## Abstract

Previous research has identified a relationship between the rate of dissipation of turbulent kinetic energy,* ε* , and the mass-transfer-limited rate of uptake by a surface, herein called the* ε* ^{1/4} law, and suggests this law may be applicable to nutrient uptake on coral reefs. To test this suggestion, nitrate uptake rate and gravitational potential energy loss have been measured for a section of Warraber Island reef flat, Torres Strait, northern Australia. The reef flat section is 3 km long, with a 3 m tidal range, and on the days measured, subject to 6 m s^{−1} tradewinds. The measured nitrate uptake coefficient,* S* , on two consecutive days during the rising tide was 1.23±0.28 and 1.42±0.52×10^{−4} m s^{−1}. The measured loss of gravitational potential energy across the reef flat,* ΔGPE* , on the same rising tides over a 178 m section was 208±24 and 161±20 kg m^{−1} s^{−2}. Assuming the* ΔGPE* is dissipated as turbulent kinetic energy in the water column, and using the* ε* ^{1/4} law, the mass-transfer-limited nitrate uptake coefficient,* S*_{
MTL
}*,* on the two days was 1.57±0.03 and 1.45±0.04×10^{−4 }m s^{−1}. Nitrate uptake on Warraber Island reef flat is close to the mass-transfer limit, and is determined by oceanographic nitrate concentrations and energy climate.

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## Acknowledgements

The generous support of the School of Mathematics, UNSW through the University Research Support Program and School travel funds is gratefully acknowledged. The authors greatly appreciated the hospitality of the Warraber people, in particular Clara Tamu and Bogo Billy, and Bill and Bev Stephens. We would like to thank Douglas Jacobs of the Torres Strait Regional Council for coordinating the project with the Warraber Council, David Terhill at CSIRO Marine Research who generously undertook the nutrient analysis, and Jean Rueger, UNSW, for the loan of surveying equipment. We would also like to acknowledge the generous help with theoretical aspects provide by Cliff Hearn and Eric Schulz, and Marlin Atkinson for inspiring this work. MB was funded by an Australian Research Council Postdoctoral Fellowship, and RB by a UNSW Goldstar grant.

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Communicated by B.C. Hatcher

## Appendix A

### Appendix A

The one-dimensional equation for depth-averaged flow across the reef flat may be written as:

where* U* is the velocity across the reef flat,* g* is the gravitational acceleration,* η* is the sea level elevation,* d* is the depth,* ρ* is the density of water,* τ*_{
w
} is the wind stress in the x-direction, and *τ*_{
b
} is the bottom stress as a result of friction. A common relationship for the bottom stress is:

where the coefficient of drag,* C* _{D}, depends on the roughness of the reef flat. The wind stress may be written in the same form:

where* ρ* _{a} is the air density (~1.2 kg m^{-3}),* C* _{a} is the drag coefficient of airflow over the ocean surface, and* w* is the wind speed.* C* _{a} depends on the roughness of the sea surface. A value of* C* _{a}=1.0×10^{−3} from offshore measurements in small seas (Smith et al. 1992) has been used.

With values typical of Warraber Island reef flat of* U* ~0.2 m s^{−1},* d* ~ 0.8 m,^{} ∂η/∂x~10^{−4} and* w* ~5 m s^{−1}, a scaling analysis of Eq. (A1) shows that the two acceleration terms are of order 10^{−5}, the wind stress terms is of order 5×10^{−5}, and the pressure gradient term is of order 10^{−3}. Thus to within 5%, the balance between pressure gradient and bottom friction terms reflects a steady-state flow in which acceleration and wind stress terms play no significant role.

The balance can be written in the form:

in which the left term represents the loss of gravitational potential energy and the right term the energy dissipation rate, where* ε* is the dissipation rate of TKE. Equation A4 can be used to obtain a value for* C* _{D}.

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Baird, M.E., Roughan, M., Brander, R.W. *et al.* Mass-transfer-limited nitrate uptake on a coral reef flat, Warraber Island, Torres Strait, Australia.
*Coral Reefs* **23, **386–396 (2004). https://doi.org/10.1007/s00338-004-0404-z

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### Keywords

- Coral reefs
- Friction
- Nutrient uptake
- Hydrodynamics
- Torres Strait
- Great Barrier Reef