## 1. Introduction

A full set of barotropic long waves trapped in the coastal ocean over a variable topography includes a zero (fundamental) mode that exists at both subinertial and superinertial frequencies and propagates with the coast on its right (left) in the Northern (Southern) Hemisphere (hereinafter, we refer to this direction as downstream) (e.g., Huthnance 1975). This zero mode resembles a Kelvin wave at lower, near- or subinertial frequencies and an edge wave (Stokes mode) at high frequencies (Mysak 1980). At the intermediate superinertial frequencies this mode becomes a hybrid Kelvin–edge wave (HKEW), as both rotational effects and the variable depth become important (Munk et al. 1970). Yankovsky (2009) showed that if the shelf is wide enough the group velocity of this hybrid mode becomes zero. A zero group velocity implies that the wave energy does not propagate along the coastline.

Semidiurnal tides represent the strongest signal in the intermediate superinertial frequency band in the subtropics and midlatitudes and often propagate along the continental margins in the form of a fundamental trapped mode, possibly as a HKEW (Munk et al. 1970). The transmission of tidal energy along the shelf can be impaired by a reduction of the group velocity, especially as it approaches zero. This effect was previously discussed for the diurnal tides off the Scottish west coast (Smith 1975). In this area, diurnal tides propagate as subinertial continental shelf waves (Cartwright et al. 1980), and the reduction of their group speed can explain the observed amplification of tidal currents, especially at the shelf break. The goal of this paper is twofold: 1) to determine characteristics of shelf geometry when the HKEW group velocity becomes zero at a semidiurnal frequency and 2) to model the evolution of a semidiurnal HKEW when it encounters topographic variations that cause the wave group velocity to become zero. The rest of the paper is organized as follows: Section 2 describes the HKEW dispersion characteristics and relates zero group velocity at a semidiurnal frequency with the shelf geometry. Section 3 presents a numerical experiment in which the incident wave encounters a variable shelf width and its downstream propagation is essentially blocked because of a zero group velocity. Section 4 compares these results with World Ocean tides exhibiting similar patterns of alongshore propagation and concludes the paper.

## 2. Dispersion characteristics of the fundamental mode

*f*-plane approximation and seeking the solution that is periodic in time and alongshore coordinate. This yields a well-known boundary problem (e.g., Huthnance 1975),subject to boundary conditionswhere

*x*and

*y*are the alongshore and offshore coordinates (Fig. 1),

*u*and

*υ*are the

*x*-axis and

*y*-axis velocity components,

*f*is the Coriolis parameter,

*η*is the free surface elevation,

*h*=

*h*(

*y*) is the water depth,

*g*is the acceleration due to gravity,

*k*is the alongshore wavenumber, and

*ω*is the wave frequency. The boundary condition for trapped waves (3) implies an exponential decay of free-surface perturbation in the offshore direction. At the finite offshore distance at which the water depth becomes constant it can be expressed asBoundary problem (1)–(2) and (4) is solved numerically by applying resonant iteration in

*ω*for a specified

*k*. In this study we utilize a computational code developed by Brink and Chapman (1987). The numerical solution contains a spurious mode at the inertial frequency (e.g., Pedlosky 2003). This spurious mode represents a certain difficulty for numerical calculations of near-inertial trapped wave modes in the stratified ocean (e.g., Dale et al. 2001). However, in our case of a barotropic ocean, only a Kelvin wave mode exists at the inertial frequency and its structure is easily distinguishable from the spurious inertial mode. Nevertheless, we did not calculate a Kelvin wave mode in the vicinity of the inertial frequency so as to avoid a numerical interference of “real” and spurious wave-mode solutions.

