Semi diurnal internal wave diffraction caused by Dixon Entrance, British Columbia

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1 This article was downloaded by: [Meteorologisk Institutt] On: 0 December 01, At: 17:10 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1079 Registered office: Mortimer House, 7-1 Mortimer Street, London W1T JH, UK Atmosphere-Ocean Publication details, including instructions for authors and subscription information: Semi diurnal internal wave diffraction caused by Dixon Entrance, British Columbia A.C. Carrasco a b, S.E. Allen a & P.H. LeBlond a a Dept. of Earth and Ocean Sciences, University of British Columbia, 670 University Boulevard, Vancouver, BC, V6T 1Z b Norwegian Polar Institute, Polarmiljøsenteret, Tromsø, N 996, Norway Published online: 1 Nov 010. To cite this article: A.C. Carrasco, S.E. Allen & P.H. LeBlond (00) Semi diurnal internal wave diffraction caused by Dixon Entrance, British Columbia, Atmosphere-Ocean, 0:, 0-, DOI: 10.17/ao.000 To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, British Columbia A.C. Carrasco*, S.E. Allen and P.H. LeBlond Dept. of Earth and Ocean Sciences, University of British Columbia, 670 University Boulevard, Vancouver BC V6T 1Z [Original manuscript received 11 September 001; in revised form March 00] ABSTRACT To develop an understanding of the complex internal tidal phenomena observed near and inside Dixon Entrance, an idealized numerical model was developed for the area, which explores the influence of various topographic features on the scattering of internal tides. The model uses a non-linear, two-layered, frictionless finite difference formulation of the shallow water equations and is forced by a barotropic wave over simplified topography. It was found that the main bathymetric features responsible for the generation of semi-diurnal internal tides are the steep continental slope together with the orientation of Dixon Entrance. The prevalent baroclinic wave pattern, which is similar to the one found by Buchwald (1971), suggests that the western end of Dixon Entrance can be considered as an internal tide generation region for the open ocean. Use of the simple model allows easy identification of the generated waves. When the model is run with a non-flat channel it reproduces features observed inside Dixon Entrance. RÉSUMÉ [Traduit par la direction] Afin de comprendre les phénomènes complexes de marée interne observés dans l Entrée Dixon et à proximité, on a mis au point un modèle numérique idéalisé pour la région qui examine l influence de divers éléments topographiques sur la diffusion des marées internes. Le modèle utilise une formulation à différences finies sans frottement, non linéaire et à deux couches pour résoudre les équations de Saint-Venant. Il est forcé par une onde barotrope sur un relief simplifié. On a constaté que le talus continental abrupte et l orientation de l Entrée Dixon sont les principales caractéristques bathymétriques à l origine des marées internes semi-diurnes. La configuration d ondes baroclines prédominante, qui est semblable à celle qu a reconnue Buchwald (1971), laisse croire que l extrémité occidentale de l Entrée Dixon peut être considérée comme une zone de génération de marée interne pour la haute mer. L utilisation de ce modèle simple permet de reconnaître facilement les ondes générées. Lorsqu on utilise le modèle avec un chenal dont le fond n est pas plat, il reproduit les éléments observés dans l Entrée Dixon. 1 Introduction Dixon Entrance is a coastal oceanic region that, together with Hecate Strait and Queen Charlotte Sound, separates the Queen Charlotte Islands from the mainland on the west coast of Canada (Fig. 1). Tides in these coastal oceanic regions are mixed, predominantly semi-diurnal, and co-oscillate with tides in the adjoining North Pacific Ocean (Thomson, 1981). A notable feature of the region is the very steep continental slope and narrow continental shelf. The 1000-m depth contour lies within km of most of the coastline of the Queen Charlotte Islands. To the north, the continental shelf reaches a width of roughly 0 km at N latitude. There it merges with the broader continental shelf west of Dixon Entrance and the continental slope becomes less steep. Further north, the shelf widens to approximately 100 km. Surface tides in these northern waters of British Columbia, Canada, have been simulated by Foreman et al. (199), Ballantyne et al. (1996) and Cummins and Oey (1997) among others. These numerical models have used realistic topography, bottom friction and tidal forcing, giving a very good fit to the data available for a large portion of the British Columbia coast. The finite element models used (Foreman et al., 199; Ballantyne et al., 1996) allowed varying grid size which is a distinct advantage in this highly irregular region and the finite difference three-dimensional model (Princeton Ocean Model) allowed for the presence of internal tides (Cummins and Oey, 1997). Although numerical models have predicted a strong semidiurnal surface signal in the western end of Dixon Entrance (Ballantyne et al., 1996; Cummins and Oey, 1997), observed *Corresponding author s current affiliation: Norwegian Polar Institute, Polarmiljøsenteret, N-996 Tromsø, Norway; ana@npolar.no ATMOSPHERE-OCEAN 0 () 00, 0 Canadian Meteorological and Oceanographic Society

