## Abstract

In boundary areas of the World Ocean, a semidiurnal tide propagates in the form of a Kelvin wave mode trapped by the coastline. Over wide continental shelves, the semidiurnal tide is no longer a pure Kelvin wave but attains features of a zero-mode edge wave. As a result, the wave structure and the alongshore energy flux concentrate over the continental shelf and slope topography and become very sensitive to the variations of shelf geometry. When a semidiurnal Kelvin wave encounters alongshore changes of the shelf width, its energy scatters into other wave modes, including internal waves. A particularly strong scattering occurs on wide shelves, where Kelvin wave structure undergoes significant modifications over short alongshore distances. These dynamics are studied using the Regional Ocean Modeling System (ROMS). This study found that when the alongshore energy flux in the Kelvin wave mode converges on the shelf, the offshore wave radiation occurs through barotropic waves, while for the divergent alongshore energy flux, internal waves are generated. Under favorable conditions, more than 10% of the incident barotropic Kelvin wave energy flux can be scattered into internal waves. For the surface-intensified stratification mostly the first internal mode is generated, while for the uniform with depth stratification, multiple internal modes are present in the form of an internal wave beam. A nondimensional internal wave scattering parameter is derived based on the theoretical properties of a Kelvin wave mode, bottom topography, and stratification.

## 1. Introduction

Generation of internal tides is considered to be an important mechanism for barotropic tidal energy dissipation (Baines 1982; Egbert and Ray 2001; Simmons et al. 2004). Recently, this process has been extensively studied in the deep ocean, away from oceanic lateral boundaries [see review by Garrett and Kunze (2007)]. In the open ocean, barotropic tidal waves propagate over irregular bottom topography (e.g., submarine ridges) in such a manner that tidal currents cross isobaths and induce vertical displacements of isopycnals, which result in the energy conversion from a barotropic mode into baroclinic modes. For instance, Falahat et al. (2014) estimated such a conversion for the global ocean using a semianalytical approach, although their analysis was limited by a 400-m isobath, that is, excluded areas of the shelf break and the upper continental slope. Buijsman et al. (2015) parameterized this conversion in the global, barotropic tidal model by applying a linear internal wave drag in water depths exceeding 1000 m and found a significant improvement of tidal prediction in the open ocean.

Tidal dissipation on shelves is attributed primarily to the effects of bottom friction (e.g., Egbert and Ray 2001) while studies of internal wave contribution to tidal dissipation have shifted to the deep ocean, stimulated primarily by global estimates of barotropic to baroclinic tidal energy conversion. However, there is a long record of theoretical, modeling, and observational studies of internal wave generation at the continental shelf break. A good review of earlier works on this topic can be found, for instance, in Huthnance (1995). Shelf break regions, especially those of wide continental shelves, have long since been recognized as hot spots for internal tide generation (e.g., Baines 1982).

Over the continental margins, barotropic tidal waves propagate along the coastline as Kelvin waves, the lowest (zero) mode in the set of long-wave basin modes. In a Kelvin wave, currents are aligned with isobaths, especially over the upper continental slope where the bathymetric gradients are maximal. This polarization of tidal currents along the continental slope generally limits the internal wave generation. Exceptions occur when continental slopes contain bathymetric features such as submarine canyons, banks, or promontories that intercept the alongshore propagation of a Kelvin wave. Energetic internal waves associated with submarine canyons cutting through continental slopes are well documented (e.g., Kunze et al. 2002; Vlasenko et al. 2016). Kurapov et al. (2003) demonstrated that on the narrow shelf off Oregon where the barotropic to baroclinic tidal energy conversion is generally weak, enhanced conversion rates occur around a coastal bank.

Another situation supporting internal tide generation at the shelf break and over the continental slope occurs when the continental shelf is wide, and this situation is the topic of our study. A zero long-wave mode at semidiurnal frequency trapped over a wide shelf is no longer a pure Kelvin wave but attains features of a zero-mode edge wave (the latter is often referred to as a Stokes mode; e.g., Liu et al. 1998). The resulting hybrid Kelvin edge (HKE) wave mode is characterized by reduced phase speed as well as by concentration of the wave structure and the alongshore energy flux over the variable depth of the continental shelf and slope (e.g., Ke and Yankovsky 2010; Zhang and Yankovsky 2016, hereinafter referred to as ZY16).

When a zero mode propagates from a narrow to a wide shelf (or vice versa), the adjustment of its structure to the changing shelf width requires a very substantial cross-isobath energy flux, sometimes comparable with the alongshore energy flux over the continental shelf and slope (ZY16). Potentially, this can lead to the generation of intense internal tides. In reality, these dynamics are more complex than the incident mode adjustment because the incident zero mode scatters its energy into other available wave modes. The scattered wave field can include both barotropic and baroclinic modes, radiating offshore or remaining trapped over the continental shelf and slope. Energy partitioning between different modes depends on how well the structures of wave modes available in the scattering region fit the incident wave mode structure, so that a continuity of pressure and velocity fields is maintained through the scattering region, a region of topographic perturbations in the coastal waveguide.

There are several regions in the World Ocean where the shelf width reaches or exceeds 200 km and where the processes discussed in this study can operate. Examples include the West European shelf (e.g., the Celtic Sea), the Patagonian shelf in the South Atlantic Ocean, the East and South China Seas, the Grand Banks of Newfoundland, the Australian North West shelf, and some parts of the Arctic Basin. Most of these areas exhibit energetic tidal dynamics including semidiurnal tidal species.

