Numerical investigation of oscillations within a harbor of constant slope induced by seafloor movements

Ocean Engineering 38 (2011) 2151–2161

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Numerical investigation of oscillations within a harbor of constant slope induced by seafloor movements Gang Wang a,b,n, Guohai Dong c,nn, Marc Perlin d, Xiaozhou Ma c, Yuxiang Ma c a

State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China College of Harbour, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China c State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023 China d Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA b

a r t i c l e i n f o

abstract

Article history: Received 24 December 2010 Accepted 27 September 2011 Editor-in-Chief: A.I. Incecik Available online 21 October 2011

A numerical model, which can simulate wave generation and propagation is developed to simulate oscillations induced by seafloor movements inside a harbor of constant slope, and once verified and then validated through comparison with experimental results, the numerical results are used to examine the analytic solutions presented in Wang et al. (Wang, G., Dong, G., Perlin, M., Ma, X., Ma, Y., 2011. An analytic investigation of oscillations within a harbor of constant slope. Ocean Engineering 38, 479–486). Small-scale seafloor movement usually induces small longitudinal oscillations, but evident larger transverse oscillations. These transverse oscillations are sensitive to the location of the moveable seafloor. The numerical result of each transverse eigen frequency compares well with the theoretical solution; in addition the spatial structure of each mode is also well-captured by the theory. Furthermore, evident/larger longitudinal oscillations induced by large-scale seafloor movements are simulated, and the numerical resonant frequencies agree favorably with the analytical solutions. These longitudinal oscillations are sensitive to the horizontal location of the moveable seafloor. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Harbor resonance Oscillations Boussinesq equations Wave generation Numerical experiments

1. Introduction Based on the linear shallow-water approximation, the companion paper (Wang et al., 2011) presented formulary descriptions of longitudinal oscillations inside a harbor with constant slope open to the sea, and further showed that there could be several transverse oscillation modes existing in the harbor when the width of the harbor is on the order of the incident wavelength, which can be described by Confluent Hypergeometric Function of the Second Kind based on a liner, weakly dispersive, Boussinesq-type equation extended from shallow water equations by modifying the offshore velocity. The purpose of this paper is to examine the putative range of the results presented in the previous paper, and to further investigate, which oscillation modes can be induced readily. A variety of dynamic forcings can induce significant oscillations within a harbor. These external forcings include tsunamis originating from distant earthquakes, short wave groups, infragravity waves and impact waves induced by submarine landslides

n Corresponding author at: State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China. Tel.: þ 86 25 83786982; fax: þ86 25 83787706. nn Corresponding author. E-mail addresses: [email protected], [email protected] (G. Wang), [email protected] (G. Dong).

0029-8018/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2011.09.033

or failures of structures near the harbor (Kulikov et al., 1996; Dong et al., 2010a, b). This paper numerically investigates oscillations induced by seafloor movements within the harbor. Impulsive waves generated by seafloor movement have been modeled conventionally by the shallow water equations. Lynett and Liu (2002), Ataie-Ashtiani and Jilani (2007), and Fuhrman and Madsen (2009) developed higher-order Boussinesq-type models to accurately capture the nonlinear effects as well as the frequency dispersion of the landslide-generated waves. Certainly, the simulations by these extremely attractive approaches require large computational effort. In harbor wave studies, the oscillation wavelengths are much larger than the water depth; hence shallow water is assured. Therefore a weakly dispersive Boussinesq model is adequate for these problems. To improve the computational efficiency and also guarantee sufficient accuracy for our study, a second-order dispersive Boussinesq-type model, that can describe the generation and propagation of earthquake- and landslide-induced tsunamis, is developed. Its application is validated by comparing results with a set of laboratory experiments in Section 2. In Section 3 numerical investigation is presented on the oscillations induced by seafloor movements. Effects of variations of the moveable seafloor’s maximum displacement, location and its velocity, on the oscillation pattern also are examined in this section. Conclusions are presented in Section 4.

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G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

2. Numerical model

0

u0t0 þ eu0 r u0 þ

e 0 0 0 0 w uz0 ¼ r p0 , h rz0 r eZ0 m2

ð2:3Þ

2.1. Governing equations and numerical procedure

ew0t0 þ e2 u0 r0 w0 þ

Consider a three-dimensional wave field with surface elevation Z (x, y, t), at time t, propagating over a variable water depth h (x, y, t). A Cartesian coordinate system (x, y, z) is adopted, with z measured positive upwards from the still-water level. Introducing the characteristic water depth hc as the vertical length-scale, the characteristic length of the typical wavelength lc as the horizontal length-scale, and the characteristic wave amplitude ac as the scale of the wave motion, we can define the following dimensionless variables: ðx,yÞ ðx ,y Þ ¼ , lc 0

