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Meanders and Eddies from Topographic Transformation of Coastal-Trapped Waves

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  • 1 Department of Mathematics, University College London, London, United Kingdom
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Abstract

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.

Corresponding author address: Jamie Rodney, Department of Mathematics, Gower St., London, WC1E 6BT, United Kingdom. E-mail: jamier@math.ucl.ac.uk

Abstract

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.

Corresponding author address: Jamie Rodney, Department of Mathematics, Gower St., London, WC1E 6BT, United Kingdom. E-mail: jamier@math.ucl.ac.uk
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