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  • View in gallery

    The occurrence of lee waves is indicated with ωlee − 2ωtide = N0α − 2ωtide > 0 (shaded area) for (a) semidiurnal and (b) diurnal tides. The depth is contoured every kilometer. The straight black dashed line is the 2D modeling transect. The black circle marks station N2 at the west ridge.

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    (a) Bathymetry of the west and east ridges in central Luzon Strait. The white dashed line marks the transect used for the 2D modeling, and the solid white square the N2a and N2b mooring locations. Depth contours are the white and black lines. The 0-m contour is the thick black line. (b) Realistic bathymetry along the transect for double ridge (reWE, gray solid line), single east ridge (reE, black dashed line), and reW (black solid line). The criticality γ2 and γ1 for semidiurnal and diurnal tides is plotted below the depth profile: γ ≤ 0.5 is white (subcritical), 0.5 < γ ≤ 1.5 is gray (near critical), and γ > 1.5 is black (supercritical).

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    Model–data comparison at station N2a when semidiurnal tides dominate; observed (a) zonal velocities (positive is to the right) and (b) dissipation estimated from overturns. The density of an ocean at rest is contoured in vertical increments of 100 m. (d),(e) The model results; (c) the time-mean dissipation; (f) the barotropic velocities; and (g) the depth-integrated dissipation. The black diamonds plotted on the top axes mark the times of snapshots presented in Figs. 5 and 6.

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    As in Fig. 3 but at station N2b when diurnal tides dominate. The black diamonds plotted on the top axes mark the times of snapshots presented in Fig. 6.

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    Snapshots of (a)–(d) zonal velocities (positive is to the right) and (e)–(h) dissipation during accelerating and decelerating semidiurnal barotropic eastward flow at the west ridge for the double ridge case. The times and barotropic velocities are above the top panels; “max ebb” refers to maximum eastward barotropic flow. The times of the snapshots are also marked in Fig. 3. The gray square marks the N2 site. The density of an ocean at rest is contoured in vertical increments of 50 m.

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    Snapshots of zonal velocities (positive is to the right) 1.3 h after maximum eastward barotropic flow at the west ridge for (a) N2a and the double ridge (reWE), (b) N2a and the single ridge reW, (c) N2b and the double ridge, (d) N2b and the single ridge reW. (e)–(h) The dissipation for the four cases. The times of the snapshots are also marked in Figs. 3 and 4. The gray square marks the N2 site. The density of an ocean at rest is contoured in vertical increments of 100 m.

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    The time-mean (a) horizontal baroclinic kinetic energy and (b) dissipation for the double ridge case. (c) The time-mean and depth-integrated dissipation. (d) The cumulative dissipation, normalized by the total dissipation. The numbers are normalized dissipation computed over areas bounded by the dots. The strongest dissipation occurs at the tallest subridge on the east ridge.

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    (a) Barotropic velocity at the east ridge for the double ridge and single ridge cases plotted on top of each other. The survey periods N2a and N2b are marked by the black horizontal bars. (b) Unfiltered dissipation integrated over both ridges for the double ridge (reWE) (gray), single east ridge (reE) (black), and single west ridge (reW) (dashed line) cases. Low-passed dissipation integrated (c) over both ridges for all cases, (d) over the east ridge for the double ridge and east ridge cases, and (e) over the west ridge for the double ridge and west ridge cases. The cutoff frequency is 2 days.

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    (a) The barotropic velocity at the east ridge for the double ridge case. Low-passed (b) conversion, (c) flux divergence, and (e) the ratio of the dissipation over conversion for the double ridge (reWE) (gray), single east ridge (reE) (black) and single west ridge (reW) (dashed line) cases. The energy terms are volume-integrated over both ridges. (e) The amplification (7) for the low-passed and volume-integrated dissipation, conversion, and divergence. Constructive double ridge interference with Ψ > 0 occurs when semidiurnal tides dominate.

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    The differences between the double ridge and single west ridge cases for (b) the daily-mean depth-integrated dissipation , (d) the amplitude Aub of the semidiurnal (D2) zonal bottom velocities averaged over the bottom 200 m, and (f) the amplitude of the diurnal (D1) zonal bottom velocities. (c),(e),(g) The differences between the double ridge and single east ridge cases. The black lines in the left panels are the deep water amplitudes of the zonal barotropic velocities × 200. Zero velocities are at y axis. (a) The double ridge bathymetry and subridges are marked by a line and shading.

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    (a) The amplification (7) of the conversion for the double-ridge knife edge model as a function of the ridge separation distance ΔX and west ridge height for a constant east ridge height , where the maximum water depth H = 3699 m. The dashed and dotted lines mark the superposition of beams after an odd number of surface and bottom reflections, respectively. The lines with circles and crosses indicate beam superposition after an even number of surface and bottom reflections. Zonal velocity snapshots for (b) a resonant case with an attractor, (c) Luzon Strait parameters and semidiurnal tides, and (d) Luzon Strait parameters and diurnal tides. The west ridge height is 0.52H in (b) and 0.16H in (c),(d). The ridge separation distance is 0.5λ in (b), 0.64λ in (c), and 0.26λ in (d); λ is the horizontal first-mode wavelength. The three cases corresponding to (b)–(d) are also marked in (a).

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    Internal wave beams are indicated by the baroclinic zonal velocity amplitudes of the double ridge case for (a) a semidiurnal spring tide (yearday ~238) and (b) a diurnal spring tide (yearday ~246). Superposed are the theoretical beams (white dashed lines). The black square in (a) marks where the downward and upward beams from the western subridge on the east ridge cross. The phase differences Δϕ between velocities of the single-east and single west ridge cases for (c) semidiurnal and (d) diurnal tides: Δϕ is colored for −90 < Δϕ < 90, indicating the velocities are approximately in phase.

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    The amplification of tidal-mean and volume-integrated (a) conversion, (b) flux divergence, and (c) dissipation for the realistic model as a function of the ridge separation distance ΔX and west-ridge height for a constant east-ridge height: ΔX for the diurnal (semidiurnal) cases is normalized by λ1 = 370 km (λ2 = 147 km). The diagonal lines are adapted from Fig. 11a. The amplification values for each case are plotted. The bold values are for the realistic double ridge case during (semi) diurnal tides. For both semidiurnal and diurnal tides, simulations are performed with ΔX = 94.4 km and 94.4 ± 25 km and hWR = 1636 m, 2136 m, 2636 m, and 3236 m (z = −2063 m, −1563 m, −1063 m, and −463 m). For semidiurnal tides, additional cases are run for ΔX = 94.4 km and 94.4 ± 20.9 km and hWR = 3349 m (z = −350 m).

  • View in gallery

    The diagnosed dissipation computed from the total energy balance 〈Dv,E〉 vs the reported dissipation computed from the mixing scheme 〈Dv〉 for various cases in hydrostatic (H, gray markers) and nonhydrostatic (NH, black markers) mode for a tidal barotropic flow with an amplitude of 0.1, 0.2, and 0.3 m s−1 at the ridge, with a horizontal grid size of 30 and 100 m, and for two cases with a constant viscosity of Az = 10−5 m2 s−1; that is, the vertical mixing scheme is not used (NM). The units are not shown in the legend.

  • View in gallery

    Snapshots of dissipation at the Gaussian ridge during eastward barotropic flow at yearday 233.77 for (a) the hydrostatic (hyd) case with Δx = 100 m, (b) the nonhydrostatic (nonhyd) case with Δx = 100 m, (c) the hydrostatic case with Δx = 30 m, and (d) the nonhydrostatic case with Δx = 30 m. The density of an ocean at rest is contoured in vertical increments of 50 m. (e) The barotropic flow at the ridge crest is the same for the four cases. (f) The dissipation for the four cases, volume integrated over the ridge. The circles indicate the time of the snapshots.

