On the second approximation to mass transport in the bottom boundary layer

Abstract

The second approximation is obtained for the mass transport velocity within the oscillatory bottom boundary layer beneath sinusoidal progressive and standing waves of finite amplitude. This approximation includes a simple new term, which essentially ensures continuity of the vertical gradient of mass transport at the edge of the layer and is of third-order in the perturbation (or wave-slope) parameter. For long progressive waves in conditions of zero net mass flow, the term represents a moderate reduction in mass transport at the edge of the layer, compared with the first approximation of Longuet-Higgins. For standing waves of arbitrary length, the mass transport is reduced (increased) far from (near) the bottom, except near nodal locations where an increase (a reduction) is predicted. The proposed correction to the first approximation yields clearly improved results when compared with appropriate experimental evidence. Deficiencies in the higher-order theories of Sleath and Isaacson for propagating waves are disclosed.