3 06 / A.C. Carrasco, S.E. Allen and P.H. LeBlond 6 o N Clarence Strait Fig. 1 o N o N Dixon Entrance o W 1 o W surface currents exceed those estimated from the models (Crawford et al., 1998). The local Princeton Ocean Model gave better agreement suggesting that the region is strongly influenced by internal tides. Crawford et al. (1998) argued that the discrepancy between observed and predicted currents might be due to the lack of horizontal density variation in the models, as well as insufficient resolution in the horizontal. Altimeter observations (Cummins et al., 001) showed a large region offshore of Dixon Entrance to be strongly influenced by semi-diurnal internal tides. This feature is consistent with the results of the local Princeton Ocean Model (Cummins and Oey, 1997) which predicted a large offshore flux of baroclinic energy radiating from Dixon Entrance. A large baroclinic signal has been identified in the western end of Dixon Entrance and just offshore by models and observations but it is not clear how such internal waves arise and propagate. The purpose of this paper is to elucidate the basic dynamics of the internal tide generation with a simpler approach. Internal tides have been recognized as internal waves excited at or near tidal periods. An explanation of their existence 00 Learmonth Bank Queen Charlotte isl. Hecate Strait o W Queen Charlotte Sound Part of the north coast of British Columbia, Canada. Dixon Entrance is the east-west oriented coastal oceanic region north of the Queen Charlotte Islands. Depth contours are in metres. is that energy from barotropic tides is scattered to internal tides by bottom roughness (Hendershott, 1981). Steep continental slopes, in conjunction with strong stratification, are well known as likely places for the generation of high amplitude internal waves (e.g., Wunsch, 197; Baines, 198). An observational study by Carrasco (1998) showed that the western end of Dixon Entrance contained two potential generation regions for semi-diurnal internal tides. These are the shelf break and Learmonth Bank (Fig. 1). The generation of internal waves in the ocean can be affected by many interrelated factors including non-linearities in the governing equations, spatially varying tidal-current strength and phase, spatially and temporally varying stratification, boundary-layer and turbulent effects, overturning waves and associated mixing, and background residual currents (Lamb, 199). We present results of some numerical experiments on the wave generation process, where only a few of the aforementioned factors are taken into account. The model used is frictionless, non-linear, two-layered, finite difference and forced by a long barotropic Kelvin wave over idealized

4 Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, B.C./ 07 topographies. Dixon Entrance is modelled as a zonal channel closed at its eastern end and connected to a flat, open ocean by a steep slope at the western end. The goal is to develop an understanding of the generation and scattering of internal tides, by different topographic features, near and inside Dixon Entrance with a simplified model inspired by the analytical work of Buchwald (1971). Buchwald (1971) used a Green function to study the linear barotropic diffraction of waves originating at the mouth of an infinitely long channel connected to a deep, flat ocean. In his case, the forcing at the mouth simulated a uniform barotropic tide height that went up and down at a certain frequency. In our case the long barotropic Kelvin wave produces, at the entrance of the channel, a vertical shear which initiates the baroclinic channel-diffraction process. Results from a simple channel are presented first. More complex baroclinic wave patterns generated in channels with additional bathymetric features are then considered. Description of the model a The Code The numerical model is a discretization of the shallow water equations using a leap-frog scheme in time (Gill et al., 1986). It conserves both total enstrophy and energy with variable bottom topography (Arakawa and Lamb, 1981). Stratification is two layers, each with a constant density (Allen, 1988). The primitive shallow water equations for a two-layer, inviscid, non-linear flow are: η + t x Du Dt + f = g η η v g 1, x x Dv fu Dt (( ) ) + = g η η g 1, y y (( ) ) = (1a) (1b) H + η u H + η v 0 (1c) y Du1 Dt f 1 = g η v 1, x Dv 1 + fu1 = g 1, Dt η y η1 η + t t x + y (( H1 + η1 η) u1) (( H1 + η1 η) v1) = 0 (1d) (1e) (1f) where η is the interfacial displacement, η 1 is the sea surface displacement, H and H 1 are the undisturbed depths of the lower and upper layers, respectively, (u,v ) and (u 1,v 1 ) are the horizontal velocities in the lower and upper layers, respectively, f is the Coriolis parameter, g is the gravitational acceleration and g = g(ρ ρ 1 ) / (ρ 1 + ρ ) is the reduced gravity. The domain is rectangular and the grid spacing, δd, is chosen to be a small fraction of the Rossby radius. The horizontal velocity for each layer, and the layer thicknesses h 1 = η 1 η + H 1 and h = η + H are calculated on a square grid. The upper layer variables are defined at the same horizontal positions as the lower layer variables. A staggered grid, the C grid, is used (Arakawa and Lamb, 1981). The code has no explicit viscous or frictional damping; however, there is some implicit numerical damping. The Arakawa-Lamb scheme conserves energy exactly. However, to keep the leap-frog solutions similar, every 101 steps they are averaged and then a 1/ Euler step forward and backward is taken. This process generates a weak implicit numerical damping. The advantage of this explicit scheme is its accuracy and the fast execution per time step. A disadvantage is the short time step required for stability. b Topography The model topography approximates Dixon Entrance and the adjacent continental shelf in the Pacific Ocean at. N, 11. W (i.e., with f = s 1 ). To analyse the impact of bathymetric features on the scattering of internal tides, the model was run with three different topographies. In each case Dixon Entrance is assumed to be a zonal channel closed at its eastern end and connected to a flat, open ocean at the western end by a steep slope. For simplicity Hecate Strait and Clarence Strait are ignored. We closed the channel based on the observation of Crawford et al. (1998) who indicated the presence of a very narrow and shallow, north-south oriented reef structure, Celestial Reef, (not shown) that could be regarded as a wall that partially closes Dixon Entrance at its eastern end. The sharp coastline bend at the north-western side of the Queen Charlotte Islands and the adjacent steep continental slope (Fig. 1) inspired modelling the western end of Dixon Entrance in this way. In the base case, Dixon Entrance is assumed to be flat bottomed. In the second case Learmonth Bank is added. Finally, in the third case, a shoaling bottom on both sides and towards the east is added inside Dixon Entrance. The simulation of the offshore region is the same in all three cases. c Boundary Conditions The free-slip boundary condition was applied at all solid vertical walls. For the north and west open boundaries a wave radiation criterion was applied that consisted of applying the Sommerfeld conditions separately to the barotropic and baroclinic variables (LeBlond and Mysak, 1978; Fig. ). d The Two-layer Approximation Dixon Entrance is stratified year round (Dodimead, 1980; Huggett et al. 199) with a broad pycnocline from 0 m to 100 m depth. To approximate the stratification using two layers, the first vertical mode for the horizontal velocity was estimated from averaged density profiles. The depth at which this function changes sign (the velocity changes direction) was chosen as the undisturbed interfacial depth of the two-layer