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Double-Ridge Internal Tide Interference and Its Effect on Dissipation in Luzon Strait

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  • 1 Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey
  • | 2 University of Victoria, Victoria, British Columbia, Canada
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Abstract

Luzon Strait between Taiwan and the Philippines features two parallel north–south-oriented ridges. The barotropic tides that propagate over these ridges cause strong internal waves and dissipation. The energy dissipation mechanisms and the role of the baroclinic wave fields in this dissipation are investigated using numerical simulations with the Massachusetts Institute of Technology general circulation model (MITgcm). The model is integrated over two-dimensional configurations along a zonal transect at 20.6°N for a maximum duration of a spring–neap cycle. Nearly all dissipation occurs at the steep ridge crests due to high-mode turbulent lee waves with horizontal scales of several kilometers and vertical scales of hundreds of meters. The spatial structure and timing of the predicted velocities and dissipation agree with observations and confirm the existence of these lee waves. The lee wave strength is greatly affected by the internal waves generated at the other ridge. When semidiurnal barotropic tides are dominant, the internal wave beams from both ridges nearly superpose after one surface reflection. The remotely generated internal waves from both ridges are therefore in phase with each other and the barotropic tides at the ridges. The barotropic-to-baroclinic energy conversion, energy flux divergence, ridge top velocities, and dissipation are stronger compared to the sum of the single east ridge and single west ridge cases. When diurnal tides are dominant, the wave fields are more out of phase and the conversion, divergence, and dissipation are less than or equal to the single ridge cases combined.

Corresponding author address: Maarten C. Buijsman, Geophysical Fluid Dynamics Laboratory, 201 Forrestal Road, Princeton, NJ 08540-6649. E-mail: maarten.buijsman@noaa.gov

Abstract

Luzon Strait between Taiwan and the Philippines features two parallel north–south-oriented ridges. The barotropic tides that propagate over these ridges cause strong internal waves and dissipation. The energy dissipation mechanisms and the role of the baroclinic wave fields in this dissipation are investigated using numerical simulations with the Massachusetts Institute of Technology general circulation model (MITgcm). The model is integrated over two-dimensional configurations along a zonal transect at 20.6°N for a maximum duration of a spring–neap cycle. Nearly all dissipation occurs at the steep ridge crests due to high-mode turbulent lee waves with horizontal scales of several kilometers and vertical scales of hundreds of meters. The spatial structure and timing of the predicted velocities and dissipation agree with observations and confirm the existence of these lee waves. The lee wave strength is greatly affected by the internal waves generated at the other ridge. When semidiurnal barotropic tides are dominant, the internal wave beams from both ridges nearly superpose after one surface reflection. The remotely generated internal waves from both ridges are therefore in phase with each other and the barotropic tides at the ridges. The barotropic-to-baroclinic energy conversion, energy flux divergence, ridge top velocities, and dissipation are stronger compared to the sum of the single east ridge and single west ridge cases. When diurnal tides are dominant, the wave fields are more out of phase and the conversion, divergence, and dissipation are less than or equal to the single ridge cases combined.

Corresponding author address: Maarten C. Buijsman, Geophysical Fluid Dynamics Laboratory, 201 Forrestal Road, Princeton, NJ 08540-6649. E-mail: maarten.buijsman@noaa.gov

1. Introduction

When tidal barotropic energy is converted to baroclinic energy at supercritical topography such as ocean ridges and shelf breaks, energy is lost to local turbulent mixing and dissipation. Klymak et al. (2008) observed large convective overturn and dissipation in areas with high strain and low velocity shear above the crest of the supercritical Hawaiian Island Ridge. In a two-dimensional (2D) numerical modeling study, Legg and Klymak (2008) attributed the overturning to high-mode hydraulic jumps or lee waves that form during downridge flow on the ridge slope. As the tide relaxes, the lee waves form into turbulent bores that propagate upslope and are advected over the ridge crest. These lee waves are the most dominant dissipation and mixing mechanisms at the crests of super critical ridges in high-resolution 2D models (Cummins 2000; Legg and Klymak 2008; Warn-Varnas et al. 2010; Buijsman et al. 2010a; Klymak and Legg 2010). Although low-mode lee waves have been observed in fjords (Farmer and Smith 1979) and coastal seas (Chereskin 1983), there are few direct observations of high-mode lee waves at steep ocean ridges (e.g., by Alford et al. 2011).

A tidal lee wave is an internal wave that is arrested by the barotropic tide. It has a phase speed cn, where n is the mode number, that is equal to the barotropic flow Um over a ridge. All higher baroclinic modal waves are arrested and therefore dissipate (Klymak et al. 2010a). Lee waves have a dispersion relation , where ωlee is the frequency of the lee wave, λx is the horizontal wavelength, the vertical wavelength, N0 is the depth-independent buoyancy frequency, and α the topographic slope (Klymak et al. 2010b,a). Turbulent lee waves are expected to form when the duration of half a tidal cycle is long enough to allow the lee wave to grow to at least one vertical wavelength (ωlee > 2ωtide, where ωtide is the tidal frequency), which requires the ridge slope to be sufficiently supercritical (α > 2ωtide/N0) (Legg and Klymak 2008; Klymak et al. 2010a).

Luzon Strait between Taiwan and the Philippines (Fig. 1) comprises two parallel ridges, the Hengchun or west ridge (WR) and the Lanyu or east ridge (ER), where barotropic-to-baroclinic conversion is large. This causes strong nonlinear internal waves (solitons), modified by rotation, that propagate into the South China Sea and the Pacific Ocean. They have been extensively observed (Ramp et al. 2004; Zhao and Alford 2006; Alford et al. 2010; Ramp et al. 2010) and modeled (Jan et al. 2008; Helfrich and Grimshaw 2008; Farmer et al. 2009; Warn-Varnas et al. 2010; Buijsman et al. 2010a,b; Vlasenko et al. 2010; Zhang et al. 2011; Simmons et al. 2011). These studies have shown that the nonlinear internal waves evolve from linear internal tides through nonlinear steepening.

Fig. 1.
Fig. 1.

The occurrence of lee waves is indicated with ωlee − 2ωtide = N0α − 2ωtide > 0 (shaded area) for (a) semidiurnal and (b) diurnal tides. The depth is contoured every kilometer. The straight black dashed line is the 2D modeling transect. The black circle marks station N2 at the west ridge.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

The two ridges generate wave fields, and—depending on the tidal frequency, ridge height, and separation distance—baroclinic waves may be enhanced through increased barotropic-to-baroclinic energy conversion and linear superposition. Jan et al. (2008) computed the baroclinic energy divergence, conversion, and dissipation in Luzon Strait for semidiurnal and diurnal tides with a three-dimensional (3D) numerical model. The dissipation decreased more during semidiurnal than diurnal tides when the west ridge was removed. They argued that the double-ridge internal wave interference strongly affects the dissipation. Farmer et al. (2009) found that the semidiurnal internal wave beams that radiate upward from both ridges nearly superpose after one surface reflection. This superposition occurred most clearly in the northern part of the strait, “suggesting the potential for resonance leading to enhanced generation of internal tides.” Farmer et al. also noticed that this superposition does not occur for diurnal beams. With a 2D numerical model of an idealized Gaussian double ridge system, Buijsman et al. (2010b) found that westward baroclinic energy fluxes and conversion are sensitive to the ridge height and separation distance between the two ridges. The fluxes and conversion are strongly enhanced when semidiurnal tides dominate for a separation distance of 90–100 km, which is less than the first-mode wavelength of about 140 km. Echeverri and Peacock (2010) used the Green function approach and found that the constructive and destructive interference of beams radiating from the double ridge along 2D transects is very sensitive to the bathymetry. This sensitivity has been explored further with a double-ridge knife edge model by Klymak et al. (2012, hereafter KBLP). Alford et al. (2011) observed dissipation rates at the west ridge at station N2 in the central Luzon Strait (Fig. 1) that were stronger during semidiurnal than diurnal tides, although the diurnal tides featured stronger barotropic velocities. They speculated that the observed dissipation and velocities may be attributed to nonlinear high-mode hydraulic jumps. Although most of these studies have noticed a difference in wave generation and dissipation between diurnal and semidiurnal tides, a clear explanation has not yet been presented.

The potential occurrence of lee waves in Luzon Strait is indicated with ωlee − 2ωtide > 0 in Fig. 1, where ωlee is computed in a Wentzel–Kramers–Brillouin (WKB) scaled ocean with a depth-independent buoyancy frequency (Althaus et al. 2003). Note that barotropic velocities should be across the ridge in order for lee waves to form. According to this simple relation, lee waves are more widespread when diurnal tides dominate. Although turbulent lee waves are predicted to occur at the west ridge near station N2 at 20.6°N when diurnal tides dominate, they are not predicted to occur when semidiurnal tides dominate. This contrasts with the aforementioned measurements by Alford et al. We will show that this discrepancy results from ignoring the effect of the remote ridge in the simplistic prediction.

