Abstract
Although infrequent, volcanic tsunamis have accounted for almost 20% of those killed by volcanic eruptions since 1000 A.D (Mastin and Witter in J Volcanol Geotherm Res 97(1–4):195–214. https://doi.org/10.1016/S0377-0273(99)00174-2, 2000; Latter in Bull Volcanol 44(3):467–490. https://doi.org/10.1007/BF02600578, 1981). The destructive potential and unpredictability of such waves highlight the need to improve our understanding of the phenomena. Underwater eruptions are one of the source mechanisms that can generate volcanic tsunamis. For example, the 1952 explosions of Myojinsho volcano which destroyed a surveying boat of the Japan Hydrographic Department, killing 31 people, also generated a series of tsunamis. Wave gauges at Hachijo Island, 130 km from the volcano, recorded waves of up to 92 cm, which provide us with one of the few wave gauge observations of a tsunami generated by a volcanic eruption. The present work is based on the mathematical model of Duffy (J Volcanol Geotherm Res 50(3):323–344. https://doi.org/10.1016/0377-0273(92)90100-R, 1992), who analysed surface waves generated by an instantaneous underwater point-source volcanic explosion. We extend this work by modifying the source model to include time dependence and compare the efficiency of wave generation of the following sources: the instantaneous source used by Duffy (1992), a source with exponentially decreasing intensity of the explosion, and two time-constrained sources—a source with a defined sharp end to the explosion, and a source that ceases with a cavity at the free surface. The shorter the characteristic time of the exponential explosion, the more like the original Duffy model the resulting waves are. Waves resulting from time-constrained explosions exhibit a beating behaviour. The duration of the explosion modulates the generated wave field while the average wave amplitudes remain the same. Conversely, the geometric parameters of the explosion affect the wave amplitudes. The beating behaviour of our second time-constrained model bears a strong resemblance to that observed in tsunami records from the 1952 Myojinsho explosions. Thus, we apply the model to these events. The predicted leading wave is absent in the recorded data, but the trailing waves were calculated accurately, particularly in terms of the wave phasing and the lengths of the wave groups. We infer that the absence of this leading wave could be due to the mathematical formulation of the problem and it reflects how gas-rich explosions produce sea-surface displacements. The results suggest that the beating behaviour observed in the wave gauges records could have been a consequence of time constraints on the duration of a volcanic explosion.
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Abbreviations
- r :
-
Radial coordinate
- z :
-
Vertical coordinates
- \({\mathbf {u}}\) :
-
Velocity field
- \(\phi \) :
-
Velocity potential
- \(\psi \) :
-
Displacement potential
- t :
-
Time
- \(\delta ( )\) :
-
The Dirac delta function
- \(\phi _0(t)\) :
-
Source strength as a function of time
- \(\eta \) :
-
Free surface elevation
- d :
-
Water depth
- g :
-
Gravitational acceleration
- \(r_0\) :
-
Vent size
- \(z_0\) :
-
Depth of the explosion
- \(S_0\) :
-
Amplitude of the displacement source
- V :
-
Volume of the explosion
- S(r, t):
-
Source function
- G(r, z, t):
-
The Green’s function
- k :
-
Wave number; Hankel-transformed r coordinate
- p :
-
Laplace-transformed t variable
- H( ):
-
The Heaviside step function
- \(\omega \) :
-
Angular frequency
- \(J_0\) :
-
The Bessel function of the first kind of order 0
- \(J_1\) :
-
The Bessel function of the first kind of order 1
- \(\alpha \) :
-
Parameter describing time-scale of the exponential source
- T :
-
Duration of the explosion
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Acknowledgements
This research was funded by the Marsden Fund Council, Royal Society Te Ap\({\bar{a}}\)rangi Grant Number 17-NIW-017 awarded to NIWA.
Funding
This study is a part of a NIWA led project titled: Volcanoes can make waves too: a new understanding of tsunamis generated by volcanic eruptions, funded by the Royal Society of New Zealand Marsden Fund no 17-NIWA-017. Leaders are Emily Lane with William Power as co-PI.
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Lipiejko, N., Whittaker, C.N., Lane, E.M. et al. Tsunami Generation by Underwater Volcanic Explosions: Application to the 1952 Explosions of Myojinsho Volcano. Pure Appl. Geophys. 178, 4743–4761 (2021). https://doi.org/10.1007/s00024-021-02857-1
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DOI: https://doi.org/10.1007/s00024-021-02857-1