Effect of Sea Level Rise on Nearshore Waves and Wave Energy
Article information
Abstract
Sea level rise presents significant challenges for the stability and sustainability of coastal regions. This study examined how sea level rise affects nearshore waves and wave energy. Previous research derived the changes in wave characteristics by expressing sea level rise as a relative change in water depth. Based on this general framework, earlier work was extended by incorporating wave energy and related parameters into the analysis and providing an intuitive way to examine the wave responses across varying depths, beach slopes, and degrees of nonlinearity. The results suggested that under sea level rise, the wave height decreases in the shoaling and transition zones but increases in the surf zone. Nevertheless, sea level rise leads to an overall increase in wave height in strongly nonlinear wave fields, where wave breaking occurs farther offshore. Wave energy displays similar patterns but with an amplified magnitude, while energy flux is influenced further by group velocity. The bottom orbital velocity decreases in deeper water but increases in shallow regions, with the transition zone between decreasing and increasing values shifting offshore as the nonlinearity intensifies. The framework offers a practical tool for understanding a nearshore wave transformation and supports improved hazard assessment and coastal planning under future climate change.
1. Introduction
Climate change caused by global warming has emerged as one of the most pressing environmental challenges in recent decades, with profound implications for the stability and sustainability of coastal regions (Alexandra et al., 2020; Bilskie et al., 2014; Cheon and Suh, 2016; Houghton et al., 1996, 2001; Kim and Suh, 2018; Marland et al., 2003; Marchetti, 1977; Schneider and Chen, 1980; Shen et al., 2016; Solomon et al., 2007; Stern et al., 2006; Stocker et al., 2013; Testut et al., 2016). Rising atmospheric and oceanic temperatures have accelerated the thermal expansion of seawater and the melting of land ice, resulting in a persistent increase in global mean sea level. Sea level rise has been widely recognized as one of the most critical consequences of climate change, directly threatening coastal zones through beach erosion, saltwater intrusion into groundwater, coastal flooding, ecosystem imbalance, habitat loss, and damage to infrastructure. The international scientific community, led by the Intergovernmental Panel on Climate Change (IPCC), has provided authoritative projections of future sea level rise. The IPCC Sixth Assessment Report (AR6) projects an increase of 0.38 m under strong mitigation (SSP1-1.9) to 0.77 m under high-emission scenarios (SSP5-8.5) by 2100, with the potential for even larger rises under low-probability ice-sheet instability scenarios (Fox-Kemper et al., 2021). Recent studies (e.g., Dangendorf et al., 2019; IPCC, 2019; Palmer et al., 2020) further indicate an acceleration in sea level rise, with increases exceeding 1 m by 2100. Sea level rise introduces additional uncertainties. For example, it reduces the freeboard between high-water levels and established flood thresholds, increasing uncertainty in the estimates of future coastal flood hazards. Therefore, incorporating these uncertainties regarding climate change is essential for robust coastal risk assessments and adaptation planning (Arns et al., 2017; Boumis et al., 2023; Moftakhari et al., 2024; Nicholls et al., 2021).
In addition to rising sea levels, climate change is expected to alter the wave conditions, including the wave height, period, and direction. Studies using global and regional climate model outputs (Debernard et al., 2002; Wang and Swail, 2006; Hemer et al., 2010, 2013a; 2013b; Mori et al., 2010, 2013; Shimura et al., 2011; Lim et al., 2013; Casas-Prat et al, 2024) predict that the mean wind speed and average wave height may decrease by 5%–10%, while extreme waves generated by tropical cyclones could increase by more than 20% in some regions by the end of the century. Around the Korean Peninsula, these projections indicate that extreme summer wave heights associated with tropical cyclones will likely increase, whereas winter wave heights from extratropical storms will decrease. These findings are consistent with observed impacts, including shoreline retreat and sand loss at Anmok Beach in Gangneung and Gimnyeong Beach in Jeju, as well as frequent overtopping and inundation of coastal infrastructure in Haeundae (Busan) and Sorae Port (Incheon). Increasing exposure of breakwaters and seawalls to waves exceeding their design thresholds and recurrent flooding of low-lying hinterlands further demonstrate how sea level rise and wave climate change together are intensifying the coastal hazards and disrupting coastal communities.
