• Alford, M. H., J. A. MacKinnon, Z. Zhao, R. Pinkel, J. Klymak, and T. Peacock, 2007: Internal waves across the Pacific. Geophys. Res. Lett., 34, L24601, https://doi.org/10.1029/2007GL031566.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Apel, J. R., L. A. Ostrovsky, Y. A. Stepanyants, and J. F. Lynch, 2007: Internal solitons in the ocean and their effect on underwater sound. J. Acoust. Soc. Amer., 121, 695722, https://doi.org/10.1121/1.2395914.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buijsman, M. C., Y. Kanarska, and J. C. McWilliams, 2010: On the generation and evolution of nonlinear internal waves in the South China Sea. J. Geophys. Res., 115, C02012, https://doi.org/10.1029/2009JC005275.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buijsman, M. C., S. Legg, and J. Klymak, 2012: Double-ridge internal tide interference and its effect on dissipation in Luzon Strait. J. Phys. Oceanogr., 42, 13371356, https://doi.org/10.1175/JPO-D-11-0210.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Carter, G. S., O. B. Fringer, and E. D. Zaron, 2012: Regional models of internal tides. Oceanography, 25 (2), 5665, https://doi.org/10.5670/oceanog.2012.42.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Egbert, G. D., and S. Y. Erofeeva, 2002: Efficient inverse modeling of barotropic ocean tides. J. Atmos. Oceanic Technol., 19, 183204, https://doi.org/10.1175/1520-0426(2002)019<0183:EIMOBO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garrett, C., and E. Kunze, 2007: Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech., 39, 5787, https://doi.org/10.1146/annurev.fluid.39.050905.110227.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gerkema, T., and J. T. F. Zimmerman, 2008: An introduction to internal waves. Lecture Notes, Royal NIOZ, 207 pp.

  • Hall, R. A., and G. S. Carter, 2011: Internal tides in Monterey submarine canyon. J. Phys. Oceanogr., 41, 186204, https://doi.org/10.1175/2010JPO4471.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1984: On the semidiurnal internal tide at a shelf-break region on the Australian North West Shelf. J. Phys. Oceanogr., 14, 17871799, https://doi.org/10.1175/1520-0485(1984)014<1787:OTSITA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1985: A comparison of semidiurnal internal tides from different bathymetric locations on the Australia North West Shelf. J. Phys. Oceanogr., 15, 240251, https://doi.org/10.1175/1520-0485(1985)015<0240:ACOSIT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1988: Physical oceanography of the Exmouth Plateau region, North-western Australia. Mar. Freshwater Res., 39, 589606, https://doi.org/10.1071/MF9880589.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1996: A numerical model of internal tides with application to the Australian North West Shelf. J. Phys. Oceanogr., 26, 2137, https://doi.org/10.1175/1520-0485(1996)026<0021:ANMOIT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., E. Pelinovsky, T. Talipova, and B. Barnes, 1997: A nonlinear model of internal tide transformation on the Australian North West Shelf. J. Phys. Oceanogr., 27, 871896, https://doi.org/10.1175/1520-0485(1997)027<0871:ANMOIT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., P. G. Chatwin, and P. Craig, 2001: Internal tide observations from the Australian North West Shelf in summer 1995. J. Phys. Oceanogr., 31, 11821199, https://doi.org/10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Horn, D. A., J. Imberger, and G. N. Ivey, 2001: The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech., 434, 181207, https://doi.org/10.1017/S0022112001003536.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jones, N. L., and G. N. Ivey, 2017: Internal waves. Encyclopedia of Maritime and Offshore Engineering, J. Carlton, P. Jukes, and Y. S. Choo, Eds., Wiley, https://doi.org/10.1002/9781118476406.emoe089.

    • Crossref
    • Export Citation
  • Kelly, S. M., and J. D. Nash, 2010: Internal-tide generation and destruction by shoaling internal tides. Geophys. Res. Lett., 37, L23611, https://doi.org/10.1029/2010GL045598.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kelly, S. M., N. L. Jones, and J. D. Nash, 2013: A coupled model for Laplace’s tidal equations in a fluid with one horizontal dimension and variable depth. J. Phys. Oceanogr., 43, 17801797, https://doi.org/10.1175/JPO-D-12-0147.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kerry, C. G., B. S. Powell, and G. S. Carter, 2013: Effects of remote generation sites on model estimates of M2 internal tides in the Philippine Sea. J. Phys. Oceanogr., 43, 187204, https://doi.org/10.1175/JPO-D-12-081.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., M. H. Alford, R. Pinkel, R. C. Lien, Y. J. Yang, and T. Y. Tang, 2011: The breaking and scattering of the internal tide on a continental slope. J. Phys. Oceanogr., 41, 926945, https://doi.org/10.1175/2010JPO4500.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., M. Buijsman, S. Legg, and R. Pinkel, 2013: Parameterizing surface and internal tide scattering and breaking on supercritical topography: The one-and two-ridge cases. J. Phys. Oceanogr., 43, 13801397, https://doi.org/10.1175/JPO-D-12-061.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., L. K. Rosenfeld, G. S. Carter, and M. C. Gregg, 2002: Internal waves in Monterey submarine canyon. J. Phys. Oceanogr., 32, 18901913, https://doi.org/10.1175/1520-0485(2002)032<1890:IWIMSC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kurapov, A. L., G. D. Egbert, J. S. Allen, R. N. Miller, S. Y. Erofeeva, and P. M. Kosro, 2003: The M2 internal tide off Oregon: Inferences from data assimilation. J. Phys. Oceanogr., 33, 17331757, https://doi.org/10.1175/2397.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Legg, S., and K. M. Huijts, 2006: Preliminary simulations of internal waves and mixing generated by finite amplitude tidal flow over isolated topography. Deep-Sea Res. II, 53, 140156, https://doi.org/10.1016/j.dsr2.2005.09.014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lerczak, J. A., C. D. Winant, and M. C. Hendershott, 2003: Observations of the semidiurnal internal tide on the southern California slope and shelf. J. Geophys. Res., 108, 3068, https://doi.org/10.1029/2001JC001128.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marshall, J., C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res., 102, 57335752, https://doi.org/10.1029/96JC02776.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nagai, T., and T. Hibiya, 2015: Internal tides and associated vertical mixing in the Indonesian Archipelago. J. Geophys. Res. Oceans, 120, 33733390, https://doi.org/10.1002/2014JC010592.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nash, J. D., E. Kunze, J. M. Toole, and R. W. Schmitt, 2004: Internal tide reflection and turbulent mixing on the continental slope. J. Phys. Oceanogr., 34, 11171134, https://doi.org/10.1175/1520-0485(2004)034<1117:ITRATM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nash, J. D., M. H. Alford, and E. Kunze, 2005: Estimating internal wave energy fluxes in the ocean. J. Atmos. Oceanic Technol., 22, 15511570, https://doi.org/10.1175/JTECH1784.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nash, J. D., S. M. Kelly, E. L. Shroyer, J. N. Moum, and T. F. Duda, 2012: The unpredictable nature of internal tides on continental shelves. J. Phys. Oceanogr., 42, 19812000, https://doi.org/10.1175/JPO-D-12-028.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Oke, P. R., and P. Sakov, 2008: Representation error of oceanic observations for data assimilation. J. Atmos. Oceanic Technol., 25, 10041017, https://doi.org/10.1175/2007JTECHO558.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ponte, A. L., and B. D. Cornuelle, 2013: Coastal numerical modelling of tides: Sensitivity to domain size and remotely generated internal tide. Ocean Modell., 62, 1726, https://doi.org/10.1016/j.ocemod.2012.11.007.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rayson, M. D., N. L. Jones, and G. N. Ivey, 2012: Temporal variability of the standing internal tide in the Browse Basin, Western Australia. J. Geophys. Res., 117, C06013, https://doi.org/10.1029/2011JC007523.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stewart, K. D., A. M. Hogg, S. M. Griffies, A. P. Heerdegen, M. L. Ward, P. Spence, and M. H. England, 2017: Vertical resolution of baroclinic modes in global ocean models. Ocean Modell., 113, 5065, https://doi.org/10.1016/j.ocemod.2017.03.012.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Van Gastel, P., G. N. Ivey, M. J. Meuleners, J. P. Antenucci, and O. Fringer, 2009: The variability of the large-amplitude internal wave field on the Australian North West Shelf. Cont. Shelf Res., 29, 13731383, https://doi.org/10.1016/j.csr.2009.02.006.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhao, Z., and M. H. Alford, 2009: New altimetric estimates of mode-1 M2 internal tides in the central North Pacific Ocean. J. Phys. Oceanogr., 39, 16691684, https://doi.org/10.1175/2009JPO3922.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zilberman, N. V., J. M. Becker, M. A. Merrifield, and G. S. Carter, 2009: Model estimates of M2 internal tide generation over Mid-Atlantic Ridge topography. J. Phys. Oceanogr., 39, 26352651, https://doi.org/10.1175/2008JPO4136.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zilberman, N. V., M. A. Merrifield, G. S. Carter, D. S. Luther, M. D. Levine, and T. J. Boyd, 2011: Incoherent nature of M2 internal tides at the Hawaiian Ridge. J. Phys. Oceanogr., 41, 20212036, https://doi.org/10.1175/JPO-D-10-05009.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • View in gallery

    Bathymetry for the model domain on the NWS (standard case). The black dashed box is the smaller model domain that does not include the western portion of the Exmouth Plateau. Triangles represent the locations of the six moorings used to validate the 3D model and calculate the tidal energy. The red dashed line indicates the slice used to investigate the negative conversion values.

