Generation Mechanism of Higher Mode Nondispersive Shelf Waves by Wind Forcing

Transcription

1 Journal of Oceanography Vol. 49, pp. 535 to Generation Mechanism of Higher Mode Nondispersive Shelf Waves by Wind Forcing YUTAKA ISODA Department of Civil and Ocean Engineering, Ehime University, Bunkyo 3, Matsuyama 790, Japan (Received 26 October 1992; in revised form 24 March 1993; accepted 24 March 1993) The higher mode predominance in the current velocity fields associated with windinduced shelf waves in the nondispersive regime is studied with a special attention to the effect of the geographical boundary, e.g. wide strait or wide bank areas. The effect of such large topographic change is represented by wind forcing with a finite dimension near the geographical boundary. The time development processes of the wind-induced shelf waves is examined in the context of an initial-value problem, where a spatially finite wind stress is applied at t = 0. Various modes of shelf waves excited at the boundary start propagating simultaneously and develop monotonically within the forcing region. After the passage of such wave, the energy of wind is used to maintain the attained equilibrium condition, i.e. the steady shelf circulation. The current evolution of the lower mode is restricted to the earlier stage because of the large propagation speed. In contrast, the higher mode waves can travel slowly within the forcing region so that the kinetic energy is supplied from wind stress for a long time before the equilibrium condition is established. Consequently, the observation at the fixed point near the geographical boundary would show that the higher mode waves gradually dominate as time goes on, i.e. for the long-term forcing. 1. Introduction It is well known that the shelf waves are generated by wind stress over the continental shelf and slope in the world oceans (see Mysak, 1980 for detailed reviews). Adams and Buchwald (1969) firstly demonstrated that the primary generation mechanism of shelf waves was the alongshore component of wind stress. Recently, it becomes clear that the wind-induced shelf waves have the unique property such that the higher modes are frequently dominant in the current velocity fields. Accordingly, theoretical studies for the higher mode predominance of shelf waves have been carried out from the following two directions. One is that the dispersive effects may influence the response of the higher mode waves, i.e. waves with shorter wavelength. In particular, an importance of the higher mode with zero group velocities has been often indicated, e.g. off Oregon coast (Cutchin and Smith, 1973), in the Florida Straits (Brooks and Mooers, 1977), off North Carolina (Brooks, 1978) and on Scottish shelf (Gordon and Huthnance, 1987). The other is to study the generation of the higher mode waves in the nondispersive regime, i.e. waves with larger wavelength. Actually, the higher modes of nondispersive shelf waves (hereinafter referred to as non-dsws) have been observed in the following shelf areas, e.g. off the Pacific coast of North Japan (Kubota et al., 1981; Kubota, 1982, 1985), on the East Australian shelf (Church et al., 1986), off South Africa (Schumann and Brink, 1990), off the west coast of New Zealand (Madelein et al., 1991) and off San in coast in Japan (Yanagi et al., 1984; Isoda et al., 1992). Table 1 is a summary of their observational evidences

