Scattering of Semidiurnal Internal Kelvin Wave at Step Bottom Topography

Journal of Oceanography, Vol. 61, pp. 59 to 68, 005 Scattering of Semidiurnal Internal Kelvin Wave at Step Bottom Topography YUJI KAWAMURA 1 *, YUJIRO KITADE and MASAJI MATSUYAMA 1 Tokyo University of Fisheries, Konan, Minato-ku, Tokyo 108-8477, Japan Faculty of Marine Science, Tokyo University of Marine Science and Technology, Konan, Minato-ku, Tokyo 108-8477, Japan (Received 1 November 003; in revised form 11 March 004; accepted 3 April 004) A three-dimensional, multi-level model was used to study the energy dissipation of semidiurnal internal Kelvin waves due to their interaction with bottom topography. A simplified topography consisting of a channel with an additional shallow bay was used to clarify the wave s scattering process. When the first mode semidiurnal internal wave given at an open boundary arrives at the bay mouth, higher-mode internal waves are generated at a step bottom of the bay mouth. As a result, the energy of the first mode internal Kelvin wave is effectively decayed. The decay rate of the internal Kelvin wave depends on both the width and length of the additional bay. The maximum decay rate was found when a resonance condition occurs the bay, that is, the bay length is equal to a quarter of wave length of the first mode internal wave on the shallow region. The decay rate in the wide bay cases is higher than that in a narrow case, due to a contribution from the scattering due to the Poincaré wave that emanates from the corners of the bay head. The decay rate with the additional bay is 1.1 1.8 times that of the case without the additional bay. The decay rate due to the scattering process is found to be of the same order as that of the internal and bottom friction. Keywords: Internal Kelvin wave, scattering, internal seiche, decay rate, resonance. 1. Introduction Internal tides are caused by the interaction of barotropic tides with bottom topography such as a shelf break or a sea mount in a stratified ocean (e.g. Baines, 198). Generally, the generated internal tidal waves are considered to decay due to vertical mixing or internal and bottom friction within a relatively short distance from the generating region. Internal tidal waves having a beamlike structure require a large number of vertical modes in the generating region. As the wave propagates away from the generation region, the higher mode internal waves decay rapidly (Prinsenberg and Rattray, 1975). Craig (1991) reported that the internal wave energy is dissipated by bottom friction and an eddy viscosity, which is approximately proportional to the inverse of the vertical mode number. On the other hand, Müller and Xu (199) explained theoretically that the scattering of an internal * Corresponding author. E-mail: kawamura@dpc.ehime-u.ac.jp Present address: Center for Marine Environmental Studies, Ehime University, -5 Bunkyo-cho, Matsuyama 790-8577, Japan. Copyright The Oceanographic Society of Japan. wave on a bump contributes to the energy dissipation of the incident wave. The scattering process of internal waves is considered to play an important role in vertical mixing (e.g. Gilbert and Garrett, 1989). The internal tides generated at the shelf edge mostly decay before reaching the coast over a wide continental shelf region, so that the internal tides have the characteristics of a progressive wave over the wide shelf region (e.g. Sherwin, 1988). On the other hand, in a narrow shelf with a steep coast, as seen along the Japanese coast, the internal tides have been found to have the character of a standing wave, as shown by current