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    (a),(b) The initial model temperature and salinity profiles. (c) The profile of buoyancy frequency. The initial stratification for the 2D simulations only extended to 1000 m (indicated by dashed line). (d) The profile of eigenfunction W (blue) and its vertical gradient (black) based on the initial stratification in the 2D simulations. Note that the y axis in (a)–(d) has a logarithmic scale. (e) A snapshot of the internal tide horizontal velocity field with the idealized 2D bottom topography following the hyperbolic tangent function at T = 20 h. For this example, the arrow represents the incoming baroclinic forcing with the amplitude of 4 cm s−1 on the left boundary. The colored contours over the slope represent the baroclinic velocities induced by the interaction between the barotropic tides (1 cm s−1 on the left and 5 cm s−1 on the right) with the bottom topography.

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    (a)–(f) Comparison of M2 barotropic current ellipses between the model predictions (black lines), observations (blue lines), and TPXO8-atlas data (red lines) at different moorings. The scale of the ellipse is shown on the right. (g),(h) Scatterplots comparing the M2 baroclinic tidal amplitude and phase between 3D model predictions and observations at six mooring sites.

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    (a) Spring tide cycle–averaged conversion rates in standard case with the Exmouth Plateau. Vectors mean the depth-integrated tidal-averaged energy flux of baroclinic tides. Three red boxes represent the main generation sites of internal tides. Triangles represent five moorings. (b) The case excluding the western portion of the Exmouth Plateau. (c) Conversion rate difference CExP between the standard case and model run excluding the Exmouth Plateau.

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    Semidiurnal coherent conversion at the mooring sites (a) NICOP, (c) PIL200, (e) PIL100, (g) N3C, and (i) N2C during spring tides. Bottom pressure perturbation (black) and vertical barotropic velocity (blue) at the moorings (b) NICOP, (d) PIL200, (f) PIL100, (h) N3C, and (j) N2C. Solid and dotted lines indicate observed and modeled results, respectively. Note that the magnitude of the y axis in (g) and (i) is one order smaller than in (a), (c), and (e).

  • View in gallery

    (a) Tidal-averaged conversion rates and (b) cosine function of the phase difference between the bottom pressure perturbation and vertical barotropic velocity along the cross section for the 3D case (black lines, location shown in Fig. 1) and the 2D case (blue lines) with single M2 tidal forcing. (c) Bottom topography (black solid line) and topographic critical parameter γ (dashed line).

  • View in gallery

    (a) Snapshots of pr for cases with 4 cm s−1 velocity (Fr = 0.017) on the left boundary at times from 0 to 100 h. The black dashed line is the edge of continental shelf. Blue lines indicate the kinematic prediction of pr. (b) The black solid line represents numerically predicted phase speed while the blue dashed line is theoretical linear phase speed. (c) The bar graph shows the propagation time of numerically predicted RITs across a single grid point (100 m); the blue dotted line is the accumulated propagation time along the model domain. (d) The cosine function of numerically predicted θbc,r [refer to Eq. (3)] is shown with the black line, while the cosine function of theoretical θth is the blue line. (e)–(h) As in (a)–(d), but for the cases with initial velocity amplitudes of 20 cm s−1 (Fr = 0.084).

  • View in gallery

    The 4 cm s−1 baroclinic tide amplitude case with baroclinic tide phase θp,r ranging from 0.25π to 2.00π at an interval of 0.25π. (a)–(h) Cycle-averaged Cr over the slope for different θp,r. Black and blue lines represent model estimates and kinematic predictions, respectively.

  • View in gallery

    Parameter Cr at the location x = 405 km with the baroclinic forcing phase ranging from 0 to 2.00π and amplitude ranging from 2 to 12 cm s−1. Markers and lines are numerical modeling and kinematic predictions, respectively. The two x axes are the phase of the RITs (θp,r and θ), while the two y axes are Cr and nondimensional Cr scaled by the local conversion rate Cl in the case of only barotropic tidal forcing.

  • View in gallery

    (a) Bottom topography over the shelf (L is the shelf length). Blue triangles represent five locations shown in panels below. (b) Nondimensional Cr at location x/L = 0.45 as a function of the RIT wave Froude number. Dots and dashed lines represent the numerical model and kinematic predictions, respectively. Different colors represent different phases of the RITs. (d),(f),(h),(j) As in (b), but at locations x/L = 0.5, 0.55, 0.6, and 0.65. (c) Gray shading is the deviation of Cr between the numeric model and kinematic predictions nondimensionalized by local conversion rate Cl for different phases. Black lines represent phase-averaged error. Dotted rectangles indicate the small Froude number zone (Fr < 0.05). (e),(g),(i),(k) As in (c), but at locations x/L = 0.5, 0.55, 0.6, and 0.65.

  • View in gallery

    (a) Bottom topography over the shelf. (b) Histograms are the horizontal distribution of Cr over the shelf in the case with low Froude number (Fr = 0.017). Magenta, cyan, yellow, and gray colors represent cases with the remote phase θp,r of 0.50π, 1.00π, 1.50π, and 2.00π, respectively. (c) As in (b), but for high Froude number (Fr = 0.169).

  • View in gallery

    (a) A snapshots of pr for the case with RIT amplitude of 4 cm s−1 (Fr = 0.017) at the time of t = 96 h. Slopes with topographic critical parameter γ of 0.5, 1.0, and 2.0 are shown in the upper, middle, and bottom panels. Blue dotted lines indicate arrival locations and amplitudes of RITs over the slope. (b) As in (a), but for 20 cm s−1 RIT amplitude (Fr = 0.084). (c) Bottom topography with subcritical, critical, and supercritical slopes.

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The Effects of Remote Internal Tides on Continental Slope Internal Tide Generation

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  • 1 Oceans Graduate School and Oceans Institute, University of Western Australia, Perth, Western Australia, Australia
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Abstract

Internal tide generation at sloping topography is nominally determined by the local slope geometry, density stratification, and tidal forcing. Recent global ocean models have revealed that remotely generated internal tides (RITs) can also influence locally generated internal tides (LITs). Field measurements with through-the-water column moorings on the southern portion of the Australian North West Shelf (NWS) suggested that RITs led to local regions with either positive or negative barotropic to baroclinic energy conversion. Three-dimensional numerical simulations were used to examine the role of RITs on local internal tide climatology on the inner slope and shelf portion of the NWS. The model demonstrated the principle remote generation site was the western portion of the offshore Exmouth Plateau. Extending the model domain to include this offshore plateau region increased the local net energy conversion on the inner shelf by 13.5% and on the slope by 8%. Simulations using an idealized 2D model configuration aligned along the principal direction of RIT propagation demonstrated that the sign and magnitude of the local energy conversion was dependent on the distance between the remote and local generation sites, the phase difference between the local barotropic tide and the RIT, and the amplitude of both the local barotropic tide and the RIT. For RITs with a low-wave Froude number (Fr < 0.05), where Fr is the ratio of the internal wave baroclinic velocity to the linear wave speed, the conversion rates were consistent with kinematic predictions based on the phase difference only. For stronger flows with Fr > 0.05, the conversion rates showed a nonlinear dependence on Fr.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yankun Gong, yankun.gong@research.uwa.edu.au

Abstract

Internal tide generation at sloping topography is nominally determined by the local slope geometry, density stratification, and tidal forcing. Recent global ocean models have revealed that remotely generated internal tides (RITs) can also influence locally generated internal tides (LITs). Field measurements with through-the-water column moorings on the southern portion of the Australian North West Shelf (NWS) suggested that RITs led to local regions with either positive or negative barotropic to baroclinic energy conversion. Three-dimensional numerical simulations were used to examine the role of RITs on local internal tide climatology on the inner slope and shelf portion of the NWS. The model demonstrated the principle remote generation site was the western portion of the offshore Exmouth Plateau. Extending the model domain to include this offshore plateau region increased the local net energy conversion on the inner shelf by 13.5% and on the slope by 8%. Simulations using an idealized 2D model configuration aligned along the principal direction of RIT propagation demonstrated that the sign and magnitude of the local energy conversion was dependent on the distance between the remote and local generation sites, the phase difference between the local barotropic tide and the RIT, and the amplitude of both the local barotropic tide and the RIT. For RITs with a low-wave Froude number (Fr < 0.05), where Fr is the ratio of the internal wave baroclinic velocity to the linear wave speed, the conversion rates were consistent with kinematic predictions based on the phase difference only. For stronger flows with Fr > 0.05, the conversion rates showed a nonlinear dependence on Fr.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yankun Gong, yankun.gong@research.uwa.edu.au

1. Introduction

The interaction of the oscillatory surface (i.e., barotropic) tide with sloping topography in the density-stratified interior of the ocean produces internal (i.e., baroclinic) tides (e.g., Garrett and Kunze 2007). The resulting baroclinic perturbations can then propagate for large distances (over 1000 km) from the generation site (Zhao and Alford 2009) with typically minimal energy loss (Alford et al. 2007; Kerry et al. 2013). Remotely generated internal tides (hereinafter RITs) from multiple far-field locations can thus interact with locally generated internal tides (hereinafter LITs), resulting in a complicated internal tide climatology. Here we explore the interaction between RITs and LITs.