0

z z ¼ , hc

ðu,vÞ ðu0 ,v0 Þ ¼ pffiffiffiffiffiffiffiffi , e ghc

0

w0 ¼

pffiffiffiffiffiffiffiffi ghc t t ¼ , lc 0

w pffiffiffiffiffiffiffiffi , ghc

ðe=mÞ

h h ¼ , hc 0

Z0 ¼

e2 0 0 0 w wz0 ¼ ep0z0 1, h rz0 r eZ0 m2

ð2:4Þ

Here u0 ¼ ðu0 ,v0 Þ is the dimensionless horizontal velocity vector, r0 ¼ ð@=@x0 ,@=@y0 Þ is the dimensionless horizontal gradient vector, and the subscripts denote partial derivatives. The kinematic and dynamic boundary conditions on the free surface z0 ¼ eZ0 can be expressed as w0 ¼ m2 ðZ0t0 þ eu0 r0 Z0 Þ p0 ¼ 0

p p ¼ rgac

ð2:5Þ

0

Z

The kinematic boundary condition at the seafloor z0 ¼  h0 is ð2:1Þ

ac

0

where g is the gravitational acceleration, (u, v, w) is the water particle velocity vector, p is the pressure, and r is the fluid density. The small parameters e ¼ac/hc and m ¼ hc/lc, respectively, measuring nonlinearity and frequency dispersion have been introduced. Assuming that the fluid is inviscid and incompressible, the wave motion can be described by Euler’s equations in dimensionless form as

m2 r0 u0 þ w0z0 ¼ 0, h0 r z0 r eZ0

0

w0 ¼ m2 ðu0 r Þh 

m2 0 h0 e t

ð2:6Þ

Assuming further that the flow is irrotational gives the irrotationality condition u0y0 v0x0 ¼ 0,

r0 w0 u0z0 ¼ 0

ð2:7Þ

Strictly following Nwogu’s (1993) derivation procedure and retaining terms to O(e) and O(m2), we obtain a set of Boussinesqtype equations that facilitate a movable bottom, i.e. boundary conditions for movement of the seafloor in the two horizontal

ð2:2Þ

0.08

0.08

(x-xs)/hs=20

(x-xs)/hs=0

0.06

0.04

0.04

0.02

0.02

0.00

0.00

η/hs

0.06

-0.02

-0.02 0

14

28

42

56

70

0

84

14

28

t(g/hs)0.5

42

56

70

84

t(g/hs)0.5

Fig. 1. Comparison between Hammack’s (1973) experimental data (dots) for an impulsive seafloor upheaval (xs/hs ¼ 12.2, zs/hs ¼0.1, b ¼8.6) and the numerical simulation (solid line). The downstream distance is indicated on each time series.

0.02

0.00

0.00

-0.02

-0.02

-0.04

-0.04

η/hs

0.02

-0.06

-0.06

(x-xs)/hs=0

-0.08

(x-xs)/hs=20

-0.08 0

14

28

42 t(g/hs)0.5

56

70

84

0

14

28

42

56

70

84

t(g/hs)0.5

Fig. 2. Comparison between Hammack’s (1973) experimental data (dots) for an impulsive seafloor down thrust (xs/hs ¼ 12.2, zs/hs ¼  0.1, b ¼ 13.7) and the numerical simulation (solid line). The downstream distance is indicated on each time series.

G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

 0

h0 0 þ z0a r0 r0 ðh u0a Þ þ t ¼0

directions:  z02 h02 0 0 0 0 a  h r ðr ua Þ Z þ þ r ðh þ eZ Þua þ m r e 2 6   

0 h 1 0 0 0 0 0 h r r ðh u0a Þ þ ht0 ¼0 þ z0a þ 2 e 0 t0

1

0 ht0

0

0

0

0

0



0

u0at0 þ r Z0 þ eðu0a r Þu0a þ m2

2

@ @t 0



0

2153



ð2:9Þ

e

ð2:8Þ

z02 a r 0 ðr 0 u0 0 Þ at 2

where u0 a is the horizontal velocity vector at an arbitrary elevation z0 ¼z0 a (x0 , y0 , t0 ), and neither is to be confused with subscripts denoting differentiation. As a check, if the bottom boundary condition is now taken for a fixed arbitrary seafloor, and h0 t0 ¼0, the above equations are identical to Eqs. (25a) and (25b) derived by Nwogu (1993). As the Boussinesq equations reduce

Fig. 3. Definition sketch of the harbor and computational domain. (a) plan view, (b) elevation view and (c) computational domain.

1.6

-1.6 0

20

40

60

80

100 Time (s)

120

140

160

180

200

0.0 0.0

0.4 0.6 Frequency (Hz)

0.671

0.223 0.2

0.757

0.1

0.586

-0.8

0.409

0.0

0.296 0.317

Amplitude/ζs

η/ζs

0.2

0.180

0.8

0.125

0.3

0.8

Fig. 4. Time histories of the free surface elevations and their attendant amplitude spectra at the corner (2.5,  2.5 m) generated by an impulsive downthrow of the region A.

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G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

the three-dimensional problem to a two-dimensional one, a major limitation is that they are only applicable to relatively shallow water depths. The elevation of the velocity variable z0a is a free parameter and is chosen to extend the applicability of the equations to relatively deep water. Assuming a stationary seafloor, z0a is recommended to be evaluated as z0a ¼  0.531h0 by Nwogu (1993). The Boussinesq Eqs. (2.8) and (2.9) are solved on staggered grids to eliminate the abrupt changes introduced by imposed, impulsive seafloor motions. As the dispersive terms embedded in the Boussinesq equations may have similar mathematical form with the lower-order terms associated with truncation errors of the discretized equations, higher-order schemes, both in time and space, are needed. The first-order spatial derivatives are discretized to O((Dx)4) using five-point finite-differencing, and a fourthorder predictor-corrector method originally developed by Wei and Kirby (1995) is employed to perform time updating. All boundaries are treated as perfectly reflecting vertical walls. ‘‘Sponge layers’’ following Wei and Kirby (1995) are used at the seaward boundary to allow reflected waves to leave the computational domain without causing spurious oscillations.

displacement, and the parameter b governed the seafloor’s velocity. We consider his experiments on an undisturbed depth hs ¼0.05 m. The computations were conducted with grid size Dx¼0.01 m and time step Dt ¼0.005 s. The length of the simulated flume was 20 m, and the downstream end was treated as a fully reflecting wall. The simulations were discontinued before any reflections occurred. As the ability of Nwogu’s (1993) Boussinesq equations to predict wave transformations has been well documented, more attention is paid to comparing time series of surface elevations near the source region to examine the numerical model’s capability to describe the wave generation process. Figs. 1 and 2 show comparisons between the numerical results and the experimental data. The agreement between the experimental and numerical data indicates that, although the numerical model is weakly nonlinear, as the magnitude of the seafloor movement (zs/hs ¼0.1) is small, the model describes the wave generation and propagation process well, although the high frequency oscillations have slight amplitude discrepancies and phase shifts.