This contribution extends the ZY16 results to the stratified ocean in order to identify conditions when internal waves become an important component of the scattered wave field. The rest of the paper is organized as follows: Section 2 describes a scattering process of the HKE mode over alongshore topographic irregularities and derives a parameter characterizing the HKE wave scattering into internal waves. Section 3 presents the results of numerical experiments for the internal wave generation by a semidiurnal HKE mode encountering changes of the shelf width in the alongshore direction, while section 4 discusses results, relates them to specific regions of the World Ocean, and concludes the paper.

## 2. Scattering of a semidiurnal Kelvin-like wave mode

We consider a zero barotropic mode trapped by the coastline and the continental shelf topography and propagating with the coast on its right (left) in the Northern (Southern) Hemisphere. This mode is commonly referred to as a Kelvin wave, but its properties somewhat differ from the analytical Kelvin wave solution over a flat bottom, and at higher frequencies this mode translates into a zero-mode edge wave (e.g., Mysak 1980; Ke and Yankovsky 2010; ZY16). We use a well-known boundary problem described, for instance, by Huthnance (1975) and Mysak (1980) in order to obtain a dispersion curve and a wave structure for the Kelvin-like zero mode in the presence of the continental shelf topography. We utilize a two-slope bottom geometry (Fig. 1) where the *x* axis coincides with the coastal wall and points in the direction of the Kelvin wave propagation (hereinafter referred to as a downstream direction), the *y* axis points offshore, and the water depth *h*(*y*) monotonically increases offshore until it reaches a constant value of the open-ocean *h*_{3}. The solution is obtained numerically using a computer code written by Brink and Chapman (1987). The wave frequency *ω* is found as the problem’s eigenvalue for a specified *h*(*y*) and alongshore wavenumber *k*, and then the corresponding free-surface perturbation *η*(*y*) is calculated as an eigenvector.

When continental shelf topography changes alongshore, the incident wave mode structure adjusts to these changes, and, if the changes are substantial, the incident wave energy scatters into other modes available at this frequency such that the continuity of the flow field is preserved and the boundary conditions at the coast and on the bottom are satisfied. It follows from the orthogonality relation for trapped modes (Huthnance 1975) that for a specified wave frequency the wave energy flux can be separated into individual mode contributions. For this reason, the strength of the trapped wave scattering is often expressed as a fraction of the incident wave energy flux transferred into other propagating wave modes. This modal approach was utilized in scattering problems of subinertial coastally trapped waves (e.g., Webster 1987; Wilkin and Chapman 1987). In addition, evanescent modes can be generated within the scattering region. Evanescent modes exponentially decay away from their source and do not contribute to the energy flux, but they produce a more complex wave field within the scattering region (e.g., Wilkin and Chapman 1987; Yankovsky and Chapman 1996).

We illustrate the effect of shelf geometry on the zero-mode properties at semidiurnal frequency by comparing depth profiles with two shelf widths *L*_{1}: 150 and 300 km. Otherwise, the depth profiles are identical with the following metrics (Fig. 1): the width of the continental slope *L*_{2} is 100 km, the coastal wall depth *h*_{1} is 5 m, the shelf break depth *h*_{2} is 100 m, and the open-ocean depth *h*_{3} is 2000 m. Dispersion diagrams of superinertial, trapped wave modes for these depth profiles were presented and discussed in ZY16. The important result is that at semidiurnal frequency only a zero, Kelvin-like mode exists, while other wave modes exhibit a low-wavenumber cutoff at frequencies higher than the semidiurnal one. In other words, they merge with the continuum of inertial–gravity waves (often referred to as Poincaré waves) above the semidiurnal frequency. Poincaré waves are not trapped and can radiate energy offshore. For this reason, only a zero mode propagating in the direction of the Kelvin wave is shown in Fig. 1 for each depth profile.

Cross-shore structure of *η*, corresponding alongshore barotropic velocity *U*, and alongshore energy flux *F*_{x} for both values of *L*_{1} are presented in Fig. 2, where *F*_{x} is defined as

Here, *g* is the acceleration caused by gravity, and *ρ* is the seawater density. Energy fluxes in Fig. 2c are normalized in such a way that their integrals over the entire width of the numerical domain are the same. In such a case, the amplitude of *η* at *x* = 0 km for *L*_{1} = 300 km should exceed the amplitude of *η* for *L*_{1} = 150 by a factor of ~7.8.

The semidiurnal wave corresponding to the 150-km-wide shelf in general resembles a Kelvin wave mode, for example, its phase speed *C* = *ω*/*k* is relatively high, close to the long gravity wave phase speed in the open-ocean ; see the dispersion diagram in Fig. 1. Also, its free-surface perturbation extends offshore over 2000 km, exhibiting a large scale for the offshore decay (comparable to the open-ocean Rossby radius of deformation; Fig. 2a). However, shelf topography does induce some changes compared to the analytical Kelvin wave solution over a flat bottom. The most important difference is that the alongshore velocity reverses its direction over the shelf compared to the deep ocean. The wave structure over a 300-km-wide shelf resembles an edge wave; its phase speed is much lower, while the wave structure is trapped over the variable topography and does not extend into an open ocean of constant depth.

If we now assume that the shelf width *L*_{1} changes in the alongshore direction, the cross-shore structure of the propagating wave should adjust, resulting in the alongshore energy flux on the shelf becoming divergent. For instance, the transition of *L*_{1} from 300 to 150 km in the downstream direction (narrowing shelf) results in alongshore energy flux convergence on the shelf (cf. the energy flux structures in Fig. 2c). This convergence induces compensating offshore energy flux across the shelf break and over the continental slope. The opposite, onshore energy flux across the shelf break develops when the shelf widens from 150 to 300 km in the downstream direction. These dynamics were studied in detail by ZY16 for the barotropic mode. They show that strong scattering of the incident semidiurnal HKE mode into radiating Poincaré waves occurs when the cross-isobath energy flux over the slope is offshore. In this situation, scattering can exceed 50% of the incident energy flux. In the case of the onshore energy flux over the continental slope, the scattering is significantly lower, within 10%–20%.

