**4.2 Hydrodynamic modelling**

A hydrodynamic model, RMA-10, was utilized to simulate the depth-averaged velocity field of the fore-reef and back-reef along with the shoreline flow under wind and tidal conditions typical of the Jamaican south-east coastal area. RMA-10 is a three-dimensional finite element model for simulation of stratified flow in bays and streams (King, 2005). The primary features of RMA-10 are the solution of the Navier-Stokes equations in three-dimensions; the use of the shallow-water and hydrostatic assumptions; coupling of advection and diffusion

The Hydrodynamic Modelling of Reefal Bays –

maximum the final overtopping formula becomes:

= influence factor for roughness

= influence factor for slope

where it was reduced to 0.0015.

*Y* = wind y component

tested for significance using the t-test.

expectations of large changes.

or three-hour time block.

tidal cycles, and

**4.3 Gyre analysis** 

where:

*f* 

*b* 

Placing Coral Reefs at the Center of Bay Circulation 161

as -5.4 Pa s. The turbulent exchange coefficient of the z direction shear of the x and y direction flow was set at 0.44 Pa s. The turbulent diffusion coefficient associated with the x and y directions were set at 2.11 m2 s-1 and that associated with the z direction set at 0.21 m2 s-1. The Chezy coefficient of 0.029 m0.5 s-1 was used for all nodes except at the shoreline

Particular conditions at the Hellshire coastline led to adding a third variable, Y, to account for the diurnal effect of the wind regime. It was found that emanation of the land-breeze significantly reduced wave heights and caused more variation in the flow over the reef than predicted by the Van de Meer calculations. This variable Y is a function of the southward wind flow and leads to a large reduction in the *q* value once the land-breeze emanates. At a

<sup>3</sup> <sup>0</sup> <sup>0</sup>

*<sup>q</sup> <sup>R</sup> <sup>Y</sup> gH H*

*q* = average wave overtopping discharge (m3 s-1 m-1) *g* = acceleration due to gravity (m s-2) *Hm*0 = significant wave height (m) *Rc* = free crest height above still water line (m)

<sup>1</sup> 0.2 exp( ) exp 2.3 ( )( )

*m fb m*

Comparisons between RMA Model results and field-collected current measurements were

The model mesh was built using assemblages of two-dimensional triangular and quadrilateral elements. The software RAMGEN (King, 2003), a graphics based pre-processor for RMA-10, was used to form the grid and create the interface file that the RMA software utilized. The regional mesh covered the entire south-east coastal shelf including Kingston Harbour to the north, and had two open boundaries - one at the east side and the other south-west. Courser elements (>1 km2) were created for the offshore shelf areas. Elements were more refined (<100 m2) closer to the shoreline or in areas where there were

Individual particles were tracked based on the velocity distribution used by the RMATRK software (King, 2005). This application is designed to track particles released into a surface water system that have been simulated with the RMA-10 model. It transports discrete objects through a surface water system defined by the RMA-10 finite element grid. Time steps were set up so that track increments were drawn for every six minutes in a one-hour

The horizontal expansion and contraction of gyres were measured to quantify the extent of

 Hourly plotted tracks: where new particles were introduced in the same positions at the beginning of each hour for as long as the duration of one and a half tidal cycles, Three-hourly tracks: where new particles were introduced in the same positions at the beginning of each three-hour time block for as long as the duration of one and a half

bay fluctuation. Tracks produced by the RMATRK model were of three categories:

*c*

  (2)

of temperature, salinity and sediment to the hydrodynamics; the inclusion of turbulence in Reynolds stress form; horizontal components of the non-linear terms; and vertical turbulence quantities are estimated by either a quadratic parameterisation of turbulent exchange or a Mellor-Yamada Level 2 turbulence sub-model (Mellor & Yamada, 1982). Computations in the model are based on the Reynolds form of the Navier-Stokes equations for turbulent flows and employ an iterative process that solves simultaneous equations for conservation of mass and momentum. RMA-10 requires the input of nodal x, y and z data depicting sea floor bathymetry, parameters for roughness and eddy viscosity, and boundary conditions of flow discharge. The iterative process computes nodal values of water surface elevation, flow, depth and layered horizontal velocity components or vertically averaged velocity components if this option is used.

