Wave dynamics alteration by discontinuous flexible mats of artificial seagrass can support seagrass restoration efforts

Seagrass restoration can be promoted through the use of artificial seagrass (ASG). However, there is no guideline for ASG design, which requires a sound understanding of the inherent hydrodynamics in a submerged environment. Present know-how primarily stems from idealized ASG attached to a fixed bed. To develop accessible field deployment for restoration, anchored prototype scale ASG mats (coconut mesh) were proposed and tested under differing wave conditions. The aim of this study was then to: 1) analyze hydrodynamic interaction of ASG mats; and 2) assess the suitability of contemporary predictive hydrodynamic models. Velocity structure and wave propagation were measured around one and two ASG mats (separated by a 2-m gap). The mats reduced orbital velocities by up to 16% (2 mats), whereby the average reduction of all tested vegetated conditions was low (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<10\%$$\end{document}<10%) compared to the non-vegetated conditions. Velocities increased above the ASG, with the gap enhancing velocity (up to 11%) instead of attenuating it. Wave decay followed an exponential decrease, further enhanced by the second mat. Current models did not capture the induced hydrodynamics for the full range of wave conditions tested, with the second mat increasing uncertainties. Wave decay models generally overestimated wave attenuation (up to 30%), except for longer wave periods. Nevertheless, for the full range of conditions, the models provide accurate insight into the expected magnitude of attenuation under field conditions. It is speculated that mat flexibility affects the surrounding hydrodynamics through inherent motion, with the gap contributing to the uncertainties.


Raúl Villanueva * , Maike Paul & Torsten Schlurmann
Seagrass restoration can be promoted through the use of artificial seagrass (ASG).However, there is no guideline for ASG design, which requires a sound understanding of the inherent hydrodynamics in a submerged environment.Present know-how primarily stems from idealized ASG attached to a fixed bed.To develop accessible field deployment for restoration, anchored prototype scale ASG mats (coconut mesh) were proposed and tested under differing wave conditions.The aim of this study was then to: 1) analyze hydrodynamic interaction of ASG mats; and 2) assess the suitability of contemporary predictive hydrodynamic models.Velocity structure and wave propagation were measured around one and two ASG mats (separated by a 2-m gap).The mats reduced orbital velocities by up to 16% (2 mats), whereby the average reduction of all tested vegetated conditions was low ( < 10% ) compared to the non-vegetated conditions.Velocities increased above the ASG, with the gap enhancing velocity (up to 11%) instead of attenuating it.Wave decay followed an exponential decrease, further enhanced by the second mat.Current models did not capture the induced hydrodynamics for the full range of wave conditions tested, with the second mat increasing uncertainties.Wave decay models generally overestimated wave attenuation (up to 30%), except for longer wave periods.Nevertheless, for the full range of conditions, the models provide accurate insight into the expected magnitude of attenuation under field conditions.It is speculated that mat flexibility affects the surrounding hydrodynamics through inherent motion, with the gap contributing to the uncertainties.Seagrass meadows have experienced great losses in recent decades, mostly due to human development 1,2 .They represent important coastal ecosystems that provide services to the environment and to human populations 3 , including fish habitat, livelihood for coastal communities, and carbon sequestration 4 .Seagrasses alter their environment drastically by reducing wave energy and current velocity 5 , lowering turbidity by increasing sedimentation 6,7 , and stabilizing the coastline by reducing erosion [8][9][10] .This makes these ecosystems important actors in coastal protection schemes and priority targets in conservation and restoration within the contemporary key concept of nature-based solutions 11,12 .Several methods of seagrass restoration exist 13 , with no go-to method unanimously considered the most efficient.Single-shoot transplantation techniques, for example, have been shown to be successful 14 ; however, they can also be expensive and time intensive.A widely accepted notion is that seagrasses provide themselves with the means of survival through positive feedback mechanisms 15 .From an ecosystem engineering perspective, this means that seagrasses modify their local environmental dynamics (e.g.flow-induced energy flux and mass transport) in such a way that ensures survival and promotes further proliferation.Preliminarily, this can be externally achieved through shelter provided by other structures 16 .Building on this, Carus et al. 17 proposed the use of biodegradable mats of artificial seagrass (ASG) to serve as shelter for real seagrass and thus promote growth.ASG would then provide the protection that spawning seedlings need by emulating seagrass coastal protection services.

List of symbols
The proposed mats must be anchored to the seabed as they undergo hydrodynamic loading.Considering the inherent costs of field applications, these anchors should be discrete (i.e. a fixed number of punctual anchors), while able to resist the highest expected hydrodynamic loads.Further, to adapt to the highly dynamic environment, both mats and ASG should be flexible.Usage of flexible mats in the context of restoration makes sense, as these can be manufactured extrinsically, easily transported and deployed in the field; however, this flexibility also means that their interaction with the surrounding environment is complex.Presently, most understanding of flow-vegetation interaction focuses on shoots fixed to rigid, non-mobile base layers.Nonetheless, the importance of plant flexibility was rapidly recognized as flow-vegetation interaction research began, with concepts like relative plant velocity 18 (plant sway with water particle motion reduces drag, thus reducing flow attenuation) and the monami phenomenon (coherent motion of the meadow alters the in-canopy and wake velocity structure, see e.g. 19) showing that flexibility should not be neglected.A flexible mat then means that the whole meadow comes into motion, limited only by the amount of anchoring points used.
To date, other than scour protection studies with commercial motivation 20 , there is no research regarding flexible, discretely anchored mats under marine conditions.Moreover, current predictive models of flowvegetation interaction have been validated for either idealized laboratory conditions 21,22 (i.e.fixed vegetation and uniformly anchored rigid base layers), or natural vegetation in the field 23 .The experiments described in this study revolved around the applicability of state-of-the-art flow-vegetation models to predict wave dynamics around flexible mats under oscillatory flow conditions.The loads undertaken by the anchoring points was investigated by Villanueva et al. 20 .Interestingly, the study found that about 95% of wave-induced drag on the flexible ASG mats could be explained by existing drag formulations based on meadow morphology and incident hydrodynamics.A similar effect may then be expected for both the velocity structure along the water column and the wave height evolution along the meadow.Specifically, we a) analyze the effect of flexible ASG, discretely anchored to a sand bed via a flexible base layer, on wave-induced flow velocities and wave propagation; b) compare this with the status quo of wave-vegetation interaction research; and c) discuss the suitability of currently accepted models applied to fully flexible anchored mats intended for field applications.

