## Abstract

Internal gravity waves propagating in a uniformly stratified ocean are scattered on reflection from a rough inclined boundary. The boundary is inclined at angle *α* to the horizontal and the roughness is represented by superimposed sinusoidal ripples of amplitude *A* and steepness *S* with crests inclined at angle *ζ* to a horizontal line on the slope. The incident internal waves, of amplitude *a* and steepness *s*, propagate in a constant stability frequency ocean toward the boundary at azimuthal angle *θ*_{I} and at an angle, *β* > *α,* to the horizontal. They have a wavelength *R* times that of the sinusoidal roughness. Effects of the earth’s rotation are neglected. To order *aS,* there are three scattered components: the “primary” component, identical to that reflected from a plane slope with inclination *α,* and scattered “sum” and “difference” components. These scattered components have wavenumber components in the plane of the slope equal to the sum and difference of the internal wave’s wavenumber components and those of the sinusoidal topography. It is found that the directional scatter, measured as the angle between the primary reflected wavenumber component on the inclined plane and that of the scattered wave components, is particularly large for topographic angle *ζ* near 90°, and when the incident wave’s azimuth angle *θ*_{I} is near zero or 180°. The wavenumbers of the scattered waves, especially those of difference scattered waves, may be an order of magnitude larger than the primary reflected waves when *R* is large, in particular when the internal wave upslope wavenumber at zero azimuth exceeds the wavenumber of the roughness. The amplitude of the scattered waves is generally largest for “sum” scattered waves and when *ζ* and *θ*_{I} are small. The amplitude of the largest scattered waves increases as *α* increases from 2.5° to 7.5°, typical of oceanic slopes, with *β* − *α* maintained constant (at 5°) and 0.05 ⩽ *R* ⩽ 0.8. The shear in the scattered waves scales with the steepness of the roughness, and is of comparable size to that in the reflected waves when *R* is large (i.e., typically *R* ≳ 0.4 with *α* = 5°); scattering of the longer incident internal waves by topography contributes to the mean magnitude of the (variable) shear near sloping boundaries. The flux of energy carried by the scattered waves as a fraction of the flux incident on the boundary is largest when *R* is smallest and when *β* is near *α.* In consequence the effect of scattering by rough topography is likely to contribute to the processes dominant when short waves are incident on a sloping boundary at, or near, critical slope (when *α* = *β*). The ratio of the scattered flux to that of the primary reflected wave component increases by factors of 3 to 7 as *α* increases from 2.5° to 7.5° with *β* − *α* = 5° and 0.05 ⩽ *R* ⩽ 0.8; a greater proportion of the incident flux is scattered on the larger slopes. The likelihood of resonant interactions is enhanced by topographic scattering; not only is it possible for the incident and reflected waves to interact resonantly, but the incident and scattered waves may also interact. Conditions for resonance depend on *α, β, θ*_{I}, *ζ,* and *R.* Resonance becomes less likely as *R* increases with other values kept constant, and impossible at second order when *β* > 30°.

*Corresponding author address:* Dr. Steve Thorpe, Bodfryn, Glanrafon, Llangoed, Anglesey LL 58 8PH, United Kingdom.

Email: s.a.thorpe@bangor.ac.uk