The barotropic-to-baroclinic energy conversion rate C and the baroclinic energy flux Fbc quantify the generation and propagation characteristics of internal tides, respectively. The conversion rate is given by C=(pwbt)|z=H (Kurapov et al. 2003; Kelly and Nash 2010). Here, p and wbt represent the baroclinic pressure perturbation and the vertical component of the barotropic velocity, respectively. The depth-integrated baroclinic energy flux is defined as Fbc=H0(pubc)dz, in which ubc represents the horizontal baroclinic velocity (note that bold variables indicate a vector quantity throughout the paper).

Positive conversion rates are common in regional numerical models, representing conversion from barotropic to baroclinic tides. As well as regions of positive conversion rate, regions with negative values of conversion rate have been found in many regional models (e.g., Nagai and Hibiya 2015; Kerry et al. 2013; Ponte and Cornuelle 2013; Zilberman et al. 2009). Negative conversion rates occur when the phase difference between the local pressure perturbation and the barotropic vertical velocity is larger than 90° on the bottom, and this occurs when RITs make a significant contribution to the local conditions (Kelly and Nash 2010). While negative conversion rates arise in numerical models, they have rarely been observed in field measurements. In the present work, we examine the occurrence of negative conversion rates via both observational data, at mooring stations on the southern portion of the Australian North West Shelf (NWS), as well as with numerical modeling.

The influence of RIT signals is often seen in continental shelf observations (Lerczak et al. 2003; Nash et al. 2004), producing changes in the local internal tide field (Kelly and Nash 2010). To investigate the effect of RITs, Kerry et al. (2013) compared results from three numerical simulations, each with different domain sizes, and concluded that in the absence of RITs, domain-integrated conversion rates were 11% greater at the Luzon Strait and 65% greater at the Mariana Arc. Based on a two-dimensional (2D) Massachusetts Institute of Technology General Circulation Model (MITgcm), Buijsman et al. (2012) and Klymak et al. (2013) both demonstrated the conversion rates varied with the distance between two adjacent knife ridges. However, the contribution of both the remote and local tidal characteristics, and the distance between generation sites, on local conversion rates over the slope and shelf has not yet been quantitatively examined. In energetic shelf locations (e.g., the NWS), this process can become further complicated by the evolution of large-amplitude linear RITs into nonlinear internal waves (NLIWs) as they move onto the shelf (Holloway et al. 1997), significantly influencing the local internal tide climatology.

There are two main goals in this paper: 1) quantifying the baroclinic energy distribution and examining the spatial distribution of negative conversion rates on the southern NWS and 2) examining the effects of varying phase and amplitude of the RITs on the local internal tide climatology. The structure of this paper is as follows: in section 2 the relevant theoretical framework of internal tides is introduced in detail; in section 3 we present our study site, field mooring stations, model configurations, and data analysis methodology; and in section 4 we describe the internal tidal energetics, the phase difference mechanism leading to negative conversion rates, and the linear effects of the RITs. In section 5 we discuss the effects of nonlinearity on the RITs and the steepening processes. Finally, in section 6 we summarize the key results from this work.

2. Theoretical framework

The hydrostatic perturbation pressure p associated with an internal tide-induced vertical isopycnal displacement η can be calculated for a particular stratification profile N(z) from
0=1ρ0pz+N2η.
Integrating Eq. (1) with depth, and satisfying the condition of zero depth-average pressure perturbation, Kunze et al. (2002) showed
p(z)=ρ0[z0N2(z)η(z)dz1HH0z0N2(z)η(z)dzdz].
As p has both local (pl) and remote contributions (pr), the total conversion rate C can be decomposed into a local conversion rate Cl and a remotely influenced conversion rate Cr. As the phase difference between pl and wbt is close to zero, Cl will always be positive, but Cr can be either positive or negative.
To derive an expression for Cr, we assume a remote generation site and a local generation site separated by a distance L, and consider the barotropic tide at a single frequency ω. Since the RIT takes time Tp to arrive at the local generation site, the resulting pressure perturbation
pr=Pr|x=Lcos(ωtθbc,r),θbc,r=(θp,r+TpTbt2π).
Here θp,r is the phase of RITs at the remote generation site; Tbt is the barotropic tide period and the propagation time is Tp=L/cp, where cp is the mode-1 linear phase speed. Parameter Pr|x=L is the amplitude of the remotely influenced pressure perturbation at the local generation site. The vertical barotropic velocity is wbt=(UbtH)cos(ωtθbt,l), in which Ubt is the amplitude of the horizontal barotropic velocity and H is the topographic slope. Therefore, the tidal cycle-average Cr is
Cr=1T0T(prwbt)dt=12Pr|x=L(UbtH)cos(θ),
in which θ=(θbc,rθbt,l) is the phase contribution of the RIT. The pressure amplitude is related to the horizontal velocity amplitude P=ρ0cU, in which c is the group speed for the baroclinic tides (Kelly et al. 2013). In addition, U can be obtained from eigenfunctions Wn(z) by solving the Sturm–Liouville equation (e.g., Apel et al. 2007) based on the hydrostatic approximation
d2Wn(z)dz2+N(z)2cp2Wn(z)=0,Wn(0)=Wn(H)=0.
The linear phase speed cp=ω/kh, where kh is the horizontal wavenumber. If the buoyancy frequency N is independent of z, Eq. (5) then has the normalized analytical solution
W(z)=sin(Nzcp).
Based on the continuity equation, the horizontal baroclinic velocity
ubc=Ucos(Nzcp)sin(khxωt),
in which the amplitude of the seabed horizontal velocity U is proportional to N/ω. Thus, the remotely influenced pressure perturbation amplitude at the local generation site is given by
Pr|x=L=δPr|x=R,
where the coefficient δ=(clNl)/(crNr). The kinematic prediction of Cr in Eq. (4) can thus be rewritten as
Cr=δ2Pr|x=R(UbtH)cos(θ).
We can see that Cr depends on both the amplitudes and phase differences of RITs and the local barotropic tides. We compare the kinematic-predicted Cr to numerical model solutions to quantitatively estimate the role of RITs on local energy conversion below.

3. Methods

a. Study site

The study site is located offshore of the Pilbara region on the southern portion of the NWS (Fig. 1). This is a region with a ubiquitous and energetic internal tide, generated by the interaction between the strong semidiurnal surface tides and sloping bottom topography over the continental slope and shelf (Holloway 1984, 1985; Van Gastel et al. 2009; Jones and Ivey 2017). Exmouth Plateau is an elongated plateau in water depths of a 1000 m or more located offshore of the Pilbara region, which has steep slopes on the western portion (Fig. 1).

Fig. 1.
Fig. 1.

Bathymetry for the model domain on the NWS (standard case). The black dashed box is the smaller model domain that does not include the western portion of the Exmouth Plateau. Triangles represent the locations of the six moorings used to validate the 3D model and calculate the tidal energy. The red dashed line indicates the slice used to investigate the negative conversion values.

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

Previous studies (Holloway et al. 2001; Van Gastel et al. 2009) modeled the internal wave field of the southern NWS with a limited spatial model domain, excluding the Exmouth Plateau, and concluded that the continental slope and inshore shelf with water depths of 500 m or less were the main generation sites. However, the western portion of the Exmouth Plateau also has critical or near critical slope conditions, thereby the region is also likely an important generator of internal tides (as demonstrated in section 4b). The southern NWS is therefore an ideal site to examine the influence of RITs on LITs.

b. Field observations

Moorings were deployed in the Pilbara region from 2011 to 2012 to quantify the regional internal tide dynamics on the southern NWS. Four of these moorings and additional observational data from the PIL200 and PIL100 moorings of the Integrated Marine Observing System (IMOS) were used to determine the regional internal tide characteristics (see blue triangles in Fig. 1). Coincident through-water-column profiles of velocity and density were available at five of these moorings. Note that N4B was only used for model validation of the currents.