In this study, we extend results of ZY16 to the stratified ocean. When stratification is present, scattering can also occur into internal waves, both radiating (baroclinic Poincaré modes) and trapped over the continental slope topography (e.g., Dale et al. 2001). Because of their smaller spatial scales, internal waves can potentially provide a better fit between the incident and transmitted HKE waves than the barotropic mode. Below, we formulate simple scaling arguments, which define favorable conditions for the incident barotropic HKE mode scattering into internal waves. The energy conversion from barotropic into baroclinic motion per unit mass is

where *ρ*′ is the density perturbation (associated with the vertical displacement of isopycnals) relative to the unperturbed density profile *ρ*_{0}(*z*), and *W* is the vertical velocity associated with the barotropic flow field. The vertical coordinate *z* points upward, originating at the unperturbed free surface. Vertical barotropic velocity arises from the cross-isobath flow and is defined at the bottom as *W* = *Vα*, where *α* is the bottom slope. The term *V* can be estimated as follows: we assume that the cross-shore integral of *F*_{x} remains the same in upstream and downstream segments of model topography (as shown in Fig. 2c), and we use this constant energy flux condition to normalize *U* and *η*. For any isobath *h** (with corresponding offshore coordinate *y**) we now estimate the alongshore volume transport *Q* between the coast and this isobath:

Volume transport across the given isobath *h** in the scattering region is given by the difference in *Q* between the downstream and upstream segments Δ*Q* = *Q*_{dn} − *Q*_{up}. Then the cross-isobath barotropic velocity *V*(*h**) is scaled as

In defining (3) and (4), we assume that the alongshore scale of the topographic change is small compared to the HKE wavelength and acts as a discontinuity in the shelf width at least for the longer wave over the narrower shelf. Furthermore, as the alongshore scale for the wave adjustment, we choose Δ*L*_{1}, the change of the shelf width. The latter is based on the assumption that the wave will spread radially from the topographic corner at the shelf width discontinuity so that the alongshore and cross-shore scales for wave adjustment will be similar. The density perturbation in (2) can be scaled as

Combining (2) through (5), using *f*^{−1} as the time scale and *ωη* as the scale for *W*, yields a nondimensional function *S*_{i} describing a potential scattering of the barotropic HKE mode into internal waves:

where *N*(*z*) is the buoyancy frequency defined as *N*^{2} = −(*g*/*ρ*_{0})(*dρ*_{0}/*dz*). The scale for *W* comes from the Kelvin wave solution along the straight coastline, where the maximum vertical velocity occurs at the surface near the coastal wall and is associated with the alongshore flow convergence. In *S*_{i}, both *η*(*y* = 0) and *α* are chosen for the segment of the uniform shelf topography (either upstream or downstream) where the HKE mode group velocity *C*_{g} = ∂*ω*/∂*k* is lower (typically, a wider shelf). This segment of topography is where the energy buildup and the associated wave radiation occurs, as the ZY16 results have demonstrated.

A vertical profile of *S*_{i} for the full range of isobaths can be easily estimated for the topographic transition zone (a scattering region) connecting two alongshore-uniform depth profiles. Some examples of scattering regions implemented in this study are shown in Fig. 3 and are discussed in more detail in the following section. The estimation is based on theoretical wave structures of the HKE wave for two adjacent depth profiles and *N*(*z*). While the overall magnitude of *S*_{i} is expected to be the primary factor determining the incident HKE wave scattering into internal waves, two other factors can be important for the baroclinic Poincaré wave radiation. First, we expect that the generation of a particular internal wave mode will depend on how well its structure fits the vertical profile of *S*_{i}. Second, in order for those modes to be radiated offshore, their wavelength should be larger than the horizontal scale of *S*_{i}, in this case the width of the continental slope. If, on the other hand, the radiating wavelength is small compared to the *S*_{i} horizontal scale, the cross-shore structure of the generated internal wave is likely to be evanescent, exponentially decaying offshore.

## 3. Numerical experiments

### a. Model configuration

We utilize the Regional Ocean Modeling System (ROMS; Song and Haidvogel 1994; Shchepetkin and McWilliams 2005), which solves nonlinear, shallow-water equations for a stratified fluid on an *f* plane under the Boussinesq approximation. Vertical turbulent diffusion and viscosity are parameterized with the Mellor–Yamada 2.5 closure scheme (Mellor and Yamada 1982), where the background vertical mixing coefficient is set to 1 × 10^{−5} m^{2} s^{−1} for momentum and 1 × 10^{−6} m^{2} s^{−1} for density. In the horizontal, advection of both momentum and density is approximated with the third-order upstream scheme that produces some numerical damping (e.g., Haidvogel and Beckmann 1999) such that no horizontal eddy viscosity is required in our numerical experiments.

In this study, the HKE wave scattering occurs over the shelf where its width changes with respect to the alongshore coordinate. Examples of such topographic forms utilized in numerical experiments are shown in Fig. 3; in each case, two shelves with uniform, alongshore topographies but with different widths *L*_{1} are linked through the scattering region where the shelf width smoothly transitions from its upstream to downstream value. The shelf width within the scattering region is defined as

where subscripts up and dn refer to the upstream and downstream segments of the shelf topography, respectively; *X*_{up} is the alongshore coordinate where the scattering region begins; and *L*_{s} is the alongshore length of the scattering region set to 150 km in all but one cases where it is reduced to 100 km. The shelf is referred to as widening if its width increases in the downstream direction and as narrowing if the shelf width decreases. Furthermore, we distinguish between the shelf width changing by a coastline, when the coastline changes its orientation while the continental slope remains straight, and by a slope, when the coastline remains straight (Fig. 3) while the continental slope undulates alongshore.

