Editorial Type:
Article
Article Type:
Research Article

A Coupled Model for Laplace's Tidal Equations in a Fluid with One Horizontal Dimension and Variable Depth

Samuel M. Kelly The Oceans Institute, and School of Environmental Systems Engineering, University of Western Australia, Crawley, Western Australia, Australia

Search for other papers by Samuel M. Kelly in
Current site
Google Scholar
PubMed Close
,
Nicole L. Jones The Oceans Institute, and School of Environmental Systems Engineering, University of Western Australia, Crawley, Western Australia, Australia

Search for other papers by Nicole L. Jones in
Current site
Google Scholar
PubMed Close
, and
Jonathan D. Nash College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

Search for other papers by Jonathan D. Nash in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Aug 2013
DOI:
https://doi.org/10.1175/JPO-D-12-0147.1
Page(s):
1780–1797
Restricted access

Abstract

Tide–topography interactions dominate the transfer of tidal energy from large to small scales. At present, it is poorly understood how low-mode internal tides reflect and scatter along the continental margins. Here, the coupling equations for linear tides model (CELT) are derived to determine the independent modal solutions to Laplace's Tidal Equations (LTE) over stepwise topography in one horizontal dimension. CELT is (i) applicable to arbitrary one-dimensional topography and realistic stratification without requiring numerically expensive simulations and (ii) formulated to quantify scattering because it implicitly separates incident and reflected waves. Energy fluxes and horizontal velocities obtained using CELT are shown to converge to analytical solutions, indicating that “flat bottom” modes, which evolve according to LTE, are also relevant in describing tides over sloping topography. The theoretical framework presented can then be used to quantify simultaneous incident and reflected energy fluxes in numerical simulations and observations of tidal flows that vary in one horizontal dimension. Thus, CELT can be used to diagnose internal-tide scattering on continental slopes. Here, semidiurnal mode-1 scattering is simulated on the Australian northwest, Brazil, and Oregon continental slopes. Energy-flux divergence and directional energy fluxes computed using CELT are shown to agree with results from a finite-volume model that is significantly more numerically expensive. Last, CELT is used to examine the dynamics of two-way surface–internal-tide coupling. Semidiurnal mode-1 internal tides are found to transmit about 5% of their incident energy flux to the surface tide where they impact the continental slope. It is hypothesized that this feedback may decrease the coherence of sea surface displacement on continental shelves.

Corresponding author address: Samuel Kelly, 7-321, Department of Mechanical Engineering, MIT, 77 Massachusetts Ave., Cambridge, MA 02139. E-mail: samkelly@mit.edu

Abstract

Tide–topography interactions dominate the transfer of tidal energy from large to small scales. At present, it is poorly understood how low-mode internal tides reflect and scatter along the continental margins. Here, the coupling equations for linear tides model (CELT) are derived to determine the independent modal solutions to Laplace's Tidal Equations (LTE) over stepwise topography in one horizontal dimension. CELT is (i) applicable to arbitrary one-dimensional topography and realistic stratification without requiring numerically expensive simulations and (ii) formulated to quantify scattering because it implicitly separates incident and reflected waves. Energy fluxes and horizontal velocities obtained using CELT are shown to converge to analytical solutions, indicating that “flat bottom” modes, which evolve according to LTE, are also relevant in describing tides over sloping topography. The theoretical framework presented can then be used to quantify simultaneous incident and reflected energy fluxes in numerical simulations and observations of tidal flows that vary in one horizontal dimension. Thus, CELT can be used to diagnose internal-tide scattering on continental slopes. Here, semidiurnal mode-1 scattering is simulated on the Australian northwest, Brazil, and Oregon continental slopes. Energy-flux divergence and directional energy fluxes computed using CELT are shown to agree with results from a finite-volume model that is significantly more numerically expensive. Last, CELT is used to examine the dynamics of two-way surface–internal-tide coupling. Semidiurnal mode-1 internal tides are found to transmit about 5% of their incident energy flux to the surface tide where they impact the continental slope. It is hypothesized that this feedback may decrease the coherence of sea surface displacement on continental shelves.

Corresponding author address: Samuel Kelly, 7-321, Department of Mechanical Engineering, MIT, 77 Massachusetts Ave., Cambridge, MA 02139. E-mail: samkelly@mit.edu
Save
Cover Journal of Physical Oceanography
Journal of Physical Oceanography

  • Alford, M. H., and Z. Zhao, 2007: Global patterns of low-mode internal-wave propagation. Part II: Group velocity. J. Phys. Oceanogr., 37, 1849–1858.

    • Search Google Scholar
    • Export Citation

  • 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, doi:10.1029/2007GL031566.