5 08 / A.C. Carrasco, S.E. Allen and P.H. LeBlond gridy Fig. RADIATION RADIATION (000 m) FORCING gridx FREE SLIP FREE SLIP 60 km system. Based on this depth and on the reduced gravity values estimated from Conductivity-Temperature-Depth (CTD) data, we selected g = m s and H 1 = 100 m as appropriate for spring conditions. Unless specified otherwise, spring conditions are shown. e Forcing To simulate the barotropic tides the model was continuously forced at the southern open boundary by a barotropic Kelvin wave travelling north (Platzman, 1979; Foreman et al., 199). Such a wave has the following surface perturbation at the southern boundary: ζ = ζ O sin(ωt θ i )exp( x/r) () where R = C o / f is the barotropic Rossby radius (C o is the barotropic phase velocity in the deep ocean and f is the Coriolis parameter), ω is the M frequency and the values of the amplitude ζ O and the phase θ i are chosen based on observations (Thomson, 1981). Results a Base Case In this run a basic topography with a grid is used (Fig. ). The grid spacing δd is.6 km. Dixon Entrance is modelled as a flat channel 00 m deep connected by a very steep continental slope to a flat (000 m deep) open ocean. The width of the channel is 60 km. The amplitude and phase of the barotropic Kelvin wave are ζ O =. m and θ i = 1.9 at point number 1 in Fig.. Its wavelength is λ M = 6 km. The entire computational domain embraces 60 0 km. The part of the domain shown covers 9 km in the x direction and 7 km in the y direction. As the long barotropic Kelvin wave travels north it generates, perpendicular to its direction of propagation, a complex 1 LAND LAND (00 m) 11.8 km A plan view of the topography used for the base case. The points indicate the locations at which time series were stored. The conditions for the boundaries are indicated. baroclinic pattern that can be associated with internal waves near the mouth of the channel. Contour plots of the interface displacement η and the baroclinic velocity, u bc (x,y) = u 1 u and v bc (x,y) = v 1 v (see Appendix for details), every t = τ where τ is the semi-diurnal period (τ = 0.17 day) are shown in Figs, and respectively. First we focus on the response inside the channel. At the entrance to the narrow channel the long barotropic Kelvin wave generates eastward propagating baroclinic waves; some trapped to the southern wall inside the channel. The velocity pattern formed by the internal waves is associated with both baroclinic Kelvin and plane Poincaré waves. By t = τ the baroclinic Kelvin waves have already arrived at the end of the channel and turned around. In this reflection process, the Poincaré mode is also established. The evidence of its presence is the positive-negative pattern of v bc across the channel at t = 6τ (Fig. ) and the weak asymmetry of the two-kelvin wave pattern η and u bc (Figs and ). That is, there exists an amphidromic region along the central channel instead of a straight line of amphidromic points (Brown, 197). Asymmetry in the two-kelvin wave pattern may also occur due to damping at the head of the channel (Hendershott and Speranza, 1971), which is not the case here. The Kelvin wave progressing down one side of the channel takes some time to cross the end of the channel before returning along the opposite side which causes it to change phase (Figs and ; Taylor, 191). When the waves turn at right angles crossing the end of the channel they produce a larger rise and fall at the corners, Taylor (191), (see η ). The approximate amplitudes, wavelengths and velocities of this pattern are listed in the third (western side) and fourth (eastern side of the channel) columns in Table 1. Output variables from different models are listed in Table 1. The variables estimated are listed in the first column. The values in the second column are from a linear analytical model of waves in a flat channel of dimensions comparable to Dixon Entrance. The approximate variables in the following four columns are from the base case run. The remaining four columns are from the run with a bank. After Taylor (191), the scattering problem of normal modes in ocean channels has been studied analytically by many authors, for example Hendershott and Speranza (1971), Brown (197) and Ripa and Zavala-Garay (1999). Using linear theory, we estimated the possible semi-diurnal internal waves in Dixon Entrance by representing it as an east-west oriented flat channel, closed at its eastern end, at a fixed latitude (i.e., with f = s 1 ) and continuously stratified. The solution to the linearized equations of motion was separated into vertical and horizontal dependencies. For the vertical part a mean vertical density profile was specified from an ensemble of spring observations. Density was assumed to be constant in the horizontal. Only the first vertical mode was taken into account. For the horizontal part, a channel of width km and length of 1 km was considered. It was assumed that a Kelvin wave with phase velocity C 1 (where 1 stands for the first vertical mode) with a maximum coastal amplitude of

6 Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, B.C./ 09 t = τ t = τ t = 6τ t = 10τ t = 8τ Fig. Contours of the interface height η for the base case every τ where τ = 0.17 d. The contour interval is. m for t = τ and m for the other times. The dotted lines represent negative values, the dashed lines zero and the continuous lines positive values.

7 10 / A.C. Carrasco, S.E. Allen and P.H. LeBlond t = τ t = τ t = 6τ t = 10τ t = 8τ Fig. Contour of the baroclinic velocity u bc for the base case every two periods (τ). The contour interval for t = τ and t = τ is 0.0 m s 1 and for the other times is 0.0 m s 1.

8 Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, B.C./ 11 t = τ t = τ t = 6τ t = 10τ t = 8τ Fig. Contour of the baroclinic velocity v bc for the base case every two periods (τ). The contour interval for t = τ and t = τ is 0.0 m s 1 and for the other times is 0.0 m s 1.