2 536 Y. Isoda and the possible theoretical explanation for each of them. It is found that, in most cases, the current fluctuations are mainly caused by the 2nd mode waves, though the proposed possible explanation differs depending on each shelf area. The important property for the observed non-dsws is that the origin of their propagation can be clearly detected because the phase velocity is equal to the group velocity. Figure 1 shows the shelf bathymetry taken up in Table 1 and the wave propagation feature inferred from the analysis using coastal sea level data or current data at some alongshore points. Each map is drawn in a circle with 2000 km in diameter, which roughly coincides with the spatial scale of an anticyclone moving at mid-latitude. These continental shelf and slope region possess a geographical boundary of the deeply incised coastline or rapid change of shelf topography. From a view point of such boundary within the forcing of an anticyclone, it can be expected that the actual forcing Table 1. A summary of the evidence for the higher mode predominancy. Observed shelf region Analyzed current data Theoretical summary for the possible cause of higher mode predominancy in current off the Pacific coast Kubota et al. (1981) 3rd mode wave with zero group velocity of North Japan 2nd and 3rd mode; generated by broad-band wind forcing 0.8 to 1.4 m s 1 2nd mode wave generated depended on the particular cross-shelf topography with appropriate width of shelf slope in the East Church et al. (1986) (eigenvalue) Scattering of energy from a 1st mode to Australian shelf 1st mode (10%) 3.0 m s 1 higher-order modes 2nd mode (37%) 1.8 m s 1 or The amplitude of each mode is determined by the ratio of the width of the Bass Strait to wavelength of shelf waves off South Africa Schumann and Brink (1990) Resonance associated with the near 2nd mode; 8.0 m s 1 agreement in propagation speeds between free-shelf waves and wind system off the west coast of Madelein et al. (1991) As the same case of East Australia, the New Zealand on-offshore flow structure of importance of the Cook Strait as possible 1st and 2nd mode source for the shelf waves off San in coast Yanagi et al. (1984) Present study in Japan 2nd mode; 1.0 m s 1 Isoda et al. (1992) on-off shore flow structure of 1st mode (2.5 day period) and 2nd mode (5.5 day period)

3 Generation Mechanism of Higher Mode Nondispersive Shelf Waves 537 Fig. 1. Bottom topography and coastal geometry in the world s shelf regions where non-dsws were detected by observed evidence of sea level and current data. The black areas show the depth is less than 200 m. The contours for 1000 m and 2000 m depths are indicated by solid lines. White circles show the current observation points. Dashed line and arrow show the propagated distance and the direction of observed non-dsws. The mark shows the start point of wave propagation which has been detected mainly by sea level data analysis. would work with the limited spatial scale of weather system. Then, it is found that the observed shelf waves start propagating from these geographical boundaries; e.g. the Tsushima/Korea Strait in Japan (Isozaki, 1968; Isoda et al., 1991), the Tsugaru Strait in Japan (Isozaki, 1969), the Agulhas Bank in South Africa (Schumann and Brink, 1990), the Cook Strait in New Zealand (Madelein et al., 1991) and the Bass Strait in Australia (Church and Freeland, 1987). In the present study, non-dsws generation by the wind forcing with a finite dimension near the geographical boundary is studied theoretically and the result is applied to investigate the possibility of the 2nd mode wave predominance in our study area off San in coast. 2. Theoretical Model with the Local Wind Forcing 2.1 General formulation and solution We follow Gill and Schumann (1974) and consider a nondivergent motion trapped near the

4 538 Y. Isoda coast over the shelf topography spreading only in the region y > 0, i.e. H = H(y), and a coastline which, at least locally, may be taken to be straight. We use the linearized shallow-water wave equation governing an inviscid, homogeneous and incompressible fluid under the hydrostatic approximation, with the bottom friction and internal dissipative forces being neglected. Although the horizontal extent of the wind forcing is limited due to the geographical boundary, it is considered that this forcing region still has a larger horizontal scale compared to the shelf width (L). In this case, alongshore wind stress is assumed to be independent of y and the response is nondispersive, allowing the use of the long-wave approximation. Then, the governing vorticity equation and the boundary conditions are t y 1 H ψ y f dh ψ H 2 dy x = dh dy ψ = 0 at y = 0, Xx,t, H 2 respectively, where ψ is the stream function defined by ψ y = 0 at y = L 1 u = 1 H ψ y, v= 1 ψ H x ( 2) u and v are the depth-averaged components of velocity in the x (alongshore) and y (cross-shelf) directions respectively; t is time; H(y) is the water depth; X(x, t) is the alongshore component of wind stress and f is the Coriolis parameter which is assumed constant. We consider the impulse forcing at t = τ (>0), i.e. X(x, τ)δ(t τ), where δ is the Dirac delta function. If U(x, y, t τ) is the partial oscillation in response to the impulse forcing, the whole oscillation at t can be obtained by integrating U(x, y, t τ) with respect to τ from τ = 0 to τ = t, t = Ux, y,t τ ψ x, y,t dτ ( 3) 0 where U is the impulse response function. The advantage of this mathematical formulation using this function is that the physical characteristics of wind-induced non-dsws can be described without assuming a traveling wave-like solution in advance. The vorticity equation and the boundary conditions for the oscillations in response to the impulse forcing is written as t y 1 H U y f dh U H 2 dy x = dh dy U = 0 at y = 0, Xx, ( τ)δ ( t τ), H 2

