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    Location of T-TIDE field site: (a) Energy flux in a regional numerical simulation. Color is the absolute value of the flux; arrows show its direction. The magenta box indicates the numerical modeling domain used in this paper. (b) Detail of the bathymetry on the Tasman slope. The magenta lines indicate 100 km in the x direction and 300 km in the y direction in the modeling domain used in this paper. (c) The stratification used for all runs in this paper, taken from the World Ocean Atlas at 44°S, 152°E (Boyer et al. 2013). The steps are due to linear interpolation of the density between the atlas depths.

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    (a) Energy flux representation of the analytical response to the forcing used to drive the models used in this paper. Two mode-1 internal wave sources are located to the south east (blue lines). The model domain is rotated 12° from geographic so the shelf break approximately lies along x = 0. The outer model domain is the rectangle with the thick green line. The inner green rectangle denotes 1-km lateral resolution. Outside this rectangle, resolution increases 3.5% per cell to a maximum of 5 km in the second largest rectangle, and outside this rectangle, grid size increases to a maximum of 10 km. The sponge layer is indicated by the largest thin, green rectangle. The 250- and 3000-m isobaths are contoured, though the bathymetry is not used in the analytical response. Arrows show the direction of the energy flux and are scaled by its strength. (b) Energy flux of analytical response of energy reflecting from a wall at x = 0, north of y = 0, in a flat-ocean domain. (c) Phase of reflected response.

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    (a)–(d) Surface x-direction velocity for four snapshots. (a) The initial conditions (slightly modified after a tidal cycle) and (d) the steady state. Gray contours are depths at 3000, 2000, 1000, and 250 m. (e)–(h) Depth-integrated baroclinic energy flux at the same time periods, with arrows indicating direction.

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    Energy budget over the nineteenth tidal cycle of (a) barotropic to baroclinic conversion, (b) baroclinic energy flux convergence −∇Fbc, (c) rate of change of baroclinic energy, and (d) residual representing the dissipation in the model D = −∇Fbc + Conversion − dE/dt. The green box is the region for the energy time series (Fig. 5b).

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    (a) Integral in x to 80 km offshore of the energy terms in Fig. 4 for the REAL case (averaged over twentieth tidal cycle). Note that the barotropic–baroclinic term (red) is of the same order as the baroclinic convergence (blue) and the residual dissipation (gray) for most of the slope. (b) Energy budget time series for the REAL case, tidally averaged, where time is normalized by T = 12.4 h, between y = 0 and 400 km. There is still a small, residual increase in the energy with time (purple), representing the accumulation of high-mode energy in the region. Net barotropic–baroclinic conversion (red) is small and negative, indicating a small net loss to the barotropic tide in this region. The bulk of the budget is the balance between baroclinic flux convergence (blue) and the residual dissipation (gray). (c) Net flux (averaged over twentieth tidal cycle) in the box defined by 0 < x < 80 km and 0 < y < 400 km. Green is the value for the net flux (no modal decomposition). Blue bars are the modal decomposition. There is a net incoming flux in mode 1 and net reflecting fluxes in higher modes (primarily modes 2–4).

  • View in gallery

    Cross sections of topographies from y = 50 km.

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    Energy flux for six geometries at tidal cycle 20. Gray depth contours are −3000, −2000, −1000, and −250 m. Arrows indicate the direction of energy flux. See Fig. 6 for bathymetry cross sections at y = 50 km.

  • View in gallery

    Mode-1 decomposition of energy fluxes averaged for the twentieth tidal cycle for the SHELF experiment. (a) Incoming energy flux calculated from the NO TOPO simulation (Fig. 7a). Note that the shelf bathymetry is contoured on this plot (gray lines, and green land), but this bathymetry was not part of this simulation. The green line demarks the region over which the energy budget in the inset was integrated. The green diamonds are the location of a synthetic mooring, and the arrow indicates the estimated incoming flux from a plane wave fit over the three moorings of the total simulation (see text). (b) Reflected energy flux calculated from the difference between the velocities and displacements of the total simulation [in (d)] and the incoming [in (a)]. Blue arrow is the outgoing flux from a plane wave fit over the mooring array from the total simulation. (c) Energy flux cross terms between the incoming and outgoing waves. (d) Total simulation from the SHELF case (Fig. 7d).

  • View in gallery

    Energy partition as in Fig. 8, taking the ROUND RISE simulation as the incoming wave and SHELF/RISE simulation as the total wave field.

  • View in gallery

    Energy partition as in Fig. 8, taking the RISE simulation as the incoming wave and REAL simulation as the total wave field.

  • View in gallery

    (a) Energy budget from 80 km × 80 km control volumes along the continental slope from the twentieth tidal cycle of the REAL simulations. The incoming flux (red) is compared to the reflected (blue) and the cross terms (purple). (b) The reflection coefficients. The blue line is R1, the mode-1 reflectivity in the 80-km control volumes along slope from the nonlinear simulation. The black line is the mode-1 reflectivity from a linear model (Kelly et al. 2013a), the next four gray lines are the cumulative sum of modes 2 to 5, and the last gray line is the sum of all the reflected energy. These do not sum up to one because the linear model has some viscosity that removes some high-mode energy. The dashed lines are the mean of R1 from the linear model if a constant average is taken (light blue, dashed) and if it is weighted by the diffracted beam strength (purple, dashed).

  • View in gallery

    Barotropic to baroclinic conversion averaged over the eighteenth tidal period for different shelf widths (a) 100, (b) 70, (c) 40, and (d) 20 km. Arrows are barotropic flux vectors. Note how the along-slope barotropic flux is almost entirely confined to conversion dipoles along slope.

  • View in gallery

    Along-slope spectra of across-slope velocity (thick lines) for the three narrowest slopes in Fig. 12 from velocities on the shallow shelf in these simulations. The thin lines are resonance curves, formed from the cross-slope equations of motion assuming harmonic motion in time and along slope. As the along-slope wavenumber is varied, resonant modes have a stronger response. What are plotted are arbitrary units for the three slope geometries. The deep-water, mode-1 Rossby radius of deformation (R = c1/|f|) is 28 km, and the wavelength of a mode-1, deep-water Kelvin wave is 125 km.

  • View in gallery

    Terms from the energy budget for 40-km-wide slope. The incoming and reflected energy fluxes are computed at x = 40 km, and the conversion term is integrated from the shelf to x = 40 km.

