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Sensitivity of internal wave energy distribution over seabed corrugations to adjacent seabed features

Published online by Cambridge University Press:  04 July 2017

Farid Karimpour ,
Ahmad Zareei ,
Joël Tchoufag  and
Mohammad-Reza Alam Open the ORCID record for Mohammad-Reza Alam [Opens in a new window]
Farid Karimpour
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Ahmad Zareei
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Joël Tchoufag
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Mohammad-Reza Alam*
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
*
†Email address for correspondence: [email protected]
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Abstract

Here we show that the distribution of energy of internal gravity waves over a patch of seabed corrugations strongly depends on the distance of the patch to adjacent seafloor features located downstream of the patch. Specifically, we consider the steady state energy distribution due to an incident internal wave arriving at a patch of seabed ripples neighbouring (i) another patch of ripples (i.e. a second patch) and (ii) a vertical wall. Seabed undulations with dominant wavenumber twice as large as overpassing internal waves reflect back part of the energy of the incident internal waves (Bragg reflection) and allow the rest of the energy to transmit downstream. In the presence of a neighbouring topography on the downstream side, the transmitted energy from the patch may reflect back; partially if the downstream topography is another set of seabed ripples or fully if it is a vertical wall. The reflected wave from the downstream topography is again reflected back by the patch of ripples through the same mechanism. This consecutive reflection goes on indefinitely, leading to a complex interaction pattern including constructive and destructive interference of multiply reflected waves as well as an interplay between higher mode internal waves resonated over the topography. We show here that when steady state is reached both the qualitative and quantitative behaviour of the energy distribution over the patch is a strong function of the distance between the patch and the downstream topography: it can increase or decrease exponentially fast along the patch or stay (nearly) unchanged. As a result, for instance, the local energy density in the water column can become an order of magnitude larger in certain areas merely based on where the downstream topography is. This may result in the formation of steep waves in specific areas of the ocean, leading to breaking and enhanced mixing. At a particular distance, the wall or the second patch may also result in a complete disappearance of the trace of the seabed undulations on the upstream and the downstream wave field.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 824 , 10 August 2017 , pp. 74 - 96
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© 2017 Cambridge University Press 

