Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
www.elsevier.com/locate/soildyn
A note on the effects of nonuniform spreading velocity of submarine
slumps and slides on the near-®eld tsunami amplitudes
M.D. Trifunac*, A. Hayir 1, M.I. Todorovska
Department of Civil and Environmental Engineering, University of Southern California, KAP 210 Vermont Avenue, Los Angeles, CA 90089-2531, USA
Accepted 9 February 2002
Abstract
The effects of variable speeds of spreading of submarine slides and slumps on near-®eld tsunami amplitudes are illustrated. It is shown that
kinematic models of submarine
pslides and slumps must consider time variations in the spreading velocities, when these velocities are less
than about 2cT, where cT ˆ gh is the long period tsunami velocity in ocean of constant depth h. For average spreading velocities greater
than ,2cT, kinematic models with assumed constant spreading velocities provide good approximation for the tsunami amplitudes above the
source. q 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Tsunami modeling; Tsunami with variable speed of spreading; Slides; Slumps
1. Introduction
Submarine slides and slumps can generate large tsunami,
but usually result in more localized effects than the tsunami
caused by earthquakes [9]. Nevertheless these localized
waves can produce large run-up along the coast, especially
in narrow bays and fjords [2,4±6].
Previous studies of tsunami generation have considered
impulsive and spreading kinematic models of moving
impermeable ocean bottom. For `fast' movement (1±
3 km/s), caused by most earthquakes, the interaction of
water waves and of the tectonic motion may be neglected,
and it can be assumed that the initial surface elevation of the
water is the same as the displacement of the ocean bottom. It
has also been assumed that the movement of the ocean
bottom can be approximated by uplift or by depression
spreading with constant velocity in one direction [7,8,10],
and in two directions [11,12]. For tsunami generated by
submarine slides and slumps, it becomes necessary to
consider coupling of the slide motion and of the water
waves, because the slide duration is `long' (spreading velocities are `low', e.g. 0.001±0.10 km/s; [9]), relative to the
duration of typical earthquake sources.
Jiang and Le Blond [3] investigated coupling of a
submarine slide and the surface water waves it generates.
They found that the two major parameters that determine the
* Corresponding author. Tel.: 11-213-740-0570; fax: 11-213-744-1426.
E-mail address: [email protected] (M.D. Trifunac).
1
On leave from Istanbul Technical University, Istanbul, Turkey.
interaction between the slide and the water waves are the
density of sliding material and the depth of initiation of
the slide. For denser slides in shallow water, the interactions
are small and the amplitudes of the generated water waves
are large. Jiang and Le Blond [3] also show typical time
dependent changes of the Froude number (Fr ˆ cR =…gh†1=2 ;
where cR is the frontal speed of the slide, h is the depth of
ocean and g is acceleration due to gravity) during the slumping and sliding process. It ®rst increases to reach a maximum as the slide accelerates, and then gradually decreases
as the slide comes to rest.
The purpose of this paper is to illustrate the nature
and the extent of variations in the tsunami waveforms
caused by simple time variations of the frontal velocity
of spreading, cR, for two-dimensional kinematic models
of slides and slumps, and to compare the results with
those for the slides spreading with constant velocity [10]
cR. It will be shown how the p
changes
of cR in time act to

reduce focusing, when cR , gh: The problem is solved
by transform method (Laplace in time and Fourier in
space), with the forward and inverse Laplace transforms
computed analytically, and the Fourier transforms
computed by Fast Fourier Transform. Realistic simulation of water waves generated by submarine slumps and
slides, considering all relevant parameters and physical
properties, can be obtained only by numerical modeling.
The usefulness of analytical and semi-analytical solutions for slides and slumps spreading with constant
and with variable velocities lies in their ability to
display the physical nature of the problem, directly in
0267-7261/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0267-726 1(02)00016-7
168
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
Fig. 1. Cross-section of Model 1.B from Trifunac et al. [10], representing a slide or a slump spreading downhill and uphill with constant velocities cR and cL ; at
time t ˆ t p (the time when the spreading stops). Distance LR ˆ cR tp is the characteristic length of the slide.
terms of the model parameters, and in providing a basis
for independent veri®cation of approximate numerical
algorithms.
