Undular bore development over coral reefs

Abstract

A series of laboratory experiments were designed for a schematized 1/20 fringing reef. Both regular and bichromatic wave experiments were carried out. The focus lies on the regular waves experiments. Six regular wave experiments were carried out. During the first two experiemnts, cnoidal waves were generated with typical swell scales, while the latter four experiments involved waves with scales similar to infragravity waves. High-resolution collocated measurements of surface elevation and velocity allowed proper separation of incoming and outgoing wave components at 11 positions along the reef. This made a detailed analysis of wave transformation over the reef flat and run-up measurements on the beach possible. A collocated decomposition method has been developed which takes into account the effects of nonlinearity and dispersion, and can be used for deforming waves on the reef flat. The incoming and outgoing waves are given in terms of discharge or free surface elevation. For all wave conditions, undular bores develop over the reef flat and well-developed undulations are observed at the beach toe. The processes leading to the development of the undulations are however quite different for the different wave conditions. For the swell cases, the wave front steepens considerably on the fore-reef slope and the waves subsequently break violently at the reef crest. The breaking bore then decays progressively until it stops breaking and start forming undulations at mid reef. The IG-scale waves stay relatively symmetric about the vertical during the shoaling process on the fore-reef. The front only becomes steeper on the reef flat, and does not break turbulently for all IG-scale cases. For some cases the steepening process continues and is directly followed by formation of undulations. For the more energetic IG-scale cases the front first breaks turbulently. The rate at which the waves steepen on the reef flat depends, as expected, on the wave height and wave length. This is in concordance with the theory of Stoker; larger and shorter waves steepen faster than smaller and longer waves. Consequently, shorter shallow water waves starts forming undulations earlier on and feature more developed undulations when they arrive at the beach. The cross-shore evolution of the total variance clearly reflects these differences in behaviour. The cases involving strong breaking (i.e. the swell cases) show a rapid decrease of the variance at the outer reef flat, while the IG wave cases involving no breaking experience a slow decay of energy, most probably due to bottom and wall friction. The leading undulation is generally the largest and propagates faster than the trailing undulations due to amplitude dispersion. The leading undulation progressively separates from the rest of the wave train, and ultimately resembles a solitary wave when it arrives at the toe of the beach. Nonlinear steepening is associated with generation of higher harmonics at a multiple of the primary frequency. The undulations are also higher harmonics, but form a secondary peak instead a tail of decreasing energy in the spectrum. This secondary peak shifts towards lower frequencies as the undular bore travels over the reef. Moreover, the reflection coefficient is larger for a larger offshore period and wave height. The relative magnitude of HF bulk energy with respect to energy in the LF band near the beach differs per experiment. For the swell waves and IG-scale waves with smaller periods, the major part of LF energy is transferred to the HF band due to undular bore formation. For the cases that involve wave breaking, energy dissipation is, at least initially, the main mechanism for energy loss in the LF band. However, as energy decay continues and the bore becomes weaker, radiation of energy at the front by means of formation of undulations can take over this role. The undulations break while running up the beach. The signature of the first undulations is clearly visible in the run-up signal. It is shown that the leading undulation governs maximum run-up. More specifically, we demonstrate that empirically derived formulas for the run-up of breaking solitary waves are able to predict the maximum run-up accurately, when the characteristics of the leading undulation at the beach toe are used. Two types of empirical formulations were presented, of which only requires a single parameter to be estimated from data. This empirical formula, formally valid for solitary waves, can be used for run-up estimation of undular bores. Given the current fit, maximum run-up can be estimated for a given reef if the beach slope is known.