
5. Acknowledgments
Support was provided by the Michigan Tech’s Earth, Planetary, and Space Sciences Institute 259
(EPSSI). Support for the Shocktube Facility was provided by the National Science Foundation 260
(EAR-1250153). The authors thank Jeffrey Allen for the technical discussions regarding the 261
experiments; REL, Inc. of Calumet, Michigan, USA for design, fabrication and technical support 262
of the shock tube; and the thorough reviews of David Fee and an anonymous referee which 263
resulted in an improved manuscript. Data are archived at Michigan Tech’s HazHub archive: 264
http://www.mtu.edu/geo/research/hazard-data. 265
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1 2 3 12 3 4
150 µs 350 µs
150 µs 350 µs
a) b)
c) d)
1
2a
2c
3
4 5
6
78
2b
50 µs 350 µs
1 2 12 2c 3 4 5 6 7
8
50 µs 250 µs
PressureSensor
a) b)
c) d)
0 5 10 151
1.05
1.1
1.15
1.2
1.25
x [cm]
M
1st
2nd
5171 [kPa]
Pressure Sensor