A Stability Model for Sea Cliffs Considering the Coupled Effects of Sea Erosion and Rainfall

Abstract

This study proposed a sea cliff stability model that accounted for the coupled effects of sea erosion and rainfall, offering an improved quantitative assessment of the toppling risk. The approach integrated the notch morphology (height and depth) and rainfall infiltration to quantify stability, validated by field data from six toppling sites near Da’ao Bay, where the maximum erosion distance error between model predictions and measurements ranged from 0.81% to 48.8% (with <20% error for Sites S2, S3, and S4). The results indicated that the notch morphology and rainfall exerted significant impacts on the sea cliff stability. Site S4 (the highest site) corresponded to a 17.5% decrease in K per 0.1 m notch depth increment. The rainfall infiltration reduced the maximum stable notch depth, decreasing by 8.86–21.92% during prolonged rainfall. This model can predict sea cliff stability and calculate the critical notch depth (e.g., 0.56–1.22 m for the study sites), providing a quantitative framework for coastal engineering applications and disaster mitigation strategies under climate change scenarios.

1. Introduction

Nearly one-fourth of the world’s population lives close to coastal cliffs [1], which make up 52–80% of the world’s coastline [2]. Sea cliffs’ instability threatens human activities and safety. Specifically, it endangers coastal infrastructure and adversely impacts public property, recreational resources, safety, and critical transportation routes [3,4]. Therefore, understanding the mechanisms governing cliff stability is essential for effective disaster mitigation and coastal management.

Coastal cliff mass movements can generally be categorized into falls, topples, slides, and flows. These movements are primarily driven by the basal erosion caused by waves, as well as by rainfall, which increases the soil moisture, augments its weight, and reduces shear strength [5,6]. A notch, a distinctive geomorphic feature shaped by the wave activity, critically affects the cliff stability [7,8]. Its gradual expansion weakens the overall cliff stability, often precipitating large-scale toppling failures [9,10].

In recent years, various methods have been developed to assess the instability hazards of coastal cliffs. References [11,12,13,14] used statistical methods, and an attempt was made to predict future unstable areas based on a determination of the contribution of the factors affecting past instabilities. This method is implemented by determining past positions of cliffs by comparing aerial photographs and/or satellite images at different times. Reference [15] proposed a Cliff Stability Index based on hydraulic and geomorphological characteristics to evaluate coastal cliff erosion risks. Similarly, reference [16] employed experimental and numerical modeling to analyze wave pressures on troughs beneath sea cliffs and revealed that an increased trough depth and height amplify wave-induced instability. Likewise, reference [17] investigated the interaction between the wave action and sea troughs along ancient Baltic Sea cliffs, emphasizing the significant influence of the trough morphology on cliff erosion and stability. Reference [18] combined two methodologies, CoastSnap NE participatory monitoring and photogrammetry using unmanned aerial vehicles (UAVs), which made it possible to identify and account for cliff movement events.

Reference [19] highlighted the synergistic interaction of the rainfall and wave action, which significantly accelerates coastal cliff erosion, and underscored the importance of long-term monitoring data for stability assessments. Reference [20] employed 3D modeling to analyze rock cliffs in the Adriatic region of Italy, demonstrating the profound impact of landslides induced by heavy rainfall and wave erosion on cliff stability. Similarly, reference [21] identified the rainfall and wave interaction as a primary driver of cliff destabilization and landslides in Svalbard, Norway.

Reference [22] focused on local erosion and landslides at the microlevel to analyze instability along the Basque coast of Spain. They identified landslides and erosional processes as critical contributors to cliff destabilization, exacerbated by rainfall and wave activity. Reference [23] examined coastal cliffs in Brazil, illustrating that basal erosion is crucial in destabilization because rainfall and wave forces intensify erosion at the cliff base, thereby increasing landslide risks. Experimental and numerical studies have yielded additional theoretical insights. Reference [24] demonstrated that the combined effects of coastal scouring and the abrasive action of marine troughs significantly weaken cliff stability.

Collectively, these findings highlight that the morphology of notches and the synergistic effects of the rainfall and wave action accelerate cliff erosion and heighten toppling failure risks, offering methodological guidance for stability assessments. Quantifying these factors remains essential; however, the combined effects of notch development and rainfall infiltration on the toppling risk have not been sufficiently quantified in the previous research, making it difficult to predict cliff instability under varying environmental conditions.

