Application of laboratory experiments to assess the error introduced by the imposition of “wall” boundary conditions in shelf models

Abstract

For mathematical and numerical reasons the physical boundary condition of regularity of solutions (i.e. finite amplitude) at the singular point of vanishing water depth (the shoreline) imposed on wavelike solutions on the continental shelf is replaced by “no normal flow” condition in analytical and numerical studies of continental shelf waves. This “no normal flow” condition that circumvents the mathematical subtlety associated with the singularity of the equations at the shoreline applies only to channel problems where a wall bounds the flow on the shoreline side. To assess the ramifications of this simplification on the phase speeds and radial (cross-shore) structure of the solutions, data from laboratory experiments are compared with predictions of models that employ the two boundary conditions. The phase speed and radial structure are measured in experiments carried out in a turntable with linearly sloping bottom in which the mean water depth vanishes on the shallow side and waves with known frequencies are generated at a point along the perimeter. The dispersion relation and wave’s radial structure are estimated by following particles floating in the water; the measured dispersion relation agrees well with that predicted by a theory that employs the shelf boundary condition and disagrees with the prediction of a theory that employs the channel conditions. This disagreement between the dispersion relation predicted by the channel theory and that predicted by the shelf theory is relevant to typical frequencies that are observed on continental shelves. Although the shelf break condition does not affect the dispersion relation, better agreement is found between the predicted and measured velocity structure when a flat bottom is assumed there instead of a wall.