Hydrodynamics of Coral Reefs -
ANRV294-FL39-03 ARI 12 December 2006 6:0 Figure 1 Coral reef on the north shore of Moorea, F.P. (Photo by A. Santoro.) Figure 2 (a) As seen by X-ray tomography: Stylophora pistillata, low velocity morphology (Chang et al. 2004, Reidenbach et al. 2006a). (b) Flow inside the coral shown in (a) as measured by Magnetic Resonance Velocimetry (MRV). This is a horizontal plane taken near mid-height using the MRV technique of Elkins et al. (2003). Image courtesy of S. Chang and C. Elkins. www.annualreviews.org ��� Hydrodynamics of Coral Reefs 39 Annu. Rev. Fluid Mech. 2007.39:37-55. Downloaded from arjournals.annualreviews.org by Prof Stephen Monismith on 01/20/07. For personal use only.
ANRV294-FL39-03 ARI 12 December 2006 6:0 using both real coral specimens and arrays of cylindrical rods, finding that much of the flow that approaches branching corals is diverted around and over the coral. As expected, coral geometries with denser branching tend to divert more flow to the exterior, whereas coral geometries with sparser branching allow more flow through the interior (Reidenbach et al. 2006a). Examining flows inside the coral colony has proved difficult since the outer branches block both optical and acoustic access to the interior. However, recent work (Elkins et al. 2003) (Figure 2b) using magnetic resonance velocimetry (MRV), which is not obstructed by the solid matrix, reveals the complex details of the interaction of wakes from upstream branches encountering downstream branches that leads to blocking of interior flows (S. Chang & C. Elkins, unpublished data). The flow through the coral colony is different in the presence of waves. In exper- iments with several species and morphologies of branching coral, Reidenbach et al. (2006a) confirmed the steady flow behavior discussed above by looking at the dissolu- tion of small gypsum plugs (a proxy for velocity) placed throughout the coral colony interiors. When waves were added, the velocities inside the colony were inferred to be similar to those outside the colony, leading to a wave enhancement of interior mass transfer. Lowe et al. (2005b,c) confirmed this behavior for arrays of cylinders serving as a simple model of a coral, showing that the enhancement depends on the Keulegan-Carpenter number (Dean & Dalrymple 1991), Kc = Uw ��S , (1) where Uw is the wave orbital velocity, �� is the frequency, and S is the cylinder spacing. Kc represents the ratio of the orbital excursion to object geometry, in this case the cylinder spacing. When Kc is large, the flow is drag dominated and velocities inside the colony are much lower than in the free stream. In contrast, when Kc is small, the interstitial flow is inertia dominated and interior velocities more nearly match exterior ones, and total mass transfer is enhanced over that of steady flows. Unlike engineered structures, corals are living structures that respond to flow in complex ways that can alter their physical structure. For example, flow variations inside the colony may induce localized calcification and a specific preferential growth form (Kaandorp et al. 2003, 2005 Kaandorp & Kubler �� 2001 Lesser et al. 1994), or may lead to branch orientation that optimizes nutrient uptake or prey capture (Helmuth & Sebens 1993). Veron & Pichon (1976) show examples of the resulting flow-related intraspecific plasticity variability. Amatzia Genin of the Steinitz Lab in Israel has carried out simple experiments demonstrating the effect of flow on morphology in which coral of a given species and flow environment is exposed in situ to higher flow velocities using underwater pumps (see Reidenbach et al. 2006a). The effect of this flow change is to make the coral skeleton almost uniformly thicker, thus reducing the size of the spaces between the branches. For branching corals this connection should be complicated by the fact that the distribution and rates of mass transfer for flow through and over corals depends on the details of separation from the branches and subsequent reattachment on downstream branches. Separated and reattached flows cause large variations in local heat (and 40 Monismith Annu. Rev. Fluid Mech. 2007.39:37-55. Downloaded from arjournals.annualreviews.org by Prof Stephen Monismith on 01/20/07. For personal use only.
ANRV294-FL39-03 ARI 12 December 2006 6:0 mass) transfer such that heat transfer and momentum can be decoupled (Vogel & Eaton 1985). Using lattice Boltzmann methods to solve the complicated interior flow problem, and assuming that local growth is limited by diffusion to the structure (diffusion limited accretion), Kaandorp and colleagues (Kaandorp et al. 2003, 2005 Kaandorp & Kubler 2001) predict structures that are quite similar in appearance to branching corals. However, although the qualitative appearance is correct, there is no doubt that the devil is in the details. For example, to correctly predict mass transfer of nutrients to the corals, the thin diffusive boundary layers on the coral branches must be resolved. For most nutrients of interest, which have Schmidt numbers Sc = ��/D that are O(1000), scalar boundary layers can be expected to be a factor of Sc1/3 ��� 10 times smaller than any local viscous boundary layers (see e.g., Kays & Crawford 1993). For example, for a 1 cm diameter (d) branch with a flow (U) of 10 cm/s, the viscous boundary layer thickness will be ��v ��� (��d/U)1/2 ��� 0.03 cm, and so a numerical model of mass transfer to this branch would need to resolve layers that are 3 �� 10���3 cm thick. Thus, a grid for a typical coral colony with a linear dimension of 10 cm would have of the order of 30003 ��� 1010 grid points. Even with immersed boundary methods (Chang et al. 2004, Iaccarino & Verzicco 2003) this is still out of reach, although it does seem that computing mass transfer for Sc ��� 1, which would only require 107 grid points, may be attainable and interesting. However, there is also biology to consider! The fact that the observed change in branch thickness with increased flow is nearly uniform suggests the possible impor- tance of internal translocation of nutrients, photosynthates, etc. in the tissue layer that connects the polyps of a given colony (Rinkevich & Loya 1983). Were there no translocation, presumably the geometric modifications would be entirely localized, as modeled by Kaandorp and colleagues. Thus, it seems that to produce a ���virtual coral��� would require coupling the challenging flow model with appropriate spatially explicit models of the biology of the polyps. In any case, although a convincing de- termination of the mechanism(s) involved remains to be produced, the fact that flow velocity affects morphology seems incontrovertible. BOUNDARY LAYER FLOW OVER REEFS: FLOWS AT THE 1- TO 10-M SCALE At scales of 1 to 10 m, the most obvious physical feature of coral reefs is that they are remarkably rough, having bottom drag coefficients, CD, that are typically ten times larger (or more) than the canonical value of 0.0025 found for muddy or sandy sea beds (Lugo-Fernandez et al. 1998b, Roberts et al. 1975). Baird & Atkinson (1997) report values of equivalent sand grain roughness, ks = 0.28 m, for flows in a flume filled with coral skeletons, although McDonald et al. (2006) found that, in part, this large apparent roughness was due to the fact that laboratory experiments in flows of limited depth force significant amounts of flow through the corals rather than over it as with deeper flows. Measurements of wave damping over the reef flat at Kane���ohe Bay reported in Falter et al. (2004) and Lowe et al. (2005a), as well as measure- ments of wave setup reported in Tait (1972) and Gerritsen (1980), gave similar values. www.annualreviews.org ��� Hydrodynamics of Coral Reefs 41 Annu. Rev. Fluid Mech. 2007.39:37-55. Downloaded from arjournals.annualreviews.org by Prof Stephen Monismith on 01/20/07. For personal use only.