Coastal structures and infrastructures are susceptible to changes in sea level and wave conditions (Crooks et al. 2018; Kuwae and Hori 2019; Kuwae and Crooks, 2021). Earlier studies evaluated the effects of sea level rise on the structural performance (e.g., Chini and Stansby, 2012; Klein et al., 1998; Lee et al., 2013; Okayasu and Sakai, 2006; Reeve et al., 2008; Reeve, 2010; Stern et al., 2006; Suh et al., 2012; Suh et al., 2013; Sutherland & Wolf, 2002; Takagi et al., 2011; Torresan et al., 2008; Wigley, 2009), but most were limited to specific sites and fixed assumptions of sea level change. As a more general framework, Townend (1994) expressed sea level rise as a relative change in water depth and derived the corresponding changes in wave characteristics, which were then used to evaluate the structural responses. This method, based initially on regular wave theory, was later extended by Cheon and Suh (2016) to irregular wave conditions observed in the field and refined further by Kim and Suh (2018), who incorporated sea level rise and offshore wave height changes associated with projected shifts in wave climate.
The present study extended the work of Cheon and Suh (2016) by incorporating wave energy and related characteristics into the analysis. Wave energy plays a pivotal role in shaping coastal systems because it governs sediment transport, shoreline change, and the distribution and resilience of marine vegetation. Several studies have shown that hydrodynamic forcing by waves and currents strongly influences the seagrass distribution, colonization depth, and spatial heterogeneity, while excessive exposure can lead to physical disturbances or adaptive growth responses (Chambers, 1987; Duarte et al., 1997; Fonseca and Bell, 1998; Koch, 2001; Koch et al., 2006; Peralta et al., 2006; Stevens and Lacy, 2012). The present study provides a more comprehensive understanding of the effects of sea level rise on coastal dynamics by integrating wave energy into the framework.
Cheon and Suh (2016) quantified the relative changes in wave characteristics under sea level rise, but their results were not presented in a unified format. In particular, for the key variable, significant wave height, the results were not shown directly as a figure of the relative change in wave height against the relative change in water depth. Instead, they provided figures containing intermediate values used to calculate the wave height variations. This makes it difficult to intuitively capture the influence of sea level rise on nearshore waves across varying depths and relative water depth changes. This limitation is addressed by presenting the relative changes in wave characteristics within a unified framework using contour diagrams. In these diagrams, the horizontal axis denotes the relative change in water depth, while the vertical axis denotes the relative depth, enabling a clear visualization of how sea level rise alters wave transformations under a variety of depth conditions. These diagrams provide engineers with a practical tool to evaluate the effects of sea level rise on the nearshore wave characteristics without requiring detailed knowledge of the governing equations.
2. Method
2.1 Outline
The variables after and before sea level rise and before sea level rise are denoted with and without a prime (′), respectively, while the relative change in each variable is denoted by a lowercase letter. The effects of sea level rise on nearshore wave characteristics were evaluated by adopting the relative change in water depth before sea level rise (D) and after sea level rise (D′) as the primary governing parameter, expressed as d = D′/D. For the analysis, a planar beach with a constant slope is assumed. Following the framework of Cheon and Suh (2016), the wavelength and refraction coefficient were calculated using regular wave formulae, but with the significant wave period and principal wave direction of random directional waves. Unless stated otherwise, all wave parameters refer to significant wave values, including the significant wave period and significant wave height. The shoaling coefficient was evaluated using a nonlinear shoaling formula for irregular waves, while the significant wave height was estimated using Goda’s (Goda, 1975) approximate expression. Subsequently, the relative changes in wave energy characteristics were derived, including wave energy density, wave energy flux, and bottom orbital velocity.
