A Numerical Study of Solitary Wave Processes over Idealized Atolls

Abstract

In this study, a Boussinesq-type wave model, namely FUNWAVE-TVD, was employed to explore solitary wave processes over coral atolls in two horizontal dimensions. First, a typical solitary wave propagation process over an idealized atoll in a field scale is simulated and analyzed. Then the effects of reef flat water depth, reef flat width, reef surface roughness, fore-reef slope, and lagoon water depth on the distribution of maximum surface elevations over atolls are investigated. Moreover, the effect of a channel on the reef flat is also studied. It is found that during solitary wave propagation, the coral reefs of an atoll can provide effective shelter for the lagoon inside; however, there will be an area of wave height enhancement near the lagoon edge at the lee side of an atoll. The maximum surface elevations over the entire atoll increase significantly with the rise in reef flat water depth, or reduced reef flat width and reef surface roughness, while the lagoon water depth and fore-reef slope have minimal influence. As the reef flat water depth increases or the reef surface roughness decreases, the extent of the wave height enhancement area at the lee side also undergoes an expansion. The presence of a channel in the reef flat mainly leads to two regions of increased wave height. The more the position of the channel deviates from the front of the atoll, the smaller the increase effect and range of the two regions will be. As the channel width increases, the increase effect and range of the two regions will also increase.

1. Introduction

Following our previous study on the solitary wave propagation around reef-fringed islands [1], this paper focuses on the solitary wave propagation over atolls. Atolls are coral reefs distributed like rings in the open ocean and are generally found in the tropical and subtropical regions of the Pacific and Indian oceans. Atolls are valuable land resources in the ocean, while different from a reef-fringed island; the central part of a typical atoll is an enclosed or semi-enclosed lagoon, not an emersed island. The semi-enclosed lagoon is usually connected to the open sea through natural or artificial channels in the reef flat, forming a reef–lagoon–channel system [2]. This channel can provide access for ships to enter and exit the lagoon, somehow making the atoll become a natural harbor. According to Darwin’s theory, atolls are formed after the sink of the central island caused by earthquakes, ocean current, volcanic eruptions, and so on. However, so far, there is still no consensus about the cause of atoll formations.

As described in our previous study [1], the function of coral reefs in alleviating tsunami hazards has aroused increasing attentions [3,4,5,6,7] since the Indian Ocean Earthquake of 2004, especially under the circumstances of global warming and sea level rise [8]. A series of experimental [9,10] and numerical [11,12,13] studies have been conducted to better understand how hydrodynamic and morphological factors related to coral reefs affect tsunami hazards. A solitary wave paradigm is adopted in these studies to mimic tsunamis because solitary waves are believed to effectively model some important aspects of the coastal effects of tsunamis and the leading wave of a tsunami often appears as a solitary wave with a stable form [14,15]. However, so far, existing studies about solitary wave–reef interactions mainly focus on horizontal one-dimensional (1DH) wave processes, ignoring the two-dimensional complexity of nearshore topographies [16]. Through these 1DH studies, conclusions can be drawn for fringing reefs that are extensively distributed along continental coasts while for isolated reef-fringed islands and atolls that are plentiful in the open ocean, the longshore wave processes that may complicate the distribution of tsunami disasters cannot be considered. Therefore, to fill the gap about 2DH solitary wave processes over three-dimensional coral reef topographies, our previous study [1] performed a numerical study on the solitary wave processes resulting from combined refraction and diffraction over idealized circular reef-fringed islands, and a systematic investigation was carried out on the effects of key parameters on the maximum solitary wave run-up heights around the island. Some unique conclusions were drawn and can provide coastal engineers and managers with a better understanding of the role of coral reefs in two horizontal dimensions. Using the same numerical method, this study further investigates the solitary wave processes over idealized atolls as the second part.

This paper is presented as follows: In Section 2, a brief introduction to the numerical method employed in this study is provided. In Section 3, a typical solitary propagation process over an idealized atoll in a filed scale is simulated and analyzed. In Section 4, the effects of the basic morphological and hydrodynamic parameters on maximum surface elevations over atolls are studied. In Section 5, the effects of the channel position and width are studied. Finally, the main conclusions obtained from this research are provided in Section 6.