9 / A.C. Carrasco, S.E. Allen and P.H. LeBlond TABLE 1. Approximate variables from Linear theory inside the channel, from the base case run and from the run with a bank. Here A stands for amplitude and λ for wavelength. The subscripts k, p and cp stand for Kelvin, Poincaré and cylindrical Poincaré waves, respectively. C 1 is the phase velocity. u bc M and v bc M are the maximum absolute value of the east-west and north-south baroclinic velocities. The values with the subscript N indicate that the waves are travelling north. The superscript c indicates an entrance corner location. The last line has the estimated values of the baroclinic energy at M radiated offshore. base case with bank inside the deep ocean inside the deep ocean channel near entrance channel near entrance LINEAR west east off coast west east off coast A k (m) N N λ k (km) 0 1 N 1 N A p (m) λ p (km) A cp (m) λ cp (km) C 1 (m s 1 ) u bc M (cm s 1 ) 0 c 8 1 c v bc M (cm s 1 ) 11 6 c c 18 bcl. energy (W) m travels into the channel with the southern wall to the right of its direction of propagation. It was found that the first free Poincaré mode was excited together with a reflected Kelvin wave and a set of evanescent Poincaré modes trapped to the closed end of the channel. The phase velocity, amplitudes and wavelengths of these free waves are summarized in the second column of Table 1 for comparison with values obtained with the numerical model, given in the third column. The maximum distance of influence of the evanescent Poincaré modes at the closed end was km. The processes occurring at the entrance to the channel make the Poincaré mode stronger than predicted by linear theory since its wavelength is shorter and its amplitude much larger (Table 1). Now we analyse what occurs outside the channel. At the same time that the baroclinic Kelvin and Poincaré waves are generated at the mouth of the channel, fronts moving westward, away from the entrance of the channel in the deep ocean are also generated (Figs, and at t = τ). These transient waves carry energy with them, so for any finite region energy is lost through the sides by radiation of short Poincaré waves and the energy left is that associated with a solution that tends to oscillate about an equilibrium state. Such an equilibrium state consists of Kelvin waves propagating along the coast (towards the north) together with long cylindrical Poincaré waves radiating into the deep ocean (Buchwald, 1971; Figs, and at t = 10τ). The cylindrical Poincaré waves can been seen in all the variables, since they have an east-west and north-south component, while their connection to the northward Kelvin waves is most clearly seen in the north-south velocity (Fig. ). From the v bc plots (Fig. ) it can be seen that the Poincaré wave inside the channel co-oscillates with the cylindrical Poincaré wave outside the channel. This makes the Poincaré mode inside the channel stronger than predicted by linear theory. When the Kelvin waves inside the channel reach the end of the northern wall, they turn the corner and travel northwards trapped to the shelf coast feeding the wave pattern already formed in that region. At the entrance corners to the channel, the velocities reached their peak values: u bc at the southern corner and v bc at the northern one. These values are indicated in Table 1 with a superscript c. These peak values are associated with the abrupt turning of the Kelvin waves. Since the system is continuously excited by long barotropic Kelvin waves, baroclinic waves are constantly radiating from the channel, making it act as a point source of internal tides for the open ocean. This is consistent with the altimeter observations reported by Cummins et al. (001). They showed a broad region of semi-diurnal baroclinic signal just offshore of Dixon Entrance. Although the wave pattern found near the entrance to the channel is similar to the one found by Buchwald (1971) there are differences. In Buchwald s case the channel was too narrow to allow plane Poincaré modes to enter the channel. A second difference is that Buchwald used an infinitely long channel, so there were no reflected waves. In our case the reflected waves, generated because the channel is closed, intensify the wave pattern formed outside the channel. The frequency of the baroclinic waves generated is mainly M as can be seen from the figures since a new wave is generated each period. Harmonic analysis was performed at the locations marked in Fig. in order to quantify the energy at M. Outside the channel approximately 80% of the total baroclinic signal is associated with the M frequency and inside the channel from 60 to 70%. The leakage of energy to higher harmonics, mainly M (period of 6.1 hrs) and M 6 (.1 hrs), is due to the effect of the non-linear advection terms in the momentum equations (Holloway, 1996). At the entrance to the channel, where the generation of internal waves occurs, the signal at M becomes significant and because of the absence of dissipation in the model its presence is felt wherever the internal waves propagate. The ratio of the signal at M to that at M ranges from 0. to 0.. In the actual data, the signal at M does not have such a large contribution (Carrasco, 1998).

10 Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, B.C./ 1 gridy gridx Fig. 6 View from above (top) and three-dimensional view of the topography (bottom) with a bank. The bank is 1.8 by 18. km at its base and 18. by.6 km at the top. The top is 1 m from the sea surface. The generation process in the channel/deep ocean system is triggered by the interaction of the barotropic wave with the slope at the mouth of the channel and, of course, by the presence of stratification. Once the internal waves are generated they are diffracted by the presence of the channel. b Addition of a Bank To study the influence of a bank on the internal wave pattern the topography presented in Fig. 6 was used. The idea was to represent Learmonth Bank, the bank located at the eastern end of Dixon Entrance (Fig. 1). Learmonth Bank rises to within m of the surface and is about 19 km long and about 10 km wide. The same forcing and stratification as in the previous case were used. Contour plots of the interface η and the velocities, u bc, v bc at t = 10τ are shown in Fig. 7. The inclusion of the narrow bank at the entrance to the channel modifies the pattern of v bc in the central part of the channel producing larger magnitudes than in the previous case (compare with Fig. ). Near the bank, larger velocity values than in the previous case are reached (see Table 1). However, the Kelvin wave pattern associated with u bc inside the channel has remained essentially unchanged. A possible explanation is that the bank is located in the region where the Fig. 7 From top to bottom: interface elevation η and velocities u bc and v bc at t = 10τ for the topography with a bank. The contour intervals are m, 0.0 m s 1 and 0.0 m s 1 respectively.