In all model runs, *h*_{1} and *h*_{2} are set to 5 and 100 m, respectively, throughout the model domain. Two values of the open-ocean depth *h*_{3} are used: 2 (in the majority of model runs) and 4 km. Depth changes linearly from *h*_{1} to *h*_{2} over the continental shelf of width *L*_{1} and from *h*_{2} to *h*_{3} over the continental slope of width *L*_{2}. For *h*_{3} = 2 km, *L*_{1} is set to 150 and 300 km, while for *h*_{3} = 4 km, *L*_{1} is 200 and 300 km. In most cases, *L*_{2} is set to 100 km, and for *h*_{3} = 2 km, two additional values of *L*_{2} are used in the downstream segment of model topography: 75 and 50 km. The location of the scattering region *X*_{up} differs for widening shelves (*X*_{up} = 500 km) and for narrowing shelves (*X*_{up} = 750 km) because a wider shelf has a lower *C*_{g} of the HKE mode (Fig. 1), and most of the scattering effects are observed on the shelf with lower *C*_{g}. Thus, the model domain with a narrowing shelf requires a longer upstream segment of the shelf topography (where *L*_{1} = 300 km). In all cases *f* is 10^{−4} s^{−1}, which corresponds to midlatitudes.

We apply several density profiles with both constant and surface-intensified stratification: (i) linear density change from surface to bottom by 2 kg m^{−3} for *h*_{3} = 2 km and by 4 kg m^{−3} for *h*_{3} = 4 km, which corresponds to constant *N* = 3.09 × 10^{−3} s^{−1}; (ii) linear density change from surface to bottom by 6 kg m^{−3} for *h*_{3} = 2 km with corresponding *N* = 5.36 × 10^{−3} s^{−1} (these linear density profiles are referred to as *l*2, *l*4, and *l*6, respectively); and (iii) surface-intensified stratification with a net density change through the whole water column of 2 (3) kg m^{−3} for *h*_{3} = 2 (4) km defined as

Here, *h*_{pycn} = 600 (1000) m is the depth of pycnocline for *h*_{3} = 2 (4) km, Δ*ρ*_{1} = 0.5 (2) kg m^{−3}, and Δ*ρ*_{2} = 1.5 (1) kg m^{−3} for *h*_{3} = 2 (4) km. These surface-intensified density profiles are referred to as *s*2 and *s*3, respectively.

The model domain is a square basin 2000 km × 2000 km with a spatial resolution of 4 km in horizontal and 25 *s*-coordinate grid cells in vertical for most cases. For model runs with *h*_{3} = 4 km, as well as for the widening shelf with *l*6 stratification and *L*_{2} = 50 km in the downstream segment of the bottom topography, the resolution is increased to 2 km in horizontal and 40 cells in vertical. The time step is set to 5 s for two-dimensional (vertically integrated) simulations and to 300 s (150 s for 2-km horizontal resolution) for three-dimensional simulations.

The southern boundary is closed and represents a slippery coastal wall of a constant depth *h*_{1}. Other boundaries are open. At the western boundary, the incident semidiurnal HKE wave mode is specified as a free-surface perturbation (shown in Fig. 2a) oscillating with a 12-h period as a sine function. A corresponding normal to boundary barotropic velocity (*U* component) is derived from a simplified momentum balance. The nominal amplitude of *η* at the coast is set to 0.06 m for *L*_{1} = 150 km (widening shelf) and 0.2 m for *L*_{1} = 300 km (narrowing shelf). It should be noted that such an amplitude is roughly an order of magnitude smaller than typical M_{2} amplitudes found on wide oceanic shelves. This is done to reduce frictional and other nonlinear effects on the wave dynamics and to simplify the model result analysis and interpretation.

Free-surface perturbations are radiated through the northern boundary at the long gravity wave phase speed (Chapman 1985), while for the eastern boundary, where the water depth is variable, they are radiated at the same speed as in the adjacent interior grid cells (Orlanski 1976). Vertically integrated velocity is radiated through both the northern and eastern boundaries by using the Orlanski-type boundary conditions for both normal and tangential velocity components. The same radiation conditions are applied for depth-dependent velocity components on all three open boundaries as well as for density perturbations at the western and eastern boundaries. No gradient boundary condition is applied for density at the northern boundary. Shear stress at the bottom is parameterized using the quadratic law for near-bottom velocity with the bottom drag coefficient *C*_{d} set either to 2 × 10^{−3} (standard configuration) or to 5 × 10^{−4} (low friction configuration).

The model starts from rest and is run for 200 h. The incident wave amplitude is ramped up linearly over 36 h (three wave periods) to its nominal value. It takes approximately 100 h for the barotropic mode to develop a periodic regime on the continental shelf and slope within the entire model domain. After approximately 150–160 h, the effects of boundary conditions start propagating into the model interior so that the overall quality of the numerical solution somewhat deteriorates by the end of model runs. Table 1 summarizes the configurations of all model runs discussed below.

### b. Model results

For the interpretation of model results, we calculate wave properties (Table 2) and eigenvectors for the vertical velocity component *w* (Fig. 4) of baroclinic Poincaré waves in the ocean of constant depth *h*_{3} described by

Here, *C*_{i} refers to the internal wave phase speed in a nonrotating ocean, and *n* = 1, 2, 3… is the vertical mode number. For constant *N*, boundary problem (9)–(10) yields an analytical solution, while for variable *N*(*z*), this problem is discretized in finite differences and is solved numerically with *k*_{n} as an *n*th eigenvalue for specified *N*(*z*) and *ω*.