Unlike prior models focusing solely on wave erosion or rainfall, this study examined the geological structure of sea cliffs, the morphological features of notches, and the effects of rainfall infiltration to investigate how these factors interact to influence sea cliff stability. It aimed to elucidate the mechanisms by which notches and rainfall affect cliff stability and develop a corresponding stability evaluation model. Field data from six sites along the Da’ao Bay coastline were used to evaluate the model’s predictive performance. The findings are expected to provide theoretical and technical guidance for designing and implementing coastal protection projects, thereby mitigating the risk of natural disasters, such as toppling failures, and ensuring the sustainable development of coastal areas.

2. Study Area

The study area is focused on the sea-cliff-dominated coastline stretching from Liushui Town to Shilou Village in Da’ao Bay, located in northeastern Pingtan Island, Fujian Province, China, within the northwestern Taiwan Strait (Figure 1). This region experiences a mid-subtropical oceanic monsoon climate, characterized by frequent typhoons and gale-force winds. Average annual temperatures range from 19.0 to 19.8 °C, with an annual rainfall between 900 and 1200 mm.

In this study, toppling Sites S1–S6 were analyzed. The Da’ao Bay coastline runs approximately east–west in a crescent shape, featuring Yanshan-age granite headlands at its eastern and western extremities and a southward-curving terrace-type coast in the center. The sea cliffs primarily comprise wind-deposited sand, old red sand, and weathered granite residual soil.

Coastal soils are structurally weak, whereas hydrodynamic forces are particularly strong. Intense typhoon waves, heavy precipitation, and substantial erosion collectively cause severe coastal retreat each year. Between 1961 and 1983, the average annual coastal erosion rate was 1.25 m/a, rising to 1.46 m/a from 1983 to 2009 [25].

2.1. Toppling Characteristics

Based on the field investigation, Sites S1–S6 along approximately 1 km of coastline in the study area are analyzed. The coastal cliffs vary in height, ranging from a few meters to tens of meters, and their surfaces are almost completely vertical. The profile of the sea cliffs was observed using a laser rangefinder (Figure 2a). Prior to the measurement, a tripod was first set up on the beach in front of the cliff face. The legs of the tripod were inserted into the beach’s sandy soil for stabilization and then adjusted to the horizontal position. (Figure 2b). To measure the local morphology of the notch, the tilt angle of the laser rangefinder was adjusted by 0.5° per increment. The results showed that the notch exhibited a V-shaped profile (Figure 2d), where s represents the horizontal distance from the reference point, and h denotes the height of the sea cliff (Figure 2c).

Toppling materials accumulate at the base of the cliff: most are loose, a few are massive due to high soil cohesion, and some toppling blocks roll down for long distances. The toppling surface is generally vertical, with some appearing only at the top of the cliff and others extending throughout the entire cliff face. Some blocks fall from the top of the cliff, whereas others collapse from the entire cliff face. Some toppling material is transported by seawater, forming small scarps at the foot of the slope (Figure 2e).

2.2. Evolution Process of Cliff Toppling

The field survey results revealed shear failure at the sea cliffs toppling in the selected study area. Shear failure refers to the sliding of the toppling mass along the failure surface under the influence of the weight. The failure surface consists of tensile fracture surfaces and shear surfaces. Failure initiates from the cracks developed at the top of the cliff. As the cracks propagate downward, the resistance of the failure surface to the shear failure weakens until shear sliding occurs.

This form of damage begins with the formation of a notch at the base of the cliff. Under typhoon conditions, water levels rise in front of the cliff, and powerful breaking waves impact the base of the cliff. The forces acting on the cliff include the cyclic wave impact pressure, eddy current forces due to changes in the flow direction, and the abrasive forces of mud and sand carried by the waves. These forces cause erosion at the base of the cliff, resulting in the formation of a notch (Figure 3a).

As the notch develops, the overlying soil at the top of the cliff behaves like a cantilever beam under the influence of the weight. The top of the cliff experiences tensile stress, whereas the lower part is under compression. The upper part of the sea cliff is soaked and softened by rainfall infiltration, increasing its weight and reducing the cohesion of the cliff. When the maximum tensile stress at the cliff top exceeds the tensile strength of the soil, a vertical crack forms (Figure 3b). As the crack extends downward, the sliding force at the upper part of the cliff exceeds the resisting force, rendering the cliff unstable. The toppling body then undergoes shear failure along the vertical plane under the influence of the weight, leading to cliff toppling (Figure 3c).

Toppling material accumulates at the base of the slope, temporarily protecting the sea cliffs. However, due to disturbances, the toppling material is weaker than the undisturbed soil, making it more prone to erosion. Consequently, the base of the toppling material becomes susceptible to scarps or small notches (Figure 3d). As the toppling material erodes, the upper part moves downward, reducing the accumulation height, and eventually, all the toppling material is transported by waves (Figure 3e). This point marks the end of the shear failure process, and the next phase of damage begins (Figure 3f).

