The form of Tsunamis generated in coastal regions

Dynamics of Atmospheres and Oceans, 9 (1985) 39-48

39

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

THE FORM OF TSUNAMIS GENERATED IN COASTAL REGIONS

C L A U D E F. NOISEUX

Dioision of Applied Sciences, Harvard Unioersity, Cambridge, Massachusetts 02138 (U.S.A.) (Received March 29, 1984; revised August 10, 1984; accepted August 27, 1984)

ABSTRACT Noiseux, C.F., 1985. The form of Tsunamis generated in coastal regions. Dyn. Atmos. Oceans, 9: 39-48. The deep ocean waveform that results from a seismic event localized near a coastline is investigated using a simple model for the near-shore topography and the ground motion. It is found that the variable depth underlying the ground upthrust substantially enhances the distortion of the emerging Tsunami. An appropriate initial-value problem is solved numerically, from which an empirical formula is inferred that provides an approximate assessment of the deep ocean waveshape.

1. I N T R O D U C T I O N

The shape of the surface wave (Tsunami) formed by an undersea earthquake can vary greatly according to the character and location of the initiating ground motion. If the bottom motion occurs in the deep ocean, the coarse features of the waveform will evolve according to the analyses of Kajiura (1963) and Carrier (1971). In particular, it has been demonstrated by Carrier (1971) that significant distortions of the initial wave arise only when the travel path is so long (in an appropriately scaled sense) that dispersion becomes important. When the ground motion takes place near the shore, however, the underlying topography will immediately distort the wave, thereby producing a significantly different "initial condition" for the deep water propagation to distant land masses. The principal objective here is to characterize, via non-dispersive linear shallow water theory, the general form of the deep ocean wave that results from a simple type of ground displacement confined to the continental shelf and slope geometry shown in Fig. 1. Postulating a piecewise linear bottom topography, the (one space dimension)initial-value problem is solved for instantaneous piecewise linear displacements of the ocean floor. Though a 0377-0265/85/$03.30

© 1985 Elsevier Science Publishers B.V.

40 numerical procedure is generally required as part of the calculation, a simple empirical formula is inferred that adequately describes the salient features of the leading wave as a function of parameters governing the ground motion and the bottom topography. 2. MODEL PROBLEM AND SOLUTION The conservation laws, in the linear shallow water approximation with ground motion, lead to an equation for the waveheight ,/(~, f), given by [^ ^ 377] ] 3~ [ h 0 ( x )

1 32~/_

1 327/0

- 2 g -3f

g

The symbols used in (1) are defined in Fig. 1, with positive values of 7/o(X, t) indicating an upward motion of the ocean floor. Shortly, (~,f, ho) will be rendered dimensionless by use of the s h e l f / s l o p e length L, and the deep ocean depth H.The ground motion is taken to be instantaneous and occurring at t = 0, with no initial elevation or motion of the free surface, i.e. , / ( ~ , 0 - ) = --~ ( x , 0 - ) = ,/0 ( ~ , 0 - ) =

(~,0-) = 0

(2)

In an actual Tsunamogenic process, the seismic event usually takes place with a period of the order of seconds, whereas the durable ocean waves that ensue have a period of the order of at least tens of minutes. As a consequence, the detailed time history of the earthquake has little bearing on the formulation of the longer ocean waves that are the main concern here. Furthermore, since at most only the gross features of the topography and the

)-X 0

Fig. 1. Continental shelf and slope geometry.

A

41

earthquake upheaval are known in actual practice, the ocean floor displacement are modelled using piecewise linear functions of i, The tational advantages provided by these simplifications far outweigh in generality, since these restrictions on h, and n,, ensure that the transform in time of (l), using (2), given by:

and its compuany loss Laplace

can be solved analytically. It should be noted that the Laplace transform of (1) with no ground motion [qO( R,i) = 0] but with an initial elevation of the free surface [n(Z,O-) # 0, ni(&O-) = 0] leads again to (3) provided that s or n,,(-l;.,i) = n(&O) *H(i). The equivalence between a step 7)0 = 77(&O_)/ function (in i) ground motion and an initial displacement of the surface is an artifact of the hydrostatic assumption in the linear, shallow water theory. The governing equation is cast in dimensionless form by using the length scales shown in Fig. 1. The three part description of A,(_?), together with values of H = 4 km, L = 200 km and i2, = 100 km, is chosen to represent the coarse features of the continental shelf and slope bordering the Alaskan coastline (Tuck, 1979). Dimensionless variables x, t, and h are introduced by defining

?=xL, &,(a)=H.h(x)andi= so that (3) (with s also scaled so that st is dimensionless) [

~b)i,]

x

-

becomes

s2i = -4x,0)