Theoretical background
The interaction between submerged vegetation and hydrodynamics has been widely studied, with a great deal of focus given to unidirectional flow [24][25][26][27] .Interest in submerged macrophytes and their effect on oscillatory flow, i.e. waves, gained momentum with the concept of energy dissipation 28 .It was noted that wave decay took place in areas where either changes in morphology were visible, or where submerged vegetation were present.It became clear that the latter have a complex interaction with hydrodynamics, partly observable through wave decay, but also penetrating the water column and affecting the velocity structure of the oscillatory (orbital) flow.Nevertheless, despite increased research surrounding this interaction, predictive methods to describe wave decay and oscillatory flow attenuation tend to be set-up-specific, hence delivering differing and even conflicting results.Supplementary Table S1 presents a summary of relevant studies applying different methodologies to investigate wave-vegetation interaction.With different target parameters, the obtained results and conclusions can vary greatly, even for similar input conditions and experimental set-ups.

Wave decay
Wave energy dissipation has predominantly been the starting point for any formulation of wave decay 28 .The energy dissipation formulation is based on the steady conservation of energy flux 29 ∂Ec g /∂x = −ε D , where E = (1/8)ρgH 2 is the energy density, ρ the water density, g the gravitational constant, H the wave height, c g the group velocity, x the distance within the vegetation, and ε D is the vegetation-induced rate of energy dissipation.Water waves propagating through submerged and emergent vegetation lose energy by performing work on the vegetation stems, which directly results in smaller wave heights 28 .
Based on bottom interaction, wave decay was considered to be exponential through the wave decay ratio H = H 0 exp(−k i x) , based on the real part of the complex wave number ( k = k r − ik i ) and calculated through the solution to the dispersion relation 30 .However, for vegetated areas, a non-exponential wave decay was also proposed, with the following equation still widely used today 28 : where H 0 is the incident wave height at the leading edge of the meadow, H(x) is the wave height at distance x from the leading edge and in the direction of wave propagation, and β is the so-called damping coefficient.β has been modified extensively from its original proposition 28 depending on different boundary conditions, but can be expressed in general terms for flexible meadows through Eq. ( 2): where N is the shoot areal density, b v the plant width, C D the drag coefficient, d the water depth and l e the vegetation effective length.l e refers to the upright length of a rigid meadow for which a flexible meadow of canopy height h c will exert an equivalent force on flow 31,32 .This occurs due to reconfiguration of the plant caused by flow (steady or unsteady), such that the height of the meadow becomes lower than h c , in turn reducing drag.Losada et al. 33 extended the original formulation 28 using the reconfigured meadow height instead of the upright length, successfully predicting decay within flexible canopies.This length, however, needs to be actively measured, which means more complicated set-ups, especially for field applications.The use of l e allows for the implementation of the model for flexible canopies without the need to measure the actual plant reconfiguration. (1) C D b v NH 0 k sinh 3 kl e + 3 sinh kl e (sinh 2kd + 2kd) sinh kd www.nature.com/scientificreports/Note that Eq. ( 2) makes use of the drag coefficient C D and the geometric and spatial properties of the dis- sipating mechanism (i.e.submerged vegetation).Boundary reflection (e.g.end of a flume in a laboratory) and morphological features cause modulation of waves so that in any particular scenario, C D needs to be calibrated through experimentation 29 .Nonetheless, the calibrated C D will still vary greatly depending on a myriad of factors, such as load type and intensity (pure waves vs. currents vs. a combination of both), plant flexibility and morphological characteristics, and whether plant motion is taken into account 34 .Experimentally calibrated values have been commonly related to either the Reynolds Number ( Re = ub/ν , with u the velocity, ν the kinematic viscosity, and b the characteristic length) or the Keulegan-Carpenter Number ( KC = uT/b , with T the wave period), with studies showing that the latter is well suited for low-energy (inertia-dominated) conditions and the former for turbulent areas [35][36][37] .Experimental calibration of C D based on Re has been commonly given in general terms by: where n i are constants that depend on the fitted experimental conditions.Notice, however, that the range of validity of Re for any fitted case may vary depending on the trialed conditions 38,39 .Initially, values of C D between 0.1-1 were employed based on the value for a rigid cylinder; nevertheless, the inclusion of vegetation motion 18 and the use of in-canopy velocities to calculate Re 21 have led to a common use of values between 2-3, reaching up to O(2) for low Re.Re may also differ depending on the chosen input parameters, with arguments for the use of top-of-canopy velocity against in-canopy, as the latter is impractical to measure 33 , and using l e to incorporate flexibility in the calibration of C D : Regarding calibration based on KC, a formulation of C D based on the calibration for flat plates was proposed and validated for application with flexible submerged vegetation under both unidirectional currents and oscillatory flow 22 : where the minimum threshold of 1.95 was found to accurately describe drag induced by flexible blades for the unidirectional limit.Eq. ( 5) has been successfully applied to calculate drag forces on anchors under pure wave conditions 20 and wave damping under combined waves and currents 40 .Other KC-based formulations of C D following the form of Eq. ( 3) have been successfully validated to obtain a bulk C D for regular and irregular waves with model and real vegetation 36 .

Oscillatory flow
The velocity structure within the water column can be decomposed into three parts 21 : steady flow, oscillatory or time-varying, and turbulent.The instantaneous velocity at an arbitrary point in space and time can then be written as: where U i is the instantaneous velocity, U c the steady flow component, U w the oscillatory component and U ′ the turbulent component.The steady current U c observed under oscillatory flow was originally related to the mass transport velocity and solved analytically in terms of Stoke's Stream Function 41 .Non-linearity of the oscillatory flow, caused in part by near-bed viscosity, can be captured through higher order solutions.This non-linearity means that the vertical and horizontal components of 2D wave motion are not 90 • out of phase, as suggested by linear wave theory, resulting in a non-zero wave stress analogous to the turbulent Reynolds stress 31 .The mass transport velocity (here U c ) of a progressive wave within the water column has been solved analytically by solving the stream function 41 (Eq.( 7)).Eq. ( 7) is valid for small values of mass transport velocity compared to the orbital velocity.
where a is the wave amplitude, ω the wave angular frequency, and z the height from the bed.
The oscillatory component of the velocity is more complex due to the cyclic nature of motion, where inertia plays an important role.Linear and higher order wave theories can predict these oscillatory velocities accurately; however, flow-vegetation interaction additionally requires: (1) an understanding of the geometric properties of the vegetation, given by the vegetation element frontal f and planar p proportion to the surface area: (3) where S is the average separation between shoot central axes in both planar directions (x-y), and t v is the plant thickness; and (2) a scaling of these geometric properties with respect to the hydrodynamic environment, given by the canopy shear length scale L s and the drag length scale L d 21 : where C f is the friction coefficient.Lowe et al. 21proposed an analytical model to estimate flow within a canopy based on incident hydrodynamics and canopy geometric properties.The model balances the acceleration and force terms given by the set of Eqs. ( 8) and ( 9) and the oscillatory velocity U w .They then developed a non-dimensionalized form of the model in terms of the wave orbital excursion A rms ∞ to determine the relative magnitude of each of the terms: where C M is the inertia coefficient, and A rms ∞ is based on the root mean square free stream velocity U rms ∞,w (canopy unaffected).The asterisk in Eq. ( 10) indicates the velocity and time parameters non-dimensionalized through their product with (U rms ∞,w ) −1 and ω , respectively.The over-hat represents the canopy-integrated values ( z = [0 : h c ] ).The attenuation of in-canopy oscillatory velocity was then set as the ratio of canopy-averaged root mean square wave velocity and the corresponding free stream velocity: The dimensionless terms in Eq. (10) 1), whence the relative importance of each term is determined.Depending on the hydrodynamic conditions, α w can be dominated by one or more of these terms, which can then reduce the solution of Eq. ( 10) to disambiguate different flow conditions 21 : canopy-independent ( α w = 1 ), inertia-dominated ( ), and unidirectional limit or current-dominated (