The topography of the continental shelf and slope is approximated as a two-slope depth profile adjacent to a deep ocean of a constant depth (Fig. 1). The depth profile is characterized by the width of the shelf, slope, and deep ocean (*L*_{1}, *L*_{2}, and *L*_{3}, respectively) as well as by the depth of the coastal wall, shelf break, and deep ocean (*h*_{1}, *h*_{2}, and *h*_{3}, respectively). The offshore boundary condition (4) is applied at *y* = 1.5(*L*_{1} + *L*_{2}) km. The spatial resolution Δ* _{y}* in the finite-difference approximation is set to 1 km. We conduct a sensitivity study of the fundamental-mode dispersion characteristics by systematically varying the shelf and slope widths

*L*

_{1}and

*L*

_{2}, as well as the shelfbreak and deep-ocean depths

*h*

_{2}and

*h*

_{3}. The coastal wall

*h*

_{1}= 5 m represents a nearshore zone boundary at which the wave-breaking and turbulent stresses dominate, and it is kept constant for all cases.

Our study indicates that the dispersion curve of a zero mode is far more sensitive to the variations of a shelf width *L*_{1} than of a slope *L*_{2}. Hence, we first show how the dispersion diagram of a zero mode depends on the shelf width (Fig. 2) while other metrics of the depth profile are held constant: *h*_{2} = 100 m, *h*_{3} = 2000 m, and *L*_{2} = 100 km. Two values are used for the Coriolis parameter: *f* = 10^{−4} s^{−1} (representing midlatitudes) and *f* = 6.59 × 10^{−5} s^{−1} (representing the subtropics).

*h*

_{3}because of the earth’s rotation described by the dispersion relationAt high frequencies/wavenumbers, it resembles a zero edge-wave mode (often called a Stokes mode) formed by long-wave refraction over a sloping bottom. The Stokes-mode dispersion relation for the linear depth profile

*h*=

*αy*is (e.g., Kurkin and Pelinovsky 2002)At the intermediate frequencies/wavenumbers, the dispersion curves become horizontal if the shelf is sufficiently wide, that is, the group velocity

*C*= ∂

_{g}*ω*/∂

*k*becomes very small or even zero (Fig. 2). At these intermediate frequencies, the zero mode is a hybrid Kelvin–edge wave, as both the earth’s rotation and depth variations are important (Yankovsky 2009).

Zero-mode dispersion curves corresponding to different values of *L*_{1} converge at both high and low frequencies and exhibit the widest separation at the intermediate frequencies/wavenumbers, where *C _{g}* is minimal or close to zero (Fig. 2). At high frequencies the wave structure concentrates relatively close to the coastline and is insensitive to the shelfbreak position. Since we apply a vertical wall at the coast, the wave dispersion in our case will differ from (6) at high frequencies and will asymptotically approach the phase speed

*C*= (

*gh*

_{1})

^{1/2}over gently sloping topography. Indeed, the convergence of dispersion curves for different

*L*

_{1}at high frequencies is evident in Fig. 2. At low frequencies, the wave extends offshore well beyond the area of the continental shelf and slope, and its phase speed is close to that of a Kelvin wave. Only at the intermediate wavenumbers (

*k*∼ 10

^{−6}–10

^{−5}m

^{−1}) is the offshore extension of the wave structure comparable to the continental shelf and slope width, so that the phase speed becomes sensitive to the position of the shelf break. The group velocity becomes close to zero at a semidiurnal frequency (wave period is 12 h) when the shelf width is ∼250–300 km, and this property weakly depends on the Coriolis parameter (cf. left and right panels in Fig. 2).

