Geophys. J. R . astr. SOC.(1984) 77,29-41
Tsunami generation: a comparison of traditional and
normal mode approaches
Robert P. Comer
* Department ofEarth and Planetary Sciences,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Received 1983 April 25
Suxmary. Techniques to model tsunami generation by earthquakes may be
divided into two broad categories, according to whether the ocean and solid
earth are fully or partially coupled. The former category includes normal
mode techniques while the latter is comprised of more Qraditionalapproaches,
including those in which the hydrodynamic equations for the ocean are
solved subject to an inhomogeneous time-dependent boundary condition at
the ocean floor. We compare representatives of each approach, using a
solution for the flat earth tsunami mode excitation due to a point moment
tensor earthquake source and a more traditional model in which the ocean
floor boundary condition is derived from the static response of an elastic halfspace to a point moment tensor. The results are nearly identical and extend
to finite sources according to linear superposition, thus establishing the
practical equivalence of the two conceptually divergent techniques and
assuring the general validity of models with partial coupling.
Eitroiluction
The many models which have been applied to study the generation of tsunamis by
earthquakes may be clearly divided into two kinds. The more realistic are those in which the
ocean and solid earth are coupled together to form a single system in which tsunami waves
are excited by an earthquake source in the solid earth. They may be referred to as ‘fully
coupled’, and this category includes the model by Yamashita Sr. Sato (1974), the spherical
earth normal mode model by Ward (1980, 1981, 1982a, b), and the flat earth normal mode
model by Comer (1982a, 1984). The vast majority of models that have been applied to
tsunami generation belong to the other category, however, which we may call the ‘partially
coupled’ approach. Generally, in a partially coupled model, some pattern of static
deformation of the ocean bottom in the tsunami source region is assumed, assigned a timehistory, and used to drive wave motion in an ocean which has an otherwise rigid bottom.
The solid earth drives the ocean but the ocean has no influence on the solid earth, hence the
coupling is partial.
*Present address: Seismological Laboratory, California Institute of Technology, Pasadena, California
91125, USA.
30
R. P. Comer
The purpose of this paper is to present a direct comparison of a traditional, partially
coupled tsunami generation model to a fully coupled model. The latter is formally more
correct, but we seek to determine whether it is in practice more correct; that is, whether
the two approaches give practically equivalent results or not, and hence whether the partially
coupled model is reasonably valid. Since a considerable amount of work has been done with
partially coupled models, and since they permit the inclusion of ocean depth variations (e.g.
Ando 1982), which none of the existing fully coupled models allow, this is an important
question.
To represent fully coupled models we choose the one described by Comer (1 984), which
we will henceforth refer to as Paper 1. In Paper 1 , the far-field excitation problem for the
tsunami normal mode of a flat ocean-solid earth system is solved, for a point moment
tensor earthquake source. There are many variations of the partially coupled approach and
before settling on one we describe some specific examples.
The most realistic partially decoupled models start with a buried fault, use theoretical
results for dislocations in an elastic medium to obtain the static displacement of the ocean
floor, then assign some spatially uniform time-history to the vertical displacement. The
time-dependent displacement of the ocean floor drives the tsunami; mathematically the
process is formulated through a boundary condition at the ocean floor. For example, both
Ando (1975, 1979, 1982) and Aida (1978) modelled tsunami generation by using
Mansinha & Smylie’s (1971) results, for the static displacement field of a uniform rectangular inclined fault in an elastic half-space, with a ramp function time-history. Ando (1975,
1979, 1982) numerically solved the non-dispersive hydrodynamic equations for a variable
depth ocean in which the time derivative of the vertical ocean floor displacement appears
as a source term. Aida (1978) did likewise but also included non-linear terms. Hwang &
Divoky (1970) took an approach similar to Aida’s (1978) in attempting to synthesize the
1964 Alaska earthquake tsunami but used directly the static vertical displacements reported
by Plafker (1969) rather than calculations based on a fault model, and used a step function
of time smoothed with a sin’ taper rather than a ramp function.
There exist modifications of the method described above in which the hydrodynamic
equations are taken to be homogeneous and an initial static displacement of the ocean
surface is assumed that is equal to the static vertical displacement obtained from the fault
model. Abe (1978a, b) took this approach using Mansinha & Smylie’s (1971) results and
numerically solving linear non-dispersive hydrodynamic equations with homogeneous
boundary conditions. Aida (1969) followed a similar procedure but based the initial
ocean surface deformations on inverse refraction diagrams rather than seismic fault models.