Methodology
To study the response of flexible mats of artificial seagrass in a marine environment, a series of experiments were carried out under controlled laboratory conditions.Initially, different mats and anchor configurations were tested, with mat mechanical performance, flow interaction, and the resulting loads on anchoring points scrutinized 20 .
Here, we focus on the velocity structure and wave height evolution.The compound set of experiments was carried out at the Schneiderberg Wave Flume (WKS) at Ludwig Franzius Institute of Leibniz University Hannover.
The WKS is a large-scale wave flume with a length of 110 m, a width of 2.2 m, and a depth of 2 m.Waves are generated by an electrical paddle-type wave-maker with a maximum paddle stroke of 1.8 m and wave height generation of up to 0.5 m.Most flow-vegetation physical experiments are carried out in scaled-down form for practical reasons, with wave heights < 20 cm and periods < 2 s (Supplementary Table S1).Here, the large scale of the flume allowed us to test prototype-sized mats 17,20 , which provides insight into the potential hydrodynamic response of the mats under field conditions.Typical near-bed velocities found around seagrass-prone areas are usually lower than 0.5 m s −142,43 ; consequently, velocities within this range were sought and input wave conditions selected by calculating the expected near-bed velocity based on linear wave theory.This resulted in wave periods from 2-5 s and wave heights up to 33 cm.Three different water depths were selected, leading to 12 different wave conditions (Table 1).Note that the near-bed velocities in Table 1 were used as estimates of the expected velocities, and are given only as reference.The true nonlinear velocities were measured and are presented within the results.Further, to support the transmission of these conditions to other laboratory and field experiments, the dimensionless parameters kd and Ursell Number ( U R = H 2 /d 3 , where is the wavelength) are also given in Table 1.The tested wave trains consisted of 60 regular waves for each of the conditions tested.

Experimental set-up
A mobile sand bed (homogeneously graded quartz of d 50 = 0.19 mm, particle density of 2.65 g cm −3 , and bulk density of 1.45 t m −3 ) was constructed on top of the concrete bed of the flume.The sand bed began 62.65 m from the idle paddle position, had a length of 10.5 m, a width equal to that of the flume, and a depth of 10 cm (Fig. 1).2.7 cm-thick plywood panels were installed below the sand bed to facilitate instrument and anchor mounting.
The sand bed was preceded by a 1:30 plywood ramp and succeeded by a 1:10 gravel ramp.The far end of the flume was equipped with an artificial beach consisting of an aluminum stepped slope and industrial foam to enhance wave absorption.x = 0 was aligned with the beginning of the sand bed; the x-axis ran parallel to the flume and was positive in the direction of wave propagation, the y-axis ran the cross-section with 0 at the right flume wall (with respect to the direction of wave propagation), and the z-axis along the vertical with 0 at the sand bed.
Prototype ASG mats were built to simulate mats that could be deployed in the field for restoration purposes 17 .For this, 2x2-m mats were assembled using rolls of coconut mesh of different compositions 20 .Individual ASG stems were then fixed to this hybrid coconut mat to create a fully flexible ASG mat.Polyamide cable ties (PA, density δ PA = 1.13 g cm −3 and flexural rigidity EI PA = 800 N mm 2 ) of length h c = 250 mm, width b v = 4.8 mm, and thickness t v = 1.36 mm were used as a seagrass surrogate.Density effects were not a focal point of this study, therefore, a constant shoot density of N = 400 m −2 was chosen, which resulted in a shoot separation of S = 5 cm and a frontal area per canopy volume of 1.92 m −1 ( b v /S 2 ) 44 .The chosen ASG had low flexibility and the meadow density was kept low compared to real meadows to test the hydrodynamic effect of a stiffer, sparse meadow.Nonetheless, it has been shown that PA reconfigures with hydrodynamic loading while simultaneously affecting the flow field around a single stem 45 .The variation of depth resulted in different submergence ratios h c /d = [0.3,0.4, 0.5] (Table 1).
To quantify wave decay, a total of 8 Ultralab Ultrasonic Sensors (USS) with a resolution of 0.2 mm were used to measure the water level fluctuations η(t) at different positions along the x-axis (Fig. 1).The position of the devices are enumerated (starting with the sand bed), with positions 1-4 placed immediately in front of and immediately behind the ASG mats to directly measure the effect of the mats on wave height evolution and velocity structure.Position 5 (last USS in x) was located 1.52 m before the end of the sand bed.Three USS were positioned in front of the sand bed as control points (positions 6-8) for the input wave conditions (Table 1), with positions 7 and 8 varying depending on the wave condition used to cover different points along one wavelength.An HBM signal amplifier bundled with the software Catman was used to synchronously measure water level fluctuations, pressure, and forces at afrequency of 100 Hz.Wave orbital velocities were recorded simultaneously by means of a trigger connected to the HBM.Four downward-looking Nortek Vectrino+ Acoustic Doppler Velocimeters (ADV) were used to measure the orbital velocities at 50 Hz.The u, v, and w components of the  measured velocities were respectively aligned with the x, y, and z components of the set-up.Throughout this paper, specific devices and their position will be referred to by the given acronym directly followed by the position number (e.g.ADV1 corresponds to the ADV position 1, in front of the first ASG mat, Fig. 1).The four ADVs were aligned in z with each other and in x with USS1-USS4.
For each wave run (WR), the 8 USS and 4 ADVs recorded simultaneously.To build the orbital velocity profiles, the ADVs were displaced vertically ( z i in Fig. 1) and the measurements repeated for the corresponding WR.Finally, this was repeated for three different cases: 1) no vegetation, as control; 2) under the presence of 1 ASG mat, starting at x = 2 m; and 3) with 2 ASG mats with a 2-m gap between them.This resulted in a total of 168 wave run measurements.