2. Brief Introduction of the Numerical Method

This study still employed the Boussinesq-type model, FUNWAVE-TVD [17] (Version 3.0), to simulate 2DH tsunami-like solitary wave propagation over atolls. The governing equations and numerical scheme have been introduced in our previous study [1]. In FUNWAVE-TVD, a more complete set of fully nonlinear Boussinesq equations proposed by Chen [18] are adopted as the governing equations. Reformulated to a compact form, the governing equations are solved by a hybrid finite difference–finite volume TVD scheme with a third-order Strong Stability-Preserving Runge–Kutta time stepping. A quadratic friction term that incorporates the Manning coefficient is added to the momentum equation to model bottom friction. Based on the hybrid numerical scheme, one unique advantage of FUNWAVE-TVD, relative to conventional Boussinesq models, for studies of wave hydrodynamics over sharply varying reef topographies is that wave breaking can be treated as shock waves without adding an additional empirical breaking model. Empirical breaking models, for instance, the eddy-viscosity type [19] and the surface roller type [20], usually have several empirical parameters that need to be adjusted for the specific topography. For example, in order to apply the Boussinesq wave model coupled with an eddy-viscosity breaking model, Yao et al. [21] performed a series of experiments and based on the experimental data, values of parameters in the eddy-viscosity model for sharply varying coral reef topographies in their study were obtained. On the other hand, in FUNWAVE-TVD, there is only one criterion (the ratio of surface elevation to water depth) to determine the onset of wave breaking and the threshold of this criterion is set to be 0.8. During the numerical simulation, when the criterion exceeds 0.8, Boussinesq equations are switched to Nonlinear Shallow Water Equations by discarding the dispersive terms and then energy dissipation associated with wave breaking is automatically modelled through the ability of Nonlinear Shallow Water Equation with a TVD scheme. The threshold has been proved to be valid over various bathymetries [17], including coral reef topographies for regular [21] and irregular waves [22], and this wave breaking treatment greatly improves the universality of FUNWAVE-TVD. As for solitary wave processes over coral reef topographies, Ning et al. [12] and Liu et al. [13] performed a series of experiments on solitary wave propagation over an idealized two-dimensional reef-beach system and used the experimental data to validate the present model. It has been demonstrated with an appropriate Manning coefficient value, that the present model can effectively reproduce the wave shoaling, breaking, and run-ups reasonably over fringing reefs without tuning the threshold of the breaking criterion.

Moreover, the momentum equations of FUNWAVE-TVD include the vorticity term. This also improves the ability of the model in predicting 2DH wave processes. Compared to the widely used experimental data of Briggs et al. [23], our previous study [1] clearly shows FUNWAVE-TVD can predict the 2DH solitary wave propagation and run-ups around a three-dimensional circular island, especially the surface elevations and run-ups at the lee side, more accurately than the previous version of FUNWAVE [24]. Since the present model can reasonably predict solitary wave processes over sharply varying coral reef topographies and three-dimensional topographies, this study directly used FUNWAVE-TVD to study 2DH solitary wave processes over coral atolls.

3. Typical Propagation Process over an Idealized Atoll

In this section, an idealized circular atoll was designed according to an actual atoll in a numerical basin and the typical solitary wave propagation process, presented as time-varying surface elevations, current field, and distribution of maximum surface elevations, was investigated in a field scale.

As seen in Figure 1, the numerical basin is 14 km × 14 km and the internal wave maker was set at x = 3.5 km. An idealized circular atoll whose center is located at x = 7 km, y = 7 km was implemented. The profile of the atoll, such as the diameter of the reef edge circular and lagoon edge circular, was estimated based on the prototype, Alifu Atoll (Figure 2) in the northern part of Maldives. Alifu Atoll is a typical coral atoll with a deep lagoon in the middle. Measured by Google Earth, the lengths of the reef edge (the outer white line in Figure 2) and lagoon edge (the inner red line in Figure 2) are about 6700 m and 5000 m, respectively. Taking these lengths as the circumferences of the reef edge and lagoon edge circulars in Figure 1, the radiuses of the reef edge circular and lagoon edge circular (w_L) are 1100 m and 800 m, respectively. The reef flat water depth (h_r) and fore-reef slope (cot θ) were set to be 2 m and 4 based on common coral reef parameters reported in the existing literature [25]. The back-reef slope is groundless and set to be 1:4. Based on the above parameters, the reef flat width (w_r) is about 300 m also conforming to the range of reef flat width reported in the literature [25]. The water depth of the lagoon (h_L) is set to be 12 m based on the investigation [26] about the geological properties of atolls in the South China Sea. The same as the previous study, the Manning coefficient n₂ was set to be 0.09 [27] to characterize the surface roughness of healthy and prosperous coral reefs for the fore-reef slope, reef flat, and back-reef slope. The Manning coefficient n₁ for the basin and lagoon bottoms was set to be 0.02 (sand). The numerical test conditions of this section are listed in

Table 1. Numerical conditions for solitary wave propagation over an idealized atoll.

