J. Phys. Earth, 28, 531-541, 1980

PROPAGATION OF ON A LINEAR SLOPE BETWEEN TWO FLAT REGIONS. PART I EDGE

Hiroshi ISHII* and Kuniaki ABE** * Observation Center for Earthquake Prediction , Faculty of Science, Tohoku University, Sendai, Japan **Niigata Branch , Faculty of Dentistry, Nippon Dental University, Niigata, Japan (Received August 11, 1980; Revised December 18, 1980)

An edge wave traveling along a with a sloping region sand- witched by a flat shelf and an bottom is investigated theoretically by esti- mating the relation and the amplitude distribution versus the distance from the coast. The results are discussed in comparison with a step shelf model. It is demonstrated that dispersion curves shifted toward a long period range with decrease of a dip angle of a sloping region and that the minimum point of a group velocity curve becomes less clearer in the case of a sloping model. It is found that the maximum amplitude of the Kamchatka tsunami observed in Japan can be accounted for by an edge wave propagated along the shelf. The result is useful for predicting arrival time of the maximum amplitude for a distant tsunami with an oblique incidence to the continental shelf.

1. Introduction

Edge wave has so far been studied by many investigators. For examples, SEZAWAand KANAI(1939) pointed out an analogy between edge wave on a shelf with a step and Love wave in seismology. REID (1958) estimated the effect of force on an edge wave on a sloping shelf. NAKAMURA(1962) theo- retically investigated generation of edge wave on a continental shelf for a cylindrical source and obtained a theoretical marigram. HATORIand TAKAHASI(1964) applied the dispersion relation by NAKAMURA(1962) to the Iturup tsunami of 1963. AIDA et al. (1968) investigated an edge wave traveling along a curved shelf experimentally. MUNK et al. (1964) discussed trapped and leaky modes on the continental shelf off California using the shelf profile approximated by a series of steps. Generally a typical continental shelf is approximated by a linear slope and a fairly uniform profile over a long distance. Thus a linear sloping model is considered to be ap- propriate for a typical shelf like the Pacific shelf of north east Japan. Therefore, it is of significance to obtain analytical solutions for a linear sloping model between two flat regions. Such analytical solutions make it easy to investigate the property of the model and to compare theoretical results to observed ones.

531 532 H. ISHII and K. ABE

2. Theory

Let us express a continental shelf with a sloping bed as shown in Fig. 1 in which Cartesian coordinates (x,y,z) are used. The y-axis is taken parallel to the coast line, while the x-axis is directed off shore with the origin at the margin of the shelf. The z-axis is taken vertically upward. The depths of the shelf, sloping bed and outer ocean are denoted by h1, h2, and h3, respectively. The depths h1 and h3 are assumed to be constant and h2 is assumed to be expressed by:

Fig. 1. Coordinate system and a model of continental shelf.

h2=h1+xtanθ, in which θ is the dip angle of a sloping bed. The widths of the shelf and the sloping bed are denoted by l and a, respectively.

It is known in the theory of long wave that the surface elevations ζ1, 2, 3 above the mean level satisfies.

(1)

where t and g represent the time and the acceleration of gravity, respectively. In these equations suffixes 1, 2 and 3 indicate the quantities for the shelf, sloping bed and outer ocean, respectively.

After eliminating time factor eiωt, which is introduced by the assumption of purely periodic motion, a set of solutions are assumed in the form of plane as (2)

(3)

(4) where Propagation of Tsunami on a Linear Slope between Two Flat Regions 533

In these expressions c, ω, k, Ln(2λ) and F(-n:1:2λ) are the longshore phase veloc- ity, angular frequency and wave number, Laguerre polynomial and confluent hypergeometric function, respectively. Elevation ζ1 represents the waves reflected at the coast and the margin of the shelf, propagating toward the positive y direction while ζ2 is considered to be a kind of the stationary wave on the linearly sloping bed. The first term in ζ2 is equivalent to the solution derived by REID (1958) et al. and the second term is the one diverging when the depth is 0. Elevation ζ3 corresponds to the wave decaying in the outer ocean. Amplitudes A1, A2, B1, B2 and C are the constants to be determined from the boundary conditions. The boundary conditions are derived from the continuities of elevation and the flow normal to the y-axis at x=0 and a. At the coast the flow normal to the coast line is zero because a total reflection is assumed. These are written as

(5)

(6)

(7)

When formulae (2), (3) and (4) are substituted into (5), (6) and (7), five simul- taneous equations are obtained including five unknown parameters A1, A2, B1, B2 and C. A dispersion equation is derived from the existence condition of a non- trivial solution of the equation. The equation is written as follows:

(8)

where

(9)

(10) (11)

(12)

in which we use the Laguerre polynomial and the confluent hypergeometric function are 534 H. ISHII and K. ABE