  • View in gallery

    Spatial shape of modes picked out from the resonant searching technique (as shown in Fig. 13) for the 70-, 40-, and 20-km-wide slopes.

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Reflection of Linear Internal Tides from Realistic Topography: The Tasman Continental Slope

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  • 1 School of Earth and Ocean Sciences, and Department of Physics and Astronomy, University of Victoria, Victoria, Canada
  • 2 University of Alaska Fairbanks, Fairbanks, Alaska
  • 3 Large Lakes Observatory, and Department of Physics, University of Minnesota Duluth, Duluth, Minnesota
  • 4 Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
  • 5 College of Ocean and Atmospheric Science, Oregon State University, Corvallis, Oregon
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Abstract

The reflection of a low-mode internal tide on the Tasman continental slope is investigated using simulations of realistic and simplified topographies. The slope is supercritical to the internal tide, which should predict a large fraction of the energy reflected. However, the response to the slope is complicated by a number of factors: the incoming beam is confined laterally, it impacts the slope at an angle, there is a roughly cylindrical rise directly offshore of the slope, and a leaky slope-mode wave is excited. These effects are isolated in simulations that simplify the topography. To separate the incident from the reflected signal, a response without the reflector is subtracted from the total response to arrive at a reflected signal. The real slope reflects approximately 65% of the mode-1 internal tide as mode 1, less than two-dimensional linear calculations predict, because of the three-dimensional concavity of the topography. It is also less than recent glider estimates, likely as a result of along-slope inhomogeneity. The inhomogeneity of the response comes from the Tasman Rise that diffracts the incoming tidal beam into two beams: one focused along beam and one diffracted to the north. Along-slope inhomogeneity is enhanced by a partially trapped, superinertial slope wave that propagates along the continental slope, locally removing energy from the deep-water internal tide and reradiating it into the deep water farther north. This wave is present even in a simplified, straight slope topography; its character can be predicted from linear resonance theory, and it represents up to 30% of the local energy budget.

Denotes Open Access content.

Corresponding author address: Jody Klymak, University of Victoria, 3800 Finnerty Road (Ring Road), Victoria BC V8P 5C2, Canada. E-mail: jklymak@uvic.ca

Abstract

The reflection of a low-mode internal tide on the Tasman continental slope is investigated using simulations of realistic and simplified topographies. The slope is supercritical to the internal tide, which should predict a large fraction of the energy reflected. However, the response to the slope is complicated by a number of factors: the incoming beam is confined laterally, it impacts the slope at an angle, there is a roughly cylindrical rise directly offshore of the slope, and a leaky slope-mode wave is excited. These effects are isolated in simulations that simplify the topography. To separate the incident from the reflected signal, a response without the reflector is subtracted from the total response to arrive at a reflected signal. The real slope reflects approximately 65% of the mode-1 internal tide as mode 1, less than two-dimensional linear calculations predict, because of the three-dimensional concavity of the topography. It is also less than recent glider estimates, likely as a result of along-slope inhomogeneity. The inhomogeneity of the response comes from the Tasman Rise that diffracts the incoming tidal beam into two beams: one focused along beam and one diffracted to the north. Along-slope inhomogeneity is enhanced by a partially trapped, superinertial slope wave that propagates along the continental slope, locally removing energy from the deep-water internal tide and reradiating it into the deep water farther north. This wave is present even in a simplified, straight slope topography; its character can be predicted from linear resonance theory, and it represents up to 30% of the local energy budget.

Denotes Open Access content.

Corresponding author address: Jody Klymak, University of Victoria, 3800 Finnerty Road (Ring Road), Victoria BC V8P 5C2, Canada. E-mail: jklymak@uvic.ca

1. Introduction

Energy is lost from the surface tide when it interacts with topography and, in the deep ocean, is largely redistributed as an internal tide. The fate of the internal tide is unclear but depends on the dominant wavelengths that are forced. The internal tide response to topography that is subcritical to the internal tide frequency is likely dominated by higher vertical modes and is thought to break via wave–wave interactions relatively close to the topography (i.e., Polzin 2009; St. Laurent and Garrett 2002). Steeper supercritical topography, while exhibiting significant local dissipation, tends to radiate a large fraction of the internal tide away from the topography as low-mode waves (i.e., at Hawaii; Klymak et al. 2006; Carter et al. 2008). Given that a significant fraction of the internal tide energy is generated at steep topography (Legg and Klymak 2008) and that the distribution of the mixing it eventually drives has impacts on understanding the distribution of ocean properties and the strength of the overturning circulation (i.e., Melet et al. 2013), it is desirable to understand where and how the radiated energy dissipates.

One candidate sink for the low-mode internal tide is scattering and dissipation from continental slopes. These slopes are known to be hotspots of turbulent mixing from the few observational studies to date (Nash et al. 2007; Klymak et al. 2011; Martini et al. 2013). However, these studies have also demonstrated some of the difficulties in tracking internal tide energy on these slopes. Net internal tide fluxes are relatively straightforward to measure, but ideally we would like to separate the incident and reflected fluxes if a parameterization of turbulence on the slope is to be made, since the incident fluxes are what drive the turbulence. The reflectivity of a continental slope is the ratio of the reflected energy flux to the incident energy flux:
e1
where F is depth integrated for a two-dimensional budget or depth line integrated for a three-dimensional one. Even simple two-dimensional linear models of reflection indicate that determining the reflectivity will be challenging, with reflection coefficients strongly depending on the modal content and phases of the incident internal tides (Klymak et al. 2011) and the local surface tide (Kelly and Nash 2010). These linear models have been used globally to estimate reflection coefficients for the mode-1 tides on realistic continental slope bathymetries (Kelly et al. 2013b,a), but these calculations assume the incoming tide is known and that the topography is relatively homogenous over a distance similar to the mode-1 horizontal wavelength.

Determining the incident flux Fin from field data, and even from a numerical model with sufficient complexity, is not trivial. In two dimensions, or with simple plane wave geometries, it is straightforward to fit incident and reflected plane waves to recover the desired reflection coefficient. In the real ocean, even if tidal signals can be separated from confounding influences, internal tides are often laterally inhomogeneous and form lateral beams (in xy; Rainville et al. 2010) that make plane wave fits difficult from a finite array of moorings; for instance, a mooring array could be located more in the incoming beam than in the reflected, leading to an exaggeration of the computed energy convergence. Plane wave fits to satellite altimetry tracks are promising but will also suffer from a lack of fidelity if the internal tides are inhomogeneous on the scale of the plane wave fits (Zhao and Alford 2009). In the model, high-resolution temporal and spatial information makes it possible to separate signals spectrally according to their direction of propagation (i.e., using a Hilbert transform; Mercier et al. 2008), but this method works best if there are no boundaries and the signals at the edges of the model domain can be tapered to reduce Gibbs ringing, neither of which are applicable in the near field of a continental slope.

