The sampling site (30°51’N, 121°55’E) is located at the Nanhui Foreland salt marshes on the seaward side of a seawall ( Figure 1A ). The area behind the seawall was reclamation area which had been wetlands before 2001(Tian et al., 2016). This area has a mixed semidiurnal tide with a mean tide tidal range of 3.2 m and a maximum tidal range of over 4 m during spring tides (Zhu et al., 2014). The sampling site is submerged in water at high tide and exposed to air at low tide. With the spreading of the invasive species S. alterniflora, the native species S. mariqueter has shrunk to a small region ( Figure 1A ). To preserve biodiversity and improve the resilience of the coastal ecosystem, ecological measures are being conducted to restore S. mariqueter (Yuan et al., 2022). S. mariqueter is usually observed to start to flatten in December and last until January at the end of the growing season. We collected S. mariqueter samples on Jan 4 ^th 2021 when most S. mariqueter are flattened. Almost all of the S. mariqueter stems were broken at 10 cm from the base, resulting in flattened stems above the breaking point ( Figure 1B ). Three 25 × 25 cm² quadrats of S. mariqueter were collected for measurements. The measured plant density (N) was 2148 ± 414 stems/m², calculated from the average number of shoots over the three quadrats. All of the stems were cut from the base and taken to the lab to measure the geometrical and mechanical properties.

The stem length (l) was measured from total 30 stems from the three quadrats with 10 stems for each quadrat. The measured stem length ranged from 21.4 to 44.3 cm with an average of 34.2 cm. The cross section of S. mariqueter was triangular and oriented in a random direction. The height of each side of the cross-section was measured using a caliper. As the cross-section is close to an equilateral triangle, the mean height at three sides is defined as the section height (h_s ), such that the second moment of area is I = 3 h s 4 / 54 . The measured section height ranged from 1.27 to 1.89 mm with an average of 1.56 mm. The elastic modulus (Young’s modulus, E) was measured with 20 specimens from 20 stems by a three-point bending test (Vuik et al., 2018). The measured E ranged from 1.9 to 7.2 GPa with an average of 3.9 GPa. The measured values are summarized in Table 1 and also compared with the values in literature.

As it is difficult to keep the flattening state and bioactivity of stems by moving S. mariqueter in the flume, we used dynamically similar model vegetation to explore the effects of stem flattening on wave attenuation. The field measurements of the wave attenuation by S. mariqueter will be documented in another paper in preparation, where the plant properties and wave conditions were measured from May to November 2021 covering the whole growing season of S. mariqueter. The field measurements included plant properties and wave data from May to November 2021 across the growing season of S. mariqueter. To model the vegetation, hollow polypropylene (PP) tubes with circular cross-section were selected based on the dynamic similarity. The model was full scale. The outer diameter of the model stem was 1.8 mm, which was comparable to the measured section height (1.27-1.89 mm) and side lengths of S. mariqueter. The wall thickness of the model stem was 0.4 mm and the elastic modulus was 1.739 GPa, yielding a flexural rigidity (EI) of 876 N·mm², which was slightly larger than the maximum measured value of 676 N·mm² for S. mariqueter. The EI of the model vegetation is 30% larger than the measured maximum EI of S. mariqueter. The wave attenuation by the model vegetation is estimated to be larger than the real S. mariqueter. However, the effects of EI on wave attenuation become less sensitive when the vegetation is very flexible. For example in Zhu et al. (2021), the wave energy dissipation decreased by 6.7% when EI decreased by 76%. The mass density of the model stem was 0.92 g/cm³. As the PP tubes were filled with air, the effective mass density of PP tube was 0.64 g/cm³, which was larger than the measured value of 0.35 ± 0.05 g/cm³, yielding a smaller buoyancy. However, according to the theory in Henderson (2019), the effects of buoyancy on stem motion were much smaller than stiffness and therefore ignorable. Although there are some other parameters governing the motion of stems (Zhu et al., 2021), these parameters are not as important as the stiffness and buoyancy for the wave conditions in this study and therefore not discussed in detail here. The designed vegetation length (l) was 50 cm ( Figures 2A, C ), slightly longer than the maximum measurement of 44.3 cm. To represent flattened vegetation, the model vegetation was folded at 10 cm above the base ( Figures 2B, D ), which was comparable to the folding point at 13.3 ± 2.6 cm observed in the field. To keep the same configuration, the vegetation was folded towards the direction of wave propagation after finishing the experiments for standing vegetation. Due to buoyancy, the horizontal part of the vegetation curves upward at 20° with the horizontal line ( Figure 2D ). The model vegetation was installed in and evenly distributed on a 5 mm-thick acrylic plate, which was fixed at the bottom of the flume. There are 4455 stems over 9 acrylic plates, covering a 4.5 m-long area along the flume ( Figure 2E ). The plant density was therefore 1980 stems/m², similar to the measured plant density in this study and literature (e.g. Ysebaert et al., 2011; Ge et al., 2018).