All of the moorings were subsurface taut line moorings with buoyancy pulling upward on a wire against an anchor to suspend sensors at selected depth intervals over the whole water column. Current magnitude and direction were recorded with upward looking acoustic Doppler current profilers (ADCPs; Teledyne RDI), and density was measured with a variety of instruments (Sea-Bird Electronics SBE37 and SBE39 and VEMCO Minilog II-T) at different vertical resolutions on each mooring (see details in Table 1). Tidal energy conversion rates and the baroclinic energy fluxes were calculated from the density and velocity measurements to determine the generation and propagation processes of the internal tides. According to the Nash et al. (2005) criteria, our sample rates were high enough and we had sufficient coverage of sensors near-surface and near-bottom to guarantee the accuracy of the tidal energy calculations.

Table 1.

The position, vertical layout, and sampling regime of the six moorings (PIL200, NICOP, PIL100, N2C, N3C, and N4B) in this study. All ADCPs were upward looking. Note that no thermistors were deployed at the mooring N4B. ASB = above seabed.

Table 1.

c. Numerical simulations

A three-dimensional (3D) hydrostatic numerical model, the MITgcm (Marshall et al. 1997), was used to calculate the internal tide climatology in the Pilbara region. In addition, we used the model in nonhydrostatic mode for 2D vertical slice experiments with idealized tidal forcing and initial conditions to investigate the role of RITs on LITs, in particular the role of incoming wave phase and amplitude on the shelf barotropic–baroclinic energy conversion.

1) 3D model configuration

Three-dimensional experiments were implemented to compute the internal tide climatology in the Pilbara region. We used Geoscience Australia bathymetry data with a horizontal resolution of 250 m. The horizontal grid spacing was 1000 m in both the longitudinal and latitudinal directions. The standard model domain consisted of 888 × 630 grid cells. To satisfy the mode-1 to mode-3 vertical resolution requirements, model vertical layers were spaced in accordance with the hyperbolic tangent function (Stewart et al. 2017) given by
Δz(n)=Δzmaxtanh(nπshHmax)+Δzmin,
where Δz is the thickness of each layer; n is the layer number (138 layers); Δzmin and Δzmax were set as 1 and 100 m, respectively. In the Pilbara region, the maximum depth Hmax was 6000 m and sh=1.0 is a dimensionless scaling parameter for adjusting the steepness of the hyperbolic tangent function. The initial model stratification was derived from the Ocean Forecasting Australia Model (OFAM; Oke and Sakov 2008) by temporally and spatially averaging output from April 2012, resulting in horizontally uniform temperature and salinity initial conditions (Figs. 2a–c).
Fig. 2.
Fig. 2.

(a),(b) The initial model temperature and salinity profiles. (c) The profile of buoyancy frequency. The initial stratification for the 2D simulations only extended to 1000 m (indicated by dashed line). (d) The profile of eigenfunction W (blue) and its vertical gradient (black) based on the initial stratification in the 2D simulations. Note that the y axis in (a)–(d) has a logarithmic scale. (e) A snapshot of the internal tide horizontal velocity field with the idealized 2D bottom topography following the hyperbolic tangent function at T = 20 h. For this example, the arrow represents the incoming baroclinic forcing with the amplitude of 4 cm s−1 on the left boundary. The colored contours over the slope represent the baroclinic velocities induced by the interaction between the barotropic tides (1 cm s−1 on the left and 5 cm s−1 on the right) with the bottom topography.

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

As the semidiurnal barotropic tides are dominant in the Pilbara region, the model was driven by M2 and S2 tides on the boundaries with values taken from the Oregon State University TOPEX/Poseidon Solution (TPXO8-atlas data) with 1/30° resolution (Egbert and Erofeeva 2002). A 50-km-wide sponge layer was imposed on each lateral boundary to absorb internal tides and avoid reflection back to the inner region. We ran a 35-day simulation covering a complete spring and neap tide cycle. Quasi-steady conditions occurred after 10 days, so the model results were analyzed over the remaining 25 days. We applied constant horizontal and vertical eddy viscosity and diffusivity coefficients as Ah = 10 m2 s−1; Aυ = 10−4 m2 s−1; Kh = 10 m2 s−1; Kυ = 10−5 m2 s−1 to eliminate gridscale instability (Legg and Huijts 2006; Nagai and Hibiya 2015) and parameterized the bottom stress using a quadratic law with Cd = 2.5 × 10−3.

To evaluate the influence of the western portion of the Exmouth Plateau on the local internal tide climatology on the inner shelf, we undertook two numerical simulations with different computational domains (Fig. 1). The smaller domain did not include the western portion of the Exmouth Plateau. We used the same initial stratification, boundary forcing, viscosity and diffusivity coefficients, and spatial resolution in both model runs.

2) 2D model configuration

The 2D MITgcm experiments were employed to examine the effect of varying amplitude and phase of the RITs on the local energy conversion over the slope and shelf. A horizontal grid scale of 100 m and a domain of 600 km were used for the 2D experiments. The vertical layers were also set following the hyperbolic tangent function profile, with layer thicknesses ranging from 1 m near the surface to 20 m near the seabed (maximum 1000 m). An idealized hyperbolic tangent bottom topography (Fig. 2e) was used:
h(x)=12(H0Hs)tanh(xxsS)12(H0+Hs),
where H0 is the maximum water depth and xs and Hs are the location and water depth on the shelf, respectively. The shelf width S was selected to ensure a critical condition existed at one point on the slope. Values of other 2D model parameters are summarized in Table 2. The 2D model was driven by a barotropic tide on both lateral boundaries, and the baroclinic RIT was imposed at the left boundary and given by
Uleft(z)=Ubc(z)cos(ωs2tθbc)+Ubt,leftcos(ωs2tθbt),Uright=Ubt,rightcos(ωs2tθbt),
Table 2.

Parameters in the 2D nonhydrostatic MITgcm experiments.

Table 2.

The amplitudes of the barotropic velocities (Ubt,left, Ubt,right) were set at the boundaries and the phases were the same value to ensure mass conservation in the domain. A 20-km-wide sponge boundary condition was imposed at both lateral boundaries. We varied the RIT amplitude Ubc(z) and phase θbc in a series of idealized 2D runs (BT-BC Exps and High-BC Exps, Table 3). For comparison, we completed a case with no RIT (BT-only Exp) and another with a more accurate representation of the across-shore topography (Real Exp, along red dashed line in Fig. 1).

Table 3.

Barotropic and baroclinic tidal forcing and bottom topography for the 2D model runs.

Table 3.

d. Data analysis

1) Modal decomposition

We decomposed the velocity and buoyancy fields of the RITs into different vertical modes (Gerkema and Zimmerman 2008). In terms of one tidal constituent (ω) in a continuously stratified ocean, the baroclinic wavefield can be written (e.g., Gerkema and Zimmerman 2008; Buijsman et al. 2010) as the sum of vertical modes (e.g., vertical velocity w):
w(z,t)=n=1Anexp(iωt)Wn(z).
For mode n, An and Wn are the vertical amplitude and eigenfunction, respectively. The eigenfunctions were obtained by numerically solving the normal mode equation [Eq. (5)]. We calculated W1 to define the vertical structure of the RITs (Fig. 2d). Following Buijsman et al. (2010), we normalized the eigenfunctions for vertical velocity and obtained the eigenfunctions for the horizontal velocity and buoyancy as
Un(z)=AnikndWn(z)dz,Bn(z)=AniN2(z)ωWn(z).
Thus, the vertical profile of the RITs was constructed based on the initial stratification.

2) Temporal decomposition

Surface tides are highly predictable using the sum of known harmonics (sinusoidal function). Using the same model, internal tides can be divided into two components as well. The first can be interpreted as a series of sinusoidal constituents at the tidal frequencies, which are termed the “coherent constituents” (Nash et al. 2012). For the second component, the amplitudes and phases of the motions can be modulated in time. As this portion of the internal tide behavior is unpredictable via harmonic analysis, these are termed “incoherent constituents.”

Since the Pilbara region is dominated by semidiurnal surface tides, the coherent internal tides showed high spectral peaks near the M2 and S2 potentials (not shown). We extracted the contributions to tidal variables, phase locked to the surface tide, in the following way. Three-dimensional model and observed data were first bandpass filtered with the semidiurnal cutoff period (10–14 h). We fit tidal variables (e.g., ubc and p) to the M2 and S2 constituents using a least squares minimization technique. The raw time series was separated into the coherent component, and the remaining portion defined as the incoherent component. Note that since the 2D idealized cases were driven by S2 barotropic and/or baroclinic tides, we omitted the filtering calculations and assumed the 2D model outputs were all coherent constituents at the S2 frequency.