The region considered here is the Tasman continental slope, the focus of a concentrated internal tide field experiment: Tasman Tidal Dissipation Experiment (T-TIDE). As preliminary work, it has been sampled continuously by gliders for a number of months in 2012 and 2013 (Johnston et al. 2015). The gliders were piloted to form an antenna over which internal plane wave fits were made. These efforts show a standing wave pattern, with amplitudes and phases as one would expect for an internal waves incident on the slope from the southeast where internal tides are expected to be generated from the Macquarie Ridge (Fig. 1a). The amplitudes of the interfering waves were such that the reflectivity is predicted to be high on this slope, with estimates of 0.7 to 1.0 from the arrays (Johnston et al. 2015). The gliders also picked up a 100-km wavelength wave propagating along slope toward the north, a finding we isolate and discuss below.

Fig. 1.
Fig. 1.

Location of T-TIDE field site: (a) Energy flux in a regional numerical simulation. Color is the absolute value of the flux; arrows show its direction. The magenta box indicates the numerical modeling domain used in this paper. (b) Detail of the bathymetry on the Tasman slope. The magenta lines indicate 100 km in the x direction and 300 km in the y direction in the modeling domain used in this paper. (c) The stratification used for all runs in this paper, taken from the World Ocean Atlas at 44°S, 152°E (Boyer et al. 2013). The steps are due to linear interpolation of the density between the atlas depths.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

Here, we run numerical simulations that are meant to represent a mode-1 internal M2 tide incident on the Tasman slope, east of Tasmania. The simulations are only forced by the incident internal tide, and there is no barotropic forcing anywhere in the domain, allowing the reflection signal to be isolated. After discussing the model setup (section 2), we briefly consider the response this forcing has on the slope (section 3) and compute and energy budget of the complete response. To separate the physics of the reflection, we then simplify the geometry (section 4), both geometrically, and by removing parts of the topography. This technique allows us to separate incident and reflected signals from the total response without appeal to plane wave fits. We end with a discussion of the results (section 5) where we note the applicability of two-dimensional reflection models and discuss the leaky slope waves evident in the simulations. We conclude with a summary (section 6).

2. Model setup

a. Basics

The numerical model used here is the MITgcm (Marshall et al. 1997), visualized using the Python scientific stack (Hunter 2007; van der Walt et al. 2011). The setup is very similar to Buijsman et al. (2014), with the model run in hydrostatic mode, background (isotropic) diffusivities and viscosities of 10−5 m2 s−1, and enhanced diffusivity and viscosity in regions of temporarily unstable stratification (Klymak and Legg 2010). A second-order, flux-limiting temperature advection scheme is used that results in some numerical dissipation and diffusion. Sensitivity tests were run with weaker forcing, and the fraction of energy dissipated in the model did not change, indicating that the dissipation highlighted below is dominated by numerical dissipation due to the lack of lateral resolution (1 km) rather than explicit viscosities. Dissipation is not the main focus of this paper, and we are conducting ongoing research using finer resolutions for more focused efforts dealing with turbulence on the slope.

Topography is from a dataset that combines Smith and Sandwell (1997) and multibeam data from Australian surveys (Whiteway 2009; Fig. 1b). For this paper, we use a Cartesian coordinate system centered at 44°S, 148°E, with y pointing 12° east of geographic north (magenta lines, Fig. 1). This coordinate system is close to cross slope in the x direction and is used for conceptual convenience. The simulations are run on an f plane (f = −10−4 s−1).

A 1-km lateral resolution is used along the continental slope (Fig. 2a, smallest inset green box). Resolution is expanded by 3.5% per grid cell beyond the 1-km-resolution region to a maximum of 5 km in the second largest inset box (Fig. 2a); this keeps the resolution over the Tasman Rise and the rest of the continental slope at least 5 km. Farther out, the grid spacing is again increased at 3.5% per grid cell until a maximum grid cell size of 10 km is reached.

Fig. 2.
Fig. 2.

(a) Energy flux representation of the analytical response to the forcing used to drive the models used in this paper. Two mode-1 internal wave sources are located to the south east (blue lines). The model domain is rotated 12° from geographic so the shelf break approximately lies along x = 0. The outer model domain is the rectangle with the thick green line. The inner green rectangle denotes 1-km lateral resolution. Outside this rectangle, resolution increases 3.5% per cell to a maximum of 5 km in the second largest rectangle, and outside this rectangle, grid size increases to a maximum of 10 km. The sponge layer is indicated by the largest thin, green rectangle. The 250- and 3000-m isobaths are contoured, though the bathymetry is not used in the analytical response. Arrows show the direction of the energy flux and are scaled by its strength. (b) Energy flux of analytical response of energy reflecting from a wall at x = 0, north of y = 0, in a flat-ocean domain. (c) Phase of reflected response.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

Vertical resolution is approximately stretched so dz ~ 1/N, where N2(z) = −(g/ρ0)(/dz) is the vertical stratification; 200 vertical grid cells are used for these simulations. The vertical stratification is from the World Ocean Atlas for the Tasman Sea just offshore of Tasmania (Fig. 1c; Boyer et al. 2013) and is assumed to be initially laterally constant in the domain. This precludes any mesoscale effects, which are believed to be important in this area and are the subject of future work.

b. Forcing

To simplify the generation problem, we apply an analytical forcing to our model that is meant to represent a simplified version of the regional simulation pictured in Fig. 1a. This is composed of two line sources at approximately the location of the Macquarie Ridge (Fig. 2a). The initial conditions contain this wave field and sponges in the southern and eastern boundaries are forced with it. The forcing is similar to that suggested by Rainville et al. (2010), except here our line source is digitized as a number of discrete point sources along the line, and their response in the domain is summed. The mode-1 pressure anomaly is given by
e2
where ai is the amplitude of the ith source, is the distance to the source, and |k| is the absolute value of the mode-1 wavenumber
e3
where ω is the frequency of the tide, f is the Coriolis frequency, and ce is the eigenspeed of the vertical mode equation:
e4
Here, ψ(z) is the eigenfunction that sets the shape of the vertical mode, and the boundary conditions are /dz = 0 at z = 0 and −H, where H is the water depth. For convenience, we normalize ψm(z) so that
e5
Horizontal velocities can be linearly decomposed by these shapes, as can the pressure signal.
To compute the wave field, the horizontal velocity components are derived from the internal wave consistency relations:
e6
e7
where kx = |k| cos(θi) and ky = |k| sin(θi) are calculated from the angle to each element of the line sources θi = arctan[(y − yi)/(x − xi)].