The laboratory experiments were conducted in the 80 m-long, 1 m-wide, and 1.8 m-high wave flume at Hohai University in Nanjing, China. In common condition at Nanhui, the mean water depth was from 0.58 to 1.13 m, with wave height from 0.06 to 0.15 m and wave period from 0.8 s to 2.8 s (Liu et al., 2021). So the wave conditions in the present study can represent most wave periods and wave heights under common conditions. In the present study, the water depth (h) was from 0.5 m to 0.6 m, such that both standing and flattened vegetation were completely submerged. Due to the limitation of flume and wave paddle, the water depth cannot be over 0.7 m. The regular incident wave height (H_{I 0}) was from 0.05 to 0.15 m. The wave period (T) was from 1.2 to 3 s, yielding wavelength (λ) of 2.05 to 6.4 m, where λ=2π/k with k the wave number. The wave number is determined from the dispersion relation (2π/T)²=gk tanh kh with g the gravitational acceleration (Dean and Dalrymple, 1991). The meadow length was 4.5 m, covering 0.7 to 2.2 wavelength. The designed wave conditions are summarized in Table 2 .

The wave height over the meadow was measured by 15 wave gauges. A reference gauge was placed 50 cm upstream in front of the leading edge. Other gauges were fixed across the meadow with 10-30 cm intervals depending on the wavelength. The sampling time was 5 min at a rate of 1 kHz, including 100-250 waves. The data were extracted after 3 mins, when the measured waves are at steady state. According to Dalrympleet al. (1984), the wave height decays as,

where H(x) is the local wave height at x m from the leading edge ( Figure 2A ), H_{I 0} is the incident wave height at x = 0, and β is wave damping coefficient. The wave damping coefficient is fitted based on the measured wave heights along the meadow using the methods in Appendix A .

β is more related to wave height which had a wide range in the experiment. The wave attenuation under different wave conditions can be more significantly shown with β. The measured wave damping coefficients for both standing vegetation (β_S ) and flattened vegetation (β_F ) are shown in Table 3 . To investigate the characteristics of the wave attenuation capacity of vegetation in different wave conditions, β was presented as a function of H_{I 0}, h, and λ as shown in Figure 3 . Obviously, β_F is smaller than β_S for all the tested cases with β_F/β_S ranged from 33.6% to 72.4%, indicating that the wave attenuation of flattened vegetation is smaller than that of standing vegetation. Nevertheless, β_F varies with wave conditions in a similar pattern to β_S ( Figure 3 ).

As shown in Figure 3A , both β_S and β_F increase with H_{I 0}, indicating that both standing and flattened vegetation can damp more wave energy in larger waves. However, the wave attenuation reduces with increasing water depth ( Figure 3B ). As water depth rises, the wave orbital velocity decreases, yielding a smaller drag and therefore reducing wave attenuation. Associated with the water level rise, the wave energy also moves upward because the wave energy concentrates near the water surface and decays along water depth. Consequently, less wave energy is damped by the more deeply submerged flattened vegetation such that β_F drops more dramatically than β_S ( Figure 3B ). For instance in Figure 3B , β_S dropped 16.7% from 0.066 m^–1 to 0.056 m^–1 while β_F dropped 36.8% from 0.038 m^–1 to 0.024 m^–1 when the water depth increased by 20% from 0.5 m to 0.6 m. The wave damping coefficient does not show a significant change with λ in these experiments ( Figure 3C ).