  • Arbic, B. K., A. J. Wallcraft, and E. J. Metzger, 2010: Concurrent simulation of the eddying general circulation and tides in a global ocean model. Ocean Modell., 32, 175–187.

    • Search Google Scholar
    • Export Citation

  • Baines, P. G., 1982: On internal tide generation models. Deep-Sea Res., 29, 307–338.

  • Bell, T. H., 1975: Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech., 64, 705–722.

  • Bluteau, C. E., N. L. Jones, and G. N. Ivey, 2011: Dynamics of a tidally-forced stratified shear flow on the continental slope. J. Geophys. Res.,116, C11017, doi:10.1029/2011JC007214.

  • Bühler, O., and M. Holmes-Cerfon, 2011: Decay of an internal tide due to random topography in the ocean. J. Fluid Mech., 678, 271–293.

    • Search Google Scholar
    • Export Citation

  • Buijsman, M. C., J. C. McWilliams, and C. R. Jackson, 2010: East–west asymmetry in nonlinear internal waves from Luzon Strait. J. Geophys. Res.,115, C10057, doi:10.1029/2009JC006004.

  • Carter, G. S., and M. A. Merrifield, 2007: Open boundary conditions for regional tidal simulations. Ocean Modell., 18, 194–209.

  • Carter, G. S., and Coauthors, 2008: Energetics of M2 barotropic-to-baroclinic tidal conversion at the Hawaiian Islands. J. Phys. Oceanogr., 38, 2205–2223.

    • Search Google Scholar
    • Export Citation

  • Chapman, D. C., and M. C. Hendershott, 1981: Scattering of internal waves obliquely incident upon a step change in bottom relief. Deep-Sea Res., 11, 1323–1338.

    • Search Google Scholar
    • Export Citation

  • Colosi, J. A., and W. Munk, 2006: Tales of the venerable Honolulu tide gauge. J. Phys. Oceanogr., 36, 967–996.

  • Dushaw, B. D., and P. F. Worcester, 1998: Resonant diurnal internal tides in the North Atlantic. Geophys. Res. Lett., 25, 2189–2192.

  • Echeverri, P., and T. Peacock, 2010: Internal tide generation by arbitrary two-dimensional topography. J. Fluid Mech., 649, 247–266.

  • Echeverri, P., T. Yokossi, N. J. Balmforth, and T. Peacock, 2011: Internal tide attractors between double ridges. J. Fluid Mech., 669, 354–374.

    • Search Google Scholar
    • Export Citation

  • Egbert, G. D., R. D. Ray, and B. G. Bills, 2004: Numerical modeling of the global semidiurnal tide in the present day and in the last glacial maximum. J. Geophys. Res.,109, C03003, doi:10.1029/2003JC001973.

  • Garrett, C., and E. Kunze, 2007: Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech., 39, 57–87.

  • Gayen, B., and S. Sarkar, 2010: Turbulence during the generation of internal tide on a critical slope. Phys. Rev. Lett.,104, 218502, doi:10.1103/PhysRevLett.104.218502.

  • Gerkema, T., 2011: Comment on “Internal tide energy over topography” by S. M. Kelly et al. J. Geophys. Res.,116, C07029, doi:10.1029/2010JC006611.

  • Griffiths, S. D., and R. H. J. Grimshaw, 2007: Internal tide generation at the continental shelf modeled using a modal decomposition: Two-dimensional results. J. Phys. Oceanogr., 37, 428–451.

    • Search Google Scholar
    • Export Citation

  • Hall, R. A., and G. S. Carter, 2011: Internal tides in Monterey Submarine Canyon. J. Phys. Oceanogr., 41, 186–204.

  • Hendershott, M. C., 1981: Long waves and ocean tides. Evolution of Physical Oceanography: Scientific Surveys in Honor of Henry Stommel, B. A. Warren and C. Wunsch, Eds., The MIT Press, 292–341.

  • Iwamae, N., and T. Hibiya, 2012: Numerical study of tide-induced mixing over rough bathymetry in the abyssal ocean. J. Oceanogr., 68, 195–203.

    • Search Google Scholar
    • Export Citation

  • Jayne, S. R., and L. C. St. Laurent, 2001: Parameterizing tidal dissipation over rough topography. Geophys. Res. Lett., 28, 811–814.

  • Johnston, T. M. S., M. A. Merrifield, and P. E. Holloway, 2003: Internal tide scattering at the Line Islands Ridge. J. Geophys. Res.,108, 3365, doi:10.1029/2003JC001844.

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

  • Kelly, S. M., and J. D. Nash, 2011: Reply to comment by T. Gerkema on “Internal tide energy over topography.” J. Geophys. Res.,116, C09002, doi:10.1029/2011JC007040.