2. Mathematical model
We consider a kinematic tsunami source model which is a
generalization of Model 1.B in Trifunac et al. [10], shown in
Fig. 1. It represents a mass movement triggered at x ˆ 0 and
at time t ˆ 0; and spreading bilaterally. The accumulation
zone spreads downhill (in the positive x-direction) with
frontal velocity cR ; and the zone of depletion spreads downhill with velocity cC and uphill with velocity cL (this model
is meaningful if cR . cC ). The ®nal uplift of the accumulation zone is z0 and the ®nal depression of the depletion zone
is z1 ; both assumed to be constant. It is assumed that the
displacement of a point on the slide occurs instantaneously
once the disturbance has reached that point. It is also
assumed that the mass of the slide is conserved (A1 ˆ A2
in Fig. 1) which implies
12b
z1 ˆ
z
…1†
g1b 0
where b ˆ cC =cR and g ˆ cL =cR : The slide stops spreading at
time t ˆ tp : In the ®nal con®guration (Fig. 1), the length of the
accumulation zone is LR ˆ cR tp when cR is constant, used as
characteristic length of the model. The width of the slide is W.
The uplift of the ocean ¯oor, z…x; y; t†; for t # tp and
Fig. 2. Variable velocity of spreading c(t) and its piecewise constant
approximation over N time intervals.
assuming cR ; cC and cL are constant, is
8
2z1 ; 2cL t # x , 0 and 2 W=2 # y # W=2
>
>
>
>
>
< 2z1 ; 0 # x , cC t and 2 W=2 # y # W=2
z…x; y; t† ˆ
>z ;
>
cC t # x # cR t and 2 W=2 # y # W=2
0
>
>
>
:
0;
otherwise
…2†
and for t $ tp is same as for t ˆ tp : This source process is
equivalent to one that is a superposition of three simple
spreading uplifts z…I† …x; y; t†; z…II† …x; y; t† and z…III† …x; y; t†; as
those in Todorovska and Trifunac [7], subject to coordinate
transformation. The ®rst uplift has amplitude z0 and starts
spreading at x ˆ 0 in the positive x-direction with velocity
cR ; the second one has amplitude 2…z0 1 z1 † and starts
spreading also at x ˆ 0 in the positive x-direction, but
with velocity cC , cR ; and the third one has amplitude
2z1 and starts spreading also at x ˆ 0; but in the negative
x-direction and with velocity cL : This is expressed by
z…x; y; t† ˆ z…I† …x; y; t† 1 z…II† …x; y; t† 1 z…III† …x; y; t†
z…I† ˆ z0 H…t 2 x=cR †;
0 # x # cR tp ˆ LR and 2 W=2 # y # W=2
z…II† ˆ 2…z0 1 z1 †H…t 2 x=cC †;
0 # x # cC tp and 2 W=2 # y # W=2
…3†
…4a†
…4b†
Fig. 3. Variable c(t) adopted for the examples in this paper (ramp up for
0 # t # t1 ; constant for t1 # t # t2 ; and ramp down for t2 # t # tp ).
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
169
Fig. 4. Test of convergence of h=z0 for N ˆ 15; 35 and 39 segments.
Fig. 5. Normalized tsunami waveforms h=z0 along y ˆ 0; at time t ˆ t p (the time when the spreading of the slide stops), for cL =cR ˆ 0:3 and cC =cR ˆ 0:1 and for
eight values of cR =cT between 0.5 and 20. The ocean depth is h ˆ 2 km; and the slide has width W ˆ 50 km and characteristic length LR ˆ 50 km:
170
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
Fig. 6. Same as Fig. 5 but for cL =cR ˆ 0:3 and cC =cR ˆ 0:
Fig. 7. Same as Fig. 5 but for cL =cR ˆ 2 and cC =cR ˆ 0:5:
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
171
Fig. 8. Left: an enlarged area from Fig. 5 for cR =cT ; i.e. c=c
T ˆ 0:7; 0.8 and 0.9 showing peak amplitudes of the leading tsunami propagating in the positive xdirection. Right: comparison of the peak amplitudes for the Cases 1, 2 and 3 with those for a simple dislocation with constant velocity of spreading from
Todorovska and Trifunac [7] with W=L . 1 and W=L . 0:5:
Fig. 9. Normalized tsunami waveforms h=z0 along y ˆ 0; at selected instants of time t # tp and for cL =cR ˆ 1 and cR =cT ˆ 1: The ocean depth is h ˆ 2 km; and
the slide has width W ˆ 50 km and characteristic length LR ˆ 50 km: The solid lines show results for the constant velocity Model 1.B, and the dashed lines
show results for Cases 1, 2 and 3 of the variable velocity model.