11 1 / A.C. Carrasco, S.E. Allen and P.H. LeBlond magnitude of Kelvin waves is negligible, that is beyond an internal Rossby radius (around 10 km) from the walls. Outside the channel the presence of the bank increases the magnitude of both u bc and v bc by about 10%. The interface η does not have noticeable changes. From a harmonic analysis (performed at the same locations as in the previous case) we find that at the locations very close to the mouth of the channel, which now includes the bank, M currents reach values of about 6% of those at M. Elsewhere the ratio of the signal at M to that at M ranges from 0. to 0. as in the previous case. In this case, at some locations, the baroclinic velocities have larger magnitudes than the barotropic. The barotropic run was carried out for the model without stratification (not shown). In summary the presence of the bank intensifies the Poincaré modes, inside and outside the channel, (both amplitudes and velocities are increased) and produces stronger nonlinearities in its immediate neighbourhood. The wave pattern in the other regions is similar to the base case. The largest increase is seen in the cross-channel velocity v bc (Table 1). For the base case and the case with a bank, different conditions were also modelled (not shown) by modifying g (from 0.01 to 0.0 m s ) as well as H 1. It was found that as long as there were two layers inside the channel the main wave pattern described above is robust. c Baroclinic Energy Fluxes Estimates of the baroclinic energy fluxes provide insight into the propagation and magnitude of the internal tide. The semidiurnal vertically-integrated energy flux vectors (see Appendix for details) calculated at the locations shown in Fig., for the base case (i.e., with no bank) and for the case with the bank, are shown in Fig. 8. These clearly show energy spreading from the entrance to the channel. Inside the channel the Kelvin waves entering, near the southern wall, produced the most energetic region. In the open ocean the energy radiates as cylindrical Poincaré waves to the west and as Kelvin waves along the northern coast. This is consistent with Cummins and Oey s (1997) description of a region of strong internal tidal influence emanating offshore from Dixon Entrance. The largest divergence occurs near the entrance, where the bottom depth changes abruptly. This is consistent with the leading order expression of the energy source term S in Eq. (8): which relates production of baroclinic energy with a mean correlation between η and the barotropic current over a region of variable topography. Some divergence of baroclinic energy can also be detected in areas away from the main generation region, both inside the channel and in the open ocean. Although an indication that all assumptions (see Eq. (9)) may not be completely satisfied, these divergences are relatively modest; generation and spreading of energy from the channel entrance dominates. The introduction of the bank produces an increase in magnitude of the energy flux at both sides of the channel entrance together with a slight change in direction. It also produces an gridy gridy Fig. 8 gridx gridx 1000 W m W m 1 Baroclinic energy-flux vectors at M near the mouth of the channel for the base case (top) and for the case with a bank (bottom). increase in the integrated baroclinic energy radiating offshore (see Table 1). The total flux radiating into the open ocean for the case with the bank is W which is comparable to the estimate of Cummins and Oey (1997), W. For the region outside the channel the new features can be associated with an interference diffraction pattern caused by the two entrances produced by the presence of the bank. Each entrance can be seen as a weak point source of cylindrical waves but since they are so close the wave pattern presents small variations with respect to the previous case. If the bank were enlarged towards the eastern end of the channel then the two new entrances could have been seen as two strong point sources and a more symmetric structure would have been found. Figure 9 shows values, for the case with the bank, of the semi-diurnal baroclinic and barotropic energy at the time series locations. The presence of the baroclinic scattering produces gradients in the barotropic energy that are absent in the unstratified case (Fig. 9b versus Fig. 9c). When the model was run with no stratification there was no diffraction process

12 a) Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, B.C./ 1 gridy Baroclinic (KE + PE)*0.01 { N/m} b) c) gridy gridy gridx gridx Barotropic (KE + PE)*0.01 { N/m} Barotropic (KE + PE)*0.01 { N/m} (without stratification) gridx Fig. 9 a) Values of the baroclinic energy. b) Values of the barotropic energy. c) Values of the barotropic energy for the unstratified case.

