Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T20:15:13.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Baroclinic tidal conversion: note on a paper of L.R.M. Maas

Published online by Cambridge University Press:  12 August 2022

Carl Wunsch Open the ORCID record for Carl Wunsch [Opens in a new window]  and
Jared Wunsch Open the ORCID record for Jared Wunsch [Opens in a new window]
Carl Wunsch*
Affiliation:
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA
Jared Wunsch
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Maas (J. Fluid Mech., vol. 684, 2011, pp. 5–24) showed that, for an oscillating two-dimensional barotropic tide flowing over sub-critical topography of compact support, some topographic forms existed that produced non-radiating baroclinic disturbances. The problem is related to ‘stealth’ and ‘cloaking’ problems. Here Maas's result is derived using a simpler approach, not involving complicated mappings, but formally restricted to perturbation topography. Wider results come from the discussion of nearly compact support topographic disturbances provided by Schwartz functions with weak high-wavenumber radiation and by exploiting both a known functional equation formulation and Fourier methods. The problem is extended to disturbances on uniform slopes. A variety of non-radiating topographies can be found, although they are mathematically delicate and unlikely to be found in nature. Topography with weak radiation at high wavenumber is a much wider class of structures. Application of these solutions would lie with the ability to estimate dissipation over and near the topography from motions observed at a distance.