and temperature measurements at moored stations (Matsuyama and Teramoto, 1985; Okazaki, 1990). The internal tides here can be considered to reflect at the coast with less dissipation (Kitade and Matsuyama, 1997). The semidiurnal internal tide in Uchiura Bay was reported to be predominant in summer and early autumn, and to be resonant with the internal seiche of the bay, as revealed by long-term temperature measurements (Matsuyama, 1985a) and numerical experiments (Matsuyama, 1985b). Recently, the scattering of a semidiurnal internal tide has been discovered from the analysis of current and temperature data in Uchiura Bay. 59

More than 10 percent of semidiurnal internal tide energy was estimated to be transferred to the higher mode internal wave (Kawamura et al., 00). This result leads us to expect that the scattering is an important process in the dissipation of the energy of internal tides in a narrow shelf coastal region. In this study we focus on the scattering process and energy dissipation of a semidiurnal internal tide in a coastal region. Semidiurnal internal tides have been frequently observed as an internal Kelvin wave in a channel, inlet and narrow bay (e.g. Webb and Pond, 1986). We selected a channel with a small bay having a step bottom at the bay mouth, so the scattering of the internal Kelvin wave occurs at the step.. Numerical Experiments A simplified model consisting of a linear channel with an additional bay was used to clarify the process by which an internal tide scatters at the step bottom, as shown in Fig. 1. The width and length of the channel, which has a water depth of 100 m, are 0 km and 60 km, respectively. The three-dimensional level model used permits exploration of the detailed vertical structure of the internal waves. Under the Boussinesq and hydrostatic approximations, the equations of motion, continuity and density for an incompressible fluid in the f-plane are written as follows: Fig. 1. Model configuration. where u, v and w are the eastward (x), northward (y), and vertical upward (z) velocity components, respectively, t the time, f the Coriolis parameter (8.36 10 5 rad s 1 ), p the perturbation pressure, g the gravitational acceleration (9.8 m s 1 ), A h and A v the coefficients of the horizontal and vertical viscosity, respectively, K h and K v the coefficients of the horizontal and vertical diffusivities, respectively, ρ the density, ρ 0 the basic density field, and ρ the perturbation density, ρ = ρ 0 + ρ. The symbol δ is the instantaneous convective adjustment parameter, used to maintain stable stratification in the model (e.g. Suginohara, 198). It is defined as follows: u + + + u u v u w u t x y z fv = + + 1 p u u + u Ah Av x, ρ x y z 0 () 1 ρ 1 for δ = > 0 z ρ 0 for 0. z ( 6) v + + + u v v v w v t x y z + fu = + + 1 p v v + v Ah Av y, ρ x y z 0 ( ) p = ρ g, () 3 z u x v w + + = 0, y z ( 4 ) ρ + ρ + ρ + ρ + ρ0 u v w w t x y z z = K h + ρ ρ + Kv ρ δ, x y z ( ) 5 The bottom boundary conditions are adopted as follows, ρ z = 0, at z = h ( 7 ) u v Av, CU B B ub, vb at z h 8 z z = ( ) = () where U B = (u B + v B ) 1/ is the current speed, with u B and v B are the eastward and northward components of velocity just above the sea bottom in the x and y directions, respectively, and C B is the coefficient of bottom friction (C B = 0.006). Equations (1) (5) are approximated by finite difference equations. The centered difference was applied to spatial difference using the Arakawa C grid. A leap-frog scheme was applied for time difference, and the Eulerbackward scheme was applied every 0 time steps to avoid numerical instability. 60 Y. Kawamura et al.