Numerous studies have been conducted to quantify the wave damping coefficient (β) of standing vegetation based on the formula in (Dalrymple et al., 1984), which is given by,

The challenge is to calibrate the bulk drag coefficient C_D for a vegetation meadow. Conventionally, C_D is fitted as a function of Reynolds number (Re) or Keulegan–Carpenter number (KC) (e.g., Mendez and Losada (2004); Anderson and Smith (2014); Hu et al. (2014); Ozeren et al. (2014); van Veelen et al. (2020)). As Re and KC do not include vegetation rigidity (EI), these formulas are not applicable to other vegetation with different flexibilities. Alternatively, Luhar and Nepf (2016) proposed a technique that considers the effects of blade flexibility by using effective blade length (l_e ), which is defined as the length of a rigid blade that dissipates the same wave energy as the flexible blade with the original length (l). The effective blade length is usually fitted as a function of the Cauchy number (Ca) and the ratio of the blade length to wave excursion (L) (Luhar and Nepf, 2016; Lei and Nepf, 2019). Zhu et al. (2023) demonstrated that the combination of C_D –Re relation and effective plant height (EPH) can provide high accuracy in predicting wave attenuation in salt marshes. In the meantime, to avoid the uncertainties in using the formulas of bulk drag coefficient or effective blade length, Maza et al. (2022) developed a parameter, hydraulic standing biomass (HSB), to fit the wave damping coefficient, where HSB is defined as a function of the meadow mean height, standing biomass, and incident flow characteristics.

Unlike standing vegetation, flattened vegetation is composed of two parts: the vertical part and the horizontal part. The vertical part has no free end and the horizontal part is not clamped. Thus, the flattened vegetation cannot be simplified as a cantilever beam such that the CaL-based scaling law for effective blade length (Luhar and Nepf, 2016; Lei and Nepf, 2019) is not applicable to flattened vegetation because that CaL-based scaling law is derived from the static forcing balance between drag and blade stiffness by assuming the blade is a cantilever beam (Luhar and Nepf, 2016).

As it is challenging to define an effective blade length for flattened vegetation due to its complicated morphology, we developed a dimensionless wave attenuation indicator (WAI, -) to formulate β_S and β_F , inspired from Maza et al. (2019; 2021; 2022). The wave attenuation indicator is defined based on the characteristics of β_S and β_F with respect to H_{I 0}, h, and λ and given by

Note that the first two terms of the right hand side of equation (3) reflect wave attenuation in shallow water waves since the wave damping coefficient in shallow water waves is proportional to H_{I 0} l/h ² (Dalrymple et al., 1984; Zhu et al., 2021).Therefore, we add a term 1/tanh kh such that WAI can be used for a wider range of wave conditions. As WAI approaches 0, β should be 0. Thus, we use an exponential form β=aWAI^b to generate empirical formulas for the wave damping coefficients for both standing and flattened vegetation. The sample size was 7.

The β_S has the following relation with WAI,

with R ² = 0.789 and the p-value of 0.008 ( Figure 4A ). The relation between β_F and WAI is

with R ² = 0.757 and the p-value of 0.011 ( Figure 4B ). The ratio β_F/β_S also shows a relation with WAI and given by

with R ² = 0.603 and the p-value of 0.04 ( Figure 4C ). The empirical formulas (4) and (5) provide a simple way to predict the wave attenuation by standing and flattened vegetation. The results are believed to be applicable to the vegetation species that have similar dynamics of the model vegetation.

Most previous research focused on modeling wave attenuation by standing vegetation (e.g. Dalrymple et al., 1984; Kobayashi et al., 1993; Mendez and Losada, 2004; Zhu et al., 2020b; Zhu et al., 2022). Although there are some wave attenuation models for flattened vegetation (Vuik et al., 2018), or the horizontal part of mangrove roots (Suzuki et al., 2019), they assumed rigid vegetation without swaying in waves, and therefore are not applicable for flattened flexible vegetation. Due to the complicated physic processes induced by the blade sheltering and interaction from horizontal stems ( Figure 2D ), it is difficult to develop a new sophisticated wave attenuation model, particularly for flattened flexible vegetation. A simple technique to predict the wave attenuation by flattened vegetation is modifying an existing wave attenuation model for standing flexible vegetation by using a factor such as equation (6). To assess this idea, we selected the newest analytical wave attenuation model developed by Zhu et al. (2022) for standing flexible vegetation, which considered drag, inertia force, and the effects of higher-order blade motions. The inputs of Zhu et al. (2022) model included hydrodynamic parameters (water depth, wave height, wave period), plant properties (stem mass density, stem length, stem rigidity, stem cross-section dimensions, stem Young’s modulus, canopy density, and meadow length) and hydrodynamic coefficients. As the model vegetation is a flexible cylinder, the formulas for drag coefficient and added mass coefficient in Hu et al. (2021) were used is study. After obtaining the damping coefficient for standing vegetation (β_s), the damping coefficient for flattened vegetation is calculated from β F = β S × β F β S with β_F/β_S given by equation (6).

