Skip to main content
×
Home

Tidally generated internal-wave attractors between double ridges

  • P. ECHEVERRI (a1), T. YOKOSSI (a1), N. J. BALMFORTH (a2) and T. PEACOCK (a1)
Abstract

A study is presented of the generation of internal tides by barotropic tidal flow over topography in the shape of a double ridge. An iterative map is constructed to expedite the search for the closed ray paths that form wave attractors in this geometry. The map connects the positions along a ray path of consecutive reflections from the surface, which is double-valued owing to the presence of both left- and right-going waves, but which can be made into a genuine one-dimensional map using a checkerboarding algorithm. Calculations are then presented for the steady-state scattering of internal tides from the barotropic tide above the double ridges. The calculations exploit a Green function technique that distributes sources along the topography to generate the scattering, and discretizes in space to calculate the source density via a standard matrix inversion. When attractors are present, the numerical procedure appears to fail, displaying no convergence with the number of grid points used in the spatial discretizations, indicating a failure of the Green function solution. With the addition of dissipation into the problem, these difficulties are avoided, leading to convergent numerical solutions. The paper concludes with a comparison between theory and a laboratory experiment.

Copyright
Corresponding author
Email address for correspondence: paulae@alum.mit.edu
References
Hide All
Balmforth N. J., Ierley G. R. & Young W. R. 2002 Tidal conversion by subcritical topography. J. Phys. Oceanogr. 32, 29002914.
Balmforth N. J. & Peacock T. 2009 Tidal conversion by supercritical topography. J. Phys. Oceanogr. 39, 19651974.
Balmforth N. J., Spiegel E. A. & Tresser C. 1995 Checkerboard maps. Chaos 5 (1), 216226.
Bell T. H. 1975 Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech. 67, 705722.
Echeverri P., Flynn M. R., Winters K. B. & Peacock T. 2009 Low-mode internal tide generation by topography: an experimental and numerical investigation. J. Fluid Mech. 636, 91108.
Echeverri P. & Peacock T. 2010 Internal tide generation by arbitrary two-dimensional topography. J. Fluid Mech. 659, 247266.
Garrett C. & Kunze E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 5787.
Grisouard N., Staquet C. & Pairaud I. 2008 Numerical simulation of a two-dimensional internal wave attractor. J. Fluid Mech. 614, 114.
Hazewinkel J., Van Breevoort P., Dalziel S. B. & Maas L. R. M. 2008 Observations on the wavenumber spectrum and evolution of an internal wave attractor. J. Fluid Mech. 598, 373382.
Hurley D. G. & Keady G. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution. J. Fluid Mech. 351, 119138.
Jan S., Lien R. C. & Ting C. H. 2008 Numerical study of baroclinic tides in Luzon Strait. J. Oceanogr. 64, 789802.
Lam F. P. A. & Maas L. R. M. 2008 Internal wave focusing revisited; a reanalysis and new theoretical links. Fluid Dyn. Res. 40, 95122.
Llewellyn Smith S. G. & Young W. R. 2003 Tidal conversion at a very steep ridge. J. Fluid Mech. 495, 175191.
Maas L. R. M. 2005 Wave attractors – linear yet nonlinear. Intl J. Bifurcation Chaos 15, 27572782.
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, 557561.
Maas L. R. M. & Lam F. P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 141.
Manders A. M. M., Maas L. R. M. & Gerkema T. 2004 Observations of internal tides in the Mozambique Channel. J. Geophys. Res. 109, C12034.
Nycander J. 2006 Tidal generation of internal waves from a periodic array of steep ridges. J. Fluid Mech. 567, 415432.
Ogilvie G. I. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbances. J. Fluid Mech. 543, 1944.
Ogilvie G. I. & Lin D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610 (1), 477509.
Peacock T., Echeverri P. & Balmforth N. J. 2008 An experimental investigation of internal tide generation by two-dimensional topography. J. Phys. Oceanogr. 38, 235242.
Pétrélis F., Llewellyn Smith, S. G. & Young W. R. 2006 Tidal conversion at a submarine ridge. J. Phys. Oceanogr. 36, 10531071.
Pingree R. D. & New A. L. 1989 Downward propagation of internal tide energy into the Bay of Biscay. Deep Sea Res. A 36, 735758.
Rieutord M., Georgeot B. & Valdettaro L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103144.
Rieutord M. & Valdettaro L. 1997 Inertial waves in a rotating spherical shell. J. Fluid Mech. 341, 7799.
Rieutord M. & Valdettaro L. 2010 Viscous dissipation by tidally forced inertial modes in a rotating spherical shell. J. Fluid Mech. 643, 363394.
Rudnick D. L., Boyd T. J., Brainard R. E., Carter G. S., Egbert G. D., Gregg M. C., Holloway P. E., Klymak J. M., Kunze E., Lee C. M., Levine M. D., Luther D. S., Martin J. P., Merrifield M. A., Moum J. N., Nash J. D., Pinkel R., Rainville L. & Sanford T. B. 2003 From tides to mixing along the Hawaiian Ridge. Science 301, 355357.
St Laurent L. C. & Garrett C. 2002 The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr. 32, 28822899.
St Laurent L. C., Stringer S., Garrett C. & Perrault-Joncas D. 2003 The generation of internal tides at abrupt topography. Deep Sea Res. I 50, 9871003.
Tang W. & Peacock T. 2010 Lagrangian coherent structures and internal wave attractors. Chaos 20, 017508.
Tilgner A. 1999 Driven inertial oscillations in spherical shells. Phys. Rev. E 59, 17891794.
Wunsch C. 1969 Progressive internal waves on slopes. J. Fluid Mech. 35, 131141.
Zhao Z., Klemas V., Zheng Q. & Yan X. H. 2004 Remote sensing evidence for baroclinic tide origin of internal solitary waves in the northeastern South China Sea. Geophys. Res. Lett. 31, L06302.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 174 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd November 2017. This data will be updated every 24 hours.