3. Methods

3.1. Rainfall Infiltration Model

Rainfall infiltration in sea cliffs was distinctly categorized into the free infiltration stage (Figure 4a) and the ponding infiltration stage (Figure 4b). Sea cliff soil’s water content distribution was separated into three layers: natural, transitional, and saturated [26].

During the development of cracks on sea cliffs, rainfall preferentially infiltrates through established fracture networks, exhibiting dual-phase transport mechanisms: (1) vertical percolation along primary crack apertures and (2) lateral diffusion through fracture sidewalls into adjacent soil matrices. For ease of calculation, it is assumed that the seepage effect along the cracks is the same as the infiltration effect on the sea cliff’s surface.

During rainfall, water begins to accumulate on the surface of the sea cliffs once the cumulative infiltration of the soil exceeds a critical threshold, as described by [27]. This rate can be calculated using the following formula:

$$I_p={\displaystyle \frac{\left({\theta }_s-{\theta }_i\right)S_f}{\cos \mathsf{\beta }\frac{R}{K_s-1}}}$$

(1)

where $I_p$ represents the cumulative infiltration amount before water accumulation, $R$ represents the rainfall intensity, and $K_S$ is the saturated permeability coefficient of the soil. Parameters ${\theta }_s$ and ${\theta }_i$ indicate the saturated water content and natural moisture content of the soil, respectively. Additionally, $S_f$ refers to the generalized matric suction of the soil at the wetting front, which can be determined using the following formula [28].

$$S_f\approx {\displaystyle \frac{3\lambda +2}{3\lambda +1}}{\psi }_b$$

(2)

where ${\psi }_b$ represents the air intake value of the soil, $K_r$ denotes the relative saturated permeability coefficient at the wetting front, and $\lambda$ refers to the fitting coefficient that characterizes the soil pore size distribution.

Parameter $t_p$ indicates the time when water first began to accumulate at the top of the sea cliffs.

$$t_p={\displaystyle \frac{I_p}{R\cos \beta }}$$

(3)

Based on the onset of water accumulation, the infiltration boundary conditions can be classified as follows:

(1) For a rainfall duration of $t<t_p$, all rainwater seeps into the soil, and the infiltration region stays unsaturated. The flow rate controls the infiltration boundary during this time, and the associated infiltration control equation is written as follows [29]:

$$f=R\cos \beta =K\left(\theta \right)+{\displaystyle \frac{K_s\left[{\psi }_r\left(\theta \right)-{\psi }_r\left({\theta }_i\right)\right]}{z_f}}$$

(4)

where $f$ represents the soil infiltration rate, $\theta$ denotes the soil moisture content, ${\psi }_r\left(\theta \right)$ indicates the relative suction of the soil at the wetting front, and $K\left(\theta \right)$ is the soil permeability coefficient [30]. $z_f$ denotes the wetting front depth. The calculation expressions are as follows:

$${\psi }_r\left(\theta \right)=S_e^{3+1/\lambda }{\displaystyle \frac{{\psi }_b}{3\lambda +1}}$$

(5)

$$K\left(\theta \right)=K_s{S}_e^{3+2/\lambda }$$

(6)

where ${\theta }_r$ represents the residual water content of the soil, and $S_e$ denotes the relative water content of soil, given by $S_e=\left(\theta -{\theta }_r\right)/\left({\theta }_s-{\theta }_r\right)$. The corresponding wetting front depth is derived as follows:

$$z_f={\displaystyle \frac{Rt\cos \beta }{\theta -{\theta }_i}}$$

(7)

By substituting Equation (7) into Equation (4), the resulting nonlinear equation is as follows:

$$f=R\cos \beta =K\left(\theta \right)+{\displaystyle \frac{K_s\left[{\psi }_r\left(\theta \right)-{\psi }_r\left({\theta }_i\right)\right]\left(\theta -{\theta }_i\right)}{Rt\cos \beta }}$$

(8)

The soil moisture content for various rainfall durations can be determined by solving the nonlinear equation presented in Equation (8):

$$t={\displaystyle \frac{K_s\frac{{\psi }_b}{3\lambda +1}\cdot \frac{{\left(\theta -{\theta }_r\right)}^{3+1/\lambda }-{\left({\theta }_i-{\theta }_r\right)}^{3+1/\lambda }}{{\left({\theta }_s-{\theta }_r\right)}^{3+1/\lambda }}\left(\theta -{\theta }_i\right)}{(R\cos \beta -K_s{\left(\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}\right)}^{3+2/\lambda })R\cos \beta }}$$

(9)

Additionally, the wetting front depth of the sea cliffs for various rainfall durations can be calculated using Equations (7) and (9).

