# Hydrodynamic Modeling of a Reef-Fringed Pocket Beach Using a Phase-Resolved Non-Hydrostatic Model

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Study Site and Field Observations

_{p}> ~10 s, see Figure 2a) that occurred between 30 June to 3 July 2017 during which time the significant wave heights H

_{s}recorded by AWACs ranged between 2.4 and 5.0 m (mean Hs 3.4 m, Figure 2a) and the hourly-mean water levels ranged from −0.15 to 0.19 m.

#### 2.2. Numerical Model Description

_{f}is a dimensionless friction coefficient. For the turbulent stresses, separate eddy viscosities are used for horizontal and vertical mixing see [32] for details. The horizontal eddy viscosities in this work were computed from a Smagorisnky type formulation (with default parameters), and a constant vertical viscosity was used to enhance vertical mixing (1 × 10

^{−5}m

^{2}/s).

#### 2.3. Model Set-Up

_{s}from ~2–5 m) and water levels (Figure 2a–d). In this study we used hourly directional wave spectra estimated from the AWAC at C0, which was imposed uniformly along the offshore open boundary. The wave spectra measured at AWAC C0 were back-refracted and de-shoaled to the location of the wavemaker following the approach of [33]. Based on the directional wave spectra, short-crested sea states were generated in the model using a weakly reflective wavemaker including a second order correction to include bound infragravity waves [25].

_{y}and L

_{x}are the cross-shore and long-shore lengths of model domain, respectively (Figure 3). The model was assumed to be in ‘steady state’ when these variables fluctuated about a steady mean see [33], for detailed explanation.

#### 2.4. Model Performance

_{mod}is a model prediction, X

_{obs}is an observation, and overbars indicate averaging over the 17 scenarios. WS was computed at each site for various bulk wave and flow parameters for all 17 hourly scenarios (N = 17), with the over-bar indicating averaging over the 17 scenarios. Good model performance is considered when WS > 0.6, moderate performance when WS is 0.3 to 0.6, and poor when WS < 0.3. In addition to the WS, we also computed the root mean square error (RMSE) and the mean Bias to assess the model performance,

## 3. Results

#### 3.1. Model Calibration

_{s}

_{0}= 3.14 m with a T

_{p}

_{0}= 13.4 s at the offshore boundary). The calibration process focused on assessing the performance to variations in the empirical breaking parameter α [34] and bottom friction coefficient c

_{f}based on the comparison of the modeled and observed waves, mean water levels and currents at the 22 instrument sites. Changing the breaking parameter, α, over a range from 0.04 to 3 did not impact the model skill significantly (not shown), so we used the default value of α = 0.6 within SWASH. We next focused on the sensitivity of the model output to the uniform bottom friction coefficient (c

_{f}) to reproduce the hydrodynamic parameters within the model domain. While other drag formulations are available in SWASH (e.g., Manning), we adopted a constant friction coefficient that is commonly applied, including in reef environments, e.g., [27,38]. We varied the c

_{f}over a range from 0.01 to 0.08 across the domain in intervals of 0.01 for the first eight simulations and higher intervals 0.02 from 0.08 to 0.16 and 0.05 from 0.16 to 0.3. As we describe in Section 4, we also attempted a spatially-varying c

_{f}but did not find improved results for the c

_{f}values considered.

_{f}initially increased the results in general improved for wave heights, setup and currents, reached an optimum level of performance, and then often deteriorated for higher values of c

_{f}. Across the range of c

_{f}values considered, the overall model errors across all variables (SS and IG wave heights, setup, and currents) were lowest using a c

_{f}= 0.05 (Figure 6). The estimated c

_{f}value is within the range of other studies comparing from reef sites typically of the order 0.01–0.1, e.g., [9,39,40,41]. Thus, the c

_{f}value of 0.05 was subsequently applied to the full suite of scenarios.