5 Generation Mechanism of Higher Mode Nondispersive Shelf Waves 539 Integrating (4) with respect to t from t = τ ε to t = τ + ε produces y U y = 0 at y = L. ( 4 ) y 1 U x, y,0 H = dh dy Xx, ( τ) ( 5) H 2 whereas the motion is assumed to be at rest for t < τ, so that U(x, y, t τ) = 0 for t < τ. Free oscillation equation is given from (4) such that t y 1 H U y f dh U H 2 dy x = 0. ( 6) Consequently, the problem is now reduced to the initial-value problem, namely, to solve free wave equation (6) under the initial condition (5). The response at t can be calculated by using the relation (3). The solution of U can be expressed in the form; U = U n x, y,t τ = φ n x,t τ F n y n=1 n=1 ( 7) where F n (y) is a complete set of free wave eigenfunctions of n-th mode. From (6) and (7), F n satisfies the equation for the eigenvalue problem d dy 1 H df n dy + f c n F n H 2 dh dy = 0 8 where c n is the phase velocity (or the group velocity) of the n-th mode free wave. It can be easily shown that the function F n satisfies the orthogonal relation; L dh 1 dy H F 2 n F m dy = δ 0 mn 9 where δ mn is the Kroneker delta function. Next, the wave propagation equation is also found from (6) and (7). 1 φ n + φ n c n t x Substituting (7) into (5), and integrating from y = 0 to y = L give = 0. ( 10 )

6 540 Y. Isoda where φ n ( x,0) = Q n Xx, ( τ) ( 11) Q n ~ ( df n ( 0) / dy) 1 ( 12) since H(0) >> H(L) in general. Q n is the energy flux ratio of the impulse forcing X(x, τ) to the n-th mode and affects the growth rate of non-dsws directly. The magnitude of Q n depends on the cross-shelf mode structure of F n near the coast. This means that the forcing energy is effectively supplied to the fluid at the coast (y = 0) and is mostly transferred to the lowest mode which has no node in on-offshore direction. An elementary solution of (10) describes the propagation of waves with the coast on the right in the northern hemisphere ( f > 0) φ kn ( x,t τ) = B n ( k)sink{ x c n ( t τ) }. 13 The general solutions can be calculated by integrating with respect to k such that φ n ( x,t τ) = φ kn dk. 14 By taking the Fourier transform of (11) with respect to x and comparing with (13) at t = τ, the function B n (k) can be obtained as follows 0 B n ( k) = 2 π Q n Xx, ( τ) sinkxdx 15 0 where B(k) = B n (k)/q n shows the wavelength distributions which are determined by the spatial scale of the wind forcing. Consequently, the current amplitude φ n (x, t) can be written in the form; φ n ( x,t) = B n k 2.2 Solution for the constant wind forcing We consider the shelf topography with an exponential profile t 0 0 sink{ x c n ( t τ) }dkdτ. ( 16) = h 0 e2λy Hy h 0 e 2λL 0 y L. 17 L < y For the shelf off San in coast (along the line-a in Fig. 6(a)), typical values of these parameters are h 0 = 50 m, λ = km 1, L = 160 km. For the depth profile (17), the offshore current structure is easily obtained from (8) under the boundary condition at the coast, F n = 0 at y = 0, and matching condition at the shelf edge, df n /dy = 0 at y = L, such that