In this paper, we apply the hydrostatic Massachusetts Institute of Technology general circulation model (MITgcm) in a 2D (independent of the y coordinate) setup with realistic bathymetry to investigate the cause of these dissipation processes near the ridges and the apparent discrepancy with the linear theory at the west ridge. The effect of the double ridge, the possibility for constructive and destructive interference between the internal waves generated at the two ridges, and the consequences for the dissipation are investigated. Moreover, we study how this dissipation varies during a spring–neap cycle and how much energy is locally dissipated versus how much is radiated away.

In the next section, we present the model setup, model cases, and the baroclinic energy equation used to diagnose the double ridge interaction. In section 3, the modeled hydraulic jumps are compared with observations at the west ridge, and the most important terms in the energy balance are analyzed. In the discussion section, the double-ridge internal tide interference is examined using the double-ridge knife edge model of KBLP, and the sensitivity of the interference to different bathymetry configurations is tested.

2. Methodology

a. Model setup

The numerical model MITgcm (Marshall et al. 1997) is applied in a 2D configuration. The model features a subgrid-scale scheme that computes vertical viscosities and diffusivities by Thorpe sorting unstable density profiles (Klymak and Legg 2010). Klymak and Legg showed that, when mixing is due to breaking internal waves, this scheme works better for closing energy budgets than shear-driven mixing schemes. The background horizontal and vertical viscosities (diffusivities) are 10−2 (10−4) and 10−5 (10−5) m2 s−1. The model features 2800 horizontal grid cells and 260 vertical levels. The central domain with the two ridges has Δx = 100 m. The grid is linearly stretched to Δx ≈ 5 km toward the east and west boundaries. The top 2000 m has Δz = 10 m and below this depth the grid is linearly stretched to Δz ≈ 50 m at z = −3900 m. The model is forced at the east and west boundaries with barotropic tides and the north and south boundaries are periodic. The velocities and temperature are quadratically nudged over 100 cells near the east and west boundaries to the barotropic zonal velocities and a time-invariant vertical temperature profile, respectively. We apply free slip boundaries at the surface and bottom. The vertical resolution is insufficient to resolve the frictional bottom boundary layer associated with a no-slip boundary condition. The model is run without any bottom drag. Sensitivity runs indicate that the bottom drag has a relatively small contribution to the total dissipation (<15%).

We have tested the sensitivity of the dissipation to the hydrostatic and nonhydrostatic modes, different horizontal grid sizes, and barotropic forcings, in a computationally less expensive idealized Gaussian ridge scenario with a smaller domain size, as described in the appendix. These tests show that, for a resolution of Δx = 100 m, the hydrostatic simulations report the same dissipation as the nonhydrostatic runs. Nonhydrostatic runs are only an improvement over hydrostatic runs when smaller-scale turbulent motions are resolved, that is, for horizontal grid scales smaller than the buoyancy length scale (Waite 2011), which is not the case at Δx = 100. Hence, for our more expensive, realistic domain, we choose to carry out all runs in hydrostatic mode since nonhydrostatic runs are more expensive and provide no improvement at this resolution. The wall-clock time is about 5–6 h for 3.83 days of hydrostatic simulation when the model is run on 28 nodes of the Da Vinci super computer of the U.S. Navy.

The model uses realistic topography merged from high-resolution gridded multibeam data with a resolution of ~300 m and Shuttle Radar Topography Mission 30plus (SRTM30plus) data from the Smith and Sandwell database with a resolution of ~1 km (Smith and Sandwell 1997). Away from the double ridge the model depth is a constant 3000 m. The density stratification is derived from data collected in between the ridges (Alford et al. 2011). In the model, the density is only a linear function of temperature because most of the variability in density is due to temperature at this location (Jan et al. 2008). A thermal expansion coefficient of 2 × 10−4 kg m−3 K−1 is used. The tidal forcing comprises amplitudes and phases of zonal barotropic velocities for eight tidal frequencies. The amplitudes and phases are extracted from the Ocean Topography Experiment (TOPEX)/Poseidon 7.2 (TPXO7.2) tidal inversion model (Egbert et al. 1994) at a location east of the east ridge in 3500-m water depth and at the same latitude as the 2D transect. The velocity amplitudes are scaled to the model depth of 3000 m at the east and west boundaries. The phases at both boundaries are the same. Hence, the depth-integrated zonal flow is independent of x. This standing wave assumption is justified because velocity time series computed from TPXO7.2 amplitudes and phases extracted at both sides of the double ridge are lagged by at most 30 min.

The model is run along a transect at 20.6°N with realistic (prefix re) bathymetries with either a single west ridge (reW), a single east ridge (reE), or a double ridge (reWE) (Fig. 2b). This transect is chosen because of the availability of observations of strong dissipation at the west ridge at this latitude. Along the transect, the west ridge features two subridges that are uniform in height and shape over a north–south distance of about 10 km. At the east ridge the transect runs through a gap between the Batan Islands and crosses three subridges. These subridges are intersected by deep east–west channels (Fig. 2a). The west and east ridges farther to the north and south of the transect feature a different bathymetry. The distinct double and triple subridges are not present elsewhere. The west ridge is taller to the north of the transect, whereas the east ridge becomes much wider to the south.

Fig. 2.
Fig. 2.

(a) Bathymetry of the west and east ridges in central Luzon Strait. The white dashed line marks the transect used for the 2D modeling, and the solid white square the N2a and N2b mooring locations. Depth contours are the white and black lines. The 0-m contour is the thick black line. (b) Realistic bathymetry along the transect for double ridge (reWE, gray solid line), single east ridge (reE, black dashed line), and reW (black solid line). The criticality γ2 and γ1 for semidiurnal and diurnal tides is plotted below the depth profile: γ ≤ 0.5 is white (subcritical), 0.5 < γ ≤ 1.5 is gray (near critical), and γ > 1.5 is black (supercritical).

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

The double ridge and single ridge cases are run for 16 days over a spring–neap cycle from 21 August (yearday 232) to 6 September 2010 (yearday 248). This period covers the 24-h stations N2a and N2b by Alford et al. (2011). Additional cases of 3.8 days are run during periods with mainly semidiurnal tides (yeardays 237.63–241.46) and diurnal tides (yeardays 243.17–247.00) to test the sensitivity of the double ridge interference to different west-ridge heights and ridge separation distances.

b. Baroclinic energy equation

In the following, we diagnose the most important terms in the baroclinic potential and kinetic energy equation. The baroclinic energy equation is obtained after partitioning the energy equation into a barotropic and baroclinic equation. The depth-integrated baroclinic energy equation reads (Cummins and Oey 1997; Kurapov et al. 2003; Kang and Fringer 2012)
e1
where the overbar indicates depth integrated. The first term of (1) is the tendency of the kinetic energy KE = ½(1/2)ρ0(u2 + υ2 + w2) and the linear available potential energy APE = g2ρ2/(2ρ0N2) in which u′, υ′, and w are the baroclinic horizontal and vertical velocities along the x, y, and z coordinates; the perturbation density ρ′(x, y, z, t) = ρ(x, y, z, t) − ρb(z) in which ρb is the time-invariant background density and ρ0 is the reference density; g is the gravitational acceleration; and the buoyancy frequency . The horizontal baroclinic velocities are computed as u′ = uU, where the barotropic velocity U = (U, V) is the depth mean of (u, υ). The second term is the energy flux divergence. The depth-integrated energy flux is
e2
where KE0 = ρ0(Uu′ + ′) (Kang and Fringer 2012) and Kh and Ah are the horizontal viscosity and diffusivity. The flux has contributions from pressure work (first term), energy advection (second to fourth terms), and horizontal diffusion (last two terms). The third term of (1) is the depth-integrated barotropic-to-baroclinic energy conversion
e3
where p′(z = −H) is the perturbation pressure at the bottom, H is water depth, and W = −U · H is the vertical barotropic velocity. Generally, is a source term, but it can be a sink when p′(z = −H) and W are out of phase, for example, due to remotely generated internal waves. The fourth term of (1) is the dissipation
e4
where Ax is the vertical eddy viscosity that is estimated with the vertical mixing scheme; Ah is constant in space and time. The dissipation term is the conversion of kinetic energy to heat. In (1) is the diapycnal mixing term. It increases the background potential energy and is a sink term for the APE (Kang and Fringer 2012). We do not compute this term.
If we time average (1) over several tidal cycles and integrate between x1 and x2, the new balance is
e5
where is a residual term that includes the diapycnal mixing term, a small contribution from the tendency term due to time-averaging dynamics that are not in a complete steady state, numerical dissipation and diffusion, and inaccuracies due to the offline computation. We compute all terms in (5) between x1 = 365.3 km and x2 = 576.3 km for the double ridge case after a quasi-steady state is reached and normalize them by the largest term, the barotropic-to-baroclinic energy conversion. The values for the flux divergence, dissipation, and residual term are 74.4%, 17.3%, and 8.3%, respectively. The tendency term contributes 2.0% and the numerical dissipation and diffusion, diapycnal mixing, and computational errors combined contribute 6.3%. If we assume this latter number is mainly due to irreversible mixing, the mixing efficiency can be as large as 6.3/17.3 = 0.36. The divergence of the pressure flux is much larger (73.1%) than the divergence of the advection of energy (1.2%) and the diffusive flux (0.0%). In the remainder of the paper, we only discuss the largest three terms: the conversion, the pressure flux divergence, and the dissipation. For the sake of brevity, we write the depth-integrated flux divergence as and the depth- and x-integrated conversion and dissipation as Cv and Dv.