172
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
Fig. 10. Same as Fig. 9 but for selected instants of time t $ tp :
z…III† ˆ 2z1 H…t 2 x 0 =cL †;
x 0 ˆ 2x; 0 # x 0 # cL tp and 2 W=2 # y # W=2
…4c†
The generalization of Model 1.B considered in this paper
is that the velocities of spreading vary with time. In the
numerical calculations, the true variation is discretized
over N time intervals, with constant velocity approximation within each interval, as shown in Fig. 2 (c…t† here
refers to velocity of spreading in general). In the examples shown in this paper, c…t†; linearly increasing from
zero to c for 0 # t # t1 ; constant for t1 # t # t2 and linearly decreasing to zero for t1 # t # tp ; is assumed, as
shown in Fig. 3. The average spreading velocity c for
this case is
p
1 Zt
1
c ˆ p
c…t†dt ˆ c…1 1 t2 =tp 2 t1 =tp †
t 0
2
…5†
The variability of the velocity of spreading can also be
expressed as function of x, where
x…t† ˆ
Zt
0
c…t†dt
…6†
In our selection of the spreading velocities, we were
guided by the time dependent changes of Froude number
described by Jiang and Le Blond [3] in their numerical
experiments on slide and slump motion. To keep the
number of parameters small, only the time function for
cR is speci®ed, and cC and cL are expressed as factors of
cR (hence having same type of time dependence).
A linearized shallow water solution (for water depth
much smaller than the tsunami wavelength) for the tsunami
generation and propagation is obtained by the Fourier±
Laplace transform [1,7] de®ned by
~ s† ˆ
f…k;
Z1
21
eik2 y
Z1
21
eik1 x
Z1
0
e2st f …x; y; t†dt dx dy …7†
In the transform space, the water elevation is
~ s† ˆ
h …k;
~ s†
s2 z …k;
1
2
2
s 1 v cosh kh
…8†
where
v2 ˆ gk tanh kh
…9†
and v is circular frequency of the wave motion. For constant
spreading velocities, the forward and inverse Laplace transforms can be computed analytically or from tables.
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
173
Fig. 11. Normalized tsunami waveforms h=z0 along y ˆ 0; at selected instants of time t # tp and for cL =cR ˆ 0:3 and cR =cT ˆ 1: The ocean depth is h ˆ 2 km;
and the slide has width W ˆ 50 km and characteristic length LR ˆ 50 km: The solid lines show results for the constant velocity Model 1.B, and the dashed lines
show results for Cases 1, 2 and 3 of the variable velocity model.
The Laplace±Fourier transform of z…x; y; t† in Eq. (3) is
~ s† ˆ
z …k;
ZW=2
2 W=2
2
2
Z LC
0
eik2 y dy
eik1 x
Z0
2 LL
e
ZLR
0
Z1
ik1 x
x=cC …x†
eik1 x
Z1
x=cR …x†
z0 e2st dt dx
Zt
…z0 1 z1 †e2st dt dx
Z1
2 x=cL …x†
z1 e
2st
dt dx
cz …t†dt;
z ˆ R; C; L
(10)
xz;n ˆ
…11†
For piecewise constant approximation of the velocities of
spreading in N intervals, the integrals in x in Eq. (10) can be
broken over N segments, and
~ s† ˆ
z …k;
ZW=2
2 W=2
eik2 y dy
N ZxR;n
X
nˆ1
xR;n 2 1
eik1 x
Z1
x=cR;n
z0 e2st dt dx
xC;n 2 1
e
Z2 xL;n 2 1
2 xL;n
ik1 x
!
Z1
x=cC;n
e
ik1 x
…z0 1 z1 †e
2st
dt dx
!
Z1
2 x=cL;n
z1 e
2st
dt dxŠ
(12)
where
p
0
2
where
Lz ˆ
2
ZxC;n
n
X
mˆ1
cz;m …tm 2 tm21 †;
z ˆ R; C; L
…13†
Substitution in Eq. (3) of the relationship between z1 and z0
from Eq. (1) and integration gives
~ s† ˆ z0 2 sin k2 W=2 KR 2 1 1 g bKC 2 1 2 b gKL
z …k;
k2
g1b
g1b
…14†
where
Kz ˆ
N
X
‰e2…s2ik1 cz;n †xz;n21 =cz;n 2 e2…s2ik1 cz;n †xz;n =cz;n Šcz;n
;
s…s 2 ik1 cz;n †
nˆ1
…15a†
z ˆ R; C
174
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
Fig. 12. Same as Fig. 11 but for selected instants of time t $ tp :
!