The resulting incoming wave field (Fig. 2a) has a beam of energy flux that radiates northwest and is relatively tightly focused. The interference pattern creates a null to the south and north and a secondary beam that radiates due west. This schematic agrees with more realistic regional tidal models (H. Simmons 2016, unpublished manuscript), and the amplitude of the beam was tuned to give approximately 2 kW m−1 incident at Tasmania. Note this is less than estimates from altimetry and numerical simulations and is purposely low to keep the runs as linear as possible. The initial condition is applied uniformly through the domain, regardless of bathymetry, so there are some start-up transients as the proper baroclinic flow develops.

The sponge regions on the southern and eastern boundaries are forced with this forcing. The northern and western boundaries are sponges where the velocity is slowly dropped to zero and the stratification relaxed to the initial stratification (Fig. 2a, green rectangles). Our main focus is the area from y = 0 to 400 km, so the boundaries are sufficiently far that small residual reflections do not affect the response.

The ideal response off the Tasman topography would be as a plane wave reflection from a wall at x = 0 km (i.e., Johnston et al. 2015). Here, we have a relatively confined beam, but we can make a start by considering the reflection of the beam from a wall at x = 0 km for y > 0 km (Figs. 2b,c) using the method of images with identical line sources mirrored about the y axis and their phase shifted by 180°. The reflection pattern that sets up is not entirely regular but has some straightforward features. The incoming beam impacts the wall at approximately 30°. The horizontal wavelength of an M2 internal tide is 178 km, so the standing wave in the x direction will have a wavelength 178/cos(30) ≈ 200 km and in the y direction will have a wavelength of approximately 350 km. These spatial scales are readily apparent in the analytical forcing despite the nonplane wave character of the idealized forcing (Fig. 2c). Note that the standing energy flux (Fig. 2b) has peaks and nulls in absolute value, with the peaks having large flux to the north. The peaks are every half cross-slope wavelength (i.e., 100 km). The nulls have weak southward energy flux (though it is difficult to discern from the subsampled arrows in the plot).

3. Realistic model simulation

The response of the forcing in the most real bathymetry motivates the more idealized experiments that follow. From the initial forcing (Fig. 3a), a complex wave field develops with clear scattering from the Tasman Rise, the shelf, and numerous, small inhomogeneities on the seafloor (Figs. 3b–d). Looking along slope, the phase of the velocity signal can be seen changing approximately every 200 km, and it changes approximately every 100 km offshelf, similar to what we expect from an oblique standing wave (cf. Fig. 3h to Fig. 2b). However, the pattern is complicated, with offshore peaks at approximately the correct spacing offshore, but not lining up in the north–south direction. There are inhomogeneities in the energy that are not accounted for by a simple two-wave model.

Fig. 3.
Fig. 3.

(a)–(d) Surface x-direction velocity for four snapshots. (a) The initial conditions (slightly modified after a tidal cycle) and (d) the steady state. Gray contours are depths at 3000, 2000, 1000, and 250 m. (e)–(h) Depth-integrated baroclinic energy flux at the same time periods, with arrows indicating direction.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

A time-averaged energy balance is performed online in the model using the terms outlined in Kang and Fringer (2012) and Kang (2011). The energy balance vertically integrated can be schematized as
e8
where Ebc is the depth-integrated baroclinic energy density, and −∇HFbc is the depth-integrated baroclinic energy flux convergence, including both the pressure work term and the nonlinear advection of energy (which is small in our runs). All quantities are averaged over an M2 tidal period. Conversion is a complex term representing transfer from barotropic motions to baroclinic [Kang 2011, (5.102)] and includes the barotropic heaving of the water column, the density anomaly, and a nonlinear horizontal advection term. These nonlinear terms can be nontrivial in real bathymetry (Buijsman et al. 2014). The conversion term is positive if the barotropic tide loses energy and the baroclinic tide gains energy. Dissipation is computed here as the residual and includes dissipation due to explicit viscosity, numerical dissipation, and bottom drag.

Of note in the energy calculation is that the largest local term in the energy budget is an alternating pattern of barotropic–baroclinic conversion at the shelf break, mostly balanced by baroclinic flux convergences and divergences (Fig. 4). The importance of the barotropic–baroclinic term can also be seen by considering the x integral of the energy budget from x = −50 to 100 km (Fig. 5). Recall that the simulations have no barotropic forcing. This coupling is driven by the interaction of the internal tide and the shelf corner at y = 0 and takes the form of a leaky, superinertial slope wave (see section 5).

Fig. 4.
Fig. 4.

Energy budget over the nineteenth tidal cycle of (a) barotropic to baroclinic conversion, (b) baroclinic energy flux convergence −∇Fbc, (c) rate of change of baroclinic energy, and (d) residual representing the dissipation in the model D = −∇Fbc + Conversion − dE/dt. The green box is the region for the energy time series (Fig. 5b).

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

Fig. 5.
Fig. 5.

(a) Integral in x to 80 km offshore of the energy terms in Fig. 4 for the REAL case (averaged over twentieth tidal cycle). Note that the barotropic–baroclinic term (red) is of the same order as the baroclinic convergence (blue) and the residual dissipation (gray) for most of the slope. (b) Energy budget time series for the REAL case, tidally averaged, where time is normalized by T = 12.4 h, between y = 0 and 400 km. There is still a small, residual increase in the energy with time (purple), representing the accumulation of high-mode energy in the region. Net barotropic–baroclinic conversion (red) is small and negative, indicating a small net loss to the barotropic tide in this region. The bulk of the budget is the balance between baroclinic flux convergence (blue) and the residual dissipation (gray). (c) Net flux (averaged over twentieth tidal cycle) in the box defined by 0 < x < 80 km and 0 < y < 400 km. Green is the value for the net flux (no modal decomposition). Blue bars are the modal decomposition. There is a net incoming flux in mode 1 and net reflecting fluxes in higher modes (primarily modes 2–4).

