Abstract
The generation of surface waves by seafloor displacement is a classic problem that arises in the study of tsunamis. The generation of waves in a two-dimensional domain of uniform depth by uplift or subsidence of a portion of a flat bottom boundary has been elegantly studied by Hammack (Tsunamis: a model of their generation and propagation, Ph.D. thesis, California Institute of Technology, 1972), for idealized motions. The physical problem of tsunami generation is more complex; even when the final displacement is known from seismic analysis, the deforming seafloor includes relief features such as mounts and trenches. Here, following Kajiura (J Oceanogr Soc Jpn 28:260–277, 1972), we investigate analytically the effect of bathymetry on the surface wave generation, by solving the forced linear shallow water equation. While Kajiura’s geometry consisted of a step-type bottom bathymetry with a rectangular uplift to understand the effect of the continental shelf on tsunami generation, our model bathymetry consists of an uplifting cylindrical sill initially resting on a flat bottom, a geometry which helps evaluate the effect of seamounts on tsunami generation. We find that as the sill height increases, partial wave trapping reduces the wave height in the far field, while amplifying it above the sill.
Similar content being viewed by others
References
Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions (Dover, 1965).
Bartholomeusz, E.F. (1958), The reflexion of long waves at a step, Math. Proc. Cambridge, 54:106–118.
Didenkulova, I., Nikolkina, I., Pelinovsky, E., and Zahibo, N. (2010), Tsunami waves generated by submarine landslides of variable volume: analytical solutions for a basin of variable depth, Nat. Hazard Earth Sys., vol. 10, 2407–2419.
Di Risio, M., De Girolamo, P., Bellotti, G., Panizzo, A., Aristodemo, F., Molfetta, M. G. and Petrillo, A. F. (2009) Landslide-generated tsunamis runup at the coast of a conical island: New physical model experiments, J. Geophys. Res., 114(C01009).
Dutykh, D, Dias, F. and Kervella, Y. (2006) Linear theory of wave generation by a moving bottom, C. R. Acad. Sci. Paris, Ser. I, 343:499–504.
Dutykh, D., Poncet, R. and Dias, F. (2011), The VOLNA code for the numerical modelling of tsunami waves: generation, propagation and inundation, Eur. J. Mech. B/Fluids, 30:598–615.
Fritz, H.M., Kongko, W., Moore, A., McAdoo, B., Goff, J., Harbitz, C., Uslu, B., Kalligeris, N., Suteja, D., Kalsum, K., Titov, V.V., Gusman, A., Latief, H., Santoso, E., Sujoko, S., Djulkarnaen, D., Sunendar, H. and Synolakis, C.E. (2007), Extreme runup from the 17 July 2006 Java tsunami, Geophys. Res. Lett., 34:L12602.
Hammack, J.L. (1972), Tsunamis - A Model of Their Generation and Propagation, PhD thesis, California Institute of Technology.
Hammack, J.L. (1973), A note on tsunamis: their generation and propagation in an ocean of uniform depth, J. Fluid Mech., 60:769–799.
Hill, E.M., Borrero, J.C., Huang, Z., Qiu, Q., Banerjee, P., Natawidjaja, D.H., Elosegui, P., Fritz, H.M., Suwargadi, B.W., Pranantyo, I.R., Li, L., Macpherson, K.A., Skanavis, V., Synolakis, C.E. and Sieh, K. (2012), The 2010 Mw 7.8 Mentawai earthquake: Very shallow source of a rare tsunami earthquake determined from tsunami field survey and near-field GPS data, J. Geophys. Res.-Sol. Ea., 117(B6):B06402.
Jamin, T., Gordillo, L., Ruiz-Chavarria, G., Berhanu, M., and Falcon, E. (2013), Generation of surface waves by an underwater moving bottom: Experiments and application to tsunami modeling, submitted.
Kajiura, K. (1963), The leading wave of a tsunami, Bulletin of Earthquake Engineering Research Institute, 41:535–571.
Kajiura, K. (1972), The directivity of energy radiation of the tsunami generated in the vicinity of a continental shelf, Journal of the Oceanographical Society of Japan, 28:260–277.
Kanamori, H. (1972), Mechanisms of tsunami earthquakes, Physics of the Earth and Planetary Interiors, 6:346–359.
