Testing a Model of Wind-Driven Waves in a Low-Energy Environment with High Spatial Resolution Data Free

Most marine and oceanographic wind-driven wave model studies have been conducted in moderate- to high-energy systems. This has been critical for understanding how waves can affect transportation and infrastructure. As our understanding of coastal processes and sea-level rise become more nuanced, it is valuable to bring a similar knowledge to lower-energy systems. Improved understanding of wind-driven waves in low-energy systems will help planning for sea-level rise, long-term erosional processes, vegetative shoreline protection, and the design of living shorelines.

Empirical wave-spectrum models describe how wind-driven wave energy is distributed over frequencies. Pierson and Moskowitz (1964) developed the first spectral wave model that assumed fully developed seas. The Joint North Sea Wave Observation Project (JONSWAP) wave spectrum was developed after the Pierson and Moskowitz (1964) model to depict wave growth in fetch-limited conditions and is based on data collected from the North Sea (Hasselmann et al., 1973). Although waves in the North Sea may be fetch-limited, wave energy in this region is still relatively high, with significant wave height (Hs) typically >1 m and frequency <1 Hz. The JONSWAP has been used and evaluated worldwide (Mazzaretto, Menédez, and Lobeto, 2022). No spectral models have been specifically designed for exceptionally low-energy systems.

Various definitions have been proposed for high vs. low-energy waves. These definitions may apply to either an event (such as a storm) or an environment (that consistently has high- or low-energy waves). Jackson et al. (2002) suggested that low-energy locations can be defined, where Hs should be <0.25 m during nonstorm events, Hs should be <0.5 m during strong onshore winds, and the width of the beach face should be <20 m in microtidal environments. Fairley et al. (2020) classified enclosed and semi-enclosed seas and sheltered ocean coasts as low-energy systems.

Numerical wave models—such as Steady State Spectral Wave (STWAVE; Massey et al., 2011), WAve Modeling (WAM; Wave Modeling Group, 1988), WAVEWATCH (Tolman, 2009), and Simulating WAves Nearshore (SWAN; Booij, Holthuijsen, and Ris, 1996)—are all based on these empirical spectral models. Numerical wave models calculate the evolution of wave-energy spectra according to physical laws. The SWAN model uses the JONSWAP spectral model and is specifically designed for nearshore environments. The SWAN model simulates wave propagation in time and space based on physical processes such as wave generation, wave interactions, shoaling, refraction, and breaking. Documentation of the SWAN model states that it is limited to use in systems where frequency is ≥1 Hz (SWAN Team, 2006). The SWAN and other numerical models have been used in low-energy environments, such as the Mississippi Sound (Rogers, Hwang, and Wang, 2003), Chesapeake Bay (Lin, Sanford, and Suttles, 2002), coastal ponds in Louisiana (Osorio et al., 2022), fringing reefs (Filipot and Cheung, 2012), and Lake Uberlingen (Seibt et al., 2013). Currently, no spectral models or numerical models are designed specifically for low-energy environments, where frequency may be ≥1 Hz.

All models should be tested against measured data to ensure the reliability of model results. One limitation in wave modeling is the lack of spatially robust wave measurements for model testing. Because of the high cost of most wave gauges, studies usually deploy only a few gauges (Xi et al., 2019). Emerging advances in technology, such as inexpensive wave gauges, provide the opportunity for more comprehensive datasets for comparing model results with measured data (Temple et al., 2020; Virden et al., 2023).

This study tests a numerical wave model in an exceptionally low-energy system and evaluates its ability to simulate waves with small periods and low heights. The model will be tested using a high resolution–measured wave dataset. Results will compare modeled and measured significant wave height, mean period (Tm), and peak period (Tp) throughout the spatial domain. These results will indicate the feasibility of using the SWAN model in a low-energy environment. Recommendations will be made regarding the use of SWAN in low-energy environments.

The study area is Back Bay, Biloxi, Mississippi, United States, within the northern Gulf of Mexico. This bay has a rectangular shape, with a total area of 32.87 km² and a perimeter of 174.12 km. Three sides of Back Bay are confined by shoreline, and one side is open to the northern Gulf of Mexico. The water depth varies from 0 to 12 m. Most of the area is shallow, with deeper regions occurring in the navigation channel. Back Bay has more than six marsh islands that are covered by vegetation and wetlands.