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

The time series of the energy terms integrated in the volume bounded by x = 0 to 80 km and y = 0 to 400 km demonstrates that the barotropic-to-baroclinic term is relatively small when averaged, with a small loss of energy from the baroclinic tide to the barotropic in the integral region (Fig. 5b). The model is largely in a steady state by tidal cycle 15, with some residual oscillations in dE/dt and the flux convergence. The oscillations were not explicitly examined but likely result from imperfect sponges. The large-scale baroclinic energy changes do not change the dissipation residual very much, which is relatively constant after five tidal cycles. To put the 50 MW of dissipation into context, the initial energy that comes in the east and south sides of this analysis box in the initial conditions is 315 MW, so the model is dissipating about 17% of the incoming energy. However, note that the dissipation is not the focus of these model runs or of this paper. The forcing here is approximately a factor of 3 lower than the real forcing, so it is likely the fraction of dissipation at this site is higher if real forcing is used.

The majority of the energy budget is in the first vertical mode (Fig. 5c). Net fluxes in the region directly offshore of the shelf break (0 < x < 80 km and 0 < y < 400 km) are composed of substantial mode-1 energy converging on the slope (95-MW net) and some reflected energy escaping in higher modes (28.3 MW, mostly in modes 2–4). The 95-MW net flux is made up of the incoming and reflected mode-1 energy, and separating those terms is the subject of the next section. There is some incoming higher-mode energy as well due to scattering from the Tasman Rise, but as we will also show below, this is minor. The spatial pattern (not shown) of the mode conversion at the continental slope indicates hotspots for conversion. Modes 2 and 4 have a hotspot of conversion near y = 250 km, and mode 3 has a hotspot at y = 325 km.

4. Simplified geometries

To help tease apart the effects of the Tasman Rise and the nonuniform slope, we carry out a few simplified experiments (Figs. 6, 7). The REAL case is the one discussed above (Fig. 7f). The NO TOPO case has no topography at all (Fig. 7a), just the beam being forced at the south and east boundaries and (mostly) absorbed at the west and north. RISE was run with the real bathymetry west of x = 70 km (Fig. 7e). Three idealized geometries simplify the physics even more: the SHELF case has a supercritical, two-dimensional continental slope running north from y = 0 km (Fig. 7d). ROUND RISE is a 1700-m-tall, cylinder-shaped bump with radius of 50 km centered at approximately the same location as the Tasman Rise, with no shelf to the west (Fig. 7b). The simplified slope and the rise are both used in the SHELF/RISE case (Fig. 7c).

Fig. 6.
Fig. 6.

Cross sections of topographies from y = 50 km.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

Fig. 7.
Fig. 7.

Energy flux for six geometries at tidal cycle 20. Gray depth contours are −3000, −2000, −1000, and −250 m. Arrows indicate the direction of energy flux. See Fig. 6 for bathymetry cross sections at y = 50 km.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

a. Shelf-only configuration

The simplest topography is the SHELF configuration (Fig. 7d). Here, we have a response that is quite similar to the analytical response calculated above (Fig. 2b). The only difference between these two cases is the narrow shelf west of x = 0 km and the continental slope instead of a wall. The interference pattern between the incoming wave and the reflected wave is clear in this plot, with the same characteristic length scales as above and a slight bending of the response due to the radial spreading of the beam.

The goal of this paper is to determine the amount of reflectivity of the continental slope. This is a hard number to determine in a complicated geometry and naturally depends on the region of integration. For the SHELF configuration, the situation is relatively simple, and we use it to illustrate the numerical technique used below. The signal in the full simulation is assumed to consist of an incoming signal and a reflected signal, so we can decompose the east–west velocity amplitude of the first vertical mode (for example) as
e9
e10
e11
where is the (total) complex amplitude of the M2, mode-1 east–west velocity of the simulation, which include the reflection, of the incoming signal, and of the reflected signal. We assume for this example that the incoming signal is given by the NO TOPO simulation, and is from the SHELF simulation. The reflected signal is simply the difference of these two. This method has been used by Hall et al. (2013) for a two-dimensional flow. Here, it is an absolute necessity because of the complicated three-dimensional topography.
To compute an energy budget, the energy fluxes are calculated from the decomposed signals as
e12
e13
e14
e15
The cross terms are not negligible for any realistic forcing and indeed give rise to the interference patterns seen above (Nash et al. 2004; Martini et al. 2007).

The total response (Fig. 8d) consists of the incoming response (Fig. 8a), the reflected signal (Fig. 8b), and substantial cross terms (Fig. 8c). The cross terms are mostly perpendicular to the direction of reflection (i.e., parallel to the slope) and alternately flux energy to the north and south every half cross-slope wavelength. Combined, these three components give the total flux with net fluxes to the north in alternating peaks every full offslope wavelength.

Fig. 8.
Fig. 8.

Mode-1 decomposition of energy fluxes averaged for the twentieth tidal cycle for the SHELF experiment. (a) Incoming energy flux calculated from the NO TOPO simulation (Fig. 7a). Note that the shelf bathymetry is contoured on this plot (gray lines, and green land), but this bathymetry was not part of this simulation. The green line demarks the region over which the energy budget in the inset was integrated. The green diamonds are the location of a synthetic mooring, and the arrow indicates the estimated incoming flux from a plane wave fit over the three moorings of the total simulation (see text). (b) Reflected energy flux calculated from the difference between the velocities and displacements of the total simulation [in (d)] and the incoming [in (a)]. Blue arrow is the outgoing flux from a plane wave fit over the mooring array from the total simulation. (c) Energy flux cross terms between the incoming and outgoing waves. (d) Total simulation from the SHELF case (Fig. 7d).

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

The reflected response (Fig. 8b) shows approximately what we would expect with energy being radiated to the northeast. There is some concentration of this energy at y ≈ 75 km and y ≈ 225 km because of coupling with a partially trapped slope wave. This coupling causes a redistribution of the reflected energy, focusing it approximately every along-slope wavelength of the slope wave (we show in section 5 that this wavelength changes as the slope geometry changes).

Performing this analysis for the lowest 10 modes, we arrive at an energy budget for the slope in the green box in the figures (0 < y < 400 km, and x < 80 km; Fig. 8, inset budgets). Note that we assume the flux through x = 0 is zero. With this calculation, we see that 408 MW is incident on the slope in mode 1. There is also a net flux of 50 MW into this region from the cross terms. This is a redistribution of energy from north of our box into the box. There is a net convergence of this cross-term energy because there is dissipation in the box; in a purely inviscid solution, this term should balance to zero over a closed box. If we extend the integration farther north, the cross-term flux drops to zero.

