This paper describes how topographic variations can transform a small-amplitude, linear, coastal-trapped wave (CTW) into a nonlinear wave or an eddy train. The dispersion relation for CTWs depends on the slope of the shelf. Provided the cross-shelf slope varies sufficiently slowly along the shelf, the local structure of the CTW adapts to the local geometry and the wave transformation can be analyzed by the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) method. Two regions of parameter space are straightforward: adiabatic transmission (where, at the incident wave frequency, a long wave exists everywhere along the shelf) and short-wave reflection (where somewhere on the shelf no long wave exists at the incident frequency, but the stratification is sufficiently weak that a short reflected wave can coexist with the incident wave). This paper gives the solutions for these two cases but concentrates on a third parameter regime, which includes all sufficiently strongly stratified flows, where neither of these behaviors is possible and the WKBJ method fails irrespective of how slowly the topography changes. Fully nonlinear integrations of the equation for the advection of the bottom boundary potential vorticity show that the incident wave in this third parameter regime transforms into a nonlinear wave when topographic variations are gradual or into an eddy train when the changes are abrupt.

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Alongshore variations in coastline curvature or offshore depth profile can create localised regions of shelf wave propagation with modes decaying outside these regions. These modes, termed localised continental shelf waves ( CTWs) here, exist only at certain discrete frequencies lying below the maximum frequency for propagating shelf waves. The purpose of this paper is to obtain these frequencies and construct, both analytically and numerically, and discuss CTWs for shelves with arbitrary alongshore variations in offshore depth profile and coastline curvature. If the shelf curvature changes by a small fraction of its value over the shelf section of interest or a alongshore perturbation in offshore depth profile varies slowly over the same length scale then CTWs can be constructed using WKBJ theory. Two subcases are described: (i) if the propagating region is sufficiently long that the offshore structure of the CTW varies appreciably alongshore then the frequency and alongshore structure are found from a sequence of local problems; (ii) if the propagating region is sufficiently short that the alongshore change in offshore structure of the CTW is small then the alongshore modal structure is given in an explicit uniformly valid form. A separate asymptotic theory is required for curvature perturbations to shelves that are otherwise straight rather than curved. Comparison with highly accurately numerically determined CTWs shows that both theories are extremely accurate with the WKBJ theory having a significantly wider range of applicability and remaining accurate even when the underlying shelf curvature is small. An idealised model for the generation of CTWs is also suggested. A localised time-periodic wind stress generates an evanescent continental shelf wave in the far-field of a localised mode where the coast is almost straight and the response on the shelf is obtained numerically. If the forcing frequency is close to that of an CTW then the wind stress excites energetic motions in the region of maximum curvature, creating a significant localised response far from the forcing region.

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