We examined the process by which the internal Kelvin wave scatters at the step topography of the bay. The first mode internal Kelvin wave was given at the western boundary. The current structure of the first mode internal Kelvin wave given at the western boundary is then: y U = U 0 exp z sin t, V 0, 9 λ ψ σ I () = ( ) where U and V are the eastward and northward currents, respectively, U 0 the amplitude of the eastward current, λ I the internal radius of deformation, ψ(z) the vertical dependence of horizontal current and σ the frequency of the wave. The constant buoyancy frequency (N =.0 10 rad s 1, N = (g/ρ 0 ) ρ 0 / z) was given for the basic density field. The vertical dependence, ψ(z), is then expressed as cosm(h + z), where m is the vertical wave number. The amplitude and frequency were set to U 0 = 5 cm s 1, and σ = 1.45 10 4 rad s 1 (semidiurnal period), respectively. A sponge condition at the eastern boundary allows the scattered waves to propagate out of the computational domain with little reflection. The slip condition was applied along the rigid northern and southern boundaries (Fig. 1). The grid size was taken as 500 m horizontally, and 5 m vertically. The other parameters were as follows; A h = 50.0 m s 1, A v =.6 10 4 m s 1, K h = 1.0 m s 1 and K v = 1.0 10 5 m s 1. The initial condition was given as no motion and no displacement. The numerical computation proceeded from an initial state of rest up to 7 hour later (t = 7 hours) with a time step of 0.5 second in all cases. The numerical experiments were performed under the different topographic conditions in the bay, as shown in Table 1. 3. Internal Wave Scattering Process The propagation of internal Kelvin waves through a channel was first investigated using a typical model with an additional bay, Case D3. The bay is a square with a side of 5.0 km, and the water depth is 50 m, i.e., a half the depth of water in the channel. The model also has a step bottom at the bay mouth. Figure shows the horizontal distributions of perturbation density at.5 m depth and of horizontal velocity at.5 m depth from 9 to 31 hours from the initial time, taken every hours. Both depths were chosen as representative of middle and upper layers in the bay, and the middle layer density and upper layer velocity were also chosen to be suitable to gain an understanding of the characteristic of the internal waves. The internal Kelvin wave given at the western boundary reaches the west side of the bay mouth at t = 11 hours. A part of the internal wave is then incident into the Table 1. Topographic conditions. Case A Case B Case C Case D A B C1 C C3 C4 C5 C6 C7 D1 D D3 D4 D5 D6 D7 L: Length of additional bay (km) 0 5.0 1.0 3.5 5.0 7.0 10.0 1.5 15.0 1.0 3.5 5.0 7.0 10.0 1.5 15.0 W: Width of additional bay (km) 0 5.0.5.5.5.5.5.5.5 5.0 5.0 5.0 5.0 5.0 5.0 5.0 h: Depth of additional bay (m) 0 100 50 50 50 50 50 50 50 50 50 50 50 50 50 50 Case E Case F E1 E E3 E4 E5 E6 E7 F1 F F3 F4 F5 F6 F7 L: Length of additional bay (km) 1.0 3.5 5.0 7.0 10.0 1.5 15.0 1.0 3.5 5.0 7.0 10.0 1.5 15.0 W: Width of additional bay (km) 10.0 10.0 10.0 10.0 10.0 10.0 10.0 15.0 15.0 15.0 15.0 15.0 15.0 15.0 h : Depth of additional bay (m) 50 50 50 50 50 50 50 50 50 50 50 50 50 50 Scattering of Semidiurnal Internal Kelvin Wave at Step Bottom Topography 61

Fig.. Distribution of horizontal velocity in the upper layer (.5 m depth) and perturbation density in the middle layer (.5 m depth) in Case D3. Contour interval is 0.004 kg m 3 for perturbation density. bay along the western side (t = 13 hours). The variations of the density distribution in the bay indicate cyclonic propagation of internal waves from 1 to 31 hours. The other part of the internal wave propagates eastward though the channel with the characteristic of the internal Kelvin waves. Figure 3(a) shows the distribution of current ellipses for a semidiurnal period calculated from the current data from t = 4 to t = 48 hours at both.5 m and 47.5 m depths in Case D3. The current ellipses at.5 m depth in the channel are mostly a straight line, parallel to the coast, except near the bay. The current ellipses near the bay mouth form an ellipse with the major axis direct against the bay mouth for internal wave propagation into the bay. In the bay, the current axis at.5 m depth is parallel to 6 Y. Kawamura et al. the bay axis, except at the bay head, and its amplitude gradually decreases from the bay mouth to the head. The current amplitude at 47.5 m depth in the channel is very small in comparison with that at.5 m depth except near the bay. However, the current amplitude at 47.5 m depth in the bay is approximately equal to that at.5 m depth. The significant feature is a steep increase in the current amplitude near the bay mouth at 47.5 m depth, i.e., corresponding to the bottom layer in the bay. Figure 3(b) shows the tidal current ellipses for a semidiurnal period at both.5 m and 47.5 m depths in Case B. The scale of the additional bay in Case B is the same as that in Case D3 except for the water depth, that is, the bottom is flat throughout the region in Case B. The remarkable difference between Case D3 and Case B