2.2. Validation of the numerical model To ensure that the present model can be applied to the generation and propagation of water waves induced by seafloor deformation and movement, simulations are performed, and the numerical results are compared with experimental data of impulsive seafloor motion from Hammack (1973). In Hammack’s (1973) experiments, the upstream end of the flume was a vertical wall, where waves were generated by the seafloor moving in the positive or negative vertical direction; an energy dissipation system was located at the downstream end to absorb waves and fluid motion. The water depth was initially hs, and the seafloor moved according to hðx,tÞ ¼ hs 7 zs ½1 expðbtÞHðx2s x2 Þ

ð2:10Þ

where H was the Heaviside step function, xs was the length of the moveable seafloor section of the flume, zs was the maximum

3. Oscillations induced by seafloor movements A series of numerical experiments are conducted to investigate oscillations inside a harbor of constant slope induced by impulsive seafloor movement. As indicated in Fig. 3, the backwall of the harbor runs in the y direction; x¼0 where the extended virtual bottom and mean sea level intersect, and x increases offshore. The axis z is positive upward from the still water level. The floor of the open sea is horizontal, and a rectangular harbor with a constant slope s¼tan y is located at x¼d. As evident oscillations usually occur in a long harbor, the harbor length L should be much longer than its width 2b. Meanwhile, as transverse oscillations are also investigated in our simulations, 2b must not be too small. An L¼20 m long and 2b¼5 m wide rectangular harbor with slope s¼1:30 located at d¼2.5 m is considered. The water depth at the backwall h0 is 0.083 m.

Table 1 Comparison of transverse oscillations between analytical solutions and numerical results. The subscript t represents theoretical solutions of Wang et al. (2011), while n represents numerical results. Mode (n, m)

k (m  1)

ft (Hz)

at

fn (Hz)

an

Error 9ft  fn9 /ft  100%

(1, (1, (1, (1, (1, (2, (2, (2, (2, (2, (2, (3, (3, (3, (3, (3, (3, (3, (4, (4, (4, (4, (6, (7, (8,

0.628 0.628 0.628 0.628 0.628 1.257 1.257 1.257 1.257 1.257 1.257 1.885 1.885 1.885 1.885 1.885 1.885 1.885 2.513 2.513 2.513 2.513 3.770 4.398 5.027

0.126 0.181 0.217 0.245 0.268 0.225 0.299 0.384 0.414 0.441 0.464 0.320 0.406 0.461 0.504 0.540 0.570 0.596 0.413 0.685 0.714 0.738 0.592 0.679 0.764

 1.040  2.691  4.097  5.413  6.677  1.943  3.861  6.852  8.209  9.513  10.778  2.807  4.909  6.573  8.082  9.499  10.853  12.159  3.645  12.043  13.378  14.672  5.258  6.036  6.796

0.125 0.180 0.214 0.244 0.272 0.223 0.296 0.378 0.409 0.436 0.458 0.317 0.409 0.458 0.504 0.534 0.562 0.586 0.409 0.674 0.699 0.720 0.586 0.671 0.757

 1.028  2.677  4.063  5.406  6.714  1.924  3.832  6.800  8.153  9.468  10.715  2.785  4.909  6.573  8.082  9.444  10.769  12.057  3.613  11.931  13.228  14.486  5.203  5.967  6.730

0.79 0.55 1.38 0.41 1.49 0.89 1.00 1.56 1.21 1.13 1.29 0.94 0.74 0.65 0.00 1.11 1.40 1.68 0.97 1.61 2.10 2.44 1.01 1.18 0.92

0) 1) 2) 3) 4) 0) 1) 3) 4) 5) 6) 0) 1) 2) 3) 4) 5) 6) 0) 5) 6) 7) 0) 0) 0)

G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

The computational domain is 2.5rxr30 m and 6.5r yr6.5 m as indicated in lowest panel of Fig. 3. The grid size is Dx¼ Dy¼0.05 m, and the time step is Dt¼0.01 s. Three sponge

Oscillations within the harbor of constant slope induced by seafloor movements are investigated numerically. The seafloor movement can be induced by local earthquakes, adjacent underwater landslides or dock failures. For simplicity, the moveable seafloor is described by hðx,y,tÞ ¼ sx þ zs expðbtÞHðx1 ,x2 ; y1 ,y2 Þ

x1 rx r x2 and y1 r yr y2

1,

when

0,

elsewhere

Amplitude/h0

Hðx1 ,x2 ; y1 ,y2 Þ ¼

ð3:2Þ

Obviously, the section of the moveable seafloor rises or falls in an exponential manner with a constant seafloor slope, and a positive value of zs represents a downthrow movement whereas a negative zs represents an upheaval. Certainly, the seafloor movement is characterized by the amplitude zs, the parameter b, the location of the up/down thrust, and the area over which it occurs. In the following, the effect of variations in the seafloor movement on oscillations is examined.