Data processing and analysis
All recorded data were imported and processed in matlab (R2022a).Water level fluctuations for all USS data were standardized (mean = 0) around the still water level using the detrend function of matlab.The original time series was curated by removing the initial incoming and outgoing waves corresponding to the wave generator ramp time.Self-cross-correlation and a Fast Fourier Transform were used to calculate the period T of the incoming waves over the sand bed.The calculated T did not vary from the input T (Table 1) by more than 2%; therefore, from here on, the tabulated input values are used for simplicity.Zero-up-crossing was used to identify the first full wave of the curated time series, after which 10 waves were extracted to obtain a time series with a length equal to 10T.This was done to decrease the effect of wave reflection from the far end of the flume.The 10-wave time series window was then averaged into a single representative wave for the respective wave run using a phase averaging technique 46 .The maximum and minimum values of η(t) for each wave run were then calculated from the phase-averaged wave.Table 2 shows the measured wave heights at the leading edge ( H 0 ) and the ratio H/H 0 for positions 2 and 4, i.e. just behind each ASG mat.
The raw velocity data were preprocessed using the acceleration thresholding method 47 .As current models focus on the dominant velocity component, i.e. the horizontal component u, this study thenceforth focuses solely on this component.To obtain a clear and homogeneous minimum and maximum value of the oscillating horizontal orbital velocity, a fourth-order zero-phase digital filter was utilized to remove any residual spikes in the data.Phase-averaging was used to find the maxima and minima of wave velocities ( U max w , U min w ) for each wave run.Fig. 2 shows an example of data processing for one ADV within one wave run.The resulting phaseaveraged wave was then used to calculate the spatial phase-averaged steady current U c and the root mean square wave velocity U rms w utilizing Eqs. ( 13) and ( 14) (Table 2), respectively, with ϕ the phase 31 .
Note that in Eq. ( 14), U rms w represents the phase-averaged root mean square wave velocity, whereby recently, other authors have opted for the calculation of the wave velocity amplitude, obtained by multiplying the arguments within the square root by 2 40,48 .Throughout this paper, the wave velocity amplitude refers to the minimum and maximum of the measured phase averaged wave velocity ( U min w and U max w , respectively).U rms ∞,w was taken from the measurements at position 1 (leading edge of the meadow) utilizing the fitted velocity following Eq.( 15), with b 1 and b 2 as fit coefficients 49 .
To obtain comparable in-canopy root mean square velocities, the fitted profiles were integrated along the canopy layer: Canopy flow attenuation α w was then calculated utilizing Eq. (12).Predicted values of α w were obtained by solving Eq. ( 10), with Eq. ( 11) used to calculate A rms ∞ .The friction coefficient C f was previously calculated during the analysis of anchor forces 20 , with a general formulation as a function of U w obtained as a result (Eq.( 17)).Note that previous studies have used a constant C f of O(−2) (commonly 0.01 49 ).Finally, C M was obtained following where k m is the added mass and can be estimated as k m = b v /t v for rectangular cross-sectional shapes 49 .This resulted in a value of 4.53 for our experiments.
For the calculation of wave decay, the measured wave height of USS1 was set as H 0 .The ratio of the average measured wave height to H 0 at each further position was then calculated.Eq. ( 1) was fitted to the calculated H/H 0 ratios as a function of x for cases with one and two ASG mats separately.β was then obtained from the fitted curve.
For comparison with existing models, Re was calculated through Eq. ( 4) using the canopy-integrated velocities (Eq.(16a)).l e was calculated based on the scaling parameter CaL 22 , where Ca is the Cauchy Number and L the relative velocity between blade and water (for details see Villanueva et al. 20 ).C D was then calculated utilizing Eq. ( 3) for pure wave conditions ( n 1 = 0.08 , n 2 = 50000 and n 3 = 2.2) 33 .Finally, Eq. ( 18) 33 , a modification of Eq. ( 2), was used to calculate β .
Eq. (18a) is a versatile formulation that can be applied under wave-current conditions.U c was calculated utilizing Eq. ( 7) (Table 2); note, however, that the lack of an additional, externally input steady current means that U c is orders of magnitude lower than U w and represents only the wave-induced underlying current.This reduces Eq. (18a) to a form analog to Eq. ( 2).Nevertheless, Eq. ( 18) yielded higher β values than those of Eq. (2) and were more comparable to the measured (fitted) β.
Figure 2. Processing of the orbital velocity time series for WR1, no artificial vegetation, ADV position 4 (see Fig. 1 and Table 1).(a) shows the raw data record of the ADV with the 10-wave window that was extracted for post-processing.(b) shows a 3-wave window comparing the raw data (gray, thick line), the despiked data after 47 (red, medium thick line), and the filtered data (thin, solid black line).(c) shows the resulting Phase Averaged wave calculated from the 10-wave window.

Results
Table 2 shows the results of the measurements and the respective calculations.Measurements of WR3 ( H = 0.19 m and T = 4 s) for both 1 and 2-mat configurations presented results that clearly indicated errors in measurements and were therefore omitted from the results presented below.The wave conditions chosen fell within the intermediate water regime, close to the transition to shallow water regime following 0.003 ≤ d/gT 2 ≤ 0.08.Bed interaction and flume dimensions under the chosen wave conditions resulted in vertically asymmetric waves, most of which fell under the category of Stoke's second and third-order theory, and those with T ≥ 3 s and H > 0.1 m transitioning to cnoidal waves.This asymmetry could be observed in η(t) and U w .The measured H thus represents the sum of the maxima and minima of the phase-averaged wave.For the orbital velocities, the rms-velocity U rms w was used to obtain the results described below unless otherwise specified.