(2) When the rainfall duration is $t\ge t_p$, water starts to build up on the cliff’s surface. The infiltration region gets saturated when the intensity of the rainfall is above the soil’s permitted penetration rate, and the water head controls the infiltration border [31]. The two stages of rainfall infiltration are ponding infiltration and free infiltration. The following formulas can be used to determine the infiltration rate and cumulative infiltration [26]:

$$f_t=\left\{\begin{array}{c}R\cos \beta ,t\le t_p\\ K_s\left(\cos \beta +{\displaystyle \frac{S_f}{z_f}}\right),t>t_p\end{array}\right.$$

(10)

$$I={{\displaystyle \int }}_0^t{f}_{\left(t\right)}\mathrm{d}t$$

(11)

Based on Equation (11), the cumulative infiltration amount can be determined, and subsequently, the depth of the wetting front within the saturated infiltration zone becomes

$$z_f={\displaystyle \frac{I}{{\theta }_s-{\theta }_i}}$$

(12)

The depth of the saturated layer is $z_s$, whereas the depth of the transitional layer is $z_t$ (Figure 5). To precisely describe the distribution of water content in the transitional zone, an elliptic curve distribution is employed due to its high computational precision. As a result, the water content distribution inside the transitional layer is likewise described by the elliptic curve.

The following is an expression for the soil’s final water content distribution:

$$\theta \left(z\right)=\left\{\begin{array}{l}{\theta }_s,0\le z\le z_s\\ {\theta }_i+\left({\theta }_s-{\theta }_i\right)\sqrt{\left(1-{\displaystyle \frac{{\left(z-z_s\right)}^2}{z_t^2}}\right)},z_s\le z\le z_h\\ {\theta }_i,z>z_h\end{array}\right.$$

(13)

where $z_h$ represents the total depth of the infiltration area, which can be calculated using the following equation:

$$z_h=z_s+z_t$$

(14)

As rainfall infiltrates, the depths of the saturated, transitional, and natural layers undergo dynamic changes. Experimental results [32] revealed a proportional relationship between the depth of the transitional layer and the total infiltration depth. Moreover, a strong linear correlation exists between this ratio and the depth of the entire infiltration area, as expressed by the following equation:

$$z_t=\eta z_h$$

(15)

$$\eta =\mathrm{a}{z}_h+\mathrm{b}$$

(16)

where $\eta$ represents the ratio of the transitional layer to the depth of the whole infiltration zone. Coefficients $\mathrm{a}$ and $\mathrm{b}$ are −0.003 and 0.8712, respectively.

During transitional layer correction, the cumulative infiltration amount must remain unchanged. The value of $I_2$ can be calculated using Equations (7) and (11), where $I_1=I_2=I_M$. These represent the cumulative infiltration after the addition of the transitional layer, and their values should equal the sum of the saturated layer infiltration $I_s$ and the transitional layer infiltration $I_t$.

The infiltration amount in the saturated layer is given by $I_s=\left({\theta }_s-{\theta }_i\right)z_s$. Due to the elliptical curve distribution of the transitional layer, the infiltration amount is equal to one-fourth of that of the elliptical area, expressed as $I_t=0.25\mathsf{\pi }\left({\theta }_s-{\theta }_i\right)z_t$. Therefore, the total cumulative infiltration amount can be expressed as:

$$I_1=\left({\theta }_s-{\theta }_i\right)z_s+0.25\mathsf{\pi }\left({\theta }_s-{\theta }_i\right)z_t$$

(17)

$$I_2=R\cos t_p+K_s\cos \beta \left(t-t_p\right)+{\displaystyle \frac{K_s{S}_f\left({\theta }_s-{\theta }_i\right)}{Rt\cos \beta }}\ln {\displaystyle \frac{t}{t_p}}$$

(18)

By combining Equations (14)–(18), $z_h$, $z_s$, and $z_t$ can be calculated, allowing the water content distribution to be calculated across the entire sea cliff’s soil layer.

3.2. Stability Analysis Model

The calculation model for cliff toppling assumed that shear failure results from the sliding of rock masses along the failure surface under gravitational forces. This failure surface consists of both tensile fracture and shear components. Failure initiates at cracks formed at the cliff top, and as these cracks propagate downward, the failure surface’s resistance to shear failure gradually weakens until shear sliding occurs. The formation of a notch causes the overlying sea cliffs to resemble a cantilever beam. The weight of the overlying soil generates a bending moment on the vertical plane behind the notch, inducing substantial tensile stress at the cliff top. Cracks appear near the summit of the cliff when this tensile stress is greater than the soil’s tensile strength. Given that the formation of these cracks initiates shear failure, the cantilever beam model is employed for analysis.

