Abstract
Motivated by shallow ocean waves propagating over coral reefs, we investigate the drift velocities due to surface wave motion in an effectively inviscid fluid that overlies a saturated porous bed of finite depth. Previous work in this area either neglects the large-scale flow between layers (Phillips in Flow and reactions in permeable rocks, Cambridge University Press, Cambridge, 1991) or only considers the drift above the porous layer (Monismith in Ann Rev Fluid Mech 39:37–55, 2007). Overcoming these limitations, we propose a model where flow is described by a velocity potential above the porous layer and by Darcy’s law in the porous bed, with derived matching conditions at the interface between the two layers. Both a horizontal and a novel vertical drift effect arise from the damping of the porous bed, which requires the use of a complex wavenumber k. This is in contrast to the purely horizontal second-order drift first derived by Stokes (Trans Camb Philos Soc 8:441–455, 1847) when working with solely a pure fluid layer. Our work provides a physical model for coral reefs in shallow seas, where fluid drift both above and within the reef is vitally important for maintaining a healthy reef ecosystem (Koehl et al. In: Proceedings of the 8th International Coral Reef Symposium, vol 2, pp 1087–1092, 1997; Monismith in Ann Rev Fluid Mech 39:37–55, 2007). We compare our model with field measurements by Koehl and Hadfield (J Mar Syst 49:75–88, 2004) and also explain the vertical drift effects as documented by Koehl et al. (Mar Ecol Prog Ser 335:1–18, 2007), who measured the exchange between a coral reef layer and the (relatively shallow) sea above.
Highlights
The importance of wave-driven flows and their role in mass transport within the world’s oceans has long been recognised on a macroscopic scale, carrying waste and driftwood [25]
Our work provides a physical model for coral reefs in shallow seas, where fluid drift both above and within the reef is vitally important for maintaining a healthy reef ecosystem (Koehl et al In: Proceedings of the 8th International Coral Reef Symposium, vol 2, pp 1087–1092, 1997; Monismith in Ann Rev Fluid Mech 39:37–55, 2007)
It has been shown that an analysis of surface gravity waves can be extended to a system where the fluid sits atop a saturated porous bed, incorporating the damping effect of this bed on the fluid motion
Summary
The importance of wave-driven flows and their role in mass transport within the world’s oceans has long been recognised on a macroscopic scale, carrying waste and driftwood [25]. In many oceanographic contexts, it is not realistic to treat the water as a layer of fixed depth D with an impenetrable boundary at z = −D – perhaps the simplest example of this is the case of shorelines where the sea overlies a saturated bed of sand, and is known to induce some flow within the sand (as discussed by Phillips [20]) Another such potential extension is to a coral reef underlying the ocean. We discuss the applicability of the model to complicated real-world coral reefs, and compare the predictions of our model with measurements made in the field
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