3. Results

a. Internal wave regime in Luzon Strait

Following Garrett and Kunze (2007), we classify the internal wave regime in Luzon Strait using the nondimensional parameters for criticality, topographic Froude number, and tidal excursion length. The criticality
e6
is the ratio between the bathymetry slope and the internal wave beam angle and is computed for the transect in Fig. 2b, where f is the Coriolis frequency and ωtide = 1.41 × 10−4 s−1 (7.29 × 10−5 s−1) is the semidiurnal (diurnal) tidal frequency. All subridges are supercritical with γ > 1, permitting the formation of downward internal tide beams and turbulent lee waves.

For the semidiurnal (diurnal) tides, the topographic Froude number Frt = U0/(Nrhr) = 0.004 (0.008), where U0 = 0.05 m s−1 (0.1 m s−1) is the amplitude of the far-field barotropic current, hr = 2500 m is the height of the ridge, and Nr = 0.005 s−1 the value of the buoyancy frequency at the crest. The flow is relatively unaffected by the topography for Frt ≫ 1, and breaking lee waves do not occur, whereas small-scale hydraulic effects such as nonlinear breaking lee waves occur for tall and steep topography when Frt ≪ 1.

The excursion length δ = U0/(tide) = 0.04 (0.14) for the semidiurnal (diurnal) tides, where L = 10 km is a horizontal topographic length scale. Stationary, but not necessarily breaking, lee waves occur for δ ≫ 1, whereas mainly linear internal tides at the forcing frequency are generated for δ ≪ 1.

In summary, the ridges in Luzon Strait fall into regime 4 of Garrett and Kunze (2007), which features both nonlinear breaking lee waves as well as linear internal tide beams. We will show that these beams play an important role in the double-ridge internal wave interference.

b. Model–data comparison

The results of the 16-day double ridge case reWE are compared with observations at the west ridge. Lowered ADCP (LADCP) and CTD data were collected at a number of stations spanning the ridges in Luzon Strait in August and September 2010 as part of the Internal Waves in Straits Initiative (IWISE) of the Office of Naval Research. The velocity and density data and dissipation rates estimated from density overturns at station N2 were kindly provided by M. H. Alford and others (2010, personal communication) for these comparisons; see Alford et al. (2011) for more details of these data and methods. Of all sites where data were collected, site N2 at the east slope of the easternmost subridge of the west ridge (Fig. 2) featured the strongest dissipation when semidiurnal tides dominate (yeardays 239.04–240.37, station N2a) and diurnal tides (yeardays 245.25–246.43, station N2b). (The survey durations are also in Fig. 8a.)

The spatial structure and timing of the modeled velocities, isotherm oscillations, and dissipation are similar to the observations during a period with strong semidiurnal tides in Fig. 3 and diurnal tides in Fig. 4 at site N2. During strong eastward barotropic flow at the west ridge (Figs. 3f and 4f), the downslope bottom velocities are enhanced and the isotherms are depressed. As the barotropic flow relaxes, the dissipation peaks and overturns of up to 500 m in height occur in Figs. 3b and 4b. As the flow switches to the west, the depressed isopycnals rebound upslope (e.g., at t = 239.6 and 240.1 days in Fig. 3). This rebound is weaker in the model than in the data, in particular during N2b. During these periods with westward flow, the dissipation is small at the east slope of the west ridge.

Fig. 3.
Fig. 3.

Model–data comparison at station N2a when semidiurnal tides dominate; observed (a) zonal velocities (positive is to the right) and (b) dissipation estimated from overturns. The density of an ocean at rest is contoured in vertical increments of 100 m. (d),(e) The model results; (c) the time-mean dissipation; (f) the barotropic velocities; and (g) the depth-integrated dissipation. The black diamonds plotted on the top axes mark the times of snapshots presented in Figs. 5 and 6.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

Fig. 4.
Fig. 4.

As in Fig. 3 but at station N2b when diurnal tides dominate. The black diamonds plotted on the top axes mark the times of snapshots presented in Fig. 6.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

While the spatial structure and timing of the model results and observations are similar, there are differences in the magnitudes of the modeled and observed velocities, isotherm oscillations, and dissipation. For example, the predicted dissipation during N2b (Fig. 4c) is larger than during N2a (Fig. 3c). This contrasts with the observations. Moreover, the modeled barotropic velocities in Figs. 3f and 4f are a factor of 2 smaller than the observed velocities. We have also compared the modeled barotropic velocities with observed velocities at station LS04 on the west slope of the east ridge (R. Pinkel and J. Klymak 2010, personal communication) and find good agreement in the timing and magnitude of the barotropic velocities (not shown). TPXO barotropic velocities are also extracted in the trench between the west and east ridges (not shown). Although the timing is correct, the TPXO velocities are 1.5 times larger than the predictions. A reason for these differences is that in our model the depth-integrated barotropic flow is constant along the x coordinate, whereas this is likely not the case in the 3D reality. An incorrect representation of the barotropic velocities can affect the magnitudes of the dissipation more than the conversion and flux divergence because the dissipation scales with u3, while the conversion and divergence scale with u2 (Klymak et al. 2010a). In additional model runs, in which the modeled barotropic velocities are matched with the observed velocities at the west ridge, the spatial structure is different and the dissipation is more than an order of magnitude larger than the observed dissipation during diurnal and semidiurnal tides (results not shown). This is an indication that, in addition to local barotropic velocities, the dissipation is also affected by remotely generated internal waves. Despite these shortcomings, we believe that our 2D model is a valuable tool to gain insight into the hydraulics.

We infer from snapshots of modeled zonal velocities and isotherms in Figs. 5 and 6 that the observed velocities and overturns during semidiurnal and diurnal tides can be attributed to lee waves. A lee wave, although not yet very turbulent, is set up during accelerating eastward flow (Fig. 5a). During maximum eastward flow (Fig. 5b) an arrested bore forms near x = 446 km, which features convective instabilities that are taller than 400 m. During decelerating flow (Fig. 5c) one vertical lee wavelength is visible with strong bottom attached downslope flow and nearly stagnant water above it. The detached tail of the lee wave east of N2 becomes oscillatory with large turbulent overturns. Simultaneously, the upper part of the turbulent lee wave transitions into a westward and upward oriented beamlike feature over the ridge crest in Fig. 5d that will be advected westward by the accelerating westward flow. Similarly, the turbulence that is present during the accelerating eastward flow on t = 239.76 days near z = −1.2 km in Fig. 5a is a remnant of a lee wave that was created during the previous westward flow.

Fig. 5.
Fig. 5.

Snapshots of (a)–(d) zonal velocities (positive is to the right) and (e)–(h) dissipation during accelerating and decelerating semidiurnal barotropic eastward flow at the west ridge for the double ridge case. The times and barotropic velocities are above the top panels; “max ebb” refers to maximum eastward barotropic flow. The times of the snapshots are also marked in Fig. 3. The gray square marks the N2 site. The density of an ocean at rest is contoured in vertical increments of 50 m.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

Fig. 6.
Fig. 6.