ζT/ζs

0.04

0.00 -0.4

-0.3

-0.2

-0.1

0.0 ζs/h0

0.1

0.2

0.3

0.4

Fig. 6. Amplitudes of several primary oscillation components at the corner (2.5,  2.5 m) induced by the movement of region A with various amplitudes, zs.

0.24 mode (1, 0) f = 0.125Hz

0.15

mode (2, 0) f = 0.223Hz

0.18

0.10

0.12

0.05

0.06

0.00

0.00

0.100

0.128 mode (2, 1) f = 0.296Hz

0.075 ζT/ζs

0.06

mode (1, 0) mode (1, 1) mode (2, 0) mode (2, 1) mode (3, 0)

0.02

0.20

mode (3, 0) f = 0.317Hz

0.096

0.050

0.064

0.025

0.032

0.000

0.000

0.048

0.056 mode (4, 0) f = 0.409Hz

0.036 ζT/ζs

f = 0.125Hz f = 0.180Hz f = 0.223Hz f = 0.296Hz f = 0.317Hz

0.08

ð3:1Þ

where the Heaviside step function is defined as (

2155

mode (6, 0) f = 0.586Hz

0.042

0.024

0.028

0.012

0.014

0.000

0.000

0.056 0.024 mode (7, 0) f = 0.671Hz

ζT/ζs

0.042

mode (8, 0) f = 0.757Hz

0.018

0.028

0.012

0.014

0.006 0.000

0.000 0.0

0.1

0.2

0.3

0.4

0.5 0.6 (x-d)/L

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5 0.6 (x-d)/L

0.7

0.8

0.9

1.0

Fig. 5. Comparisons of low-mode transverse oscillation profiles along the sidewall of the harbor induced by an impulsive exponential downthrow of A: analytic solutions (solid curve), numerical results (dotted curve).

G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

layers of 3 m width are placed around the harbor to absorb the energy radiated by the piston motion at the entrance. The model is run for at least 400 s for each simulation prior to encountering any stability problems. As our study focuses on small oscillations compared with water depth, energy transfers through nonlinear interactions in the quasisteady state between different components can be neglected and each oscillation component exists independently. The oscillation components are revealed by the amplitude spectra at the corner (2.5,  2.5). Whether the component is longitudinal or transverse is revealed from its spatial structure. For longitudinal oscillation, there is no variation across the harbor and the motions vary in parallel with the sidewall. However, the transverse oscillations are two dimensional. For the specific mode (n, m), there are n node lines along the backwall and m node lines along the sidewall. The spatial variation of each component along the backwall (x¼2.5,  2.5ryr2.5 m) is used to confirm it as longitudinal or transverse. If the motion is transverse, it is further used to identify the value of n corresponding to mode (n, m). The spatial variation of each component along the sidewall (2.5rxr22.5, y¼  2.5 m) is used to identify the mode of longitudinal oscillations or the value of m corresponding to mode (n, m) for transverse oscillations. The spectral analysis of the simulated free surface elevations is carried out with the time segment of 50–377.68 s (time interval Dt¼0.01 s, hence the total number of temporal points¼214), as the oscillations have reached a quasi-steady state at this period. 3.1. Specified movement amplitude, zs Oscillations induced by an impulsive downthrow of the region 2.5 mrx r3.5 m and 2.5 ryr  1.5 m (marked as A in Fig. 3a) with parameters b ¼8.0 and zs/h0 ¼ 0.2 are investigated. The time history of the free-surface elevation at the corner (2.5,  2.5) is plotted in Fig. 4. The water surface initially moves to a negative maximum displacement and remains stationary for a short time, and then rapidly reaches a positive maximum. Subsequently, it rapidly returns to the still-water level and remains stationary until 10 s when the energy reflected by the backwall and sidewalls returns. Oscillations reach a quasi-steady state at around t¼50 s. The oscillation frequency components are revealed by the amplitude spectrum shown in the right panel of Fig. 4, and their properties can be further identified from their spatial structure. There are small longitudinal oscillations in the harbor, which correspond to the components lower than 0.125 Hz in the amplitude spectrum. The wave fields are dominated by transverse oscillations with small number, m corresponding to mode (n, m), such as mode (1, 0) (i.e. 0.125 Hz), mode (2, 0) (i.e. 0.223 Hz), mode (2, 1) (i.e. 0.296 Hz), mode (3, 0) (i.e. 0.317 Hz), mode (4, 0) (i.e. 0.409 Hz), mode(6, 0) (i.e. 0.586 Hz) and mode (7, 0) (i.e.

0.671 Hz), etc. Table 1 compares most transverse oscillation modes between the theoretical solutions and the numerical results. As these transverse motions are types of low-mode standing edge waves (see the companion paper Wang et al., 2011) and their energy is trapped strictly near the inner end of the harbor, which is in the range of validity of the theory, very good agreement is found between the numerical results and the analytic solutions. Furthermore, Fig. 5 presents the amplitude variation of each component along the sidewall, and good agreement between the simulation result and the theoretical outcome for each oscillation mode is seen. Oscillations inside the harbor of constant slope induced by an impulsive exponential upthrust or downthrow of A are simulated by changing the value of zs with the same value of b. It is shown that the primary oscillation components are similar to each other; that is, there are small longitudinal oscillations but evident transverse oscillations with small number m corresponding to mode (n, m), with the only difference the magnitude of each oscillation component. Fig. 6 presents the relation for the amplitudes of several primary oscillation components versus zs/h0. Wave amplitudes are approximately linearly proportional to the parameter zs for small seafloor movement (9zs/h09o0.1), and they are essentially independent of the direction of seafloor movement (i.e. they are even functions for zs/h0 o0.1). The growth rate of wave amplitudes decreases with the absolute value of zs for larger seafloor movement, and some components’ amplitudes even decrease with 9zs9. A case in point is mode (2, 0); its amplitude decreases with zs when zs/h0 40.33 for the downthrows and zs/h0 o 0.3 for the upheavals. The amplitude of each oscillation component is asymmetric for oscillations induced by large