Fig. 3. Distributions of tidal ellipse in the upper and middle layers. (a) Case D3. (b) Case B. Fig. 4. Variations of vertical distribution for the perturbation density, and the eastward and northward components of velocity along Line A in Case D3. Contour interval is 0.015 kg m 3 for perturbation density and 0.5 cm s 1 for current velocities. Scattering of Semidiurnal Internal Kelvin Wave at Step Bottom Topography 63

Fig. 5. (a) Variations of vertical distribution for the northward component of velocity along Line A in Case D3. Contour interval is 0.5 cm s 1 for current velocities. Broken lines show the characteristic slope for semidiurnal internal wave. (b) Schematic view of internal wave, i.e., scattering wave, emanating from the edge of step bottom. is found in the current distribution at 47.5 m depth in the bay. The tidal ellipse at 47.5 m depth is comparable to that at.5 m depth in Case D3, whereas the former is very small in comparison with the latter in Case B. This implies that, in Case D3, the strong currents at 47.5 m depth are generated by the first mode internal Kelvin wave at the step bottom of the bay mouth. Figure 4 shows the vertical section of perturbation density, and the eastward and northward components of velocity along Line A (Fig. 1) from 1 to 1 hours, taken every 3 hours in Case D3. The process by which the higher mode internal wave is generated is clarified by the sequential patterns of the vertical section of the velocity and density across the step bottom. At t = 1 hours, the propagation of the internal Kelvin wave is recognized through the east-west currents in the upper and lower layers in the channel, respectively, and the negative density perturbation through the water column. At t = 15 hours, the density perturbation is negative in the section and the maximum value appears near the step bottom at the bay mouth. The eastward component of velocity at t = 15 hours becomes strong in the channel, and the eastward and westward currents show a maximum at the sea surface and bottom just outside the bay with step bottom, respectively. At t = 1 hours, the patterns of the eastward component of velocity and density perturbation in the channel are the inverse of those at t = 15 hours. These distributions of the density perturbation and currents clearly show the eastward propagation of the first mode internal Kelvin wave with semidiurnal period. Furthermore, the pattern of northward current differs from that of the eastward current. As the first mode internal Kelvin wave reaches Line A, it is found that the northward current with small vertical scale is caused near the edge of the step bottom (t = 15 hours). Figure 5(a) shows the vertical distribution of the northward component of velocity along Line A (Fig. 1) from 18 to hours in Case D3. The maximum current appears in the upper and/or lower layers in the bay, whereas a beam-like structure of the current is clearly found in the channel. The strong southward current region emanating from the edge of the step bottom spreads downward to the north and gradually shifts upward with time. The patterns of the northward component of velocity at t = 1 hours are almost the inverse of those at t = 15 hours (shown in Fig. 4). These features indicate the vertical propagation of internal wave, i.e., downward energy propagation and upward phase propagation, as shown in Fig. 5(b). Such a beam-like structure of the internal wave (hereafter we call this a scattering wave ) can be explained by the superposition of numerous internal modes under continuous density stratification (Rattray et al., 1969). 4. Modal Structure of Scattering Wave The distributions of the northward component of velocity associated with the scattering wave in the channel indicate the contribution of numerous internal modes, as mentioned in the previous section. The current data were decomposed into vertical mode of the semidiurnal internal wave to clarify the modal structures of the scattering wave. Assuming non-viscosity and non-diffusion, the solution of the linearized wave equation for the vertical displacement is expressed as φ = Φ (z) Z (x,y) e iσt where Φ and Z are the vertical and horizontal dependence, respectively, 64 Y. Kawamura et al.