Figure 5 shows estimates of *S*_{i} profiles for most of the topographies and stratifications considered in this study. In all cases it reaches a maximum at the shelf break and quickly decays with depth (offshore). For this reason, we select the maximum value of *S*_{i} as a parameter determining the possible strength of scattering into internal waves.

We start with three cases of the continental shelf widening by slope and with *L*_{2} = 100 km (model runs 1–3). The barotropic mode solution is shown in Fig. 6 for the *l*2 stratification, where the internal wave generation is weakest. The phase propagation along the coastline (Fig. 6a) is in agreement with theoretical values of the HKE mode phase speed inferred from the dispersion diagram for both the upstream and downstream segments of the shelf topography, where the shelf width is uniform alongshore and the solution can be expressed as a sum of normal modes. Within and downstream of the scattering region (500 < *x* < 1300 km) the wave amplitude is amplified, indicating a multimode regime; however, this amplification is not particularly strong. Since no trapped modes apart from the HKE mode exist at this frequency, the scattering can occur only into Poincaré modes radiating offshore. The instantaneous distribution of *η* (Fig. 6b) shows that the wave structure concentrates over the shelf and slope topography on a wider shelf, in agreement with the modal wave structure shown in Fig. 2. The amplification of *η* downstream of the scattering region implies that a substantial part of the integral alongshore energy flux is conserved, even if a fraction of it radiates offshore or is damped by the bottom friction. The horizontal distribution of *F*_{x} (Fig. 6c) exhibits a transition from a Kelvin wave to an edge wave pattern from upstream to downstream segments of the model domain. Energy flux values on the shelf in the downstream segment of topography increase by an order of magnitude within 200 km from the coast and reverse to negative farther offshore, near the shelf break. This pattern agrees with the theoretical energy flux structure in Fig. 2c. A strong divergence of the alongshore energy flux occurs on the shelf within the scattering region, resulting in the compensating onshore energy flux component (Fig. 6d). While the strongest onshore energy flux occurs in the vicinity of the scattering region, it extends downstream for ~500 km.

The internal wave field in the presence of constant stratification *l*2 and *l*6 (model runs 1 and 2) exhibits a multimode regime with similar spatial patterns, although wave amplitudes are higher when the stratification is stronger (*l*6), both in terms of density and velocity perturbations. For this reason, only the latter case of the two is shown in Fig. 7. Internal waves radiate offshore at a nearly normal to isobaths direction (Fig. 7a), and the area of their generation extends downstream from the scattering region over several hundred kilometers, matching onshore barotropic energy flux in Fig. 6d. In Fig. 7a, the horizontal plane is selected at *z* = −397 m, where vertical profiles of *w* are close to their maxima for both modes 2 and 3, but at only ~50% for mode 1 (Fig. 4b). Hence, density perturbations associated with modes 2 and 3 (Figs. 7a,b) are near their maxima with respect to the vertical coordinate, while somewhat reduced for mode 1. The phase diagram of density perturbation (Fig. 7b) as well as the vertical structure of cross-shore velocity (Fig. 7c) both indicate the presence of multiple internal wave modes forming internal wave beams off the continental slope. The internal wave dispersion effects are clearly visible in Fig. 7, where the first mode with the highest *C*_{g} propagates farthest offshore and is identifiable by its phase speed (Fig. 7b), simple vertical structure of horizontal velocity with only one nodal line at 1000 m (Fig. 7c; *y* > 1000 km), and wavelength of ~200 km (Fig. 7a; Table 2). However, its velocity amplitude is small (Fig. 7c), which is readily explained by a poor match of *S*_{i}(*z*) and the first-mode vertical structure. Both second and third modes have higher amplitudes, but their maximum effect is confined within ~300 km from the edge of the continental slope by 150 h and continues to slowly spread offshore with time.

For the same topography, but with surface-intensified stratification *s*2, predominantly the first internal wave mode is generated (Fig. 8), while the contribution of higher modes is negligible. For the horizontal plane, we select *z* = −471 m, which is very close to the maximum of the vertical profile of *w* for the dominant first mode (Fig. 4c). However, selection of other *z* levels (not shown) does not show significant contribution of higher modes. The wavelength remains constant in Fig. 8a and is approximately 100 km, showing a good agreement with the theoretical first-mode wavelength (see Table 2). The nodal line for the cross-shore velocity component [corresponding to the maximum of *w*(*z*)] is at ~500 m, in good agreement with the vertical modal structure (Fig. 4c) for *s*2 stratification. The baroclinic velocity now is surface intensified. Density perturbations propagate offshore at a speed very close to the predicted first-mode phase speed (Fig. 8b); a minor discrepancy might arise from the fact that the wave front is not precisely aligned with the *x* axis but is tilted slightly downstream (at *x* = 900 km, where the phase diagram is obtained). That is, the internal waves radiate not strictly offshore but with some downstream component in their wave vectors.

We repeat the previously shown model runs but with the shelf width changing by a coastline (model runs 4 and 5). The internal wave radiation remains similar over the topography with a straight continental slope. The only difference is that the wave fronts also become straight, reflecting the continental slope geometry, a site for their generation. For demonstration, we present an example with *s*2 stratification, where the wave field has the simplest structure (Fig. 9a).

Next, we discuss model runs 18 through 21, where the shelf topography is reversed and the shelf narrows (Fig. 3c) from *L*_{1} = 300 km to *L*_{1} = 150 km. Compared to corresponding cases with a widening shelf, two differences arise: (i) the radiation of internal waves now occurs from the scattering region and the upstream segment of the shelf topography (where *L*_{1} = 300 km), and (ii) the internal wave amplitude is significantly weaker than for the widening shelf. These features are seen in the example shown in Fig. 9b, the case with the narrowing by slope topography and the *l*6 stratification (model run 19). The wave crests still originate from the wider shelf, which is now the upstream segment of the coastal topography (*x* < 750 km). For comparison, we use the same range for density in Fig. 9b’s color bar as in Fig. 7.

