Skip to main content Accessibility help

Topographies lacking tidal conversion

  • Leo R. M. Maas (a1)


The consensus is that in a stratified sea a classical model of tidal flow over irregular but smooth topography necessarily leads to the generation of internal tides, regardless of the shape of the topography. This is referred to as tidal conversion. Here it is shown, however, that there exists a large class of topographies for which there is neither tidal conversion nor any scattering of incident internal waves. This result is obtained in a uniformly stratified, rigid-lid sea using a barotropic tide that, owing to its large horizontal scale, is supposed to be simply a mass-conserving, periodic back-and-forth flow. The baroclinic response at the tidal frequency is, upon non-dimensionalizing and stretching of coordinates, determined by a standard hyperbolic boundary value problem (BVP). We here solve this hyperbolic BVP by mapping a domain of complicated, yet a priori unknown shape, onto a uniform-depth channel for which the same hyperbolic problem is known to display neither conversion of the barotropic tide nor scattering of internal wave modes. The map achieving this is required to satisfy hyperbolic Cauchy–Riemann equations, defined as analogues of the Cauchy–Riemann equations that are used in solving elliptic problems. Mapping the rigid-lid surface in the original Cartesian frame onto a rigid-lid surface in the transformed frame, this map is solved in terms of one arbitrary function. Each particular function defines a new topographic shape that can be computed a posteriori. The map is unique provided the Jacobian of transformation does not vanish, which is guaranteed for subcritical bottom topography, whose slope is everywhere less than that of the characteristics. For topographies that can thus be mapped onto a channel, tidal conversion and scattering are absent. Examples discussed include the (classical) wedge, a (near-Gaussian) ridge, a continental slope and (near) sinusoidal topographies.


Corresponding author

Email address for correspondence:


Hide All
1. Baines, P. G. 1973 The generation of internal tides by flat-bump topography. Deep-Sea Res. 20 (2), 179205.
2. Baines, P. G. 1982 On internal tide generation models. Deep-Sea Res. 29 (3), 307338.
3. Balmforth, N. J., Ierley, G. R. & Young, W. R. 2002 Tidal conversion by subcritical topography. J. Phys. Oceanogr. 32 (10), 29002914.
4. Bühler, O. & Holmes Cerfon, M. 2011 Scattering of internal tides by random topography. J. Fluid Mech. 638, 526.
5. Cacchione, D. A., Pratson, L. F. & Ogston, A. S. 2002 The shaping of continental slopes by internal tides. Science 296, 724727.
6. Craig, P. D. 1987 Solutions for internal tidal generation over coastal topography. J. Mar. Res. 45, 83105.
7. Garrett, C. & Gerkema, T. 2007 On the body-force term in internal-tide generation. J. Phys. Oceanogr. 37 (8), 2172.
8. Gerkema, T. 2002 Application of an internal tide generation model to baroclinic spring–neap cycles. J. Geophys. Res. 107 (C9), 31243131.
9. Gerkema, T. 2011 Comment on ‘Internal-tide energy over topography’ by Kelly et al. J. Geophys. Res. (in press).
10. Griffiths, S. D. & Grimshaw, R. H. J. 2007 Internal tide generation at the continental shelf modeled using a modal decomposition: two-dimensional results. J. Phys. Oceanogr. 37, 428451.
11. Howard, L. N. & Yu, J. 2007 Normal modes of a rectangular tank with corrugated bottom. J. Fluid Mech. 593, 209234.
12. Llewellyn Smith, S. G. & Young, W. R. 2002 Conversion of the barotropic tide. J. Phys. Oceanogr. 32 (2), 15541566.
13. Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Intl J. Bifurcation Chaos 15, 27572782.
14. Maas, L. R. M. 2009 Exact analytic self-similar solution of a wave attractor field. Physica D 238 (5), 502505.
15. Maas, L. R. M. & Lam, F. P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 141.
16. Maas, L. R. M. & Zimmerman, J. T. F. 1989 Tide-topography interactions in a stratified shelf sea. Part II. Bottom-trapped internal tides and baroclinic residual currents. Geophys. Astrophys. Fluid Dyn. 45, 3769.
17. Maas, L. R. M., Zimmerman, J. T. F. & Temme, N. M. 1987 On the exact shape of the horizontal profile of a topographically rectified tidal flow. Geophys. Astrophys. Fluid Dyn. 38, 105129.
18. Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.
19. Motter, A. E. & Rosa, M. A. F. 1998 Hyperbolic calculus. Advances in Applied Clifford Algebras 8, 109128.
20. Müller, P. & Liu, X. 2000 Scattering of internal waves at finite topography in two dimensions. Part I. Theory and case studies. J. Phys. Oceanogr. 30 (3), 532.
21. Ou, H. W. & Bennett, J. R. 1979 A theory of the mean flow driven by long internal waves in a rotating basin, with application to Lake Kinneret. J. Phys. Oceanogr. 6, 11121125.
22. Ou, H. W. & Maas, L. R. M. 1986 Tidal-induced buoyancy flux and mean transverse circulation. Cont. Shelf Res. 5, 611628.
23. Pétrélis, F., Llewellyn Smith, S. & Young, W. R. 2006 Tidal conversion at a submarine ridge. J. Phys. Oceanogr. 36 (6), 10531071.
24. Sandstrom, H. 1976 On topographic generation and coupling of internal waves. Geophys. Astrophys. Fluid Dyn. 7, 231270.
25. St Laurent, L., Stringer, S., Garrett, C. & Perrault-Joncas, D. 2003 The generation of internal tides at abrupt topography. Deep-Sea Res. (I) 50 (8), 9871003.
26. Vlasenko, V., Stashchuk, N. & Hutter, K. 2005 Baroclinic Tides: Theoretical Modeling and Observational Evidence. Cambridge University Press.
27. Watson, G. N. 1966 A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.
28. Wunsch, C. 1968 On the propagation of internal waves up a slope. Deep-Sea Res. 15, 251258.
29. Wunsch, C. 1971 Note on some Reynolds stress effects of internal waves on slopes. Deep-Sea Res. 18 (6), 583591.
30. Yu, J. & Howard, L. N. 2010 On higher order Bragg resonance of water waves by bottom corrugations. J. Fluid Mech. 659, 484504.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Topographies lacking tidal conversion

  • Leo R. M. Maas (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.