Fig. 6. (a) Model configuration. Shaded area shows an area estimated energy density. (b) Distribution of the energy density of northward current for each vertical mode in Case D3. The energy density for each distance is averaged within the shaded area. and σ the frequency. The vertical dependence can be expressed as follows (e.g. Phillips, 1977), Fig. 7. (a) Model configuration. The kinetic energy density of the northward component of velocity is averaged in each Regions P and Q. (b) Kinetic energy density of the northward current in Cases C1 C7. Φ N σ + k z σ f Φ = 0. ( 10) Using the surface and bottom boundary conditions, Φ = 0 at z = 0 and z = h, the profile of vertical mode is sin(nπz/ h), where n is the mode number. The modal structure of horizontal velocity is derived from Φ and the continuity equation (4). The amplitude and phase of each mode are estimated to fit data of the vertical displacement and horizontal velocity at each grid by the least squares method. The kinetic energy of the northward component of velocity for n-th mode through the water column, KE n, is calculated by KE n 1 0 d = V Φ ρ 0 n dz h dz 1 0 = ρ0vn cos nπz/ h dz, h ( ) ( 11) where V n is the amplitude of northward component of velocity for n-th mode. The energy density was calculated to investigate the modal distribution of the scattering wave emanating at the edge of the step bottom. The energy density for the lowest 5 modes at each grid was calculated and is shown in the shaded region in Fig. 6(a). Figure 6(b) shows the cross-channel variation of the energy density for each mode averaged over the bay width, i.e., mean energy density between Line D and Line E. The maximum of kinetic energy density for the first mode is found at about 11.5 km (8.5 km from the northern boundary) and is caused by interference between the scattering wave of the first vertical mode and the reflected one at the northern boundary. The effect of reflection at the northern boundary for the second mode is also found at about 16 km (4 km from the northern boundary). Although the reflection effect at the northern boundary appears in the variation of the first and second modes, the energy density gradually decreases from the step bottom at the bay mouth and the energy density for the higher mode rapidly decreases in comparison with that for the lower modes. The decrease of the energy density is considered to be caused by both radial propagation and energy decay of the scattering wave. For the higher mode, the gradient of energy density with distance becomes steeper with increasing mode number, which implies that the energy decay for the higher mode internal wave is faster than that for the lower mode one. The energy dissipation of the internal Kelvin wave propagating through the channel is thus intimately connected with the generation of the higher mode internal waves, i.e., scattering waves at the bay mouth. 5. Dissipation of Internal Kelvin Waves The internal Kelvin waves propagating through the channel contribute to the generation of the scattering internal waves at the steep bottom of the additional bay mouth, that is, the energy of the former waves is trans- Scattering of Semidiurnal Internal Kelvin Wave at Step Bottom Topography 65

ferred to the latter. The ratios of the energy transfer and dissipation of internal Kelvin wave were considered to depend on the length of the additional bay, as expected from a result of a numerical experiment using a two-dimensional level model conducted by Kitade et al. (00). The relation between the length of the bay and kinetic energy of the northward component of velocity was then investigated in Region P and Region Q, shown in Fig. 7(a). Figure 7(b) shows the kinetic energy density of the northward component of velocity in Cases C1 C7. In these cases the numerical experiments were performed using various bay lengths from 1.0 km (C1) to 15.0 km (C7) with a bay width of.5 km and water depth of 50 m, as indicated in Table 1. The width of the bay is shorter than the internal radius of deformation (4.5 km) at 50 m depth. The energy density in both Region P and Region Q shows two peaks at the bay length of 3.5 km and 10.0 km. From the density stratification and water depth, the wave-length of the first mode internal wave with semidiurnal period is 13.7 km in the bay. Thus, one-fourth and three-fourths of the wavelength are about 3.5 km and 10.0 km, respectively. These results suggest that the internal wave regenerated at the bay mouth is resonant with the longitudinal internal seiche in the bay. Furthermore, the energy density in Region P and Region Q can be regarded as typical of internal waves in the bay and