The land surrounding the Bay has elevations ranging from 0 to 18 m above sea level. The northern area features rolling topography, and the southern area is low and flat. The area encompasses the small city of Biloxi with a population of approximately 50,000. Topography around Back Bay was not considered as influencing wind or waves.

Measured data used in this study included wave data, bathymetric data, and meteorological data. Wave and meteorological data were collected over 5 days from 29 August to 3 September 2019.

Do-it-yourself (DIY) wave gauges, 22 in total, were deployed throughout the study area to measure water level using pressure sensors (Table 1; Temple et al., 2020; Virden et al., 2023). Each gauge was placed 30 cm from the bottom. Gauges were placed in depths ranging from 1.2 to 3.2 m below the water surface (Table 1). The tidal range during the time of deployment was 0.76 m. Gauges sampled with a frequency of 10 Hz. The DIY wave gauge is equipped with a digital pressure sensor that offers an accuracy of 14.3% full scale and a resolution of up to 20 Pascals (Pa). The DIY gauge serves as a cost-effective option for various applications. In a study by Temple et al. (2020), the DIY gauge’s performance was assessed through controlled laboratory wave-channel tests and real-world field trials, demonstrating alignment with commercial devices such as the Nortek Aquadopp and RBR Solo3 D. Average differences between the DIY gauge and commercial gauges were close to zero, with 95% of discrepancies remaining within 663 Pa, equivalent to <1 cm of static water depth. Although less precise than commercial alternatives, this level of accuracy is sufficient for many practical and research applications.

Pressure time series from each wave gauge was processed using the MATLAB toolbox OCEANLYZ (Karimpour and Chen, 2017). The data were corrected for pressure changes from tidal fluctuations. This correction was achieved by splitting the pressure data into hydrostatic (wind wave) and dynamic (tidal) components. The dynamic pressure was then corrected using the pressure response factor (Karimpour and Chen, 2017). Spectral analysis was used to create time series datasets of significant wave height, mean wave period, and peak wave period every 30 minutes.

Bathymetric data were obtained from the National Oceanographic and Atmospheric Administration (NOAA, 2024). The data have a spatial resolution of 3 m. The horizontal and vertical accuracy of this bathymetry data are 100 cm and 50 cm, respectively. Measured meteorological data for this study included time series data of wind speed, wind direction, and sea-level pressure. The NOAA station (station ID: 74768613820) provided the time series of meteorological data for the model. The NOAA station is located on Keesler Airforce Base, Biloxi, on the southern side of the study area with latitude of 30.41667, longitude of 88.91667. The wind sensor was placed at an elevation of 10 m. This station recorded hourly wind speed, wind direction, and sea-level pressure. Wind speed and direction were combined to return the wind velocity in x (U) and y directions (V) in meters per second (Figures 1 and S1).

In this study, SWAN + Advanced Circulation (ADCIRC) model was used to simulate the behavior of wind-driven waves. A third-generation numerical model, SWAN uses the spectrum concept to solve the energy-balanced equation. The SWAN model uses fetch length, along with wind speed and direction, to determine the energy transferred from wind to waves, which influences wave height and period. The model can be implemented for varying scales and depths from deep to shallow water. The SWAN model simulates wind-wave generation in time and space, accounting for the following shallow water physics: refraction, diffraction, shoaling, bottom friction, depth-induced wave breaking, white-capping, and nonlinear wave-wave interactions such as triads and quadruplets. The SWAN model supports the surface-water modeling system mesh generator and ADCIRC for reading unstructured mesh.

Surface Modeling and Simulation was used to develop the mesh (Zundel, Jones, and Kraus, 2004); it allows the user to visually generate an unstructured mesh according to the complex shape of the study. SMS uses the finite element technique in geographical space to design triangle cell meshes with various sizes. The ADCIRC was used to read the unstructured mesh and implement SWAN (Luettich, Westerink, and Scheffner, 1992). Coupling ADCIRC with SWAN gives the advantage of running models on an unstructured mesh and using SWAN in the nonstationary mode, leading to simulation of temporally dynamic waves (Dietrich et al., 2012). The ADCIRC model was not used to simulate currents.

The two-dimensional horizontal grid comprised 50,790 nodes and 94,827 elements (Figure 2). Closed boundaries included coastlines and islands. The open boundary, on the east side of the study area adjacent to the Mississippi Sound, assumed a steady state for water-surface elevation. The model was run in the nonstationary mode, using default physics for lake environments. Diffraction was turned off because of the low variation in wave height.