7 Generation Mechanism of Higher Mode Nondispersive Shelf Waves 541 F n ( y) = A n e λy sinβ n y where, tanβ n L = β ( n / λ ), c n = 2λf /β ( 2 n + λ 2 ). The dispersion relation for the lowest three modes are plotted in Fig. 2. The solid (dotted) lines show the nondispersive (dispersive) case. When the wavelength is larger than about 500 km, most of the shelf waves off San in coast considered are nondispersive. The amplitude of F n are determined by the normalization (9) so that [ / { λl( β 2 n + λ 2 ) + λ 2 }] 1/2. A n = h 0 β n 2 + λ 2 The growth rate Q n is given by (12), Q n ~ A n c n 2 β n /h 0 f 2. The values of c n and Q n in the case of San in coast are given in Table 2 for n = 1, 2 and 3. Q n of the 2nd and 3rd modes are 39% and 24% of that of the 1st mode, respectively, so that the higher mode predominance cannot be explained by the growth rate. Fig. 2. The dispersion curves for the lowest three modes of shelf waves in the nondispersive (solid lines) and dispersive (dotted lines) cases. ω, frequency; f, the Coriolis parameter; k, alongshore wavenumber; h, depth (200 m) and g, gravity acceleration.

8 542 Y. Isoda Table 2. The coefficients of c n and Q n for n = 1, 2 and 3. n c n (m s 1 ) Q n ( 10 2 ) Over the shelf off San in coast, the weather systems generally consist of anticyclones moving eastward. Then, the edge of wind forcing is fixed relative to a geographic feature, i.e. the Tsushima/Korea Strait (see Fig. 1). In the present theoretical study, the effect of this large topographic change is represented by wind forcing with a finite dimension near the strait, so that direct effect of variations in topography is not considered. Furthermore, the propagation speed of forcing is assumed to be much larger than that of the generated waves, so that the resonances between the wind forcing and the generated waves do not occur (Isozaki, 1968; Yanagi et al., 1984). To show the situation more clearly, we assume a simple forcing that the spatially limited wind stress is applied parallel to the coast suddenly at t = 0, Xx,t = X a = constant for 0 < t and for 0 < x < x 0. Here, we adopt x 0 = 300 km which corresponds to the distance between two characteristic geographic features, namely, the Tsushima/Korea Strait and the Oki Islands (see Fig. 6(a)). The wavelength distributions B(k) generated by the above local wind forcing is; Bk = 2X a ( coskx 0 1) /kπ ( 18) which is sketched in Fig. 3. The wavelength of most energetic generated wave is 2x 0 (=600 km), namely, twice the forcing scale. Figure 4 shows time evolution of φ n (n = 1, 2 and 3) with no initial perturbation. Various modes of non-dsws excited at the boundary of the forcing region start propagating simultaneously with corresponding propagation velocities c n. The current amplitude φ n of each mode monotonically increases at the growth rate Q n within the forcing region. It is clear that φ n in a final state becomes larger as mode number increases, though the growth rate is smaller. The time which takes for the amplitude of each mode wave to reach the maximum depends on the parameter 2x 0 /c n, e.g. 1.2, 5.8 and 13.8 day for the 1st, 2nd and 3rd mode, respectively. This suggests that the forcing period determines which mode of non-dsws will be dominant. After the passage of each mode, the equilibrium condition is attained within the forcing region. As time t, the vorticity equation (1) is simplified to β T v = d X dy H where β T = f(dh/dy)/h is the topographic-β. Namely, the curl of the net external wind force vanishes with the cross-isobath transport. The fluid particles then move along isobaths.

9 Generation Mechanism of Higher Mode Nondispersive Shelf Waves 543 Fig. 3. The distributions of wavelength for the waves generated by a local forcing with a spatial scale x 0. The sketches are based on (18) for B(k), where k is the wave number. Fig. 4. Time evolutions of φ n for n = 1, 2 and 3 after the sudden onset of local wind forcing.