Kanamori, H. and Kikuchi, M. (1993), The 1992 Nicaragua earthquake: a slow tsunami earthquake associated with subducted sediments, Nature, 361:714–716.
Kânoğlu, U. and Synolakis, C.E. (1998), Long wave runup on piecewise linear topographies, J. Fluid Mech., 374:1–28.
Kerr, R.A. (2005), Model shows islands muted tsunami after latest Indonesian quake, Science, 308(5720):341.
Kervella, Y., Dutykh, D. and Dias, F. (2007), Comparison between three-dimensional linear and nonlinear tsunami generation models, Theor. Comput. Fluid Dyn., 21:245–269.
Knowles, J.K. (1966), On Saint-Venant’s principle in the two-dimensional linear theory of elasticity, Arch. Rat. Mech. Anal., 21:1–22.
Lautenbacher, C.C. (1970), Gravity wave refraction by islands, J. Fluid Mech., 41:655–672.
Levin, B.V., Nosov, M.A. Physics of Tsunamis (Springer, 2008).
Lin, I.-C. and Tung, C.C. (1982) A preliminary investigation of tsunami hazard, Bulletin of the Seismological Society of America, 72(6):2323–2337.
Liu, P.L.-F., Lynett, P. and Synolakis, C.E. (2003), Analytical solutions for forced long waves on a sloping beach, J. Fluid Mech., 478:101–109.
Longuet-Higgins, M. S. (1967), On the trapping of wave energy round islands, J. Fluid Mech., 29:781–821.
Mei, C. C., The applied dynamics of water waves (World Scientific, 1989).
Okada, Y. (1992), Internal deformation due to shear and tensile faults in a half-space, Bull. Seism. Soc. Am., 82:1018–1040.
Okal, E.A. and Synolakis, C.E. (2007), Far-field tsunami hazard from mega-thrust earthquakes in the Indian Ocean, Geophysical Journal International, 172:995–1015.
Renzi, E. and Sammarco, P. (2010), Landslide tsunamis propagating around a conical island, J. Fluid Mech., 650:251–285.
Sammarco, P. and Renzi, E. (2008), Landslide tsunamis propagating along a plane beach, J. Fluid Mech., 598:107–119.
Stoker, J.J. Water Waves (Wiley, 1957).
Synolakis, C.E., Liu, P.L.-F., Carrier, G. and Yeh, H. (1997), Tsunamigenic seafloor deformations, Science, 278(5338):598–600.
Tinti, S., Bortolucci, E. and Chiavettieri, C. (2001), Tsunami excitation by submarine slides in shallow-water approximation, Pure Appl. Geophys., 158, 759–797.
Tuck, E. O. and Hwang, L.-S. (1972), Long wave generation on a sloping beach, J. Fluid Mech., 51:449–461.
Zhang Y. and Zhu, S. (1994), New solutions for the propagation of long water waves over variable depth, J. Fluid Mech., 278:391–406.
Acknowledgments
This work was funded by EDSP of ENS-Cachan, the Cultural Service of the French Embassy in Dublin, the ERC under the research project ERC-2011-AdG 290562-MULTIWAVE, SFI under the programme ERC Starter Grant-Top Up, Grant 12/ERC/E2227 and the Strategic, and Major Initiatives scheme of University College Dublin.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: Contour Integral Evaluation
Here we will evaluate the complex integrals that appear in the inverse Laplace transform. The procedure followed is almost, but not fully standard, and is similar to (Renzi and Sammarco 2010, Appendix B).