2.2 Wave Characteristics
2.2.1 Wavelength
The wavelengths corresponding to the significant wave period of irregular waves, before and after sea level rise, were calculated using the linear dispersion relations in Eqs. (1) and (2), respectively:
The relative change in wavelength is then given by
where the wave period T is assumed to remain constant during nearshore propagation. For irregular waves, however, longer components are influenced more strongly by shoaling and refraction than shorter components. Thus, the significant wave period may vary considerably. Consequently, the significant wave period before and after sea level rise could differ in reality. Nevertheless, following Cheon and Suh (2016), this study neglects such variations and uses the same period T in Eqs. (1) and (2). Future work will handle the changes in waves due to climate change more explicitly (Kim and Suh, 2018).
2.2.2 Refraction coefficient
Snell’s law relates the deepwater wave direction. α0 and the principal wave direction of irregular waves α (i.e.,
From these equations, the relative change in refraction coefficient is obtained as follows:
where
2.2.3 Shoaling coefficient
The linear shoaling coefficient from linear wave theory is given by
The relative change in the linear shoaling coefficient is then expressed as
This study implemented the nonlinear shoaling coefficient suggested by Kweon and Goda (1996), based on Shuto (1974) and Iwagaki et al. (1982). The nonlinear shoaling coefficients before and after sea level rise are given by
and the relative change in the nonlinear shoaling coefficient is given by
where
2.2.4 Wave height
The significant wave height was adopted as a representative value of the irregular wave height, and it was estimated using the formula proposed by Goda (1975):
where
Fig. 1 shows the wave transformation zones (Eq. 13) as waves with H0/L0 = 0.01 and T = 10 s propagate from deep to shallow water on planar beaches with slopes of 1/100 and 1/10. The significant wave height increases rapidly as the relative depth decreases because of shoaling, where the wave energy is compressed in shallower water. Growth continues until a maximum is reached, after which breaking and dissipation dominate in the surf zone. For a steeper slope (m = 1/10), the maximum wave height is higher, suggesting that steeper profiles intensify shoaling and concentrate wave energy over a shorter cross-shore distance. In the transition zone, waves are on the verge of breaking, and the wave height remains relatively constant with depth. This behavior results from approximating the gradual variations of the wave height with a smooth curve.
Different wave transformation zones where H0/L0 = 0.01 and T = 10 s (Solid: m = 1/100; Dashed: m = 1/10)
The zonal classification was determined solely by the corresponding water depths in this study because deepwater steepness and beach slope remain unchanged before and after sea level rise. As a result, sea level rise may or may not cause zone shifts; if such a shift occurs, locations originally within the surf or transition zones may move into the transition or shoaling zones, respectively. Cheon and Suh (2016) derived the relative change in wave height by accounting for this zone shift under sea level rise:
The opposite shift may occur if wave climate change is also considered. For example, if the relative increase in offshore wave height exceeds that of the water depth, the locations in the shoaling or transition zones may instead move into the transition or surf zones, respectively, even under sea level rise (Kim and Suh, 2018).
2.3 Wave Energy–Related Characteristics
The wave energy is a key driver of coastal processes, directly influencing sediment transport, shoreline evolution, and nearshore hydrodynamics. It also governs the distribution of seagrass and other marine vegetation through its control of the orbital velocities and associated physical forcing (Chambers, 1987; Duarte et al., 1997; Fonseca and Bell, 1998; Koch, 2001). Near-bed orbital velocity and wave energy flux are fundamental parameters in sediment-transport and hydrodynamic models because they quantify wave-induced forcing on the seabed and the transfer of energy through the water column. This study establishes a framework to assess how sea level rise modifies the coastal wave dynamics by integrating these wave energy–related parameters into the analysis.