(13)

(14) respectively, their derivatives with dash and

λ1=2kh1/tanθ, (15)

λ2=2K(a+h1/tanθ). (16)

Transcendental equation (8) including polynomials (13) and (14) is solved numerically, so that an ω versus c relation can be determined. The c-ω curve thus obtained is called the dispersion curve for the phase velocity. The longshore group velocity U can then obtained by making use of the following formula:

(17)

The U-ω curve is called the dispersion curve for the group velocity. The dispersion curves represent the propagation property of the edge wave excited. The ampli- tude distribution against the distance from the coast can also be obtained by solving simultaneous equations (5), (6) and (7) for A1/C, A2/C, B1/C and B2/C, and substituting c-ω values obtained by the dispersion relation into them.

3. Dispersion Relation

The dispersion curves are illustrated for the various sloping angles in Fig. 2. The curve for dip angle 90° was obtained by NAKAMURA (1962). Parameters h1, h3 and l are fixed to be 0.6, 6.0 and 50km, respectively. Only the fundamental mode is plotted and the other higher modes are not considered in this paper, because only waves having a long period are to be dealt with here. In the figure, c and ω are normalized as c/c1 and ωl/c1(=γ) respectively. It is seen that the phase velocity curve takes on a value for γ=0 and approaches asymp- totically to 1 for short-wave limit γ→ ∞. From c2/c1 to 1 the curves decrease monotonically and the point of inflection moves toward γ=0 along with a decrease of the sloping angle. It is found that the result for 90° agrees with the limiting case when the sloping angle is increased. The differences between the limiting case and the sloping one are obtained as

-3% for 15°

-10% for 5 .7°

-37% for 1 .8°, Propagation of Tsunami on a Linear Slope between Two Flat Regions 535

Fig. 2. Dispersion curves of the fundamental mode edge wave for various dip angles. Phase (c) and group (U) velocities are shown by the solid and broken lines, respec- tively. when γ=1 is assumed.

It is seen that the group velocity curve takes on a value for γ=0, and approaches asymptotically to 1 for short-wave limit γ=∞, A striking feature is the occurrence of a minimum value. It is found that the position of the minimum shifts towards a small value of γ with decrease of a dip angle. It is also shown that the case for 90° is the limiting case when the dipping angle is increased. The relative differences between the limiting case and the sloping ones for γ=1 are expressed as follows:

-12% for 15°

-35% for 5 .7°

-61% for 1 .8°.

The relative difference is larger than that of the phase velocity.

4. Comparison with Step Models

It is interesting to investigate the role of the width of a flat shelf on the edge wave propagation. The dispersion curves for step models with three different 536 H. ISHII and K. ABE

Fig. 3. Comparison of sloping models with step models of various widths. Phase (c) and group (U) velocities are shown with the solid and broken lines, respectively. For both the models the depths h1 and h3 are fixed to be 0.6 and 6.0km, respectively. widths are compared with those of the sloping models having a gentle slope amounting to only 1.8°, such a slope model corresponding to the case off north east Japan so called the Sanriku District. The sloping case is characterized by the constants of 0.6km, 6.0km, 1.8° and 50km for h1, h3, θ and l, respectively. The dispersion curves for various cases are shown in Fig. 3. Les us take the follow- ing three cases for step model. Case II, for which the shelf takes the shortest width amounting to 50km, is the limit of the above sloping model when the slope angle is increased. For Case IV, the shelf width which takes on a value amounting to 220km, the largest value among the models considered, is equal to the horizontal width of sloping shelf plus that of flat shelf which amounts to 50km. The shelf width for Case III is given by the sum of width of flat shelf, i.e. 50km, and the equivalent width defined by the following equation:

(18) Propagation of Tsunami on a Linear Slope between Two Flat Regions 537

The right side of (20) expresses the travel time through the sloping region. It is found that the best fit among these step models is the case of 91km (III). It is apparent that sloping models produce fairly gentle variation of dispersion curves for the phase and group velocities, so that it is no easy matter to identify the value of ω at which the group velocity is minimized. We can also compute the variation of water elevation versus distance from the coast on utilizing the dispersion relation. The distribution of water elevation is

presented in Fig. 4 for the two values of γ of minimum group velocity in the case of the sloping and the step models. Case I is employed for the sloping model and Case III for the step model. An amplitude pattern of cosine type is seen for the flat shelf and that of exponential decay type for the outer ocean. These two models are similar in their dispersion relations as shown in Fig. 3. Nevertheless, the sloping model has an amplitude ratio larger than that of the step model. It is of much interest that such a difference is obtained in the amplitude ratio.

Fig. 4. Amplitude of the fundamental-mode edge wave along the x-axis for the slop- ing model (above) and the step model (below).