Most of the incoming energy reflects back out of the box (Fig. 8b), with the bulk remaining in mode 1 and some scattering to higher modes. This scattered energy radiates to the northeast (not shown). The mode-1 reflection is affected by the slope wave that transfers energy to and from the barotropic tide along the slope, resulting in nulls and peaks in the mode-1 reflection.

b. Tasman Rise and simplified shelf

The Tasman Rise has a profound effect on the energy that impacts the continental slope (Figs. 7e,f). The incoming beam is almost 500 km wide at x = 0 if there is no Tasman Rise but breaks into three narrower beams when there is a Tasman Rise (Fig. 7e). Upstream of the rise, the effect consists of somewhat less energy propagating westward, with an interference pattern toward the east indicating some back reflection.

This pattern can be explained in terms of diffraction of the internal tide beam from a deep obstacle (i.e., Johnston and Merrifield 2003). There is a downwave concentration of energy along the seamount’s axis, a null, and sidelobes to the north and south. In this case, the incident beam is of comparable size to the obstacle, leading to an asymmetry and a stronger lobe to the north than south.

Most of the response due to the Tasman Rise can be modeled simply as a cylindrical obstacle in the beam (Figs. 7b,c). Here, our obstacle is 1800 m high in 5000 m of water and has a radius of 50 km (Fig. 6). This captures most of the features of the actual Tasman Rise, despite not having a shallow spire in the center and being slightly smaller than the real rise. The differences make the simplified response have weaker nulls, and the whole response is directed a bit farther north than the real rise. Adding the shelf (Fig. 7c) yields a response that bears substantial similarity to the REAL forcing case.

Decomposing into an incoming and reflected signal (Fig. 9) demonstrates the effect of the Tasman Rise on the response. Less energy is incident on the control volume, largely because the diffraction redirects some of that energy to the north of y = 400 km. There is a strong reflection of energy where the main diffraction lobe reflects from the slope (Fig. 9b) and a smaller maximum just to the north (y = 250 km) due to the along-slope wave that is strummed. There is a reflection farther north where the northern lobe of the diffraction pattern reflects.

Fig. 9.
Fig. 9.

Energy partition as in Fig. 8, taking the ROUND RISE simulation as the incoming wave and SHELF/RISE simulation as the total wave field.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

The incoming energy has some more high-mode content due to scattering at the cylindrical rise (Fig. 9a), though it is still 95% mode 1. The reflection is almost 80% mode 1, with some scattering to higher modes. The net flux shows approximately 15% of the incoming energy is dissipated at the shelf.

c. Real case

The REAL forcing is similar, if more complex (Fig. 10). The simulation using the bathymetry in the RISE ONLY case (Fig. 7e) is used as the incoming energy flux, and the REAL (Fig. 7f) case is the total. Compared to the cylindrical rise, the real Tasman Rise creates a sharper diffraction pattern and more back reflection. However, the REAL simulation has many of the same features as the SHELF/RISE simulation (Fig. 7c).

Fig. 10.
Fig. 10.

Energy partition as in Fig. 8, taking the RISE simulation as the incoming wave and REAL simulation as the total wave field.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

Slightly less incoming energy passes into the control volume (Fig. 10a) because the diffraction by the real Tasman Rise is sharper than the cylindrical rise. As for the cylindrical rise case, there is some incoming, higher-mode energy due to forward scattering, though again over 95% is mode 1. Reflection is concentrated near y = 125 and 450 km, associated with the diffraction nodes, with about 85% in mode 1 (Fig. 10b). Dissipation is less than 25% of the incoming energy (Fig. 10d).

5. Discussion

a. Estimating reflection coefficients

A major goal of this effort is estimating the fraction of the incoming tide that is reflected by the Tasman continental slope to come up with a reflectivity coefficient. Here, we discriminate between the mode-1 reflection, R1 = Fref,1/Fin,1, and the total reflection into all the modes, RT = Fref/Fin. Evaluating these coefficients is less straightforward than it may sound because it is difficult to separate the incoming from reflected signal in complicated geometry, even in a fully resolved numerical model, let alone in observations. Above, we used an integrated measure, comparing the incoming flux from a model with no continental slope to one with a continental slope and integrating the fluxes over a control volume from y = 0 to 400 km. This control volume was an arbitrary choice but yielded reflectivities of mode-1 internal tide R1 = 0.65 and the total internal tide of RT = 0.76 (Fig. 10).

The T-TIDE field effort deployed a three-point mooring array to quantify the wave field offshore of the continental slope. Determining reflectivity from such a mooring array is significantly complicated by three-dimensionality and along-slope variability. From the mooring array in Fig. 10, the reflectivity is R1 = 0.6/1.3 = 0.46, a significant underestimate. The reason for this should be relatively clear from looking at Figs. 10a and 10b; the mooring array nicely captures the northward diffracted ray but catches some of the reflected pattern from the main beam to the south. There are significant interferences in the reflected patterns (Fig. 10b) because the reflected pattern is a complicated superposition of the cylindrically spreading reflections along the slope.

Determining the reflectivity as a function of along-slope direction y is difficult. A simple one-dimensional comparison of onslope and offslope fluxes does not yield useful results because the reflection from any given point on the slope radiates cylindrically, so it is necessary to integrate over volumes. Here, we take the same approach as used in the previous section (i.e., Fig. 10) but integrate over smaller control volumes (80 km in y and between 0 and 80 km in x) to see the reflectivity as a function of y (Figs. 11a,b). The incoming flux every 80 km shows the diffracted beam pattern with a maximum net incoming flux at y = 120 km (Fig. 11a, red line) and a secondary peak to the north at about 440 km. The net reflectivity from these boxes ranges from 0.8 to a low of almost zero at y = 280 km, where the slope is less steep (Fig. 11b, solid blue line). Note an uncertainty in the flux decomposition associated with the flux in the cross terms (Fig. 11a, purple line). This term does not balance to zero and forms a significant part of the energy budget over such small control volumes. It cannot be uniquely decomposed into either the incoming or reflected energy terms and so remains as an uncertainty.

Fig. 11.
Fig. 11.