13 / A.C. Carrasco, S.E. Allen and P.H. LeBlond gridy Fig gridx Topography inside the channel used in the shoaling bottom case. View from above (top) and three-dimensional perspective (bottom). The printed depths are in metres. The location where time series were stored are marked with dots, for the energy calculations, and with squares for the comparison with data. at the entrance to the channel. Although the values of the baroclinic velocity at the M frequency are, at some locations, close to or greater than the barotropic velocity magnitudes, the baroclinic energy is about two orders of magnitude smaller than the barotropic energy. It ranges from 1 to % of the barotropic energy. While the energy flux pattern is very similar for the base case and for the case with the bank, the presence of the bank increased the baroclinic energy flux and diffracted it slightly north in the offshore region. d Run with a Shoaling Bottom Channel Some bathymetric features inside the channel are modelled. On the western side of Graham Island (the northernmost island of the Queen Charlotte Islands) and over the banks are shallow regions that are adjacent to deep regions (Fig. 1). These abrupt changes are modelled as steep slopes in a smaller grid than in the previous cases. The topographic features in the region of interest are presented in Fig. 10. The grid in the entire domain is with a grid spacing of δd =.6 km. The width of the channel is 7. km and the length is 1.6 km. The barotropic amplitude and phase near the coast at the southern boundary were ζ O = 1.9 m and θ i =. The entire computational domain embraces km. The part of the domain shown in Fig. 10 covers 7 km in x and 0 km in y. Figure 11 shows the structure of the interface displacement η, the velocities u bc and v bc at t = 10τ. Inside the channel the velocities increased in the narrow passage and an intense cross-channel flow developed near the bank. In this case the Kelvin waves have an approximate wavelength of 7 km, a phase velocity of 1. m s 1, and they oscillate at M frequency. The clear two-kelvin wave pattern found in a flat channel no longer exists. However, the regions of positive and negative values of v bc and of u bc near the walls can still be associated with Kelvin and Poincaré waves plus the appearance of short waves related to topographic features. The new short wavelengths generated in this run range from 60 to km, therefore the spatial resolution needed to follow their behaviour properly must be as small as 1. km. Indeed Crawford et al. (1998) suggested that bathymetric features of order 1 km can drastically modify the semi-diurnal internal tide inside Dixon Entrance. From the harmonic analysis performed at the locations marked in Fig. 10 it was found that there is considerable energy transfered to M and M 6, therefore these topographic waves are mainly a product of non-linear interactions. The semi-diurnal baroclinic energy fluxes offshore from the channel for the sloping-bottom case are very similar to those for the case with the bank and an otherwise flat channel (Fig. ). In the open ocean the baroclinic energy magnitude has slightly decreased near the entrance of the channel. The total flux radiating into the open ocean is W. Inside the channel, in the non-flat case, the energy flux increased in magnitude at the southern wall because of the narrow passage to the south of Learmonth Bank. In summary the addition of a non-flat channel greatly changes the wave pattern found inside the channel but outside the channel the same main features found in the case with just a bank remained. Inside the channel the dynamics of the internal waves involve several different frequencies and more wavelengths. However, there are still some features associated with the Kelvin-Poincaré wave system. Outside the channel the energy flux features are still associated with cylindrical Poincaré waves with a slight northern component just as in the case with the bank. Comparison with data A quantitative comparison between the baroclinic field estimated from current mooring data and the baroclinic field obtained when the model was run with a non-flat channel is made. Modelled values of baroclinic velocities at the locations marked with squares in Fig. 10 are used. The idea is to represent the line described by the moorings QF1 D1 indicated in Fig. 1. The data used in this study consist of two sets of current-meter records from several moorings. The data were taken during the North Coast Oceanic Dynamics experiment from April 198 to June 198 (Huggett et al., 199).

14 Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, B.C./ 17 Fig. 11 From top to bottom: interface elevation η and velocities u bc and v bc at t = 10τ for the topography shown in Fig. 10. The contour intervals are: m, 0.0 m s 1 and 0.0 m s 1 respectively.

15 18 / A.C. Carrasco, S.E. Allen and P.H. LeBlond o N gridy 0' 0 QF1 D D 0 D10 D D09 D1 0 0 o N 1000 W m 1 1 o W 0' 1 o W 0' 11 o W 0' Fig. gridx Baroclinic energy-flux vectors at M near the mouth of the channel for the non-flat topography. Two periods were considered: the first one, which we called (SP), is about 18 days duration, from April to October 198, and the second, (SU), about 19 days from October 198 to July 198. The extraction of the baroclinic tidal currents from the raw time series was done by subtracting barotropic modelled currents from observed currents at the M frequency (Marsden, 1986). To obtain the total signal at M a standard harmonic analysis by blocks was performed on the velocity records (Foreman, 1978). The length of the blocks was chosen to be 8 days, because this is the shortest period in which the separation of the M constituent from its neighbours, S and N, is ensured. A halfblock overlap was selected and a new time series of amplitudes and phases every 1 days at M was thus obtained. The time series of harmonic amplitudes and phases represent a superposition of barotropic and baroclinic components. The barotropic coefficients that were subtracted were obtained from the finite element model developed by Foreman et al. (199). The modelled barotropic currents are assumed to have a small error compared to the errors associated with the observations. The major contributor to the surface elevation is the barotropic component and Foreman s model reproduced the sea surface elevations within a very reasonable range (.0 cm for the amplitude and 6.0 for the phase). These modelled barotropic currents are calculated satisfying volume conservation in the region taking into account variations in the bottom topography. We obtained a time series of the baroclinic velocity field (every 1 days) along this central line of moorings. Figures 1 and 1 represent the observed averaged baroclinic velocities for the first period (SP) and second period (SU), respectively. From the results of the numerical model the M baroclinic velocity components at each layer are obtained by subtracting the harmonically analysed barotropic component from the total. The barotropic component was obtained when the model was run without stratification. Fig. 1 Locations of the moorings. The mooring D has data only from the first period (April Oct. 8), D has data from the second period (Oct. 8 July 8) and the rest have data from both periods. Depth contours are in metres. Two conditions were modelled, the spring condition with g = m s and H 1 = 100 m and the summer condition with g = 0.0 m s and H 1 = 0 m. Figures and 17 represent the spring and summer modelled conditions and can be compared to Fig. 1 (period SP) and Fig. 1 (period SU), respectively. From the data analysis we found that the eastwest and north-south baroclinic currents have similar magnitudes in both seasons. The modelled baroclinic velocity magnitudes are larger in the summer than in spring. The data shows stronger semidiurnal flow in the lower layer than predicted by the model. This might be due to the lack of finite horizontal density gradients as well as missing topographic features in the model, (Crawford et al., 1998). The phases show a change of 180 in the vertical which is reproduced by the model. In the horizontal, the wave pattern is undetectable from this data distribution. The distance between stations is around 0 km, which is close to the Kelvin wavelength expected. At the locations D, D10, D1 and D09 the upper baroclinic velocities represented 60% of the total velocity and were larger than the barotropic by %. In the spring case the baroclinic velocity magnitudes represented 0% of the total velocity and were everywhere smaller than the barotropic by about 0%. What the model can reproduce is: 1) at each season the model magnitudes of the cross-channel velocity are similar to the longitudinal velocity; ) the phases show a change of 180 in the vertical; and ) in the summer the baroclinic velocity magnitudes in the upper layer are similar to those found from the data analysis. A direct evaluation of agreement with the data is not possible because of the limitations of the model. Table presents velocity magnitudes of observed and modelled results. The summer conditions are better modelled than the spring conditions. Conclusions The model results suggest that, as a first approximation, the western end of Dixon Entrance can be considered to be a