Snapshots of zonal velocities (positive is to the right) 1.3 h after maximum eastward barotropic flow at the west ridge for (a) N2a and the double ridge (reWE), (b) N2a and the single ridge reW, (c) N2b and the double ridge, (d) N2b and the single ridge reW. (e)–(h) The dissipation for the four cases. The times of the snapshots are also marked in Figs. 3 and 4. The gray square marks the N2 site. The density of an ocean at rest is contoured in vertical increments of 100 m.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

In the single west ridge (reW) case, during maximum eastward semidiurnal barotropic flow in Fig. 6b, the lee wave at the west ridge is much weaker than in the double ridge case at the same time in Fig. 6a. This illustrates the effect of the remote internal waves from the east ridge on the lee wave formation. Similarly, the lee wave is also weaker and less turbulent in the single west ridge case in Figs. 6d,h than in the double ridge case in Figs. 6c,g when diurnal tides dominate, but the difference is less striking.

The linear theory discussed in the introduction predicts the vertical lee wavelength well in the single west ridge case when diurnal tides dominate (Fig. 6d). Using the formulas for λz and ωlee, Umax = 0.15 m s−1, N0 = 0.005 rad s−1, and λx = 2000 m, we find λz = 475 m, Tlee = 2π/ωlee = 3.7 h, and mode number n = 16 (all higher modes are trapped). In Fig. 6d, about 1.5 vertical wavelengths are visible. The period of Tlee = 3.7 h is small enough to allow the lee wave to fully develop in half a diurnal cycle.

c. Dissipation in a spring–neap cycle

First, we consider the spatial variability of dissipation. The time-mean baroclinic horizontal kinetic energy 〈HKE〉 = 〈½(1/2)ρ0(u2 + υ2)〉 and the time-mean dissipation for the double ridge case are shown in Fig. 7. The kinetic energy is large at the surface, near the subridges, and along beams. The strongest dissipation occurs at the subridges and the dissipation quickly drops by several orders of magnitude away from the subridges in Figs. 7b,c. This is also illustrated with the normalized cumulative depth-integrated dissipation in Fig. 7d, which increases at each subridge in a steplike fashion. Although the tallest subridge on the east ridge contributes about 38% to the total dissipation, the contribution of the other subridges cannot be discounted. The localized dissipation at these subridges is mainly due to high-mode turbulent lee waves in the model results.

Fig. 7.
Fig. 7.

The time-mean (a) horizontal baroclinic kinetic energy and (b) dissipation for the double ridge case. (c) The time-mean and depth-integrated dissipation. (d) The cumulative dissipation, normalized by the total dissipation. The numbers are normalized dissipation computed over areas bounded by the dots. The strongest dissipation occurs at the tallest subridge on the east ridge.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

The temporal variability in the volume-integrated dissipation shows stark differences between the double ridge (reWE), single east ridge (reE), and single west ridge (reW) cases. The dissipation is volume integrated about both ridges (Figs. 8b,c), the east ridge (Fig. 8d), and the west ridge (Fig. 8e). The x coordinates that bound the ridges are 365.3, 473.3, and 576.3 km. The peaks in the total dissipation in Fig. 8b occur for the strongest barotropic velocities in a tidal cycle in Fig. 8a. The amplitudes of the velocities and dissipation reflect the spring–neap cycle. To better illustrate the differences between the cases, the dissipation is low-pass filtered with a cutoff frequency of 2 days (Figs. 8c–e). In contrast to the single ridge cases, the low-passed dissipation for the double ridge case does not portray a strong spring–neap variability in Figs. 8c–e. It is nearly constant in time. While the total low-passed dissipation of the double ridge and single east ridge cases is about the same during strong diurnal tides (yeardays 232–236 and 243–248) in Fig. 8c, the total low-passed dissipation of the double ridge case is about twice as large as that of the single east ridge case during strong semidiurnal tides (yeardays 236–243). The dissipation integrated over the east ridge for the double ridge and single east ridge cases reveals a similar temporal variability in Fig. 8d except that the dissipation at the east ridge during diurnal periods is smaller in the double ridge case. In contrast, the dissipation integrated over the west ridge is larger during the entire spring–neap cycle in the double ridge case compared to the single west ridge case (Fig. 8e). However, the increase in dissipation in the double ridge case relative to single west ridge case is larger when semidiurnal tides dominate on yearday 240 (a factor of 8) than when diurnal tides dominate on yeardays 235 and 246 (a factor of 3). In the double ridge case, the dissipation at the west ridge equilibrates slower (2 days in Fig. 8e) than at the east ridge (0.5 days in Fig. 8d). This difference in response is an indication that the dissipation at the west ridge is not only affected by the local barotropic forcing but also by the internal waves generated at the east ridge, which take some time to cross the trench between the ridges. It takes a diurnal vertical mode-1 wave about 0.25 days and a mode-8 wave about 2 days to cross the ~90 km ridge separation distance.

Fig. 8.
Fig. 8.

(a) Barotropic velocity at the east ridge for the double ridge and single ridge cases plotted on top of each other. The survey periods N2a and N2b are marked by the black horizontal bars. (b) Unfiltered dissipation integrated over both ridges for the double ridge (reWE) (gray), single east ridge (reE) (black), and single west ridge (reW) (dashed line) cases. Low-passed dissipation integrated (c) over both ridges for all cases, (d) over the east ridge for the double ridge and east ridge cases, and (e) over the west ridge for the double ridge and west ridge cases. The cutoff frequency is 2 days.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

d. The amplification of the flux divergence, conversion, and dissipation

As for the dissipation, the conversion and flux divergence are volume integrated over the west and east ridges for the double ridge and single ridge cases (Fig. 9). Similar to the dissipation for the double ridge case (Fig. 8c), the conversion 〈Cv〉 (Fig. 9b) and flux divergence (Fig. 9c) are larger than in the single east ridge case during periods with stronger semidiurnal tides, while they are smaller or equal when diurnal tides dominate.

Fig. 9.
Fig. 9.

(a) The barotropic velocity at the east ridge for the double ridge case. Low-passed (b) conversion, (c) flux divergence, and (e) the ratio of the dissipation over conversion for the double ridge (reWE) (gray), single east ridge (reE) (black) and single west ridge (reW) (dashed line) cases. The energy terms are volume-integrated over both ridges. (e) The amplification (7) for the low-passed and volume-integrated dissipation, conversion, and divergence. Constructive double ridge interference with Ψ > 0 occurs when semidiurnal tides dominate.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

The time-mean westward and eastward fluxes for the double ridge case (single east ridge case) are −25 and 18 kW m−1 (−21 and 24 kW m−1) when semidiurnal tides dominate and −17 and 43 kW m−1 (−30 and 50 kW m−1) when diurnal tides dominate. In the double ridge case the diurnal fluxes are greatly reduced compared to the single east ridge case. Three-dimensional model results and observations by Alford et al. (2011) show that the westward semidiurnal fluxes are also larger than the westward diurnal fluxes in the double ridge case.

The fraction of the converted energy that is locally dissipated, q = 〈Dv〉/〈Cv〉, gives insight into the spatial distribution, local and remote, of tidally driven mixing (Fig. 9e). In the single west ridge and single east ridge cases it is constant in time at ~13%, whereas in the double ridge case at the end of the semidiurnal period it peaks at 23%; that is, more of the converted energy is dissipated when semidiurnal tides dominate in the double ridge case compared to the single ridge cases. Jan et al. (2008) and Alford et al. (2011) applied 3D models of Luzon Strait and computed and 39%, respectively, where is based on the pressure flux. Our local dissipation fraction differs from theirs because the three dimensionality may affect the wave generation and dissipation, the higher west ridge in the northern Luzon Strait may dissipate more energy than the west ridge at the 2D transect, and they used coarse-resolution models that do not resolve the lee waves and that have different subgrid-scale dissipation schemes. The local dissipation fraction may also be underestimated if the barotropic velocity is underestimated since dissipation scales like u3. Moreover, the dissipation computed as may overestimate the real dissipation because of the omission of small terms such as advection and tendency (in our 2D runs they add up to 3% of the conversion).

The difference in response between diurnal and semidiurnal tides is illustrated with the amplification factor Ψ in Fig. 9e. The amplification for conversion, for example, is the difference between the conversion for the double ridge case and the sum of the conversion for the single ridge cases, normalized by sum of the single ridge cases. It reads
e7
where subscripts W, E, and WE refer to the single west ridge, single east-ridge, and double ridge cases, respectively. Similarly, the amplification can also be computed for the divergence ΨΔF and the dissipation ΨD. When the conversion of the double ridge case is larger than the sum of the single ridge cases, constructive interference occurs (Ψ > 0). Destructive interference occurs for Ψ < 0. The amplification of the dissipation in Fig. 9e is much larger than that of the conversion and flux divergence when semidiurnal tides dominate. This difference is because the dissipation scales with u3, whereas the conversion and flux divergence scale with u2. When diurnal tides are dominant, Ψ ≲ 0.