In the stability calculation, the point of action of the sliding weight was assumed to remain unchanged after rainfall. The shear failure calculation model simplifies the redistribution of stress during this adjustment. Furthermore, wave forces also influence the stability of the sea cliffs. When waves approach the sea cliffs, the cliff experiences wave pressure, which generally enhances stability. Conversely, when waves recede from the cliff or the wave notch is formed, the cliff experiences wave tension, which undermines its stability. However, the magnitude of this tensile force could not be determined by calculation in the model, and no relevant empirical data are available for calibration. Therefore, in the shear failure calculation model, only the effect of the weight of the cliff was considered; wave forces were excluded.

At the top of the cliff, the bending moment causes the most tensile stress. In the schematic of sea cliff’s shear damage (Figure 6b), $\beta$ represents the slope of the cliff-front beach, whereas $H_c$ denotes the height of the sea cliffs. Parameters $d_w$ and $d_n$ refer to the height and depth of the notch, respectively. The cantilever body above the notch is labeled ABCD, and the sliding force acting on it, induced by weight, is denoted by $F$. The variable $y$ represents the crack depth, and O marks the neutral point where tensile and compressive stresses are zero. Additionally, ${\sigma }_{t,\max }$ indicates the maximum tensile stress, and ${\sigma }_{c,\max }$ refers to the maximum compressive stress.

The vertical plane behind the notch experiences a bending moment due to the weight of the ABCD section above it. The top part of the cliff experiences tensile stress as a result of this bending moment, whereas the lower half experiences compressive stress.

If the tensile stress exceeds the tensile strength of the soil, cracks will form at the top of the cliff. The relevant derivation process is described in Appendix A.

$$y=z_0\left(1-{\displaystyle \frac{s_t}{{\sigma }_{t,max}}}\right)$$

(19)

where $z_0$ represents the depth of tensile stress, and $s_t$ represents the tensile strength.

Rainfall reduces the tensile strength of the soil, whereas the formation of cracks further decreases the shear strength resistance along the sliding surface. When the sliding force generated by the weight of the cantilever body ABCD exceeds the shear resistance force provided by the shear strength of the surface BC soil, sliding occurs. The safety factor $K$ of sea cliff stability is expressed as

$$K={\displaystyle \frac{{\tau }_{cr}A}{F}}$$

(20)

In Equation (20), ${\tau }_{cr}$ represents the shear strength of soil. According to Coulomb shear strength theory, ${\tau }_{cr}=c+{\sigma }_n\tan \phi$, and for the vertical failure surface, the normal stress ${\sigma }_n$ is equal to 0 [33]; therefore, ${\tau }_{cr}=c$.

When the safety factor $K$ is less than 1.0, the sea cliffs undergo vertical shear failure; otherwise, the sea cliff is stable. Make safety factor $K=1$, and solving $d_n$, the erosion distance can be obtained.

(1)

In the first stage of rainfall, ${\sigma }_{t,max}\ge s_t$:

$$F={\displaystyle \frac{1}{2}}\left(2H_c-d_w-d_n\tan \beta -4z_f\right)d_n\gamma +2z_f{d}_n\left(r_d+\theta r_w\right)$$

(21)

$$K={\displaystyle \frac{c\left(H_c-d_n\tan \beta -y-z_f\right)+c_0{z}_f}{F}}$$

(22)

where $c_0$ represents the cohesion of unsaturated soil in the infiltration area of the first stage of rainfall.

(2)

In the second stage of rainfall,

$$F_1=\left(r_d+{\theta }_s{r}_w\right)z_s{d}_n$$

(23)

$$F_2={\displaystyle \frac{\mathsf{\pi }}{4}}{r}_d{z}_t{d}_n+{\displaystyle \frac{\mathsf{\pi }}{4}}{\theta }_i{r}_w{z}_t{d}_n+{\displaystyle \frac{\left({\theta }_s-{\theta }_i\right)r_w{z}_t{d}_n}{3}}$$

(24)

$$F_3=\left(r_d+{\theta }_i{r}_w\right)\times \left[\left(2H_c-d_w-d_n\tan \beta \right){\displaystyle \frac{d_n}{2}}-z_s{d}_n-0.25\mathsf{\pi }{d}_n{z}_t\right]$$

