On group velocity and spatial damping of diurnal continental shelf waves

Jan Erik H Weber,Eli Børve

doi:10.1016/j.csr.2021.104630

Abstract

Diurnal continental shelf waves (CSWs) are studied theoretically for an idealized shelf topography. Wave attenuation is caused by the exchange of fluid on the sloping shelf with an inner region through a permeable coastline. As an example, we consider the region outside Lofoten-Vesterålen in north Norway. Here CSWs with diurnal tidal frequencies are possible in a small wave number range centered around zero group velocity. A previous investigation with a Robin condition (a weighted combination of Dirichlet and Neumann conditions) at the permeable boundary has shown that the spatial damping coefficient becomes infinitely large when the group velocity of the CSWs approaches zero. Here we demonstrate that this is not a result of the mathematical formulation, but reflects a physical reality. We show this by modelling the highly convoluted inner archipelagic region as a series of densely packed vertical Hele Shaw cells. By comparing the two ways of describing a permeable coastal boundary (Robin/Hele Shaw), we may express the Robin parameter in terms of the physical parameters (permeability, eddy viscosity) that characterize the flow on the inner porous shelf. The radiation stresses that drive the Lagrangian mean currents are the same in the two cases. This means that the spatial mean current distribution over the sloping shelf becomes unaltered when we compare the Robin case and the porous inner shelf case.

Highlights

It is well known that oscillating fluxes through straits may generate continental shelf waves (CSWs); see e.g. Buchwald and Kachoyan (1987), Middleton (1988), Morrow et al (1990) for the generation of CSWs along the Australian shelf from oscillating motion in the Bass Strait
We demonstrate that a non-zero velocity normal to the coastline inevitably will lead to a spatial damping of the CSWs
The spatial damping coefficient for a CSW on a shelf where the inner archipelago is modelled as a porous medium, is proportional to the in verse group velocity

Summary

Introduction

It is well known that oscillating fluxes through straits may generate continental shelf waves (CSWs); see e.g. Buchwald and Kachoyan (1987), Middleton (1988), Morrow et al (1990) for the generation of CSWs along the Australian shelf from oscillating motion in the Bass Strait. We demonstrate that a non-zero velocity normal to the coastline inevitably will lead to a spatial damping of the CSWs. In a recent paper (Weber and Børve, 2021) this is done by applying a Robin condition (Gustafson, 1998) at the coastal boundary. If the plates are sufficiently close, we can average the pressure-driven quadratic flow between them and obtain a mean friction force that is proportional to the (negative) mean velocity In this way, the fjord system acts as a macroscopic porous medium, obeying Darcy’s law (Bear, 1972) in the direction normal to the coast. By matching the mean velocity and the pressure at the boundary between the permeable inner shelf and the inviscid outer CSW region, we find the complex dispersion relation for the CSWs, yielding the spatial damping rate.

Mathematical formulation

The sloping shelf region

The permeable inner shelf

Matching conditions and the dispersion relation

The mean drift over the shelf

Concluding remarks