16 Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, B.C./ 19 east-west amplitude (cm s 1 ) east-west phase (degrees) QF1 D D10 D0 D09 D Fig north-south amplitude (cm s 1 ) north-south phase (degrees) From the data analysis: the two upper panels present the averaged values of the east-west velocity U b and phase φ b, and the two lower panels, the averaged values of the north-south velocity V b and phase θ b in an east-west section along Dixon Entrance for the first period (SP). The lowest line is the water depth. The dots mark the location of the instruments. east-west amplitude (cm s 1 ) QF1 D D10 D0 D09 D east-west phase (degrees) north-south amplitude (cm s 1 ) north-south phase (degrees) Fig. 1 As in Fig. 1 but for the second period (SU).

17 0 / A.C. Carrasco, S.E. Allen and P.H. LeBlond east-west amplitude (cm s 1 ) QF1 D D10 D0 D09 D east-west phase (degrees) north-south amplitude (cm s 1 ) north-south velocity phase (degrees) Fig Spring modelled values of baroclinic velocities along an east-west vertical section. The location of these points is indicated in Fig. 10. The thickness of the upper layer H 1 is 100 m and g = m s. The two upper panels represent the amplitude and phase of the east-west baroclinic velocity and the two lower panels represent the north-south baroclinic velocity. The values were printed at the middle depth of each layer. The lines indicate the actual and the modelled water depth. east-west amplitude (cm s 1 ) QF1 D D10 D0 D09 D east-west phase (degrees) north-south amplitude (cm s 1 ) north-south velocity phase (degrees) Fig. 17 As in Fig. but the summer conditions were modelled. The thickness of the upper layer H 1 is 0 m and g = 0.0 m s.

18 Semi-diurnal Internal Wave Diffraction Caused by Dixon Entrance, B.C./ 1 TABLE. source of semi-diurnal internal cylindrical Poincaré waves to the open ocean as well as coastal Kelvin waves travelling north in the open ocean and Kelvin and Poincaré waves entering the channel. This wave pattern seems to be permanent and can explain the strong semi-diurnal currents found in the real data. The bathymetric feature responsible for the generation of internal tides is the steep slope together with the orientation of Dixon Entrance. The presence of the bank at the entrance increases the magnitude of the baroclinic velocities across the channel but it does not considerably alter the wave pattern found in its absence. However, when the channel has a shoaling bottom the wave pattern inside the channel changes greatly. The presence of a bank at the entrance to the channel as well as a sloping bottom inside the channel produce only moderate changes in the pattern of the cylindrical waves in the open ocean. Many physical effects are missing in these idealized calculations: horizontal and vertical density variations, horizontal diffusion and bottom friction as well as many topographic features such as Clarence Strait and Hecate Strait. Admittedly some of the simplifying assumptions made in this study render the results incomplete. However, when a complex topography and a minimum grid size of.6 km are used, the model reproduces some features from observations inside the channel. Also, towards the open ocean the radiating wave pattern produced in this highly idealized model largely coincides with altimeter observations presented by Cummins et al. (001) and the model findings of Cummins and Oey (1997). So we conclude that this model embraces most of the dynamics of the internal tides in this region. Appendix a Baroclinic Signal The currents generated in each layer, U 1 = (u 1,v 1 ) and U = (u,v ), contain both a barotropic and a baroclinic component. The velocity in the upper layer is: and in the lower where Velocity range and mean for the upper layer from the model results and from the data analysis. Units are cm s 1. spring summer total mean total mean range upper layer range upper layer Obs. u Obs. v 18. Mod. u Mod. v U 1 = U 0 + H * (u bc, v bc )/(H 1 + H ) () U = U 0 H 1 * (u bc, v bc )/(H 1 + H ) U 0 = (H 1 * U 1 + H * U )/(H 1 + H ) () is the barotropic velocity vector and u bc (x,y) = u 1 u v bc (x,y) = v 1 v. () can be thought of as the amplitude of the baroclinic mode. As an aid to interpreting the model results, u bc and v bc are presented. Time series of these baroclinic velocity components are tidally analysed at selected locations. b Internal Wave Energetics If the advective flux energy is neglected the equation for the baroclinic perturbation energy can be written as where t ( KE + PE) + J = S, 1 KE = H u bc + bc ρ v is the perturbation kinetic energy density term and is the perturbation potential energy. The last term on the lefthand side in Eq. (6) is the divergence of the vector (6) J = (ρ ρ 1 )gη H(u bc v bc ), (7) which represents the flux of baroclinic energy given in units of W m 1. The term on the right-hand side is: H S = 1 H1 + H ( ) 1 PE = ( ρ ρ1) gη 0 ( ρ ρ1) gηu H, and involves the conversion between barotropic and baroclinic energy. These equations are derived under the following assumptions i) η 1 << η, ii) η 1 << H 1, iii) η << H, iv) η << H 1.(9) Assumptions i) and ii) are valid to within % for internal tides, and so are excellent approximations. Assumption iii) is valid within 10 0% and assumption iv) is the most tenuous, the error may be as large as 0%. In a steady state and in the absence of sources or sinks of energy, S is zero and the vector J is non-divergent. When J is divergent, at least one of the two remaining terms in Eq. (6) is non-zero. When S is non-zero, within the linear framework, there is a transfer of energy from barotropic to baroclinic modes. As the leading term Eq. (8) indicates this transfer occurs in regions of variable topography. The horizontal velocity (u bc, v bc ) is given by Eq. (), the barotropic current U 0 by Eq. () and the depth H by (8)