a scattering wave in the channel, respectively. Thus, the energy peaks in Region Q imply that the scattering wave is also strengthened by the resonance in the bay. The energy of an internal Kelvin wave propagating through the channel is partly transferred at the step bottom region to the internal wave in the bay and to the scattering wave in the channel. The energy transfer depends strongly on the horizontal and bottom topography (Fig. 7(b)), so we investigated the energy dissipation of the internal Kelvin wave due to scattering at the step bottom region. We estimated the magnitude of mean energy flux r F (= p r u ), where p is the pressure and r u the eastward component of velocity in the channel (e.g. Gill, 198). The eastward currents across Line F and Line G are assumed to be only due to the eastward propagating internal Kelvin wave. The magnitude of mean energy flux is calculated at Line F and Line G, shown in Fig. 1. The magnitude of mean energy flux in Case A, the case of a channel without the additional bay, was chosen as the reference of the magnitude of mean energy flux in each case. The estimated magnitude of mean energy flux in Case A is 1.7 10 4 (J m s 1 ) at Line F, 8.8 10 4 (J m s 1 ) at Line G. Thus, the magnitude of mean energy flux deceases about 35 percent by a distance of 30 km from Line F to Line G, and the energy dissipation in Case A is due to the internal and bottom friction. The difference of the magnitude of mean energy flux of the internal Kelvin wave in each case was normalized by F A, and the decay rate DR, Fig. 8. Decay rate of the energy flux for each case. was defined as follows: DR F, 1 = ( ) where F is the difference of the magnitude of energy flux between at Line F (F W ) and at Line G (F E ), and F A = 4.4 10 4 (J m s 1 ) is the difference of the magnitude of energy flux in Case A. Figure 8 shows the decay rate plotted against the bay length. The decay rate is estimated as 1.1 to 1.8 in the cases of a step bottom at the bay mouth, i.e., Case C to Case F, and is the largest in Case F and the smallest in Case C. It then tends to increase with the width of the bay (W). In comparison with the cases of the same bay width, the decay rate shows a maximum when the bay length is 3.5 km. The decay rate peaks when the model has a bay length of 10 km only in cases with a narrow bay width, i.e., Case C (.5 km) and Case D (5.0 km). 6. Discussion The decay of the internal Kelvin wave is indicated to depend on the topographic condition of the bay under the density stratification described in the previous section. We now discuss the physical process of the scattering of the internal wave in relation to its dependence on the topography. As shown in Fig. 4, the density perturbation is largest in the middle layer, while the horizontal velocity is largest in the upper and lower layers in the additional bay. These results imply the dominance of first mode internal wave in the bay. This wave is generated at the bay mouth by the internal Kelvin wave propagating along the channel, and has the characteristic of a standing wave along the bay axis, as shown in Fig., since the phase difference between northward current and perturbation density is about 90 in the bay. When the bay width is much narrower than the internal radius of deformation, the relation between the period, T, for the internal seiche of the bay and the bay length, L, is given as follows, F A 66 Y. Kawamura et al.

Fig. 9. Schematic view of propagation process of the internal wave in narrow (a) and wide (b) bays. L= T ( ma 1)c 4, (13) where ma is the along-bay mode number without the rotation effect, and c the phase velocity of first mode internal wave in the bay (LeBlond and Mysak, 1978). In this case, the first mode internal wave with semidiurnal period is amplified by the resonance when the period of the internal seiche of the bay is near the semidiurnal period. The bay length for the resonance of the semidiurnal internal wave is estimated to be 3.4 and 10.3 km by substituting the phase velocity of 0.3 m s 1 in Eq. (13). As shown in Fig. 7(b), amplification of the internal wave in the bay has been found in Case C (L = 3.5 km) and Case C5 (L = 10 km). When the bay is wide, the rotational effect plays an important role in the propagation and reflection of the internal wave in the bay. In this model, the internal wave generated at the bay mouth propagates toward the bay head with the characteristic of an internal Kelvin wave and reflects at the bay head; the internal Poincaré wave is generated at the bay head (Brown, 1973). The dispersion relation of internal waves in a rectangular bay with uniform water depth is given as