!#
xz;n21
xz;n21
2 iv sin v t 2
2k1 cz;n cos v t 2
cz;n
cz;n
"
!
xz;n21
ik1 xz;n
£H t2
2e
k1 cz;n ek1 …tcz;n 2xz;n †
cz;n
!
xz;n
2k1 cz;n cos v t 2
cz;n
!#
!)
xz;n
xz;n
2iv sin v t 2
H t2
; Z ˆ R; C
cz;n
cz;n
and
KL ˆ
N
X
‰e2…s1ik1 cL;n †xL;n21 =cL;n 2 e2…s1ik1 cz;n †xL;n =cL;n ŠcL;n
…15b†
s…s 1 ik1 cL;n †
nˆ1
~ s† in Eq. (8) gives
Substitution of z …k;
~ s† ˆ
h …k;
s2
1
2 sin k2 W=2
z0
2
2
cosh
kh
k2
s 1v
#
"
11g
12b
£ KR 2
K 2
K
g1b C
g1b L
…16†
~ s†; h~ …k;
~ t†; can be
The inverse Laplace transform of h …k;
evaluated analytically, as follows
(
1
2 sin k2 W=2 21
s2
~
h~ …k; t† ˆ
z0
L
cosh kh
k2
s2 1 v2
!)
11g
12b
…17†
KR 2
K 2
K
g1b C
g1b L
with
(
L21
ˆ
(18a)
and
21
L
ˆ
s2
KL
s2 1 v2
N
X
nˆ1
s2
Kz
s2 1 v2
N
X
nˆ1
)
"
(
icz;n
ik1 xz;n21
e
k1 cz;n ek1 …tcz;n 2xz;n21 †
v2 2 k12 c2z;n
icL;n
xL;n21
2ik1 xL;n21
e
ik1 cL;n cos v t 2
cL;n
v2 2 k12 c2L;n
xL;n21
xL;n21
1v sin v t 2
2 exp ik1 cL;n t 2
cL;n
cL;n
x
x
H t 2 L;n21 1 e2ik1 xL;n ik1 cL;n cos v t 2 L;n
cL;n
cL;n
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
175
Fig. 13. Comparison of tsunami waveforms h=z0 for the constant velocity Model 1.B [10] and for Cases 1, 2 and 3 of the variable velocity model, all computed
at time t ˆ tp ; and for h ˆ 2 km; LR ˆ W ˆ 50 km; cR =cT ˆ 1; cL =cR ˆ 0:3 and cC =cR ˆ 0:
x
1v sin v t 2 L;n
cL;n
!)
xL;n
£H t2
cL;n
!
(
2 exp ik1 cL;n
x
t 2 L;n
cL;n
!)#
(18b)
~ t† by
Finally inverse Fourier transform is performed on h~ …k;
Fast Fourier Transform to obtain h…x; y; t†:
1 Z1 2ik2 y 1 Z1 2ik1 x ~
h…x; y; t† ˆ
e
e
h~ …k; t†dk1 dk2
2p 2 1
2p 2 1
…19†
176
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
Fig. 14. Same as Fig. 13, but for cL =cR ˆ 2 and cC =cR ˆ 0:5:
~ t† is
The long wavelength limit of h~ …k;
~ t† ˆ 0
lim h~ …k;
k!0
3. Numerical results
…20†
The other limits are almost the same as for the Model B
[10].
We illustrate results for three cases of trapezoidal
cR …t†; as shown in Fig. 3, de®ned by parameters
a 1, a 2 and a 3, which are equal to the duration of
each segment normalized by the total duration of the
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
Fig. 15. Same as Fig. 13, but for cR =cT ˆ 0:5 cL =cR ˆ 2 and cC =cR ˆ 0:5:
177
178
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
Fig. 16. Normalized positive (top) and negative (bottom) tsunami peak
amplitudes, hR;max =z0 and hL;min =z0 ; versus cT =cR for rectangular slides
with areas 10 £ 10, 50 £ 50 and 100 £ 100 km 2, for cL =cR ˆ 0:3 and
cC =cR ˆ 0:1; for ocean depth h ˆ 2 km; and for Model 1.B in Trifunac et
al. [10] and Cases 1, 2 and 3 of the variable velocity model.
slide
a1 ˆ t1 =tp
a2 ˆ …t2 2 t1 †=tp
a3 ˆ …tp 2 t2 †=tp
…21†
The three cases are:
Case1 :
a1 ˆ 0:6;
a2 ˆ 0:35;
a3 ˆ 0:05
Case2 :
a1 ˆ 0:5;
a2 ˆ 0:45;
a3 ˆ 0:05
Case3 :
a1 ˆ 0:4;
a2 ˆ 0:55;
a3 ˆ 0:05:
Fig. 4 shows the effects of the discretization on the
accuracy of the normalized wave amplitudes h=z0 : The
different curves correspond to N ˆ 15; 35 and 39 intervals. It is seen that the results for N ˆ 35 and 39 are
essentially the same, and therefore we adopt N ˆ 39 for
all subsequent examples in this paper.