Results of measured and modeled Hs, Tm, and Tp are provided in this section. Wave statistics are presented in tabular, graphical, and mapped formats. Time series of modeled and measured Hs, Tm, and Tp are shown in the supplementary material (Figures S2–S4).

Average measured Hs was low and varied between 0.01 and 0.15 m, with an average of 0.04 m (Table 2). The highest Hs measurements (0.06 to 0.15 m) were recorded at gauges G05, G08, G09, G13, G20, and G31. Gauges G05, G08, and G09 were situated on the eastern side of the study area where fetch lengths in the southern direction are the highest. G31 is an outlier that measured the highest Hs but is located in a fairly enclosed area. Conversely, the gauges recording the lowest Hs (0.01 to 0.02 m) included G01, G03, G17, G21, G28, G33, G36, G37, and G40. All gauges that measured the lowest average Hs were located on the central or western side of the study area, where the site is more enclosed.

Average modeled Hs ranged from 0.02 and 0.06 m, with an average of 0.05 m at the different gauges (Table 2). This shows that the model was generally able to simulate the range of Hs measured by the gauges; however, no correlation was found between higher measured Hs and higher modeled Hs (Figure 3a). This indicates that although the model was able to simulate the average Hs throughout the study domain, it was not able to differentiate specific areas within the domain that have higher or lower Hs.

The RMSE between modeled and measured Hs ranged between 0.01 and 0.11 with an average of 0.04 m (Table 2). The ARE ranged from 0 and 3, with an average of 1. These two statistics indicate that the error in simulated Hs was similar in magnitude to the average Hs measurements. The bias ranged between −0.1 and 0.04, with an average of 0.01. A strong correlation is found between measured Hs and model bias, and the model consistently overpredicted smaller wave heights and underpredicted larger wave heights (Figure 3d).

Figure 4 maps mean modeled Hs across the study area. The map shows that most of the highest modeled wave heights were on the east side in more open areas where fetches are higher. Conversely, the narrower portion of the Bay had lower Hs. The areas with lowest Hs occurred near shorelines and islands. The spatial distribution of RMSE and relative error (RE) showed no obvious trend (Figure 5).

Measured Tm varied between 0.4 and 2 seconds, with an average of 0.8 seconds (Table 3). Recall that the SWAN documentation states that the model should not be used in systems where T is ≤1 second, therefore many of these measurements are outside the range of the model’s intended use. Also recall that one of the objectives of this study is to test SWAN in very low-energy environments as there are currently no physics-based models available for simulating wind-driven waves in these systems. The longest Tm measurements (>1 s) were recorded at gauges G07, G08, G09, G12, and G17, all of which are located on the eastern portion of the study area where the domain is more open.

Modeled Tm ranged between 0.85 and 1 second, with an average of 0.95 seconds at the different gauges (Table 3). The modeled range of Tm shows that the simulations produced a very narrow range of Tm and that the model was not able to represent the shorter (0.4 s) or longer (2 s) Tm measurements. There was also no correlation between higher measured Tm and higher modeled Tm (Figure 6a). However, the range of modeled Tm (0.85 − 1 s) is similar to the average measured Tm measured over the study area (0.8). As with Hs, this indicates that although the model was not able to differentiate specific areas within the domain that have higher or lower Tm, it was able to simulate the average Tm throughout the study domain.

The RMSE between modeled and measured Tm ranged between 0.3 and 1.2 seconds, with an average of 0.6 seconds (Table 3). The ARE ranged between 0.1 and 1.5 seconds, with an average of 0.6 seconds. This indicates that the error in simulated Tm was, on average, smaller than the magnitude of the average Tm measurements. The bias ranged between −1.1 and 0.6 m with an average of 0.2. A strong correlation was found between measured Tm and model bias (Figure 6d), and the modeled overpredicted smaller wave periods and underpredicted larger wave periods.

Figure 7 maps mean modeled Tm across the study area. The map shows that most of the longest wave periods were on the NE side in more open areas where fetches are higher. Conversely, the narrower portion of the Bay had lower Tm. The spatial distribution of RMSE and RE showed no obvious trend (Figure 8).

Measured Tp varied between 1.1 and 7.6 seconds with an average of 2.9 seconds (Table 3). The longest Tp measurements (>3 s) were recorded at gauges G03, G06, G07, G08, G09, G21, and G37. Gauges G03, G06, G07, G08, G09 were all located on the eastern portion of the study area where the domain is more open. Note that gauges G37 and G21 were outliers where Tp both measured greater than 6 seconds, their locations were not located in the eastern portion of the study area, and their measurements of Tm were <1 second.