2.3.1 Wave energy
Wave energy consists of potential energy from free-surface displacement and kinetic energy from particle motion. Using the velocity potential from linear wave theory, the depth-integrated, time-averaged total energy per unit surface area before and after sea level rise is given by Eqs. (15) and (16), respectively:
The relative change in total energy per unit surface area was obtained as the square of the relative change in wave height because all variables except wave height, including seawater density ρ, remain unchanged under sea level rise:
2.3.2 Wave energy flux
The wave energy flux represents the rate at which the wave energy is transported shoreward per unit width and is a key parameter in internal wave energetics for identifying energy sources, wave propagation, and energy dissipation mechanisms (Munk and Wunsch, 1998). The wave energy flux, including terms up to the second order in wave height, can be expressed as the product of the wave energy and group velocity:
Accordingly, the relative change in energy flux is obtained as the product of the relative changes in wave energy and group velocity:
2.3.3 Bottom orbital velocity
The bottom orbital velocity, defined as the near-bed particle velocity induced by waves, is a critical wave parameter that governs bed shear stress, sediment resuspension, erosion, and the evolution of seabed morphology. The values before and after the sea level rise are evaluated by
The bottom velocities are directly proportional to the wave height and depend inversely on the water depth, and the relative change in the bottom orbital velocity is then given by
3. Results and Discussion
3.1 Outline
The equations derived in the previous section were applied to evaluate the effect of sea level rise on nearshore waves and wave energy under various conditions. The analysis was performed for a set of representative scenarios with a fixed significant wave period of T = 10 s and the relative water depth D/L0 ranging from 10−3 to 1 to clearly illustrate the trends. The relative change in water depth d was examined over the range from 1 to 1.5. The reference case corresponds to d = , which represents the condition without sea level rise, for which all relative change values are unity. With the wave period fixed, the input variables for the relative change in wavelength l are the water depth D (or equivalently the relative water depth D/L0) and the relative change in water depth d (see Eq. (3)). Accordingly, the effects of sea level rise on the relative change in wavelength l can be visualized using contour diagrams, where the horizontal axis denotes the relative change in water depth D/L0, and the vertical axis denotes relative depth d. For other wave characteristics, additional input parameters are required in addition to D/L0 and d (for example, α0 is an additional input parameter for the relative change in refraction coefficient kr). A consistent and unified format can be maintained across all analyses by fixing such additional parameters. In all figures presented in this study, the red contours and lines indicate relative changes greater than unity; the blue contours and lines represent values less than unity, and the white contours and black lines correspond to no change under sea level rise (i.e., unity).
3.2 Wave Characteristics
3.2.1 Wavelength
The relative change in wavelength under sea level rise is always greater than unity. In deep water (D/L0 > 0.3), the wavelength remains relatively unchanged even when d = 1.5, as shown in Fig. 2; thus, the relative change in wavelength stays close to unity across a range of d values considered. In contrast, in shallower water (D/L0 < 0.1), the relative change increases rapidly with rising water depth, reaching approximately 20% when d = 1.5. These results suggest that the effect of sea level rise on wavelength variation is significant primarily in shallow-water regions, while it can be neglected in deeper water.
Relative change in wavelength as a function of d and D/L0
3.2.2 Refraction coefficient
As discussed earlier, Eq. (6) expresses that the deepwater wave direction α0 also influences the relative change in the refraction coefficient. In this study, the values of α0 ranging from 20° to 60° are considered. Even at the maximum relative change in water depth (i.e., d = 1.5), the increase in the refraction coefficient is less than 1% for incident angles below 20° (see Fig. 3). For α0 = 20°, the refraction coefficient remains essentially unity across all depth conditions, indicating the negligible effects of sea level rise. In contrast, for α0 = 60°, a significant increase in refraction coefficient is observed in intermediate water depths (0.1 < D/L0 < 0.3). These results suggest that sea level rise reduces the relative influence of the seabed, causing waves to propagate less perpendicularly toward the shoreline. This effect becomes more pronounced as the deepwater incident angle increases.
Relative change in the refraction coefficient as a function of d and D/L0: (a) ksi = 20°; (b) ks = 40°; (c) H0/L0 = 60°
3.2.3 Shoaling coefficient
The relative change in the linear shoaling coefficient is expressed as the square root of the inverse of that in group velocity. Therefore, it depends only on d and D/L0, as shown in Eq. (8). On the other hand, the changes in the nonlinear shoaling coefficient under sea level rise are associated with variations in the deepwater wave steepness H0/L0 due to Γ in Eq. (11). H0/L0 = 0.1, 0.01, and 0.001 are considered when examining this effect. Fig. 4(a) presents the relative change in the linear shoaling coefficient, while panels (b), (c), and (d) of Fig. 4 present the relative changes in the nonlinear shoaling coefficient as the deepwater wave steepness increases. For linear waves, the shoaling coefficient shows little variation, but under nonlinear conditions, the change becomes more pronounced as the nonlinearity increases. Specifically, the nonlinear shoaling coefficient decreases after sea level rise, yielding relative change values of less than one, and the dark blue region with values below 0.4 expands with increasing nonlinearity, or equivalently, as the deepwater wave steepness increases. This effect, however, influences the wave height variation only outside the surf zone, particularly within portions of the transition and shoaling zones.