4. Results

a. M2 surface and internal tide validation

We computed the barotropic current ellipse from the observed depth-averaged velocity at the six different mooring stations on the continental shelf (Figs. 3a–f). The M2 tidal ellipses were the main focus as they dominate in the Pilbara region (Holloway 1988). The predicted M2 tidal ellipses from the 3D model were generally in good agreement with the field measurements and TPXO8-Atlas dataset, reproducing the major features of the local barotropic tides. The discrepancy between the model, observations and the TPXO8 solution at N3C likely occurred due to inadequately resolved bathymetry in this region.

Fig. 3.
Fig. 3.

(a)–(f) Comparison of M2 barotropic current ellipses between the model predictions (black lines), observations (blue lines), and TPXO8-atlas data (red lines) at different moorings. The scale of the ellipse is shown on the right. (g),(h) Scatterplots comparing the M2 baroclinic tidal amplitude and phase between 3D model predictions and observations at six mooring sites.

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

The model-predicted M2 baroclinic velocity amplitude was compared with field measurements at the six mooring stations for the upper, middle, and bottom depths (Figs. 3g,h). The correlation coefficients for the amplitude and phase between the predicted and observed values were both greater than 0.8. In general, M2 tidal features were well predicted in the 3D model giving us confidence in the internal tide calculations presented below.

b. Internal tide energetics

In the 3D model, large positive conversion rates (~1 W m−2) demonstrated that internal tides were generated in regions A, B, and C (Fig. 4a). During the spring tide period, domain-integrated conversion rates were 500, 370, and 260 MW in regions A, B, and C, respectively.

Fig. 4.
Fig. 4.

(a) Spring tide cycle–averaged conversion rates in standard case with the Exmouth Plateau. Vectors mean the depth-integrated tidal-averaged energy flux of baroclinic tides. Three red boxes represent the main generation sites of internal tides. Triangles represent five moorings. (b) The case excluding the western portion of the Exmouth Plateau. (c) Conversion rate difference CExP between the standard case and model run excluding the Exmouth Plateau.

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

There were also a number of regions where negative conversion rates occurred (Fig. 4a), with values locally as large in magnitude as the positive conversion rates seen elsewhere. The negative conversion rates were mainly found in the inshore regions (e.g., regions B and C) between the 200- and 400-m isobaths, where the PIL200 mooring was located. The model run with a smaller domain, excluding the Exmouth Plateau, demonstrated that large positive and negative conversion rates still occurred in regions B and C, with a similar pattern to the standard large domain case (Fig. 4b). However, in the large domain case net energy conversion integrated over regions B and C increased by 13.5% (50 MW) in the region B and 8% (20 MW) in the region C when the western portion of the Exmouth Plateau was included. Exmouth Plateau thus constructively influences the conversion rate on the southern NWS continental slope.

During the spring tide period, strong baroclinic energy fluxes (>2.0 KW m−1) emanated from regions B and C, coincident with larger conversion rates (vectors in Fig. 4a). Internal tides mainly propagated in the offshore direction from these regions and the energy fluxes indicated a complicated pattern ~150 km from the major generation site (region B), suggesting the formation of standing internal tides due to the interaction between RITs and LITs (e.g., Rayson et al. 2012). However, in the small domain case the energy fluxes primarily propagated northwestward from region B and then diminished at the west boundary (Fig. 4b), with no suggestion of standing internal tides.

To compare the conversion rate distribution in the large and small domain cases, we calculated the difference between them, and this conversion difference was denoted by CExP (Fig. 4c). The magnitude of CExP was comparable to the conversion rate (~1.0 W m−2) for the standard case. Alternating positive and negative bands of CExP can be seen along the isobaths over the slope and perpendicular to the southwest side of the Exmouth Plateau (region A in Fig. 4a), a result of the constructive and destructive interference caused by the varying phase difference between the RITs and the LITs throughout the domain [Eq. (9)]. Along the 800-m isobath, the alternating positive and negative CExP regions are ~50 km long, approximately half the wavelength of the mode-1 baroclinic tide in this water depth.

c. Negative conversion rates

The semidiurnal coherent conversion rate was estimated from both observations and the model output at five sites. We compared observations and the model at two locations on the 200-m isobath (NICOP and PIL200) and three locations on the 100-m isobath (PIL100, N3C and N2C); the model identified a range of positive and negative conversion rates across these locations. Due to the different deployment periods of the five moorings, model comparisons were undertaken over different time periods (note variable x axes in Fig. 5). Semidiurnal p and wbt were used to calculate the semidiurnal-coherent conversion rates.

Fig. 5.
Fig. 5.

Semidiurnal coherent conversion at the mooring sites (a) NICOP, (c) PIL200, (e) PIL100, (g) N3C, and (i) N2C during spring tides. Bottom pressure perturbation (black) and vertical barotropic velocity (blue) at the moorings (b) NICOP, (d) PIL200, (f) PIL100, (h) N3C, and (j) N2C. Solid and dotted lines indicate observed and modeled results, respectively. Note that the magnitude of the y axis in (g) and (i) is one order smaller than in (a), (c), and (e).

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

At the NICOP site (Figs. 5a,b), the phase difference between p and wbt was always much less than 90°, resulting in positive conversion rates in agreement with the model predictions. At the PIL200 site (Figs. 5c,d), p and wbt were nearly 180° out of phase, leading to negative conversion rates that were again in agreement with the model predictions.

At the 100-m contour, the PIL100 and N2C sites had a phase difference between p and wbt ranging from 90° to 180°, resulting in conversion rates that periodically crossed the zero point. However, both the model results and field data demonstrated the time-averaged conversion rates were positive at these two moorings. A large difference between the model and observations occurred at N3C. Note that the conversion magnitude was one order smaller than most other sites. This disagreement may come from two sources. First, the 3D numerical model does not predict the tidal climatology well at N3C (see Fig. 3) and second; the observed conversion rate may not be predicted well due to a low vertical resolution of the thermistors and inadequate sampling near the surface, as well as the overall short record length leading to errors in the temporal decomposition.

The 2D model (Real Exp) was run with realistic topography along the selected cross section identified from the 3D model (Fig. 1). A small-amplitude barotropic M2 tide was used to force the model to maintain linear conditions (refer to Real Exp in Table 3), resulting in different conversion magnitudes to the 2D and 3D cases. There was no RIT in this model run. The slope criticality parameter γ (ratio of local wave characteristic slope to bottom slope) along this cross section indicated that near-critical slopes occur at 25 km < x < 55 km (Fig. 6c), coincident with the region of large positive conversion (Fig. 6a) and zero phase difference [cos(θ)1] (shown in Fig. 6b). The differences in θ between the 2D and 3D cases resulted from additional remote generation sites in the 3D case influencing the phase of p.

Fig. 6.
Fig. 6.

(a) Tidal-averaged conversion rates and (b) cosine function of the phase difference between the bottom pressure perturbation and vertical barotropic velocity along the cross section for the 3D case (black lines, location shown in Fig. 1) and the 2D case (blue lines) with single M2 tidal forcing. (c) Bottom topography (black solid line) and topographic critical parameter γ (dashed line).

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

The strongest positive conversion site was located approximately 35 km from the strongest negative conversion site (Fig. 6a). The mode-1 phase speed cbc=1.5 m s−1 between 300- and 500-m water depth, while the barotropic phase speed was cbt=gH=70 m s−1. The propagation time from the main generation site to the site with negative conversion was thus approximately 6.5 h and 8 min for the baroclinic and barotropic tides, respectively. The time lag between the mode-0 and mode-1 tides was thus nearly half of the semidiurnal period, and resulted in p and wbt being out of phase at this site, leading to negative conversion.

d. Effect of RITs

The 3D model revealed that the western portion of the Exmouth Plateau was a significant remote generation site that influenced local net energy conversion on the inner shelf and slope. To study the interaction of RITs with LITs, we implemented a series of 2D idealized cases with the amplitude of the baroclinic forcing ranging from 2 to 40 cm s−1 and the phase (θp,r) ranging from 0.25π to 2.00π (BT-BC Exps and High-BC Exps, Table 3).

1) Propagation process

We first illustrate the spatial variation of the RIT during the propagation process using snapshots of pr for different RIT amplitudes of 4 and 20 cm s−1, but both with a constant phase difference of 0.00π (Figs. 7a,e). The numerically predicted pr was calculated as the difference between BT-BC Exps and BT-only Exp, while the kinematic model predicted pr was obtained from Eq. (9).

Fig. 7.
Fig. 7.

(a) Snapshots of pr for cases with 4 cm s−1 velocity (Fr = 0.017) on the left boundary at times from 0 to 100 h. The black dashed line is the edge of continental shelf. Blue lines indicate the kinematic prediction of pr. (b) The black solid line represents numerically predicted phase speed while the blue dashed line is theoretical linear phase speed. (c) The bar graph shows the propagation time of numerically predicted RITs across a single grid point (100 m); the blue dotted line is the accumulated propagation time along the model domain. (d) The cosine function of numerically predicted θbc,r [refer to Eq. (3)] is shown with the black line, while the cosine function of theoretical θth is the blue line. (e)–(h) As in (a)–(d), but for the cases with initial velocity amplitudes of 20 cm s−1 (Fr = 0.084).