Modeled Tp ranged between 0.8 and 1.1 seconds, with an average of 1.05 seconds at the different gauges (Table 4). The modeled range of Tp shows that the simulation produced an especially narrow range of Tp and that the model was not able to represent the longer Tp measurements. The range of modeled Tp (0.83 − 1.1 s) was lower than the average measured Tp measured over the study area (0.8). There was also no correlation between higher measured Tp and higher modeled Tp (Figure 9a). The RMSE between modeled and measured Tp ranged between 0.9 and 8.9 seconds, with an average of 2.9 seconds (Table 4). The ARE ranged between 0.1 and 0.9 seconds, with an average of 0.6 seconds. This indicates that the error in simulated Tp was, on average, smaller than the magnitude of the average Tp measurements. The bias ranged between −6.6 and −0.1 m, with an average of −1.1. Bias was negative for all the gauges indicating that the model always underpredicted Tp.

Figure 10 maps modeled Tp across the study area. Similar to Tp, the map shows that most of the longest peak wave periods were on the NE side in more open areas where fetches are higher. Conversely, the narrower portion of the Bay had lower Tp. The spatial distribution of RMSE and RE showed no obvious trend (Figure 11).

These results indicate that in this low-energy system, SWAN exhibits better performance in simulating Hs and Tm compared with Tp and consistently underestimates Tp, Hs, and Tm. Other studies where numerical models were used to simulate waves in low-energy systems found similar findings (Table 5). In these studies, the average period ranged between 0.4 and 5 seconds, and Hs ranged between 0.02 and 0.8 m. The models are all better at simulating height compared with period. As a group, the studies indicate the overall usefulness of applying numerical models in low-energy environments.

These findings do not override the original recommendations made by the model developers. Regular applications of the model should still be limited to use in areas where frequency is >1 second. Extreme care should be used in areas where frequency is <1 second. Based on the findings from this study, when SWAN is used in low-energy environments the following recommendations are suggested:

1. Comparing measured and modeled wave height and frequency at numerous locations is highly recommended in low-energy systems.

2. In low-energy systems, SWAN may be used to represent spatial averages of Hs and Tm. SWAN should not be used to compare differences between Hs or Tm in specific areas within the study domain.

3. Modeled Tp may not be accurate in low-energy environments.

Several limitations of this study should be acknowledged. First, the study was conducted over a period of only 5 days. Second, because of the gauge design, all gauges were placed in shallow water (<3 m). Third, water-piercing wave gauges would have improved the accuracy wave measurements. Fourth, only one wind gauge was used to represent winds over the entire study site. Fifth, the study did not account for waves created by boat wakes. Areas within the study site experience recreational boating traffic (Virden, 2021). The model did not simulate waves produced via boat wake. Measurements did not differentiate between wind-driven waves and boat wakes. Finally, the model did not account for wave attenuation via emergent or submerged vegetation.

The study measured wave period and height at 22 gauges over a period of 1 week. Average measured Hs was 0.04 m, average measured period was 0.79 seconds, and average peak period was 2.85 seconds. This comprehensive spatial dataset was used to evaluate the performance of the SWAN model in simulating small- and high-frequency waves. A summary of the results from these measurements and testing are as follows.

This study was conducted in an exceptionally low-energy system. The average measured Tm (0.8 m) was below the SWAN threshold of 1 m. This is outside the range suggested for SWAN use. The average measured Hs was also very low (<0.2 m).

Average measured wave height was 0.04 m, average wave period was 0.8 seconds, and average peak wave period was 2.9 seconds. Average modeled wave height was 0.05 m, average modeled wave period was 1.0 second, and average modeled peak wave period was 1.0 second. Thus, the model was able to represent the spatially averaged wave conditions throughout the study area for Hs and Tm, but not for Tp.

The model was able to simulate the range of measured Hs. Model ranges of Tm and Tp were narrower than the measured ranges of Tm and Tp. There was no correlation between modeled and measured Hs, Tm, or Tp at the individual gauges. The model simulated higher and longer Hs, Tm, and Tp in the more open, eastern portion of the study domain. Other studies in low-energy environments report similar findings.

Funding for this project was generously provided by Mississippi State University. This research project was also supported by the intramural research program of the U.S. Department of Agriculture, National Institute of Food and Agriculture (Hatch Accession Number 7004342).