#### 3.2. Model Application

_{f}) based on a typical (moderate) wave condition, we evaluated the model for all 17 scenarios over the 4-day period. Below, the predicted SS and IG wave heights, current velocities (U and V vectors, as well as the magnitude) and wave setup are compared against the observations. Similar to the analysis in the previous section, the results were grouped into 3 different zones of offshore, channels and lagoon. We also divided the lagoon into northern, central and southern areas based on the position of each instrument within the lagoon relative to the reefs (see yellow dashed lines in Figure 1a). The northern portion of the lagoon is deeper and relatively exposed to offshore waves (compared to the other parts) with scattered rocks near the shoreline (Figure 4). The central area is located behind two offshore reefs and influenced by strong currents due to wave breaking on the reefs. The southern area is the most protected part of the lagoon and typically has low SS wave energy and weak currents due to protection from the rocky headland.

#### 3.2.1. Wave Height and Setup

#### 3.2.2. Depth-Averaged Currents

#### 3.3. General Circulation Dynamics

## 4. Discussion

_{f}values to IG waves and mean currents has been identified in a number of reef hydrodynamic studies, e.g., [18,44,45]. In the present study, the best model performance for IG waves occurred when a uniform c

_{f}of 0.05 was applied across the entire domain (Figure 6). While SS waves, IG waves and setup were predicted with moderate to good skill, predictions of the mean wave-driven current were often less accurate. To investigate if predictions of the currents could be improved by using a spatially-uniform c

_{f}, we considered a simulation with a spatially variable c

_{f}to investigate its impact on the skill of the modeled currents (Figure 12a). The c

_{f}values of different locations within the domain were determined from visual estimation of aerial images. Sand was assigned c

_{f}= 0.002 based on [46], 0.02 for macro-algae vegetation, e.g., [47] and 0.05 for reef, e.g., [48]. We also identified some areas dominated by a mix of sand and vegetation, in which we defined the c

_{f}value of 0.01. The offshore forcing for this simulation was the moderate wave scenario with H

_{s}= 3.4 m with a T

_{p}= 13.4 s at AWAC C0. We compared the result of bulk waves at SS-IG bands, and mean currents (magnitudes and directions) at the 22 observed points, between the spatial c

_{f}and the uniform c

_{f}scenarios. At the offshore sites, there were effectively no differences between the compared variables of both scenarios, that also suggested the bottom friction has no impact on the waves and currents at deep water. However, in shallower depths (i.e., channels and lagoon), the spatially-varying c

_{f}simulations exhibit higher magnitudes of all modeled waves heights and currents than that of the uniform c

_{f}(Figure 12b–e). Overall, using the spatially variable c

_{f}resulted in worse model performance overall than using a spatially-uniform c

_{f}(Figure 12). While it is possible a different set of combinations of c

_{f}values may have resulted in better performance, the computational effort in obtaining the best possible combination of c

_{f}values would likely outweigh the incremental improvement in model performance.

## 5. Conclusions

_{f}= 0.05, was found to best reproduce the observations, which is within the range of value found in many other reef studies. An attempt to use spatially variable friction coefficients representing the different bottom types found at the site (sand, aquatic vegetation, reef) did not improve results. In general, the model performed best in areas where the most accurate bathymetry was available (closest to survey lines), indicating that in complex nearshore reef environments such as at Gnarabup Beach, very high resolution bathymetry (order 1–10 m survey resolution) is necessary to properly resolve the nearshore hydrodynamics, independent of physical processes incorporated within the model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{s}of 3.14 m and T

_{p}of 13.4 s with JONSWAP spectrum that corresponded to the average wave height during the observation period. The RMSE of modeled significant wave height (Figure A1a) and cross-shore velocity (Figure A1c) parameters between 1m grid size and coarser grids showed the difference between SWASH output with 2 m and that of 3 m was not significant. i.e., less than 1 cm and 1 cm/s for Hs and cross-shore currents respectively, yet the error considerably increased when we used grid size ≥4 m. Hence, we considered cross-shore grid resolution of 2 or 3 m to be suitable for the study.

_{s}and velocity components in 2D to a reference grid of 1 × 1 m. For this purpose, we carried out 7 simulations with similar model setting to that of the 1D model in different sizes (cross-shore size of 1–3 m and longshore size of 2–4 m, see Figure A1d) and compare the result among them. The result showed that for all the 2D grids, the errors between the tested grids and the reference of 1 × 1m for Hs and current magnitude were not higher than ~0.05 m and ~0.04 m/s respectively (Figure A1b,d). We also identified grid sizes with a larger ratio between the cross- and alongshore dimension (i.e., 1 × 3 and 2 × 4 m) had larger errors. Despite relatively small differences among the tested 2D grid resolutions, there was a large difference among the simulation time needed to finish the simulations as shown in Figure A2.