follows: K = σ f mc π, ghn W (14) where K is the along-bay component of wavenumber, mc the cross-bay mode number, W the width of the bay, and h n the equivalent depth for the n-th vertical mode (LeBlond and Mysak, 1978). The propagation condition with a characteristic of internal Poincaré waves in a rectangular bay, K > 0, is given by W > mc π ghn = Wcr, σ f (15) Fig. 10. (a) Kinetic energy density of northward current (the sum of the second to fifth mode energy density) in each case (cases C F) averaged in the shaded area. The energy density (TE) kg m 3 is plotted in logarithmic levels. L1 and L are one-quarter and three-fourths of the wavelength for the first vertical mode internal wave in the bay, respectively. where Wcr is the critical value of the bay width. The width, Wcr, for the first vertical mode internal wave with the semidiurnal period is estimated as 8.3 km for mc = 1 and 16.6 km for m c =. Thus, the characteristic of the internal wave in the bay depends on the bay width. When W > Wcr, the internal wave in the bay behaves as the internal Poincaré wave or internal Kelvin wave, but when W < Wcr it behaves only as the internal Kelvin wave. When W < 8.3 km in this model, the internal wave with the first vertical mode behaves as the internal Kelvin wave and the internal Poincaré modes generated at the bay head to satisfy the boundary condition are evanescent and decay rapidly near the bay head. Figs. 9(a) and (b) show the schematic views. The scattering wave, having a beam-like structure, consists of a large number of vertical internal modes. However, the northward component of velocity for the first internal mode around the bay mouth is not only due to the scattering wave but also the Poincaré wave, which is caused to satisfy the boundary condition (as shown Fig. 3). Therefore the sum of the kinetic energy density (TE) from second to fifth vertical modes is calculated as the index of the scattering wave energy (Fig. 10(a)). When the bay width is narrow, as in Case C (W =.5 km) and Case D (W = 5.0 km), the energy density has double peaks Scattering of Semidiurnal Internal Kelvin Wave at Step Bottom Topography 67

with a bay length of 3.5 km and 10.0 km. On the other hand, when W > W cr, as in Case E (W = 10.0 km) and Case F (W = 15.0 km), the energy peak is singular with the bay length of 3.5 km. In the case W > W cr and L > 7 km, energy density is greater than that in the case of W < W cr and no energy peak was found in the case of L = 10 km. Thus, these results indicate that the additional contribution of Poincaré waves generated at the corners of the bay head are consistent with the propagation process of an internal wave in the bay, as shown in Fig. 9. 7. Summary The energy dispersion of a semidiurnal internal Kelvin wave at a shelf edge has been investigated using a three-dimensional multi-level model. A simplified topography, i.e., a rectangular channel with an additional bay located at the center of the southern coast, was used to clarify the scattering process. As the semidiurnal internal Kelvin wave given at the western boundary reaches at the bay region, higher modes of internal waves are generated at the step bottom of the bay mouth. Since the higher-mode internal waves decay quickly from the northern edge of the bay, the internal Kelvin wave is effectively decayed by the scattering process. The decay rate of the internal Kelvin wave due to the scattering process changes with the width and length of the bay. The decay rate is calculated to be 1.1 1.8 times that of the case without the bay. These rates indicate decay due to scattering, and these effects are of the same order as the internal and bottom friction. In the narrow bay case, the decay rate is found to be a maximum when the bay length is equal to one-quarter of the wavelength for the first vertical mode of the internal wave in the bay, implying that the energy of the internal seiche in the bay is effectively transferred to the higher mode internal wave in the channel region by resonant interaction. Further, the decay rate in the wide bay case is found to be higher than that in the narrow bay case, because of the added contribution of a Poincaré wave propagating from the corners of the bay head. Acknowledgements Numerical experiments in the present study were conducted on the VX-E in Tokyo University of Fisheries. This study was partially supported by a Grant-in-Aid for Science Research (B) (Project No. 13556030) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. References Baines, P. G. (198): On internal tide generation models. Deep- Sea Res., 9, 307 338. Brown, P. J. 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