JFM classification

Geophysical and Geological Flows: Ocean circulation Geophysical and Geological Flows: Stratified flows Geophysical and Geological Flows: Topographic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 946 , 10 September 2022 , A47
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aczél, J. 1966 Lectures on Functional Equations and Their Applications. Academic Press.Google Scholar
Bahret, W.F. 1993 The beginnings of stealth technology. IEEE Trans. Aerosp. Elec. Syst. 29, 1377–1385.CrossRefGoogle Scholar
Baines, P.G. 1973 Generation of internal tides by flat-bump topography. Deep-Sea Res. 20, 179–205.Google Scholar
Balazs, N.L. 1961 On the solutions of the wave equations with moving boundaries. J. Math. Anal. Appl. 3, 472–484.CrossRefGoogle Scholar
Balmforth, N., Ierley, J.G.R. & Young, W.R. 2002 Tidal conversion by subcritical topography. J. Phys. Oceanogr. 32, 2900–2914.2.0.CO;2>CrossRefGoogle Scholar
Beckebanze, F. & Keady, G. 2016 On functional equations leading to exact solutions for standing internal waves. Wave Motion 60, 181–195.CrossRefGoogle Scholar
Bracewell, R.N. 1978 The Fourier Transform and Its Applications. McGraw-Hill.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.CrossRefGoogle Scholar
Cheney, E.W. 2001 Analysis for Applied Mathematics. Springer.CrossRefGoogle Scholar
Colin De Verdiere, Y. & Saint–Raymond, L. 2020 Attractors for two-dimensional waves with homogeneous hamiltonians of degree 0. Commun. Pure Appl. Maths 73, 421–462.CrossRefGoogle 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
Dyatlov, S. & Zworski, M. 2019 Microlocal analysis of forced waves. Pure Appl. Anal.SEP 1, 359–384.CrossRefGoogle Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu Rev. Fluid Mech. 39, 57–87.CrossRefGoogle Scholar
Gordon, C. & Webb, D. 1996 You can't hear the shape of a drum. Am. Sci. 84, 46–55.Google Scholar
Greenspan, H.P. 1963 A string problem. J. Math. Anal. Appl. 6, 339–348.CrossRefGoogle Scholar
Hazewinkel, J., Tsimitri, C., Maas, L.R.M. & Dalziel, S.B. 2010 Observations on the robustness of internal wave attractors to perturbations. Phys. Fluids 22, 107102.CrossRefGoogle Scholar
Hurley, D.G. 1972 General method for solving steady-state internal gravity wave problems. J. Fluid Mech. 56, 721–740.CrossRefGoogle Scholar
Kac, M. 1966 Can one hear the shape of a drum? Am. Math. Mon. 73, 1–23.CrossRefGoogle Scholar
Kadec, M., Bückmann, T., Schittny, R. & Wegener, M. 2015 Experiments on cloaking in optics, thermodynamics and mechanics. Phil. Trans. R. Soc. Lond. A 373, 20140357.Google Scholar
Liang, X. & Wunsch, C. 2015 Note on the redistribution and dissipation of tidal energy over mid-ocean ridges. Tellus A 67, 1–9.CrossRefGoogle Scholar
Lighthill, M.J. 1958 Fourier Analysis and Generalized Functions. Cambridge University Press.Google Scholar
Llewelyn Smith, S.G. 2011 A conundrum in conversion. J. Fluid Mech. 684, 1–4.CrossRefGoogle Scholar
Llewellyn Smith, S.G. & Young, W. 2002 Conversion of the barotropic tide. J. Phys. Oceanogr. 32, 1554–1566.2.0.CO;2>CrossRefGoogle Scholar
Maas, L.R.M. 2011 Topographies lacking tidal conversion. J. Fluid Mech. 684, 5–24.CrossRefGoogle Scholar
Maas, L.R.M. & Harlander, U. 2011 Tide topography interaction? In Proceedings of the 7th International Symposium on Stratified Flows, Rome, Italy, August 22–26, pp. 1–8. U. Utrecht.Google Scholar
Magaard, L. 1962 Zur. Berechnung interner Wellen in Meeresräumen mit nicht-ebenen Böden bei einer speziellen Dichteverteilung. Open Access Kieler Meeresforschungen 18, 161–183.Google Scholar
Manton, M.J. & Mysak, L.A. 1971 Construction of internal wave solutions via a certain functional equation. J. Math. Anal. Appl. 35, 237–248.CrossRefGoogle Scholar
Morozov, E.G. 2018 Oceanic Internal Tides: Observations, Analysis and Modeling. Springer International.CrossRefGoogle Scholar
Nie, Y.H., Chen, Z., Xie, J., Xu, J., He, Y. & Cai, S., 2019 Internal waves generated by tidal flows over a triangular ridge with critical slopes. J. Ocean Univ. China 18, 1005–1012.CrossRefGoogle Scholar
Pétrélis, F., Llewellyn Smith, S. & Young, W. 2006 Tidal conversion at a submarine ridge. J. Phys. Oceanogr. 36, 1053–1071.CrossRefGoogle Scholar
Poincaré, H. 1885 Sur l’équilibre d'un masse fluide animée d'un mouvement de rotation. Acta Math. 7, 259–380.CrossRefGoogle Scholar
Ray, R.D. & Mitchum, G.T. 1997 Surface manifestation of internal tides in the deep ocean: observations from altimetry and island gauges. Prog. Oceanogr. 40, 135–162.CrossRefGoogle Scholar
Sandstrom, H. 1975 On topographic generation and coupling of internal waves. Geophys. Fluid Dyn. 7, 231–270.CrossRefGoogle Scholar
Sneddon, I.N. 1972 The Use of Integral Transforms. McGraw-Hill.Google Scholar
Stratton, J.A. 1941 Electromagnetic Theory. McGraw-Hill Book Company, Inc.Google Scholar
Wunsch, C. 1969 Progressive internal waves on slopes. J. Fluid Mech. 35, 131–145.CrossRefGoogle Scholar
Zhao, Z. 2017 The global mode-1 S2 internal tide. J. Geophys. Res.: Oceans 122, 8794–8812.CrossRefGoogle Scholar
Zhao, Z., Alford, M.H., Girton, J.B., Rainville, L. & Simmons, H.L. 2016 Global observations of open-ocean mode-1 M2 internal tides. J. Phys. Oceanogr. 46, 1657–1684.CrossRefGoogle Scholar