Isobath variation and trapping of continental shelf waves

Abstract

Since Trösch (Proceedings of the 4th International Conference on Applied Numerical Modeling, Tainan, Taiwan, 1984 (ed. H. M. Hsia, Y. L. Chou, S. Y. Wang & S. J. Hsieh) Science and Technology Series, vol. 63, 1986, pp. 307–311. American Astronautical Society) found trapped sub-inertial oscillations in computations of low-frequency variability in the Lake of Lugano, models of trapping have generally considered evenly spaced isobaths parallel to shorelines with approximate boundary conditions at any shelf–ocean boundary. Here an asymptotic analysis for slowly varying topography and accurate spectral computations demonstrate trapping on non-rectilinear shelves. It is shown that changes in any of three factors, isobath curvature, distance from the coast and the shelf-break, and the slope at the shelf-break, are sufficient on their own to give trapping. Continental shelves that abut smoothly onto the open ocean are considered thus avoiding the shelf–ocean boundary condition approximation and allowing the accuracy of previous approximations to be assessed.

Trapped continental shelf waves with a free surface

“…It is well known that in the rigid-lid limit (Kaoullas & Johnson 2012; Rodney & Johnson 2015) alongshore variations in the topography can lead to CSWs being trapped both away and along the coast. Here, a regular asymptotic expansion method has been presented to investigate whether similar shelf geometries can still support tCSWs when the rigid-lid assumption is relaxed.…”

“…Unlike the rigid-lid case (Kaoullas & Johnson 2012;Rodney & Johnson 2015), the bracketed term in (4.18) is non-zero; this proves crucial below. At the next order…”

“…the maximum frequency of propagation as a function of the alongshore wavenumber. Expansion (4.5) differs from those in Postnova & Craster (2008) and Kaoullas & Johnson (2012), which are expansions in powers of the reciprocal of the frequency. The strategy here is to obtain a hierarchy of equations where, to leading order, the cross-stream structure of the modes for a rectilinear shelf are determined, and to second order, the mechanisms of trapping are given by a second-order ordinary differential equation for f 0 (ξ ) (see below).…”

“…at the shelf–ocean boundary, however, for simplicity we will use a smooth topography function and extend the flat ocean sufficiently for offshore modes to decay. It is shown in Kaoullas & Johnson (2012), that for the rigid-lid case, locally changing the shelf slope or the distance of the shelf break from the coast can lead to trapped CSWs. Thus for the free-surface case consider the (rather more general) depth profile where, is the distance of the shelf break from the coast, is the slope of the shelf and the depth at the coast (for sufficiently large ).…”

“…Examples of shelf-coastline geometries that admit tCSWs (i.e. submerged ridges and bays) were shown with asymptotic and numerical methods in Kaoullas & Johnson (2012). Changes in coastline curvature (while isobaths remained parallel to the coast) have also been shown by Johnson, Levitin & Parnovski (2006) rigorously to admit tCSWs under (weak) geometric assumptions provided the shelf bends towards the coast, i.e.…”