(a) Energy budget from 80 km × 80 km control volumes along the continental slope from the twentieth tidal cycle of the REAL simulations. The incoming flux (red) is compared to the reflected (blue) and the cross terms (purple). (b) The reflection coefficients. The blue line is R1, the mode-1 reflectivity in the 80-km control volumes along slope from the nonlinear simulation. The black line is the mode-1 reflectivity from a linear model (Kelly et al. 2013a), the next four gray lines are the cumulative sum of modes 2 to 5, and the last gray line is the sum of all the reflected energy. These do not sum up to one because the linear model has some viscosity that removes some high-mode energy. The dashed lines are the mean of R1 from the linear model if a constant average is taken (light blue, dashed) and if it is weighted by the diffracted beam strength (purple, dashed).

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

In two dimensions, the fraction of the tide reflected into mode 1 (and higher) can be predicted from linear theory using the method described by Kelly et al. (2013a) of matching Laplacian tidal solutions at discrete steps on a discretized topography. If the tide is obliquely incident on the slope, there can be substantial differences in the reflected tide (Kelly et al. 2013b). If we run these solutions for the Tasman slope with an incident angle of 30°, the reflectivity into mode 1, R1 is similar to the numerical simulation (Fig. 11b, thick black line). The predicted reflectivity is greater for most of the ridge, but the null at y = 250 km is captured.

The REAL simulation has a mode-1 reflectivity of R1 = 0.65. A naive average of the reflectivity from the linear model between y = 0 and 400 km yields 〈R1〉 = 0.71. However, that does not take into account the varying strength of the incoming diffracted beam, which is stronger where the reflectivity is higher. Weighting by the incoming beam strength, the reflectivity averages 〈R1beam = 0.8 and is substantially larger than in the numerical simulations.

An attempt has been made to estimate reflectivity from this site from autonomous gliders surveys (Johnston et al. 2015). First, the gliders saw a substantial concentration of energy shoreward of the Tasman Rise. This is a feature of the model and is clearly explained by the diffraction of energy by the Tasman Rise (Fig. 10).

For the region in the lee of the Tasman Rise, Johnston et al. (2015) estimate a reflectivity of the mode-1 internal tide of between 0.8 to 1.0 by fitting plane waves to the velocity and displacement amplitudes and phases. If we confine our incoming versus outgoing energy budget to the region 80 < y < 200 km, representative of their Spray 56 deployment, we calculate a reflectivity of 0.7, which is lower than their lowest estimate of 0.8 and much lower than their high estimate of 1.0. A second deployment, Spray 55, covered more of the slope (up to y = 300 km). In this domain, they estimate a reflectivity of 0.6. This is in agreement with the numerical simulation, which achieves the same result from 0 < y < 300 km.

The directions of wave propagation fit from the glider data are not in agreement with the model. The fits to the Spray 55 data show incoming energy at between 125° and 145°, which is similar to the model. However, the reflection is slightly south of due east (0° to −30° geographic), whereas the numerical model is definitely to the northeast far from shore. An explanation is evident from close inspection of Fig. 10b between the Tasman Rise and the continental slope, where the glider spent the most time. At this location, the offshore energy flux is almost exactly in the x direction (−12° geographic) in agreement with the glider observations.

Finally, one of the gliders (Spray 56) picked out a northward-propagating disturbance along the continental slope with wavelength of 100 km. This wavelength matches the wavelength of the slope wave seen in the real simulations (Figs. 4a,b). Interestingly, they only picked this wavelength out in vertical displacement data, not in velocity.

b. Slope wave importance and dynamics

The structure of the barotropic-to-baroclinic conversion on the slope is an intriguing feature of these simulations and appears in regional simulations (H. Simmons 2016, unpublished manuscript) and the glider data (Johnston et al. 2015). Here, it shows up most clearly in the SHELF simulations because of the simplified bathymetry. However, it is also clear in the REAL simulation (Fig. 4a). This slope wave redistributes energy in the reflected baroclinic response (Fig. 8), taking a relatively homogenous incoming energy source and focusing the reflection every 200 km or so along slope.

This wave is a slope mode that is strummed by the incident internal tide at the corner of the topography (x = 0, y = 0); a long slope without the corner does not excite this wave nor does an internal tide coming directly from the east and hitting the topography at a normal angle. The along-slope wavelength is independent of the incident, along-slope wavelength in the open water (tested by changing the angle of the incident tide; not shown) and is a robust feature of the slope shape. A sensitivity experiment that varied the continental slope widths demonstrates that narrower slopes strum longer along-slope waves (Fig. 12).

Fig. 12.
Fig. 12.

Barotropic to baroclinic conversion averaged over the eighteenth tidal period for different shelf widths (a) 100, (b) 70, (c) 40, and (d) 20 km. Arrows are barotropic flux vectors. Note how the along-slope barotropic flux is almost entirely confined to conversion dipoles along slope.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

These waves are superinertial and are an example of partially trapped slope waves (Dale and Sherwin 1996; Dale et al. 2001). While analogous to coastally trapped waves, the superinertial response is not freely propagating and instead can exchange energy with the open ocean. We compare the wavelength of the slope waves (Fig. 13, thick lines) to the empirical modes predicted from linear theory (Dale et al. 2001). The procedure solves for the response of the flow in the coastal bathymetry due to forcing with varying along-slope wavelengths. Resonant, along-slope wavelengths lead to a much stronger response (appendix and Fig. 13, thin lines). The along-slope wavelength of the resonant modes in the linear calculation agrees quite well with the wavelengths of the fully nonlinear solutions. Narrower continental slopes yield longer along-slope wavelengths, and the spatial modes that correspond to the peaks are similar to deep-ocean mode 1 off the slope.

Fig. 13.
Fig. 13.

Along-slope spectra of across-slope velocity (thick lines) for the three narrowest slopes in Fig. 12 from velocities on the shallow shelf in these simulations. The thin lines are resonance curves, formed from the cross-slope equations of motion assuming harmonic motion in time and along slope. As the along-slope wavenumber is varied, resonant modes have a stronger response. What are plotted are arbitrary units for the three slope geometries. The deep-water, mode-1 Rossby radius of deformation (R = c1/|f|) is 28 km, and the wavelength of a mode-1, deep-water Kelvin wave is 125 km.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

The slope wave is an important term in the local energy budget when compared the incoming and reflected energy fluxes (Fig. 14). The incoming energy peaks at 1.2 kW m−1 and the reflected energy is of a similar magnitude but with oscillations at twice the wavelength of the slope wave. The integrated barotropic–baroclinic conversion is as high as 0.4 kW m−1 coastline and leads to 0.4 kW m−1 peak-to-peak oscillation in the reflected energy (it is not 0.8 kW m−1 peak to peak because the reflected energy spreads spherically by x = 40 km, where the reflected flux is evaluated; Fig. 8). This turns even a relatively straight slope into a series of internal tide absorbers and radiators, leading to 100-km-scale inhomogeneity in the reflected internal tide. There is a wave–wave interaction between the internal tide and the slope wave that leads to this alternating pattern, the exact character and strength of which will be the subject of future research.