Wave decay
Wave evolution for the control experiments (i.e. in the absence of ASG) showed that ramp-induced shoaling increased the wave height at USS1 relative to USS6 by around 3%.Viscous dissipation caused by the walls and bed could also be observed through wave decay within the control experiments.The average rate of wave decay between positions 1 and 5 for all wave runs of the control experiments was 1.9±1.5% m −1 .The incident wave heights from all measurements with ASG ranged from 5-31 cm in front of the sand bed (USS6) and 4.5-29 cm at the leading edge of the first meadow (USS1).The ratio H/H 0 was close to unity at the positions in front of the sand bed, with a mean of 0.95±0.23 for all runs with ASG, while at positions above the sand bed, this lowered to 0.84-0.9±0.22 (positions 2-5).The rate of wave decay above the sand bed was 2.6±1.9%m −1 under the presence of a single ASG mat, and 3.1±2.4%m −1 when both ASG mats were present.The decay rate was highest between positions 1 and 2, at 5.6 and 7.2% m −1 for 1 and 2 mats, respectively, suggesting that the first mat relative to the wave propagation direction has a more marked effect on wave decay.Fig. 3 shows the average wave height ratio evolution ( H/H 0 ) along x for 6 different runs segregating between one and two-mat configurations.The average of ratios for all runs with ASG is presented to encompass all measurements.The damping coefficient β was calculated by fitting Eq. ( 1) to the wave height data, resulting in values ranging between -0.005-0.06m −1 and -0.012-0.06m −1 for 1 and 2-mat configurations, respectively (Table 2).The set of 1-mat experiments averaged β = 0.0161 ± 0.018 m −1 , whereas for the 2-mat experiments β = 0.021 ± 0.023 m −1 .A second mat then enhanced wave decay, with β for the 2-mat configuration being on average 30% higher than for the 1-mat configuration.Table 2. Calculated Parameters.H 0 represents the incident wave height, i.e. at the leading edge of the meadow (position 1, Figure 1).Non-zero numeric subscripts of H and U indicate the device position.WR3 was omitted due to anomalies in the measurements.Although β shows that wave decay is augmented by the mats, the variability of the values (Table 2) indicates that the input parameters have a meaningful effect on wave decay.As shoot and base layer morphology were not modified during the experiments, the incident wave conditions represent the governing variables.Analysis of the relationship between β and the incident wave height H 0 showed no correlation for the wave conditions tested here.A simple linear regression between the water depth d and β also showed no significant correlation between both variables ( p > 0.05 ).However, a light tendency of β decreasing with increasing d was observable, showing a low rate of change of -0.0048 per 10 cm of added water depth for the 1-mat experiments and -0.0019 dm −1 for 2 mats.Regardless, the low rate and correlation indicate that the submergence ratios trialed here, i.e. h c /d = [0.3,0.4, 0.5] , had little effect on wave attenuation for the wave conditions trialed.
In contrast, analysis of the relation between the wave period T and β showed that β increased exponentially with increasing T. The relationship between β and T makes it obvious that β is analogously sensitive to the wave- length .Fig. 4 shows the change of β with respect to T for both mat configurations.An exponential fit for each set of experiments was done to showcase the effect of T on wave decay, with the respective fits given in Eq. set 19.
Note, however, that the variance of the fitted β with respect to T is still relatively high, with R 2 < 0.5 for both cases.The limited data and variability of the calculated β produce fits with a high prediction variability within the confidence intervals, as shown in Fig. 4. Therefore, it is important to keep in mind that although Eqs.(19a,b) may provide insight into an increasing damping coefficient with respect to T, this is explicitly valid for wave decay above meadows of similar geometric and mechanical characteristics and wave conditions to those trialed here, i.e. within the range 1 < T < 6 s.Wave Decay evolution for different wave runs.Data points show the average and standard deviation of H/H 0 for each USS over the sand bed and at position 6 in front of the sand bed.H 0 is taken from USS1 at the leading edge of the meadow.Fits follow Eq.(1) with solid line representing runs with one mat (subscript M1) and dashed line runs with two mats (subscript M2).The corresponding resulting β is given for each run.Shaded areas show ramp and ASG meadows (not to scale).  5yielded values between 1.95-2.96(mean = 2.4).
Fig. 5 shows C D plotted as a function of Re l e .Results from other studies with different experimentally fitted C D = f (Re) following Eq.( 3) are shown for comparison.For the present set of experiments, the corresponding fit is shown resulting in the coefficients shown in Eq. ( 20) ( R 2 = 0.79 ).The different results show how sensitive C D is to the choice of characteristic length and input velocity, directly reflected in the variation of the scale of Re.ASG mats (M1 and M2, respectively).Lines show fit for each configuration (Eq.( 19)) and the corresponding confidence bounds.5)) as function of Re l e (Eq.( 4)) and resulting fit (solid black line, Eq. ( 20)).Dotted line shows the fit for C D as a function of the stem-Re.Results of Eq. ( 3) using the coefficients: [ n 1 = 0.08, n 2 = 50000, and n 3 = 2.2] 33 , [ n 1 = 0.40, n 2 = 4600, and n 3 = 2.9] 50 , and [ n 1 = 0.1, n 2 = 925, and n 3 = 3.16] 23 are given for comparison.
The modeled damping coefficient β mod was then calculated based on C D (Eq. ( 5)) following Eq. ( 2).Fig. 6 shows the comparison between the modeled values and the values obtained from measurements ( β obs ).A linear fit between modeled values and measurements showed that the model underestimates the damping coefficient by 52% (based on the slope of the linear fit, no intercept).A comparison with modified forms of Eq. ( 2) is also shown in Fig. 6: Eq. ( 18) 33 overestimated β obs , with β mod = 1.15(±0.32)βobs , while a further modification 32 (discussed later) yielded β mod = 1.30(±0.21)βobs .Both models underestimated β obs for wave runs with the longest periods, i.e.T = 5 s, and wave heights above 0.1 m.A similar pattern was observable with the model after the original formulation (Eq.( 2)), where a 1:1 comparison between modeled and measured values showed that the β mod underpredicts β obs for all but one of the wave runs with T ≥ 4 s.