Case No.	H (m)	hd (m)	wr (m)	wL (m)	hr (m)	hL (m)	cot θ	n2	n1
1A	3	62	300	800	2	12	4	0.09	0.02

, the incident wave height (H) and deep-water depth (h_d) were specified as 3 m and 62 m, respectively. Then the height–depth ratio (ε = H/h_d = 0.048), which serves as a measure of the nonlinearity of the incident solitary wave, falls within the range of the experiments conducted by Briggs et al. [23]. For Case 1A, the total simulation time was set as 400 s and the time interval for data output was 1 s.

Figure 3 shows the computed water surface elevations and current field at different moments for Case 1A. As seen in Figure 3, when t = 75 s, the solitary wave has reached the reef flat of the atoll with an increased wave height as a result of the shoaling effect. The current velocity over the reef flat is clearly larger than the one in other areas. When t = 120 s, a part of the water body enters the lagoon directly while the other parts diffract along both sides of the atoll. During the diffraction process, the current velocity on the reef flat points towards the lagoon, indicating that part of the water body of diffraction waves continues to enter the lagoon. When t = 195 s, the water body in the lagoon continuously propagates towards the lee of the atoll. Diffraction waves outside the lagoon continue to propagate along the atoll and start to be trapped at the lee side of the atoll. When t = 220 s, trapped diffraction waves converge and collide at the lee side, generating a current opposite to the direction of the incident wave. The water body in the lagoon continues to propagate towards the lee of the atoll. When t = 260 s, the water body of trapped diffraction waves collides with the water body propagation in the lagoon, generating large surface elevations. When t = 300 s, the water body has fallen back after collision. It can be seen that, when solitary waves propagate over the atoll, there will be two wave height enhancements at the lee side of the atoll. One (the first) is produced by the collision of trapped diffraction waves and the other (the second) is produced by the collision of the water body in the lagoon and the water body of diffraction waves trapped by the atoll.

Figure 4 further shows the distribution of relative maximum surface elevations (the ratio of maximum surface elevation η_max to the incident wave height H) over the entire atoll for Case 1A. As seen in this figure, under the cover of coral reefs, the wave heights in the lagoon are significantly lower than those outside the lagoon. The relative maximum surface elevations are less than 0.5 in most areas of the lagoon, which means the attenuation of solitary wave heights can reach more than 50% in the lagoon, and coral reefs around the atoll can provide protection against tsunami hazards for ships in the lagoon. The second wave height enhancement appears near the lagoon edge and the enhancement effect of it is obviously greater than the first wave height enhancement.

4. Influences of Morphological and Hydrodynamic Parameters

After studying the typical propagation process of solitary waves over an idealized atoll, a series of numerical experiments were then designed and performed based on the atoll profile in Section 3. By checking the distribution of maximum surface elevations over atolls, the effects of basic morphological and hydrodynamic parameters associated with atolls on solitary wave propagation were systematically investigated.

The numerical experimental conditions of this section are listed in

Table 2. Numerical experiment conditions of Section 4.

Group No.	H (m)	hd (m)	wr (m)	hr (m)	hL (m)	cot θ	n2
G1A	3	62	300	1~6	12	4	0.09
G2A	3	62	50~800	2	12	4	0.09
G3A	3	62	300	2	12	4	0.02~0.09
G4A	3	62	300	2	12	4~18	0.09
G5A	3	62	300	2	7~22	4	0.09

. As seen in Table 2, five groups were designed with only one parameter changing in each group. The values of the unchanged parameters in all groups are set based on Case 1A. In addition to the parameters (i.e., the reef flat water depth, reef flat width, fore-reef slope, and reef surface roughness) determining the state of coral reefs, this study further considers the change of the lagoon water depth. The varying ranges of the parameters determining the state of coral reefs were set based on Quataert et al. [25] and Gelfenbaum et al. [27], including the effect of sea level rise by 2100 ad [28,29]. The varying range of the lagoon water depth is set based on the investigation [26] of the atolls in the South China Sea as well.