(4)

where the right-hand side is now to be interpreted as the equivalent initial surface displacement. The prescribed function q(x,O) is assumed to be nonzero only within the region 0 < x c 1, and composed of line segments p,x+7,

(i=1,2...N)

linking the node shown in Fig. 2.

points

x(i)

with

ri = pNx(N)

+ rN = 0. An example

h =l Fig. 2. Initial surface displacement.

is

42 The depth function is given by

(aax h(x)=t12(x-1)+l

if

x
x(J)x(N)=I

where x(J) (= Yq/L in Fig. 1) represents the end of the continental shelf. The general solution of (4) over each interval defined by the node points x(i) may be written

gli(x,s)=CiIo for l _ < i _ < J

~-7(oqx)1/2

+ DiKo 2s (a,x)l/2 +--77--+7(ixix+ri)

[O<_x<_x(J)]

(5)

and

f 2s[ } + DiK°{ }2S7 [Ot2(x-1)+l]l/2 o r - ~i(x,s)= cIi o~-dTta2(x-1)+l]'/2__ + a2t~, + l ( t , , x + r, ) s3 s

forJ+l<_i<_N [X(J)<_x<_l] where s is the Laplace transform variable, and I 0 and K 0 the modified Bessel functions. The boundary conditions appropriate to this model are that D~(s) = 0, in order that the solution remain bounded at x = 0, and that at each node x(i) the adjacent solutions be matched using the two conditions

d~i

d~i+l

dx dx Also, since there is no reflection of the wave once it begins its journey over the deep ocean region, Tl(x,s) must take the form

~ ( x , s ) = [ A ( s ) e - q e -s~x-~)

(6)

in x > 1, representing a right-moving wave. The internal matching at the node points x(i), together with the boundary conditions at x = 0 and x = 1, uniquely specifies the functions Di(s), Cl(s ) and A(s) in (5) and (6). In this study numerical results will be generated for values of x >__ 1, hence only A(s) in (6) needs to be specified. The process by which A(s) is determined is straightforward though quite cumbersome. Therefore only the final formulas will be recorded. Introducing the notation

( [ OtlX(i) ] 1/2

h(i)=lla2[x(i)-I ] !

i
J+ l

43

and the definitions

pi=-~lh(i ) -2Sh(i)

qi -- ol 2

the term

i<~J J+ l < i < N ,

A(s)e -s

[A(s) e-~]

in (6) may be written

1 g " 2SD = 2/ ' t 2 3 1 o t I I ( p j ) ( O t l - - O t 2 ) + - - p f j E

f(l)

(7)

l=1

where

D = [KI(qN) - Ko(qN)] [ Io (qJ)I1(PJ) -- 11(q~) to (p~)]

+[lo(

q~ ) + 11( q~)]

[ Ko (qj)11(pj) + KI( qg )Io( pj )]

-j F(1)=p,, l~'-l~'+a[--s-la(pt)-h(1)lo(P,)] } Ol1

T(l)=qt.

/xt-s 7#'+' [ --~-Kl(ql)+h(l)Ko(ql) a2 1)

R(l) = q,

"1--"1+1 ~2 s2 [ 711(q,)

-h(l)lo(qt)] )

and the convention/~N+I = 0 is used. Note that if the upthrust is confined to the continental shelf [x < x(J)], then /~g+l = #J+2... = ~N+I = 0, SO that ET(/) = ER(/) = 0 in (7). The inversion of (6), using (7), gives the waveform that emerges in x > 1 as

~l(x,t ) = ~

s )e-'] e -'t(x- a)-tlds

(8)

The integration is undertaken numerically using the Dubner-Abate method (Dubner and Abate, 1968), and this process has been carried out for a variety of simple upthrust profiles ,/(x,0). A selection of these results is given in Figs. 3-6. In each of the figures the deep ocean waveform ,/(x > 1,t0) is shown together with the generating ground motion ,/(x < 1,0). The horizontal scale is given in units of the shelf/slope length L, so that a unit distance is traversed in unit time. Also, 7/(x > 1,t0) is given a three part description,

44 ~(x,O) -- -- -- approximate

2"0 I1.0

formula

~-Smeters~

numerical result

oo

;

-1ot.

1.s

z'.o

z'5

3'.o

t I = tf

/~,~,-~'~ ~ 0 ,5i

31s

,

,

,.o

,s

7

~.o

-

-

T

5.5

~

6.0

,

\,

,

65

~.o

7.5

8o

t 2 = tf +7 1,0

T

~ -~.°~X._Jli o

1.

. --

/

3.o

3.s

,o

,5

~.o

~.5

6.0

65

,.o

75

8o

2O

t 3 = t2+7 1.0

T ~.5

zo

2.s

3.0

3.5

4o

.5

so

55

eo

6.5

7o

~ 7 5

80

-1.0

Fig. 3. Deep ocean waveform due to ~q(x,0).