Two-dimensional depth-averaged approximations were used for the Hellshire bays' simulations. Depth-averaged results are appropriate given the shallowness of the reefal bay and the knowledge that this usually presents a well-mixed system. Boundary conditions were entered into RMA-10 using a list of nodes defined as flow continuity checks simulating flow over the reefs and also used to specify initial values of salinity concentration (36.0 ppt), temperature (28.0 °C) and suspended sediment concentration (2.0 gL-1) conditions along the model east and west open boundaries. Boundary conditions were also read from a wind velocity and direction file derived from wind data. This was input as hourly averaged wind velocity and direction and allowed the model to read dynamic wind conditions useful in examining the influence of a diurnal wind regime. Boundaries were also subject to a tidalgraph of hourly tidal elevation data for interpolation.

Reef parabola were represented by continuity lines where hydrograph data of dynamic flow over the reef were interpolated. Flow over the reef was calculated as hourly-averaged values using the wave run-up and overtopping Van der Meer equations (Van der Meer, 2002) as a base. Wave overtopping is the average discharge per linear meter of width, *q*, and is calculated in relation to the height of the reef crest line. The final flow value Q used in the hydrograph file is given as the length of the reef parabola long axis multiplied by the average discharge *q*. For breaking waves (*b<sup>0</sup>* ≤ 2), wave overtopping increases for increasing breaker parameter *<sup>0</sup>*. Assumptions are made of a fully developed wave at the reef crest and so the incident wave height is used. Determination of correct wave period for heavy wave-breaking on a shallow fore-shore is neglected here as this requires complex wave transformation Boussinesq models (Nwogu et al., 2008) and lies beyond the objectives of this study. Instead, an average value for the wave period is used. Other influences are included in the general formula such as roughness on the reef slope and the reef slope itself (considered here to be equal to or steeper than 1:8 close to the reef crest). The wave overtopping formula is given as exponential functions with the general form:

$$
\eta = a \cdot \exp(b.R\_c) \tag{1}
$$

The coefficients *a* and *b* are still functions of the wave height, slope angle, breaker parameter and the influence factors of reef roughness and slope; *Rc* being the free crest height above still water line. Wave heights used varied around the predicted value of 0.48 m but were not simulated for extreme events (<1-year event occurrence). A set of turbulent exchange, turbulent diffusion and Chezy coefficients was applied at all nodes. The turbulent exchange coefficient associated with the x and y direction shear of the x and y direction flow was set as -5.4 Pa s. The turbulent exchange coefficient of the z direction shear of the x and y direction flow was set at 0.44 Pa s. The turbulent diffusion coefficient associated with the x and y directions were set at 2.11 m2 s-1 and that associated with the z direction set at 0.21 m2 s-1. The Chezy coefficient of 0.029 m0.5 s-1 was used for all nodes except at the shoreline where it was reduced to 0.0015.

Particular conditions at the Hellshire coastline led to adding a third variable, Y, to account for the diurnal effect of the wind regime. It was found that emanation of the land-breeze significantly reduced wave heights and caused more variation in the flow over the reef than predicted by the Van de Meer calculations. This variable Y is a function of the southward wind flow and leads to a large reduction in the *q* value once the land-breeze emanates. At a maximum the final overtopping formula becomes:

$$\frac{q}{\sqrt{gH\_{m0}^3}} = 0.2 \cdot \exp(Y) \cdot \exp\left[-2.3 \frac{R\_c}{H\_{m0}} \frac{1}{(\chi\_f)(\chi\_b)}\right] \tag{2}$$

where:

