Wave‐driven flow over shallow reefs

Long‐term (1 month) observations of waves and currents over a natural reef are presented which show a strong correlation between offshore rms incident wave height and cross‐reef currents at subtidal frequencies. The energy spectrum of the cross‐reef currents shows a significant peak at twice the semidiurnal tidal frequency, while the spectrum of sea surface elevation over the reef flat shows no corresponding peak. Furthermore, experimental results reported by Gourlay (1993) show setup over the reef occurs in the absence of a beach, and the cross‐reef transport decreases with an increase in the sea surface slope across the reef flat due to an increase in setup at the top of the reef face. Analytic solutions for flow forced by wave breaking over an idealized reef explain the above features of cross‐reef flows in the absence of a beach. Through the surf zone on the reef face the cross‐reef gradient in the radiation stress due to wave breaking is partitioned between balancing an offshore pressure gradient associated with setup over the reef and forcing a mean flow across the reef. Over the reef flat, where the depth is constant, there is no forcing due to wave breaking and the flow is driven by a pressure gradient which results from the setup through the surf zone. The magnitude of the setup through the surf zone is such that the transport across the reef flat matches the transport through the surf zone which is forced by the gradient in the radiation stress. Solutions are presented for general reef geometry, defined by the reef width and slope of the seaward reef face, and incident wave forcing, defined by the depth at the breakpoint and the depth of water over the reef. As the depth over the reef goes to zero, the solutions converge to the plane beach solutions described by Longuet‐Higgins and Stewart (1964), wave setup is maximized, and the cross‐reef transport is zero. In other cases the relative magnitudes of the setup and the cross‐reef transport depend on the geometry of the reef and the incident wave forcing.