Figs. 5±12 show variations of h=z0 with the model parameters. They show h=z0 versus x and along y ˆ 0; for ocean
depth h ˆ 2 km …cT ˆ 0:14 km=s†; W ˆ 50 km and LR ˆ
50 km: The solid lines correspond to Model 1.B in Trifunac
et al. [10], with constant velocities of spreading, equal to the
average computed by Eq. (5), and the dashed lines correspond to Cases 1, 2 and 3.
Figs. 5±7 show how h=z0 changes with respect to cR =cT ;
They show results at t ˆ tp for chosen cL =cR ;
when cR ˆ c:
and cC =cR and for eight values of cR =cT between 0.5 and 20
(cL =cR ˆ 0:3 and cC =cR ˆ 0:1 in Fig. 5, cL =cR ˆ 0:3 and
Fig. 17. Comparison of selected peak tsunami amplitudes from Fig. 16
(plotted versus L=h or LE =h) with peak amplitudes hR;max =z0 for slides
from Todorovska and Trifunac [7] spreading with constant velocity cR ˆ
cT and with W=L . 1 and W=L ˆ 0:5:
cC =cR ˆ 0 in Fig. 6, and cL =cR ˆ 2 and cC =cR ˆ 0:5 in Fig.
7). It is seen that the largest near-®eld amplitudes of h=z0
occur for cR =cT , 1 and cL =cR , 1:
Fig. 8 (left) shows the leading tsunami peaks, propagating
T ˆ 0:7;
in the positive x-direction, at time t ˆ tp and for c=c
0.8 and 1.0. The numerical values of the normalized peak
amplitudes, hmax =z0 ; are also shown. The right half of Fig. 8
shows the peak values plotted versus L/h, where h is the
depth of ocean and L represents that part of LR (Fig. 1)
during which cR is constant and equal to c (Figs. 3 and 4).
The peak amplitudes corresponding to Model 1.B are
plotted at L=h ˆ 50=2; for cC ˆ 0; 0.1cÅ and 0.5cÅ. It is seen
that as L ! LR ; hmax =z0 approaches the estimates for W=L .
1 from Todorovska and Trifunac [7]. For Cases 1, 2 and 3,
the peak values are plotted as open circles in three groups,
T ) ˆ 0.7, 0.8 and 1.0.
corresponding to cR/cT (i.e. c=c
Adjacent to each point is the value of the constant velocity
c (during time interval t2 2 t1 ) in terms of cT. It is seen that
for each case hmax =z0 increases as c ! cT ; and the peaks
approach the estimates of hmax =z0 for W=L . 1 in Model
1.B.
Figs. 9±12 show how h=z0 evolves in time. Waveforms
for selected cL =cR ; cC =cR and cR =cT are shown in
time windows 0:1 # t=tp # 1 and 1 # t=tp # 5: In all
T ˆ 1: In Figs. 9 and 10,
four ®gures, cR =cT ˆ 1; i.e. c=c
M.D. Trifunac et al. / Soil Dynamics and Earthquake Engineering 22 (2002) 167±180
cL =cR ˆ 1; cC =cR ˆ 0 and t/t p is, respectively, in the intervals 0:1 # t=tp # 1 and 1 # t=tp # 5: In Figs. 11 and 12,
cL =cR ˆ 0:3; cC =cR ˆ 0:3 and t/t p is again in the intervals
0:1 # t=tp # 1 and 1 # t=tp # 5: It is seen that for most
slides with downhill motion (which in most cases would
correspond to the direction away from coastline), the slides
that originate near the foot of the slide and then spread uphill
will produce the largest run-up.
Figs. 13±15 show top views of the tsunami waveforms at
time t ˆ tp ; for Model 1.B and for the three cases of variable
velocity model. Fig. 13 corresponds to Fig. 6 with cR =cT ˆ
1; while Figs. 14 and 15 correspond to Fig. 7 for cR =cT ˆ 1
and cR =cT ˆ 0:5; respectively.