We will show that applying a given static displacement pattern instantaneously through an
inhomogeneous ocean floor boundary condition is equivalent to applying it through a static
(i.e. aq/at = 0 at t = 0) ocean surface intial condition, except that in the latter case the
pattern should be spatially filtered through multiplication by l/cosh kh in the wavenumber
domain. Kajiura (198 l), who specified ocean surface initial conditions to estimate tsunami
energy, has taken note of this fact. As he states, the approximation made by neglecting this
filtering is better for large events with much long-wavelength energy (so that cosh kh is
generally close to 1) than for small ones. An additional variation of the basic decoupled
method appears in works by Abe (1973) and Fukao & Furumoto (1975), where amplitudes
in tsunami source areas are estimated from fault models, then translated to coastal sites with
simple corrections for geometric spreading and shoaling.
The kind of decoupled approach addressed principally in this paper is the more physical
sort in which an inhomogeneous ocean floor boundary condition is imposed. With respect
to this, the technique of specifying an ocean surface initial condition is simply an additional
approximation that is well understood. Our development of a partially coupled tsunami
Approaches to tsunami generation
31
generation model may be outlined as follows. Initially we consider the response of a flat
ocean to arbitrary motions of an otherwise rigid bottom. Then a spatially uniform step
function or ramp function is assigned to govern the time dependence of the displacement.
In order to have results that are valid in the far-field we use the linear dispersive hydrodynamic equations. To enable a straightforward comparison to the results of Paper 1 we use
a point moment tensor earthquake source, so that the dependence of the tsunami o n most
of the source parameters is linear, and if equivalence is established for a point source it
should hold for extended sources also, by principles of linear superposition. Thus for the
vertical ocean floor displacement we use the static response of an elastic half-space to a
buried point moment tensor source, rather than a finite fault. Finally, comparisons to the
normal mode synthesis of far-field tsunami waveforms are made both analytically and
numerically.
Tsunami response to an arbitrary ocean floor deformation
Using integral transforms it is straightforward to obtain the response to motion of the ocean
floor of a flat, uniform-depth, ideal, incompressible ocean subject to a uniform gravitational
acceleration g , as long as the hydrodynamic equations and boundary conditions are
linearized. Hammack (1973) treated this problem in the case of one-dimensional wave
propagation by using a Laplace transform in time and a Fourier transform in the horizontal
spatial coordinate. Lee & Chang (1980) generalized the solution to include both horizontal
dimensions, taking a Laplace transform in time and a two-dimensional Cartesian Fourier
transform in the horizontal coordinates, We will follow the same approach but instead of
Cartesian coordinates we use cylindrical coordinates where, as in Paper 1 , the origin is at the
ocean floor and the z-axis is positive downward. Thus a 2-D Fourier transform pair over the
horizontal coordinates has the form
The motivation for using cylindrical coordinates is that in the next section we will transform
the static response of an elastic half-space to a buried point moment tensor source which
contains only the harmonics with I m I 2.
We begin by letting l(r, 4, t ) denote the vertical displacement of the ocean floor. We take
to be positive for displacement upward (in the negative z-direction) and assume that = 0
for t < 0. Assuming irrotational flow we can use the formulation of Paper 1 except that x
and y are transformed to r and 4 and the linearized boundary condition at the ocean floor
is now
-
acqaz iZ
=o = a t p t
(3)
where CP denotes the velocity potential. The velocity potential also satisfies the continuity
equation (1 -5) and the free surface condition (1 -1 1). (Equation numbers prefaced by ‘1 -’
are from Paper 1 .)
Using ‘ A ’t o denote functions in the transform domain we have
dQ exp(-im@)
La
@ ( r , $ , z , t ) Jrn(kr)rdr
(3)
32
R . f.Comer
from the Laplace transform operator
ds exp(-st)
JOm
and the forward transform given by (lb). Transforming (1-S), (1--11) and ( 2 ) we have,
respectively,
The solution to (4) is easily found to be
6m(k,z, s) = sg,(k,
k
s) [(s2 sinhkh +gk coshkh) coshkz
+ gk
+ (s2 cosh kh
sinh k h ) sinh kh]/(s2cosh kh
From the transform of (I-lo), e m ( k , s)= (s/g)&,,(k,
+ gic sinh kh).
(51
4,s) and (5) we find
where o2= gk tanh k h , corresponding to (1-22). The coefficient s2/[(s2 + w’) cosh kh]
was also found by Lee & Chang (1980) and is clearly independent of the form used for the
2-D Fourier transform.