0.32

f = 0.125Hz f = 0.180Hz f = 0.223Hz f = 0.296Hz f = 0.317Hz

0.24 Amplitude/ζs

2156

mode (1, 0) mode (1, 1) mode (2, 0) mode (2, 1) mode (3, 0)

0.16

0.08

1

2

3

4

5 

6

7

8

9

Fig. 8. Amplitudes of several primary transverse oscillation components at the corner (2.5,  2.5 m) induced by downthrow movement at A as a function of different values of the parameter b.

0.18

0.757

0.671

0.586

0.06

0.409

-0.4

0.12 0.296 0.317

0.0

0.18

Amplitude/ζs

η/ζs

0.4

0.223

0.125

0.8

0.00

-0.8 0

25

50

75

100 Time (s)

125

150

175

200

0.0

0.1

0.2

0.3 0.4 0.5 Frequency (Hz)

0.6

0.7

0.8

Fig. 7. Time history of the free surface elevation and the amplitude spectrum at the corner (2.5,  2.5 m) generated by downthrow movement at A with b ¼1.0.

G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

seafloor movement with opposite direction. For example, mode (3, 0) increases monotonically with 9zs9 for downthrow movement (zs 40), but it decreases with 9zs9 for upheaval movement (zs o0) when zs/h0 o  0.2. Furthermore, the largest oscillation mode for significant downthrow is mode (3, 0); however it is mode (1, 0) for upheaval movement. 3.2. Effects of the seafloor’s vertical velocity This section examines the effects of the moveable seafloor’s velocity by changing the value of the parameter b while retaining the same maximum displacement zs/h0 ¼0.2. Fig. 7 presents the free surface time series and its corresponding amplitude spectrum at the corner (2.5,  2.5) induced by the movement of A with b ¼ 1.0. It is noted that oscillations induced by the seafloor shift with lower velocity are similar to the case with higher velocity, the primary difference being the amplitude for each oscillation mode. For very rapid seafloor movement (i.e. b ¼8.0), the initial water surface displacement is similar to the shape of

2157

the seafloor deformation and the maximum amplitude at the corner (2.5, 2.5) is approximately zs. For the slower seafloor movement (i.e. b ¼1.0), as the kinetic energy transferred from the seafloor to the water decreases and as there is more time for the wave energy to propagate from the source, the maximum amplitude of the deformed water surface should be and is smaller than zs. Due to gravitational forcing, the induced water surface displacement falls first and then rises. As this process repeats, waves are generated and radiated. These waves are then reflected repeatedly by the harbor’s boundaries and entrance. After a long modulation period, oscillations inside the harbor reach a quasisteady state. As the oscillations are small compared to the water depth, wave motions are trapped essentially inside the harbor, and the energy radiated through the entrance is very small. Hence the ‘‘damping’’ is very slow in the quasi-steady state. Fig. 8 depicts the influence of the seafloor’s velocity (i.e. b variation) on several primary oscillation modes induced by the movement of A at the corner (2.5,  2.5). The magnitude of each mode increases with the parameter b, though their growth rates

0.36

-1.2 100

125

150

175

200

0.504

0.671

0.586

0.409

0.317

0.0

0.1

0.2

0.0

0.1

0.2

-0.6

0.27 0.18 0.09

0.7

0.8

0.00 175

200

0.6

0.2 0.0 -0.2 -0.4

0.039 0.026 0.013

0.18

Amplitude/ζs

0.4

0.4

0.214 0.244

0.052

0.3

0.5

0.6

0.7

0.8

0.699 0.72

150

0.562 0.586

125

0.623

100

0.504 0.534

75

0.409

50

0.378

25

0.436 0.458

0

0.272 0.296

-1.2

0.6

0.586

0.0

0.5

0.296

Amplitude/ζs

0.6

0.4

0.223

0.36

0.3

0.757

75

0.504

50

0.409

25

1.2 η/ζs

0.09 0.00

0

η/ζs

0.18

0.223

-0.6

0.125

0.0

0.27

0.027 0.067

η/ζs

0.6

0.180

Amplitude/ζs

1.2

0.000 0

25

50

75

100 Time (s)

125

150

175

200

0.0

0.1

0.2

0.3 0.4 0.5 Frequency (Hz)

0.6

0.7

0.8

Fig. 9. Time histories of the free surface elevations and amplitude spectra at the corner (2.5,  2.5 m) generated by impulsive exponential downthrows at B (the upper panel), C (the middle panel) and D (the lower panel).

Fig. 10. Transverse oscillation modes of the harbor.

