The near-field tsunami amplitudes caused by submarine landslides and slumps spreading in two orthogonal directions

Ocean Engineering 33 (2006) 654–664 www.elsevier.com/locate/oceaneng

The near-field tsunami amplitudes caused by submarine landslides and slumps spreading in two orthogonal directions Abdul Hayir* Division of Mechanics, Civil Engineering Department, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey Received 17 December 2004; accepted 25 May 2005 Available online 10 October 2005

Abstract The aim of this study is to present the solutions for the near-field tsunami amplitudes caused by submarine landslumps and slides spreading in two orthogonal directions. A linearized shallow water wave theory is derived. The transform techniques (Fourier and Laplace transform) are used for the solution of Laplace equation. The results show that if the ratio of the velocties is v1/v2Z0.1, the numerical results are almost the same as the values obtained for one dimensional movement of the slumps and slides. But, when the ratio of the velocties is v1/v2Z1, obtained normalized peak amplitudes, hmax/z0 are smaller than the numerical values for one dimensional solution. It is concluded that normalized peak amplitudes for the models are small because of the interaction of the velocities. Numerical examples are presented for various parameters. q 2005 Elsevier Ltd. All rights reserved. Keywords: Tsunamis; Near-field tsunami amplitudes; Wave focusing; Landslides and slumps; Slow earthquakes; Laplace transform; Fourier transform; Fast Fourier transform; Laplace equation

1. Introduction Tsunamis are dispersive gravity water waves, and at long periods have phase velocity pffiffiffiffiffi cT Z gh where h is the ocean depth and g is the acceleration due to gravity. A vertical motion of the seafloor generating tsunami during earthquakes has been considered in * Corresponding author. Tel./fax: C90 212 285 6587. E-mail address: [email protected]

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.05.010

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literature (Hammack, 1973). While pioneering studies suggest that submarine slumps and slides may also generate tsunami (Gutenberg, 1939), only recently it has been recognized that those generation mechanisms are common, and that their study is important for the estimation of the hazards from inundation. The effects of source directivity and wave focusing on amplitudes of tsunami generated by slowly spreading uplift of the sea floor, using a simple kinematic source model are studied (Todorovska and Trifunac, 2001). The results show that amplitude amplification of up to an order of magnitude occurs in the direction of source propagation when the wave velocity of the source is close to the long period tsunami velocity. This amplification occurs above the source, progressively, as the source evolves by adding uplifted fluid to the fluid displaced previously by uplifts of preceding source segments. This amplification is larger for a wider source compared to the depth of water, and is the largest for an infinitely wide source. In addition to this, tsunami generated by submarine slumps and slides using more detailed kinematic source models are examined (Trifunac et al., 2001a,b, 2002a,b, 2003a,b; Todorovska et al., 2001, 2002, 2003; Trifunac and Todorovska, 2002; Hayir, 2003, 2004a,b). The purpose of those studies are to add to further understanding of the nature of wave forms of tsunami in the near-field of such processes. All studies above include one-dimensional movement of the submarine landslides and slumps. In this study, submarine landslides and slumps are spreading in two orthogonal directions, so this model is more extended and generalized problem, and it might cover the former studies. The problem is solved by transform method with the forward and inverse Laplace transforms computed analytically, and Fourier transforms computed by FFT (Trifunac et al., 2002a).

2. Mathematical model and problem formulation The model is bounded at the top by the free surface bounded at the bottom by a solid boundary and unbounded in the direction of wave propagation, i.e KN!x!N and KN!y!N as in Fig. 1. Initially the fluid is at rest with the free surface and solid boundary defined by zZ0 and yZKh, respectively. For tO0 the solid boundary is allowed to move in a prescribed manner given by zZKhC zðx; y; tÞ such that limjxj/N;jyj/N zðx; y; tÞZ 0. The resulting deformation of the free surface, which is to be determined, is given by zZh(x,y;t). If it is assumed that the fluid is incompressible and the flow irrotational a velocity potential f(x,y;t) is known to exist such that the fluid velocity vector can be expressed as vZPf. Hence, from the continuity equation (Hammack, 1973), V2 fðx; y; z; tÞ Z 0

(1)

subject to the following three boundary conditions fz ðx; y; z; tÞKht ðx; y; tÞ Z 0 at z Z 0

(2a)

ft ðx; y; z; tÞKghðx; y; tÞ Z 0 at z Z 0

(2b)

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Fig. 1. Movement of the slumps and slides in two orthogonal directions until it stops (a) in the plane (b) in the space.

and fz ðx; y; z; tÞKzt ðx; y; tÞ Z 0 at z Z 0

(2c)

A linearized shallow water solution can be obtained by the Fourier–Laplace transform defined as

 k; ð sÞ Z zð

WZv ð 2 t2

eik2 y

0

8 LZv t < ð1 :

0

2N 3 9 ð = eik1 x 4 eKst z0 ðHðtKx=v1 Þ C HðtKy=v2 ÞÞ5dt dxdy ;