Relative change in the linear and nonlinear shoaling coefficients as a function of d and D/L0: (a) ksi; (b) ks when H0/L0 = 0.001; (c) ks when H0/L0 = 0.001; (d) ks when H0/L0 = 0.1
3.2.4 Wave height
In addition to the variables on the axes (d and D/L0), the wave height also depends on the beach slope m as well as deepwater wave steepness H0/L0. Two slope values were considered when examining the influence of these additional parameters, m = 1/10 and 1/100, together with H0/L0 = 0.1, 0.01, and 0.001 in Fig. 5.
Relative change in wave height as a function of d and D/L0: (a) H0/L0 = 0.001 and m = 1/100; (b) H0/L0 = 0.01 and m = 1/100; (c) H0/L0 = 0.1 and m = 1/100; (d) H0/L0 = 0.001 and m = 1/10; (e) H0/L0 = 0.01 and m = 1/10; (f) H0/L0 = 0.1 and m = 1/10
For weakly and moderately nonlinear waves (i.e., H0/L0 = 0.001 and 0.01), the relative change in wave height remains close to unity in non-shallow water (D/L0 > 0.05), irrespective of the beach slope, as shown in panels (a), (b), (d), and (e) of Fig. 5. Nevertheless, pronounced variations occur in shallow water: (1) the wave height decreases under sea level rise in the shoaling and transition zones, whereas (2) wave height increases after sea level rise in the surf zone, where wave height is directly proportional to the depth. These changes intensify with a larger sea level rise. For the steeper slope (m = 1/10), the maximum wave heights are larger because of stronger shoaling and energy concentration over a shorter cross-shore distance. As a result, the transition from decreasing to increasing relative wave height occurs closer to shore on steeper profiles. The overall trend is consistent with a small net reduction in wave height because the region of decreasing relative change in wave height is more extensive than the region of increase (Mori et al., 2010; Shimura et al., 2011).
For strongly nonlinear waves (i.e., H0/L0 = 0.1 in panels c and f of Fig. 5), wave height variation is largely governed by processes in the shoaling and transition zones, with breaking occurring farther offshore. Consequently, the nearshore wave height increases overall despite the substantial decrease in the nonlinear shoaling coefficient under sea level rise (Fig. 4). The result is consistent with previous studies showing that extreme waves generated by tropical cyclones can increase by more than 20% in certain regions (Lim et al., 2013; Mori et al., 2010; Shimura et al., 2011).
3.3 Wave Energy–Related Characteristics
The wave height is a key factor controlling wave energy-related characteristics. Accordingly, the figures in this section for these characteristics were generated using the same set of conditions and are presented in a consistent format as in Fig. 5, namely, with two slope values, m = 1/10 and 1/100 and three deepwater wave steepness values, H0/L0 = 0.1, 0.01, and 0.001.
3.3.1 Wave energy
Fig. 6 shows the relative changes in wave energy under sea level rise. The overall patterns follow the same tendencies as the wave height results but exhibit stronger variations because the wave energy is proportional to the square of wave height (Eq. 17). Consequently, the decreases in the shoaling and transition zones and increases in the surf zone are more pronounced. For strongly nonlinear cases (H0/L0 = 0.1), where wave breaking occurs farther offshore, sea level rise leads to a clear overall increase in nearshore wave energy. These results indicate greater wave energy exposure in nearshore regions after sea level rise, which is consistent with future projections of wave energy under climate change, particularly during extreme events such as typhoons (Lim et al., 2013). In addition, steeper slopes concentrate energy over a shorter cross-shore distance.