1.1 The Far Field \((r > 1)\)
Consider the integral
where
Here we replaced \(\hat{f}_{tt}\) with its expression. In the following, \(r\) and \(c_d\) are regarded as parameters, while \(c\) is a positive real constant. A branch cut on the negative real axis is introduced to avoid multivaluedness of \(K_0\). Consider a large semi-circular contour \(C_R\) on the half plane:
In the complex domain \(\Omega , \hat{\zeta }(r; \omega )\) is an entire function of \(\omega \). \(K_0\) has a pole at \(\omega = 0 \not \in \Omega \) and \(\hat{f}_{tt}\) has a pole at \(\omega = -\gamma \not \in \Omega \), since \(\gamma > 0\). The denominator in the expression for \(\beta _2\) does not have any zeros in the same complex domain (Fig. 7). Since no poles are found for \({\hat{\zeta}}(r; \omega )\), then \({\oint _\Gamma {\hat{\zeta}}(r; \omega ) e^{\omega t} \mathrm {d}\omega = 0}\) according to Cauchy’s theorem, where \(\Gamma \) is a closed circuit as depicted in Fig. 6. For large \(\omega \), the modified Bessel functions of the first and second kind can be approximated to the leading order by (Abramowitz and Stegun 1965):
Substitution of the above expressions inside (Eq. 47) yields
By taking the contour \(\Gamma \subseteq \Omega \) (Fig. 6) for \(t > 0\), we have
Since, for \(\omega \in \Omega \), the integrand \({\hat{\zeta}}(r; \omega )\) has no poles, application of Cauchy’s theorem to the first integral gives
For finite \(t\), the second integral becomes
Analogously, one can find that the third integral \(\int _{\Gamma ^-} {\hat{\zeta}}(r; \omega )\ e^{\omega t}\ \mathrm {d}\omega = 0\). Now, consider the transform \(\omega = e^{-i\pi /2} s\), which is a rotation of \(+90^{\circ }\) about the origin. The integral equation (52) becomes
In the limit \(\epsilon \rightarrow 0\), the new integration path becomes slightly deformed due to the pole of \(K_0\) at \(\omega = 0\) (Fig. 8). After the transform, the branch cut is now along the negative imaginary axis. Therefore,
For small argument \(x\), the modified Bessel functions of the first and second kind can be approximated to leading order (Abramowitz and Stegun 1965) as
By substituting the above expressions into \({\hat{\zeta}}(r; -is)\), we find
Making use of the parametric transform \(s = \delta e^{i\phi }, \ \phi \in (0, \pi )\) and letting \(\delta \rightarrow 0\), we find
Hence
1.2 The Near Field \((r < 1)\)
Consider the integral
where
Again, here \(r\) and \(c_d\) are regarded as parameters. In the same \(\Omega \) plane as before, \({\hat{\zeta}}(r; \omega )\) is an entire function of \(\omega \). \(K_{0,1}\) have a pole at \(\omega = 0 \not \in \Omega \) and \(\hat{f}_{tt}\) has a pole at \(\omega = -\gamma \not \in \Omega \). The denominator of \(\alpha (\omega )\) is the same as the denominator of \(\beta _2(\omega )\) and has no zeros in the complex domain \(\Omega \). Since no poles are found for \({\hat{\zeta}}(r; \omega )\), then, according to Cauchy’s theorem, \({\oint _\Gamma {\hat{\zeta}}(r; \omega )\ e^{\omega t}\ \mathrm {d}\omega = 0}\), where \(\Gamma \) is a closed circuit as depicted in Fig. 6. Substitution of the leading order approximation for \(I_{0,1}\) and \(K_{0,1}\) when \(\omega \) is large yields
For \(t > 0\), we take the contour \(\Gamma \) as before:
The application of Cauchy’s theorem to the first integral resulted in \({\oint _\Gamma {\hat{\zeta}}(r; \omega )\ e^{\omega t}\ \mathrm {d}\omega = 0}\). By following the same procedure as in our far-field analysis, we can prove that
Therefore,
with \(\epsilon \) small. By doing the same rotation as before, \(\omega = e^{-i\pi /2} s\), the previous expression becomes
The new integration path in the limit \(\epsilon \rightarrow 0\) is deformed due to the pole at \(s = 0\) as shown in Fig. 8, which yields
In order to evaluate the integral on the semicircle \(C_\delta \), we need to evaluate the behavior of \({\hat{\zeta}}(r; -is)\) for small arguments, using the small argument approximations for \(I_{0,1}\) and \(K_{0,1}\) (Eq. 57):
Consequently, making use of the parametric transform \(s = \delta e^{i\phi }, \ \phi \in (0, \pi )\) and letting \(\delta \rightarrow 0\) yields
Hence,
Appendix 2: Solution with Fourier Transform
Even though the preferred integral transform for transient phenomena is the Laplace transform, since in our case the motion starts from rest and returns asymptotically to rest again, we could also apply the Fourier transform. Its advantage is that it does not require the cumbersome contour integral evaluation for the inverse transform. Starting from Eq. (8), we apply the Fourier transform pair
The resulting governing equation in transformed space is
where
As before, we split the fluid domain into two subregions, namely the near field where \(r^* < r_c^*\) or \(r < 1\) and the far field where \(r^* > r_c^*\) or \(r > 1\). We solve separately in each subregion the forced long-wave equation (74), and then we match the solutions at the common boundary \(r = 1\).