Velocity structure
The maximum measured orbital velocities U max w increased proportionally with the increase of H, as is expected from wave theory, with no correlation to either T or d found for the range of conditions trialed.Comparison of U max w with the maximum velocities measured in the opposite direction (i.e.U min w ) showed that the non-zero net momentum transport in the direction of wave propagation (i.e.Stoke's Drift) was present regardless of the number of mats (0, 1 or 2) and ADV position (1-4).ASG thence did not alter the overall mass transport in the direction of wave propagation, demonstrated by the ratio U max w /U min w , which, averaging for each set of measurements, yielded 1.34 ± 0.22 for the control experiments, and 1.36 ± 0.19 and 1.37 ± 0.23 for the one and two-mat configurations, respectively.Note that Stoke's Drift may be an artefact of the wave flume; however, its analysis is out of the scope of this study and is therefore not discussed further.
Fig. 7a shows the range of U w at each ADV position for WR1 ( H = 0.11 m, T = 2 s), whereby the aforemen- tioned asymmetry becomes greater for higher T as non-linearity effects intensify.The canopy-integrated velocities ranged from 0.35-0.53m s −1 for M0, 0.34-0.48m s −1 for M1, and 0.33-0.46m s −1 for M2.The highest velocities in the direction of wave propagation were measured exiting the meadows, i.e. at positions 2 and 4, while in the opposite direction, the highest velocities were at position 1, i.e. exiting the first meadow at the leading edge.Overall, WR8 ( H = 0.22 m, T = 5 s, and d = 0.63 m) recorded the highest velocities.Fig. 7b shows U c (z) (Eq.( 13)), whose magnitude was predominantly within the range of O(−4) and O(−5) .The highest values were observed for M0 ( O(−3) ), though the near-bed velocities of both ASG experiments likewise reached O(−3) .At the leading edge of the meadow, a peak in the positive direction could be observed within z ≤ 10 cm for WR2, 5, 9, and 10.The theoretical model 41 (Eq.( 7)) predicted values in the same order of magnitude of the measurements for the lowest energy runs (i.e.H < 0.1 m).However, neither the shape of the profile nor the highest magnitudes were captured well by the model which overestimated U c (z) for H > 0.2 m by up to O(1).Fig. 7c shows the profiles of U rms w (z) .The dashed line was fitted as a function of z at position 1 (Eq. ( 15) and is thence shown at each further x-position for reference.U rms w (z) decreased with increasing distance along x, with changes along x becoming more conspicuous as the velocities increased.To better visualize the effect of the meadows on U rms w , the percentage change in velocity relative to position 1 was calculated and is shown in Fig. 8.The velocity profiles show that the vegetation induces skimming flow, which is the result of discontinuity in the drag force  www.nature.com/scientificreports/and mass balance along the meadow.Skimming flow has been shown for wind profiles around urban canopies 51 and submerged vegetation as a function of wave conditions 52 .
Analogous to H/H 0 (Fig. 3), Fig. 8 shows the evolution of U rms w (z) along x with respect to position 1 (i.e.U rms ∞,w ), calculated as: Negative values represent velocity reduction, while positive values indicate an increase.The values are averaged from all wave runs, with deviations depicted by vertical bars.The deviation ranged between 8-20%, except at position 2 for M2, where the deviation reached on average 30% at all z i .Deviations > 20% were also measured at z 4 at all x-positions for M1, and just over 20% at z 1 for M2 at positions 3 and 4. The reduction of U rms w along x ranged from 2-17.5% and was more prominent directly behind the ASG mats.Increases in U rms w ranged from 0.5-11%, with the highest values ( ≥ 10% ) in the region above the meadow ( z 4 , z 5 ), which indicates an increase in velocity resulting from the presence of ASG.For 2 mats, there was an increase (7-7.7%) in the near-bed velocities within the gap, whereas with 1 mat, the same value (7%) was observed at position 3, but rather increasing from a reduction ( −15.6% ) at position 2. With increasing distance behind the first mat, U rms w decreased again, regardless of mat configuration.However, the second mat enhanced this reduction, which was additionally accentuated by the proximity to the bed, with M2 reducing 10, 7.5, and 5.7% more than M1 for z 1 , z 2 , and z 3 , respectively.This tendency to increase after the first mat and gradually reduce as waves go through the second mat was observed along the full transect for M2.For M1, the velocities sink directly behind the ASG mat and subsequently set up immediately afterward, a behavior that is only prominently observed within the canopy height.Above the canopy, the velocities for M1 remain on average relatively stable.
The flow attenuation parameter α w is analogous to the percentage change shown in Fig. 8, depicting the evo- lution of Ûrms w (Eq.( 16)).Ûrms w ranged from 0.04-0.29 m s −1 for M0, and from 0.04-0.25 m s −1 for M1 and M2.The maximum value was consistent at all ADV positions for M1, but lowered to 0.22 m s −1 for M2 at position 4, showing a reduction of Ûrms w along x.Table 2 shows α w calculated between positions 1 and 2, however, like U rms w (x, z) , α w displayed variation along x.Increases in Ûrms w (x) were recorded, leading to α w > 1 , thus indicating  15).(b) Steady current component of flow U c calculated from measurements using Eq.(13); dashed line represents Eq. ( 7).(c) Root mean square velocity calculated from measurements and phase-averaged using Eq. ( 14); fitted line from Eq. (15).Shaded areas show ASG at full height h c .flow enhancement instead of attenuation.The variability exhibited by the spread of the velocity data (Fig. 8) and α w indicates that the incident wave conditions greatly alter the velocity structure (as vegetation parameters did not vary).To assess the effect of the incident hydrodynamics, α w was calculated at each x-position with respect to Ûrms ∞,w , subsequently plotting against the different instances of H, T, and d separately (Fig. 9).Within the uncertainties, the calculated values of α w (x) generally did not show a clear trend toward gain or loss depending on H, T, or d.An exception can be observed for the averaged values with the addition of a second mat, especially between positions 1 and 2; however, uncertainties increased by up to 40%.For the averages, it can be seen that α w decreases with increasing H, going from 1.26±0.52 to 0.90±0.1 for waves >15 cm, indicating that flow is more readily attenuated for higher waves (i.e. higher velocities).Similar to β , α w increased from 0.83±0.14for h c /d = 0.5 to 1.22±0.46for h c /d = 0.3 (2 mats), denoting less attenuation for increased depths (i.e.decreased submergence ratio).On the other hand, attenuation decreased, i.e. higher α w , as T increased, going from 0.90±0.05for T = 2 s to 1.17±0.43for T = 5 s.These tendencies were reflected less as the distance along x increased, with the differ- ences between 1 and 2 mats becoming less obvious.
Modeled flow attenuation Fig. 10 shows the empirical values of α w plotted against the ratio A rms ∞ /S along with the solution to the model in Eq. (10).The wave conditions and vegetation characteristics trialed here led to a range of A rms ∞ from 3-22 cm for M0 and 3.2-20 cm for M1 and M2.With a constant shoot separation S = 5 cm, this resulted in the range of A rms ∞ /S of roughly 0.6-4.This range puts the conditions trialed here at the interface between the general flow and inertia-dominated regimes 21 .The wide spread of α w presented above makes it obvious that the model does not agree with a wide range of the conditions trialed here.Fig. 10a shows α w between positions 1 and 2 distinguished by the number of mats and the measured wave height.It can be seen that the model generally performs better for greater wave heights ( H > 0.15 m), with lower wave heights both over and underpredicting α w .As before, this was accentuated for M2.
Fig. 10b shows α w at all positions.The differentiation between 1 and 2 mats reveals that over and underprediction by the model is more likely to occur under the presence of a second mat, with only 36% of the M2 runs falling within ±10% of the modeled α w compared to 55% for M1.Fig. 10c shows the solution to Eq. ( 10) from studies with different input hydrodynamic conditions and vegetation parameters for comparison with the present set of experiments.For reference, the average value of the set of wave runs for M1 and M2 are shown, where it can be observed that although the set of experiments agrees well with the model for the average (especially for 1 mat), the uncertainty is high for varying hydrodynamic conditions.