  • Kelly, S. M., J. D. Nash, and E. Kunze, 2010: Internal-tide energy over topography. J. Geophys. Res.,115, C06014, doi:10.1029/2009JC005618.

  • Kelly, S. M., J. D. Nash, K. I. Martini, M. H. Alford, and E. Kunze, 2012: The cascade of tidal energy from low to high modes on a continental slope. J. Phys. Oceanogr., 42, 1217–1232.

    • Search Google Scholar
    • Export Citation

  • Klymak, J. M., R. Pinkel, and L. Rainville, 2008: Direct breaking of the internal tide near topography: Kaena Ridge, Hawaii. J. Phys. Oceanogr., 38, 380–399.

    • Search Google Scholar
    • Export Citation

  • Klymak, J. M., M. H. Alford, R.-C. Lien, Y. J. Yang, and T.-Y. Tang, 2011a: The breaking and scattering of the internal tide on a continental slope. J. Phys. Oceanogr., 41, 926–945.

    • Search Google Scholar
    • Export Citation

  • Klymak, J. M., S. Legg, and R. Pinkel, 2011b: A simple parameterization of turbulent tidal mixing near supercritical topography. J. Phys. Oceanogr., 40, 2059–2074.

    • Search Google Scholar
    • Export Citation

  • Klymak, J. M., M. Buijsman, S. Legg, and R. Pinkel, 2013: Parameterizing baroclinic internal tide scattering and breaking on supercritical topography: The one- and two-ridge cases. J. Phys. Oceanogr., 43, 1380–1397.

    • 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, 1890–1913.

    • Search Google Scholar
    • Export Citation

  • Kurapov, A., G. 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, 1733–1757.

    • Search Google Scholar
    • Export Citation

  • Laplace, P.-S., 1776: Recherches sur plusieurs points du systme du monde. Mem. Acad. Roy. Sci. Paris,89, 177–264.

  • Legg, S., 2004: Internal tides generated on a corrugated continental slope. Part I: Cross-slope barotropic forcing. J. Phys. Oceanogr., 34, 156–173.

    • Search Google Scholar
    • Export Citation

  • Legg, S., and A. Adcroft, 2003: Internal wave breaking at concave and convex continental slopes. J. Phys. Oceanogr., 33, 2224–2246.

  • Legg, S., and K. M. H. Huijts, 2006: Preliminary simulations of internal waves and mixing generated by finite amplitude tidal flow over isolated topography. Deep-Sea Res., 53, 140–156.

    • Search Google Scholar
    • Export Citation

  • Llewellyn Smith, S. G., and W. R. Young, 2002: Conversion of the barotropic tide. J. Phys. Oceanogr., 32, 1554–1566.

  • Maas, L. R. M., D. Benielli, J. Sommeria, and F. P. A. Lam, 1997: Observation of an internal wave attractor in a confined, stably stratified fluid. Nature, 388, 557–561.

    • Search Google Scholar
    • Export Citation

  • MacKinnon, J. A., and K. B. Winters, 2005: Subtropical catastrophe: Significant loss of low-mode tidal energy at 28.9°. Geophys. Res. Lett.,32, L15605, doi:10.1029/2005GL023376.

  • Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997: A finite-volume, incompressible Navier-Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102 (C3), 5753–5766.

    • Search Google Scholar
    • Export Citation

  • Martini, K. I., M. H. Alford, E. Kunze, S. M. Kelly, and J. D. Nash, 2011: Observations of internal tides on the Oregon continental slope. J. Phys. Oceanogr., 41, 1772–1794.

    • Search Google Scholar
    • Export Citation

  • Martini, K. I., M. H. Alford, E. Kunze, J. D. Nash, and S. M. Kelly, 2013: Internal bores and breaking internal tides on the Oregon continental slope. J. Phys. Oceanogr., 43, 120–139.

    • Search Google Scholar
    • Export Citation

  • McCreary, J., 1981: A linear stratified ocean model of the equatorial undercurrent. Philos. Trans. Roy. Soc. London, 298, 603–635.

  • Miles, J. W., 1974: On Laplace's tidal equations. J. Fluid Mech., 66, 241–260.

  • Moum, J. N., D. Caldwell, J. D. Nash, and G. Gunderson, 2002: Observations of boundary mixing over the continental slope. J. Phys. Oceanogr., 32, 2113–2130.

    • Search Google Scholar
    • Export Citation

  • Müller, P., and X. Liu, 2000: Scattering of internal waves at finite topography in two dimensions. Part I: Theory and case studies. J. Phys. Oceanogr., 30, 532–549.

    • Search Google Scholar
    • Export Citation

  • Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res., 45, 1977–2010.