5. Application to the Actual Tsunami Data

The results presentedin this paper were applied to accounting for the Kam- chatka tsunami of Nov. 5, 1952. The maximum amplitude wave arrived atthe 538 H. ISHII and K. ABE

Fig. 5. Observation points, and the source area illustrated by the rectangle of KANA- MORI'Sfault model (1976). The chain line shows a base line of the epicentral distance.

Fig. 6. of the cross-section of the real shelf and the sloping model used.

coast of north east Japan at 3-10hr after the initial arrival. The observation points face to the continental shelf of north west Pacific ocean, so that the present results are suitably applied. The observation points employed here are shown in Fig. 5. They are Kushiro, Miyako, Ayukawa, Onahama and Mera from north to south. The bathymetry of the real shelf and the idealized one are compared in Fig. 6. This is the case off Miyako. We employ this model with h1 nearly equal to the average value through the propagating path, since the minimum group velocity is controlled by h1, the depth of the shelf. It may be said, to a high degree of approxi- mation, that the case of dip angle 1.8° as computed in Fig. 2 corresponds to this model. Although north east Japan has a very irregular coast line with bays and peninsulas, the assumption of straight coast line seems to hold good when the ex- tremely long wavelength of the tsunami is taken into account. The original records necessarily contain the ocean and bay super- posing on the tsunami. The former is excluded by the use of the tide table and the latter with a period shorter than 30min by smoothing the original records. The records thus filtered are shown in Fig. 7. From top to bottom in the figure, the records are shown at receptive observation points according as the distance from the origin along the curved shelf, while the abscissa indicates the travel times from the origin of the mainshock. The solid line and dotted one show the travel time curves of the edge wave propagating from the epicenter and the fault margin, Propagation of Tsunami on a Linear Slope between Two Flat Regions 539

Fig. 7. Time histories observed for the Kamchatka tsunami of 1952, after excluding the ocean tide and the component having a period shorter than 30min. From top to bottom the records at observation points are arranged according as the distances from the origin along the curved shelfs. The abscissa is the lapse time measured from the origin time of the mainshock. Both the solid and broken lines show the travel time curve of 65m/sec in which the former is assumed to be the epicenter and the latter is assumed to be a fault marginal one with the epicentral distance of 610km. respectively. The positions of the epicenter and the fault assumed are shown in Fig. 5. Both the travel times are associated with a constant velocity of 65m/sec which corresponds to the minimum group velocity in the dispersion curve computed 540 H. ISHII and K. ABE for the profile off Miyako as shown in Fig. 6. Since the tsunami energy is trans- ported with a group velocity, it is expected that a large amplitude wave propagates with the minimum group velocity. It is found that the maximum phases observed at Ayukawa and Onahama can be explained by the wave propagating with a velocity equal to the minimum group velocity from the epicenter. Particularly, the wave train of Onahama shows a dispersive property, in which the maximum phase is attained through a gradual increase of amplitude. For the other obser- vation points we cannot find the maximum amplitude phase at the expected travel time. These observation points are located on a curved shelf, where the maximum phase may not well develop. AIDA et al. (1968) demonstrated an amplitude insta- bility of the trapped mode on a curved shelf from a model experiment. Next, the dispersion data are plotted along with the theoretical curve in Fig. 8. From this figure it is found that some phases including the maximum amplitude agree well with the theoretical curve. Therefore, it is recognized that the 4-9th wave trains propagate as the edge wave. It is concluded that the maximum phases observed at Onahama and Ayukawa are identified as the edge wave gener- ated at the epicenter and propagating along the shelf.

Fig. 8. Observed group velocities for various observation points and the computed group velocity curve shown by a solid line.

6. Conclusion

The problem of edge wave propagating on a sloping shelf sandwitched by a flat shelf and ocean bottom is theoretically investigated using an analytical expres- sion derived. The dispersion relation and the amplitude distribution versus the Propagation of Tsunami on a Linear Slope between Two Flat Regions 541 distance from the coast are computed for the model of the Sanriku coast. It is clarified that dispersion curves move toward a long period range with decreasing dip angle of sloping shelf. It is also recognized that the minimum point of a group velocity curve becomes less clearer in the case of a sloping model compared with a step model. It is found that the Kamchatka tsunami (Nov. 5, 1952) observed in north east Japan can be accounted for by an edge wave propagating along the sloping shelf. Particularly the maximum phases at Ayukawa and Onahama are well ex- plained by the theory. It is also elucidated that the assumption of the straight coast line is reasonable for explaining the fundamental mode of edge wave. It is concluded that the present theory is useful for predicting the arrival time of the maximum amplitude wave due to the edge wave for a distant tsunami.

The computation is due to the Computer System HITAC 8800-8700 of the University of Tokyo. Thanks are also due to Miss N. Sakaki for help in preparing the manuscript.

REFERENCES

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