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

For the 4 cm s−1 amplitude forcing case, the amplitude and phase of the model and kinematic predictions pr [calculated using Eq. (2)] were in good agreement in deep water, demonstrating the interaction was a linear process. For the 20 cm s−1 forcing case, model and kinematic predictions were in agreement for the early stage from 0 to 40 model hours. After 40 h (x > 300 km), the leading edge of the numerically predicted RITs steepened and became strongly nonlinear, thus deviating from the linear kinematic predictions (Fig. 7e).

2) Phase speeds and propagation time

The phase of RITs can be expressed as θbc,r=[θp,r+(Tp/Tbt)2π] [Eq. (3)], thereby the propagation time of RITs can affect the sign of Cr [Eq. (9)]. To consider the propagation time of RITs, we first estimated both the linear and numerical model phase speeds. The mode-1 linear phase speed cth was derived from Eq. (5), which was approximately 2.4 m s−1 in deep water (shown in blue dashed lines in Figs. 7b,f). We then traced the troughs of RITs and calculated the numerically predicted phase speed cmod (shown in black lines in Fig. 7b). For the 4 cm s−1 amplitude forcing case, cmod varied slightly with distance, but kept in close agreement with cth. In contrast, the deviation between cmod and cth was significant for the 20 cm s−1 amplitude case.

Based on the numerically predicted phase speeds in deep water, the RIT propagation time per grid was obtained over the model domain (Figs. 7c,g). The arrival time of RITs was ~40 and ~38 h for the 4 and 20 cm s−1 amplitude cases, respectively. The cosine function of the kinematically predicted θth agreed well with numerical prediction (θbc,r) for the entire model domain for the 4 cm s−1 amplitude case. In contrast, for the 20 cm s−1 amplitude case, cos(θth) and cos(θbc,r) deviated and had different signs at x = 350 km. These deviations resulted from the steepening process of large-amplitude RITs, which are discussed below (section 5b).

3) Phases and amplitudes

The phase of the RIT (θp,r) also contributes to the sign of Cr. We use the case with an RIT amplitude of 4 cm s−1 as an example. When the phase of the RIT was varied, Cr over the slope varied from regions with predominantly positive to predominantly negative values (e.g., Figs. 8b,f). Similar positive and negative regions of Cr occurred along the isobath on the continental slope in the 3D model (Fig. 4c). Although the center of the slope was xs = 400 km, the topography was subcritical at this location. The near-critical point on the slope was located around x = 405 km, where we find both the minimum (Fig. 8h) and maximum values of Cr (Fig. 8d). Numerical and kinematic predictions of Cr were in good agreement (blue and black lines in Fig. 8).

Fig. 8.
Fig. 8.

The 4 cm s−1 baroclinic tide amplitude case with baroclinic tide phase θp,r ranging from 0.25π to 2.00π at an interval of 0.25π. (a)–(h) Cycle-averaged Cr over the slope for different θp,r. Black and blue lines represent model estimates and kinematic predictions, respectively.

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

We compared the model Cr from the 2D idealized simulations with the kinematic predictions [Eq. (9)] at the critical point for a range of different RIT amplitudes and phases (Fig. 9). The kinematic model reproduced the simulation results well, with Cr exhibiting a sinusoidal relationship with the baroclinic phase. For the kinematic prediction, Cr should have a linear relationship with the RIT amplitude [according to Eq. (9)], which was in agreement with the model predictions when the baroclinic amplitude ranged from 2 to 4 cm s−1. However, when the amplitude increased to above 6 cm s−1, the linear kinematic model slightly overpredicted Cr at the critical point.

Fig. 9.
Fig. 9.

Parameter Cr at the location x = 405 km with the baroclinic forcing phase ranging from 0 to 2.00π and amplitude ranging from 2 to 12 cm s−1. Markers and lines are numerical modeling and kinematic predictions, respectively. The two x axes are the phase of the RITs (θp,r and θ), while the two y axes are Cr and nondimensional Cr scaled by the local conversion rate Cl in the case of only barotropic tidal forcing.

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

To compare the kinematic predictions with the model results, we added an additional x axis with θ [see Eq. (9)] and nondimensionalized the y axis with Cl, the conversion rate at x = 405 km in the BT-only case (Fig. 9). The peak of the nondimensional Cr occurred when θ was 8.0π, or four complete wave cycles. Parameter Cr thus varied sinusoidally with the phase of the RITs, and varied linearly with the amplitudes of RITs for low-amplitude forcing cases.

5. Discussion

Historically, internal tide climatology on shelf regions has been estimated using regional models, under the assumption that local internal tide generation is dominant (e.g., Holloway 1996). As model domains are increased, a significant effect of far-field generation sites on local energy conversion is often observed. Here we found that on the southern NWS, net energy conversion was 11% (0.22 GW) smaller when the model domain excluded the (remote) western portion of the Exmouth Plateau (Fig. 4b). This is the net result of a 0.34 GW decrease in positive conversion, and a 0.12 GW decrease in negative conversion. This is comparable to the total internal tide generation reduction of 13% (or 7.1 MW) when the outlying bathymetric features were excluded from a model of the Monterey Bay region (Hall and Carter 2011). In contrast, Kerry et al. (2013) estimated that, in the absence of RITs, domain-integrated conversion rates were 11% greater at the Luzon Strait and 65% greater at the Mariana Arc.

Our results demonstrate that for small-amplitude tides the phase of RITs is responsible for the changes in local energy conversion, agreeing with previous studies (e.g., Kelly and Nash 2010; Zilberman et al. 2011). However, we have demonstrated that as the amplitude of the RITs increased the changes in the local energy conversion also become dependent on the nonlinearity of the propagating RIT.

a. Remotely influenced conversion rates versus Froude number

The sensitivity of Cr to the forcing amplitude can be described by considering the Froude number (Fr)
Fr=Ubccp,
where cp is the mode-1 linear phase speed [Eq. (5)]. Five locations (x/L = 0.45, 0.5, 0.55, 0.6 and 0.65) over the shelf were selected to show the horizontal distribution of nondimensional Cr (Figs. 10a–e; both numerical and kinematic predictions) at different sites with different criticalities (blue triangles in Fig. 10a). Note that the critical point was located at x/L = 0.55. The different phases of RITs play a key role in the sign of the nondimensional Cr as well as the magnitude of Fr.
Fig. 10.
Fig. 10.

(a) Bottom topography over the shelf (L is the shelf length). Blue triangles represent five locations shown in panels below. (b) Nondimensional Cr at location x/L = 0.45 as a function of the RIT wave Froude number. Dots and dashed lines represent the numerical model and kinematic predictions, respectively. Different colors represent different phases of the RITs. (d),(f),(h),(j) As in (b), but at locations x/L = 0.5, 0.55, 0.6, and 0.65. (c) Gray shading is the deviation of Cr between the numeric model and kinematic predictions nondimensionalized by local conversion rate Cl for different phases. Black lines represent phase-averaged error. Dotted rectangles indicate the small Froude number zone (Fr < 0.05). (e),(g),(i),(k) As in (c), but at locations x/L = 0.5, 0.55, 0.6, and 0.65.

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

The subcritical topographic features occurred at the location x/L = 0.45. Taking 4 cm s−1 baroclinic tides as an example, numerical Cr at the subcritical location (x = 395 km) had a small value ~O(10−3) W m−2 and slightly diverged from the kinematic Cr (e.g., Figs. 8a,e). In addition, the topographic gradient was small, resulting in a small vertical barotropic velocity wbt and thereby a small value of local conversion Cl=plwbt (~2.0 × 10−3 W m−2). The nondimensionalized value Cr/Cl thus accentuated the magnitude of the deviation between the numerical and kinematic predictions at the subcritical locations (shown in Fig. 10b).

At the other subcritical location (x/L = 0.65) on the right portion of the shelf, the phase difference between the numerical and kinematic predictions was increasing as the RIT was steepening (e.g., Fig. 7h) and shoaling on the shelf. Therefore, a slight phase difference between the numerical and kinematic predictions can cause the opposing sign as well as a great difference in magnitude between the nondimensionalized numerical and kinematic predictions (shown in Fig. 10j).

In contrast, near-critical and critical points (x/L = 0.5, 0.55, and 0.6) indicated better agreement between the numerical and kinematic predictions, irrespective of the baroclinic phase. When Fr < 0.05, the model and the kinematic predictions agreed well, but for Fr > 0.05 the two predictions diverged. To quantitatively demonstrate the difference between the numerical and kinematic predictions, we nondimensionalized the difference of Cr with Cl (Figs. 10c,e,g,i,k). For subcritical points x/L = 0.45 and 0.65, the phase-averaged error of Cr was always smaller than 2.5 when Fr < 0.05. When Fr > 0.05, the error kept rising as Fr increased (e.g., Fig. 10c). On the other hand, the critical and near-critical points (x/L = 0.5, 0.55, and 0.6) showed much smaller deviations between the two predictions from the two methods. The phase-averaged error was smaller than 1.0 when Fr < 0.05 (Figs. 10e,g,i).