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G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

correspond to different modes, which have been identified from their spatial structure. Distinct oscillation forms induced by B and C can be explained from the relation between the moveable seafloor’s position paralleling the backwall and the antinode line running along the sidewall. There are n þ1 antinode lines and n node lines along the backwall for the transverse oscillation mode (n, m). When the seafloor is located at one of the n þ1 antinode lines, evidently various oscillations related to n are induced. As the center line of C is superposed with one of the antinode lines corresponding to n¼2, 4, 6 and 8 (see Fig. 10), the oscillations are characterized by mode (2, 0), mode (2, 1), mode (2, 4), mode (2, 8), mode (6, 0), mode (8, 0), etc. As one of the antinode lines corresponding to n¼3, 4 and 7 is located in region B, mode (3, 0), mode (4, 0), mode (3, 6) and mode (7, 0) are induced. In contrast, when the seafloor is located at one of the n node lines along the backwall corresponding to mode (n, m), oscillation modes related with this particular n are very small. One such case is mode (3, 0), which is seen clearly in the numerical experiment with B’s movement, but is very small in the case where C moves, as one of the n¼3 node lines is superposed with the center line of region C. When the region that is used to excite the oscillations is neither located at an antinode line nor a node line, transverse oscillation modes related with the particular n are induced, but their magnitudes are smaller than the case of the moveable seafloor being located at one of the antinode lines. For example, consider mode (1, 0) (i.e. 0.125 Hz); as none of the node lines or

are different, and the growth rates asymptotically approach constants for very rapid seafloor movement.

3.3. Changing the position of the seafloor movement Changing the seafloor location of the downthrow, but using the same velocity (i.e. constant b ¼8.0) and movement amplitude (i.e. constant zs/h0 ¼0.2), should generate different oscillations. The time histories of the free surface elevation and their amplitude spectra at the corner (2.5,  2.5) respectively, induced by the impulsive exponential downthrow of the regions 2.5 rxr3.5 m and  1.5ryr  0.5 m (marked as B in Fig. 3a) and 2.5rx r3.5 and 0.5 ryr0.5 m (marked as C in Fig. 3a), are presented in Fig. 9 in the upper two sets of panels. Their maximum oscillations in the quasi-steady state approximate those of A; however, the amplitude spectra reveal differences as discussed next. Though these motions are dominated by transverse oscillations with small number of m corresponding to mode (n, m), n differs. Oscillations induced by B are characterized by mode (3, 0) (i.e. 0.317 Hz), mode (4, 0) (i.e. 0.409 Hz), mode (3, 6) (i.e. 0.586 Hz) and mode (7, 0) (i.e. 0.671 Hz), while oscillations induced by C are dominated by mode (2, 0) (i.e. 0.223 Hz), mode (2, 1) (i.e. 0.296 Hz), mode (2, 4) (i.e. 0.409 Hz), mode (2, 8) (i.e. 0.504 Hz), mode (6, 0) (i.e. 0.586 Hz) and mode (8, 0) (i.e. 0.757 Hz). It should be noted that, although the components with 0.409 and 0.586 Hz are both present in cases B and C, they

0.04

0.04 mode (1, 2) f= 0.214Hz

ζT/ζs

0.03

mode (1, 3) f = 0.244Hz

0.03

0.02

0.02

0.01

0.01

0.00

0.00 0.04

0.016

mode (2, 3) f = 0.378Hz

ζT/ζs

0.012

0.02

0.008

0.01

0.004

0.00

0.000

0.030

0.024 mode (2, 5) f = 0.436Hz

ζT/ζs

0.018

mode (2, 6) f= 0.458Hz

0.024 0.018

0.012

0.012 0.006

0.006

0.000

0.000

0.032

0.032 mode (3, 5) f = 0.562Hz

0.024 ζT/ζs

mode (2, 4) f = 0.409Hz

0.03

mode (3, 6) f = 0.586Hz

0.024

0.016

0.016

0.008

0.008

0.000

0.000 0.0

0.1

0.2

0.3

0.4

0.5 0.6 (x-d)/L

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5 0.6 (x-d)/L

0.7

0.8

0.9

1.0

Fig. 11. Comparisons of large m transverse oscillation profiles along the sidewall of the harbor induced by an impulsive exponential downthrow of D: analytic solutions (solid curve), numerical results (dotted curve).

G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

the antinode lines is located within B, its amplitude induced by B is smaller than that induced by A, but much larger than that induced by C. To investigate the differences between oscillations induced by the same moveable seafloor but with various offshore locations, the seafloor location of the downthrow region was shifted along the sidewall y¼  2.5 m. Simulations were conducted in one meter increments along y¼ 2.5 m for its entire length, with the values of the other parameters retained, that is b ¼8.0 and zs/h0 ¼ 0.2. However, results are discussed in detail only for D (see Fig. 3a, 12.5rx r13.5 m and  2.5ryr  1.5 m). Oscillations induced by various sidewall positions (e.g. D) of the vertical displacement regions are distinct from those induced by different backwall positions (i.e. A, B, and C). The lowest panel of Fig. 9 presents the free-surface fluctuation and its amplitude spectrum at (2.5,  2.5) induced by vertical displacement at D. Compared to oscillations induced by the seafloor movement at