10 544 Y. Isoda Fig. 5. Time variations of φ n for n = 1, 2 and 3 under the periodical local wind forcing for the same parameter set in Fig. 4. (a) 2.5 day period (b) 5.0 day period. Consequently, the steady shelf circulation will be gradually formed under the wind forcing. It is also seen that φ n has non-zero values at x = 0. This suggests that disturbances affect in the negative direction from x = 0. In order to treat the flow at x < 0, exact response of the dispersive effect is more important (Martell and Allen, 1979). In the present study, however, such treatments have not been made because the flow structure in the forcing region for x > 0 is determined mainly by

11 Generation Mechanism of Higher Mode Nondispersive Shelf Waves 545 nondispersive waves propagating only in the positive x-direction rather than dispersive waves those can propagate in the negative direction. 2.3 Solution for the periodical wind forcing Yanagi et al. (1984) showed that the current velocity disturbance with the period of about 5 days propagated from west to east along San in coast, at the speed of about 1.0 m s 1 as the 2nd mode of non-dsws. Isoda et al. (1992) detected two kinds of mode waves by examining the observed current structures along the line-a in Fig. 6(a). One is the 2nd mode variations at the period of about 5.5 days, for which current vectors show clockwise rotation at the shoreward station and anticlockwise rotation at the seaward station. The other is the 1st mode variations at the period of about 2.5 days, for which current vectors show clockwise rotation at both stations. Then, two kinds of wind forcings with 2.5 and 5.0 day periods, i.e. X(x, t) = X a sinωt where ω = 2π/(each forcing period), are employed in order to investigate the ocean response to the realistic periodical forcing with the weather-band frequency. Figure 5 shows time variations of φ n (n = 1, 2 and 3) under the forcings with (a) 2.5 and (b) 5.0 day periods, for the same parameter set in Fig. 4. It is clear that the current amplitude φ n is very sensitive to the period of forcing. For example, the 1st mode evolves only under the forcing of 2.5 day period, whereas the 2nd mode becomes more energetic than the 1st mode under the forcing of 5.0 day period. However, the 3rd mode has not evolved in both cases because this mode cannot fully develop under the forcing period of which is less than about 5 days. These results demonstrate the 2nd mode predominance for the long-term forcing of 5.0 day period or more, simulating at least qualitatively the observed features off San in coast. Now we consider the possibility of the higher mode predominance in the current velocity fields based on the above mentioned mode characteristics. Following the sudden onset of the wind forcing parallel to the coast, various mode non-dsws are excited and start propagating from the boundary of the forcing region. In this case, the kinetic energy of wind is effectively supplied to a fluid through the coastal boundary and the growth rate in the lower mode waves tends to be larger. However, the evolution of the lower mode waves is restricted to the earlier stage because of its larger propagation speed. Under the long-term forcing, the higher mode waves gradually dominate as time goes on because they can travel slowly within the forcing region. 3. Numerical Experiment under the Realistic Topography off San in Coast In order to apply the theoretical results of the previous section to the realistic situation, it is convenient to carry out a numerical experiment. In the shelf region off San in coast, a shallow and gently sloping continental shelf extends eastward from the Tsushima/Korea Strait as shown in Fig. 6(a). The relative wide shelf region is developed from the Tsushima/Korea Strait to the Oki Islands. The bottom slope of the shelf becomes steeper to east of the Oki Islands. In this section, using a numerical model under the realistic bottom topography and periodical wind forcing, we clarify the possibility of the 2nd mode predominance of shelf waves. 3.1 Model description Model geometry and governing equations Figure 6(b) shows the model geometry. The model geometry is simplified to some extent though the fundamental features of the real topography in the western half of the Japan Sea is included. We use the linearized shallow-water equation of motion and continuity equation. In a

12 546 Y. Isoda Fig. 6. (a) The bottom topography of the Japan Sea. The line-a denote the current observation sites off San in coast by Isoda et al. (1992). (b) Basin configuration and bottom topography (thin contour lines). The dotted thick lines indicate open boundaries. The enclosed area corresponds the model domain off San in coast which is used to present the instantaneous field for current and sea level in Fig. 7. Cartesian coordinate system with horizontal axes x and y directed as shown in Fig. 6(b), these are written as u τ + fk u = g η + A h 2 u + Xx,t H, η + Hu τ = 0 u is the vertically averaged horizontal velocity with components u and v; is the horizontal differential operator; η is the sea surface elevation relative to the equilibrium level, H is the depth; A h (=10 6 cm 2 s 1 ) is the horizontal eddy viscosity; f (= s 1 ) is the Coriolis parameter at 36 N; g (=9.8 m s 2 ) is the acceleration of gravity; X(x, t) is the alongshore wind stress; k is the unit vector directed positive upward. We use a grid spacing of 20 km on a grid and a time step of 360 seconds Driving force The ocean is initially at rest. We examine the ocean response after the sudden onset of the alongshore wind stress X(x, t) = X a sinωt for the same periodical forcing as in Fig. 5. The amplitude X a is 1 dyne cm 2. The horizontal extent of forcing is limited to 1200 km (Fig. 6(b)) corresponding to that of the weather system.