The influence of other parameters of the depth profile on the zero-mode dispersion is summarized in Fig. 3. Variations of continental slope width have very little impact on the wide shelf (300 km) but become noticeable for the narrow shelf (50 km) at lower wavenumbers: for a fixed wavenumber the wave frequency (and consequently the phase speed) becomes lower over a wider continental slope. When the deep ocean depth *h*_{3} increases, the difference between the edge-wave and the Kelvin-wave phase speed also increases. This leads to a broader separation of these two asymptotic regimes in the wavenumber domain. As a result, the range of wavenumbers over which *C _{g}* is close to zero widens (cf. Figs. 3a and 3d;

*L*

_{1}= 300 km). A decrease of the open-ocean depth has the opposite effect and can even eliminate a zero group velocity altogether (Fig. 3c). Variations of the shelfbreak depth

*h*

_{2}have a somewhat similar effect: its decrease from 100 to 50 m broadens the separation between the edge-wave and the Kelvin-wave branches of the dispersion curve and thus increases the wavenumber range over which

*C*is close to zero (not shown). In contrast, the deepening of

_{g}*h*

_{2}to 200 m eliminates the zero group velocity (

*L*

_{1}= 300 km), although it still remains low for hybrid waves (Fig. 3b).

The alongshore propagation of tidal energy should be strongly affected (and possibly blocked) when a semidiurnal tidal wave in the form of a fundamental trapped mode encounters a shelf profile where the wave group velocity becomes zero. This situation is delineated in the numerical experiment discussed in the following section.

## 3. Numerical experiments

### a. Model configuration

We use the Regional Ocean Modeling System (ROMS; Song and Haidvogel 1994; Shchepetkin and McWilliams 2005) in a two-dimensional (vertically integrated) configuration. The model solves nonlinear shallow-water equations on an *f* plane. The Coriolis parameter is set to *f* = 1.0 × 10^{−4} s^{−1}. We apply a horizontal Laplacian viscosity with a constant coefficient of *A* = 50 m^{2} s^{−1}.

*x*coordinate coincides with the coastline and points in the direction of the Kelvin-wave propagation (hereinafter referred to as downstream), and the

*y*coordinate points offshore. The domain is 3450 km long (

*x*direction) and 2400 km wide (

*y*direction). The coastal wall is specified at the southern boundary (

*y*= 0 km) while other boundaries are open. The two-slope offshore depth profile is defined in the same manner as in the previous section with the following configuration:

*h*

_{1}= 5 m,

*h*

_{2}= 100 m,

*h*

_{3}= 2000 m, and

*L*

_{2}= 100 km. Upstream and downstream segments of the model topography have a uniform shelf width

*L*

_{1}set to 300 and 200 km, respectively. The shelf width

*L*

_{1}(

*x*) changes from its upstream

*L*

_{1u}to downstream

*L*

_{1d}value between

*x*= 1300 km and

*x*= 1700 km as a cosine function:

According to dispersion diagrams shown in Fig. 2a, the fundamental-mode group velocity becomes zero at *L*_{1} ≈ 280–290 km (i.e., within the area of variable shelf width in the numerical domain). The horizontal model grid is rectangular with an alongshore and across-shelf resolution of 7.5 km. A free-slip, no-normal-flow boundary condition is applied at the coastal wall. An incident wave is specified at the upstream (western) boundary (see below). The Orlanski-type radiation boundary condition is applied at the eastern boundary. At the deep-ocean (northern) boundary, perturbations are radiated with long gravity wave speed (Chapman 1985). A quadratic stress is specified at the bottom with a bottom drag coefficient of 3 × 10^{−3}.

At the initial moment the model is at rest. The model is forced by the incident wave through the upstream (western) across-shelf boundary (*x* = 0). The incident wave is a zero-mode downstream-propagating wave with a period of 12 h corresponding to *L*_{1} = 300 km. Its *η* structure is found from (1)–(4) and is specified as the upstream boundary condition varying in time as a sine function with a 12-h period. The time step is 10 s.

We use a numerical domain that is much wider than the shelf and slope width. This is necessary for two reasons. First, this allows the incident wave to decay offshore within the numerical domain so that there is no conflict between the boundary condition (4) applied for calculating the wave structure and the radiation boundary condition applied in the primitive equation model. Second, if the alongshore energy flux is indeed blocked at some location, there should be a sufficient distance for the offshore radiation of waves not impaired by the presence of boundary conditions (which are never perfectly transparent).