4. Discussion: Effect of the double ridge

In the previous section, we have demonstrated that the dissipation at supercritical ridge crests in the model and observations is due to lee waves. Moreover, we have shown that the dissipation, conversion, and flux divergence in the double ridge case are larger than the sum of the single ridge cases when semidiurnal tides dominate. The reverse is true when diurnal tides dominate. In this section, the effect of the double ridge on the conversion, divergence, and dissipation is further investigated.

a. Bottom velocities and dissipation

The spatial and temporal variability of the differences in depth-integrated daily-mean dissipation between the double ridge and single west ridge cases (Fig. 10b) and between the double ridge and single east ridge cases (Fig. 10c) is reasonably well correlated with the difference in the amplitudes of the bottom velocities (Figs. 10d–g). The amplitudes are computed by averaging the zonal velocities over the bottom 200 m. Next, we compute the semidiurnal and diurnal amplitudes by harmonically fitting either the M2 or K1 frequencies to 1.1-day windows of semidiurnal or diurnal bandpassed time series, respectively (Buijsman et al. 2010a). The 1.1-day windows are advanced by steps of 0.55 days. The M2 and K1 frequencies are used because they have the largest amplitudes in the semidiurnal and diurnal frequency bands. In this method energy in S2 (O1) affects the M2 (K1) amplitudes. As a consequence, these amplitudes portray fortnightly cycles. Finally, we compute the differences in amplitudes for each day (ΔAub). We focus on the velocity differences near the subridges where they contribute to the dissipation differences.

Fig. 10.
Fig. 10.

The differences between the double ridge and single west ridge cases for (b) the daily-mean depth-integrated dissipation , (d) the amplitude Aub of the semidiurnal (D2) zonal bottom velocities averaged over the bottom 200 m, and (f) the amplitude of the diurnal (D1) zonal bottom velocities. (c),(e),(g) The differences between the double ridge and single east ridge cases. The black lines in the left panels are the deep water amplitudes of the zonal barotropic velocities × 200. Zero velocities are at y axis. (a) The double ridge bathymetry and subridges are marked by a line and shading.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

At the west ridge, the stronger dissipation coincides with stronger semidiurnal bottom velocities in the double ridge case during the entire spring–neap cycle (Fig. 10d), whereas the diurnal bottom velocities of the double ridge are only weakly stronger than those of the single west ridge case (Fig. 10f). Similar to the west ridge, the semidiurnal bottom velocities of the double ridge case at the east ridge are larger than those of the single east ridge case during the entire spring–neap cycle (Fig. 10e). During semidiurnal spring tides these velocities are largest and coincide with the larger dissipation in the double ridge case (Fig. 10c). At the east ridge the diurnal bottom velocities of the double ridge case are smaller than in the single east ridge case, in particular during diurnal spring tides (Fig. 10g). At these times the dissipation of double ridge case is smaller than that of the single ridge case (Fig. 10c).

Larger bottom velocities at the subridges lead to larger lee waves, overturns, and dissipation. KBLP showed that internal waves with lower mode numbers, which have higher phase speeds, are arrested when bottom velocities are higher over a height that is similar to the vertical wavelength of the arrested lee wave. All higher modes are dissipated, causing the total dissipation to be higher.

b. Double ridge resonance mechanism

1) Idealized double-ridge knife edge model

We apply the inviscid double-ridge knife edge model of KBLP to gain insight in the double ridge interaction during semidiurnal and diurnal tides. The double-ridge knife edge model is an extension of the single-ridge knife edge model by St. Laurent et al. (2003) and Klymak et al. (2010a). In the knife edge model, the ridge separation distance, the water depth, the ridge heights, the depth-independent buoyancy frequency N0, and the tidal forcing are prescribed. Each knife edge ridge radiates internal waves due to the barotropic tidal forcing. At the same time, the wave generation at each ridge is affected by the incoming internal waves generated at the opposing ridge. At each ridge the horizontal and vertical velocities are decomposed into vertical modes for the transmitted, incoming, and reflected progressive waves. The velocity matching conditions at each ridge give a set of equations that can be solved for the velocity amplitudes and the phases of the transmitted, incoming, and reflected internal waves at each ridge. To make a comparison with our realistic runs, we have to WKB scale the realistic ridge heights. In our experiment, we vary the ridge separation distance ΔX and the WKB-scaled west-ridge height , while keeping the east-ridge height, buoyancy frequency, and forcing frequency constant (Fig. 11a). The east-ridge height and buoyancy frequency are based on the double ridge case. The model is also run for the single west ridge and single east ridge cases in order to compute the amplification (7).

Fig. 11.
Fig. 11.

(a) The amplification (7) of the conversion for the double-ridge knife edge model as a function of the ridge separation distance ΔX and west ridge height for a constant east ridge height , where the maximum water depth H = 3699 m. The dashed and dotted lines mark the superposition of beams after an odd number of surface and bottom reflections, respectively. The lines with circles and crosses indicate beam superposition after an even number of surface and bottom reflections. Zonal velocity snapshots for (b) a resonant case with an attractor, (c) Luzon Strait parameters and semidiurnal tides, and (d) Luzon Strait parameters and diurnal tides. The west ridge height is 0.52H in (b) and 0.16H in (c),(d). The ridge separation distance is 0.5λ in (b), 0.64λ in (c), and 0.26λ in (d); λ is the horizontal first-mode wavelength. The three cases corresponding to (b)–(d) are also marked in (a).

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

In the absence of dissipation, the flux divergence is equal to the barotropic-to-baroclinic energy conversion in the knife edge model. For comparison with the numerical model runs, we refer to conversion when discussing the flux divergence in the knife edge model. The amplification of the tidal-mean conversion ΨC, volume integrated over both ridges, shows an intricate interference pattern in Fig. 11a. Destructive double ridge interference with ΨC < 0 occurs when the beams connect the ridge tops for zero or an even number of surface or bottom reflections. The latter happens along the lines with circles, , where λ is the horizontal first-mode wavelength, and crosses, . Constructive interference with ΨC > 0 occurs when the beams connect the ridges after an odd number of reflections off the surface along the dashed line and off the seafloor along the dotted line . Resonance occurs for and ΔX/λ = 0.5. In this case the beams reflect once off the seafloor and surface and superpose to form an attractor (Maas et al. 1997; Echeverri et al. 2011), as in Fig. 11b. In this inviscid system, the velocities go to infinity along the attractor. Energy is trapped in standing waves between the ridges and the energy conversion and amplification approach infinity. For and ΔX/λ ~ 0.5, the amplification of the conversion is very sensitive to changes in the separation distance. This sensitivity is much smaller for .

The double ridge configurations based on the realistic double ridge case for the semidiurnal and diurnal tides in Figs. 11c,d are not resonant but allow for a weak constructive internal wave interference with ΨC equal to 0.15 (semidiurnal) and 0.02 (diurnal). The ridge separation distance between the tallest subridges of the double ridge case is 94.4 km. The semidiurnal and diurnal first-mode wavelengths are λ2 = 147 km and λ1 = 370 km. When semidiurnal tides are dominant, the upward-radiating beams nearly connect the ridge tops after one surface bounce, and the constructive interference is stronger than when diurnal tides dominate. Note the close proximity of case c (corresponding to the semidiurnal tide) to the dashed line in Fig. 11a. When diurnal tides are dominant, the beams do not connect nor do they have an odd number of bounces. The magnitude of the (semi-) diurnal amplification is in approximate agreement with the amplification computed for the realistic ridges in Fig. 9e.

When beams emitted from two ridges superpose after an odd number of bounces, the baroclinic wave fields from both ridges are in phase with each other and with the barotropic wave. This causes larger barotropic-to-baroclinic energy conversion and divergence and stronger velocities along the beams. Since the beams impinge at the ridge tops, the ridge-top velocities are stronger than in the single ridge case, causing stronger lee waves and dissipation.

