TY - JOUR

T1 - Decay of an internal tide due to random topography in the ocean

AU - Bühler, Oliver

AU - Holmes-Cerfon, Miranda

N1 - Funding Information:
The research reported here grew out of a summer project undertaken in collaboration with Erinna Chen and Neil Balmforth at the 2009 Woods Hole summer in Geophysical Fluid Dynamics (see Chen 2009), and we acknowledge several stimulating conversations in this regard, including several with N. Grisouard. Further stimulus was provided at the recent Banff Internal Wave meeting in April 2010 and we gladly acknowledge the organizers of that meeting. The comments of several referees significantly improved our manuscript. Financial support for this work under the United States National Science Foundation grant DMS-0604519 is gratefully acknowledged. M.H.-C. was supported in part by a Canadian NSERC PGS-D scholarship.

PY - 2011/7/10

Y1 - 2011/7/10

N2 - We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.

AB - We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.

KW - Internal waves

KW - topographic effects

KW - wave scattering

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U2 - 10.1017/jfm.2011.115

DO - 10.1017/jfm.2011.115

M3 - Article

AN - SCOPUS:79960135617

SN - 0022-1120

VL - 678

SP - 271

EP - 293

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -