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Peter Müller and Xianbing Liu

Abstract

The scattering of internal gravity waves at finite topography is studied theoretically and numerically for a two-dimensional finite-depth ocean. In Part I a single incident plane wave was considered. Here a random superposition of incident waves is considered with a spectrum derived from the Garrett and Munk spectrum. The topography is either a slope–shelf or ridge configuration with the bottom being flat in the far fields. The incident energy flux is partitioned into reflected and transmitted waves and redistributed in modenumber space. The scattering is irreversible. In frequency space topography acts like a filter. Waves with frequencies lower than the critical frequency are reflected. Waves with frequencies higher than the critical frequency are transmitted onto the shelf or across the ridge. In modenumber space, both the reflected and transmitted flux spectra are flatter than the incident spectrum, indicating a transfer from low to high modenumbers. This transfer is accompanied by an increase in the energy spectrum and, even more so, in the shear spectrum. A critical modenumber is determined such that the cumulative inverse Richardson number up to this modenumber is one. The flux scattered to modenumbers beyond this critical modenumber is calculated and is assumed to be available for internal wave induced boundary mixing. Various topographic profiles are compared. Convex profiles are more efficient than linear or concave profiles in scattering waves to high modenumbers.

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Peter Müller and Xianbing Liu

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.

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