Fig. 16 shows hR;max =z0 (top) and hL;min =z0 (bottom)
versus cT =cR for rectangular slides with areas 100 £ 100,
50 £ 50 and 10 £ 10 km 2, for cL =cR ˆ 0:3; cC =cR ˆ 0:1
and for the ocean depth h ˆ 2 km: Here hR;max =z0 is the
largest positive wave amplitude associated with the accumulation zone, and hL;min =z0 is the largest negative wave
amplitude, associated with the depletion zone and propagating in negative x-direction. It is seen that for Model 1.B the
peak of hR;max =z0 is near cT =cR ˆ 1; and the amplitudes of
hR;max =z0 resemble ampli®cation in the vibration of a singledegree-of-freedom oscillator [7], with h=…WL† acting as fraction of critical damping. For Cases 1, 2 and 3, the effective
length of sliding LE, during which the spreading velocity is
constant and equal to c (Fig. 3), is shorter than LR
(LE ˆ 12:96; 15.52 and 17.74 km, respectively, for Cases
1, 2 and 3, when LR ˆ 50 km; and LE ˆ 25:92; 31.04 and
1.38cÅ
35.48 km, when LR ˆ 100 km). Because c ˆ 1:48c;
and 1.29cÅ for Cases 1, 2 and 3, respectively, and because the
results shown in Fig. 16 are plotted versus cT =cR ; i.e. cT =c;
the peaks hR;max =z0 occur at cT =cR ˆ 1:50; 1.37 and 1.30 for
the 50 £ 50 km 2 source, and at cT =c ˆ 1:50; 1.43 and 1.32
for the 100 £ 100 km 2 source (Fig. 16 top) rather than at
cT =cR ˆ 1; as for Model 1.B.
Fig. 17 shows selected peak amplitudes of h=z0 in
the near-®eld versus L=h: These peak amplitudes were
evaluated at t ˆ tp and correspond to different cR =cT ratios.
The four sets of peak amplitudes correspond to Model 1.B
and to Cases 1, 2 and 3. The cT =cR ratios where the peaks
occur (Fig. 16) are shown adjacent to each point. Open
circles show that the hL;min =z0 peaks are larger than the
trend for constant velocity of spreading and W=L . 1 [7],
apparently because of the parts of increasing and decreasing
ramps in c ˆ c…t† (Fig. 3), which tend to extend the `equivalent' LE beyond the values computed only for the segment
when c…t† ˆ c; during time interval t2 2 t1 : All hL;min =z0
peaks are smaller than the corresponding hL;min =z0 for
Model 1.B.
It can be seen from Figs. 5±17 that the differences in
the waveforms between the results for Model 1.B and
for Cases 1, 2 and 3 are small for cR =cT . 2: For , 2 #
cR =cT # 1; and in particular for cR =cT , 1; peak amplitudes of h /z 0 are reduced signi®cantly, because the time
dependent cR results in shorter length LE of sliding with
179
constant velocity and thus smaller build up of amplitudes by focusing.
4. Discussion and conclusions
We illustrated the effects of variable sliding velocity of
submarine slides and slumps on the near-®eld amplitudes
of the resulting tsunami. The simple nature of the
assumed time dependence of these changes was motivated by the reported changes in Froude number, in
numerical simulations of slump motions, in the studies
of Jiang and Le Blond [3].
We found that the overall nature of near-®eld tsunami
amplitudes, involving directivity, focusing and strong
dependence of waveforms on the overall average speed of
slumping and sliding remains unchanged, with respect to
our previous studies, which approximated the motions of
submarine slides by kinematic models with constant velocities of spreading [10,11]. The details of the wave motions,
however, can change appreciably, when the spreading velocity cR varies with time, especially when the average spreading velocity
c is near the long period tsunami velocity
p
cT ˆ gh:
For the accumulation and depletion zones of submarine slides and slumps spreading only downhill and
away from the coast, essentially all the tsunami energy
is directed away from the coast. Large wave amplitudes
which propagate towards the coast occur mainly only
during retrogressive evolution of slides and slumps
(that is when cL ± 0).
For velocities of spreading cR and cL such that cR =cT . 2
and cL =cT . 2; the effects of variable velocities of sliding
are small for the model studied in this paper. These effects
become signi®cant for cR =cT , 2 and/or for cL =cT , 2:
Acknowledgements
During the course of this work, the second author was on
leave from the Istanbul Technical University and visiting
the University of Southern California. The ®nancial support
for his stay provided by TUBITAK-NATO is gratefully
acknowledged.
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