When the incident wave encounters a topographic variation, its structure and phase speed adjust to the changing depth profile. In addition, a fraction of the incident wave energy flux can be reflected upstream or can scatter into other modes available at the frequency of the incident wave (Wilkin and Chapman 1987). Figure 5 investigates these possibilities for trapped wave modes. The reflection and upstream propagation occur through the generation of upstream-propagating edge waves. However, it is not possible in our case since the zero and higher upstream-propagating modes are cut off above the semidiurnal frequency in the upstream segment of the model domain (*L*_{1} = 300 km). At the same time, first and higher downstream-propagating modes are also cut off above the semidiurnal frequency for both *L*_{1} = 300 and *L*_{1} = 200 km. Thus, the wave can travel alongshore through the domain while retaining its semidiurnal frequency only as a zero downstream-propagating mode.

However, the structure of this mode undergoes dramatic change from wider to narrower shelf segments (Fig. 6). In essence, the wave changes from an edge-wave-like to a Kelvin-wave-like structure when the shelf width decreases from 300 to 200 km. The free surface and alongshore velocity disturbances both extend much farther offshore on the 200-km shelf and decay on the barotropic Rossby radius scale. The velocity profile on the 200-km shelf has two nodal points (whereas a pure Kelvin wave has none), and the velocity becomes negative on the shelf (opposite to the instantaneous velocity direction in the incident wave on the 300-km shelf). The complexity of this velocity structure makes the seamless transition of the incident wave to a narrower shelf unlikely. The wave energy flux concentrates on the shelf and near the coast over the 300-km-wide shelf but shifts offshore and reaches its maximum at the edge of the continental slope in the case of the 200-km-wide shelf. This implies that the transition from the upstream to downstream wave structure will require an offshore spreading of the wave energy.

### b. Model results

The propagation of a zero wave mode through the model domain is illustrated by temporal evolution of a free surface at the coast (phase diagram; Fig. 7). The wave propagates through the upstream segment of the model domain (where the shelf width is 300 km) at a constant phase speed close to 17 m s^{−1}, which is a value obtained from the boundary problem (1)–(4) for the same frequency. This theoretical phase speed is plotted for reference as a black straight line in the phase diagram. Upon entering the domain, the incident wave maintains its analytically prescribed structure over the 300-km-wide shelf (Fig. 7, bottom panel), even though the offshore resolution is coarser in the numerical experiment relative to the boundary problem (1)–(4). Since the group velocity of the incident wave is lower than its phase speed (Fig. 5), it takes four to five wave periods for the wave amplitude throughout the upstream segment of the model domain to match the amplitude of the incident wave. By 90 h, the wave amplitude exceeds the amplitude of the incident wave just upstream of the shelf with a variable width (*x* ∼ 1100–1300 km). Subsequently, the wave amplitude increases through the entire upstream segment of the model domain. There is also a slight shift of phase and a modulation (decrease) of wave amplitude at *x* ∼ 700–800 km—especially evident after 100 h. The phase speed and amplitude modulation indicate the presence of multiple wave modes.

We conducted an additional test run to prove that this phase and amplitude modulation is indeed related with the narrowing shelf and is not induced by the adjustment of an analytically prescribed incident wave to the numerical domain. We let the same incident wave propagate through the 2300-km-long channel uniformly alongshore. The channel is 800 km wide, narrower than in the main numerical experiment because no offshore radiation is expected, but with finer offshore resolution of 2.5 km. The wave propagates through the channel without any modulation or deviation of its phase speed from the theoretical value (not shown). Its structure at *x* = 1000 km is shown in Fig. 7, and it is close to both the analytically prescribed structure at the upstream boundary and the wave structure in the main numerical experiment in the upstream segment of the channel. The only minor difference is somewhat gentler offshore decay for *y* < 250 km. This is related to the bottom friction applied in the numerical experiments, and the effect becomes more noticeable with downstream distance.