Figure 5 shows the distribution of relative maximum surface elevations over atolls as the reef flat water depth varies for Group G1A. It can be seen that, as the reef flat water depth increases, the maximum wave surface elevations over atolls increase, and the effect and range of the wave height enhancement near the lagoon edge at the lee side of atolls also increase obviously. It can be observed that, when the reef flat water depth increases to 5 m, the attenuation effect of coral reefs on solitary waves is so small that the maximum surface elevations in most areas of the lagoon are even greater than the incident wave height. Meanwhile, the maximum wave surface elevations in the wave height enhancement area near the lagoon edge are also much greater than the incident solitary wave height. This clearly reveals that sea level rise may significantly reduce the protection capability of coral reefs against tsunami hazards in the lagoon within this century.

Figure 6 shows the distribution of relative maximum surface elevations over atolls as the reef flat width varies for Group G2A. It is observed that, as the reef flat width increases, the maximum wave surface elevations over atolls will decrease. When the reef flat width increases to a certain extent (e.g., w_r = 600 m), the wave height enhancement near the lagoon edge at the lee side of atolls almost disappears as most of wave energy should be dissipated by wave breaking and bottom friction over the wide coral reefs. On the other hand, when the reef flat width is less than 200 m, the maximum surface elevations in the wave height enhancement area near the lagoon edge will exceed the incident wave height.

Figure 7 shows the distribution of relative maximum surface elevations over atolls as the reef surface roughness varies for Group G3A. As seen in Figure 7, the maximum wave surface elevations over atolls will increase as the reef surface roughness decreases. The effect and range of wave height enhancement near the lagoon edge at the lee side also increases with decreasing reef surface roughness. When n₂ is less than 0.05, the maximum surface elevations in the wave height enhancement area will exceed the incident wave height. Furthermore, when n₂ is reduced to 0.02, most areas within the lagoon have a maximum wave elevation that is roughly equal to the incident wave height, indicating that climate change-induced coral bleaching can also reduce the protective capacity of coral reefs with respect to tsunami hazards.

Figure 8 shows the distribution of relative maximum surface elevations over atolls as the fore-reef slope varies for Group G4A. It shows that as the fore-reef slope gets milder, the maximum wave elevations over the fore-reef slope at the front side will decrease a little, while the variation of the fore-reef slope exerts minimal influence on the maximum wave elevations in the lagoon and wave height enhancement area. It is noteworthy that our previous study [1] found that the run-up enhancement at the lee side of a reef-fringed island will increase when the fore-reef slope becomes gentle to a certain degree (e.g., cot θ = 20) due to more water bodies being involved in wave diffraction while a similar increase was not observed in the second wave height enhancement area of this study. This may be because for an atoll, although more water bodies are involved in wave diffraction, when the fore-reef slope becomes milder, this results in an increase in maximum surface elevations in the first wave height enhancement area, and there is also a reduction in water bodies entering the lagoon, resulting in no significant increase in maximum surface elevations in the second wave height enhancement.

Figure 9 shows the distribution of relative maximum surface elevations over atolls as the lagoon water depth varies for G5A. It shows that the maximum surface elevations over the reef flat and in the lagoon are generally not affected by the lagoon water depth. However, it is noteworthy that the lagoon water depth has some influence on the second wave height enhancement near the lagoon edge. As the lagoon water depth increases, the enhancement effect significantly decreases, but the range of the second enhancement area tends to extend along the lagoon edge. When the lagoon water depth is 7 m, the maximum surface elevations in the second wave height enhancement area are noticeably larger than the incident wave height.

5. Effects of Channel Location and Width

In addition to the above basic morphological and hydrodynamic parameters, there may be natural or artificially excavated channels in the reef flat of atolls, forming a reef–channel–lagoon system as shown in Figure 10. These channels can serve as accesses for vessels to enter and exit the lagoon. The presence of the channel can influence the wave propagation and nearshore current circulation [2,30]. Therefore, based on the numerical basin of Section 3, this section further conducts a series of parametric numerical experiments to analyze the effects of channel location and width on the solitary wave propagation over atolls. According to previous studies [2,30], the basic sectional profile of the channel is set as shown in Figure 11, where W_g represents the width of the channel and h_L represents the water depth of the channel (equal to that of the lagoon). The numerical experimental conditions of this section are listed in

Table 3. Numerical experiment conditions of Section 5.

