The Effect of Alongshore Topographic Variation and Bottom Friction on Shelf Wave Interactions

Abstract

The theory of resonant interactions between continental shelf waves developed by Hsieh and Mysak to explain aspects of the shelf wave spectra observed on the Oregon shelf by Cutchin and Smith and Huyer et al. is extended to include the effect of bottom friction and alongshore topographic variation. The model equations are derived via a multiple-scale asymptotic expansion in which it is assumed that the alongshore topography varies over a length scale over which the nonlinear interactions make an order-one contribution to the dynamics. It is shown that alongshore topographic variability leads to a wavenumber mismatch in the wave resonance conditions. It is possible to identify a purely linear and nonlinear component to the wavenumber mismatch. The linear component can be identified simply as the topographic modulation in a WKB sense of the alongshore wavenumber. The nonlinear component of the wavenumber mismatch is a cumulative effect associated with the dynamic interactions between the waves occurring over regions of alongshore topographic variability. It is shown that even after a triad of initially maximally interacting shelf waves has traversed a topographic anomaly of finite alongshore extent, the energy exchange remains permanently suppressed and does not recover to its pretopographic efficiency. For some specialized alongshore topographic variations, the interaction equations can be solved exactly. An illustrative solution is presented for an isolated topographic feature superimposed on an Adams-Buchwald exponential shelf profile. Numerical solutions are presented for the purely dissipative wave interaction problem. For realistic values of the bottom friction parameter it is possible to almost completely damp out any interaction. It is suggested that the geographically localized nature of observed interacting shelf waves may in part be due to alongshore topographic detuning of the resonance conditions or strong frictional effects.