The wave amplitude decreases dramatically downstream of *x* = 1500 km and varies within 2–3 cm in the downstream segment of the channel (*L*_{2} = 200 km) after 80 h. Even more important is that after ∼100 h there is very little or no phase propagation past the area of a variable shelf width (*x* > 1700 km). This implies that in the downstream segment the oscillations occur simultaneously along the coastline and the wave propagates at an angle that is close to normal to the coastal boundary. The phase diagram in the across-shelf direction (Fig. 8) captures this feature. We choose a line running normal to the coast at *x* = 1575 km, where the wave amplitude is still substantial. The phase leads in the vicinity of the shelf break and lags both inshore and offshore, suggesting that waves can propagate both toward the coast and offshore. It is worth mentioning that the maximum downstream energy flux shifts from nearshore to an offshore position as the shelf width narrows from 300 to 200 km (Fig. 6). Hence, the energy convergence at the shelf break can lead to wave radiation in both directions normal to the coastline. Of course the phase lag offshore the shelf break is barely noticeable on the phase diagram (Fig. 8) since the long gravity wave phase speed increases dramatically in the deep ocean.

### c. Energy flux and energy balance

*x*= 1000 km), B (

*x*= 1950 km), and C (

*y*= 200 km), shown in Fig. 4, aswhere

*F*is the alongshore (transects A and B) component and

_{x}*F*is the offshore (transect C) component of the energy flux and

_{y}*ρ*is the water density. Energy fluxes are averaged over the wave period (12 h). Of the three sections, the largest energy flux is through the upstream section A (Fig. 9, top panel). It shows some low-frequency fluctuations following the wave-amplitude modulation at this location. The energy flux through section B is small and negative (in the upstream direction). The positive energy flux through section C indicates an offshore radiation of the wave energy, consistent with the phase diagram (Fig. 7). Initially, it is smaller than the energy flux through section A at that same time, which implies energy flux convergence. However, it gradually grows and by 180 h reaches corresponding values through section A. The energy balance equation iswhereHere

*E*is the wave energy density, KE and PE refer to its kinetic and potential components, respectively, and

*R*is the residual term (dissipation and numerical errors). Integrating (9) over the control area shown in Fig. 4 yields

The first term in (10) is the rate of change of wave energy in the control area (referred to as power in Fig. 9), and the second, third and fourth terms account for the energy flux horizontal divergence. The results of integration are shown in Fig. 9 (bottom panel). The wave energy in the control area continuously grows through 100 h (which is evident in the wave-amplitude increase on the phase diagram). The growth then becomes intermittent and finally halts by 180 h. There still exists some energy flux convergence after 180 h, but it is balanced by the residual term (dissipation). This implies that the wave field has reached a quasi-periodic state within the control area.

The results of the energy-balance integration are different if the offshore section C is extended to *y* = 400 km. In this case the energy gain continues through the end of the model calculations because of wave energy flux convergence. The amplitude of radiating waves grows offshore the shelf break but still remains an order of magnitude smaller than on the shelf, and the dissipation is insufficient to balance the divergence term. We conclude that the alongshore energy flux convergence associated with a zero group velocity leads to wave amplification and offshore wave energy radiation.