Wave dynamics around flexible ASG mats
The velocity profiles showed the presence of a wave-induced current whose profile U c (z) was modified by the flexible ASG mats.For rigid surrogates, it has been shown how U c (z) shifts direction at canopy interfaces (i.e.canopy top and meadow edges) 53 , while a return current, essential for nutrient circulation within the meadow, w with respect to measurements at the leading edge of the meadow (position 1).The average percentage change of all wave runs is shown.Vertical bars at each point represent standard deviation.Markers deviate slightly from ADV position axes for clarity.Shaded areas show ASG at full height h c .Note that the lines joining each x-marker only qualitatively represent the trend between the positions, as changes in U rms w (z) are not actually linear and would require more information within each meadow to depict their true behavior.has been observed for flexible vegetation 31 .Here, these behaviors were only observed at certain interfaces, yet not as a common occurrence.Nonetheless, like previous studies 49 , the profiles of U c (z) around flexible vegetation are of similar form to non-vegetated conditions.This is because N was too low to properly modify the wave-induced current, which easily penetrated into the canopy.
The control experiments showed that the measured wave decay encompassed energy loss not only due to vegetation but also static continuous losses, i.e. losses due to wall and bed friction.The choice of H 0 in Eq. ( 1) is thus not inconsequential.Further, previous studies 29,31 emphasize the assumption that energy dissipation and the ensuing wave decay stem mostly from drag-induced forces caused by vegetation, whereby the current paradigm sets H 0 as the incident H (leading edge of meadow).Hence, neither the non-vegetated wave height nor the effect of continuous losses along x is taken into account.For comparison, β calculated based on H of the control experiments (M0) at each position (effectively removing continuous losses) ranged from 0-0.03 m −1 , or half the range presented above ( −0.01 < β < 0.06 m −1 ) .Moreover, if we take H 0 as the incident H of M0, β ranges from -0.01-0.13,demonstrating an additional effect of the ASG compared to non-vegetated areas.These differences are particularly important when intending to apply empirical models developed under laboratory conditions to field applications.
With a low shoot density, a flexible base layer, and single-stem flexible shoots, the measurements indicate that a sparse, fully flexible canopy is able to dissipate energy.This in turn could promote seagrass growth 17 .The degree of attenuation is clearly dependent on vegetation properties and incident hydrodynamic conditions and can vary greatly between set-ups.For example, Losada et al. 33 obtained a range of 0.02 ≤ β ≤ 0.32 m −1 (O (1) higher than this study) for experiments with circular patches of S. anglica and P. maritima of varying rigidity, whereby, compared to this study, density was up to 3 times higher, h c /d ≥ 0.5 (including emergent conditions), and 1.7 ≤ T ≤ 2.2 s (see Supplementary Table S1).Here, β < 0 for 3 wave runs (all with T = 3 s, Table 2), which result from H/H 0 > 1 at certain positions.Flow attenuation displayed a similar behavior ( α w > 1 ), indicating increased flow around the meadows.The wave setup and enhanced flow were certainly initiated by the obstructing structure, i.e.ASG, but could be boosted by mat motion.Furthermore, the interaction between the oscillatory flow and the meadow at position 1 could lead to an increase of Ûrms w with respect to Ûrms ∞,w .Of course, this has not been observed as a common phenomenon in other experimental studies, which suggests that the increased mat flexibility and added mat motion may be responsible for the reduced attenuation.
Moreover, with measurements taken at canopy interfaces, the measured U w is affected by a vegetated side and a non-vegetated side.The effect of the bare side is particularly evident within the gap hydrodynamics, where, within the canopy height, in-gap measured velocities varied depending on whether a second mat was present further downstream.The evolution of U rms w /U rms ∞,w shown in Fig. 8 shows how, with one mat, the attenuation of flow was enhanced directly behind the mat, and velocities increase rapidly before steering toward a steadily attenuated flow downstream.With 2 mats, however, the first mat does not have the same attenuating effect per meter, but rather present a shift of the attenuated velocities toward the end of the second meadow.This suggests that the 2 mats behave as a single meadow, with the highest attenuation at the trailing edge.Furthermore, skimming flow becomes apparent at the canopy interfaces, gradually attenuating flow further down the wave propagation direction 54 .Savio et al. 55 showed for aligned patches that skimming flow develops roughly at gap width equal to patch length, consistent with the measurements here.On the other hand, a similar study on gap hydrodynamics 56 found that wave velocities were reduced after the first meadow and returned to pre-vegetated conditions for gaps longer than 2h c .This was not corroborated here (gap length: 8h c ) as U rms w showed no attenuation within the gap, but further down the meadows.
Perhaps an interesting outcome to discuss with regard to wave decay and flow attenuation is the relationship between both.Fig. 11 shows the relation between H/H 0 and Ûrms w / Ûrms ∞,w for the complete set of experiments (all positions).A linear fit was done for the set of 66 points (4 outliers removed due to unlikelihood).Fig. 11 shows that measurements with higher wave attenuation (i.e.lower H/H 0 ) yielded lower flow attenuation (higher α w ), while higher flow attenuation resulted in lower wave attenuation.In general terms, this suggests that as velocities around the meadows are more actively reduced, wave heights are less reduced with respect to the incidence.This is consistent with Lowe et al. 57 , who found that the rate of wave dissipation increases with increasing flow through the meadow, i.e. higher α w .Therefore, as flow penetrates more readily into the meadow, wave heights are more actively reduced.

Effect of incident hydrodynamic conditions
Studies have shown that wave energy dissipation can vary depending on the wave period 38 , higher submergence ratios h c /d lead to higher wave attenuation 33,54 , and vegetation geometric characteristics (e.g.f , p ) play an essential role 58 .Here, other than the variation of h c /d , the constant vegetation traits remained independent of the fluctuating hydrodynamics.The high variability exhibited by the results thus indicates that the hydrodynamic conditions greatly affect wave and flow attenuation.Previous studies have shown that the rate of energy dissipation is highly dependent on T 10,23,57 , with shorter wave periods yielding higher wave attenuation and longer periods higher in-canopy flow attenuation.The relationship between H/H 0 and α w shown in Fig. 11 corroborates this inverse behavior.However, as shown in Fig. 9, for T ≥ 3 s and h c /d ≤ 0.4 , α w was enhanced within the gap of the 2-mat configuration compared to the 1-mat configuration, suggesting that flow enhancement is a result of the gap.This was also reflected with h c /d , where α w decreased for increasing h c /d at position 2, indicating that flow within the gap becomes proportionally lower as the meadow takes more of the water column.Moreover, connecting h c /d and T, the measurements indicate that flow penetrates more easily into the canopy for longer wave periods as the submergence ratio decreases.The inconsistency within the results suggests that the effect of the gap on flow varies with T and h c /d .Lara et al. 59 found that while turbulent diffusion was enhanced by heterogeneous meadows (i.e. with gaps), flow velocities were not enhanced by meadow patchiness.Moreover, Paul et al. 58 argued that f and p are more important than a varying submergence ratio stemming from varying d, which, for h c /d ≤ 0.4 , was also observed here.For storm conditions, Möller et al. 39 found that marshes attenuated waves regardless of the variable water depth.In-depth gap hydrodynamic studies are necessary to couple the effect of the gap with interchanging submergence ratios and varying hydrodynamic conditions.