RITs with Fr = 0.017 and 0.169 were selected to compare Cr over the slope in low- and high-wave Froude number cases (Figs. 11b,c). Both positive and negative Cr occurred over the shelf for different RIT phase. The linear RITs with low Fr remained linear during propagation, resulting in a smoothly varying pattern of Cr over the shelf (e.g., Fig. 11b). However, the RITs with high Fr nonlinearly steepened and lead to an abruptly varying pattern of Cr over the shelf (Fig. 11c).

Fig. 11.
Fig. 11.

(a) Bottom topography over the shelf. (b) Histograms are the horizontal distribution of Cr over the shelf in the case with low Froude number (Fr = 0.017). Magenta, cyan, yellow, and gray colors represent cases with the remote phase θp,r of 0.50π, 1.00π, 1.50π, and 2.00π, respectively. (c) As in (b), but for high Froude number (Fr = 0.169).

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

b. Steepening processes of RITs

As mentioned above, small-amplitude and large-amplitude RITs showed different wave shape developments during the propagating processes on the flat bottom (e.g., Fig. 7a,e), eventually inducing linear and highly nonlinear Cr (Figs. 11b,c), respectively. Hence, the purpose of this section is to quantify the differences between the steepening processes of small-amplitude and large-amplitude RITs.

To estimate the ability for an initial RIT to evolve into nonlinear internal waves (NLIWs) during a given propagation time Tp, we presented the nondimensional steepening parameter λs
λs=TpTs.
Here the steepening time Ts can be expressed as Ts~l/αA0 (Horn et al. 2001), arising from balancing the dispersion and nonlinear terms in the Korteweg–de Vries (KdV) equation. Here l and A0 are the wavelength and amplitude of the initial RIT, respectively, and α is the nonlinearity coefficient derived from the KdV equation, which is given by
α=3cpH0Wz3dz2H0Wz2dz.

For small RITs (Fr < 0.05), the steepening parameter λs was smaller than 1 over the entire model domain (600 km), hence NLIW steepening could not occur, and thus is why the linear kinematic prediction did well for this small forcing. When large-amplitude RITs (Fr > 0.08) propagated from the left boundary to the slope, however, λs was greater than 1, the RITs lost their linear sinusoidal character and the wave fronts steepened. Thus, pr loses the phase coherence (e.g., Fig. 7e) and the kinematic model could no longer accurately predict Cr.

Nonlinear effects due to different slope criticalities can also play a role on the effect of RITs (see appendix). Our results show that there is not only a linear effect of RITs on slope generation due to phase interference but also a nonlinear effect that is dependent on the RIT amplitude. Both effects should be considered when calculating barotropic to baroclinic conversion rates over topography in the real ocean.

6. Conclusions

The superposition of multiple internal tides results in a complicated internal tide climatology. The southwest part of the offshore Exmouth Plateau showed large barotropic-to-baroclinic conversion rates and, by running the 3D model in a reduced domain which excluded the Exmouth Plateau, we demonstrated the internal tides emanating from the Exmouth Plateau played a key role in the internal tide processes on the inshore continental slope and shelf. A comparison of model predictions with observational data at five moorings on the continental shelf revealed both positive and negative local barotropic-to-baroclinic conversion rates as a result of the influence of these remotely generated internal tides.

To quantitatively explain how the RITs influence the LITs on the shelf, we performed a series of 2D model runs with idealized topography and barotropic tidal forcing at a single forcing frequency. The model demonstrated how that local internal tide climatology was modulated by varying phases and amplitudes of RITs. The principle results were as follows:

  1. Phase differences between pr and local wbt induced both positive and negative conversion rates over the shelf.
  2. RITs with low Froude number (Fr < 0.05) had steepening length scales greater than the model domain length and thus retained their linear characteristics. The RITs linearly influenced the LIT generation processes, resulting in a smoothly varying pattern of Cr over the shelf (e.g., Fig. 11b), and the numerically simulated Cr were consistent with kinematic predictions.
  3. For RITs with high Froude number (Fr > 0.05), the steepening length scale was smaller than the domain length. Due to the increasing nonlinearity, the arrival time of the wave front sharply altered on the shelf, and Cr showed a strongly varying pattern over the shelf (Fig. 11c).

The local energy conversion was dependent on the distance between the remote generation site and the local generation sites and the phase relationship and the intensity of both the local barotropic tides and the RITs. Our results demonstrate the importance of considering RITs for both selection of field observation sites and model domains, in order to capture all of the significant internal tide climatology. Our results are consistent with those of Carter et al. (2012) and Kelly and Nash (2010) in demonstrating that regional models need to be either forced by global models or be nested in large regional models to accurately predict the local internal tide climatology.

Acknowledgments

This research was supported by the Australian Research Council Industrial Transformation Research Hub for Offshore Floating Facilities (IH140100012). Yankun Gong was supported by the Chinese Scholarship Council. Field data were obtained from a collaboration between the U.S. Naval Research Laboratory, the University of Western Australia, and the Australian Institute of Marine Science under the project DP 140101322 and N62909-11-1-7058. In addition, the data of PIL100 and PIL200 moorings was from the Integrated Marine Observing System—a national collaborative research infrastructure, supported by the Australian Government. The numerical experiments were supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. Observational data are available on the UWA Library Research Repository.

APPENDIX

Steepening of RITs over Slopes with Varying Criticality

To examine the effect of varying slope criticality, we considered six cases with different Froude numbers (Fr) and slope steepness. RITs with amplitudes of 4 and 20 cm s−1 were selected to vary Fr, and topographic critical parameters γ of 0.5, 1.0, and 2.0 were taken as examples of subcritical, critical, and supercritical slopes (see Fig. A1). These six cases were driven solely by RITs. Other configurations were the same as above 2D cases (shown in Table 2).

Fig. A1.
Fig. A1.

(a) A snapshots of pr for the case with RIT amplitude of 4 cm s−1 (Fr = 0.017) at the time of t = 96 h. Slopes with topographic critical parameter γ of 0.5, 1.0, and 2.0 are shown in the upper, middle, and bottom panels. Blue dotted lines indicate arrival locations and amplitudes of RITs over the slope. (b) As in (a), but for 20 cm s−1 RIT amplitude (Fr = 0.084). (c) Bottom topography with subcritical, critical, and supercritical slopes.

Citation: Journal of Physical Oceanography 49, 6; 10.1175/JPO-D-18-0180.1

For cases with low Froude number (Fr = 0.017), RITs retained their sinusoidal form over the entire model domain, regardless of the slope shape (Fig. A1). The arrival location of the RIT trough was quite consistent after the shoaling process over the subcritical and critical slopes. However, the arrival location of the RIT was slightly delayed in the supercritical case. In addition, the magnitude of RITs on the supercritical shelf was approximately 50% of the magnitude of RITs on the subcritical shelf. For an incident mode-1 internal tide transmitting upon slopes with a topographic height to water depth ratio of 0.8 (as used in this study), Kelly et al. (2013) assumed that the reflection coefficients for subcritical slopes decrease to zero, while the reflection coefficients for supercritical slopes remain nearly constant and nonnegligible. In comparison, linear model and field observations on the continental shelf of the South China Sea indicate 67% of incoming mode-1 energy is transmitted up the supercritical slope (Klymak et al. 2011). For our simulations, compared to the subcritical case (γ = 0.5), 65% and 51% of the incoming mode-1 linear internal tide was transmitted up the critical (γ = 1.0) and supercritical slopes (γ = 2.0), respectively. However, the ratio of the topographic height to the water depth is quite different between our model runs (0.8) and the Klymak et al. (2011) observations (0.5), therefore a difference is anticipated.

For cases with high Froude number (Fr = 0.084), RITs retained their sinusoidal form from x = 0–150 km, and steepened at x = 200 km, becoming increasingly nonlinear after that (Fig. A1). The RITs steepened at different rates over the different slopes. At t = 96 h, the RIT arrived at location x = 489, 482, and 480 km with pr amplitudes of 222, 156, and 137 Pa over the subcritical, critical and supercritical slope regions, respectively Thereby, 70% and 62% of the incoming mode-1 linear internal tide was transmitted up the critical (γ = 1.0) and supercritical slopes (γ = 2.0), respectively. In summary, slope criticality played a large role in the arrival location and amplitude variation of RITs.