13 Generation Mechanism of Higher Mode Nondispersive Shelf Waves Boundary conditions Along the coastal boundaries indicated by the solid lines in Fig. 6(b), slippery conditions are applied. In the vicinity of the open boundaries indicated by the broken lines, we assume the constant water depths of 100 m and 500 m, respectively. At these boundaries, we apply the Sommerfeld s radiation conditions since any disturbances can be transferred to gravity waves. 3.2 Results of the model experiment Figure 7 shows a time series of the instantaneous field for the current vectors and sea level in the cases of periodical forcing with 2.5 (a) and 5.0 (b) day periods. A time interval between two subsequent figures is a quarter of each forcing period and these four phases cover one forcing cycle. Fig. 7. Time series of the instantaneous field for the current vectors and sea level in the case of the periodical wind forcing for (a) 2.5 and (b) 5.0 day periods. A time interval between two subsequent figures is a quarter of each forcing period and these four phases cover one forcing cycle. The vortical mode waves denoted by symbols a1 and a2 (b1 and b2) show the generated 1st and 2nd shelf waves in response to the eastward (westward) wind forcing, respectively. The area between two dashed lines denotes the wide shelf region from the Tsushima/Korea Strait to the Oki Islands. The contour interval for sea level is 0.25 cm.

14 548 Y. Isoda The 1st and 2nd mode waves can be seen in both cases. These waves may be generated around the exit of the Tsushima/Korea Strait and propagate eastward along the shelf isobaths as time goes on. The horizontal scale of the generated waves is comparable to that of the wide shelf region from the Tsushima/Korea Strait to the Oki Islands, i.e. 200 km to 400 km alongshore scale. As the wind stress decreases, the features of 2nd mode waves (a2 and b2) can be clearly identified over the shelf near the exit of the Tsushima/Korea Strait. In particular, the evolution of 2nd mode waves for the long-term forcing of 5.0 day period is more significant. Such mode characteristic has been already predicted by the inviscid theory under the local wind forcing (Fig. 5). 4. Conclusion Theoretical and numerical models have been developed to explain why the higher mode non- DSWs can be dominant in a realistic situation in the ocean. In the present study, we have focused our attention on a situation where the boundary of forcing region is determined by the geographical conditions, e.g. wide strait or wide bank areas. The scenario for the generation mechanism of higher mode non-dsws is as follows. Each mode wave excited at such boundary within the forcing region starts propagating after the sudden onset of the wind forcing parallel to the coast. At that time, the wavelengths of all modes are the same, i.e. twice the limited forcing scale. The kinetic energy of winds is effectively supplied to a fluid through the coastal boundary. Accordingly, its growth rate depends on the flow structure near the coast and tends to be larger in the lower modes. Then, the current amplitudes within the forcing region increases monotonically. After the passage of each mode wave, the energy of wind is used to maintain the equilibrium condition, i.e. the steady shelf circulation, so that the acceleration of the current stops. The current evolution of the lower mode, therefore, is restricted to the earlier stage because of the large propagation speed. In contrast, the higher mode waves can travel slowly within the forcing region so that the kinetic energy is supplied from wind stress for a long time before the equilibrium condition is established. Consequently, the observation at the fixed point near the geographical boundary would show that the higher mode non-dsws gradually dominate as time goes on. Under the real topographic conditions off San in coast in Japan, the numerical model reproduces the predominance of the 2nd mode non-dsws for the long-term forcing of 5.0 day period. Acknowledgements The author wishes to express his heartfelt thanks to Prof. T. Yanagi for his continuous discussion and encouragement. The author acknowledges Dr. T. Hibiya of Hokkaido University for his comments on earlier version of this manuscript and is also much indebted to Dr. T. Takeoka and Mr. H. Akiyama for their invaluable discussion and helpful comments. The numerical experiments were carried out on a FACOM M-770 of the Computer Center of Ehime University. References Adams, J. K. and V. T. Buchwald (1969): The generation of continental shelf waves. J. Fluid Mech., 35, Brooks, D. A. (1978): Subtidal sea level fluctuations and their relation to atmospheric forcing along the Carolina coast. J. Phys. Oceanogr., 8, Brooks, D. A. and C. N. K. Mooers (1977): Wind-forced continental shelf waves in the Florida Current. J. Geophys. Res., 82, Church, J. A. and H. J. Freeland (1987): The energy source for the coastal-trapped waves in the Australian coastal experiment region. J. Phys. Oceanogr., 17,