Relative change in wave energy as a function of d and D/L0: (a) H0/L0 = 0.001 and m = 1/100; (b) H0/L0 = 0.01 and m = 1/100; (c) H0/L0 = 0.1 and m = 1/100; (d) H0/L0 = 0.001 and m = 1/10; (e) H0/L0 = 0.01 and m = 1/10; (f) H0/L0 = 0.1 and m = 1/10
3.3.2 Wave energy flux
Fig. 7 presents the relative changes in wave energy flux as a function of d and D/L0, with different combinations of the other input parameters. The overall patterns are similar to those of the wave energy, but the magnitude is further influenced by the group velocity because the energy flux is defined as f = ecg. For weakly and moderately nonlinear waves (H0/L0 = 0.001 and 0.01), the increase in group velocity in intermediate and shallow water (D/L0 < 0.1) offsets the reduction in wave energy in the shoaling and transition zones and amplifies the increase in the surf zone. In the same manner, for waves with strong nonlinearity (i.e., H0/L0 = 0.1), the relative change in group velocity remaining greater than unity provides an overall increase in energy flux similar to wave energy, but with a stronger intensity.
Relative change in wave energy flux as a function of d and D/L0: (a) H0/L0 = 0.001 and m = 1/100; (b) H0/L0 = 0.01 and m = 1/100; (c) H0/L0 = 0.1 and m = 1/100; (d) H0/L0 = 0.001 and m = 1/10; (e) H0/L0 = 0.01 and m = 1/10; (f) H0/L0 = 0.1 and m = 1/10
3.3.3 Bottom orbital velocity
Bottom wave orbital velocity and wave-motion–induced bed shear stresses are essential parameters in sediment-transport and hydrodynamic models of coastal and estuarine systems. Previous studies assessed the wave exposure of coastal vegetation using simplified wave energy estimates based on the wind speed and fetch, whereas later research incorporated wind-wave growth and applied linear wave theory to calculate seabed orbital velocities (Chambers, 1987; Fagherazzi and Wiberg, 2009; Fonseca et al., 2002; Wiberg and Sherwood, 2008). Stevens and Lacy (2012) adopted the bottom orbital velocity as a measure of wave forcing to examine the effects of the spatial distribution of wave energy on the seagrass habitat variability, finding that seagrass percentage cover and minimum depth were negatively correlated with orbital velocity.
Fig. 8 presents the relative changes in bottom orbital velocity under sea-level rise. Sea level rise reduces the near-bottom orbital velocities in these regions because orbital velocity is inversely related to water depth, and wave height remains nearly constant in intermediate and deep water for H0/L0 = 0.001 and 0.01. In contrast, in shallow water, sea level rise allows larger waves, resulting in higher near-bottom orbital velocities. With stronger nonlinearity, the transition zone between the decreasing and increasing orbital velocity shifts farther offshore. These results are consistent with Stevens and Lacy (2012), who reported lower orbital velocities for all sea level rise scenarios during high tide, while sea level rise enabled more wave energy to propagate onto the low-tide terrace during low tide. Their study, which used a coupled hydrodynamic and sediment-transport model, highlighted the implications of sea level rise for nearshore ecosystem dynamics.
Relative change in wave energy flux as a function of d and D/L0: (a) H0/L0 = 0.001 and m = 1/100; (b) H0/L0 = 0.01 and m = 1/100; (c) H0/L0 = 0.1 and m = 1/100; (d) H0/L0 = 0.001 and m = 1/10; (e) H0/L0 = 0.01 and m = 1/10; (f) H0/L0 = 0.1 and m = 1/10
4. Conclusions
Cheon and Suh (2016) reported the relative changes in the refraction coefficient using contour diagrams, as done in the present study, but showed the relative changes in other wave characteristics using different formats. For example, for the significant wave height, the results were not shown directly as the relative changes in wave height with respect to the relative changes in water depth. Instead, they provided figures of intermediate quantities (such as Γ) that were used to calculate wave height variations. This makes it difficult to intuitively assess the influence of sea level rise on nearshore waves across varying depths and relative water-depth conditions. This limitation was addressed by presenting the relative changes in all wave characteristics within a unified framework using contour diagrams.