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positions along the backwall (such as A, B, and C), the wave energy is distributed across more components. Especially evident are the modes with higher values of m corresponding to mode (n, m); these include mode (1, 3) (0.244 Hz), mode (2, 4) (0.409 Hz), mode (2, 6) (0.458 Hz), mode (3, 5) (0.562 Hz), mode (3, 6) (0.586 Hz), etc. The transverse eigenfrequencies can also be well predicted by the theory presented in Wang et al. (2011) (see Table 1). Fig. 11 compares analytic solutions with numerical results for transverse oscillation profiles along the sidewall with large m, and good agreement is obtained. As there are more transverse oscillation modes apparent, and these oscillations distribute more energy in the offshore region, the maximum fluctuation amplitude in the quasisteady state is much lower than those induced by A, B, and C. Fig. 12 compares the offshore profiles of the transverse mode (1, 3) (i.e. 0.244 Hz) induced by the same vertical movement with different offshore locations along the sidewall (y ¼  2.5 m). It is seen that the transverse oscillation mode is related closely to the seafloor’s location. There are m þ1 antinode lines and m node lines along the sidewall for the transverse oscillation mode (n, m). When the moveable seafloor is located at one of the m þ1 antinode lines, evidently the oscillation mode will be excited; whereas when it is located at one of the m node lines, the mode will be very small. 3.4. Longitudinal oscillations As shown in previous sections, there are not only transverse oscillations, but also longitudinal oscillations induced by seafloor displacements. Certainly, both responses will be excited additionally by enlarging the area of the moveable seafloor as there is more exchange of energy and momentum from the solid bottom to the water column. However, the longitudinal oscillations become dominant in oscillations induced by larger seafloor movements. To Table 2 Comparisons of longitudinal oscillations between theoretical solutions and numerical results. Again the subscript t represents analytical solutions of Wang et al. (2011), while n represents numerical results.

Fig. 12. Offshore profiles for the transverse oscillation mode (1, 3) (i.e. 0.244 Hz) induced by the seafloor movement in different offshore locations, where xc ¼ x1 þ1/2 is the offshore center of the seafloor (x1 rxrx1 þ 1,  2.5ryr  1.5 m).

Mode

ft (Hz)

fn (Hz)

Error |ft  fn| /ft  100%

1st 2nd 3rd 4th

0.025 0.064 0.106 0.149

0.027 0.067 0.110 0.153

8.000 4.690 3.770 2.680

1.0

0.12

0.09 Amplitude/ζs

η/ζs

0.5

0.027

0.0

-0.5

0.06

0.067 0.110

0.03

0.153 -1.0 0

50

100

150

200 Time (s)

250

300

350

400

0.00 0.00

0.05

0.10 0.15 0.20 Frequency (Hz)

0.25 0.30

Fig. 13. Time history of the free surface elevation and its amplitude spectrum at the corner (2.5,  2.5 m) generated by the movement of the region 2.5 r xr 3.5 m and  2.5r yr 2.5 m.

G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

investigate in particular these longitudinal motions induced by increased area and vertical movement, and compare them with the analytic solutions, oscillations are simulated for the movement of the region 2.5rxr3.5 and 2.5ryr2.5 m with parameters b ¼8.0 and zs/h0 ¼0.2. The time history of the free-surface elevation at the corner (2.5,  2.5) is plotted in the left panel of Fig. 13. The result shows a quite different pattern of free surface movement than the previous cases. The amplitude spectrum shown in the right panel of Fig. 13 reveals that these oscillation modes correspond to lower frequencies. The oscillation components corresponding to frequencies 0.027, 0.067, 0.110, and 0.153 Hz are confirmed as longitudinal oscillations from their spatial structures. Table 2 shows that these resonant frequencies are very close to those predicted by the analytical solution of Wang et al. (2011). Slight differences between them may be expected from two reasons: firstly, the assumption that the width of the harbor is much smaller than the wavelength is not strictly satisfied in the simulations; secondly, the analytic solution for longitudinal oscillations is obtained based on linear shallow-water equations while the numerical model is based on the Boussinesq equations. Although there is no variation across the harbor for the seafloor transferring its kinetic energy to the water, slight transverse oscillations can be observed if an animation is created (not shown) from the simulation data. These motions correspond to energy between 0.17 and 0.25 Hz in the spectrum of Fig. 13. In Fig. 14 the spatial structures for the first four lowest longitudinal oscillation modes (from the inner end toward the entrance) are compared for the analytic solutions and the numerical results. Very good agreement is obtained; in fact even the nodes and antinodes in each mode are well captured. Simulations are conducted with a different value of zs (while retaining the value of b) for oscillations inside the harbor of constant slope induced by an impulsive upthrust or downthrow of the same region (2.5rx r3.5;  2.5ryr2.5 m). It is shown that the primary oscillation components are mainly longitudinal, while transverse oscillations are very small. Fig. 15 presents the relation for the amplitudes of several primary oscillation components versus zs/h0. Wave amplitudes are approximately linearly proportional to the parameter zs even for rather large movements,

though their growth rates are not the same. In addition, wave amplitudes are essentially independent of the direction of seafloor movement being approximately symmetric. Effects of the moveable seafloor’s velocity on longitudinal oscillations are also examined (by changing the value of parameter b while retaining the same maximum displacement zs/h0 ¼0.2). It is noted that oscillations induced by the seafloor shift with different velocities are similar, the only difference being the amplitude for each oscillation mode. For rapid seafloor movements, as the kinetic energy transferred from the seafloor to the water is increased over those for slower seafloor movements, wave amplitudes induced by them are larger. To further investigate the effect of the moveable seafloor’s horizontal position on longitudinal oscillations, we shift the seafloor location along the sidewall keeping the other parameters unchanged (i.e. b ¼8.0 and zs/h0 ¼0.2). It is found that the longitudinal oscillations are again the first several modes, but that the magnitude of each mode is altered. There are n antinode lines and n node lines along the sidewall for the nth longitudinal oscillation mode. When the seafloor overlaps one of the antinode lines, this mode is excited;

0.050

Amplitude/h0

2160

f = 0.027Hz f = 0.067Hz f = 0.110Hz f = 0.153Hz

the 1st mode the 2nd mode the 3rd mode the 4th mode

0.025

0.000 -0.4

-0.3

-0.2

-0.1

0.0 ζs/h0

0.1

0.2

0.3

0.4

Fig. 15. Amplitudes of the first four lowest longitudinal oscillation modes at the corner (2.5,  2.5 m) induced by the movement of the region (2.5r xr 3.5;  2.5 r yr2.5 m) with various amplitudes, zs.