## 4. Discussion and summary

A fundamental (zero) wave mode trapped over the continental shelf and slope topography represents a hybrid wave combining properties of a Kelvin wave and a Stokes edge wave in a certain frequency range, when both rotational and topographic effects are important. When the shelf width is sufficiently wide, 1) the HKEWs exist within the semidiurnal tidal frequencies and 2) their group velocity can become zero. We found that the 12-h wave’s group velocity is close to zero over the shelf width of ∼300 km. Once the HKEW encounters such a shelf width, its downstream propagation is essentially blocked, resulting in strong convergence of the alongshore energy flux and the increase of tidal amplitude and energy density upstream. The continuing convergence of wave energy in the vicinity of this “choking point” can be balanced by strong tidal dissipation and/or by the offshore radiation of tidal energy. Dissipation in our case was relatively weak, which resulted in a growth of tidal amplitude through the whole upstream segment of the model domain. We applied smoothly varying topography in our model experiment. However, the presence of smaller-scale topographic and coastline irregularities in the real ocean can vastly enhance the tidal dissipation (e.g., Pinsent 1972; Mysak and Tang 1974). In our case the accumulating wave energy was released through the offshore wave radiation. The barotropic Poincaré waves at semidiurnal frequencies have a spatial scale comparable to the size of oceanic basins, and their fate should be determined in the context of the tidal dynamics of the whole basin. There is also a possibility for generation of internal waves, especially at the shelf break where the energy flux of HKEWs with zero group velocity concentrates. This possible conversion of barotropic into baroclinic tidal energy merits future investigation.

Our finding is qualitatively consistent with the dynamics of semidiurnal tides on wide shelves in the World Ocean. For example, the Patagonian shelf in the southwestern Atlantic Ocean is characterized by strong tidal currents; it is extremely wide at the latitude of the Malvinas (or Falkland) Plateau (∼50°S) and then gradually narrows northward (in the downstream direction), reaching a critical width of slightly less than 300 km at approximately 40°S. There are several examples of high-resolution numerical simulations of tidal currents on the Patagonian shelf, most notably by Glorioso and Flather (1997) and Palma et al. (2004). According to these studies, the dominant tidal harmonic is the semidiurnal lunar tide *M*_{2}. Its dynamics is summarized in Figs. 10 and 11 (Palma et al. 2004): it propagates downstream (northward) in the manner of a trapped wave mode up to ∼42°S. At this latitude the tidal amplitude is amplified while propagation becomes predominantly normal to the coastline (Fig. 10). On the shelf the phase propagates inshore (Fig. 10) while over the deep ocean (not shown here) *M*_{2} propagates eastward in this latitudinal band (e.g., Le Provost et al. 1995). The amplitude is dramatically reduced farther downstream at 40°S. The energy budget indicates strong tidal dissipation between 44° and 41°S with a negligible alongshore energy flux reaching 40°S (Fig. 11). These patterns can be explained by the results presented in our study. Of interest is that the diurnal tidal harmonic *K*_{1}, which also propagates northward along the coast, does not exhibit any blockage or amplitude dampening in the vicinity of 40°S (Glorioso and Flather 1997).

Another possible candidate for the alongshelf blockage of semidiurnal tidal energy is the East China Sea and the Taiwan Strait. The shelf width of the East China Sea rapidly narrows southwestward, in the downstream direction, from several hundred kilometers to less than 200 km at the entrance of the Taiwan Strait. As the *M*_{2} tide enters the Taiwan Strait, its amplitude is dramatically amplified (reaching 2 m). A high tidal amplitude is still observed at the southern (downstream) exit from the strait, but very little tidal energy propagates beyond that point (e.g., Jan et al. 2004; Beardsley et al. 2004). Tidal dissipation is also high throughout the Taiwan Strait. The diurnal tide *K*_{1}, on the other hand, is little affected by the presence of the Taiwan Strait as it propagates downstream in the southwestward direction (Beardsley et al. 2004). Thus, the results presented here allow easy diagnostics of the World Ocean shelf topography where the semidiurnal tidal amplitude and dissipation can be enhanced while the alongshore energy flux farther downstream is dramatically reduced.

## Acknowledgments

We thank Ricardo Matano and Elbio Palma for providing the results of their tidal simulations on the Patagonian shelf. We are also indebted to Ken Brink who kindly provided the latest version of his coastal-trapped-wave programs and to anonymous reviewers for their insightful comments and suggestions. This work was supported by the U.S. National Science Foundation through Grant OCE-0752059.

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