The effect of continental shelves on tides

Coastal tides are influenced by several factors and one of the most important of these is the character of the adjacent continental shelf. A continental margin theory is derived and used to discuss several different aspects of the effect " 'smooth" continental shelves have on tides. The main results are as follows. (a) The theory suggests, in accordance with observations, that semi-diurnal tides should be amplified on wide shelves in mid and low latitudes, but that diurnal tides should not be amplified. (b) Continental shelf tidal resonance occurs when the shelf scale 9~/(co 2 _f2 ) (~ = shelf bottom slope, co = tidal frequency) is approximately equal to the shelf width. Theoretical arguments and observation can be used to show that shelf resonance occurs, for example, along sections of the northwest Australian shelf. (c) Given the easily obtained coastal tide, theory shows that tides over the continental shelf and slope can be approximately estimated analytically. Calculations using simple prediction formulae can be made on a hand calculator. Subject to some restrictions, a simple and inexpensive method is thus available for estimating barotropic tides on continental shelves. (d) An appropriate boundary condition for global numerical tidal models, which cannot resolve the continental margin region, is derived. For the diurnal tides, the boundary condition can be well approximated by an impermeable wall condition at the deep-sea continental slope boundary. For the semi-diurnal tides, the impermeable wall condition usually, but not always, suffices ; it can break down on wide continental shelves. 1, INTRODUCTION THE INFLUENCE o f continental shelves on barotropic tides can be dramatic. On the wide Patagonian shelf off Argentina, for example, M z coastal tidal amplitudes in excess of 3.5 m are observed. This amplitude is many times that typical in the deep ocean. While amplification of semi-diurnal tides on wide shelves seems to generally occur, diurnal tides are rarely strongly amplified. Why should this be so? How wide should a shelf be before significant semi-diurnal amplification occurs? As continental shelves apparently can strongly affect the tides, what deep-sea boundary condition should global numerical tidal models, which cannot resolve the boundary topography, use? These questions have formed the prime motivation for the research presented in this paper. Some research on the influence of continental shelves on tides has been carried out previously, most notably by FLEMING (1938), REDFIELD (1958), MUNK, SNODGRASS and W1MBUSH (1970), and MILES (1972). Redfield showed that, for the M 2 tide on the east coast of the United States, there is a correlation between shelf width and the coastal tide, with tidal amplitudes generally increasing as the shelf width increases. He also, in effect, suggested that the amplification of the tide across the shelf could be described by the canal theory method of STERNECK (1915). FLEMING (1938) used a similar theory in describing the * Department of Oceanography, University of Washington, Seattle, WA 98195, U.S.A. 665 666 ALLAN J. CLARKE and DAVID S. BATT1STI tidal elevations in the Gulf of Panama. However, as the theory later in this paper shows, longshore gradients and rotation, which are omitted by canal theory, are, in general, important for describing tides on continental shelves. More recent studies by MILES (1972) and MUNK et al. (1970) have taken rotation and longshore gradients into account. Miles examined the effect of continental shelf topography on a deep-sea Kelvin wave propagating along the coastline. Munk summarized possible wave motions on continental shelves and used measurements and theory to construct, with some success, the deep-sea tidal response near the California coast. However, they did not specifically address the questions raised earlier. In this paper, a different approach is adopted. The boundary layer theory describing the approach is presented in Section 2. In following sections the theory is used and in some cases developed further, to discuss the amplification of tides on continental shelves (Section 3), to estimate tides on continental shelves quantitatively (Section 4), and to derive a deep-sea boundary condition for global numerical tidal models (Section 5). 2. A THEORY FOR TIDAL MOTIONS ON CONTINENTAL MARGINS Because motions on the continental shelf and slope regions only will be considered, direct tidal generation by the large-scale astronomical forcing will be ignored. Under this approximation, the Laplace tidal equations, with frictional effects included, are x "~B u , f v = g t l , (2.1) p H