1.1 Solution in the Transformed Space
1.1.1 The Near Field \((r < 1)\)
In the near field \(h^* = h_1^*\) and, therefore, \(h = h_1^* / h_2^* \,\,\doteq\,\, c_d\). Furthermore, since both the bathymetry and the forcing are axisymmetric, the \(\theta \)-term is zero in Eq. (74), and because the water depth is constant, its radial derivative is zero. Therefore, Eq. (74) can be simplified:
which is an inhomogeneous, second-order partial differential equation. In order to solve it, we will apply the method of variation of parameters. Thus, we first consider the relevant homogeneous equation
With the change of variables \(\chi = \omega r / \sqrt{c_d}\), the above equation becomes a standard Bessel equation of zeroth order, whose two independent solutions are the two Bessel functions \(J_0(\chi )\) and \(Y_0(\chi )\). Therefore, the general solution to Eq. (77) is
However, \(Y_0(\chi ) \approx \frac{2}{\pi } \ln (\chi )\) as \(\chi \rightarrow 0\). Therefore, boundedness of the free-surface elevation at \(r = 0\) requires \(\beta _1 = 0\). Hence,
In order to find the solution of the forced ordinary differential equation, the method of variation of parameters requires the Wronskian of the two independent solutions of the homogeneous equation (77). From Abramowitz and Stegun (1965) we have
Finally, the solution of the inhomogeneous ordinary differential equation (77) is
where
is the particular solution, \(W(\rho ) = 2/(\pi \rho )\) is the Wronskian of the two homogeneous solutions and \(\tilde{f}_{tt}(\omega ) = - i \gamma \omega /(\gamma - i \omega )\) is the forcing term. The integration yields
1.1.2 The Far Field \((r > 1)\)
As above, in the far field we have axial symmetry in the bathymetry and the water depth does not vary with \(r\) and is \(h^* = h_2^*\) or \(h = 1\). Moreover, there is no direct influence from the forcing term on the fluid motion. Consequently, Eq. (74) can be simplified:
The above equation is a standard Bessel equation of zeroth order. The two independent solutions could be again the Bessel functions \(J_0(\omega r)\) and \(Y_0(\omega r)\) or the Hankel functions of the first and second kind and order zero, \(H_0^{(1)}(\omega r)\) and \(H_0^{(2)}(\omega r)\). However, from all these options, only the Hankel function of the first kind satisfies the Sommerfeld radiation condition and thus the solution to Eq. (84) is
where \(H_0\) is the Hankel function of the first kind and order zero. The coefficients \(\alpha _1\) and \(\alpha _2\) will be obtained by the matching conditions at \(r = 1\).
1.2 Matching at \(r = 1\)
The near- and far-field solutions are matched at the common boundary \(r = 1\). We require continuity of the free-surface elevation \(\tilde{\zeta }\) and the radial fluxes \(h\, \tilde{\zeta _r}\), which yield, respectively,
By solving the linear system of Eq. (86), we obtain the expressions for \(\alpha _1\) and \(\alpha _2\) (the dependence on \(c_d\) is omitted for brevity):
By replacing the expression (Eq. 83) for \(P(r;\omega )\) and by noting that
we find
and
The expression of the transformed free-surface elevation in the near field can be further simplified:
Finally, the transformed free-surface elevation is
1.3 Wave Description
To return to the physical space, we apply the inverse Fourier transform, which is straightforward,
The last two expressions are identical to Eqs. (42) and (35), respectively.
Rights and permissions
About this article
Cite this article
Stefanakis, T.S., Dias, F. & Synolakis, C. Tsunami Generation Above a Sill. Pure Appl. Geophys. 172, 985–1002 (2015). https://doi.org/10.1007/s00024-014-1021-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00024-014-1021-6