The positive amplification for two ridges that are separated by approximately half a first-mode wavelength can be explained by only considering the generation of first-mode waves, which generally contain the most energy. However, higher modes are needed to obtain the complex interference patterns in Fig. 11a. At t = T/4, where T is the tidal period, barotropic flow U reaches its maximum value to the right and the isotherms are maximally depressed to the right of the west ridge and elevated to the left of the east ridge. At t = T/2, U = 0 and the depression and elevation waves have propagated λ/4 to the right and left, respectively, reaching the center of the basin, and the net isotherm displacement is zero. At t = ¾(3/4)T, the depression and elevation waves have propagated λ/4 to the right and left, and their trough and crest are at the location of the east ridge and west ridge, respectively. Simultaneously, U reaches its maximum flow to the left, and the isotherms of the local waves are maximally elevated to the right of the west ridge and depressed to the left of the east ridge. Hence, the local waves constructively superpose with the remote waves. The baroclinic waves are further enhanced because of ridge reflection, causing the trapping of energy and an increase in barotropic-to-baroclinic conversion (Jan et al. 2008; Kelly and Nash 2010; Buijsman et al. 2010b). For example, at the east ridge, the downslope barotropic flow (W < 0) at t = ¾(3/4)T is in phase with the remotely generated depression wave with p′(z = −H) < 0, further enhancing . A standing wave forms with the largest kinetic energy in between the ridges and the largest available potential energy at the ridges (Nash et al. 2006). This kind of first-mode resonance has been observed in the 2000-m-deep Santa Cruz Basin in the Southern California Bight in a three-dimensional model by Buijsman et al. (2012).

2) Two-dimensional realistic ridges

In the numerical model, with its realistic stratification and topography and nonlinear processes and dissipation, internal wave beams are less well defined compared to the knife-edge model. In an ocean with a flat bottom the vertical modes superpose to form beams. However, higher modes are dissipated through lee wave breaking. The stronger the forcing, the stronger the lee waves and the weaker the appearance of the internal wave beams (Buijsman et al. 2010a; Klymak et al. 2010a). The thermocline can trap the internal wave beams and scatter their energy (Gerkema 2001). Moreover, the stronger velocity shear of the higher modes allows them to be dissipated faster than the lower modes. Hence, relatively more energy is in the low modes in a nonlinear realistic simulation than in a linear inviscid knife edge model. In the realistic single east ridge case, the first mode comprises about 50% and the first three modes comprise about 70% of the horizontal kinetic energy in both diurnal and semidiurnal bands. In contrast, in the single-ridge knife edge model, the first mode contains about 27% and the first three modes contain about 37% of the total horizontal kinetic energy in the single east ridge case, when 30 modes are used.

Yet, the modeled velocity field cannot be predicted with only the barotropic mode and the first three baroclinic modes. Semidiurnal and diurnal beams are present in the model results in Figs. 12a,b. The agreement in the beam patterns between the numerical model and the knife edge model in Fig. 11 during (semi) diurnal tides is obvious, although the subridges cause parallel beams and block downward beams from adjacent subridges. When semidiurnal tides are dominant, an energy beam almost connects the central ridge on the east ridge with the eastern subridge on the west ridge after one surface bounce (Fig. 12a), whereas there is no such connectivity when diurnal tides dominate (Fig. 12b). This difference between semidiurnal and diurnal beams has also been shown in Farmer et al. (2009) and Alford et al. (2011). To have constructive interference, the wave fields from both ridges need to be in phase. We compute the semidiurnal and diurnal phases of the zonal velocities of the single west ridge and single east ridge cases and subtract them (ϕEϕW; Figs. 12c,d). When semidiurnal tides are dominant, −90° < ϕEϕW < 90° occurs along narrow beams that connect both subridges of the east and west ridges, indicating that the velocities due to the wave fields from both ridges are in phase. These patterns are identical for the density perturbation phase differences (not shown). Conversely, the diurnal velocities of the single ridge cases are not in phase at the west and east ridge tops, leading to smaller velocities and lee waves.

Fig. 12.
Fig. 12.

Internal wave beams are indicated by the baroclinic zonal velocity amplitudes of the double ridge case for (a) a semidiurnal spring tide (yearday ~238) and (b) a diurnal spring tide (yearday ~246). Superposed are the theoretical beams (white dashed lines). The black square in (a) marks where the downward and upward beams from the western subridge on the east ridge cross. The phase differences Δϕ between velocities of the single-east and single west ridge cases for (c) semidiurnal and (d) diurnal tides: Δϕ is colored for −90 < Δϕ < 90, indicating the velocities are approximately in phase.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

c. Sensitivity of the resonance to west-ridge height and separation distance

Similar to the double-ridge knife edge model, we explore the sensitivity of internal wave interference to changes in west-ridge height and separation distance for a period with mainly semidiurnal (yeardays 237.63–241.46) and diurnal (yeardays 243.17–247.00) tides. In all cases the east-ridge height is kept constant. It is similar to the height in the double ridge case reWE. The west ridge is raised or lowered by stretching the lower slopes without affecting the steepness of the slopes near the crests. For each double ridge case, additional single west ridge and single east ridge cases are run with ridge heights that are the same as in the double ridge cases. The amplification for the time-averaged and volume-integrated conversion, flux divergence, and dissipation is in Fig. 13.

Fig. 13.
Fig. 13.

The amplification of tidal-mean and volume-integrated (a) conversion, (b) flux divergence, and (c) dissipation for the realistic model as a function of the ridge separation distance ΔX and west-ridge height for a constant east-ridge height: ΔX for the diurnal (semidiurnal) cases is normalized by λ1 = 370 km (λ2 = 147 km). The diagonal lines are adapted from Fig. 11a. The amplification values for each case are plotted. The bold values are for the realistic double ridge case during (semi) diurnal tides. For both semidiurnal and diurnal tides, simulations are performed with ΔX = 94.4 km and 94.4 ± 25 km and hWR = 1636 m, 2136 m, 2636 m, and 3236 m (z = −2063 m, −1563 m, −1063 m, and −463 m). For semidiurnal tides, additional cases are run for ΔX = 94.4 km and 94.4 ± 20.9 km and hWR = 3349 m (z = −350 m).

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

Although lacking the broad range and resolution of the knife edge model, the amplification patterns for the conversion in the realistic model in Fig. 13a are in general agreement with those of the knife edge model in Fig. 11a. Destructive interference occurs for small ΔX/λ (i.e., diurnal tides), whereas constructive interference occurs for larger ΔX/λ (i.e., semidiurnal tides) and larger . Similarly, the conversion in the double ridge case is strongly amplified for . However, there are also differences, which can be attributed to the beams that radiate from the multiple subridges and to the interference of the beams with the realistic bathymetry.

The maximum in amplification for occurs at ΔX/λ = 0.64 in the realistic model versus ΔX/λ = 0.5 in the knife edge model. In the realistic model, resonance does not occur at the separation distance of 0.5λ because the western subridge on the east ridge blocks downward radiating beams from the central subridge, prohibiting the formation of an attractor with the west ridge. However, at ΔX/λ = 0.64 and an attractor has formed between the eastern subridge of the west ridge and the western subridge on the east ridge. This attractor is indicated by the beams that intersect the black square in Fig. 12a. Note that the attractor only occurs if the west ridge is raised to the black square. The separation distance between these two subridges is ΔX/λ ≈ 0.5. Hence, an attractor is expected to form according to the knife-edge model in Fig. 11b. In this resonant case the lee waves and dissipation at the lower western subridge on the east ridge are larger than at the taller central subridge. In contrast, the central subridge of the single east ridge case has the largest dissipation.

In agreement with the knife edge model, constructive interference due to the upward radiating beams in the realistic model occurs near the dashed line for in Fig. 13a. However, no constructive interference occurs due to superposing downward radiating beams near the dotted line for . An important reason for this is that the destructive interference of the multiple upward beams is larger than the constructive interference of the fewer downward radiating beams from the west and east ridges for small ΔX/λ. Moreover, the crest of the eastern subridge on the west ridge is sheltered by its lower slope from downward beams from the east ridge for small ΔX/λ (Fig. 12a).

The amplification of the flux divergence in Fig. 13b is weaker than that of the conversion. The ΨΔF has a maximum near and ΔX/λ = 0.64 and decreases for taller west ridges. This decrease is offset by an increase in the amplification of the dissipation in Fig. 13c. When , more of the converted barotropic energy is trapped between the ridges and dissipated, and less is allowed to radiate away in the double ridge case, compared to the sum of the single ridge cases. Hence, the ratio of dissipation over conversion q approaches 40% in the double ridge case.

