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Scattering of Internal Waves at Finite Topography in Two Dimensions. Part I: Theory and Case Studies

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  • 1 Department of Oceanography, University of Hawaii at Manoa, Honolulu, Hawaii
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Abstract

The scattering of internal gravity waves at finite topography in two dimensions is studied theoretically and numerically for a finite depth ocean. A formal solution is derived using a mapping function based on ray tracing. The solution satisfies radiation conditions. Energy is conserved. The incoming energy flux is redistributed in physical and modenumber space. Numerical solutions are calculated for a single plane wave propagating from the ocean side onto slope–shelf configurations where a flat shallow shelf is connected to a flat deep ocean by linear slopes, staircases, convex or concave parabolic profiles, and half-cosine slopes. The fraction of the incoming energy flux transmitted onto the shelf and reflected back to the deep ocean and the distribution of these fluxes in modenumber space are calculated. The results depend on the parameters of the incident wave and of the topography. Especially important is the distinction between supercritical topography, where the slope of the topography exceeds the wave slope, and subcritical topography. Results obtained are (i) for subcritical topography nearly all of the incoming energy flux is transmitted onto the shelf; (ii) for supercritical topography part is transmitted onto the shelf and part is reflected back to the deep ocean; the partition depends on the incident modenumber and the shelf to ocean depth ratio; (iii) for a linear slope the distribution of the transmitted and reflected fluxes in modenumber space shows peaks at modenumbers roughly consistent with reflection laws; (iv) more of the incident wave energy flux is scattered to higher than to lower modenumbers, especially for near-critical topography; (v) convex slopes are more efficient in scattering the energy flux to high modenumbers than concave slopes; (vi) major differences occur when a linear slope is represented by a series of steps, especially for subscritical topography and high incident modenumbers; and (vii) the scattering results differ in important aspects from the results obtained by reflection theory, especially for supercritical topography and low incident modenumbers. Scattering at ridge configurations is also considered. The results can be inferred from the results for the slope–shelf configurations. The extension to a superposition of incident waves with a realistic spectrum and the implications for internal wave-induced boundary mixing are treated in Part II.

Corresponding author address: Dr. Peter Müller, Department of Oceanography, University of Hawaii at Manoa, 1000 Pope Rd, MSB 429, Honolulu, HI 96822.

Email: pmuller@iniki.soest.hawaii.edu

Abstract

The scattering of internal gravity waves at finite topography in two dimensions is studied theoretically and numerically for a finite depth ocean. A formal solution is derived using a mapping function based on ray tracing. The solution satisfies radiation conditions. Energy is conserved. The incoming energy flux is redistributed in physical and modenumber space. Numerical solutions are calculated for a single plane wave propagating from the ocean side onto slope–shelf configurations where a flat shallow shelf is connected to a flat deep ocean by linear slopes, staircases, convex or concave parabolic profiles, and half-cosine slopes. The fraction of the incoming energy flux transmitted onto the shelf and reflected back to the deep ocean and the distribution of these fluxes in modenumber space are calculated. The results depend on the parameters of the incident wave and of the topography. Especially important is the distinction between supercritical topography, where the slope of the topography exceeds the wave slope, and subcritical topography. Results obtained are (i) for subcritical topography nearly all of the incoming energy flux is transmitted onto the shelf; (ii) for supercritical topography part is transmitted onto the shelf and part is reflected back to the deep ocean; the partition depends on the incident modenumber and the shelf to ocean depth ratio; (iii) for a linear slope the distribution of the transmitted and reflected fluxes in modenumber space shows peaks at modenumbers roughly consistent with reflection laws; (iv) more of the incident wave energy flux is scattered to higher than to lower modenumbers, especially for near-critical topography; (v) convex slopes are more efficient in scattering the energy flux to high modenumbers than concave slopes; (vi) major differences occur when a linear slope is represented by a series of steps, especially for subscritical topography and high incident modenumbers; and (vii) the scattering results differ in important aspects from the results obtained by reflection theory, especially for supercritical topography and low incident modenumbers. Scattering at ridge configurations is also considered. The results can be inferred from the results for the slope–shelf configurations. The extension to a superposition of incident waves with a realistic spectrum and the implications for internal wave-induced boundary mixing are treated in Part II.

Corresponding author address: Dr. Peter Müller, Department of Oceanography, University of Hawaii at Manoa, 1000 Pope Rd, MSB 429, Honolulu, HI 96822.

Email: pmuller@iniki.soest.hawaii.edu

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