0.048

0.12 1st mode f = 0.027Hz

2nd mode f = 0.067Hz

0.036

ζL/ζs

0.08 0.024 0.04 0.012 0.00

0.000 0.012

0.03 3rd mode f = 0.110Hz

4th mode f = 0.153Hz 0.008

0.01

0.004

ζL/ζs

0.02

0.000

0.00 0.0

0.1

0.2

0.3

0.4

0.5 0.6 (x-d)/L

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5 0.6 (x-d)/L

0.7

0.8

0.9

Fig. 14. Comparison of the first four longitudinal oscillation profiles along the sidewall: analytic solutions (solid curves), numerical results (dotted curves).

1.0

G. Wang et al. / Ocean Engineering 38 (2011) 2151–2161

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inner region of the harbor, and seafloor movements away from the backwall induced transverse oscillations with higher m. Transverse oscillations associated with m are generated when the moveable seafloor is located at one of the m antinode lines along the sidewall. Fluctuation patterns and primary oscillation components are not affected by the seafloor’s velocity or by its maximum displacement, though the amplitude of each mode changes with them. The numerical result of each transverse eigenfrequency is very close to the theoretical solution, and the spatial structure of each mode is also well captured by the theory. Evident longitudinal oscillations are generated by large-scale seafloor movements. These oscillations are mainly the first several lowest resonant modes and the longitudinal eigenfrequencies and the spatial structures are well captured by the theory from Wang et al. (2011). Furthermore, these longitudinal oscillations are sensitive to the horizontal location of the moveable seafloor. When the seafloor overlaps one of the antinode lines corresponding to a particular mode, this mode is evidently excited; when the seafloor is located at one of the node lines, this longitudinal mode is very small.

Fig. 16. Oscillation profiles of the third longitudinal mode (i.e. 0.110 Hz) induced by vertical movement of the seafloor with area 1 m  5 m in different offshore locations, where xc ¼x1 þ1/2 is the offshore center of the seafloor (x1 rx r x1 þ 1;  2.5r yr 2.5 m).

when the seafloor is located at one of the node lines, this longitudinal mode is very small. Fig. 16 demonstrates vividly this influence for the third longitudinal mode.

4. Conclusions A numerical model, which can simulate wave generation by seafloor movement, was developed to simulate oscillations inside a harbor of constant slope, and the numerical results were used to examine the fidelity of the analytic solutions presented by Wang et al. (2011). Small-scale seafloor movements usually excite small longitudinal oscillations but obvious transverse oscillations. These transverse oscillations are sensitive to the location of the moveable seafloor. The value of the integer n corresponding to mode (n, m) is related strongly to the seafloor’s longshore position. When the seafloor is located at one of the n þ1 antinode lines, the transverse oscillations modes (n, m) are induced. On the other hand, if the seafloor is located at one of the n node lines, the transverse oscillations related with n are very small. The value of the integer m corresponding to mode (n, m) is influenced highly by the seafloor’s offshore position. Usually, oscillation modes with lower m were evidently excited by the seafloor movement in the

Acknowledgments This research was supported financially by the National Natural Science Foundation (50921001 and 51009024) and the Specialized Research Fund for the Doctoral Program (20100041110005). References Ataie-Ashtiani, B., Jilani, A.N., 2007. A higher-order Boussinesq-type model with moving bottom boundary: applications to submarine landslide tsunami waves. Int. J. Numer. Methods Fluids 53 (6), 1019–1048. Dong, G., Wang, G., Ma, X., Ma, Y., 2010a. Harbor resonance induced by subaerial landslide-generated impact waves. Ocean Eng. 37 (10), 927–934. Dong, G., Wang, G., Ma, X., Ma, Y., 2010b. Numerical study of transient nonlinear harbor resonance. Sci. China-Technol. Sci. 53 (2), 558–565. Fuhrman, D.R., Madsen, P.A., 2009. Tsunami generation, propagation, and run-up with a high-order Boussinesq model. Coastal Eng. 56 (7), 747–758. Hammack, J.L., 1973. A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60 (4), 769–799. Kulikov, E.A., Rabinovich, A.B., Thomson, R.E., Bornhold, B.D., 1996. The landslide Tsunami of November 3, 1994, Skagway harbor, Alaska. J. Geophys. Res.Oceans 101 (C3), 6609–6615. Lynett, P., Liu, P.L.F., 2002. A numerical study of submarine-landslide-generated waves and run-up. In: Proceedings of the Royal Society of London Series A—Mathematical Physical and Engineering Sciences, vol. 458 (2028), pp. 2885–2910. Nwogu, O., 1993. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw., Port, Coastal Ocean Eng. 119 (6), 618–638. Wang, G., Dong, G., Perlin, M., Ma, X., Ma, Y., 2011. An analytic investigation of oscillations within a harbor of constant slope. Ocean Eng. 38, 479–486. Wei, G., Kirby, J.T., 1995. Time-dependent numerical code for extended Boussinesq equations. J. Waterw. Port Coastal Ocean Eng.-ASCE 121 (5), 251–261.