(25)

$$F_4=\left(z_s{d}_n+0.25\mathsf{\pi }{d}_n{z}_t\right)r_w$$

(26)

where $F_1$ represents the weight of the saturated layer of the sea cliff soil, $F_2$ denotes the weight of the transitional layer of the sea cliff soil, $F_3$ refers to the weight of the natural layer of the sea cliff soil, and $F_4$ is the rainwater weight of the rainwater infiltration along the cliff cracks. The weight of the sea cliff’s soil $F=F_1+F_2+F_3+F_4$ can be calculated, and the weight of the sea cliff’s soil can be obtained by combining Equations (23)–(26).

$$c^{\prime }={\displaystyle \frac{{\mathrm{c}}_1{z}_s+z_t\frac{{\mathrm{c}}_1+c}{2}}{z_f}}$$

(27)

where $c^{\prime }$ is the average cohesion of the soil in the infiltration area of the second stage of rainfall, and ${\mathrm{c}}_1$ is the cohesion of saturated soil.

The safety factor of the sea cliff stability model is

$$K={\displaystyle \frac{c\left(H_c-d_n\tan \beta -y-z_f\right)+c^{\prime }{z}_f}{F}}$$

(28)

when the sea cliffs crack and the safety factor is less than 1, the shear failure of sea cliffs occurs, resulting in toppling.

3.3. Parametric Selection

The size of the toppling and the morphological parameters of the cliff were measured using a laser rangefinder and tape. The slope of the beach in front of the sea cliffs is 5°, the height of the sea cliffs is 1.5–13.0 m, the toppling of S3 rolled to 7.4 m from the cliff surface, the accumulation slope is 31.4–53.2°, and the toppling of S5 is mainly loose. The toppling of S2 is mainly massive; the rest of the toppling is a mixture of loose soil and massive soil. Some of the larger massive toppling is distributed on the outside, and there are signs of outward rolling after the toppling. Notably, the size of the toppling block differs. The small side length is several centimeters, and the large side length is approximately 1 m. The notch observed in the field had a height of 0.56–1.20 m. For a certain wave, the notch height was regarded as a fixed value, so it was calculated according to 1 m.

The granite-weathered residual soil of the study area has a high clay content, placing it in a strong over-consolidated state. The old red sand, which has been compacted over an extended period, contains some clay. In contrast, the aeolian sand, located at the top, has a loose structure. Undisturbed soil samples are collected from the cliff surface at six toppling locations and transported to the laboratory to analyze their main engineering properties. Weight was measured using the ring knife method, whereas the direct shear test was used to determine cohesion and the internal friction angle. Additionally, the water content was measured using the drying method. The relevant parameters for the cliff materials are summarized in

Table 1. Related parameters of sea cliffs.

Position	Hc/m	dn/m	γ/kN·m−3	θ	c/kPa	φ/(°)
S1	7.5	0.58	18.9	0.185	14	17
S2	1.5	0.50	19.2	0.323	17	15
S3	12.0	1.23	19.1	0.204	24	12
S4	13.0	0.89	18.8	0.315	15	17
S5	5.3	0.77	18.9	0.253	23	19
S6	2.0	0.41	18.5	0.176	13	20

.

Regression analysis was used to analyze the topsoil at each study site. The shear strength index of the sea cliffs in the area first increases and then decreases with increasing water content. A segmented function was used to represent the relationship between the cohesive strength of the cliff soil and water content.

$$c=\left\{\begin{array}{c}414\theta -60.88,\theta \le 0.204\\ -55\theta +33.8,\theta >0.204\end{array}\right.$$

(29)

when the water content of cliff soil reaches 20.4%, the cohesion peaks, which is the optimal water content.

Because the main influencing factor of the shear strength of a cliff is cohesion, the influence of internal friction angle is small. Moreover, the change in internal friction angle with the change in water content is small, so the influence of internal friction angle on the shear strength of a cliff was ignored [23].

The values of the remaining calculated parameters are listed in

Table 2. Other related calculation parameters.

Design Conditions	Value
${\theta }_s$	0.5
$K_s/\mathrm{m}\cdot {\mathrm{s}}^{-1}$	1.40 × 10−6
${\theta }_r$	0.013
$R/\mathrm{mm}\cdot {\mathrm{h}}^{-1}$	5
${\psi }_b/\mathrm{kPa}$	2.752
$S_f/\mathrm{mm}$	424.3
$\lambda$	0.319

[31].

3.4. Model Validation

At the maximum notch depth, the sea cliffs topple. Therefore, the maximum groove depth of the notch represents the erosion distance of the sea cliffs caused by each toppling. To validate the model, the rainfall intensity and rainfall time were set close to 0 in the model verification to simulate the situation without rainfall. This setting allows the results of the calculation model and the actual data to be compared.