Next we consider two somewhat more specific cases for which we can obtain the inverse
Laplace transform of (6) analytically. We assume that the time and space dependences of E
are separable and that
t ( r , @ , t )= Z ( r , @ ) H ( t )
(7 a)
or
t (r, $, t ) = Z ( r , @)R (7,t )
(7b)
where Z(r, @)= E(r, @, -) is the static vertical displacement of the ocean floor, H(t) is the
step function defined by (1-74a), and R(t, 7) is the ramp function defined by (1-74b).
The assumption of separability is not very realistic, but it makes the problem far more
tractable and, as we shall see, q(r, @, t ) depends only weakly on the choice of rise time 7.
The Laplace transforms of H(t) and R(t, 7) are, respectively, l/s and [ I - e x p ( - - s ~ ) ] / ( ~ s ~ ) .
Thus the transforms of (7a) and (7b) are
im( k , s) = S - l -=m ( k )
@a)
and
2 -1%
i m ( k , S) = [ 1- ~ X (-s
P 7)I (7s
) -m ( k )
(8b)
where k,(k)
denotes the purely spatial transform of &(r, @).If (8a) or (8b) is used in (6)
the inverse transform gives
Approaches to tsunami generation
33
where
K (t, k ) =
H ( t j cos at
(lea)
cosh kh
or
K (t,k ) =
H ( t ) sin wt - H(t - T) sin o (t
~~
7)
o r cosh kii
respectively. The inverse Fourier transform follows from (la) and the inverse Laplace
transforms have been taken by inspection, using
cos at exp (-st) d t =
~
sz t o2
and
[ H ( t ) sinwt - H ( t - 7 ) sin o ( t - r ) ] exp(-st)dt =
WT
1- exp(-sT)
r(s2
to
2 )
.
Finally, before assigning a specific form to Z ( r , $), we consider a variation of the foregoing problem in which the ocean bottom is assumed rigid and stationary, so that (2) is
re?laced by a@/az = 0 at z = 0 and the initial conditions for the ocean surface
37
clt
--(r,@,O)=O
are imposed. This problem is easily solved in the spatial transform domain using (1) and
requiring that the velocity potential @ satisfy (1 -S), (1 -1 l), and the rigid-bottom condition
ust introduced. The procedure is straightforward, and yields the result that q(r, $, t) is
given by (9) with
K ( t , k ) = H ( t ) coswt.
(13)
This is identical to the result (loa) obtained by using the ocean floor boundary condition (2)
with instantaneous displacement (r = 0) except for a factor of l/cosh k h . Thus a zero-phase
spatial filtering could be applied to the ocean surface initial conditions (12) to adjust the
final result by reducing the contribution of short-wavelength components of the ocean floor
vertical displacement to the initial ocean surface displacement, as noted by Kajiura (1981).
Static displacement due to a point moment tensor source
Now we specify that E(r, 4) be the static vertical displacement at the surface of an elastic
half-space due to a buried point moment tensor source. Z(r, #) can be obtained from the
formulas given by Maruyama (1964) for the case of a Poisson solid ( A = p). An independent
derivation, valid for arbitrary X and p , is given by Comer (1982a).
L
R . P . Comer
34
Mzz
47Tpd'
Mxz
4flpd"
MYZ
4lT)~d~
x/d
Figure 1. Contour plots of static vertical surface displacements due t o point moment tensor sources in
an elastic half-space, computed from (14) with h = p . The horizontal coordinates are normalized against
the source depth d , while the displacements themselves are given in non-dimensional form. Four basic
patterns, superimposed in (14), are shown and next to each is a factor or pair of factors that determines
the magnitude and orientation of its contribution.
The result is that
Approaches to tsunami generation
35
where R = (r2 + ~ 1 ~ and,
) ” ~as in Paper 1, Mij are the static moment tensor components and
d is the source depth. The four terms of (14) represent a superposition of four basic patterns
of vertical displacement whose magnitudes and orientations are determined by the moment
tensor components. The first two patterns are axisymmetric and their magnitudes are
proportional to (M,, +Myy)/2 and Mzz. The third and fourth have two and four lobes,
respectively. Each requires two parameters to assign it a magnitude and an orientation. These
and MxY for the fourth.
depend on M,, and My, for the third, and on (M,,-Myy)/2
Fig. 1 illustrates contours of the four patterns in a dimensionless for. The six weighting
parameters are shown next to the appropriate patterns and are normalized to have units of
length. The orientations shown for the third and fourth patterns were chosen arbitrarily to
correspond to cases where M,, = 0 and M,, -Myy = 0, respectively.