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References

Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009a Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis. J. Fluid Mech. 624, 191–224.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009b Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation. J. Fluid Mech. 624, 225–253.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2010 Oblique sub-and super-harmonic Bragg resonance of surface waves by bottom ripples. J. Fluid Mech. 643, 437–447.Google Scholar
Ansong, J. K., Arbic, B. K., Buijsman, M. C., Richman, J. G., Shriver, J. F. & Wallcraft, A. J. 2015 Indirect evidence for substantial damping of low-mode internal tides in the open ocean. J. Geophys. Res.: Oceans 120, 6057–6071.Google Scholar
Baines, P. G. 1971a The reflexion of internal/inertial waves from bumpy surfaces. J. Fluid Mech. 46, 273–291.Google Scholar
Baines, P. G. 1971b The reflexion of internal/inertial waves from bumpy surfaces. Part 2. Split reflexion and diffraction. J. Fluid Mech. 49, 113–131.Google Scholar
Baines, P. G. 1998 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Balmforth, N. J. & Peacock, T. 2009 Tidal conversion by supercritical topography. J. Phys. Oceanogr. 39, 1965–1974.CrossRefGoogle Scholar
Bell, T. H. 1975a Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech. 67, 705–722.Google Scholar
Bell, T. H. 1975b Topographically generated internal waves in the open ocean. J. Geophys. Res. 80, 320–327.Google Scholar
Boczar-Karakiewicz, B., Amos, C. L. & Drapeau, G. 1990 The origin and stability of sand ridges on Sable Island Bank, Scotian Shelf. Cont. Shelf Res. 10, 683–704.CrossRefGoogle Scholar
Bragg, W. H. & Bragg, W. L. 1913 The reflection of X-rays by crystals. Proc. R. Soc. Lond. A 88, 428–438.Google Scholar
Buck, W. R. & Poliakov, A. N. B. 1998 Abyssal hills formed by stretching oceanic lithosphere. Nature 392, 272–275.Google Scholar
Bühler, O. & Holmes-Cerfon, M. 2011 Decay of an internal tide due to random topography in the ocean. J. Fluid Mech. 678, 271–293.Google Scholar
Cacchione, D. & Wunsch, C. 1974 Experimental study of internal waves over a slope. J. Fluid Mech. 66, 223–239.Google Scholar
Chang-shu, Y. & Jia-song, S. 1988 Tidal sand ridges on the East China Sea shelf. In Tide-Influenced Sedimentary Environments and Facies, pp. 23–38. Springer.Google Scholar
Couston, L.-A., Guo, Q., Chamanzar, M. & Alam, M.-R. 2015 Fabry–Pérot resonance of water waves. Phys. Rev. E 92, 043015.Google Scholar
Couston, L.-A., Jalali, M. A. & Alam, M.-R. 2017 Shore protection by oblique seabed bars. J. Fluid Mech. 815, 481–510.Google Scholar
Couston, L.-A., Liang, Y. & Alam, M.-R.2016 Oblique internal-wave chain resonance over seabed corrugations, arXiv:1604.07308.Google Scholar
Cox, C. & Sandstrom, H. 1962 Coupling of internal and surface waves in water of variable depth. J. Oceanogr. Soc. Japan 18, 499–513.Google Scholar
Eriksen, C. C. 1982 Observations of internal wave reflection off sloping bottoms. J. Geophys. Res.: Oceans 87, 525–538.Google Scholar
Fabry, C. & Pérot, A. 1897 Sur les franges des lames minces argentees et leur application a la mesure de petites epaisseurs d’air. Ann. Chim. Phys. 12, 459–501.Google Scholar
Fermi, E. & Marshall, L. 1947 Interference phenomena of slow neutrons. Phys. Rev. 71, 666.Google Scholar
Fringer, O. B., Gerritsen, M. & Street, R. L. 2006 An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator. Ocean Model. 14, 139–173.CrossRefGoogle Scholar
Fringer, O. B. & Zhang, Z. 2008 High-resolution simulations of nonlinear internal gravity waves in the South China Sea. In DoD HPCMP Users Group Conference, 2008. DOD HPCMP UGC, pp. 43–46. IEEE.Google Scholar
Garabato, A. C. N., Polzin, K. L., King, B. A., Heywood, K. J. & Visbeck, M. 2004 Widespread intense turbulent mixing in the Southern Ocean. Science 303, 210–213.Google Scholar
Gargett, A. E. & Holloway, G. 1984 Dissipation and diffusion by internal wave breaking. J. Mar. Res. 42, 15–27.Google Scholar
Garrett, C. & Munk, W. 1972 Space-time scales of internal waves. Geophys. Astrophys. Fluid Dyn. 3, 225–264.Google Scholar
Garrett, C. & Munk, W. 1975 Space-time scales of internal waves: a progress report. J. Geophys. Res. 80, 291–297.Google Scholar
Guo, Y. & Holmes-Cerfon, M. 2016 Internal wave attractors over random, small-amplitude topography. J. Fluid Mech. 787, 148–174.Google Scholar
Hecht, F. 2012 New development in FreeFem++. J. Numer. Math. 20, 251–265.Google Scholar