Predictive models of wave and flow attenuation
The experiments demonstrated that the flexible ASG mats attenuate waves and flow similarly to conventional (fixed) meadows, i.e. behavior and parameter sensibility are established comparably.However, the attenuation magnitude was generally lower than predictive models would suggest, with uncertainties increasing as a function of incident hydrodynamic conditions.As analytical models are generally validated experimentally to support the range of empirical values obtained, most models are also sensitive to input variability.Recent wave attenuation models based on the original model 28 thus further incorporate intrinsic conditions such as measured in-canopy flow and plant motion.van Veelen et al. 60 extended the model to account for vegetation flexibility and motion by including blade excursion during wave loading; they validated it for regular waves with small A ∞ over cylindrical, flexible, and near-stiff vegetation, and performed well applied to real salt marshes of different stiffness (P.maritima, S. anglica and E. athericus).Further, following the findings of Lowe et al. 57 , Lei and Nepf 32 extended Eq. ( 2) by factoring in flow attenuation, thus incorporating the effect of vegetation on in-canopy flow; they also incorporated l e to account for plant motion and consequent drag reduction (similar to 33 ), obtaining a modified Eq. (2) as β = (2/9π)C D b v NkH 0 α 3 w (J 1 /J 2 ) , where J 1 = 9 sinh(kl e ) + sinh(3kl e ) and J 2 = sinh kd(sinh(2kd) + 2kd) .Modified forms of Eq. (2) ( 32,33 ) yielded values of β of the same order of magnitude as those observed here, while Eq.(2) largely underestimated them (Fig. 6).It is not surprising that the modified formulations improved the results; however, it is interesting to note that both approaches underestimated the damping coefficient for longerperiod waves.With both models empirically validated for T < 2 s, it becomes clear that future models should focus on longer-period waves, as such conditions also lead to higher flow attenuation 57 and are more likely to dominate field conditions 23,61 .Note that shorter wave periods lead to more flow through the canopy, leading to lim T→0 α w = 1 , which reduces the equation described above to Eq. ( 2).
The measurements yielded a wide range of α w , suggesting that Eq. ( 10) is not applicable to flexible ASG mats.However, it is important to remember that α w measured here corresponds to the ratio at the edges of the meadows and not the in-canopy flow (measured through a clearing 21 ).Knowing this, it is expected to have higher values of α w than in-canopy measurements would yield.However, as the results show, α w was both over and underestimated by the model, depending on wave run and position.This leads us back to the sensitivity with regard to input conditions and the role of the gap in patch-to-patch hydrodynamics.Even though the results suggest that the model should not be used to determine the flow attenuation expected by similar flexible mats, the resulting average from the ensemble of measurements indicates that for a general range of conditions, the model can provide proper insight into the effect of the canopy.Special care should still be taken regarding gap hydrodynamics, which was shown to be unstable.
Model results proved to be sensitive to the choice of input parameters, such as C D , C f , and C M , the characteristic lengths (e.g.b v , h c , and l e ), and the resulting dimensionless quantities describing flow patterns (e.g.Re and KC).Supplementary Table S1 lists several studies that have investigated different parameters, proposing ways to calculate them depending on target conditions.Ultimately, it is important to know how a model was validated to determine which parameters are more relevant and what input they need for their calculation.For the present experiments, canopy-integrated rms-velocities were used as the characteristic velocity.The use of in-canopy velocities provides an estimation of C D which better represents vegetation-induced drag as this may also vary along z.This is especially true for higher frequency conditions beyond the shallow water regime, as the oscillatory velocity becomes more sensitive to z and the velocity at the canopy top may thus greatly differ from near-bed velocities.Note that rms-velocities are lower than the maximum velocity U max w , meaning that dependent parameters will be proportionally lower.However, U max w does not dominate the range of velocities in a temporal scale within a given period (see 20 ), especially as the frequency decreases.Thus, a better representation of the time-averaged Re or KC is given by the representative rms-velocity.

Figure 1 .
Figure 1.Schematic of the experimental set-up used for the experiments.D = water depth to the concrete flume bed; d = depth to sand bed.Shown are the device positions.USS = Ultrasonic Sensor; ADV = Acoustic Doppler Velocimeter.z i are the different ADV positions, where each of the 4 ADVs was vertically displaced to the i positions labeled, with z 1−5 = [3.3;10.5; 25.7; 35.9; 45.7] [cm].z-axis exaggerated by 4 times.All dimensions in meters.

41 Figure 3 .
Figure 3.Wave Decay evolution for different wave runs.Data points show the average and standard deviation of H/H 0 for each USS over the sand bed and at position 6 in front of the sand bed.H 0 is taken from USS1 at the leading edge of the meadow.Fits follow Eq.(1) with solid line representing runs with one mat (subscript M1) and dashed line runs with two mats (subscript M2).The corresponding resulting β is given for each run.Shaded areas show ramp and ASG meadows (not to scale).
https://doi.org/10.1038/s41598-023-46612-zwww.nature.com/scientificreports/Modeled damping coefficient The calculation of β requires an estimation of the drag coefficient C D , which, for vegetated flow, has been commonly related to the Reynolds Number Re and Keulegan-Carpenter Number KC.The range of canopyintegrated rms-velocities Ûrms w measured here yielded values of Re l e between 8300-32500 and KC between 42-260.The ratio of effective length to upright canopy height ( l e /h c ) ranged between 0.44-0.81.Values of the stem-based Re (i.e. based on b v ) were O(1) lower, ranging between 200-1200.The calculation of C D following Eq. (

Figure 4 .
Figure 4. Damping coefficient β plotted against the incident wave period T. Values are shown for 1 and 2ASG mats (M1 and M2, respectively).Lines show fit for each configuration (Eq.(19)) and the corresponding confidence bounds.

Figure 7 .
Figure 7. Velocity Structure for WR1 ( H = 0.11 m, T = 2 s).Profiles show x-z position of ADV measurements.Data points show measured values for control (no ASG mats), 1-mat and 2-mat experiments.(a) fully measured wave velocity excursion showing maxima ( U max w ) and minima ( U min w ); dashed fitted line shows theoretical profile following Eq.(15).(b) Steady current component of flow U c calculated from measurements using Eq.(13); dashed line represents Eq. (7).(c) Root mean square velocity calculated from measurements and phase-averaged using Eq.(14); fitted line from Eq.(15).Shaded areas show ASG at full height h c .

Figure 8 .
Figure 8. Percentage change in U rmsw with respect to measurements at the leading edge of the meadow (position 1).The average percentage change of all wave runs is shown.Vertical bars at each point represent standard deviation.Markers deviate slightly from ADV position axes for clarity.Shaded areas show ASG at full height h c .Note that the lines joining each x-marker only qualitatively represent the trend between the positions, as changes in U rms w (z) are not actually linear and would require more information within each meadow to depict their true behavior.

Figure 9 .Figure 10 .
Figure 9. Variation of the flow attenuation parameter α w depending on the hydrodynamic parameters and the position of measurement: wave height H, wave period T, and water depth d shown in columns 1, 2, and 3, respectively.Position of α w shown by rows (subscripts on the right indicate position).Deviation of measurement given by vertical bars.H is shown for a range of measured wave heights.

Figure 11 .
Figure 11.Relation between wave evolution H/H 0 and flow attenuation Ûrms w / Ûrms ∞,w at all positions (Pos) relative to position 1.Hollow markers indicate 2-mat configuration.Solid line represents linear fit.

Table 1 .
Wave conditions tested for each wave run (WR).*Expected values of and the maximum horizontal orbital velocity 3 cm above the bed U max b were calculated based on linear wave theory for reference.