We now return to the widening shelf configuration. In all previous cases, the bottom slope *α* of the 100-km-wide continental slope was 1.9 × 10^{−2}, which is a relatively small value compared to real oceanic shelves where *α* can reach *O*(10^{−1}). For this reason, we repeat cases with shelf topography widening by a slope, but for *L*_{2} in the downstream segment set to 75 and 50 km (model runs 6–8 and 10–11). This change in bottom geometry has very little effect on the barotropic mode adjustment but increases the possibility for the internal wave generation as illustrated by *S*_{i}(*z*), which nearly doubles for *L*_{2} = 50 km (Fig. 5). We start with the *s*2 stratification (Fig. 10). Qualitatively, Figs. 10 and 8 are very similar, except the wave amplitude is enhanced when the continental slope is steeper (note the doubled ranges for both density and velocity color bars in Fig. 10 compared to Fig. 8). The first mode still dominates the internal wave field, which is not surprising given the good fit between the first-mode vertical structure and the *S*_{i} vertical profile. The higher modes are more noticeable in the phase diagram after 120 h and in the vertical structure of velocity for *y* < 500 km compared to Fig. 8, but the differences are very minor.

More important differences arise when the *l*6 stratification is specified (model run 8). For this case the model configuration required a finer spatial resolution of 2 km in horizontal and 40 *s*-coordinate grid points in vertical, since the standard resolution applied in previous cases yielded very noisy results indicating poorly resolved dynamics. Obviously higher and more energetic internal wave modes are generated in this case. Among all model runs with *h*_{3} = 2 km, the internal wave amplitude in this case is the largest both in terms of density perturbations (Figs. 11a,b) and the baroclinic velocity (Fig. 11c). On the other hand, the internal wave far field dominated by the first mode (*y* ≥ 900 km) in this case remains similar to previous cases with *l*6 stratification. The phase diagram (Fig. 11b) shows stronger modulation of isopycnal perturbations, especially for *y* < 500 km, which is indicative of higher-mode contributions (compared to Fig. 7b). These relatively more energetic higher modes result in the formation of a “tighter” internal wave beam seen at the offshore distances 400–600 km (Fig. 11c).

To further extend the parameter space of numerical experiments, we consider the shelf widening by a slope with the 4-km-deep open ocean, *L*_{2} = 100 km, and two types of stratification: *s*3 and *l*4 (model runs 16 and 17). The HKE mode over the 300-km-wide shelf is affected by the increased *h*_{3} in such a way that its *C*_{g} at the semidiurnal frequency is reduced compared to the depth profile with *L*_{2} = 50 km and *h*_{3} = 2 km. The actual dispersion curves can be found in Ke and Yankovsky (2010) and are not presented here. This relative reduction of group velocity requires a compensating increase of the wave amplitude and as a result a stronger cross-isobath velocity estimate in (4), yielding higher values of *S* than for corresponding density profiles *l*2 and *s*2 with *h*_{3} = 2 km and *L*_{2} = 50 km. Both model runs reveal a multimode, beamlike internal wave radiation (similar to the *l*6 stratification in previous cases). In the *s*3 density profile, the pycnocline represents a smaller fraction of the total water column compared to the *s*2 case (Fig. 4) because of the deeper open ocean. The first mode does not “feel” this surface intensification of stratification and retains its maximum *w* in the middle of the water column. Hence, a multimode response similar to cases with constant *N* is required to match a surface-intensified structure of the *S*_{i} vertical profile.

The incident integral energy flux (a semidiurnal HKE mode introduced at the upstream boundary) is not the same for topographies with a widening versus narrowing shelf. For this reason, the internal wave generation needs to be quantified in terms of the incident barotropic energy flux such that previous references to the internal wave amplitude as “small” or “large” are substantiated with some numbers. As discussed in section 2, the strength or efficiency of scattering can then be determined as a fraction of the incident wave mode energy flux converted into other wave mode fluxes. For each topography we define a control area *R* demarcated by transects A, B, and C in Fig. 3. For widening (narrowing) shelf topographies, transect A is at *x* = 400 km (600 km), transect B is at *x* = 1000 km (1200 km), and transect C is at *y* = 450 km, respectively. Baroclinic energy flux radiating from *R* is the horizontal baroclinic energy flux divergence integrated over *R* and defined as

where ∇_{H} = **i**∂/∂*x* + **j**∂/∂*y*, and **i** and **j** are the unit vectors in *x* and *y* directions, respectively; **F**_{i} is the vertically integrated internal wave energy flux vector, and subscripts *x* and *y* refer to its *x* and *y* components. For practical purposes, baroclinic velocity is defined as the difference between the actual and depth-averaged velocities, and the baroclinic pressure perturbations are determined from vertical displacements of isopycnals minus *η*.

First, all energy fluxes are averaged over a wave period, and all subsequent analysis is based on the wave period–averaged time series sampled at 1-h time intervals. For the incident wave with *L*_{1} = 150 km, *F*_{x} is integrated in the offshore direction at *x* = 400 km from the coast to the point where the free-surface amplitude decays to 0.1 of its value at the coast, while for *L*_{1} = 300 km, it is integrated to *y* = 600 km at *x* = 300 km. Time series of the cross-shore integrated incident energy flux is averaged over the time interval from 80 to 140 h (100 to 160 h for *h*_{3} = 4 km) for each model run, and the result is normalized by the length of the transect such that it yields the mean incident energy flux per unit cross-shore distance (W m^{−1}) when the model is in a quasi-periodic regime. Baroclinic energy flux out of the control area is averaged over the time interval from 130 to 150 h (160 to 180 h for *h*_{3} = 4 km) and is normalized by the length of the control area open boundary. Finally, the mean baroclinic energy flux radiating from the control area is normalized by the mean incident energy flux for each model run. The resulting normalized internal wave energy radiation *R* is plotted in Fig. 12 for the widening shelf against the corresponding internal wave scattering parameter *S* = max[*S*_{i}(*z*)]. As we argued in section 2, the strength of scattering should depend not only on the magnitude of *S* but also on the availability of modes for radiation. For this reason, data points in Fig. 12 are scaled according to the number of modes in the deep ocean with wave lengths exceeding the corresponding width of continental slope *L*_{2} in the downstream segment of topography.