15 Generation Mechanism of Higher Mode Nondispersive Shelf Waves 549 Church, J. A., N. J. White, A. J. Clarke, H. J. Freeland and R. L. Smith (1986): Coastal-trapped waves on the East Australian continental shelf, Part II. Model verification. J. Phys. Oceanogr., 16, Cutchin, D. L. and R. L. Smith (1973): Continental shelf waves: Low frequency variations in sea level and currents over the Oregon continental shelf. J. Phys. Oceanogr., 3, Gill, A. E. and E. H. Schumann (1974): The generation of long shelf waves by the wind. J. Phys. Oceanogr., 4, Gordon, R. L. and J. M. Huthnance (1987): Storm-driven continental shelf waves over the Scottish continental shelf. Contin. Shelf Res., 7, Isoda, Y., T. Yanagi and H. J. Lie (1991): Sea-level variations with a several-day period along the southwestern Japan Sea coast. Contin. Shelf Res., 11, Isoda, Y., T. Murayama and T. Tamai (1992): Variabilities of current and water temperature due to meteorological disturbances on the shelf off San in coast. Bull. Coastal Oceanogr., 29, (in Japanese). Isozaki, I. (1968): An investigation on the variations of sea level due to meteorological disturbances on the coast of Japan islands (II), Storms surges on the coast of the Japan Sea. J. Oceanogr. Soc. Japan, 25, Isozaki, I. (1969): An investigation on the variations of sea level due to meteorological disturbances on the coast of Japan islands (IV), Storm surges on the Pacific and Okhotsk Sea coast of north Japan. J. Oceanogr. Soc. Japan, 25, Kubota, M. (1982): Continental shelf wave off the Fukushima coast, Part 2. Theory of their generation. J. Oceanogr. Soc. Japan, 38, Kubota, M. (1985): Continental shelf wave off the Fukushima coast, Part 3. Numerical experiments. J. Oceanogr. Soc. Japan, 41, Kubota, M., K. Nakata and Y. Nakamura (1981): Continental shelf wave off the Fukushima coast, Part 1. Observation. J. Oceanogr. Soc. Japan, 37, Madeleine, L. C., J. H. Middleton and B. R. Stanton (1991): Coastal-trapped waves on the west coast of south island, New Zealand. J. Phys. Oceanogr., 21, Martell, C. M. and J. S. Allen (1979): The generation of continental shelf waves by alongshore variations in bottom topography. J. Phys. Oceanogr., 9, Mysak, L. A. (1980): Recent advances in shelf wave dynamics. Rev. Geophys. Space Phys., 18, Schumann, E. H. and K. H. Brink (1990): Coastal-trapped waves off the coast of South Africa: Generation, propagation and current structures. J. Phys. Oceanogr., 20, Yanagi, T., Y. Isoda and N. Kodama (1984): The long period waves on the San in coast. Bull. Disa. Preven. Inst., Kyoto Univ., 27, (in Japanese).