....

approximate formula numerical result

- -

~oi

~ ' r / ( x ' O ) -~ ~ 5meters

~-~,-v ,->-. , 0

1.0

0.07~ l

'T - - I - 2.0 25

.'5

--I- --

3.0

3.5

4.0

.5

4.5

5.0

1

5.5

6.0

tl = tf

6.5

7.0

7.5

i

8.0

- 101. t2=t1+7

I'0 I ~

0.0

_l.O

I

k ~

~

.

I

~

I

.

~

I

~.~.-,-""T"---I

°.o -

I

I

I

I

I

I

~

t3=tz+7 I'0 I 0.0

I

I

-1.0

Fig. 4. Deep ocean waveform due to rt(x,0).

--I

~.o ~.~ ~.o ~.~ ~.o ,.~ ~.o

I

'

~

.

.

.

.

45

r/(x,O)

2oI

------

approximate formula numerical result

LV

- / ~

I 1

"5"

tl = tf

1.0 ~. ]5

O0

,ioi

=

--

I

I.

2.0

I

I

-

(_

I .,..,~..J~"~

.0

~

5,5

I

I

60

6.5

I ~'

ZO

I

I

Z5

8.0

t 2 =tf +7

].0

0.0

I

I

f

.j

--

F

3.5

40

4.5

5.0

5.5

6.0

6.5

70

75

8.0

- 1 0 ~

t 3 = t2+7 1.0

T

0.0~ -1.0

i

t

i

I

I

I

I

15

2.0

25

3.0

3.5

40

I ~._-..J

~

4.5

5.0

5,5

I

i

I

60

65

7,0

"~'P

~...

I

7.5

8.0

Fig. 5. Deep ocean waveform due to 7/(x,0).

I

----

approximate formula

- -

numerical result

t 5 meters

T/(x,O) I

0

1,0 0.0

~'~

I •

15

2.0

2.5

,'.'5.0

5.5

.5

1

tI = t f + 2

I

I

I

I

I

I

I

I

I

4.0

4.5

5.0

,5.5

6.0

6.5

7.0

7.5

8.0

-1.0

t2 =t t +7 "~

10 I O0

I 1.5

I 2.0

I

I

3.5

2.0

I 2.5

I 3.0

I 3.5

I 4.0

I 4.5

I 5.0

5.5

I

I 6.0

I 6.5

~ 7.0

', - - ' 7.5 8.0

"1

;

I

I

I

I

I

I

5.0

5.0

5.5

6.0

6.5

7.0

7.5

8.0

-1.0

1.01 0.0

~ I '

2.5

-1.0 -

2.0

Fig. 6. Deep ocean waveform due to ~/(x,0).

t 5 =t1+14

46 each comprising the domain 1 < x < 8, for three different values of time to=ta,tz,t 3 (the special value tf in the figures is defined in the next paragraph), chosen so that the three graphs may be concatenated, with the top graph becoming the right-most portion of a single extended profile for t o = t 3. The vertical scale is given in meters with 5 m chosen to be the standard peak value of each 7(x,0), as indicated in the figures. As anticipated, the greatest distortions to the initial waveform occur when 7(x,0) is concentrated near the coastline. A n unexpected result, however, is the magnitude of the negative portion of the waveform (e.g. Fig. 3) from the corresponding one-signal initial profile.

3. DISCUSSION AND APPROXIMATE FORMULA Certain properties of 7 ( x > 1,t) can be established without recourse to numerical methods. The wave emerging in x > 1 can be viewed as the sum of the partially reflected waves on the shelf/slope geometry, with the coastline and shelf edge as the principal sources of reflection. At t = 0 + the profile splits into two parts, 7 + and 7 - , each equal to ½7(x0,0 ), moving in opposite directions. The component initially moving toward the ocean (7 +) will form the lead wave in x > 1. During its passage over the coastal geometry the wave will be elongated owing to the differing wavespeeds implied by the depth distribution, and will be continuously diminished in amplitude due to the partial reflections. The elongation of 7 + can be readily assessed by considering the characteristics of the hyperbolic equation (1). In dimensionless form, the forward characteristic

f

x

dl

[h(t)ll/

when set equal to zero, provides the desired expression for the new position x , of a point x 0 in the initial distribution 7+(x0,0). With tf denoting the time required for the trailing edge of 7+(x0,0) to get to x = 1, the new position of a point x 0 is given by

+ ( loq

l+tf X=

a21

)(h[x(J)])l/2- l[h(1x°)]a/2}a

if x o < x ( J ) l+t r

(9)

--2 (1 _ [h(xo)l 1/2 }

~2

if x 0 > x ( Y ) Thus the horizontal extent of the lead wave can be determined, and these waves are shown in the topmost graph of Figs. 3-6, where t 1 = If.

