### Abstract

We consider whether small-amplitude topography in a two-dimensional ocean may contain internal wave attractors. These are closed orbits formed by the characteristics (or wave beam paths) of the linear, inviscid, steady-state Boussinesq equations, and their existence may imply enhanced scattering and energy decay for the internal tide when dissipation is present. We develop a numerical code to detect attractors over arbitrary topography, and apply this to random, Gaussian topography with different covariance functions. The rate of attractors per length of topography increases with the fraction of supercritical topography, but surprisingly, it also increases as the amplitude of the topography is decreased, while the supercritical fraction is held constant. This can partly be understood by appealing to Rice's formula for the rate of zero crossings of a stochastic process. We compute the rate of attractors for a covariance function typical of ocean bathymetry away from large features and find it is about 10 attractors per 1000 km. This could have implications for the overall energy budget of the ocean.

Original language | English (US) |
---|---|

Pages (from-to) | 148-174 |

Number of pages | 27 |

Journal | Journal of Fluid Mechanics |

Volume | 787 |

DOIs | |

State | Published - Dec 9 2015 |

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### Keywords

- internal waves
- nonlinear dynamical systems
- topographic effects

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Condensed Matter Physics

### Cite this

**Internal wave attractors over random, small-amplitude topography.** / Guo, Yuan; Holmes-Cerfon, Miranda.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 787, pp. 148-174. https://doi.org/10.1017/jfm.2015.648

}

TY - JOUR

T1 - Internal wave attractors over random, small-amplitude topography

AU - Guo, Yuan

AU - Holmes-Cerfon, Miranda

PY - 2015/12/9

Y1 - 2015/12/9

N2 - We consider whether small-amplitude topography in a two-dimensional ocean may contain internal wave attractors. These are closed orbits formed by the characteristics (or wave beam paths) of the linear, inviscid, steady-state Boussinesq equations, and their existence may imply enhanced scattering and energy decay for the internal tide when dissipation is present. We develop a numerical code to detect attractors over arbitrary topography, and apply this to random, Gaussian topography with different covariance functions. The rate of attractors per length of topography increases with the fraction of supercritical topography, but surprisingly, it also increases as the amplitude of the topography is decreased, while the supercritical fraction is held constant. This can partly be understood by appealing to Rice's formula for the rate of zero crossings of a stochastic process. We compute the rate of attractors for a covariance function typical of ocean bathymetry away from large features and find it is about 10 attractors per 1000 km. This could have implications for the overall energy budget of the ocean.

AB - We consider whether small-amplitude topography in a two-dimensional ocean may contain internal wave attractors. These are closed orbits formed by the characteristics (or wave beam paths) of the linear, inviscid, steady-state Boussinesq equations, and their existence may imply enhanced scattering and energy decay for the internal tide when dissipation is present. We develop a numerical code to detect attractors over arbitrary topography, and apply this to random, Gaussian topography with different covariance functions. The rate of attractors per length of topography increases with the fraction of supercritical topography, but surprisingly, it also increases as the amplitude of the topography is decreased, while the supercritical fraction is held constant. This can partly be understood by appealing to Rice's formula for the rate of zero crossings of a stochastic process. We compute the rate of attractors for a covariance function typical of ocean bathymetry away from large features and find it is about 10 attractors per 1000 km. This could have implications for the overall energy budget of the ocean.

KW - internal waves

KW - nonlinear dynamical systems

KW - topographic effects

UR - http://www.scopus.com/inward/record.url?scp=84949567204&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84949567204&partnerID=8YFLogxK

U2 - 10.1017/jfm.2015.648

DO - 10.1017/jfm.2015.648

M3 - Article

AN - SCOPUS:84949567204

VL - 787

SP - 148

EP - 174

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -