Hydrodynamic characteristics of a fringing coral reef on the east coast of Ishigaki Island, southwest Japan

Abstract

To investigate the characteristics of currents on a fringing coral reef, a field survey was conducted, mostly under weak wind conditions in summer, on the east coast of Ishigaki Island, southwest Japan, which is encompassed by well-developed fringing reefs. For the same study period, numerical simulations of the current were also performed using a shallow water turbulent flow model with high accuracy reef bathymetry data, which were estimated from high-resolution imagery obtained from satellite remote sensing. The numerical simulation results showed good agreement with the observed data and revealed that the currents have an appreciable magnitude of tide-averaged velocities, even during neap tides, which are governed mostly by wave set-up effects. The results also indicated that temporal variations in velocity and water surface elevation during a tide cycle in the reef exhibit highly asymmetrical patterns; in spring tides especially, the velocities around channels indicate rapid transitions over a short period from peak ebb flow to peak flood flow. The simulations also indicated that a big channel penetrating deeply into the reef attracts the tide-averaged mean flow, even from distant areas of the reef.

Appendix: Derivation of modified SDS-Q3D turbulent flow model equations

The SDS-Q3D model is based on the 3-D Reynolds equations with the usual gradient-diffusion-type modeling for the Reynolds stress terms and the hydrostatic assumption for the vertical distribution of pressure. The horizontal momentum equation is simply expressed as follows:

$$ \frac{{\partial {\varvec {u}}}}{{\partial t}} + {\varvec {M}}({\varvec {u}}) = 0 $$

(1)

where t is time; u, a horizontal velocity vector; and M(u), the terms except for the time derivative term. In the SDS-Q3D model, the Galerkin method is employed for the Quasi-3-D formulation.

$$ {\int\limits_{- h}^{\eta} {P_{i} (z) {\left({\frac{{\partial {\varvec {u}}}}{{\partial t}} + {\varvec {M}}({\varvec {u}})} \right)}}}{\rm d}z = 0,\quad {\left({i = 1, \ldots, N} \right)} $$

(2)

where h is water depth and η is water surface elevation. Here, Legendre polynomials are used as the vertical dependence functions P _i (z). For the vertical coordinate z, the following transformation is introduced.

$$ \tilde{z} = \frac{{2z + h - \eta}}{{h + \eta}} $$

(3)

By using this transformation, ${\tilde{z}}$ ranges from ${\tilde{z}= -1\, \hbox{at}\,z = h\,\hbox{to}\,\tilde{z}= 1\,\hbox{at}\,z = \eta}$ and the Legendre polynomials are expressed as follows:

$$ P_{1} (\tilde{z}) = 1, \quad P_{2} (\tilde{z}) = \tilde{z}, \quad P_{3} (\tilde{z}) = \frac{1}{2}{\left({3 \tilde{z}^{2} - 1} \right)}, \quad P_{4} (\tilde{z}) = \frac{1}{2}{\left({5 \tilde{z}^{3} - 3 \tilde{z}} \right)},\, \ldots $$

(4)

The momentum equation is expressed as

$$ {\int\limits_{- 1}^{\;1} {P_{i} (\tilde{z}) {\left({\frac{{\partial {\varvec {u}}(\tilde{z})}}{{\partial t}} + {\varvec {M}}({\varvec {u}}(\tilde{z}))} \right)}}}\;{\rm d} \tilde{z} = 0 $$

(5)

The horizontal velocity vector u(x, y, z, t) is expressed with the vertical profile functions P _j (z)(j =  1,..., N), each of which is multiplied with a weight function U _j (x,y,t), i.e.,

$$ {\varvec {u}}^{N} (x,y, \tilde{z},t) = {\sum\limits_{j = 1}^N {P_j(\tilde{z}) {\varvec {U}}_j(x,y,t)}} $$

(6)

The vertical velocity w(x, y, z, t) is derived from the continuity equation:

$$ w^{N} (x,y, \tilde{z},t) = - {\sum\limits_{j = 1}^N {I_j(\tilde{z})\,{\rm div}{\varvec {U}}_j(x,y,t)}} $$

(7)

where

$$ I_j\;(\tilde{z}) = {\int\limits_{- h}^{{\kern 1pt} \;z} {P_j\;(\tilde{z})\,{\rm d}z}} = \frac{{h + \eta}}{2}{\int\limits_{- 1}^{{\kern 1pt} \tilde{z}} {P_j\;(q)\,{\rm d}q}} $$

(8)

In addition, the derivative and the integrated value of P _i (z) are defined as

$$ J_j\;(\tilde{z}) = {\int\limits_{z}^{\eta} {P_j\;(\tilde{z})\,{\rm d}z}} = \frac{{h + \eta}}{2}{\int\limits_{\tilde{z}}^{1} {P_j\;(q)\,{\rm d}q}} $$

(9)

$$ D_j\;(\tilde{z}) = \frac{\partial}{{\partial z}}P_j\;(\tilde{z}) = \frac{{\partial P_j\;(\tilde{z})}}{{\partial \tilde{z}}}\frac{{\partial \tilde{z}}}{{\partial z}} = \frac{2}{{h + \eta}}\frac{{\partial P_j\;(\tilde{z})}}{{\partial \tilde{z}}} $$

(10)

Substituting Eqs. 6 and 7 into Eqs. 5, the horizontal momentum equations are derived:

$$ \begin{aligned} & {\sum\limits_j {pLiLj \cdot \frac{{\partial U_j}}{{\partial t}}}} + {\sum\limits_j {{\sum\limits_k {{\left({pLiLjLk \cdot {\left({\varvec U_j \cdot \nabla U_k} \right)} - pLiIjDk \cdot {\left({\nabla \cdot \varvec U_j} \right)} \cdot U_k} \right)}}}}}\\ &\quad = - pLi \cdot g\frac{{\partial \eta}}{{\partial x}} + \frac{2}{{h + \eta}}{\left({P_{{i\;}} (\eta) \cdot \tau sx - P_{{i\;}} (- h) \cdot \tau bx} \right)} - \frac{2}{{h + \eta}} \cdot {\left({\frac{{\partial S_{{xx}}}}{{\partial x}} + \frac{{\partial S_{{yx}}}}{{\partial y}}} \right)}\\ &\qquad + {\sum\limits_j {pLiLj \cdot \frac{\partial}{{\partial x}}{\left({2\nu _{t} \frac{{\partial U_j}}{{\partial x}}} \right)}}} - \frac{2}{3}pLi \cdot \frac{{\partial k}}{{\partial x}} + {\sum\limits_j {pLiLj \cdot \frac{\partial}{{\partial y}}{\left({\nu _{t} {\left({\frac{{\partial U_j}}{{\partial y}} + \frac{{\partial V_j}}{{\partial x}}} \right)}} \right)}}}\\ &\qquad - \nu _{t} {\sum\limits_j {{\left({pDiDj \cdot {\left({\frac{2}{{h + \eta}}} \right)}^{2} \cdot U_j - pDiIj \cdot {\left({\frac{{\partial ^{2} U_j}}{{\partial x^{2}}} + \frac{{\partial ^{2} V_j}}{{\partial x\partial y}}} \right)}} \right)}}}\\ &{\sum\limits_j {pLiLj \cdot \frac{{\partial V_j}}{{\partial t}}}} + {\sum\limits_j {{\sum\limits_k {{\left({pLiLjLk \cdot {\left({\varvec U_j \cdot \nabla V_k} \right)} - pLiIjDk \cdot {\left({\nabla \cdot \varvec U_j} \right)} \cdot V_k} \right)}}}}}\\ &\quad = -pLi \cdot g\frac{{\partial \eta}}{{\partial y}} + \frac{2}{{h + \eta}}{\left({P_{{i\;}} (\eta) \cdot \tau sy - P_{{i\;}} (- h) \cdot \tau by} \right)} - \frac{2}{{h + \eta}} \cdot {\left({\frac{{\partial S_{{xy}}}}{{\partial x}} + \frac{{\partial S_{{yy}}}}{{\partial y}}} \right)}\\ &\qquad+ {\sum\limits_j {pLiLj \cdot \frac{\partial}{{\partial x}}{\left({\nu _{t} {\left({\frac{{\partial U_j}}{{\partial y}} + \frac{{\partial V_j}}{{\partial x}}} \right)}} \right)}}} +{\sum\limits_j {pLiLj \cdot \frac{\partial}{{\partial y}}{\left({2\nu _{t} \frac{{\partial V_j}}{{\partial y}}} \right)}}} - \frac{2}{3}pLi \cdot \frac{{\partial k}}{{\partial y}}\\ &\qquad - \nu _{t} {\sum\limits_j {{\left({pDiDj \cdot {\left({\frac{2}{{h + \eta}}} \right)}^{2} \cdot V_j - pDiIj \cdot {\left({\frac{{\partial ^{2} U_j}}{{\partial x\partial y}} + \frac{{\partial ^{2} V_j}}{{\partial y^{2}}}} \right)}} \right)}}}\\ &\qquad \quad {\left({i = 1, \ldots, N} \right)}\\ \end{aligned} $$

(11)

where g is gravity acceleration; τ_s is wind stress; τ_b is bottom friction; S _xx , S _yx , S _xy , and S _yy are radiation stress; and k and ν_t are turbulent kinetic energy and eddy viscosity due to SDS turbulence. The methods for evaluating k and ν_t are found in, e.g., Nadaoka and Yagi (1998). The coefficients in Eq. 11 are defined as follows:

$$ \begin{aligned} pLi &= \left\{{\begin{array}{*{20}c} {{2\quad (i = 1)}} \\ {{0\quad (i \ne 1)}} \\ \end{array}}, \right. \quad pLiLj = \left\{{\begin{array}{*{20}c} {{2/(2j - 1)\quad (j = i)}} \\ {{\quad 0\quad \quad \quad (j \ne i)}} \\ \end{array}} \right.\\ pLiJj &= {\int\limits_{- 1}^{{\kern 1pt} \;1} {P_i\;(\tilde{z}) \cdot {\left({{\int\limits_{\, \tilde{z}}^{\,1} {P_j\,(t)\,{\rm d}t}}} \right)}\,{\rm d} \tilde{z}}}\\ pLiLjLk &= {\int\limits_{- 1}^{{\kern 1pt} \;1} {P_i\;(\tilde{z})P_j\,(\tilde{z})P_k\,(\tilde{z})\,{\rm d}\tilde{z}}}\\ pLiIjDk &= {\int\limits_{- 1}^{{\kern 1pt} \;1} {P_i\;(\tilde{z}) \cdot {\left({{\int\limits_{- 1}^{\; \tilde{z}} {P_j(t)\;{\rm d}t}}} \right)}\frac{{\partial P_k(\tilde{z})}}{{\partial \tilde{z}}}\,{\rm d} \tilde{z}}}\\ pDiDj &= {\int\limits_{- 1}^{{\kern 1pt} \;1} {\frac{{\partial P_i\;(\tilde{z})}}{{\partial \tilde{z}}}\frac{{\partial P_j\;(\tilde{z})}}{{\partial \tilde{z}}}\,{\rm d} \tilde{z}}}\\ pDiIj &= {\int\limits_{-1}^{{\kern 1pt} \;1} {\frac{{\partial P_i(\tilde{z})}}{{\partial \tilde{z}}} \cdot {\left({{\int\limits_{- 1}^{\, \tilde{z}} {P_j\;(t)}}\,{\rm d}t} \right)}\,{\rm d} \tilde{z}}}\\ \end{aligned} $$

The continuity equation is derived by depth integration from −h to η:

$$ \frac{{\partial \eta}}{{\partial t}} + {\sum\limits_{i = 1}^N {{\left({\frac{\partial}{{\partial x}}{\int\limits_{- h}^{\;\eta} {P_{i} (z)\,{\rm d}z \cdot}}\;U_i + \frac{\partial}{{\partial y}}{\int\limits_{- h}^{\;\eta} {P_{i} (z)\,{\rm d}z \cdot \;}}V_i} \right)}}} = 0 $$

(12)

As the orthogonal property of Legendre polynomials, the continuity equation is simply expressed as follows:

$$ \frac{{\partial \eta}}{{\partial t}} + \frac{\partial}{{\partial x}}(h + \eta)\;U_1 + \frac{\partial}{{\partial y}}(h + \eta)\;V_1 = 0 $$

(13)

In the present study, only one component is employed in Eq. 6, as the currents generated by tidal and wave set-up effects are principally barotropic in nature. The quasi 3-D model presented here, however, may be applied to more general cases of hydrodynamic simulations taking also wind or density current effects, or both, into account, in which case, 3-D features of currents will be important.