Response to the comment by G. Symonds, K.P. Black, and I.R. Young on "A physical derivation of nutrient-uptake rates in coral reefs: effects of roughness and waves

Abstract

The comment by Symonds et al. (2002, this volume) concerns corrections to the value of the parameter G, which is the proportionality constant between wave setup and excess wave height in Eq. (16) in Hearn et al. (2001). G was analyzed in detail by Hearn (1999) and not given any detailed treatment in our paper so as not to detract from the main theme of the effect of roughness and waves on the nutrient uptake rate constant, S. The exact value of G was not of any special concern, and we only discussed the magnitude of wave setup as a means of illustrating the role of roughness in controlling S. Furthermore, note that S is proportional to G raised to the power of 3/8 in our Eq. (19), so that even a 50% error in G produces only 20% error in the nutrient uptake rate constant, S, which is a quantity which had never previously been derived to within an order of magnitude. We discuss here some of the corrections to the Tait (1972) value for G from the perspective of our present understanding of coral reef hydrodynamics, with particular attention to Kaneohe Bay in Hawaii (Hearn and Atkinson 2000) since this reef was used as an illustration in our paper. For the simple Tait (1972) description of wave setup, the statement by Symonds et al. that there is no crossreef flow through the surf zone (so that the model does not conserve mass) is not true. Instead, Tait (1972) uses the usual hydrodynamic methodology of balancing the dominant forces. On the fore-reef these are the pressure gradient (due to wave setup) and the radiation stress due to the breaking of waves. The most significant of the forces which are neglected in the Tait (1972) solution are friction and advection, and this modifies G slightly because these forces produce some ‘set-down’ of the water surface. Frictional set-down is only significant if the slope of the fore-reef is very gentle and an upper limit for its value at Kaneohe Bay is less than 5% of the wave setup. This is based on an extreme lower limit of slope of 10 m in 1,000 m and the highest value of frictional drag coefficient of 0.1 measured by Atkinson and Bilger (1992). To emphasize this point, we did explicitly state: ‘‘there is however an approximation in (16) which assumes that the fore-reef is very steep so that some ‘setdown’ of the surface in the surf zone can be ignored; this slightly affects the value of G’’. On the Kaneohe Bay reef, advection produces an additional set-down of about 0.002 m and we chose not to comment on that very small correction (of order 1% of the wave setup). We used the original Tait (1972) result for G because the later study by Symonds et al. (1995) achieves an analytical solution by using linear hydrodynamics which are of questionable validity for coral reefs; no such assumption is made by Tait (1972). To emphasize this point, we stated in our paper: ‘‘notice that the Tait (1972) result allows for non-linear terms in the continuity equation on the fore-reef and should not be linearized’’, i.e., the wave setup should not be assumed small relative to the water depth. This approximation produces an error of about 25% which is much greater than the effect of friction. Linear hydrodynamics, like Tait (1972), neglects advection. The ratio of advective to frictional set-down (the frictional Reynolds number) is equal to bottom slope/drag coefficient. Because fore-reefs can have slopes reaching at least 0.1, and drag coefficients can be as low as 0.01 (e.g., Atkinson and Bilger 1992), there are situations in which advection is more important than friction although for most reefs, and Kaneohe Bay in particular, the effect of advection is nevertheless small. Coral Reefs (2002) 21: 319–321 DOI 10.1007/s00338-002-0239-4