  • Nansen, F., Ed., 1902: Oceanography of the North Pole Basin: Norwegian North Polar Expedition, 1893–1896: Scientific Results. Vol. 3, Fridtjof Nansen Fund for the Advancement of Science, 427 pp.

  • 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, 1117–1134.

    • 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, 1551–1570.

    • Search Google Scholar
    • Export Citation

  • Nash, J. D., E. Kunze, C. M. Lee, and T. B. Sanford, 2006: Structure of the baroclinic tide generated at Kaena Ridge, Hawaii. J. Phys. Oceanogr., 36, 1123–1135.

    • Search Google Scholar
    • Export Citation

  • Nash, J. D., M. H. Alford, E. Kunze, K. I. Martini, and S. M. Kelly, 2007: Hotspots of deep ocean mixing on the Oregon continental slope. Geophys. Res. Lett.,34, L01605, doi:10.1029/2006GL028170.

  • Newton, I., 1687: Philosophiae Naturalis Principia Mathematica. 1st ed. Royal Society, 1031 pp.

  • Niwa, Y., and T. Hibiya, 2011: Estimation of baroclinic tide energy available for deep ocean mixing based on three-dimensional global numerical simulations. J. Oceanogr., 67, 493–502.

    • Search Google Scholar
    • Export Citation

  • Pedlosky, J., 2003: Waves in the Ocean and Atmosphere. 1st ed. Springer, 260 pp.

  • Pettersson, O., 1907: Strumstudien vid osterions portar. Sven. Hydrogr. Biol. Komm. Skr.,3, 13–37.

  • Rainville, L., and R. Pinkel, 2006: Propogation of low-mode internal waves through the ocean. J. Phys. Oceanogr., 36, 1220–1236.

  • Rattray, M., 1960: On the coastal generation of internal tides. Tellus, 12, 54–62.

  • Rattray, M., J. G. Dworski, and P. E. Kovala, 1969: Generation of long internal waves at the continental slope. Deep-Sea Res., 16, 179–195.

    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and G. T. Mitchum, 1996: Surface manifestation of internal tides generated near Hawaii. Geophys. Res. Lett., 23, 2101–2104.

    • Search Google Scholar
    • Export Citation

  • Rayleigh, L., 1883: Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. London Math. Soc.,s1–15, 170–177.

  • Rayson, M. D., M. J. Meuleners, G. N. Ivey, N. L. Jones, and G. Wake, 2011: Internal tide dynamics in a topographically complex region; Browse Basin, Australian North West shelf. J. Geophys. Res.,116, C01016, doi:10.1029/2009JC005881.

  • 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, doi:10.1029/2011JC007523.

  • Rudnick, D. L., and Coauthors, 2003: From tides to mixing along the Hawaiian Ridge. Science, 301, 355–357.

  • Sheremet, A., R. T. Guza, S. Elgar, and T. H. C. Herbers, 2002: Observations of nearshore infragravity waves: Seaward and shoreward propagating components. J. Geophys. Res.,107, C83095, doi:10.1029/2001JC000970.

  • Shimizu, K., 2011: A theory of vertical modes in multilayer stratified fluids. J. Phys. Oceanogr., 41, 1649–1707.

  • Sjöberg, B., and A. Stigebrandt, 1992: Computations of the geographical distribution of the energy flux to mixing processes via internal tides and the associated vertical circulation in the ocean. Deep-Sea Res., 39, 269–291.

    • Search Google Scholar
    • Export Citation

  • Smith, W. H. F., and D. T. Sandwell, 1997: Global sea floor topography from satellite altimetry and ship depth soundings. Science, 277, 1956–1962.

    • Search Google Scholar
    • Export Citation

  • St. Laurent, L., and C. Garrett, 2002: The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr., 32, 2882–2899.

  • St. Laurent, L., S. Stringer, C. Garrett, and D. Perrault-Joncas, 2003: The generation of internal tides at abrupt topography. Deep-Sea Res. I, 50, 987–1003.

    • Search Google Scholar
    • Export Citation

  • Thomson, W., 1879: On gravitational oscillations of rotating water. Proc. Roy. Soc. Edinburgh, 10, 92–100.

  • Thorpe, S. A., 2001: Internal wave reflection and scatter from sloping rough topography. J. Phys. Oceanogr., 31, 537–553.

  • 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, 1373–1383.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., 1969: Progressive internal waves on slopes. J. Fluid Mech., 35, 131–144.

  • Wunsch, C., 1975: Internal tides in the ocean. Rev. Geophys. Space Phys., 13, 167–182.

  • Zeilon, N., 1912: On tidal boundary-waves and related hydrodynamical problems. K. Sven. Vetenskapsakad. Handlingar,4, 1–46.

  • 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, 1669–1684.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 1550 1004 63
PDF Downloads 600 110 15