(3)

0

In the transform space, the water elevation (Trodorovska and Trifunac, 2001) is

ð sÞ Z  k; hð

s2

s2  k; ð sÞ 1 zð cosh kh C u2

(4)

where u2 Z gk tanh kh

(5)

and u is the circular frequency of the wave motion. Substituting Eq. (3) into Eq. (4) gives ð sÞ Z  k; hð

z0 ðp C p2 C p3 Þ cosh kh 1

(6)

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where p1 Z

v1 ðu1 Ku2 C u3 K1Þ s2 k2 ðv2 k2 C isÞðsKiv1 k1 Þ s2 C u2

iv v ðu C u3 C u4 C u5 K2Þ s2 p2 ZK 1 2 1 ðv2 k2 C isÞðsKiv1 k1 Þ s2 C u 2 p3 Z

v1 ðu4 C u5 Ku6 K1Þ s2 2 k2 ðv2 k2 C isÞðsKiv1 k1 Þ s C u2

(7a) (7b) (7c)

and, where u1 Z eiv1 k1 t1Kst1

(8)

u2 Z eiðv1 k1 t1Cv2 k2 t2 ÞKst1 u3 Z eiv2 k2 t2 u4 Z eiv2 k2 t2Kst2 u5 Z eiv1 t1 and u6 Z eiðv1 k1 t1Cv2 k2 t2 ÞKst2 ð sÞ can be obtained analytically as follows  k; The inverse Laplace transform of hð ð tÞ Z ~ k; hð with

z0 LK1 ðp1 C p2 C p3 Þ cosh kh

(9)

h v1 iðk1 v1 t1Ck2 v2 t2 Þ ðe Keik2 v2 t1 Þ k12 v21 m1 eKik1 v1 t1 Kk22 v22 m2 eKik2 v2 t1 k2 m m  m m i Cueiut1 3 C ueKiut1 4 HðtKt1 Þ C k12 v21 m1 Kk22 v22 m2 C u 3 C u 4 2 2 2 2 (10a)

LK1 fP1 g Z

  LK1 fP2 g Z v1 v2 k1 v1 m1 Kk2 v2 m2 eiðk1 v1Kk2 v2 Þt2 C m3 eiðk1 v1CuÞt1 C m4 eiðk1 v1KuÞt1   !HðtKt1 Þ C k1 v1 m1 eiðk2 v2Kk1 v1 Þt2 Kk2 v2 m2 C eiðk2 v2CuÞt2 m3 C eiðk2 v2KuÞt2  (10b) !HðtKt2 Þ C ð1 C eik2 v2 t2 Þð2k1 v1 m1 K2k2 v2 m2 C m3 C m4 Þ h 2 2 v2  iðk1 v1 t1Ck2 v2 t2 Þ e Keik2 v2 t2 k1 v1 m1 eKik1 v1 t2 Kk22 v22 m2 eKik2 v2 t2 k1 m m  m m i Cueiut2 3 C ueKiut2 4 HðtKt2 Þ C k12 v21 m1 Kk22 v22 m2 C u 3 C u 4 2 2 2 2 (10c)

LK1 fP3 g Z

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where m1 Z

eik1 v1 t   ðk1 v1 Kk2 v2 Þ k12 v21 Ku2

(11a)

m2 Z

eik2 v2 t   ðk1 v1 Kk2 v2 Þ k22 v22 Ku2

(11b)

m3 Z

eKiut ðk1 v1 C uÞðk2 v2 C uÞ

(11c)

m4 Z

eiut ðk1 v1 KuÞðk2 v2 KuÞ

(11d)

ð tÞ numerically using fast Fourier ~ k; Inverse Fourier transforms are evaluated on hð Transform (FFT) to obtained h(x,y;t) 2 3 N N ð ð 1 Kik2 y 4 1 Kik2 y ð ~ k; tÞdk15dk2 hðx; y; tÞ Z e e (12) hð 2p 2p KN

KN

Fig. 2. Normalized positive tsunami peak amplitudes,hmax/z0, versus vT/v for the rectangular slumps and slides with 10!10, 50!50 and 100!100 km2 for ocean depth hZ2 km.