**Figure A1.**RMSE for wave height (

**a**,

**b**) and cross-shore velocity magnitude (

**c**,

**d**) of a range of grid cell sizes relative to the reference case with 1 m resolution. The left panels (

**a**,

**c**) are for the 1D case and right (

**b**,

**d**) for the 2D case.

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**Figure 1.**(

**a**) Aerial image and instrument locations (see inset for location in Australia). The red line is the position of the cross-shore transect used for the 1D model (Appendix A). Dashed yellow lines separate the lagoon into 3 area behind the offshore reefs. (

**b**) Bathymetry survey data used to generate the bathymetry grid. (

**c**) Numerical domain (the red box with solid lines, the left side of the box indicates the offshore boundary) and 1 × 1 m grid interpolated bathymetry used in the model. The red dashed box indicates the area of the model output shown in subsequent figures. Source of the aerial image is www.nearmap.com.

**Figure 2.**Hourly observation of offshore conditions at site C0: (

**a**) significant wave height, (

**b**) peak period, (

**c**) peak direction, and (

**d**) mean water level. Vertical red lines indicate the 17 scenarios simulated in SWASH.

**Figure 3.**Calculated kinetic energy (KE), potential energy (PE) and enstrophy (Z) integrated over the model domain versus simulation time.

**Figure 4.**Snapshot of (

**a**) modeled sea surface elevation, (

**b**) hourly SS significant wave height, (

**c**) IG significant wave height and (

**d**) averaged currents fields for a scenario with the average conditions during field observations (H

_{s}

_{0}= 3.14 m, T

_{p}

_{0}= 13.4 s and a water level of −0.18 m at offshore boundary). The current vectors are in logarithmic scale. Black lines at a, b and c are the depth contours.

**Figure 5.**(

**a**) Mean setup of scenario with averaged wave height during observation. (

**b**) Mean setup averaged over 17 scenarios.

**Figure 6.**The influence of friction coefficient (c

_{f}) variability to the average (across each grouping of sensors, offshore, channels, and lagoon) RMS error (

**a**,

**c**,

**e**,

**g**,

**i**,

**k**) and bias (

**b**,

**d**,

**f**,

**h**,

**j**,

**l**) of modeled hydrodynamics within the study area, see Figure 1 to refer the sensor locations. The green and red vertical lines represent the standard deviations whereas the vertical black line indicates a c

_{f}of 0.05.

**Figure 7.**Comparison between observed and modeled SS waves (

**a**–

**c**), IG waves (

**d**–

**f**), and setup (

**g**,

**h**), for the 17 scenarios using a constant bottom friction c

_{f}= 0.05. Green, black and red colors on the figures represent the respective North, Central and South parts of the lagoon (see Figure 1a). Numbers on the subplots represent average values of the respective Willmott Skills, RMSEs and Bias (refer to Table 2).

**Figure 8.**Time series comparison of SS wave (

**a**–

**d**), IG wave heights (

**e**–

**h**), and wave setup (

**i**–

**k**) between the model (circles) and observations (lines) at the various locations that represent North, Central and South part of the site.

**Figure 9.**Observations (red color) and model (black color) results of (

**a**) time-averaged current vectors computed over 17 (hour) scenarios with gray lines are contours at 5 to 15 m depth (see the labels). (

**b**) Average current magnitude and directions obtained from 17 scenarios of SWASH model. The current vectors in (

**b**) are plotted using a logarithmic scale.

**Figure 10.**Comparison between observed and modeled current magnitude and direction, with colors denoting the offshore significant wave heights at C0 (colors). Observations (

**a**,

**b**) in the channels and (

**c**,

**d**) in the lagoon. Circles and triangles on the figures distinguish observations from the Central and South regions, respectively.

**Figure 11.**Modeled mean (hourly) current velocity at South (S1), Central (C1) and North (N1) channels (refer to Figure 1a for the locations) versus observed wave height at the AWAC (C0).