In addition, the work of Cheon and Suh (2016) was augmented by incorporating the analyses of wave energy and related characteristics. The equations for the relative changes in a range of wave characteristics, including energy-related parameters such as bottom orbital velocity, were derived and visualized to understand the effects of sea level rise on nearshore waves and wave energy within a unified analytical framework. By formulating the sea level rise as a relative change in water depth and using a consistent contour-based visualization, the framework provides an intuitive means of assessing the wave responses to different sea level rise scenarios across varying depths, bed slopes, and nonlinearities.
The results showed that the relative change in wavelength under sea level rise can reach values of up to approximately 1.2 when the relative depth increases to 1.5 in shallow water, while the effects of sea level rise on wave direction (i.e., the refraction coefficient) remain negligible in most cases. Nevertheless, the increase in the refraction coefficient becomes non-negligible when waves in deep water are incident more obliquely, near 60°. Hence, sea level rise can weaken local wave refraction and may alter the overall patterns of wave transformation in nearshore regions because of the reduced influence of seabed variations in two-dimensional horizontal bathymetry.
An analysis of the shoaling coefficient showed that it decreases after sea level rise, and this reduction becomes more pronounced as the nonlinearity increases. The wave height decreases in the shoaling and transition zones but increases in the surf zone, with strongly nonlinear waves exhibiting an overall increase due to offshore breaking. Wave energy displays similar patterns but with an amplified magnitude, while the wave energy flux is further modulated by the group velocity, producing stronger overall increases under nonlinear conditions. Specifically, for strongly nonlinear cases, the relative change in energy typically exceeds 1.5 for relative changes in water depth greater than approximately 1.3, and reaches values greater than 2.0 when the relative depth increases to 1.5 in the intermediate water depth. These results indicate greater wave-energy exposure in the nearshore regions after sea level rise, consistent with future projections of amplified nearshore wave energy under climate change, particularly during extreme events. In addition, steeper slopes further intensify this response by concentrating energy over a shorter cross-shore distance. The bottom orbital velocity decreases in deeper water but increases in shallow regions because sea level rise enables larger waves to propagate shoreward, with the transition zone between decreasing and increasing velocities shifting offshore under greater nonlinearity. These findings show that sea level rise can significantly alter wave transformation processes, enhance nearshore wave energy exposure, and modify seabed forcing, with important implications for sediment transport, coastal infrastructure, and ecosystem dynamics.
As mentioned in the introduction, the present study provides a general and dimensionless framework that can be applied to a wide range of sites and scenarios. In future work, fixed values of sea level rise, for example, 0.38 m under strong mitigation (SSP1-1.9) and 0.77 m under high-emission scenarios (SSP5-8.5), may be used to generate contour diagrams of the relative changes in wave characteristics for specific sea level rise changes. In addition, the framework can be applied to site-specific analyses with assumed sea level changes to investigate the impacts on the coastal structures and local morphology in a more realistic context.
Notes
No potential conflict of interest relevant to this article was reported.
This work was supported by a Research Grant of Pukyong National University (2025).
Nomenclature
α
Angle of principal wave direction (°)
α0
Deepwater angle of principal wave direction (°)
β0
β1
= 0.52e4.2m
βmax
Γ
ρ
Density of seawater (kg/m3)
A
C
Phase velocity (m/s)
Cg
Group velocity (m/s)
D
Water depth (m)
d
Relative change in water depth
E
Wave energy density (N/m)
e
Relative change in wave energy density
F
Wave energy flux (N/s)
f
Relative change in wave energy flux
g
Gravitational acceleration (m/s2)
H
Significant wave height (m)
H0
Deepwater significant wave height (m)
Kr
Refraction coefficient
kr
Relative change in refraction coefficient
Ksi
Linear shoaling coefficient
ksi
Relative change in linear shoaling coefficient
Ks
Nonlinear shoaling coefficient
kr
Relative change in nonlinear shoaling coefficient
L
Wavelength (m)
l
Relative change in wavelength
L0
Deepwater wavelength (m)
s0
Deepwater wave steepness ( = H0/L0)
T
Wave period (s)
Ub
Bottom orbital velocity (m/s)
ub
Relative change in bottom orbital velocity (m/s)