In Fig. 12, we consider all model runs for the widening shelf with a standard bottom friction and *L*_{s} = 150 km. The scattering of the semidiurnal HKE mode into radiating internal waves increases until *S* reaches ~0.15; after that, *R* becomes nearly constant at 14%–16% of the incident energy flux. Values of *R* as a function of *S* do not collapse onto a single curve in Fig. 12. Indeed, relatively stronger scattering occurs for *l*6 stratification over narrower continental slopes of 75 and 50 km (*S* is ~0.1 and ~0.15, respectively) because there are more internal wave modes available in these cases for the offshore radiation with wavelengths exceeding or comparable with *L*_{2} (Table 2). Parameter *S* does not distinguish between shelf topographies widening by coastline and slope, but there is a tendency for shelves widening by a coastline to yield slightly lower values of *R*. However, this difference diminishes when the low bottom friction is specified. This is evident in model runs 12 versus 14 and 13 versus 15, which are not included in Fig. 12 because their values of *R* cluster very closely to corresponding cases with the standard bottom friction. Another example of the robustness of the parameter *S* comes from model run 9, where the length of the scattering region *L*_{s} is reduced to 100 km, which does not affect *S*. The resulting *R* is nearly identical to model run 2 with “conventional” *L*_{s} (Table 1).

In the cases of a narrowing shelf, the scattering into internal waves is significantly lower than for the corresponding widening shelf cases (model runs 18–21; Table 1) varying between a small fraction of a percent and one percent. To understand this result, we refer to the barotropic mode scattering discussed in detail by ZY16. For the narrowing shelf topographies used here the barotropic alongshore energy flux on the shelf is convergent within the scattering region, which leads to the compensating offshore energy flux over the continental slope. According to ZY16, this scenario produces strong offshore radiation of barotropic Poincaré waves, reaching 50% or more of the incident energy flux when their divergence parameter *D*_{e} is −1 or less. This result holds here such that the downstream energy flux associated with the HKE mode is substantially reduced past the scattering region. The definition of *S*_{i} on the other hand is based on the assumption that the incident wave mode conserves its integral *F*_{x}. We conclude that scattering into barotropic radiating modes plays a primary role in the HKE wave adjustment to the topographic variations, while the internal wave generation is additive when the water column is stratified. This implies that the internal wave generation primarily occurs when the barotropic scattering is relatively weak, that is, when the alongshore energy flux is divergent on the shelf (a widening shelf topography in our case). It should be reminded here that a widening versus narrowing shelf does not define unambiguously whether the alongshore energy flux on the shelf is divergent or convergent. Instead, the modal structures for upstream versus downstream segments of topography should be obtained and matched. For instance, different configurations of a narrowing shelf in ZY16 produced both convergent and divergent alongshore energy fluxes on the shelf within the scattering region.

Dissipation of the semidiurnal HKE wave mode through internal wave generation is not solely characterized by the radiating energy flux. The importance of the local dissipation over the sloping topography within the scattering region associated with baroclinic motions is illustrated in Fig. 13. Here, the two model runs, 2 and 8, are compared. They have different *L*_{2} in the downstream segment of topography: 100 and 50 km. Model run 8 with a steeper continental slope has the strongest internal wave radiation among all examples presented.

First, we look at wave period–averaged barotropic energy fluxes through three transects demarcating the control area (Figs. 13a,d). These energy fluxes reach a near-steady regime by 90–100 h. After that time, temporal energy variations of the barotropic mode in the control area are minimal, and the barotropic energy flux convergence should be balanced by the local energy dissipation (the latter will obviously include the internal wave generation). The corresponding baroclinic energy fluxes through the same transects indicate that radiating internal waves take a longer time to reach a near-steady regime (by 130–150 h). For the model run with *L*_{2} = 100 km, the internal wave energy flux divergence over the control area is a relatively small fraction (~15%–20%) of the barotropic energy flux convergence and hence dissipation (Fig. 13c). For the case with *L*_{2} = 50 km, the internal wave radiation is threefold higher, reaching ~7 × 10^{7} W. However, the corresponding barotropic convergence increases by more than 10^{8} W (Fig. 13f), which implies that some baroclinic energy dissipates locally (within the control area). As a result, in this last example the barotropic energy flux convergence within the control area is more than twice higher than in the other (similar) case with weaker internal wave generation.

## 4. Discussion and conclusions

Our numerical experiments demonstrate that semidiurnal Kelvin-like waves propagating over wide continental shelves can scatter a significant fraction of their energy into internal waves when encountering alongshore changes of the shelf width. We have derived a simple internal wave scattering parameter *S* based on the theoretical properties of HKE wave modes upstream and downstream of the topographic feature, stratification, and the bottom topography. We do not anticipate a linear dependence of the strength of scattering on *S* because the former depends not only on the magnitude of *S*, but also on the number of baroclinic modes available for radiation as well as on how well vertical structures of those modes fit the *S*_{i} profile. Nevertheless, the parameter *S* appears to capture the relative magnitude of the radiating internal wave energy flux reasonably well as long as the radiation of barotropic Poincaré waves is limited. In this regard, barotropic semidiurnal Kelvin wave scattering into internal waves is additive (secondary) to the barotropic mode wave field and occurs on large alongshore spatial scales typical of the barotropic mode adjustment.