The fitted β by equation (4) and (5) compared with measured β and showed good agreement with normalized root mean square error (NRMSE) of 0.07 and 0.2 for standing and flattened vegetation, respectively ( Figure 5A ), where the normalization is based on the average of measured β. Additionally, the calculated β is compared with measured β in Figure 5B . For standing vegetation, the calculated damping coefficient from the model in Zhu et al. (2022) showed a small NRMSE of 0.31 ( Figure 5B ), which is larger than that of fitted β with NRMSE = 0.07 ( Figure 5A ). The calculated β overestimated by 16% (calculated from the slope of the linear fitting in Figure 5B ). For flattened vegetation, the calculated β has a NRMSE of 0.46 ( Figure 5B ), which is larger than fitted β with NRMSE= 0.20 ( Figure 5A ). The calculated β for flattened vegetation is overestimated by 10% ( Figure 5B ). It should be noted that if the calculated β for standing vegetation is not overestimated with a slope of 1, the slope for the corresponding β for flattened vegetation would be 1.1*1/1.16 = 0.95, indicating that the β for flattened vegetation may be underestimated by 5%. This is also acceptable for engineering application, indicating the success of modifying the existing standing vegetation-based wave attenuation model for flattened vegetation.

To explore the wave attenuation potential of flattened vegetation under extreme water depth and wave height during storms, a case study was performed based on the observed wave conditions in a storm at Nanhui shore, Shanghai. The wave conditions were measured from Sept. 10 to 17 in 2021. Two TWR-2050 (RBR cooperation, Canada) wave sensors were deployed at marsh edge (s1) and 35 m inside the vegetation (s2). The wave data at s1 was used in the present study. The storm closed to Shanghai with the shortest distance 116 km in East China Sea and wind speed up to 42 m/s, classified as Severe Typhoon. The peak wave period (T_p ) ranged from 3 to 9 s. The maximum depth was 1.43 m and the significant wave height was up to 0.68 m (Li et al., 2022). The sample size was 104. The wave conditions of the experiments can cover part of storm conditions. The linear relation between water depth and wave height is

with R ² = 0.29 ( Figure 6A ). The linear relation between wavelength and wave height is

with R ² = 0.75 ( Figure 6B ). In this preliminary case study, the vegetation are assumed to have the same properties as the model vegetation in the experiments, namely, the vegetation length is l = 50 cm and the flattened vegetation folds at 10 cm above bottom, the vegetation diameter is b = 1.8 mm, and the plant density is N = 1980 stems/m². We assumed that we can use the modification factor for flattened vegetation in equation (6), that was derived for regular waves, for the irregular waves in the case study using H = H_s , and T = T_p . This seems to be a fair assumption as the significant wave height is the average of the highest 1 3 of wave height in a short-term record that is linked to the mean wave transmission, and the modification β_F /β_S is a relative factor. The water depth is designed as 0.5 m to 1.5 m such that the vegetation is fully submerged. With a given water depth, the wave height and wavelength are calculated from equation 7 and 8, respectively. The wave damping coefficients for standing vegetation and flattened vegetation are calculated from the empirical formulas 4 and 5, respectively. The flattened vegetation is composed of two parts: the vertical unfolded part and the horizontal folded part. To further understand the wave attenuation by flattened vegetation, especially the contribution of the folded horizontal part, the wave damping coefficient by the vertical unfolded part (l_v = 10 cm) is also calculated by using the equation (2). Equation (2) is sensitive to the bulk drag coefficient C_D . We selected some representative formulas for cylinder-type vegetation from literature as shown in Table 4 . The results were shown in Figure 6C .

Comparisons between the wave damping coefficient by fully standing vegetation (β_S ), flattened vegetation (β_F ), and the vertical unfolded part of the flattened vegetation (β_V ) under storm events are shown in Figure 6C . During the storm, the wave height and wavelength increase as the storm comes associated with increasing water depth (storm surge). With increasing water depth, all the wave damping coefficients β_S , β_F , and β_V decrease indicating that the wave attenuation capacity decreases as the storm strengthens. The wave attenuation by flattened vegetation drops quicker as water depth increases. As expected, β_S>β_F>β_V , indicating that standing vegetation provides the largest wave attenuation. As the vegetation breaks to be flattening, the wave attenuation decreases. However, the wave attenuation by flattened vegetation is larger than that by only the vertical part of the flattened vegetation, indicating that the horizontal part of the flattened vegetation also plays a significant role in wave attenuation and contributes to wave attenuation.