The actual measured erosion distance is 0.41–1.23 m. The maximum erosion distance of the calculation model is 0.56–1.22 m. The maximum erosion distance error between the calculated value and the measured value of the model is between 0.81% and 48.8% (Figure 7), which validates the model. The error in S2, S3, and S4 is less than 20%, whereas the error in S1, S5, and S6 is slightly larger, as the potential study area may have a longer toppling time. The lower part of the toppling is partially transported under the action of strong waves, and the upper part of the toppling moves downward, thus burying the notch. The calculated value of the maximum notch is less than the measured value, which may be due to the toppling of the sea cliffs in the study area under the influence of heavy rainfall. Rainfall increases the weight of the sea cliffs themselves while decreasing the strength of the sea cliff soil. The maximum depth of the sea cliffs is reduced, and the notch may occur in the early stage of the development of the sea cliffs, with shear failure resulting in toppling.

4. Results

4.1. Trough Height

A stability analysis model was employed to evaluate the relationship between the height of the notch and the safety stability factor of sea cliffs at six sites (S1–S6). This analysis was conducted under no-rainfall conditions and a constant trough depth.

As depicted in Figure 8, the safety stability factor generally increased with the height of the notch (0.6–1.2 m) across all sites. Notably, the influence of the notch height on the stability diminished with the increasing cliff height.

In the studied sea cliffs, Site S2 and Site S4 represent the lowest and highest cliff profiles. At Site S2 (cliff height = 1.5 m), K increased from 0.34 to 1.63 (a 379.4% increment) as the notch height increased from 0.6 m to 1.2 m (Figure 8b), corresponding to a 63.2% rise in K per 0.1 m notch height increment. At Site S4 (cliff height = 13.0 m), K increased from 0.73 to 0.75 (a 2.7% increment) as the notch height increased from 0.6 m to 1.2 m (Figure 8b), corresponding to a 0.45% rise in K per 0.1 m notch height increment.

4.2. Trough Depth

The influence of the trough depth on the sea cliff stability was examined using an analytical model at six study sites. The relationship between the trough depth and the safety factor was analyzed under the conditions without rainfall.

Figure 9 showed that the safety factor of sea cliffs generally decreased with an increasing trough depth. Notably, Sites S2 and S6, with the lowest cliff heights, experienced the most significant variations in safety factors. At Site S4 (the highest site), the safety factor (K) decreased from 1.8 to 0.8 as the notch depth increased from 0.46 m to 1.03 m (Figure 9b), corresponding to a 17.5% decrease in K per 0.1 m notch depth increment.

4.3. Rainfall Duration

Figure 10 illustrates the initial water content and the corresponding time for the water accumulation on the cliff surface when the rainfall intensity was set to 5 mm/h. A lower initial soil water content was associated with longer water accumulation times.

Under dry conditions, the critical trough depths at S1–S6 were 0.79 m, 0.56 m, 1.22 m, 0.83 m, 1.08 m, and 0.61 m (Figure 9), respectively. During prolonged rainfall, which leads to surface water accumulation, these depths reduced to 0.72 m, 0.44 m, 1.1 m, 0.82 m, 0.89 m, and 0.51 m (Figure 11). The reduction rates in maximum erosion distances before rainfall ranged between 8.86% and 21.92%, emphasizing the critical role of rainfall in accelerating cliff instability.

The impact of the rainfall duration on the sea cliff’s stability was investigated, with a focus on the changes in the safety factor due to infiltration at Sites S1–S6. Figure 12 demonstrates the inverse relationship between the rainfall duration and the safety factor. At the threshold of 1.4 for the safety factor, a prolonged rainfall infiltration led to significant reductions in stability across all locations. Sites S2, S5, and S6 experienced sharper declines than the gradual reductions observed at S1, S3, and S4.

Figure 13 illustrates the relationship between the rainfall duration and the maximum depth of the trough when the safety factor of the sea cliffs stabilized at one across the study sites (S1–S6). The results demonstrated a clear decreasing trend: as the rainfall duration increases, the maximum depth of the trough that allows the cliffs to maintain stability progressively diminished.

4.4. Rainfall Intensity

Figure 14 shows the relationship between the maximum trough depth and the rainfall intensity across varying rainfall durations. The results indicated that the maximum trough depth decreased consistently with increasing rainfall durations at all sites (S1–S6). This reduction was more rapid under higher rainfall intensities (4 mm/h and 5 mm/h) than at a lower intensity (3 mm/h).