**Figure 12.**(

**a**) Spatial variability of bottom friction (c

_{f}) which was estimated from aerial images. The comparison between data-model with uniform and spatially-variable c

_{f}for (

**b**) SS waves, (

**c**) IG waves, (

**d**) current magnitude and (

**e**) direction.

Site Label | Instrument Type | Approximate Depth (m) | Sampling Frequency (Hz) | Location |
---|---|---|---|---|

Northern area | ||||

N1 | RBR Solo | 3.5 | 2 | Lagoon |

N2 | RBR Solo | 2.5 | 2 | Lagoon |

Central area | ||||

C0 | AWAC/RBR | 21.3 | 1 | Offshore |

C1 | RDI Workhorse | 7.9 | 1 | Channel |

C2 | RBR Solo | 4.7 | 1 | Channel |

C3 | RBR Solo | 4.0 | 2 | Channel |

C4 | RDI Workhorse | 4.0 | 1 | Lagoon |

C5 | RBR Solo | 2.1 | 2 | Lagoon |

C6 | Nortek Aquadopp | 3.4 | 1 | Lagoon |

C7 | RBR Solo | 3.0 | 2 | Lagoon |

C8 | RBR Solo | 2.6 | 2 | Lagoon |

Southern area | ||||

S0 | AWAC/RBR | 19.0 | 1 | Offshore |

S1 | RDI Workhorse | 7.6 | 1 | Channel |

S2 | RBR Solo | 5.5 | 2 | Channel |

S3 | RBR Solo | 3.6 | 2 | Channel |

S4 | RBR Solo | 2.9 | 2 | Lagoon |

S5 | RBR Solo | 2.9 | 2 | Lagoon |

S6 | Nortek Aquadopp | 2.1 | 1 | Lagoon |

S7 | Nortek Vector | 3.3 | 2 | Lagoon |

S8 | RBR Solo | 2.3 | 2 | Lagoon |

S9 | Nortek Vector | 2.5 | 2 | Lagoon |

S10 | Nortek Aquadopp | 2.3 | 1 | Lagoon |

**Table 2.**Summary of Willmott Skill (WS), RMSE and Bias calculated at each sensor averaged across the 17 scenarios.

Sensor Location | SS Wave Heights | IG Wave Heights | Setup | Current Magnitude | Current Direction | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

WS [-] | RMSE [m] | Bias [m] | WS [-] | RMSE [m] | Bias [m] | WS [-] | RMSE [m] | Bias [m] | WS [-] | RMSE [m/s] | Bias [m/s] | WS [-] | RMSE [°] | Bias [°] | |