Fig. 14.
Fig. 14.

Terms from the energy budget for 40-km-wide slope. The incoming and reflected energy fluxes are computed at x = 40 km, and the conversion term is integrated from the shelf to x = 40 km.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

6. Summary

A mode-1 internal tide was launched at a variety of topographies, representing the Tasmanian continental slope. The goal was to determine the reflectivity of this slope in terms of the modal content of the reflected energy and the local dissipation. The latter is somewhat suspect in this model because of crude lateral resolution, but the REAL simulation indicated that 21% of the incoming energy was dissipated, and 65% was reflected as mode-1 energy. The incoming internal tide flux used here was weak compared to the flux modeled and inferred from altimetry in the Tasman Sea, so we expect the dissipation in more realistically forced models to increase.

Despite a simple incoming internal tide that is linear, semidiurnal, and mode 1, we have found a rich and complex response when the remote wave impacts the topography. The response can be characterized as follows:

  • diffraction of the beam by the Tasman Rise,
  • oblique reflection from the continental slope, and
  • a leaky slope wave response that redistributes reflected internal energy along slope.
Of these, perhaps only the second effect was expected before carrying out the simulations. However, as we saw above, even the reflection problem is significantly complicated in the presence of three-dimensionality.

Diffraction around underwater topography should have been expected, however, the relative depth of the Tasman Rise makes it surprising that the effect is so strong. The fact that the lateral width of the Rise is close to the wavelength of the incoming internal tide makes predicting the diffraction pattern difficult. Baines (2007) considers generation of internal tides at seamounts, but does not deal with scattering and diffraction. The problem is similar to electromagnetic waves passing through a wire, but a linear response for that problem is not trivial to compute (i.e., Bonod et al. 2005), and does not have a confined vertical mode structure as we find in the internal wave problem.

The excitation of slope waves has been explored by Dale et al. (2001). It has an important effect on the redistribution of energy along slope. The redistribution affects where high dissipation is found in the model (Fig. 4), and adds more inhomogeneity to the reflected internal tide.

The complexity grows if other real-world influences are to be accounted for. The East Australian Current flows along this slope, varying the stratification in the horizontal, provides lateral shears that can distort the internal tide response and carrying eddies that can add a strong time dependence to these effects. Even in two dimensions, the strength of the internal tide reflection can be significantly impacted by the phase of the incoming tide with other baroclinic modes (Klymak et al. 2011) or the barotropic (Kelly and Nash 2010). The simulations here exclude the local barotropic tide, so this would certainly complicate the reflected response. Finally, the internal tide used here was monotonic, whereas the real tide will also have other frequencies, most notably subinertial diurnal frequencies that will have trapped wave responses (Musgrave et al. 2016).

Regardless, it is useful to have studied the simplest response we could in this system to tease apart the dominant physics. This response is complex, and it should be clear that solely observational efforts to balance a reflection budget are going to be a challenge. Merging simulations and observations is a likely way forward in understanding the wave field in this complex slope region.

With respect to the reflection problem, the modeled slope has a relatively high reflection back into the open ocean, with as much as 65% of the incoming energy being reflected as mode 1. It is possible that higher-resolution runs will be more dissipative and that stronger forcing will lead to a higher fraction of dissipation. However, these simulations, and the results from the rest of the experiment to date (i.e., Johnston et al. 2015), indicate that bulk of the energy from the Macquarie Ridge must dissipate elsewhere.

Acknowledgments

Our thanks to conversations with Andy Dale on leaky slope waves and Qiang Li for sharing his linear code for the Lindzen-Kuo method. J. Klymak was supported by the Canadian National Science Engineering Research Council Discovery Grant 327920-2006 and by computer time from the U.S. Office of Naval Research, T. Paluszkiewicz, program officer. H. Simmons and D. Brazhnikov were supported by NSF OCE 1130048. J. MacKinnon and R. Pinkel were supported by Award NSF OCE 1129763. M. Alford was supported by NSF OCE 1129246. A website for this paper with analysis scripts, intermediate data files, and model setups is available online (at http://web.uvic.ca/~jklymak/ttide15).

APPENDIX

Appendix Slope Wave Calculation

The slope wave calculation follows the calculation made by Dale et al. (2001), where there are more details. The linear equation for the pressure perturbations are assumed to have the form , where ky is the along-slope wavenumber:
ea1
Subject to boundary conditions at the surface of ∂p/∂z = 0 and at the seafloor of
ea2
The coastal boundary condition is assumed to be a wall with no flow, while the open-ocean boundary condition consists of waves that radiate away from the slope or, if the cross-slope wavenumber kx is imaginary, of disturbances that decay with distance from the slope.

The above are discretized on a domain that is 260 km wide into 261 grid cells in x and onto a sigma coordinate with 192 vertical levels. The hyperbolic method of solution due to Lindzen and Kuo (1969) was used to solve on this domain for p for ω = 1.4f, f = 10−4 s−1, and for a sweep of ky. Under an arbitrary forcing, certain values of ky resonate and lead to stronger-amplitude responses in p, corresponding to spatial modes of the system. The numerical method is sensitive to the stratification, so we used a fit exponential of N2(z) = (2 × 10−5 s−2)ez/1000 (where z is negative downward). The scan was taken over 300 wavelengths equally spaced between 30 and 180 km.

The resulting spatial modes are similar to those in Dale et al. (2001; Fig. A1). There is a peak of amplitude on the shelf and then a second peak on the slope. As the slope gets narrower, the peak on the slope becomes broader. These shapes are the lowest modes.

Fig. A1.
Fig. A1.

Spatial shape of modes picked out from the resonant searching technique (as shown in Fig. 13) for the 70-, 40-, and 20-km-wide slopes.

Citation: Journal of Physical Oceanography 46, 11; 10.1175/JPO-D-16-0061.1

(The general code to solve this is at https://github.com/jklymak/LindzenKuo. The code used for this paper is at http://web.uvic.ca/~jklymak/ttide15/.)

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