Now, using the following Hankel transforms
JOw
d
kdk = 7
R
exp (-kd) J,-,(kr)
jOm
(1 +
exp(-kd)Jo(kr)kdk
kd)
JOm
Iff
3d3
7
=
R
6rd2
2kd exp(-kd)J1(kr) kdk = 7
R
kd exp ( A d ) J,(kr) kdk =
3r2d
~
R5
where R = (r2+ d’)”’, we rewrite (14) as
Z(r, 4) = (Mxx +Myy)’2
47rP
+ M,,
4 w
-
M,,
1-
[( L
- kd) exp(-kd)Jo(kr)
(L
+ kd)
0
exp(-kd)Jo(kr) kdk
h+P
cos q5 + My, sin q5
47rP
-
kdk
X+P
(1/2)(M,,-My,y)cos2q5
lw
2kd exp(-kd)Jl(kr)
+Mxy s i n U
(16)
kdk
exp (-kd) J,(kr) kdk .
4nP
Note that (16) has the form of the transform (la) with real trigonometric functions used
in place of complex exponentials. Thus it follows from (9) that, for Z(r, @) given by (14)
or (161,
36
R . P.Comer
t M,,
4w
-
1- [&
K(t, k )
M,, cos @ t M y , sin@
4w
x
t kd] exp ( - k d ) J&r) kdk
0
1 [A
K(t, k )
lw
-kd]
K ( t , k ) [ 2 k d ]exp(-kd)J1(kr)kdk
exp(-kd)Jz(kr)kdk.
For far-field points we can replace each of the Bessel functions in (17) by the leading term
of its asymptotic series for large real arguments, i.e. J,(x) 2 [ ~ / ( R x )I ]” cos(x-nni2-ni4).
Since J , ( x ) = - J o ( x ) for large x, the first and fourth terms of (17) can be combined, and
q(r, 4, t ) can be expressed in the form of (I-70),
where
We have now formulated a decoupled model of tsunami generation that can be checked
against the normal mode model by directly comparing their predictions of the elementary
waveforms f i , f 2 and f3, that is, comparing (19) to the inverse Fourier transform of
(1-71). This comparison is independent of the moment tensor of the source and, for a
given ocean--earth model, depends only on d, r, t and 7. Thus a wide range of cases can be
covered by checking a fairly small parameter space, as noted by Comer (1982b).
Comparison to X
A O I Tmode
~
results
It is straightforward to check the equivalence of (19) and the inverse Fourier transform
of (1-71) by numerically computing both. However, in the special case where ~ = 0 ,
this equivalence can be established analytically, using the tsunami eigenfunctions for a
homogeneous solid earth that are given in Paper 1.
Assuming S ( t ) = Y ( t ) and observing from (1-72) that N ( o ) = 0 at o = 0, we see from
37
Approaches to tsunami generation
(1-75a) that S(w) may be replaced by i/w in (1 -71). Then using (1-72) and the approximate relations
k2 c2(1 / 2 -kd) exp (- kd)
krl (d) r2(-h)
I
I
p cosh kh
I1 + 15
dr2/dz (d)r2(-h)
I1 + I ,
k2c2(1/2 + kd) exp(-kd)
21
~
~
_
_
_
_
_
_
_
p cosh kh
[drl/dz (d)-kr2(d)]r2(-h)
k2c2(2kd) exp(-kd)
c=
p cosh kh
I1 +I5
which follow from (1 -2O), (1 - 2 l), (1 -24) and (1 -30), we find by applying the operator
2ll
-=
d w exp(-iwt)
to (1 -7 1) that
1
-
fl(r’
I,
k2c( 1/2-kd) exp (-kd)
k2c(1/2
f k , t ) = - 8~
+ kd) exp(-kd)
wUcoshkh
~
0
k2c(2kd) exp(-kd)
f3(r, t )
0
+ kr + 3n/4)dw
cos(-wt
+ kr + 3n/4)dw
cos(-wt
~
8w
cos(-wt
wU cosh kh
w z c o s h k h
(21)
+ kr + 5n/4)dw.
We have also taken note that fl(r, t),f2(r, t) andf3(r, t) are real so that the real parts of their
Fourier transforms are even and the imaginary parts odd. Since U = J o / d k , k2c/w = k, and
the minus signs in front can be absorbed into the phase of the cosine factors, (21) can also
be written as
fi(r, t ) =
-
~
I
8nP 0
(1/2-kd)exp(-kd)
__
cosh kh
(&)
‘I2
cos (at - kr
(2)
+ r / 4 ) kdk
112
cos(wt-kr
+ n/4) kdk
112
cos (wt - kr n/4) kdk
Finally we note that by substituting K(k, t) from (lOa), setting X = p , and noting that in the
far field we can replace cos a t cos(kr-@o) by (1/2)cos(ot-kr + q0), (19) can be shown to
be equivalent to (22).