REFERENCES

  • Alford, M. H., J. A. MacKinnon, Z. Zhao, R. Pinkel, J. Klymak, and T. Peacock, 2007: Internal waves across the Pacific. Geophys. Res. Lett., 34, L24601, https://doi.org/10.1029/2007GL031566.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Apel, J. R., L. A. Ostrovsky, Y. A. Stepanyants, and J. F. Lynch, 2007: Internal solitons in the ocean and their effect on underwater sound. J. Acoust. Soc. Amer., 121, 695722, https://doi.org/10.1121/1.2395914.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buijsman, M. C., Y. Kanarska, and J. C. McWilliams, 2010: On the generation and evolution of nonlinear internal waves in the South China Sea. J. Geophys. Res., 115, C02012, https://doi.org/10.1029/2009JC005275.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buijsman, M. C., S. Legg, and J. Klymak, 2012: Double-ridge internal tide interference and its effect on dissipation in Luzon Strait. J. Phys. Oceanogr., 42, 13371356, https://doi.org/10.1175/JPO-D-11-0210.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Carter, G. S., O. B. Fringer, and E. D. Zaron, 2012: Regional models of internal tides. Oceanography, 25 (2), 5665, https://doi.org/10.5670/oceanog.2012.42.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Egbert, G. D., and S. Y. Erofeeva, 2002: Efficient inverse modeling of barotropic ocean tides. J. Atmos. Oceanic Technol., 19, 183204, https://doi.org/10.1175/1520-0426(2002)019<0183:EIMOBO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garrett, C., and E. Kunze, 2007: Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech., 39, 5787, https://doi.org/10.1146/annurev.fluid.39.050905.110227.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gerkema, T., and J. T. F. Zimmerman, 2008: An introduction to internal waves. Lecture Notes, Royal NIOZ, 207 pp.

  • Hall, R. A., and G. S. Carter, 2011: Internal tides in Monterey submarine canyon. J. Phys. Oceanogr., 41, 186204, https://doi.org/10.1175/2010JPO4471.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1984: On the semidiurnal internal tide at a shelf-break region on the Australian North West Shelf. J. Phys. Oceanogr., 14, 17871799, https://doi.org/10.1175/1520-0485(1984)014<1787:OTSITA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1985: A comparison of semidiurnal internal tides from different bathymetric locations on the Australia North West Shelf. J. Phys. Oceanogr., 15, 240251, https://doi.org/10.1175/1520-0485(1985)015<0240:ACOSIT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1988: Physical oceanography of the Exmouth Plateau region, North-western Australia. Mar. Freshwater Res., 39, 589606, https://doi.org/10.1071/MF9880589.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1996: A numerical model of internal tides with application to the Australian North West Shelf. J. Phys. Oceanogr., 26, 2137, https://doi.org/10.1175/1520-0485(1996)026<0021:ANMOIT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., E. Pelinovsky, T. Talipova, and B. Barnes, 1997: A nonlinear model of internal tide transformation on the Australian North West Shelf. J. Phys. Oceanogr., 27, 871896, https://doi.org/10.1175/1520-0485(1997)027<0871:ANMOIT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., P. G. Chatwin, and P. Craig, 2001: Internal tide observations from the Australian North West Shelf in summer 1995. J. Phys. Oceanogr., 31, 11821199, https://doi.org/10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Horn, D. A., J. Imberger, and G. N. Ivey, 2001: The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech., 434, 181207, https://doi.org/10.1017/S0022112001003536.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jones, N. L., and G. N. Ivey, 2017: Internal waves. Encyclopedia of Maritime and Offshore Engineering, J. Carlton, P. Jukes, and Y. S. Choo, Eds., Wiley, https://doi.org/10.1002/9781118476406.emoe089.

    • Crossref
    • Export Citation
  • Kelly, S. M., and J. D. Nash, 2010: Internal-tide generation and destruction by shoaling internal tides. Geophys. Res. Lett., 37, L23611, https://doi.org/10.1029/2010GL045598.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kelly, S. M., N. L. Jones, and J. D. Nash, 2013: A coupled model for Laplace’s tidal equations in a fluid with one horizontal dimension and variable depth. J. Phys. Oceanogr., 43, 17801797, https://doi.org/10.1175/JPO-D-12-0147.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kerry, C. G., B. S. Powell, and G. S. Carter, 2013: Effects of remote generation sites on model estimates of M2 internal tides in the Philippine Sea. J. Phys. Oceanogr., 43, 187204, https://doi.org/10.1175/JPO-D-12-081.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., M. H. Alford, R. Pinkel, R. C. Lien, Y. J. Yang, and T. Y. Tang, 2011: The breaking and scattering of the internal tide on a continental slope. J. Phys. Oceanogr., 41, 926945, https://doi.org/10.1175/2010JPO4500.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., M. Buijsman, S. Legg, and R. Pinkel, 2013: Parameterizing surface and internal tide scattering and breaking on supercritical topography: The one-and two-ridge cases. J. Phys. Oceanogr., 43, 13801397, https://doi.org/10.1175/JPO-D-12-061.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., L. K. Rosenfeld, G. S. Carter, and M. C. Gregg, 2002: Internal waves in Monterey submarine canyon. J. Phys. Oceanogr., 32, 18901913, https://doi.org/10.1175/1520-0485(2002)032<1890:IWIMSC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kurapov, A. L., G. D. Egbert, J. S. Allen, R. N. Miller, S. Y. Erofeeva, and P. M. Kosro, 2003: The M2 internal tide off Oregon: Inferences from data assimilation. J. Phys. Oceanogr., 33, 17331757, https://doi.org/10.1175/2397.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Legg, S., and K. M. Huijts, 2006: Preliminary simulations of internal waves and mixing generated by finite amplitude tidal flow over isolated topography. Deep-Sea Res. II, 53, 140156, https://doi.org/10.1016/j.dsr2.2005.09.014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lerczak, J. A., C. D. Winant, and M. C. Hendershott, 2003: Observations of the semidiurnal internal tide on the southern California slope and shelf. J. Geophys. Res., 108, 3068, https://doi.org/10.1029/2001JC001128.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marshall, J., C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res., 102, 57335752, https://doi.org/10.1029/96JC02776.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nagai, T., and T. Hibiya, 2015: Internal tides and associated vertical mixing in the Indonesian Archipelago. J. Geophys. Res. Oceans, 120, 33733390, https://doi.org/10.1002/2014JC010592.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nash, J. D., E. Kunze, J. M. Toole, and R. W. Schmitt, 2004: Internal tide reflection and turbulent mixing on the continental slope. J. Phys. Oceanogr., 34, 11171134, https://doi.org/10.1175/1520-0485(2004)034<1117:ITRATM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nash, J. D., M. H. Alford, and E. Kunze, 2005: Estimating internal wave energy fluxes in the ocean. J. Atmos. Oceanic Technol., 22, 15511570, https://doi.org/10.1175/JTECH1784.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nash, J. D., S. M. Kelly, E. L. Shroyer, J. N. Moum, and T. F. Duda, 2012: The unpredictable nature of internal tides on continental shelves. J. Phys. Oceanogr., 42, 19812000, https://doi.org/10.1175/JPO-D-12-028.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Oke, P. R., and P. Sakov, 2008: Representation error of oceanic observations for data assimilation. J. Atmos. Oceanic Technol., 25, 10041017, https://doi.org/10.1175/2007JTECHO558.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ponte, A. L., and B. D. Cornuelle, 2013: Coastal numerical modelling of tides: Sensitivity to domain size and remotely generated internal tide. Ocean Modell., 62, 1726, https://doi.org/10.1016/j.ocemod.2012.11.007.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rayson, M. D., N. L. Jones, and G. N. Ivey, 2012: Temporal variability of the standing internal tide in the Browse Basin, Western Australia. J. Geophys. Res., 117, C06013, https://doi.org/10.1029/2011JC007523.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stewart, K. D., A. M. Hogg, S. M. Griffies, A. P. Heerdegen, M. L. Ward, P. Spence, and M. H. England, 2017: Vertical resolution of baroclinic modes in global ocean models. Ocean Modell., 113, 5065, https://doi.org/10.1016/j.ocemod.2017.03.012.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Van Gastel, P., G. N. Ivey, M. J. Meuleners, J. P. Antenucci, and O. Fringer, 2009: The variability of the large-amplitude internal wave field on the Australian North West Shelf. Cont. Shelf Res., 29, 13731383, https://doi.org/10.1016/j.csr.2009.02.006.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhao, Z., and M. H. Alford, 2009: New altimetric estimates of mode-1 M2 internal tides in the central North Pacific Ocean. J. Phys. Oceanogr., 39, 16691684, https://doi.org/10.1175/2009JPO3922.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zilberman, N. V., J. M. Becker, M. A. Merrifield, and G. S. Carter, 2009: Model estimates of M2 internal tide generation over Mid-Atlantic Ridge topography. J. Phys. Oceanogr., 39, 26352651, https://doi.org/10.1175/2008JPO4136.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zilberman, N. V., M. A. Merrifield, G. S. Carter, D. S. Luther, M. D. Levine, and T. J. Boyd, 2011: Incoherent nature of M2 internal tides at the Hawaiian Ridge. J. Phys. Oceanogr., 41, 20212036, https://doi.org/10.1175/JPO-D-10-05009.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
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