Figure 15 illustrates the variation in the safety stability factor of sea cliffs as a function of the rainfall intensity and duration. Across all study sites, the safety stability factor consistently decreased with increasing rainfall durations, regardless of the rainfall intensity. The decline in the safety stability factor was notably more pronounced under higher rainfall intensities (4 mm/h and 5 mm/h).

5. Discussion

Compared to previous models that focus solely on wave-induced erosion or rainfall-triggered failures, this study integrated both processes into a single stability framework. The model demonstrated potential for coastal hazard assessments and provided a useful tool for evaluating sea cliffs’ stability under different environmental conditions. While the model performed well in the study area, some limitations remain. The effects of dynamic wave forces are not explicitly considered, and future research should integrate hydrodynamic simulations for an improved accuracy. Future work could also explore real-time meteorological data integration and the use of machine learning techniques to refine predictive capabilities and expand the model’s applicability.

5.1. The Effect of Notch Morphology on the K

This study demonstrated that the notch morphology significantly influenced the sea cliff stability. When the notch depth is constant, as the height of the notch increased, the weight of the sea cliffs decreased, which in turn reduced the sliding force of the sea cliffs. Furthermore, the tensile stress of the overlying soil was also reduced, which resulted in a decrease in the depth of the cliff-top cracks. These factors work in concert to enhance the ant sliding ability of the sea cliffs, thereby improving their overall stability. Concurrently, as the sea trough approaches the static water level, an increase in the water depth from the trough bottom will result in a corresponding rise in pressure. Consequently, when the height of the sea trough is minimal, the vertical wave force at the base of the sea trough will increase proportionately. Typically, the wave pressure peaks at the still water level and subsequently gradually decreases in either direction. Accordingly, it can be posited that when the depth of the sea trough remains unaltered, an increase in the trough height will result in a reduction in wave pressure, thereby enhancing the stability of the sea cliffs [16].

When the notch height was constant, the stability decreased with the depth of the notch. These results corroborated previous research showing that deeper notches increase the self-weight and tensile stresses of cliffs, exacerbating instability [16,24]. The notch exerted a differential effect on cliffs of varying heights. In taller cliffs, the notch morphology exhibited a weaker relative influence on stability. Notably, the impact on the stability of the cliff was inversely proportional to the geometric proportion of the notch to the cliff.

5.2. The Effect of Rainfall on the K

Rainfall significantly affected both the maximum trough depth and safety stability factor of sea cliffs, and the following key trends were observed:

Prolonged rainfall led to a consistent decline in stability across all sites, with reductions in safety factors. The maximum trough depth also decreased as the rainfall duration extended (Figure 14). Higher rainfall intensities (4–5 mm/h) accelerated the soil saturation, reducing safety stability factors more rapidly compared to lower intensities (3 mm/h) (Figure 15). These trends underlined the role of sustained rainfall as a critical destabilizing factor in sea cliff dynamics. This finding agreed with that of [2,19], who observed that sustained rainfall weakens soil strength and increases the weight of the soil, facilitating slope failure.

Interestingly, during the early stages of infiltration, cliffs under lower rainfall intensities occasionally exhibited smaller safety factors than those under higher-intensity scenarios. This phenomenon can be attributed to the relatively low initial water content of the sea cliff soil. Rainfall initially increased the shear strength of soil, which decreased subsequently. This resulted in an initial increase in the stability of the sea cliff, which then continues to decline as the rainfall persists.

6. Conclusions

This study developed an improved sea cliff stability model that integrated rainfall-induced weakening and notch morphology, addressing an aspect that has not been comprehensively considered in previous models. This model could predict the stability of sea cliffs and calculate the maximum depth of the notch.

The model was validated against field measurements from six toppling sites in Da’ao Bay, quantifying the relative influence of key parameters (notch depth, rainfall intensity, and soil strength) on the toppling risk. The notch morphology and rainfall exerted significant impacts on sea cliff stability: the effect of the notch height on stability diminished with the increasing cliff height, whereas the notch depth and rainfall emerged as the primary destabilizing factor. Specifically, at Site S4, a 0.1 m increase in the notch depth reduced the safety factor by an average of 17.5%, significantly elevating the toppling probability. The prolonged rainfall exacerbated instability through dual mechanisms—decreasing the soil shear strength and increasing the soil weight—leading to an 8.86–21.92% reduction in the maximum stable notch depth.

In conclusion, the sea cliff stability model proposed in this study demonstrated considerable effectiveness in the quantitative assessment of the coupled effects of wave erosion and rainfall. The model holds significant potential for stability prediction in the context of coastal engineering disasters and climate change scenarios.