19 / A.C. Carrasco, S.E. Allen and P.H. LeBlond H HH = 1. H1 + H The energy flux vector averaged over a tidal period (T = π/ω), is: 1 T < J > = (10) Jdt. T 0 Following Cummins and Oey (1997) the energy flux is calculated by the use of a harmonic analysis. The horizontal velocity components and interface displacement can be considered as the direction of the positive semi-major axis of the tidal ellipse. G is the phase lag of maximum current behind maximum tidal potential and η, θ η are the amplitude and phase of the interface. The components of the energy flux vector <J > are 1 < J > = gh( ρ ρ1) η u cos θη G, v sin θη G (11) ( ( ) ( )) <J > = (J x,j y ) The energy fluxes in the (x,y) coordinates of the model are: u bc = u cos(ωt G) v bc = v sin(ωt G) Jx Jx J y = cos( ψ) sin( ψ) sin( ψ) cos( ψ) Jy η = η cos(ωt θ η ) where u and v are the velocity magnitudes at frequency ω in a coordinate system (x,y ) such that the x axis is oriented in References ALLEN, S.E Rossby adjustment over a slope. PhD thesis Cambridge University, 06 pp. ARAKAWA, A. and V.R. LAMB A potential enstrophy and energy conserving scheme for the shallow water equations. Mon. Weather Rev. 109: BAINES, P.G On internal tide generation models. Deep-Sea Res. 9(No. A): BALLANTYNE, V.A.; M.G.G. FOREMAN, W.R. CRAWFORD and R. JACQUES Three-dimensional model simulations for the north coast of British Columbia. Continental Shelf Res. (No. 1): 8. BROWN, P.J Kelvin-wave reflection in a semi-infinite canal. J. Mar. Res. 1: BUCHWALDS, V.T The diffraction of tides by a narrow channel. J. Fluid Mech. 6: CARRASCO, A.C Internal Tides in Dixon Entrance. University of British Columbia Canada, Earth and Ocean Sciences. PhD thesis, 17 pp. CRAWFORD, W.R.; J.Y. CHERNIAWSKY, P.F. CUMMINS and M.G.G. FOREMAN Variability of tidal currents in a wide strait: A comparison between drifter observations and numerical simulations. J. Geophys. Res. 10: CUMMINS, P.F. and L.-Y. OEY Simulation of barotropic and baroclinic tides off Northern British Columbia. J. Phys. Oceanogr. 7: ; J.Y. CHERNIAWSKY and M.G.G. FOREMAN North Pacific internal tides from the Aleutian Ridge: Altimeter observations and modeling. J. Mar. Res. 9: DODIMEAD, A.J A general review of oceanography of the Queen Charlotte Sound - Hecate Strait - Dixon Entrance. Can. Ms. Rep. Fish. and Aquat. Sci. No 17, 8 pp. FOREMAN, M.G.G Manual for tidal currents analysis and predictions. Pacific Marine Science Report Institute of Ocean Sciences, Patricia Bay Sidney, B.C., 66 pp. ; R.F. HENRY, R.A. WALTERS and V.A. BALLANTYNE A finite element model for tides and resonance along the north coast of British Columbia. J. Geophys. Res. 98: where ψ is the inclination of the tidal ellipse. GILL, A.E.; M.K. DAVEY, E.R. JOHNSON and P.F. LINDEN Rossby adjustment over a step. J. Mar. Res. : HENDERSHOTT, M.C Long Waves and Ocean Tides. In: Evolution of Physical Oceanography, Warren, B.A. and C. Wunsch (Eds), Cambridge Massachusetts and London, England. 6 pp. and A. SPERANZA Co-oscillating tides in long, narrow bays; the Taylor problem revisited. Deep-Sea Res. 18: HOLLOWAY, P.E A numerical model of internal tides with application to the Australian north west shelf. J. Phys. Oceanogr. 6: 1 7. HUGGETT, W.S.; R.E. THOMSON and M.J. WOODWARD Data record of current observations volume XX. Dixon Entrance, Hecate Strait and the west coast of the Queen Charlotte Islands 198, 198 and 198. Institute of Ocean sciences, Sidney, B.C. Canada. 01 pp. LAMB, K.G Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge. J. Geophys. Res. 99: LEBLOND, P.H. and L.A. MYSAK Waves in the Ocean. Elsevier Science Publishers B. V., Amsterdam, The Netherlands. 60 pp. MARSDEN, R.F The internal tide on Georges Bank. J. Mar. Res. : 0. PLATZMAN, G.W A Kelvin wave in the Eastern North Pacific Ocean. J. Geophys. Res. 8: 8. RIPA, P. and J. ZAVALA-GARAY Ocean channel modes. J. Geophys. Res. 10: 1,79 1,9. TAYLOR, G.I Tidal oscillations in gulfs and rectangular basins. Proc. Lond. Math. Soc. 0: THOMSON, R.E Oceanography of the British Columbia Coast. Can. Spec. Publ. Fish Aquat. Sci. 6: 91 pp. WUNSCH, C Internal Tides in the Ocean. Rev. Geophys. Space Phys. 1: 7 18.