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3. Numerical results To understand the behaviour of the near-field tsunami amplitudes cause by movements of the slumps and slides in two orthogonal directions as shown in Fig. 1, the nine figures are presented in this article. Fig. 2 shows normalized positive tsunami peak amplitudes, hmax/z0, versus vT/v for the rectangular slumps and slides with 10!10, 50!50 and 100!100 km2 for ocean depth hZ2 km. The maximum normalized positive peak amplitude occurs for L!WZ100!100 km2 and 50!50 km2 at v/vTZ0.75, when v1/v2Z0.1. But, these amplitudes are smaller than those amplitudes obtained at v/vTZ1 for v1/v2Z1 and at duration time tZt*ZL/v 1. Fig. 3 illustrates comparison of normalized peak amplitudes,hmax/z0, for v/vTZ0.1, v/vTZ0.75 and v/vTZ1 versus L/h for hZ2 km in the case of tZt*ZL/v1, v1/v2Z1 and v1/v2Z0.1. In this figure similar results are obtained as in the former study in the case of v1/v2Z0.1 and v/vTZ0.1 (Todorovska and Trifunac, 2001). However, when the ratio of v/vT comes to one, the normalized peak amplitudes are getting smaller depending on the ratios of v1/v2 in the case of tZt*ZL/v1. Fig. 4. presents comparison of normalized peak amplitudes,hmax/z0 versus v1/v2 for v/vTZ1 in the case of L!WZ100!100, 50!50 km2 and tZt*ZL/v1. The amplitudes get lesser when the ratios of v1/v2 come to one from zero for L!WZ100!100 km2 and 50!50 km2. It is concluded that when the velocity v1 are big as much as v2, the amplitudes are taking the smallest values because of the interaction of the velocities. Fig. 5 shows tsunami waveforms, h/z0, for

Fig. 3. Comparision of normalized peak amplitudes,hmax/z0, for v/vTZ0.1, v/vTZ0.75 and v/vTZ1 versus L/h for hZ2 km in the case of tZt*ZL/v1, v1/v2Z1 and v1/v2Z0.1.

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Fig. 4. Comparision of normalized peak amplitudes,hmax/z0 versus v1/v2 for v/vTZ1 in the case of L!WZ100! 100, 50!50 km2 and tZt*ZL/v1.

two-dimensional motion of the slides and slumps in the case of L!WZ50!50 km2, hZ 2 km, vT/vZ1, v1/v2Z1 and tZt*ZL/v1. Fig. 6 presents tsunami waveforms, h/z0, for twodimensional motion of the slides and slumps in the case of L!WZ50!50 km2, hZ2 km, vT/vZ1, v1/v2Z0.5 and tZt*ZL/v1. Fig. 7 illustrates tsunami waveforms, h/z0, for twodimensional motion of the slides and slumps in the case of L!WZ50!50 km2, vT/vZ1, v1/ v2Z0.1 and tZt*ZL/v1.In Figs. 5–7, the wave focusing occurs, so it can be said that the volume of the slumps and slides are much smaller than the depletion volume of the tsunamis. Fig. 8 shows tsunami waveforms, h/z0, for two-dimensional motion of the slides and slumps in the case of L!WZ50!50 km2, vT/vZ0.1 and v1/v2Z1. Fig. 9. presents tsunami waveforms, h/z0, for two-dimensional motion of the slides and slumps in the case of L! WZ50!50 km2, vT/vZ0.1 and v1/v2Z0.1. In the Figs. 8 and 9, the dimensionless tsunami amplitude is one, i.e. the volume of the slump is preserved. If the velocities of the slumps and slides are large enough when they are compared with the velocity of the tsunamis, it can be said that the tsunami wave forms are the same as the volume of the eruptions. 4. Conclusions Two-dimensional problems are always more sophisticated than one-dimensional problems. In this paper, two-dimensional movements of the submarine landslides

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Fig. 5. Tsunami waveforms, h/z0, for two-dimensional motion of the slides and slumps in the case of L!WZ 50!50 km2, hZ2 km, vT/vZ1, v1/v2Z1 and tZt*ZL/v1.

Fig. 6. Tsunami waveforms, h/z0, for two-dimensional motion of the slides and slumps in the case of L!WZ 50!50 km2, hZ2 km, vT/vZ1, v1/v2Z0.5 and tZt*ZL/v1.

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Fig. 7. Tsunami waveforms, h/z0, for two-dimensional motion of the slides and slumps in the case of L!WZ 50!50 km2, vT/vZ1, v1/v2Z0.1 and tZt*ZL/v1.

Fig. 8. Tsunami waveforms, h/z0, for two-dimensional motion of the slides and slumps in the case of L!WZ 50!50 km2, vT/vZ0.1, v1/v2Z1 and tZt*ZL/v1.

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Fig. 9. Tsunami waveforms, h/z0, for two-dimensional motion of the slides and slumps in the case of L!WZ 50!50 km2, vT/vZ0.1, v1/v2Z0.1 and tZt*ZL/v1.

and slumps in two orthogonal directions are derived, and the near-field tsunami amplitudes are determined related to these movements. Numerical results show that obtained values in this case are smaller than the values for one-dimensional obtained in the former studies. It is important to consider the effect of the velocities when the velocities are close to each other, v1, v2 and vT. In that case wave focusing starts to affect either the amplitudes or directions of the tsunamis. This phenomenon was ignored in the previous studies. Accordingly, this study might help to explain some unpredictable outcomes of tsunamis.

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