5. Conclusions

Luzon Strait between Taiwan and the Philippines features two parallel, north–south oriented west and east ridges. The barotropic tides that propagate zonally over these ridges cause strong dissipation and internal waves. We have applied the hydrostatic MITgcm in a two-dimensional (x, z) quasi-realistic configuration to a zonal transect at 20.6°N to study the effect of the double ridge on the barotropic-to-baroclinic energy conversion, flux divergence, and turbulent dissipation. The model is integrated for a maximum duration of a spring–neap cycle (21 August–6 September 2010).

The spatial structure and timing of the modeled velocities, isotherm oscillations, and the dissipation are similar to the observations at the west ridge by Alford et al. (2011). These model runs thus suggest that the observed dissipation at this location is due to nonlinear internal hydraulic jumps or high-mode turbulent lee waves. The dissipation in the model is concentrated at the steep subridges where the lee waves occur. We attribute the differences in magnitude between the modeled and observed velocities and dissipation to three-dimensional effects.

When semidiurnal tides dominate, the predicted conversion, flux divergence, and dissipation integrated over both ridges are larger than for the single east ridge and single west ridge cases combined. The stronger dissipation at the ridges is attributed to larger ridge-top velocities that cause larger lee waves in the double ridge case. The amplification of the dissipation in the double ridge case is stronger than that of the conversion and divergence because dissipation scales with u3, whereas conversion and divergence scale with u2. When diurnal tides dominate, there is no such amplification. The diurnal and semidiurnal spring–neap cycles are out of phase for the model period in 2010 (Fig. 10b). Consequently, the effect of the tidal frequency on the amplification of the energy terms can easily be distinguished.

The inviscid double-ridge knife edge model of KBLP is applied to better understand the internal wave interference. The wave fields constructively interfere when beams from both ridges superpose after an odd number of reflections. Resonance or the trapping of energy along attractors occurs when the ridge separation distance is half of a first-mode wavelength and the sum of the WKB-scaled ridge heights is larger than the water depth. The constructive interference causes larger beam velocities and conversion.

In the MITgcm, the realistic bathymetry with its multiple subridges affects this interference. At 20.6°N, the ridge separation distance is close to half of a semidiurnal first-mode wavelength, but the ridges are not tall enough to allow for resonance. Upward semidiurnal beams from the tallest subridges on the west and east ridges almost superpose after one surface bounce, causing constructive interference. Because the diurnal beam slopes are flatter, there is little constructive interference. These findings are in agreement with studies by Jan et al. (2008), Farmer et al. (2009), Buijsman et al. (2010b), Echeverri and Peacock (2010), and Alford et al. (2011). However, this study provides a more detailed explanation of the interference mechanisms.

The importance of the beams for constructive interference is shown in a sensitivity analysis in which the west-ridge height and separation distance are varied. For semidiurnal tides and a taller west ridge, the double ridge leads to greater constructive interference, causing stronger conversion and dissipation. Future research needs to determine if the taller west ridges to the north of 20.6°N are, indeed, more likely to lead to stronger constructive interference and greater dissipation.

The west ridge changes little in the meridional direction over several kilometers at the model transect. This allows for the formation of a robust lee wave in the model, in agreement with the observations. However, the east ridge has a more three-dimensional bathymetry that may affect lee wave formation. Preliminary three-dimensional model runs show that the flow is channeled through ridge gaps, rather than over the subridges. Moreover, remote waves from generation sites to the north and south of the two-dimensional transect may also affect the local conversion and dissipation. These three-dimensional effects will be discussed in a subsequent paper.

Acknowledgments

We thank the Office of Naval Research for funding this research under ONRDC32025354. We are grateful to Matthew Alford, Jennifer MacKinnon, Jonathan Nash, and Harper Simmons for sharing the N2 station data. Moreover, Harper Simmons and Steve Ramp are acknowledged for sharing the multibeam data.

APPENDIX

Hydrostatic versus Nonhydrostatic

The sensitivity of the dissipation to hydrostaticity and resolution is explored. The MITgcm is run with a 1000-m-tall Gaussian ridge with a standard deviation of 5 km in a water depth of 2000 m. The buoyancy frequency is constant with depth (N0 = 5.2 × 10−3 s−1). Hence, the ridge crest is supercritical. We use a single-ridge configuration to allow for a faster computation compared to the realistic double ridge case. Various hydrostatic and nonhydrostatic simulations are run, with a semidiurnal barotropic forcing with an amplitude of 0.1, 0.2, and 0.3 m s−1 at the ridge and horizontal grid sizes of 30 and 100 m. In these cases the subgrid-scale scheme of Klymak and Legg (2010) is used to compute the vertical viscosity. In two additional hydrostatic and nonhydrostatic simulations the vertical viscosity is kept constant in time and space (Az = 10−5 m2 s−1). The vertical grid size and horizontal viscosity are constant in all cases (Δz = 10 m and Ah = 10−4 m2 s−1).

Similar to Klymak and Legg, we compute the terms in the total energy balance and integrate them over depth and the width of the Gaussian ridge (see their paper for details). From the tendency term and the flux divergence we compute the diagnosed dissipation 〈Dv,E〉 and compare this with the reported dissipation computed with the mixing scheme 〈Dv〉, which is similar to (4), but with the total velocities u. The diagnosed dissipation is similar for each tidal forcing regardless of the hydrostaticity, resolution, and vertical viscosity (Fig. A1). The reason for this is that the energy drop in the model is set by the need to dissipate energy from the supercritical part of the flow at the ridge to the subcritical reservoir downstream, which energy levels are set entirely inviscidly (Klymak et al. 2010b).

Fig. A1.
Fig. A1.

The diagnosed dissipation computed from the total energy balance 〈Dv,E〉 vs the reported dissipation computed from the mixing scheme 〈Dv〉 for various cases in hydrostatic (H, gray markers) and nonhydrostatic (NH, black markers) mode for a tidal barotropic flow with an amplitude of 0.1, 0.2, and 0.3 m s−1 at the ridge, with a horizontal grid size of 30 and 100 m, and for two cases with a constant viscosity of Az = 10−5 m2 s−1; that is, the vertical mixing scheme is not used (NM). The units are not shown in the legend.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

Conversely, not all simulations report the same dissipation. The simulations with a constant viscosity grossly underreport the dissipation, indicating that substantial dissipation is achieved numerically. In agreement with Klymak and Legg (2010), the differences between the hydrostatic and nonhydrostatic cases with Δx = 100 m and the same forcing are small. The reported and diagnosed dissipation in Fig. A1 are similar, as well as the spatial structure of the lee waves in Figs. A2a,b. When Δx is of the same order of magnitude as the buoyancy length scale, which is identical to the vertical lee wavelength scale , stratified turbulent motions with smaller scales are not well resolved in nonhydrostatic simulations (Waite 2011). This is the case in our nonhydrostatic simulations with Δx = 100 m. Nonhydrostatic effects become more important for Δx = 30 m. Moreover, the grid aspect ratio is closer to unity, which is preferred when modeling isotropic turbulence. The overturns and the oscillating tail of the lee wave in these nonhydrostatic high-resolution cases are better defined than in the hydrostatic and low-resolution cases (Figs. A2a–d). However, the nonhydrostatic high-resolution simulations overreport the dissipation, though by a relatively modest factor of 3 (Figs. A1, A2f). This is because the prescribed viscosity is valid for smoothing shear out over the outer scale of the turbulence, which is the scale of the lee wave. However, the nonhydrostatic high-resolution simulations also resolve turbulent shear explicitly, and thus the rate of dissipation is exaggerated. If one is interested in the detailed spatial structure and dissipation of breaking internal waves, direct numerical simulations (DNS) or large-eddy simulations (LES) are recommended (Gayen and Sarkar 2011).

Fig. A2.
Fig. A2.

Snapshots of dissipation at the Gaussian ridge during eastward barotropic flow at yearday 233.77 for (a) the hydrostatic (hyd) case with Δx = 100 m, (b) the nonhydrostatic (nonhyd) case with Δx = 100 m, (c) the hydrostatic case with Δx = 30 m, and (d) the nonhydrostatic case with Δx = 30 m. The density of an ocean at rest is contoured in vertical increments of 50 m. (e) The barotropic flow at the ridge crest is the same for the four cases. (f) The dissipation for the four cases, volume integrated over the ridge. The circles indicate the time of the snapshots.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0210.1

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