Internal wave reflection and scatter from sloping rough topography

Internal gravity waves propagating in a uniformly stratified ocean are scat tered on reflection from a rough inclined boundary. The boundary is incline d at angle alpha to the horizontal and the roughness is represented by supe rimposed sinusoidal ripples of amplitude A and steepness S with crests incl ined at angle zeta to a horizontal line on the slope. The incident internal waves, of amplitude a and steepness s, propagate in a constant stability f requency ocean toward the boundary at azimuthal angle theta (I) and at an a ngle, beta>alpha, to the horizontal. They have a wavelength R times that of the sinusoidal roughness. Effects of the earth's rotation are neglected. T o order aS, there are three scattered components: the "primary'' component, identical to that reflected from a plane slope with inclination a, and sca ttered "sum'' and "difference'' components. These scattered components have wavenumber components in the plane of the slope equal to the sum and diffe rence of the internal wave's wavenumber components and those of the sinusoi dal topography. It is found that the directional scatter, measured as the a ngle between the primary reflected wavenumber component on the inclined pla ne and that of the scattered wave components, is particularly large for top ographic angle zeta near 90 degrees, and when the incident wave's azimuth a ngle theta (I) is near zero or 180 degrees. The wavenumbers of the scattere d waves, especially those of difference scattered waves, may be an order of magnitude larger than the primary reflected waves when R is large, in part icular when the internal wave upslope wavenumber at zero azimuth exceeds th e wavenumber of the roughness. The amplitude of the scattered waves is gene rally largest for "sum'' scattered waves and when zeta and theta (I) are sm all. The amplitude of the largest scattered waves increases as alpha increa ses from 2.5 degrees to 7.5 degrees, typical of oceanic slopes, with beta - alpha maintained constant (at 5 degrees) and 0.05 less than or equal to R less than or equal to 0.8. The shear in the scattered waves scales with the steepness of the roughness, and is of comparable size to that in the refle cted waves when R is large (i.e., typically R greater than or similar to 0. 4 with alpha = 5 degrees); scattering of the longer incident internal waves by topography contributes to the mean magnitude of the (variable) shear ne ar 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 small est and when b is near a. In consequence the effect of scattering by rough topography is likely to contribute to the processes dominant when short wav es are incident on a sloping boundary at, or near, critical slope (when alp ha = beta). The ratio of the scattered flux to that of the primary reflecte d wave component increases by factors of 3 to 7 as alpha increases from 2.5 degrees to 7.5 degrees with beta - alpha = 5 degrees and 0.05 less than or equal to R less than or equal to 0.8; a greater proportion of the incident flux is scattered on the larger slopes. The likelihood of resonant interac tions is enhanced by topographic scattering; not only is it possible for th e incident and reflected waves to interact resonantly, but the incident and scattered waves may also interact. Conditions for resonance depend on alph a, beta, theta (I), zeta, and R. Resonance becomes less likely as R increas es with other values kept constant, and impossible at second order when bet a >30 degrees.