160 Hydrodynamics – Natural Water Bodies

of temperature, salinity and sediment to the hydrodynamics; the inclusion of turbulence in Reynolds stress form; horizontal components of the non-linear terms; and vertical turbulence quantities are estimated by either a quadratic parameterisation of turbulent exchange or a Mellor-Yamada Level 2 turbulence sub-model (Mellor & Yamada, 1982). Computations in the model are based on the Reynolds form of the Navier-Stokes equations for turbulent flows and employ an iterative process that solves simultaneous equations for conservation of mass and momentum. RMA-10 requires the input of nodal x, y and z data depicting sea floor bathymetry, parameters for roughness and eddy viscosity, and boundary conditions of flow discharge. The iterative process computes nodal values of water surface elevation, flow, depth and layered horizontal velocity components or vertically averaged

Two-dimensional depth-averaged approximations were used for the Hellshire bays' simulations. Depth-averaged results are appropriate given the shallowness of the reefal bay and the knowledge that this usually presents a well-mixed system. Boundary conditions were entered into RMA-10 using a list of nodes defined as flow continuity checks simulating flow over the reefs and also used to specify initial values of salinity concentration (36.0 ppt), temperature (28.0 °C) and suspended sediment concentration (2.0 gL-1) conditions along the model east and west open boundaries. Boundary conditions were also read from a wind velocity and direction file derived from wind data. This was input as hourly averaged wind velocity and direction and allowed the model to read dynamic wind conditions useful in examining the influence of a diurnal wind regime. Boundaries were also subject to a tidal-

Reef parabola were represented by continuity lines where hydrograph data of dynamic flow over the reef were interpolated. Flow over the reef was calculated as hourly-averaged values using the wave run-up and overtopping Van der Meer equations (Van der Meer, 2002) as a base. Wave overtopping is the average discharge per linear meter of width, *q*, and is calculated in relation to the height of the reef crest line. The final flow value Q used in the hydrograph file is given as the length of the reef parabola long axis multiplied by the

> *b*

reef crest and so the incident wave height is used. Determination of correct wave period for heavy wave-breaking on a shallow fore-shore is neglected here as this requires complex wave transformation Boussinesq models (Nwogu et al., 2008) and lies beyond the objectives of this study. Instead, an average value for the wave period is used. Other influences are included in the general formula such as roughness on the reef slope and the reef slope itself (considered here to be equal to or steeper than 1:8 close to the reef crest). The wave

The coefficients *a* and *b* are still functions of the wave height, slope angle, breaker parameter and the influence factors of reef roughness and slope; *Rc* being the free crest height above still water line. Wave heights used varied around the predicted value of 0.48 m but were not simulated for extreme events (<1-year event occurrence). A set of turbulent exchange, turbulent diffusion and Chezy coefficients was applied at all nodes. The turbulent exchange coefficient associated with the x and y direction shear of the x and y direction flow was set

overtopping formula is given as exponential functions with the general form:

*<sup>0</sup>* ≤ 2), wave overtopping increases for

*<sup>0</sup>*. Assumptions are made of a fully developed wave at the

exp( . ) *<sup>c</sup> q a bR* (1)

velocity components if this option is used.

graph of hourly tidal elevation data for interpolation.

average discharge *q*. For breaking waves (

increasing breaker parameter


*b* Comparisons between RMA Model results and field-collected current measurements were tested for significance using the t-test.

The model mesh was built using assemblages of two-dimensional triangular and quadrilateral elements. The software RAMGEN (King, 2003), a graphics based pre-processor for RMA-10, was used to form the grid and create the interface file that the RMA software utilized. The regional mesh covered the entire south-east coastal shelf including Kingston Harbour to the north, and had two open boundaries - one at the east side and the other south-west. Courser elements (>1 km2) were created for the offshore shelf areas. Elements were more refined (<100 m2) closer to the shoreline or in areas where there were expectations of large changes.

Individual particles were tracked based on the velocity distribution used by the RMATRK software (King, 2005). This application is designed to track particles released into a surface water system that have been simulated with the RMA-10 model. It transports discrete objects through a surface water system defined by the RMA-10 finite element grid. Time steps were set up so that track increments were drawn for every six minutes in a one-hour or three-hour time block.