We now compare numerical evaluations of f l,f2 and f3 using the methods of Paper 1 and
direct numerical integration of (19). It is interesting to note that the former method is much
faster, due to the great efficiency of the Fast Fourier Transform algorithm. Calculations
were undertaken for a 4 k m ocean overlying an elastic half-space where h = p with the
R. P. Comer
38
E
0
I
C
>
U
..
N
k
0
E
0
c
normal mode
c
0)
f3
E
0
0
-mn
9
f2
.-
v)
decoupled
0
-m
0
._
c
L
PI
>
time, s
time, s
I
I
8,000
I
I
1
8,000
10.000
12.000
I
14.000
I
I
10,000 12,000
14,000
Figure 2. J ( r , t ) , f;(r, t ) and A(r, t ) versus t for r = 2223 km ( A = 20"), T = 0, d = 10 km, and the model
parameters given in the text. Waveforms computed by the partially coupled method (numerically integrating equation 19) are shown with the corresponding normal mode results (computed using the method
of Paper 1).
normal mode
E
0
f,
I
decoupled
C
>.
h
'
U
c
p.
.
0
r
E
0
c
C
0)
f3
E
0)
0
-m
Q
f2
v)
._
U
m
.-0
c
L
0)
>
time, s
time, s
I
8.000
1
I
10,000 12,000
I
I
8.000
10,000
I
12,000
I
14.000
Figure 3. Elementary waveforms as in Fig. 2 but with
T =
120 s.
I
14,000
39
Approaches to tsunami generation
f,
time, s
time, s
I
0.000
I
1
10,000
12,000
I
I
1
8,000
10,000
12,000
I
14,000
I
14.000
Figure 4. Elementary waveforms as in Fig. 2 but with
7
= 120 s and d = 40 km.
P-velocity, S-velocity and density set to 7.1 km s-l, 4.1 km s-l and 3.1 g cm-j, respectively.
The source-receiver distance was fixed at r = 2223 km (A = 20") and two depths and rise
times were investigated.
Fig. 2 compares the results of both methods, normal mode and decoupled, for
d = 10 km and T = 0. As our analytic work anticipates, the waveforms in each pair appear
identical. Fig. 3 is similar, with d = 10 km and T = 120 s. It is clear that even for a large rise
time, a case not covered by the analytic results, the methods yield identical waveforms.
Although the definitions of rise time in the two models are not physically equivalent, since
T refers to the moment tensor time dependence in the normal mode formulation and to the
motion of the ocean floor in the decoupled model, the two rise times have comparable
effects in practice. A final illustration, Fig. 4, shows the waveforms for d = 40 km, T = 120 s,
and again the results of the two methods are indistinguishable.
Conclusions
It is clear that for comparable ocean-earth models with realistic physical parameters and
point earthquake sources the decoupled approach to tsunami generation gives results in
excellent agreement with those of the fully coupled normal mode theory. Through linear
superposition, this agreement should hold also for finite sources. It is a significant result,
confirming the validity of many of the studies cited earlier and making it clear that the
hitherto rather intuitive inferences made relating the initial motions of tsunamis to the
static co-seismic ocean floor displacements in their source regions should be generally
correct.
We note that the qualitative relation between vertical ocean floor displacement and
40
R. P. Comer
tsunami initial motion was examined previously by Yamashita gi Sato (1974). They
compared the results of their finite source, fully coupled model to static deformation
patterns calculated by Mansinha & Smylie’s (1971) method and found good agreement
between the polarity of vertical displacements and the initial tsunami polarities. The present
results show that such consistency between fully coupled and partially coupled models
extends to full tsunami waveforms.
As we have remarked, the flat earth normal mode calculations are considerably faster
than those done for the partially coupled formulation. However, this does not mean that one
should always use a normal mode formulation rather than a more traditional, partially
coupled model, because the former is limited in applicability by the necessary assumption
of a uniformly deep ocean. On the other hand, partially coupled models can be developed to
readily handle realistic ocean bathymetry through a finite difference (e.g. Aida 1978; Ando
1982) or finite element technique, although they will be slower. The problem at hand will
in practice dictate which category of tsunami generation models is appropriate.
Acknowledgments
I would like to thank Kei Aki, Kinjiro Kajiura and Stave Ward for helpful discussions, lnge
Knudson for assistance in drafting the figures, Jan Nattier-Barbaro for typing and wordprocessing, and Sharon Feldstein for administrative assistance. During the final preparation
of this paper, I had the support of a Bantrell Postdoctoral Fellowship at the California
Institute of Technology. This research was supported by the National Science Foundation,
grant EAR80-1835 1.
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