The effects of laterally sloping upper and lower boundaries on waves and instability in stratified shear flows

Abstract

The dispersion relation for internal waves travelling in the horizontal direction of a two-dimensional stratified shear flow is affected by the presence of topography, or upper and lower boundaries, which vary only in the horizontal direction normal to that of the flow. This is in accordance with Grimshaw's (1978) findings for long internal waves. Such topography, however, imposes boundary conditions which affect both the growth rates and orientation of the crests of unstable disturbances. Predictions are derived for the particular case of instability of a fluid with a thin interface separating two fluids of different densities in relative flow, which are confined in a tube with horizontal generators but which has upper and lower parallel plane boundaries inclined to the horizontal at an angle α. These predictions are compared with laboratory experiments. There is good qualitative agreement and some quantitative agreement, at least when the predicted angle, γ, between the crest lines of the disturbances and the cross-flow direction, is small. Relatively small disturbances, or billow, amplitudes are found along the centreline of the tube. At large γ, however, the disturbance crests becomes less well-ordered, with billows on one side of the tube centreline connecting with neighbouring pairs on the other, so evolving a zigzag pattern. The results have application to naturally occurring flows in estuaries and channels, and to the development of instability in the stratified atmosphere.