Kang, D. & Fringer, O. 2012 Energetics of barotropic and baroclinic tides in the Monterey Bay area. J. Phys. Oceanogr. 42, 272–290.Google Scholar
Klymak, J. M., Legg, S. M. & Pinkel, R. 2010 High-mode stationary waves in stratified flow over large obstacles. J. Fluid Mech. 644, 321–336.Google Scholar
Klymak, J. M., Pinkel, R. & Rainville, L. 2008 Direct breaking of the internal tide near topography: Kaena ridge, Hawaii. J. Phys. Oceanogr. 38, 380–399.Google Scholar
Kranenburg, C., Pietrzak, J. D. & Abraham, G. 1991 Trapped internal waves over undular topography. J. Fluid Mech. 226, 205–217.Google Scholar
Kryuchkyan, G. & Hatsagortsyan, K. 2011 Bragg scattering of light in vacuum structured by strong periodic fields. Phys. Rev. Lett. 107, 1–4.Google Scholar
Kundu, P. K., Cohen, I. M. & Dowling, D. R. 2012 Fluid Mechanics. Academic.Google Scholar
Labeur, R. J. & Pietrzak, J. D. 2004 Computation of non-hydrostatic internal waves over undular topography. In Shallow Flows, pp. 187–194.Google Scholar
Lamb, K. G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231–254.CrossRefGoogle Scholar
Ledwell, J. R., Montgomery, E. T., Polzin, K. L., St. Laurent, L. C., Schmitt, R. W. & Toole, J. M. 2000 Evidence for enhanced mixing over rough topography in the abyssal ocean. Nature 403, 179–182.Google Scholar
Li, Y. & Mei, C. C. 2014 Scattering of internal tides by irregular bathymetry of large extent. J. Fluid Mech. 747, 481–505.Google Scholar
Liang, Y., Zareei, A. & Alam, M.-R. 2017 Inherently unstable internal gravity waves due to resonant harmonic generation. J. Fluid Mech. 811, 400–420.Google Scholar
Lim, K., Ivey, G. N. & Jones, N. L. 2010 Experiments on the generation of internal waves over continental shelf topography. J. Fluid Mech. 663, 385–400.Google Scholar
Maas, L. R. M., Benielli, D., Sommeria, J. & Lam, F.-P. A. 1997 Observation of an internal wave attractor in a confined, stably stratified fluid. Nature 388, 557–561.Google Scholar
Mathur, M., Carter, G. S. & Peacock, T. 2014 Topographic scattering of the low-mode internal tide in the deep ocean. J. Geophys. Res.: Oceans 119, 2165–2182.Google Scholar
Mathur, M. & Peacock, T. 2010 Internal wave interferometry. Phys. Rev. Lett. 104, 118501.Google Scholar
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335.Google Scholar
Menard, H. W. 1964 Marine Geology of the Pacific. McGraw-Hill.Google Scholar
Mied, R. P. & Dugan, J. P. 1976 Internal wave reflexion from a sinusoidally corrugated surface. J. Fluid Mech. 76, 259–272.CrossRefGoogle Scholar
Müller, P. & Liu, X. 2000a Scattering of internal waves at finite topography in two dimensions. Part I: theory and case studies. J. Phys. Oceanogr. 30, 532–549.Google Scholar
Müller, P. & Liu, X. 2000b Scattering of internal waves at finite topography in two dimensions. Part II: spectral calculations and boundary mixing. J. Phys. Oceanogr. 30, 550–563.Google Scholar
Nicolas, A. 2013 The Mid-oceanic Ridges: Mountains Below Sea Level. Springer Science & Business Media.Google Scholar
Pietrzak, J. & Labeur, R. J. 2004 Trapped internal waves over undular topography in a partially mixed estuary. Ocean Dyn. 54, 315–323.CrossRefGoogle Scholar
Pietrzak, J. D., Kranenburg, C., Abraham, G., Kranenborg, B. & van der Wekken, A. 1991 Internal wave activity in Rotterdam waterway. J. Hyd. Engng 117, 738–757.Google Scholar
Roberts, J. 1975 Internal Gravity Waves in the Ocean. Marcel Dekker.Google Scholar
Sigman, D. M., Jaccard, S. L. & Haug, G. H. 2004 Polar ocean stratification in a cold climate. Nature 428, 59–63.Google Scholar
Simarro, G., Guillén, J., Puig, P., Ribó, M., Iacono, C. L., Palanques, A., Muñoz, A., Durán, R. & Acosta, J. 2015 Sediment dynamics over sand ridges on a tideless mid-outer continental shelf. Mar. Geol. 361, 25–40.Google Scholar
Simpson, J. H. 1971 Density stratification and microstructure in the western Irish sea. Deep Sea Res. Oceanogr. Abst. 18, 309–319.Google Scholar
Stastna, M. 2011 Resonant generation of internal waves by short length scale topography. Phys. Fluids 23, 1–10.Google Scholar
Thorpe, S. A. 1966 On wave interactions in a stratified fluid. J. Fluid Mech. 24, 737–751.Google Scholar
Wang, B., Fringer, O. B., Giddings, S. N. & Fong, D. A. 2009 High-resolution simulations of a macrotidal estuary using SUNTANS. Ocean Model. 28, 167–192.Google Scholar
Wu, Z., Jin, X., Li, J., Zheng, Y. & Wang, X. 2005 Linear sand ridges on the outer shelf of the East China Sea. Chin. Sci. Bulletin 50, 2517–2528.Google Scholar
Yu, J. & Mei, C. C. 2000 Do longshore bars shelter the shore? J. Fluid Mech. 404, 251–268.Google Scholar