North | |||||||||||||||

N2 | 0.79 | 0.13 | 0.11 | 0.85 | 0.06 | 0.04 | 0.84 | 0.05 | 0.04 | N/A | N/A | N/A | N/A | N/A | N/A |

N3 | 0.29 | 0.23 | 0.23 | 0.88 | 0.03 | 0.02 | 0.89 | 0.04 | 0.04 | N/A | N/A | N/A | N/A | N/A | N/A |

Central | |||||||||||||||

C0 | 0.92 | 0.29 | −0.02 | 0.62 | 0.08 | 0.06 | N/A | N/A | N/A | 0.47 | 0.09 | −0.07 | 0.52 | 132.85 | −65.91 |

C1 | 0.55 | 0.61 | 0.59 | 0.83 | 0.08 | 0.07 | N/A | N/A | N/A | 0.66 | 0.16 | −0.06 | 0.25 | 45.23 | −43.77 |

C2 | 0.34 | 0.37 | 0.36 | 0.9 | 0.04 | 0.03 | 0.88 | 0.04 | 0.04 | N/A | N/A | N/A | N/A | N/A | N/A |

C3 | 0.3 | 0.21 | 0.21 | 0.87 | 0.03 | 0.00 | 0.85 | 0.05 | 0.05 | N/A | N/A | N/A | N/A | N/A | N/A |

C4 | 0.85 | 0.06 | −0.04 | 0.86 | 0.03 | −0.01 | N/A | N/A | N/A | 0.54 | 0.04 | 0.00 | 0.25 | 66.41 | −58.91 |

C5 | 0.27 | 0.26 | 0.25 | 0.87 | 0.04 | 0.00 | 0.86 | 0.05 | 0.05 | N/A | N/A | N/A | N/A | N/A | N/A |

C6 | 0.29 | 0.21 | 0.20 | 0.84 | 0.04 | 0.00 | 0.88 | 0.05 | 0.04 | 0.32 | 0.03 | 0.02 | 0.78 | 131.8 | 130.55 |

C7 | 0.49 | 0.13 | 0.13 | 0.84 | 0.04 | −0.01 | 0.85 | 0.05 | 0.05 | N/A | N/A | N/A | N/A | N/A | N/A |

C8 | 0.77 | 0.06 | 0.05 | 0.82 | 0.04 | −0.01 | 0.77 | 0.07 | 0.07 | N/A | N/A | N/A | N/A | N/A | N/A |

South | |||||||||||||||

S0 | 0.96 | 0.23 | −0.04 | 0.73 | 0.09 | 0.08 | N/A | N/A | N/A | 0.46 | 0.08 | −0.07 | 0.44 | 128.55 | −105.02 |

S1 | 0.64 | 0.58 | 0.55 | 0.77 | 0.12 | 0.10 | 0.38 | 0.06 | 0.05 | 0.61 | 0.33 | −0.30 | 0.14 | 23.64 | −23.42 |

S2 | 0.85 | 0.13 | −0.05 | 0.87 | 0.04 | 0.02 | 0.83 | 0.05 | 0.05 | N/A | N/A | N/A | N/A | N/A | N/A |

S3 | 0.85 | 0.08 | 0.00 | 0.86 | 0.04 | 0.01 | 0.78 | 0.07 | 0.06 | N/A | N/A | N/A | N/A | N/A | N/A |

S4 | 0.68 | 0.08 | 0.04 | 0.81 | 0.06 | 0.04 | 0.84 | 0.05 | 0.05 | N/A | N/A | N/A | N/A | N/A | N/A |

S5 | 0.82 | 0.06 | 0.01 | 0.83 | 0.05 | 0.02 | 0.79 | 0.06 | 0.06 | N/A | N/A | N/A | N/A | N/A | N/A |

S6 | 0.71 | 0.06 | −0.01 | 0.85 | 0.04 | −0.01 | 0.87 | 0.05 | 0.04 | 0.39 | 0.11 | −0.09 | 0.57 | 103.49 | 40.19 |

S7 | 0.77 | 0.07 | −0.03 | 0.82 | 0.05 | −0.01 | N/A | N/A | N/A | 0.29 | 0.06 | −0.05 | 0.22 | 83.93 | 78.43 |

S8 | 0.8 | 0.06 | −0.01 | 0.84 | 0.05 | −0.01 | 0.76 | 0.07 | 0.07 | N/A | N/A | N/A | N/A | N/A | N/A |

S9 | 0.8 | 0.06 | −0.02 | 0.84 | 0.04 | −0.01 | 0.62 | 0.11 | 0.10 | 0.43 | 0.04 | −0.03 | 0.11 | 89.28 | 88.47 |

S10 | 0.35 | 0.18 | 0.18 | 0.72 | 0.05 | −0.02 | 0.76 | 0.07 | 0.07 | 0.35 | 0.02 | 0.02 | 0.42 | 73.29 | −62.45 |

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## Share and Cite

**MDPI and ACS Style**

Risandi, J.; Rijnsdorp, D.P.; Hansen, J.E.; Lowe, R.J. Hydrodynamic Modeling of a Reef-Fringed Pocket Beach Using a Phase-Resolved Non-Hydrostatic Model. *J. Mar. Sci. Eng.* **2020**, *8*, 877.
https://doi.org/10.3390/jmse8110877

**AMA Style**

Risandi J, Rijnsdorp DP, Hansen JE, Lowe RJ. Hydrodynamic Modeling of a Reef-Fringed Pocket Beach Using a Phase-Resolved Non-Hydrostatic Model. *Journal of Marine Science and Engineering*. 2020; 8(11):877.
https://doi.org/10.3390/jmse8110877

**Chicago/Turabian Style**

Risandi, Johan, Dirk P. Rijnsdorp, Jeff E. Hansen, and Ryan J. Lowe. 2020. "Hydrodynamic Modeling of a Reef-Fringed Pocket Beach Using a Phase-Resolved Non-Hydrostatic Model" *Journal of Marine Science and Engineering* 8, no. 11: 877.
https://doi.org/10.3390/jmse8110877