EDGE WAVES IN THE PRESENCE OF STRONG LONGSHORE CURRENTS

A form of the linear, inviscid shallow water wave equation which includes alongshore uniform, but cross-shore variable, longshore currents and bathymetry is presented. This formulation provides a continuum between gravity waves (either leaky or edge waves) on a longshore current, and the recently discovered shear waves. In this paper we will concentrate on gravity wave solutions for which V(x)/c < 1, where V(x) is the longshore current, and c is the edge wave celerity. The effects of the current can be uniquely accounted for in terms of a modification to the true beach profile, h'(x) = h(x) [1 - V(x)/C]-2, where h(x) is the true profile and h'(x) is the effective profile. This is particularly useful in conceptualizing the combined effects of longshore currents and variable bottom topography. We have solved numerically for the dispersion relationship and the cross-shore shapes of edge waves on a plane beach under a range of current conditions. Changes to the edge wave alongshore wavenumber, kappa, of over 50% are found for reasonable current profiles, showing that the departure from plane beach dispersion due to longshore currents can be of the same order as the effect of introducing nonplanar topography. These changes are not symmetric as they are for profile changes; \kappa\ increases for edge waves opposing the current flow (a shallower effective profile), but decreases for those coincident with the flow (a deeper effective profile). The cross-shore structure of the edge waves is also strongly modified. As \kappa\ increases (decreases), the nodal structure shifts landward (seaward) from the positions found on the test beach in the absence of a current. In addition, the predicted variances away from the nodes, particularly for the alongshore component of edge wave orbital velocity, may change dramatically from the no-current case. Many of the edge wave responses are related to the ratio V(max)/c, where V(max) is the maximum current, and to the dimensionless cross-shore scale of the current, \kappa\x(V(max), where x(V(max) is the cross-shore distance to V(max). This is most easily understood in terms of the effective profile and the strong dependence of the edge waves on the details of the inner part of the beach profile. Inclusion of the longshore current also has implications regarding the role of edge waves in the generation of nearshore morphology. For example, in the absence of a current, two phase-locked edge waves of equal frequency and mode progressing in opposite directions are expected to produce a crescentic bar. However, in the presence of a current, the wavenumbers would differ, stretching the expected crescentic bar into a welded bar. A more interesting effect is the possibility that modifications to the edge waves due to the presence of a virtual bar in the effective profile could lead to the development of a real sand bar on the true profile. These modifications appear to be only weakly sensitive to frequency, in contrast to the relatively strong dependence of the traditional model of sand bar generation at infragravity wave nodes.