For the widening shelf, *S* scales with the normalized radiating baroclinic energy flux until approximately 14%–16% of the incident barotropic energy flux is transferred into radiating internal waves. This appears to be an upper limit of the internal wave radiation in our cases, even if *S* continues to grow to 0.2–0.25 and beyond. However, this further increase in *S* is accompanied by the enhanced local dissipation of the barotropic mode within the scattering region in addition to the aforementioned baroclinic energy flux divergence around the perimeter of the scattering region. These results imply that the internal wave generation can be an important component in dissipation of semidiurnal tidal species in the coastal ocean.

In our numerical experiments, internal waves radiate in the direction normal to local isobaths, with a very small fraction of the baroclinic energy flux propagating along the continental slope. This obvious lack of topographically trapped internal waves can be explained as follows: These waves result from the extension of subinertial coastally trapped wave (CTW) dispersion curves into the superinertial frequency band where they become leaky (e.g., Brink 1991; Dale et al. 2001). Both the phase speed and the group velocity of these waves in the vicinity of the inertial frequency depend on the slope Burger number Bu = (*NH*/*fL*_{2})^{2} such that waves become nondispersive (baroclinic Kelvin wavelike) for Bu → ∞ (Huthnance 1978; Brink 1991). In our model cases even with the strongest stratification and steepest topography, for example, *l*6 with *L*_{2} = 50 km or *l*4, Bu is of *O*(1), and the dispersion curves of these wave modes are expected to flatten (decreasing *C* and *C*_{g}) with increasing *ω*. This implies that their alongshore wavenumbers become large when the dispersion curve of even the gravest, first CTW mode reaches the semidiurnal frequency. Our smooth topographic forms with an alongshore scale of *O*(100) km simply do not match the short wavelengths of internal-trapped waves available at semidiurnal frequency. These trapped internal waves are observed and modeled over a ragged slope topography with much smaller topographic scales (Stashchuk and Vlasenko 2017) or over a much narrower continental shelf and slope with corresponding higher Bu (Klymak et al. 2016).

Trapping of the internal wave energy over a continental slope can be particularly important in high latitudes, where the M_{2} tide becomes subinertial, and the radiation of Poincaré modes is not possible. In this situation, the incident Kelvin wave mode can scatter into internal bottom-trapped slope modes (Rhines 1970). The possible contribution of these modes into a global conversion of semidiurnal tidal energy into baroclinic motions was recently discussed by Falahat and Nycander (2015).

Two of the examples discussed in the introduction appear to be particular relevant to this study, the West European shelf and the Patagonian shelf. In both cases the M_{2} harmonic propagates as a Kelvin wave and exhibits features of the HKE mode. The latter tendency is particularly noticeable on the Patagonian shelf between 52° and 40°S where the M_{2} harmonic resembles an edge wave (e.g., Glorioso and Flather 1997; Palma et al. 2004). The Celtic Sea shelf break has one of the highest rates of semidiurnal tidal energy conversion into baroclinic modes in the World Ocean (e.g., Vlasenko et al. 2014). The observed baroclinic energy fluxes show that internal waves propagate both across the isobaths (radiating offshore and inshore) and along the continental slope (Green et al. 2008). The internal wave field in the Celtic Sea is strongly modulated by smaller-scale topographic features such as canyons and banks (e.g., Vlasenko et al. 2014, 2016), but the primary reason for high conversion rates appears to be related to strong, barotropic, cross-isobath fluxes associated with abrupt widening of the continental shelf to a more than 300-km width. Steep topographic slope at the shelf break and a relatively strong stratification are also contributing factors. The Patagonian shelf does not exhibit strong radiation of low-mode internal waves (e.g., Simmons et al. 2004), although cross-isobath barotropic energy fluxes there are strong (Palma et al. 2004). The lack of low internal wave modes can be related to weak stratification at the shelfbreak depth caused by the cooling effect of the Malvinas Current (Combes and Matano 2014). However, the presence of higher modes, especially trapped over the continental slope or refracted by the Malvinas Current, is still possible and merits further investigation.

The paper focuses on the semidiurnal frequency representing semidiurnal tidal species, by far the most energetic frequency band within the superinertial frequency range in the World Ocean. The same processes can operate at higher frequencies, for instance when a transient long-wave response to a moving storm system occurs on the continental shelf. While energetic higher-frequency events are not as ubiquitous as semidiurnal tides, they do not require shelves as wide as those considered here for internal wave generation. Indeed, the primary reason for strong cross-isobath energy fluxes is the transformation of a zero mode from a Kelvin wavelike to edge wavelike structure (or vice versa) as the wave propagates alongshore. For narrower shelves, HKE modes exist at higher frequencies than the semidiurnal one [see dispersion diagrams in Ke and Yankovsky (2010) for details]. This implies that internal waves can radiate from the shelf break in the presence of a fast moving storm when the shelf width is only 100 km or so. Overall, the results point to the importance of the shelf break on wide, continental shelves as a source of both radiating and topographically trapped internal waves.

## Acknowledgments

This study was supported by the U.S. National Science Foundation through Grant OCE-1537449. We are thankful to two anonymous reviewers for their thoughtful comments and suggestions and to Ken Brink for providing a code for computing properties of coastally trapped wave modes.

## REFERENCES

_{2}tidal energy dissipation from TOPEX/Poseidon altimeter data

*Numerical Ocean Circulation Modeling.*Imperial College Press, 318 pp.

_{2}internal tide off Oregon: Inferences from data assimilation

## Footnotes

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