47 Though the change in amplitude of ,/+ can only be calculated with precision via the numerical scheme, certain persistent trends are found in the emerging waveforms. Using eqs. 2-8, the estimate lim

(x,t)dx=

( x 0 , 0 ) d x 0 + 0(e -Re(s')t)

(10)

l--~ OQ

can be established, where s 1 is the rightmost pole in the left half s plane of (8). The exponential term in (10) represents the leading order estimate to the amount by which ~/÷ is diminished due to partial reflections. Integrating the numerical result ~t(x > 1,tf) and comparing with the net area of 7/+(x0,0) showed that in all the cases considered the discrepancy in the areas was less than 20%. This implies that a reasonable approximation to the lead wave can be obtained by assuming that r/(x,/f) maintains the geometrical shape of ,/+(x0,0), now distorted in its extent according to the formulas of (9), with amplitudes adjusted so as to conserve the area (mass) of 7/÷. Though an empirically derived formula will be given that is more accurate, it will retain this simple prescription as its leading order result. The secondary disturbances (t > tf) emerging in x > 1 did not conform to such a simple scheme. The coastal reflection of r/-(x0,0 ) and its subsequent propagation over the shelf/slope geometry did, however, produce two-signed waveforms with a pronounced dipole character when ~/(x0,0 ) is symmetrical (Fig. 6). This result is consistent with those presented by Carrier and Noiseux (1983), where the coastal reflection of obliquely incident Tsunamis has been investigated. It should be noted that in the present work, the long term solution may have only marginal significance, since the consequences of geometrical dispersion in the more properly formulated three-dimensional problem are absent. An empirical formula was obtained that describes the lead wave shape in terms of the underlying topography and the initial upthrust. The prescription proved to be rather robust with regard to reasonable changes in and 77(x 0,0), and in particular, predicted the maximum waveheights in x > 1 with an error of less than 2%. For a symmetrical ground motion ,/(x0,0 ), the lead wave shape is given approximately by

h(x)

rl(x°'O)(ha[l-2)ka/2(rl(x°'O))l/3])l/4 2 h. (11) where ~ (0 < ~ < 1) is the scaled extent of 7/(x0,0 ), h a = lfh(x)dx is the

r / ( x > 1 tf) '

average depth under ,/(x0,0), and x and x 0 are related by (9). The waveforms given by (11) are shown as dashed lines in Figs. 3-6, where the approximate results for the asymmetrical ground motions of Figs. 3-5 were derived by superposing symmetrical upthrust shapes. The bracket term in

48

(11) represents a mild embellishment of the so-called Greens formula (Lamb, 1932; Carrier, 1966), where the change in waveheight is related to the ratio of the averaged initial (h = ha) and final (h = 1) depths. The consequences of this formula (hla/4) a r e further mitigated by the factors X and / / / h a , with the latter term becoming significant when the ground motion occurs near the coast. Considering the apparent accuracy of the formula as implied by the figures, it is possible that (11) provides a reasonable a priori assessment of the enhanced distortion of a Tsunami when it originates on a coastal geometry. ACKNOWLEDGMENTS

The results recorded here were obtained while the author was a postdoctoral fellow in the Division of Applied Sciences of Harvard University. The author gratefully acknowledges the support of the National Science Foundation under Grant No. NSF-CEE81-07884, and the technical direction and advice of Professor George F. Carrier. REFERENCES Carrier, G.F., 1966. Gravity waves on water of variable depth. J. Fluid Mech., 24: 641-659. Carrier, G.F., 1971. The dynamics of Tsunamis. In: W.H. Reid (Editor), Mathematical Problems in the Geophysical Sciences, Vol. 1--Geophysical Fluid Dynamics, American Mathematical Society, New York. Carrier, G.F. and Noiseux, C.F., 1983. The reflection of obliquely incident Tsunamis. J. Fluid Mech., 133: 147-160. Dubner, H. and Abate, J., 1968. Numerical inversion of Laplace transforms by relating them to finite Fourier cosine transforms. J. Assoc. Comput. Mach., 15: 115-129. Kajiura, K., 1963. The leading wave of a Tsunami. Bull. Earthquake Res. Inst., 41: 535-571. Lamb, H., 1932. Hydrodynamics, 6th edn. Dover Publications, New York, p. 274. Tuck, E.O., 1979. Models for predicting Tsunami propagation. Proc. Of the Nat. Sci. Found. Workshop on Tsunamis May 1979, Coto de Caza, CA, pp. 43-104.