Group No.	H (m)	hd (m)	wr (m)	hr (m)	hL (m)	Wg (m)	cot θ	α (°)	n2
G6A	3	62	300	2	12	40	4	0~180	0.09
G7A	3	62	300	2	12	40~120	4	0	0.09
G8A	3	62	300	2	12	40~120	4	135	0.09
G9A	3	62	300	2	12	40~120	4	180	0.09

, where α denotes the angle between the centerline of the channel and the propagation direction of the incident wave. Group G6A was designed to check the effect of the channel position, and G7A~G9A were designed to check the effect of the channel width at different locations. According to Yao et al.’s physical model study [30], the varying range of the channel width is set to be 40~120 m, and side slopes of the channel are set to be 1:1.

Figure 12 shows the distribution of (η_max − η₀)/η₀ (here, η_max and η₀ are the maximum surface elevations with and without the presence of the channel) over atolls as the channel location changes for Group G6A. As seen in Figure 12, the presence of a channel can increase the maximum wave elevations over the atoll. The influence range of the channel are mainly two regions of increased wave height that start from both sides of the channel. Moreover, the presence of the channel also can increase the wave height enhancements at the lee side of the atoll. When the channel is positioned at α = 0°, both increased wave height regions are within the lagoon. Maximum surface elevations in these two regions are increased by 40%~80% compared to those without a channel. The further the position of the channel deviates from α = 0°, the smaller the increasing effect and influence range of the channel will be. When the position of the channel is at α = 45°, the maximum surface elevations in the two regions are increased by 30%~70%. When the position of the channel is at α = 90°, there is only one region within the lagoon where the maximum surface elevations are increased by 30%~50%. Similarly, when the position of the channel is at α = 135°, only one region is within the lagoon and the maximum surface elevations in this region are increased by 10%~20%. When the position of the channel is at α = 180°, both the regions of increased wave height are not observed. The maximum surface elevations within the lagoon are not affected by the presence of the channel.

Figure 13, Figure 14 and Figure 15 show the distribution of (η_max − η₀)/η₀ over atolls as the channel width varies when the channel is at different locations for Groups G7A~G9A. It can be seen that, as the channel width increases, the enhancement and ranges of the two increased wave height regions increase and the wave height enhancement at the lee side also increases. However, as the channel position deviates further from α = 0°, the influence of the channel width diminishes. When the channel position is at α = 0°, the maximum wave surface elevations in the two regions can be increased by up to 110% with a channel width of 120 m, 30% more than the increase with a channel width of 40 m. The ranges of the two regions also increase obviously as the channel width increases. When the channel position is at α = 135°, the maximum wave surface elevations in the two regions are increased by up to 30% with a channel width of 120 m, 20% more than the increase with a channel width of 40 m, and the ranges of the two regions also change minimally as the channel width changes. When the channel position is at α = 180°, the two increased wave height regions vanish and the presence of the channel only affects the surface elevations in the wave height enhancement area at the lee side. The maximum wave surface elevations in the wave height enhancement area can be increased by only up to 25%, just 10% more than the increase with a channel width of 40 m and the range of the wave height enhancement area remains almost unchanged.

6. Conclusions

Following the previous study on solitary wave processes around reef-fringed islands, this study continued to use the Boussinesq-type wave model to investigate 2DH solitary wave processes over idealized atolls in a field scale. The influences of the basic morphological and hydrodynamic parameters on the maximum surface elevations over atolls, especially in the lagoon, were systematically studied. Additionally, this study also considered the effects of the position and width of the channel in the reef flat. Based on the numerical experiment results, the following conclusions can be reached:

(1)

In general, the coral reefs of an atoll can provide effective shelter for the lagoon inside the atoll. During the solitary wave propagation over an atoll, diffracted waves can be trapped by the atoll at the lee side and ultimately collide with the water body propagating in the lagoon, forming an area of wave height enhancement near the lagoon edge at the lee side.

(2)

The maximum surface elevations over the entire atoll increase significantly with the rise in reef flat water depth, or reduced reef flat width and reef surface roughness, while the lagoon water depth and fore-reef slope have minimal influence. As the reef flat water depth increases or the reef surface roughness decreases, the extent of the wave height enhancement area at the lee side also undergoes an expansion.

(3)

The influence of the channel mainly presents as two regions of increased wave heights starting from both sides of the channel. The more the position of the channel deviates from the front of the atoll, the smaller the increase effect and range of the two regions will be. As the channel width increases, the increase effect and range of the two regions will also increase.